COMPOSITE BEAM THEORY WITH MATERIAL

COMPOSITE BEAM THEORY WITH MATERIAL NONLINEARITIES AND PROGRESSIVE DAMAGE

A Dissertation Submitted to the Faculty of Purdue University by Fang Jiang

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

May 2017 Purdue University West Lafayette, Indiana

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THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL

Dr. Wenbin Yu, Chair School of Aeronautics and Astronautics Dr. Vikas Tomar School of Aeronautics and Astronautics Dr. Kejie Zhao School of Mechanical Engineering Dr. Arun Prakash School of Civil Engineering

Approved by: Dr. Weinong Chen Head of the Departmental Graduate Program

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To my family.

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ACKNOWLEDGMENTS First and foremost, I owe my deepest gratitude to my advisor, Professor Wenbin Yu. The work presented in this dissertation would not have been possible without his support, encouragement, and guidance throughout the last five years at Utah State University and Purdue University. I especially appreciate the time, dedication, and patience he put into our discussions, which have always been enlightening, refreshing, and motivating. I have been truly inspired by his enthusiasm for pursuing the truth, his passion for seeking unity to systematically handle diversities, his innovation for striking for balance between practicality and rigor, and his wisdom for embracing humble boldness to learn from others and remain true to scholarship. I would like to thank my committee: Professor Vikas Tomar, Professor Kejie Zhao, and Professor Arun Prakash. Their valuable advice and comments on my research have been greatly appreciated. My special thanks go out to Professor Vikas Tomar for giving me guidance as a mentor. I would like to thank the Army Vertical Lift Research Center of Excellence at the Georgia Institute of Technology for their financial support. My special gratitude goes to Professor Dewey H. Hodges for giving me tremendous help on solving the trapeze and Poynting effects. I am thankful to Mohit Gupta and Hanif Hoseini for their warmest hospitalities when I visited Georgia Institute of Technology. I am privileged. I want to acknowledge my colleagues from the Multiscale Structural Mechanics Group at Purdue University: Dr. Tian Tang, Dr. Liang Zhang, Hamsasew Moges Sertse, Bo Peng, Haiqiang She, Ning Liu, Yufei Long, Zhenyuan Gao, Su Tian, Ankit Deo, Orzuri Rique Garaizar, Kshitiz Swaroop, Lingxuan Zhou, Albert Chang, Banghua Zhao, Xin Liu, Ernesto Camarena, Fei Tao, Khizar Rouf, and Dr. Xiuqi Lyu. It has been an extraordinarily agreeable experience working with such a talent, vibrant, and friendly group. I have been learning a lot from my group mates.

vi I am grateful to my colleagues and friends at Utah State Universty: Dr. Qi Wang, Dr. Zheng Ye, Dr. Chong Teng, Dr. Changjin Choi, Dr. Lin Zhang, Zhifen Wang, Davy Xi, Nicholas Chiew, Louis Lo, Kevin Kung, and Longze Li. I would not have survived my first year at Utah State University without them. I would like to thank my friends and family from China, who have never forgotten me for such long time no see. Last, but most importantly, I want to thank my grandparents, Xigui Jiang and Xiuying Wang, and my parents, Yang Jiang and Bing Zhang. They have always made all my dreams and wishes their own. My heartfelt gratitude goes to my wife, Shule Liu. I am truly blessed for having her at my side. To them, I owe everything I have accomplished. Thank you.

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TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview and Motivation . . . . . . . . . . . . . . . . . . . . . 1.1.1 Slender Solids and Structures . . . . . . . . . . . . . . 1.1.2 Mechanics of Structure Genome . . . . . . . . . . . . 1.1.3 Linear Cross-Sectional Analysis of Composite Beam 1.1.4 Trapeze and Poynting Effects . . . . . . . . . . . . . . 1.1.5 Nonlinear In-plane Shear . . . . . . . . . . . . . . . . . 1.1.6 Damage of Composite Materials . . . . . . . . . . . . 1.1.7 Large Bending Effects . . . . . . . . . . . . . . . . . . 1.2 Objectives and Outline . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 4 5 5 9 10 11 13

2 THEORETICAL FOUNDATIONS 2.1 Kinematics . . . . . . . . . . . 2.2 Hamilton’s Principle . . . . . . 2.3 Dimensional Reduction . . . .

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3 FINITE ELEMENT IMPLEMENTATION . 3.1 General Finite Element Formulation . 3.2 Hyperelasticity . . . . . . . . . . . . . . 3.3 Nonlinear Material with Linear Strain 3.4 Progressive Damage Constitutive Law

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23 23 26 28 30

4 TRAPEZE AND POYNTING EFFECTS . . . . . . . . . . . . 4.1 Analytical Modeling . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Isotropic Hyperelastic Material Models . . . . . . 4.1.2 Asymptotic Analysis with Green Strain Energy . 4.1.3 Asymptotic Analysis with Neo-Hookean Model . 4.1.4 Asymptotic Analysis with Hencky Strain Energy 4.1.5 Approximation of Warping Functions . . . . . . . 4.1.6 Perturbation of Warping Functions . . . . . . . . . 4.1.7 Transverse Stresses and Strains . . . . . . . . . . . 4.1.8 Strain-Dependent Beam Stiffness . . . . . . . . . . 4.1.9 Trapeze Effect: Stiffening of Torsional Rigidity .

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4.2

4.1.10 Trapeze Effect: Untwisting of Pretwisted Beam . . . . . 4.1.11 Poynting Effect: Rubber Tube . . . . . . . . . . . . . . . 4.1.12 Decoupling Thickness Ratio: Comparison with 3D FEA 4.1.13 Negative Poynting Effect . . . . . . . . . . . . . . . . . . . Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Material Models and Their Tensorial Derivatives . . . . 4.2.2 Small Strain Trapeze Effect Model . . . . . . . . . . . . . 4.2.3 Trapeze Effect: Spring Steel Strip . . . . . . . . . . . . . 4.2.4 Trapeze Effect for a Realistic Rotor Blade . . . . . . . . 4.2.5 Trapeze Effect: Composite Pipe . . . . . . . . . . . . . . 4.2.6 Poynting Effect: Circular Cross Section . . . . . . . . . . 4.2.7 Poynting Effect: Tubular Cross Section . . . . . . . . . .

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Page 47 50 51 53 54 54 57 59 62 65 67 71

5 NONLINEAR SHEAR BEHAVIOR IN COMPOSITE 5.1 Nonlinear Beam Constitutive Relation . . . . . . 5.2 Three-Dimensional Stresses . . . . . . . . . . . . . 5.3 Coupon Performance . . . . . . . . . . . . . . . .

BEAMS . . . . . . . . . . . . . . . . . .

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77 77 85 104

6 COMPOSITE BEAM DAMAGE MODEL 6.1 Isotropic Damage . . . . . . . . . . . . 6.2 Mesh Objectivity . . . . . . . . . . . . 6.3 Comparison with 3D FEA . . . . . . 6.4 Comparison with 2D FEA . . . . . . 6.5 Comparison with Experimental Data

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111 111 114 130 139 148

7 LARGE BENDING NONLINEARITIES . . . . . . . . 7.1 Extension-Bending Coupling . . . . . . . . . . . . 7.1.1 Beam Constitutive Law . . . . . . . . . . 7.1.2 One-Dimensional Beam Model . . . . . . 7.1.3 Semi-Analytical Solution versus 3D FEA 7.2 Brazier Effect . . . . . . . . . . . . . . . . . . . . .

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157 157 157 160 160 164

8 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ix

LIST OF TABLES Table

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4.1

A22 (×106 ) (elliptical section: a = 0.23 m). . . . . . . . . . . . . . . . . .

45

4.2

A22 (×107 ) (rectangular section: a = 0.23 m). . . . . . . . . . . . . . . . .

45

4.3

Nonlinear stiffness induced by pretwist k1 (elliptical section: a = 0.23 m, k1 = 0.013963 rad/m ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Nonlinear stiffness induced by pretwist k1 (rectangular section: a = 0.23 m, k1 =0.008727 rad/m ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.5

Derivatives of stress resultants with respect to beam strains of the strip.

60

4.6

Derivatives of stress resultants with respect to beam strains of the rotor blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Twisting moment (kNm) [ABS DIFF%] to preserve a tip rotation of 1 degree under various levels of axial strains for the rotor blade. . . . . .

65

Derivatives of stress resultants with respect to beam strains of the composite pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.9

Material constants for cylinder depicted in Figure 4.12. . . . . . . . . . .

68

5.1

Comparison of computation aspects. . . . . . . . . . . . . . . . . . . . . .

81

6.1

Material constants in mesh objectivity study. . . . . . . . . . . . . . . . .

117

6.2

Elastic constants considered in the validation with 3D FEA. . . . . . . .

132

6.3

Constants of CDM model in the validation with 3D FEA. . . . . . . . .

132

6.4

Computation aspects of bending examples with CDM. . . . . . . . . . .

139

6.5

Elastic properties of glass fiber-reinforced epoxy lamina. . . . . . . . . .

142

6.6

Constants of the modified Hashin damage initiation criterion evolution laws in matrix tensile damage mode. . . . . . . . . . . . . . . . . . . . . .

142

6.7

Layup and geometry of the beam coupons. . . . . . . . . . . . . . . . . .

143

6.8

Elastic properties of carbon fiber-reinforced epoxy lamina. . . . . . . . .

152

6.9

Material properties of initial normal damage thresholds of carbon fiberreinforced epoxy lamina. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

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4.7 4.8

x Table

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6.10 Material properties of initial shear damage thresholds of carbon fiberreinforced epoxy lamina. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

6.11 Material properties of damage evolution of carbon fiber-reinforced epoxy lamina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xi

LIST OF FIGURES Figure

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1.1

Testing the strength of materials by beam, taken from [1]. . . . . . . . .

2

1.2

Applications of beams in aeronautics and astronautics. . . . . . . . . . .

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1.3

Applications of beams in mechanical and civil engineering. . . . . . . . .

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1.4

Cellular structures formed by beams, taken from [9]. . . . . . . . . . . .

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1.5

Schematic of slightly twisted trapeze being restored to untwisted state by indicated force components, taken from [24]. . . . . . . . . . . . . . . . .

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1.6

Flowchart of nonlinear composite beam modeling. . . . . . . . . . . . . .

14

2.1

Schematic of undeformed and deformed beam. . . . . . . . . . . . . . . .

18

3.1

Schematic of stress-strain curve of a generalized elastic damageable material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Schematic of damage versus strain or stress of a generalized elastic damageable material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Plot of Eq. (4.76) for an elliptical section,taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Plot of Eq. (4.79) for a rectangular section, taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Plot of Eq. (4.81) for an elliptical section (F1 /EA = 0.005), taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers. . . . . . . . . . . . . . . . . . . . . .

50

u1 of the middle position of an elliptic-section beam, taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers. . . . . . . . . . . . . . . . . . . . . . . .

53

Sectional geometry of the spring steel strip. . . . . . . . . . . . . . . . . .

59

3.2 4.1

4.2

4.3

4.4

4.5

xii Figure

Page

4.6

Trapeze effect of the spring steel strip. . . . . . . . . . . . . . . . . . . . .

61

4.7

Plot of Eq. (4.131) in case of the spring steel strip. . . . . . . . . . . . . .

62

4.8

Sectional layout of the rotor blade. . . . . . . . . . . . . . . . . . . . . . .

63

4.9

3D mesh of the free end of the blade in ANSYS. . . . . . . . . . . . . . .

64

4.10 Geometry and layout of the anisotropic pipe section. . . . . . . . . . . .

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4.11 Twisting moment (lbf-in) to preserve a tip rotation of 5 degree under various levels of axial strains for the composite pipe. . . . . . . . . . . . .

67

4.12 Geometry of the cylinder section. . . . . . . . . . . . . . . . . . . . . . . .

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4.13 Data comparison of the axial tensile force for various values of the axial strain for the beam with the cylindrical section. . . . . . . . . . . . . . .

69

4.14 Data comparison of the axial twisting moment for various values of the twist for the beam with the cylindrical section. . . . . . . . . . . . . . . .

70

4.15 Data comparison of the Poynting effect for the beam with the cylindrical section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.16 Geometry of the tube section. . . . . . . . . . . . . . . . . . . . . . . . . .

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4.17 Data comparison of the torque for various values of the twist for the beam with the tubular section and Yeoh model. . . . . . . . . . . . . . . . . . .

72

4.18 Plot of VABS (Yeoh) prediction of the tangent coefficient of the twist in Figure 4.17 for the beam with the tubular section. . . . . . . . . . . . . .

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4.19 Data comparison of the Poynting effect for the beam with the tubular section and Yeoh model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.20 Plot of VABS (Yeoh) prediction of the tangent coefficient of the axial strain in Figure 4.19 for the beam with the tubular section. . . . . . . .

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4.21 Data comparison of the axial tensile force for various values of axial strain for the beam with the tubular section and Yeoh model #1. . . . . . . .

74

4.22 Plot of VABS (Yeoh, Material #1) prediction of the tangent coefficient of the axial strain in Figure 4.21 for the beam with the tubular section. . .

75

5.1

Cross-sectional geometry, layout, and mesh. . . . . . . . . . . . . . . . . .

78

5.2

Partition of the 3D [45/ − 45]s square-section composite beam. . . . . .

79

5.3

Comparison of extensional constitutive data. . . . . . . . . . . . . . . . .

80

5.4

Tangent beam stiffness predicted by VABS as a function of axial strain.

82

xiii Figure 5.5

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Reduction of the natural frequency due to pre-strain and shear nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.6

Comparison of bending constitutive data. . . . . . . . . . . . . . . . . . .

83

5.7

Comparison of twisting constitutive data. . . . . . . . . . . . . . . . . . .

84

5.8

Comparison of stress σ11 contour plots at γ = 0.01556. . . . . . . . . . . .

86

5.9

Comparison of stress σ12 contour plots at γ = 0.01556. . . . . . . . . . . .

87

5.10 Comparison of stress σ13 contour plots at γ = 0.01556. . . . . . . . . . . .

88


89


90


91

5.14 Comparison of stress σ13 curve plots at γ = 0.01556. . . . . . . . . . . . .

92


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5.17 Comparison of stress σ11 contour plots at κ2 = 2.0 rad/m. . . . . . . . . .

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103

5.26 Cross-sectional geometry and layout of the tube section. . . . . . . . . .

106

5.27 Comparison of performances of virtual beam coupons. . . . . . . . . . .

107

5.28 Local stress fields on the cross-sectional domain of ASTM D3518 virtual coupon at γ = 0.01556. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

5.29 Comparison of the in-plane stresses of ASTM D3518 virtual coupon predicted by VABS and CLPT. . . . . . . . . . . . . . . . . . . . . . . . . . .

109

xiv Figure 6.1

Page

Geometry and boundary conditions in 3D FEA of isotropic elastic damageable beam in tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

6.2

Comparison of beam tensile constitutive curves with isotropic damage.

113

6.3

Four different VABS mesh schemes in the mesh objectivity study. . . . .

115

6.4

Beam tensile constitutive law under isotropic damage driven by Eqs. (6.8), (6.11), and (6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

6.5

Beam bending constitutive laws with isotropic damage driven by Eq. (6.8).

119

6.6

Beam bending constitutive laws with isotropic damage driven by Eqs. (6.11) and (6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7

Damage contour at κ2 = 0.8 rad/m with damage driven by Eq. (6.8). . .

120

6.8

Mesh of the tube section. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

6.9

Damage field of the tube section at twist curvature κ1 = 0.019 rad/m. .

122

6.10 Cross-sectional contour of elastic equivalent shear strain. . . . . . . . . .

123

6.11 Twisting constitutive law with damage driven by Eq. (6.13). . . . . . . .

124

6.12 Warping of the damaged cross section. . . . . . . . . . . . . . . . . . . . .

125

6.13 Damage of the cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . .

125

6.14 Shear strain 2Γ12 on damaged cross section. . . . . . . . . . . . . . . . . .

126

6.15 Shear strain 2Γ13 on damaged cross section. . . . . . . . . . . . . . . . . .

126

6.16 Convergence of the energy release rate with respect to the mesh refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

6.17 Irregular mesh refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

6.18 Damage of the cross-section predicted by the mesh in Figure 6.17. . . .

128

6.19 Twisting moments on damage configuration versus small softening regulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

6.20 Twisting moments on damage configuration versus large softening regulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

6.21 Partition and mesh of the 3D composite beam. . . . . . . . . . . . . . . .

133

6.22 Comparison of beam tensile constitutive law of [0/903 ]8s coupon predicted by VABS and 3D FEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

6.23 Contour of damage dm on the small part of [0/903 ]8s coupon at γ = 0.015 in 3D FEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

xv Figure

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6.24 Comparison of matrix damage dm contour plots of [0/903 ]8s coupon at γ = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

6.25 Contour plot of Uz of [0/45/90/ − 45]8s coupon under bending predicted by 3D FEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136

6.26 Comparison of beam bending constitutive law of [0/45/90/ − 45]8s coupon predicted by VABS and 3D FEA. . . . . . . . . . . . . . . . . . . . . . . .

137

6.27 Contour of damage dm on the small part of [0/45/90/ − 45]8s coupon at κ2 = 3.6 rad/m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

6.28 Comparison of matrix damage dm contour plots of [0/45/90/−45]8s coupon at κ2 = 3.6 rad/m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

138

6.29 Cross-sectional schematic of laminate beam coupons. . . . . . . . . . . .

143

6.30 Comparison of tensile constitutive curves of [0/45/90/ − 45]2s coupon. .

144

6.31 Matrix tensile damage dmt contour of tensile simulation of [0/45/90/−45]2s coupon at axial strain γ = 0.01 by VABS. . . . . . . . . . . . . . . . . . .

144

6.32 Comparison of tensile constitutive curves of [30/60/90/−60/−30]2s coupon.

145

6.33 Matrix tensile damage dmt contour of tensile simulation of [30/60/90/ − 60/ − 30]2s coupon at axial strain γ = 0.007 by VABS. . . . . . . . . . . .

146

6.34 Comparison of tensile constitutive curves of [60/0/ − 60]3s coupon. . . .

146

6.35 Matrix tensile damage dmt contour of [60/0/ − 60]3s coupon at axial strain γ = 0.006 by VABS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

6.36 Comparison of bending constitutive curves of [0/45/90/ − 45]8s coupon.

147

6.37 Matrix tensile damage dmt contour of [0/45/90/ − 45]8s coupon by VABS at curvature κ2 = 3.6 rad/m. . . . . . . . . . . . . . . . . . . . . . . . . . .

148

6.38 Comparison of tensile constitutive curves of [30/60/90/ − 60/ − 30]2s carbon/epoxy coupon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

6.39 Damage d11 contour of [30/60/90/ − 60/ − 30]2s carbon/epoxy coupon by VABS at γ = 0.012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

154


154


155

6.42 Comparison of tensile constitutive curves of [0/45/90/−45]2s carbon/epoxy coupon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

xvi Figure

Page

6.43 Damage d22 contour of tensile simulation of [0/45/90/−45]2s carbon/epoxy coupon by VABS at γ = 0.012. . . . . . . . . . . . . . . . . . . . . . . . . .

156

6.44 Damage d33 contour of tensile simulation of [0/45/90/−45]2s carbon/epoxy coupon by VABS at γ = 0.012. . . . . . . . . . . . . . . . . . . . . . . . . .

156

7.1

Cross-sectional geometry of rubber beam. . . . . . . . . . . . . . . . . . .

161

7.2

Comparison of numerical and analytical solutions of extension-bending coupling behavior predicted by VABS. . . . . . . . . . . . . . . . . . . . .

162

7.3

Schematic of FE model of the cantilevered rubber beam. . . . . . . . . .

163

7.4

Contour plot of the displacement solution of 3D FEA for M2 = 0.01 Nm.

163

7.5

Axial strain γ induced by bending moment M2 . . . . . . . . . . . . . . .

164

7.6

Data comparison of predictions to the Brazier effect. . . . . . . . . . . .

165

7.7

Deformed tube sections with respect to bending curvatures κ2 . . . . . .

166

xvii

ABSTRACT Jiang, Fang Ph.D., Purdue University, May 2017. Composite Beam Theory with Material Nonlinearities and Progressive Damage. Major Professor: Wenbin Yu. Beam has historically found its broad applications. Nowadays, many engineering constructions still rely on this type of structure which could be made of anisotropic and heterogeneous materials. These applications motivate the development of beam theory in which the impact of material nonlinearities and damage on the global constitutive behavior has been a focus in recent years. Reliable predictions of these nonlinear beam responses depend on not only the quality of the material description but also a comprehensively generalized multiscale methodology which fills the theoretical gaps between the scales in an efficient yet high-fidelity manner. The conventional beam modeling methodologies which are built upon ad hoc assumptions are in lack of such reliability in need. Therefore, the focus of this dissertation is to create a reliable yet efficient method and the corresponding tool for composite beam modeling. A nonlinear beam theory is developed based on the Mechanics of Structure Genome (MSG) using the variational asymptotic method (VAM). The three-dimensional (3D) nonlinear continuum problem is rigorously reduced to a one-dimensional (1D) beam model and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities by exploiting the small geometric parameter which is an inherent geometric characteristic of the beam. The 2D nonlinear cross-sectional analysis utilizes the 3D material models to homogenize the beam cross-sectional constitutive responses considering the nonlinear elasticity and progressive damage. The results from such a homogenization are inputs as constitutive laws into the global nonlinear 1D beam analysis.

xviii The theoretical foundation is formulated without unnecessary kinematic assumptions. Curvilinear coordinates and vector calculus are utilized to build the 3D deformation gradient tensor, of which the components are formulated in terms of crosssectional coordinates, generalized beam strains, unknown warping functions, and the 3D spatial gradients of these warping functions. Asymptotic analysis of the extended Hamiltonian’s principle suggests dropping the terms of axial gradients of the warping functions. As a result, the solid mechanics problem resolved into a 3D continuum is dimensionally reduced to a problem of solving the warping functions on a 2D crosssectional field by minimizing the information loss. The present theory is implemented using the finite element method (FEM) in Variational Asymptotic Beam Sectional Analysis (VABS), a general-purpose crosssectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model in the form of the Euler-Bernoulli beam theory. The deformation gradient tensor is directly used to enable the capability of dealing with finite deformation, various strain definitions, and several types of material constitutive laws regarding the nonlinear elasticity and progressive damage. Analytical and numerical examples are given for various problems including the trapeze effect, Poynting effect, Brazier effect, extension-bending coupling effect, and free edge damage. By comparison with the predictions from 3D finite element analyses (FEA), 2D FEA based on plane stress assumptions, and experimental data, the structural and material responses are proven to be rigorously captured by the present theory and the computational cost is significantly reduced. Due to the semi-analytical feature of the code developed, the unrealistic numerical issues widely seen in the conventional FEA with strain softening material behaviors are prevented by VABS. In light of these intrinsic features, the nonlinear elastic and inelastic 3D material models can be economically calibrated by data-matching the VABS predictions directly with the experimental measurements from slender coupons. Furthermore, the global behavior of slender composite structures in meters can also be effectively characterized

xix by VABS without unnecessary loss of important information of its local laminae in micrometers.

xx

1

1. INTRODUCTION 1.1

Overview and Motivation

1.1.1

Slender Solids and Structures

Many structures possess a common feature which is often referred to as the slenderness, that is, one dimension is much larger than the other two. These structures with this common feature are commonly called beams, as they have been used in many fields. This unique feature of beams has historically found its wide applications. Mechanical properties of materials are often obtained by testing of coupons which are easy to be manufactured and handled. The slenderness of beams makes them be selected as perfect candidates to this goal. An early example dates back to 16th century [1]. Galileo wrote his world-famous book “Two New Sciences” in his last eight years. In this treatise, beams are used to investigate the strength of materials, as shown in Figure 1.1, which can be regarded as the first publication in the strength of the materials. Now, beam-like coupons are commonly used in the standards for testing the mechanical properties of the new materials [2–4]. Other than material characterization, beams also have many other applications in aeronautics (e.g., the helicopter rotor blade in Figure 1.2 (a)), space technology (e.g., the deployable boom in Figure 1.2 (b)), mechanical engineering (e.g., the wind turbine blade in Figure 1.3 (a)), and civil engineering (e.g., the bridge construction in Figure 1.3 (b)). Inspired by the superior mass efficiency possessed by natural cellular materials, human-made cellular solids, such as the ultra-light-weight lattice in Figure 1.4 (a), have recently become a research focus. If a structure is made up of an interconnected network of beams, as shown in Figure 1.4 (b), it falls into the category of the cellular solid [5]. When the length of the constituent micro-beams is at least one order of magnitude smaller than the macroscopic scale of the lattice, then the lattice can be treated as a mate-

2 rial with homogenized properties [6, 7]. Here only a brief survey of applications of beams is given, which provides the motivation for developing accurate models of these structures.

(a) Tensile test.

(b) Bending test.

Figure 1.1. Testing the strength of materials by beam, taken from [1].

(a) Chinook helicopter, courtesy of Boeing.

(b) Space deployable scheme, taken from [8].

Figure 1.2. Applications of beams in aeronautics and astronautics.

3

(a) Wind turbine, courtesy of Siemens.

(b) Golden Gate bridge, courtesy of Wikipedia.

Figure 1.3. Applications of beams in mechanical and civil engineering.

(a) Lattice.

(b) Unit cell.

Figure 1.4. Cellular structures formed by beams, taken from [9].

4 1.1.2

Mechanics of Structure Genome

As can be seen from the increasing interests to structured materials, it is not the intrinsic property of the single constituent, but that of the composed entirety we are usually interested in. As a result, with such increasing capabilities of engineering microstructure of materials, the traditional boundary between structure and material is quickly disappearing. Hence, analyzing these advanced materials should be integrated with structural analysis. The structural analyses are routinely carried out using the FEA with solid elements, plate or shell elements, or beam elements. The constitutive relation of the corresponding structural element is built regarding the generalized strain and stress. For isotropic homogeneous structures, material properties such as Young’s modulus and Poisson’s ratio are direct inputs for structural analysis using 3D solid elements, and these properties combined with geometric properties of the structure can be used for plate/shell/beam elements. However, such straightforwardness does not exist for structures featuring anisotropy or heterogeneity. Consider a typical composite rotor blade of length 8.6 m and chord 0.72 m with a main D-spar composed of 60 graphite/epoxy plies each with a ply thickness of 125 µm. To directly use the properties of graphite/epoxy composite plies in the blade analysis, at least one 3D solid element through the ply thickness should be used. Supposing one uses 20-noded brick elements with a 1 to 10 thickness-length ratio, it is estimated that around ten billion degrees of freedom are needed for the blade analysis. To this end, models are needed to bridge the material properties of the composite plies and the beam properties, and compute the stress fields within the original material for failure and safety prediction. The Structure Genome (SG) is defined as the smallest mathematical building block of the structure [10]. For example, the structural analysis of slender structures can use beam elements. If the beam has uniform cross-sections which could be made of homogeneous materials or composites, its SG is the 2D cross-sectional domain because the cross-section can be repeated along the beam reference line to form the

5 structure. This inspires a new perspective toward beam modeling, an important topic in structural mechanics. If the beam reference line is considered as a 1D continuum, every material point of this continuum has a 2D cross-section as its microstructure. In other words, constitutive modeling for beams can be effectively viewed as a specific application of micromechanics.

1.1.3

Linear Cross-Sectional Analysis of Composite Beam

A systematic approach for modeling composite beams has been developed by Hodges and his co-workers during the last three decades [11–22]. This approach uses the VAM [23] to rigorously split the original 3D geometrically nonlinear and materially linear problem of the slender structure into a 1D global beam analysis and a linear 2D cross-sectional analysis. The 2D cross-sectional analysis is implemented using FEM in the computer code VABS. VABS provides the constitutive relations needed for the 1D global beam analysis and recovery relations needed to compute the pointwise fields within the original 3D structure based on the global beam behavior. There also exist many nonlinear and inelastic effects in the mechanical behaviors of the slender structures. To explain these phenomena, one should explore beyond the linear constant stress-strain relation.

1.1.4

Trapeze and Poynting Effects

The term “trapeze effect” refers to a phenomenon of beams that can cause the effective torsional rigidity to change with axial force or cause a pretwisted beam to untwist when loaded by an axial tensile force [24]. The trapeze effect is so named because of the tendency of a trapeze which, when twisted slightly by an angle φ as in Figure 1.5, tends to restore itself to zero twist angle with a total restoring moment given by 2(a/2)T aφ/(4`) = T a2 φ/(4`). There are both linear and nonlinear aspects of the trapeze effect in the beam theory. The linear effect is caused by interaction between the initial twist and axial

6 force, tending to untwist the beam. For the nonlinear effect, an axial tension force tends to untwist an elastically twisted beam, thereby increasing its effective torsional stiffness. For example, in a helicopter rotor blade which is subject to tension because of rotation, the trapeze effect will lead to a slight increase of its torsional frequencies and will untwist a pretwisted blade.

Figure 1.5. Schematic of slightly twisted trapeze being restored to untwisted state by indicated force components, taken from [24].

A century has passed since the first observations of increasing torsional vibration frequency of beams under tension [25, 26]. Buckley [27] proposed an intuitive bifilar theory with the Poisson effect completely neglected. Wagner [28] extended the bifilar hypothesis into a multi-filar theory for a buckling problem. Biot [29] studied this effect with initial stress and small displacement assumptions. Goodier [30] utilized the second Piola-Kirchhoff stress (PK2) and the deformation gradient to build the stress equilibrium equations. It is noticed that Goodier named the PK2 as Trefftz stress which was differently defined and applied by Hill [31] in plasticity problems. In these theories, unnecessary assumptions on the sectional deformation make it impos-

7 sible to take the in-plane warping into account. Houbolt and Brooks [32] concluded the additional torsional rigidity results from axial force F1 to be F1 (J0 /A). J0 /A is the square of the polar radius of the gyration of the cross section about the elastic axis. This conclusion is widely accepted. Hodges and Dowell [33] obtained a similar nonlinear behavior. Borri and Merlini [34] developed a geometric cross-sectional stiffness accounting for prestress by using Saint-Venant/Kirchhoff material (St-V/K) model. Under the assumption that the prestress does not change the warping, their conclusion agrees with [32,33]. Fulton and Hodges [35,36] also derived the additional torsional stiffness as F1 (J0 /A) by neglecting the in-plane deformation. Hodges and his co-workers [16, 21] introduced a rigorous approach based on the VABS approach using St-V/K, showing good correlation with experiments in [37] for the laminated strip-like beams. Pretwisted feature produces torsion-extension coupling [38, 39]. Okubo [40, 41] solved the Navier’s equations by a set of harmonic functions. Chu [42] assumed that the fibers in a pretwisted beam formed a bunch of helices and the normal stress had the direction of these longitudinal spiraled fibers. Houbolt and Brooks [32] also adopted this assumption in deriving their extension-torsion coupling stiffness, i.e., F1 k1 J0 /A (to the first order of pretwist k1 ). This formulation is only valid for a very thin section. Shorr [43] made an expedient assumption — that the undeformed section of a pretwisted beam was not planar but was instead a curved surface defined by the Saint-Venant warping of the prismatic beam. Petersen [44] also adopted this assumption named as the “prewarping”. Washizu [45] improved the method of modeling the trapeze effect by using curvilinear coordinates. Ohtsuka [46] extended Washizu’s model into a generalized nonlinear beam theory with the numerical solution. The theory for this problem was mostly matured by the 1980s. The models developed during this period by Hodges [47], Rosen [48–51], Shield [52], and Krenk [53, 54] resulted in a comprehensive agreement with each other. Nevertheless, the in-plane warping of the cross section was tacitly bypassed by the uniaxial stress assumption. Shield [52]

8 a priori added the sectional contraction of magnitude νγ into the deformation mode, with ν as the Poisson’s ratio and γ as the elongation of the beam. During the development of VABS, the finite strain was briefly studied using StVenant/Kirchhoff material model for the problem of trapeze effect [16], in which one or more of the 1D moment strain measures could be larger than the other(s) [21]. Solid validation in [16] with experimental results from [37] shows that the StVenant/Kirchhoff model is capable of capturing the geometric nonlinearity for small strains in beams with thin cross sections. The nonlinear behavior in cylindrical tubes and circular wires was observed by Poynting [55, 56] firstly. From the similar experiments, lengthening is measured from the wires under torsion [57–60]. This phenomenon is further named as the positive Poynting effect. Freudenthal and Ronay [61] noticed that this type of behavior was caused not only by geometrical nonlinearity but also by the physical nonlinearity. The studies on the rubbers assist the investigations to this phenomenon. Rivlin and Saunders [62] observed the positive Poynting effect in the test of rubber cylinder and explained it by using Mooney-Rivlin material (M-R) model. Rivlin [63] found that the azimuthal and normal surface traction (of which the latter one is in correspondence with the elongation) must be applied to the ends of the rubber cylinder under twisting to keep it in equilibrium. The phenomenological material models, such as that introduced by Mooney [64], are taken by numerous researchers [65–70] to model the Poynting effects. Anand’s [71, 72] investigations on the use of Hencky’s Approximate Strain Energy (HASE) trigger a wide focus on this material description in the modeling of hyperelastic cylinders [73]. Bruhns et al [74] explained the Poynting effect of an elastic hollow cylinder under moderate twist κ1 using HASE. Experimental discoveries on biopolymer gels also exhibit the so-called negative Poynting effect [75,76], from which shortening is measured from the bio-cylinders under torsion. Then the mechanism of causing the difference between positive and negative Poynting effects is investigated. Zubov [77] showed that the Neo-Hookean material (N-H) model will result in the positive Poynting effect. Mihai and Goriely [68,69]

9 studied the generalized empirical inequality. It is also known that the negative Poynting effect can result from advanced isotropic homogeneous elasticity model [78], as well as anisotropy and inhomogeneity [67, 79, 80]. In addition, the so-called axial force-twist effect which behaves in the same manner as trapeze effect is also explicitly studied in [78, 80]. The positive Poynting effect is much more common in engineering structures. Jiang et al [81,82] applied various hyperelastic models to analyze both of the positive and the negative Poynting effects in beams based on VABS theory.

1.1.5

Nonlinear In-plane Shear

Fiber-reinforced plastic composites (FRP) exhibit physically nonlinear behaviors in both elastic and inelastic regions. The primary cause of the elastic nonlinearity is discovered to be the lamina shear stresses once they are relatively large compared to the longitudinal tensile stresses. In this situation, the resin matrix dominates in the mechanical performance of the composites. Consequently, because the shear stressstrain responses of polymer resins are nonlinear over entire strain range and at very low strain levels, the in-plane shear responses of FRP plies are nonlinear over the entire range examined and at the very near outset of deformation process when the elastic strains are dominant [83]. Several mathematical models have been published to describe the nonlinear stressstrain responses. Hahn and Tsai [84] developed a model by employing a plane-stress complementary energy function which contains a biquadratic term for in-plane shear stress. Stress field predicted by such a constitutive model is used to formulate the failure criterions which are successful in predicting the failure due to stress concentrations [85,86]. Another widely used model is the Ramberg-Osgood equation [87] which is also popular in metal fatigue studies. A more flexible modeling methodology is to utilize the mathematical curve fitting functions [88–90]. A comprehensive review of the nonlinear constitutive models for shear nonlinearity can be found in [91].

10 There are two main motivations to study the nonlinear shear stress-strain behaviors in composite beams. Firstly, the knowledge of the nonlinear shear stress-strain response of the composites can be obtained from the measurements of loaded slender coupons. For example, the ASTM D3518/D3518M standard test method [4] for “in-plane shear response of polymer matrix composite materials by the tensile test of ±45○ laminate” is based on the measured uniaxial force-strain response of a symmetrically ±45○ -laminated coupon. A rigorous beam model can serve as a virtual coupon to relate the uniaxial force-strain response precisely with the 3D stress and strain fields by the cross-sectional analysis. Consequently, the beam model can be used along with the data matching tools to calibrate the material constants built into the material descriptions. Secondly, the nonlinear in-plane shear responses have impacts on the 1D constitutive responses of beams. Predictions of static failure loads and natural frequencies of composites beams are affected by the predefined 3D nonlinear stress-strain models.

1.1.6

Damage of Composite Materials

Revealing damage mechanisms has been a focus of the material science for decades. Researchers working on this topic have been developing various types of models, as shown in the worldwide failure exercise (WWFE) [92]. Studying the damaged behaviors of beams is being of interest due to the applications in helicopter rotors and wind turbine blades. Ganguli et al [93] modeled cracks in the rotor blade which cause a local reduction in blade stiffness. In this work, the damaged beam stiffness is modeled by multiplying (1−D) with the elastic beams stiffness, where the damage variable D ranges from zero for no crack to one for a complete crack. With the given value of D for preselected locations on the blade reference line, the frequencies of the clamped beam are computed. Pawar and Ganguli [94,95] studied the effect of the progressive damage on the beam constitutive law focusing on the application of health monitoring for thin-walled composite rotor blades. Pollayi and

11 Yu [96] studied the effects of the matrix micro-cracking on the overall cross-sectional stiffness within VABS framework. Carrera et al [97] examined the free-vibration of damaged aircraft structures based on Carrera unified formulation. By definition, the damage is the deterioration which occurs in the material before the final collapse and failure [98]. The continuum damage mechanics (CDM) is developed to describe the damage including the initiation and the evolution of micro-defects (micro-cavities and micro-cracks) under the assumption that the solid containing these micro-defects can be considered as a continuum. Calibration of a CDM model could be based on the measurements of the macroscopic behavior of the damaged material. To this end, the foundational prerequisite of CDM is that the material degradation caused by these micro-defects can be represented by a set of continuous variables, that is, the damage variables. Kachanov [99] firstly proposed the definition of the damage variable. Rabotnov [100] introduced the fundamental concept of the effective stress. Anisotropic damage variables and corresponding evolution laws were also proposed by Sidoroff [101], Chow and Wang [102], Voyiadjis et al [103], and Lemaitre [104]. Several CDM models were proposed for fiber-reinforced composite materials by Talreja [105], Ladeveze and Le Dantec [106], Matzenmiller et al [107], Barbero and De Vivo [108], Maim´ı [109,110], and Lapcyzk and Hurtado [111]. Rigorous prediction of the impact of local material damage on the global structural behavior relies on not only the quality of the material description but also a comprehensive multiscale methodology which fills the theoretical gaps between the scales in an efficient yet high-fidelity manner. In light of this fact, the types of beam modeling methodologies which are built upon ad hoc assumptions lack such reliability in need.

1.1.7

Large Bending Effects

There are two primary nonlinear behaviors observed in beams experiencing large bending curvature, namely, the Brazier effect and extension-bending coupling effect.

12 Brazier [112] found that when one dimension of the cross section was relatively smaller than the others, then even while the strains remained very small, large displacements over the cross section might occur. Under this circumstance, the so-called Brazier effect deals with the cross-sectional in-plane ovalisation which leads to the bending capacity limit. Shell theories find their successes in this application since composite laminates are widely used in the thin-walled structures. Jamal and Karyadi [113] used the plane stress assumption to study the collapse of composite cylinder. Fuchs et al [114] classified the collapse modes of composite cylinders into three categories: the short wavelength buckling, the ovalization, and the local buckling. Corona and Rodrigues [115] analytically studied the response of long thin-walled, cross-ply composite tubes subjected to pure bending. Karam and Gibson [116, 117] analyzed the elastic buckling of a thin cylindrical shell supported by an elastic core. Tatting et al [118, 119] developed a nonlinear solution to the Brazier problem of cross-sectional flattening for infinitely long tubes under bending. Li [120] predicted the critical instability loads of orthotropic composite tubes under pure bending based on the assumption that the instability under pure bending results from the ovalization of its cross section. Harursampath and Hodges [121] obtained an asymptotically corrected model for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. Cecchini and Weaver [122] optimized the ply angles of laminate layers to resist the Brazier effect in orthotropic tubes by increasing the critical failure bending load. They also analyzed the Brazier effect for multibay airfoil sections without the consideration of the material failure [123]. Another interesting topic in beam theory is to determine the exact stationary profile of the deflection curve of a slender flexible elastic solid with given endpoint boundary conditions. This problem is named as the Elastica. James Bernoulli in 1691 firstly posed the problem and partially solved it. Daniel Bernoulli completed the general definition of Elastica. He also proposed the variational statement which was solved by Euler in 1744. Since then, many approaches have been proposed [124–

13 126]. Elastica theory historically found its application in the design of compliance mechanisms [127, 128]. It is also seen in the modern study of soft robots [129–131]. In most of the cases, the mathematical modelings are based on the assumption of inextensibility and linear elasticity. The bending stiffness is estimated as EI, where E is Young’s modulus, and I is the second moment of the cross-sectional area. When the axial load is considered the axial deformation rigidity is assumed to be EA, where A is the cross-sectional area.

1.2

Objectives and Outline Conventional beam theories accounting for nonlinear elasticity and progressive

damage are built upon unnecessary ad hoc assumptions which limit their accuracies and generalities. The focus of the present work is to bridge the theoretical gap between the local physical laws and the mechanics of slender solid composed of the materials which are governed by these laws. The ultimate goal of this research is to develop an efficient high-fidelity composite beam theory considering material nonlinearities and progressive damage. The theoretical foundation of the present beam theory is introduced in Chapter 2. As shown in Figure 1.6, the 3D continuum is rigorously reduced to a 1D beam analysis and a 2D cross-sectional analysis featuring both geometric and material nonlinearities. The theoretical foundation is formulated without unnecessary kinematic assumptions. Curvilinear coordinates and vector calculus are utilized to build the 3D deformation gradient tensor, of which the components are formulated in terms of cross-sectional coordinates, generalized beam strains, unknown warping functions, and the 3D spatial gradients of these warping functions. Asymptotic analysis of the extended Hamiltonian’s principle suggests dropping the terms of axial gradients of the warping functions. As a result, the solid mechanics problem resolved into a 3D continuum is dimensionally reduced to a problem of solving the warping functions on a 2D cross-sectional domain by minimizing the information loss.

14

3D Anisotropic Continuum Mechanics Dimensional Reduction Using VAM

Cross-Sectional Analysis (Material and Geometrical Nonlinearities)

Nonlinear Sectional Properties

Nonlinear Recovery Relations

Global Beam Analysis (Material and Geometrical Nonlinearities) Global Behavior

3D Displacement/Strain/Stress/Damage Fields

Figure 1.6. Flowchart of nonlinear composite beam modeling.

In Chapter 3, the present theory is implemented using the FEM in VABS. Finite deformation, nonlinear elastic stress-strain law, and continuum damage constitutive relations are considered for the beam model in the Euler-Bernoulli type. In Chapter 4, the VABS theory is used to investigation the trapeze and Poynting effects for prismatic and pretwisted beams under axial load and torsion. The analytical analysis is carried out, in which several hyperelastic models are taken into the consideration, and their differences are discussed in details. The limitations of the uniaxial stress assumption and St-V/K model are discussed. Compared with the widely accepted results in the literature, the present theory demonstrates improved results without introducing assumptions commonly used in other works. It is concluded that the trapeze and Poynting phenomena are governed by the material models and warping functions, and nonlinearly coupled extension and torsion can be eliminated by properly selecting the thickness-to-width ratio. Numerical examples are also used to validate the theory and the companion code. The trapeze and Poynting effects of various sample beam structures are simulated successfully, which demonstrates excellent agreement with 3D FEA. In Chapter 5, the Hahn-Tsai [84] nonlinear in-plane shear model is used to the validation by comparing the VABS results with those from 3D FEA. The ±45○ -laminated

15 coupon tensile tests are simulated. 3D local fields such as the free-edge stresses are precisely captured by the present model. Nominal stress-strain curves predicted for various composite beams with different cross-sections are compared to show the impact of the cross-sectional designs of the coupons on their performances in calibrating the material constants. In Chapter 6, the present theory is implemented to capture the high-fidelity elasticto-damaged beam constitutive behaviors by the cross-sectional analyses. The mesh objectivity feature of VABS theory on the prediction using damage softening constitutive law is investigated analytically and validated numerically. Various continuum damage models in literature are taken into the present beam model from which the predicted beam constitutive relations and local fields are compared to the data results from 3D solid element analysis, 2D plate element analysis, and experiments. In Chapter 7, the extension-bending coupling behavior of rectangular-sectioned rubber Elastica is analyzed by VABS which is also validated by comparison with 3D FEA. The Brazier effect of a long tube is also analyzed by VABS which captures the large ovalizing warping of the cross section. Finally, Chapter 8 summarizes and concludes this dissertation.

16

17

2. THEORETICAL FOUNDATIONS 2.1

Kinematics Theories for slender beams with arbitrary cross sections, undergoing small strain

and large deflection, and made of linearly elastic composite materials are well established. As pointed out in [16], the Jaumann-Biot-Cauchy strain tensor obtained through decomposition of rotation tensor is only attractive when the assumption of small local rotation is valid. To deal with fully nonlinear elastic behavior, we need to eliminate this restriction. Therefore, the deformation gradient tensor directly plays an important role in the kinematics. In Figure 2.1, e i for i=1, 2, 3 are fixed dextral, mutually perpendicular unit vectors in the absolute reference frame, and r 0 and R0 denote the position vectors of a material point on the reference line of the undeformed and deformed configurations, respectively. (Here and throughout this paper, except where explicitly indicated, Greek index, α, assumes values, 2 and 3 while Latin indices, a, b, c, d, i, j, k, l, m, n, p, and q, assume 1, 2, and 3. Repeated indices are summed over their range except where explicitly indicated.) b i and B i are the orthogonal triads attached to the cross section in the undeformed and deformed configurations, respectively. In addition, we assume that the cross section of the beam is uniform along the reference line. Then, the position vectors of material points in the undeformed and the deformed configurations, respectively, are r = r 0 + xα b α

(2.1)

R = R0 + xα B α + wi (x1 , x2 , x3 )B i

(2.2)

with R0 = r0 +u. wi represent the 3D unknown warping functions, which describe the difference between the positions of material points in the deformed beam and those

18

b3

b2

g2

g3 b 1

g1 r0

u

G1

e3 e1

B3 G2 B2 G3

R0

e2

B1

Figure 2.1. Schematic of undeformed and deformed beam.

that can be described by the deflection of the reference line and the rotation of the reference cross section. Note in Eq. (2.2), we actually express R in terms of R0 , B i , and wi , which is six times redundant. Six constraints are needed to ensure a unique mapping. These constraints are directly related with how we define R0 and B i in terms of R. Following [18], we can choose B 1 tangent to the reference line (effectively two constraints) and introduce the following four constraints on the warping functions ⟨wi ⟩ = 0

⟨w2,3 − w3,2 ⟩ = 0

(2.3)

where ⟨ ⟩ denotes the integration over the cross section. To derive a theory of the EulerBernoulli type, we define the generalized 1D beam strains by R0′ = (1 + γ)B1

(2.4)

Bi′ = (κj + kj )Bj × Bi

(2.5)

in which the upper prime denotes derivative to x1 , γ the axial strain, κ1 the twist and κα the curvature. k1 denotes the initial twist. k2 and k3 denote the initial bending curvatures.

19 In Figure 2.1, gi denote the covariant base vectors of the undeformed body which can be expressed by g1 = b1 + xα b′α = (1 − x2 k3 + x3 k2 )b1 − x3 k1 b2 + x2 k1 b3

(2.6)

gα = bα

(2.7)

The square root of the determinant of the undeformed metric tensor gij = gi ⋅ gj ,

√ g,

can be evaluated by √ g = g1 ⋅ (g2 × g3 ) = g1 ⋅ b1 = 1 − x2 k3 + x3 k2

(2.8)

The controvariant base vectors of the undeformed body g i are defined by 1 g 1 = √ b1 g x3 k 1 g 2 = b2 + √ b1 g x2 k 1 g 3 = b3 − √ b1 g

(2.9) (2.10) (2.11)

In Figure 2.1, the covariant base vectors of the deformed configuration, Gi , can be evaluated as Gi =

∂R ∂xi

(2.12)

Together with Eq. (2.5), we have G1 = [1 + γ + w1′ − (x2 + w2 )(κ3 + k3 ) + (x3 + w3 )(κ2 + k2 )]B1 + [w2′ − (x3 + w3 )(κ1 + k1 ) + w1 (κ3 + k3 )]B2

(2.13)

+ [w3′ + (x2 + w2 )(κ1 + k1 ) − w1 (κ2 + k2 )]B3 G2 = w1,2 B1 + (1 + w2,2 )B2 + w3,2 B3

(2.14)

G3 = w1,3 B1 + w2,3 B2 + (1 + w3,3 )B3

(2.15)

It is worth to note that the product of warping function with the curvatures, wi (κj + kj ), are neglected in [16] for modeling the trapeze effect. However, neglecting these terms is no longer valid for large deformations. For example, consider a rubber

20 cylinder under combined extreme twisting and axial stretching, wα κ1 will be in the same asymptotic order of xα κ1 . The deformation gradient tensor can be formulated as F = Gi g i = Fij Bi bj

2.2

(2.16)

Hamilton’s Principle The global behaviors of the slender structures are usually modeled by using the

1D beam theory, for example, the geometrically exact beam theory (GEBT) [22, 132, 133]. The intrinsic equations of motion for 1D problems are derived from Hamilton’s extended principle: ∫t

t2

1

l

∫0 [δK − δU + δW] dx1 dt = δA

(2.17)

where t1 and t2 are arbitrary fixed time, δK is the variation of the 1D kinetic energy for a fixed time, δU is the 1D internal virtual work density, δW is the 1D external virtual work density done by the distributed force and moment per unit length, and δA is the virtual action at the ends of the beam and at the ends of time interval. The bars over variations are used to indicate that the virtual quantities need not to be the variations of functions. In Eq. (2.17), δU is obtained by δU = ⟨ςij δχij ⟩

(2.18)

where ςij and χij are the components of the generalized 3D conjugate stress and strain tensors resolved into the beam cross-sectional triad. Let W denote the 3D strain energy density stored in the deformed configuration. Mathematically, W can be expressed as a function of χij . Then ςij is defined by ςij =

∂W ∂χij

(2.19)

According to [84], it is known that W can be also obtained by W = ςij χij − W ∗

(2.20)

21 where W ∗ denotes the 3D complimentary energy density. Then χij is defined as χij =

∂W ∗ ∂ςij

(2.21)

If real time is not the main factor of the 1D analysis, Eq. (2.17) becomes l

∫0 [−δU + δW] dx1 = 0

(2.22)

where δW contains the two portions δW = δW + δW ∗

(2.23)

where δW is not related to the warping function, while δW ∗ is related to the warping function.

2.3

Dimensional Reduction The static behavior of the structure is now governed by l

∗ ∫0 (δU − δW − δW ) dx1 = 0

(2.24)

Note for simplicity, we used the principle of virtual work governing the static behavior but what is derived here is directly applicable to the dynamic behavior [18] for lowfrequency vibrations. The beam cross-sectional characteristic dimension is denoted as h, and the wavelength of deformation as l, such that h/l ≪ 1. Using the bookkeeping parameter ε to express the magnitude of the maximum strain, we have ε = max(γ, hκ1 , hκ2 , hκ3 )

(2.25)

Based on the previous works on VABS [11–20], it has been shown that wi = O(hε)

(2.26)

Denote the asymptotic order of the damage and/or undamaged elastic constants as µ ¯. Then asymptotical analysis in [18] proves δU ∼ µ ¯h2 ε2

δW ∼ O (¯ µh2 ε2 )

δW ∗ ≪ δW

(2.27)

22 As a result, δW and δW ∗ in Eq. (2.24) disappear in the zeroth-order approximation which is sufficient to construct a beam theory of the EulerBernoulli type. Therefore, we have l

∫0 δUdx1 = 0

(2.28)

Recalling the unique feature of beams, it is clear that the following is still true for the situation under large warping and finite beam strains: wi′ ∼

wi h = O ( ε) ≪ ε L L

(2.29)

In light of Eq. (2.29), the wi′ terms in Eq. (2.13) can be neglected because their contributions to the δU in Eq. (2.28) are much smaller than the contributions of other terms. Neglecting these terms, one may rewrite Eq.(2.28) as δU = 0

(2.30)

23

3. FINITE ELEMENT IMPLEMENTATION In this chapter, FEM is used to provide the numerical approach to solve for unknown variables from Eq. (2.30). The general finite element formulation for the prismatic beam (ki = 0) is given in Section 3.1. With specified conjugate pairs of general 3D stress ς and strain χ, the detailed formulation is given for problems of hyperelasticity in Section 3.2, for problems of nonlinear materials with linear strain definition in Section 3.3, and for problems of progressive damage in Section 3.4.

3.1

General Finite Element Formulation To deal with beams made of general material models and subject to strains with

arbitrary magnitudes, we need to solve the minimization problem in Eq. (2.30) using a numerical technique. To this end, we need to write the general 3D strain χ in a matrix form: χ = $ + χ + (χh + χκ )w

(3.1)

where w = [w1 = [(1)

(2)

(3)

w2 T

w3 ]

(4) ] = [γ

T

(3.2) κ1

κ2

κ3 ]

T

(3.3)

In Eq. (3.1), $ is a constant array. χ is a multiplication operator matrix of and is a linear function of xα only. χh and χκ are derivative and multiplication operator matrices of w, respectively. χκ is a linear function of κi only. Note the formulation of Eq. (3.1) indicates that the general 3D strain should be able to mathematically expressed as a linear dependency of wi and wi,α . Let w be discretized using finite elements by shape function matrix S and nodal warping array V as w = SV

(3.4)

24 Substituting Eq. (3.4) into Eq. (3.1) then into Eq. (2.30), we arrive at the following nonlinear algebraic system R = ⟨[(χh + χκ )S]T ς⟩ = 0

(3.5)

where ς denotes the array of 3D stress tensor components. Standard approaches such as the Newton-Raphson method can be used to solve the nonlinear equations in Eq. (3.5) for V . Let Vo denote the initial guess, and let dV denote a correction to the Vo . Then in the neighborhood of Vo , R in Eq. (3.5) can be expanded in a Taylor series with the high order terms neglected as ∂R ) dV = 0 ∂V V =Vo

(3.6)

∂R = ⟨[(χh + χκ )S]T [D][(χh + χκ )S]⟩ ∂V

(3.7)

RV =Vo +dV ≈ RV =Vo + ( where

with [D] as the matrix condensed from the components of the fourth-order elasticity tensor which is defined by Dijkl =

∂ςij ∂χkl

(3.8)

Meanwhile, the four constraints in Eq. (2.3) can be written in matrix form as ⟨(ϑc S)⟩V = 0

(3.9)

where the operator matrix ϑc is defined by ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ϑc = ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1

0

0

0

1

0

0

0

1

0

∂ ∂x3

− ∂x∂ 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.10)

Giving in the form of stepwise updating of o by incremental d as = o + d

(3.11)

25 we can solve for dV iteratively for each loading step using ⎡ ⎤ ⎡ ⎢ dV ⎥ ⎢ ⟨[(χh + χκ )S]T ς⟩ ⎢ ⎥ ⎢ ⎥ = −⎢ ι⎢ ⎢ ⎥ ⎢ [0]4×1 ⎢ Λ ⎥ ⎢ ⎣ ⎦ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦V =Vo

(3.12)

with

⎡ ⎤ ⎢ ⟨[(χh + χκ )S]T [D][(χh + χκ )S]⟩ ⟨(ϑc S)T ⟩ ⎥ ⎢ ⎥ ⎥ (3.13) ι=⎢ ⎢ ⎥ ⟨(ϑc S)⟩ [0]4×4 ⎥ ⎢ ⎣ ⎦V =Vo and Λ in Eq. (3.12) denotes a 4 × 1 column vector consisting of Lagrange multipliers. To calculate the sectional stress resultants, define the following column vector ∂ δ¯(i) = [δi1 ∂(i)

δi2

δi4 ] , i = 1, 2, 3, 4 T

δi3

(3.14)

where δ(ij) = 1 if i = j and δ(ij) = 0 if i ≠ j, in which i, j = 1, 2, 3, 4. In addition, denote R = [R(1)

R(2)

T

R(4) ] = [F1

R(3)

M1

M2

M3 ]

T

(3.15)

We can evaluate the sectional resultant components in R corresponding to the strains as T

R(i) =

T

∂ς ∂χ ∂χ ∂W ∗ T ∂ς ∂χ ∂χ ∂⟨W ⟩ = ⟨χT +[ ] ς −( ) ⟩ = ⟨[ ] ς⟩ ∂(i) ∂χ ∂(i) ∂(i) ∂ς ∂χ ∂(i) ∂(i)

(3.16)

i = 1, 2, 3, 4 where ∂χ ∂χκ ∂V = χ δ¯(i) + SV + (χh + χκ )S , i = 1, 2, 3, 4 ∂(i) ∂(i) ∂(i)

(3.17)

It is possible to prove that the derivatives of the warping values on the beam strains do not affect the resultant R. Due to the discretization, nodal warping values are variables independent of the sectional coordinates, then V , dV , and

∂V ∂i

can be

extracted out of the sectional integration. When the solution of V is converged and satisfy Eq. (3.5), we will have T

T

T

∂V ∂V ∂V T ⟨[(χh + χκ )S ] ς⟩ = [ ] ⟨[(χh + χκ )S] ς⟩ = [ ] R=0 ∂(j) ∂(j) ∂(j)

(3.18)

As a result, T

∂χκ R(i) = ⟨[χ δ¯(i) + SV ] ς⟩ , i = 1, 2, 3, 4 ∂(i)

(3.19)

26 Therefore, when and W or W ∗ are given, iteratively solving Eq. (3.12) will provide a finite warping field for Eq. (3.19) to obtain a set of sectional force and moments. The relation between small incremental R(i) and (j) can be expressed as dR(i) = Tij d(j) , i, j = 1, 2, 3, 4

(3.20)

where the tangent beam cross-sectional stiffness Tij can be obtained as T

T

∂R(i) ∂χκ ∂V ∂χκ ∂χ Tij = = ⟨[ S ] ς⟩ + ⟨[χ δ¯(i) + SV ] [D] [ ]⟩ ∂(j) ∂(i) ∂(j) ∂(i) ∂(i)

(3.21)

i, j = 1, 2, 3, 4 Note that the operator matrix χκ is formulated to be linearly dependent on κi . Derivatives of warping values with respect to the beam strain

∂V ∂(i)

can be computed using

Eq. (3.5). Taking derivatives with respect to (i) on both sides of Eq. (3.5) gives T

∂χκ ∂χ ∂R T = ⟨[ S] ς⟩ + ⟨[(χh + χκ )S] [D] [ ]⟩ = 0, i = 1, 2, 3, 4 ∂(i) ∂(i) ∂(i) In light of the warping constraints Eq.(3.9), we have the same constraints on ⟨(ϑc S)⟩

∂V = 0, i = 1, 2, 3, 4 ∂(i)

(3.22) ∂V ∂(i)

as

(3.23)

Substituting Eq. (3.17) into Eq. (3.22) and combining Eqs. (3.22) and (3.23),

∂V ∂(i)

can be solved by ⎡ ∂V ⎤ ⎡ ∂χκ T ∂χκ ⎢ ⎥ ⎢ ⟨[ ∂ S] ς + [(χh + χκ )S]T [D] [χ δ¯(i) + ∂ SV ]⟩ ⎢ ∂(i) ⎥ ⎢ (i) (j) ⎥ = −⎢ ι⎢ ⎢ ⎥ ⎢ ⎢ Λ ⎥ ⎢ [0]4×1 ⎣ ⎦ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.24)

i = 1, 2, 3, 4 3.2

Hyperelasticity In this section, the general finite element formulation in Section 3.1 is specified

to handle the finite strains and the hyperelastic material models. Material model subroutines are used to define various hyperelastic material models in VABS code.

27 This 2D cross-sectional analysis is able to capture various nonlinear phenomena of composite beams, such as the known trapeze effects of rotor blades [21], and the Poynting effects found in slim wires [55, 56] and rubber cylinders [62, 69, 134]. The present theory in this section and the hyperelastic modeling capability of VABS code have been validated for various beams made of different hyperelastic materials and documented in [82]. Finite elasticity is governed by W which can be written in terms of invariants and/or principals of the kinematic second-order tensors or these tensors themselves. Some examples of these kinematic second-order tensors are the deformation gradient tensor F , the right Cauchy-Green tensor C = F T ⋅ F , the left Cauchy-Green (Finger) tensor B = F ⋅ F T , and the Green-Lagrange strain tensor E = 21 (C − I), where I is the second-order identity tensor. Because these kinematic second-order tensors are built upon the knowledge of the deformation gradient, F , it should be directly used as the general 3D strain definition, that is, χij = Fij

(3.25)

Consequently, χ and $ in Eq. (3.1) are respectively expressed by χ = [F11

F12

F13

F21

F22

F23

F31

F33 ]

F32

T

$ = [1 0 0 0 1 0 0 0 1]

T

(3.26) (3.27)

And the operator matrices in Eq. (3.1) are correspondingly ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ χ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1

0

0

0

0

0

0 −x3 0

0

0

0

0

x2

0

0

0

0

⎤ x3 −x2 ⎥⎥ ⎥ 0 0 ⎥⎥ ⎥ ⎥ 0 0 ⎥⎥ ⎥ 0 0 ⎥⎥ ⎥ ⎥ 0 0 ⎥⎥ ⎥ 0 0 ⎥⎥ ⎥ ⎥ 0 0 ⎥⎥ ⎥ 0 0 ⎥⎥ ⎥ ⎥ 0 0 ⎥⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ χh = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

∂ ∂x2

0

∂ ∂x3

0

0

0

0

∂ ∂x2

0

∂ ∂x3

0

0

0

0

0

0

⎤ 0 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ∂ ⎥ ∂x2 ⎥ ⎥ ∂ ⎥ ⎥ ∂x3 ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ χκ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

−κ3

0

0

0

0

κ3

0

0

0

0

0

−κ2

κ1

0

0

0

0

⎤ κ2 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ −κ1 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ (3.28) ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎦

28 Therefore, the general 3D stress χij is equivalent to the first Piola-Kirchhoff stress (PK1) which is denoted by P , that is, ςij = Pij =

∂W ∂Fij

(3.29)

Note that now the same expression of Eq. (2.18) can be also obtained by the principle of minimum elastic energy stored in the deformed solid, as shown in [82], that is, δ ⟨W ⟩ = ⟨

∂W δFij ⟩ = ⟨Pij δFij ⟩ = 0 ∂Fij

(3.30)

In light of Eq. (3.29), ς in Eq. (3.5) is then obtained by ς = [P11

P12

P13

P21

P22

P23

P31

P32

P33 ]

T

(3.31)

In addition, the fourth-order elasticity tensor D defined by Eq. (3.8) becomes Dijkl = Aijkl =

∂Pij ∂ 2W = ∂Fkl ∂Fij ∂Fkl

(3.32)

where A denotes the fourth-order first elasticity tensor. As a result, [D] in Eq. (3.7) is therefore a 9 × 9 matrix condensed from the components of tensor A.

3.3

Nonlinear Material with Linear Strain Based on the concept of decomposition of rotation tensor [11], the general 3D

strain definition can be specified by the 3D Jaumann-Biot-Cauchy strain components which are given by Γij =

1 (Fij + Fji ) − δij 2

(3.33)

where δij is the Kronecker delta. If the large local rotation is also considered for the kinematic nonlinearities, the nonlinear strain definitions [71, 72] should be applied. The expression of Eq. (3.33) means that the strain is linearly dependent on the deformation gradient. This linear strain definition enables it to be a suitable choice to our finite element formulation of strain, that is, we let χij = Γij

(3.34)

29 Consequently, χ Eq. (3.1) is expressed by χ = [Γ11

2Γ12

2Γ13

Γ22

2Γ23

Γ33 ]

T

(3.35)

and $ vanishes. The operator matrices in Eq. (3.1) are correspondingly ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 1 0 x3 −x2 ⎥ ⎢ 0 ⎢ 0 −κ3 κ2 ⎥ 0 0 ⎥⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ 0 −x 0 ⎥ 0 ⎥ 0 ⎥ 0 −κ1 ⎥⎥ ⎢ ∂x∂ 2 0 ⎢ ⎢ κ3 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ ⎥ ⎢ ⎥ ⎢ 0 x2 0 ⎢ ⎢ −κ2 κ1 0 ⎥⎥ 0 ⎥⎥ 0 ⎥⎥ ⎢ ⎢ ∂x3 0 ⎢ ⎥ χh = ⎢ ⎥ (3.36) ⎥ χκ = ⎢ χ = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎥ 0 0 ⎥ ⎢ 0 0 ⎢ 0 ∂x∂ 2 0 ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ⎥ ⎢ 0 0 ⎥ ⎢ ⎢ 0 0 ⎥⎥ 0 0 ⎥ ⎢ 0 ∂x3 ∂x2 ⎥ ⎢ ⎢ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ∂ ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎦ 0 ∂x3 ⎦ 0 0 ⎥⎦ ⎣ ⎣ 0 ⎣ 0 The stress conjugated to the small strain in linear elasticity could be the Cauchy stress which is defined by σij =

∂W ∂Γij

(3.37)

As a results, we have ∂σij ∂ 2W = (3.38) ∂Γkl ∂Γij ∂Γkl is the material stiffness tensor which possess both minor and major symDijkl = Cijkl =

where Cijkl

metries. Therefore, [D] in Eq. (3.7) is a 6 × 6 matrix condensed from the components of the fourth-order tensor C. In light of Eq. (3.37), ς in Eq. (3.5) is then obtained by ς = [σ11

σ33 ]

T

(3.39)

Both σij and Γkl are symmetric second-order tensors.

Note that in light of

σ12

σ13

σ22

σ23

Eq. (3.33), ∂Γmn 1 ∂Fmn ∂Fnm 1 = ( + ) = (δmi δnj + δni δmj ) ∂Fij 2 ∂Fij ∂Fij 2 Substitute Eq. (3.40) into Eq. (3.29), we have Pij =

∂W ∂W ∂Γmn 1 = = (σij + σji ) = σij ∂Fij ∂Γmn ∂Fij 2

(3.40)

(3.41)

which means under the small strain assumption, Pij reduces to symmetric stress tensor. Substituting Eqs. (3.41) and (3.40) into Eq. (3.32), we have Aijkl =

∂Pij ∂σij ∂Γmn 1 = = (Cijkl + Cijlk ) = Cijlk ∂Fkl ∂Γmn ∂Fkl 2

(3.42)

30 in view of Eq. (3.38). Consequently, the small strain scenario in this section is actually solvable by the algorithm in section 3.2. This algorithm will be the proper methodology for the cases of hybrid beams which are made of a mixture of rubbers, composites, metals, etc.

3.4

Progressive Damage Constitutive Law In literature, for most of the cases in progressive damage analysis of fiber compos-

ites, small strain assumptions are adapted. As a result, the algorithm in Section 3.3 is adaptable for the case of progressive damage. The damaged 3D material constitutive law reads σij = Cdijkl Γkl

(3.43)

where Cdijkl denotes the damaged fourth-order elasticity tensor. Consequently, the fourth-order elasticity tensor D defined by Eq. (3.8) can be computed by ∂Cdijmn ∂dpq ∂σij d Dijkl = = Cijkl + Γmn ∂Γkl ∂dpq ∂Γkl

(3.44)

where dij denotes the components of the internal tensorial damage variable. Each of dij takes value from zero as undamaged status to one as fully damaged status. It is worth to mention here that although the damage is expressed in the form of a secondorder tensor, the specified CDM model does not restrict to be in such a format. For example, if only one variable d is assumed for isotropic damage, Eq. (3.44) will reduce to

∂Cdijmn ∂d Γmn (3.45) ∂d ∂Γkl For another example, if scalar variables d1 and d2 are assumed denoting fiber and Dijkl = Cdijkl +

matrix damage respectively, Eq. (3.44) will become Dijkl =

Cdijkl

∂Cdijmn ∂d2 ∂Cdijmn ∂d1 + Γmn + Γmn ∂d1 ∂Γkl ∂d2 ∂Γkl

(3.46)

The generalized schematic of stress-strain curve of an elastic damageable material is shown in Figure 3.1. The corresponding damage history is plotted in Figure 3.2. Point A is the damage initiation point.

31

B

σ

A

ε O Figure 3.1. Schematic of stress-strain curve of a generalized elastic damageable material.

d 1 B

A

,σ

O Figure 3.2. Schematic of damage versus strain or stress of a generalized elastic damageable material.

In Figure 3.2, beyond point A, the damage value is not reversible. Instead, it will be the maximum in the overall loading history. If the strain and stress are never beyond the Point A, the damage value will be zero and the problem is pure elasticity. However, as long as once the loading is beyond point A, the damage will be initiated, and the constitutive curve of progressive damaged material will be irreversible, as

32 shown by the path of O-A-B in Figure 3.1. Imagine that at point B we unload the material to initial configuration and then reload the material again up to point B. During this loading-unloading-reloading process, the damage d is stored in the material as a constant and the constitutive curve of the damaged elastic material is reversible. This is because when the damage state is stable without further loading which initiates the damage evolution, the system will become conservative. In this situation, any point on the path of O-B in Figure 3.1 can be predicted by the principle of elasticity which is governed by the 3D elastic strain energy W . In Figure 3.1, W is the shaded area underneath the reversible path of O-B, so it can be expressed as 1 W = ΓT [Cd ]Γ 2

(3.47)

where [Cd ] is the 6×6 matrix condensed from the damaged elasticity tensor Cdijkl . Note that damage variables in [Cd ] are stable constants resulting from the convergence of damage evolution. Consequently, we can evaluate the damaged sectional resultant components in R corresponding to the strains using Eq. (3.19). It worth to note that the progressive damage constitutive law does not restrict to the regions of linear strain definition. For example, for thin laminates in which the trapeze effect is also interesting, we could have the 3D elastic strain energy written as 1 W = Eij Cdijkl Eij 2

(3.48)

where Eij denotes the components of the Green strain tensor E. When the large warping is considered, for example, a deployable boom structure, the widely used logarithmic strain tensor H can be adapted, so that 1 W = Hij Cdijkl Hij 2

(3.49)

where Hij denotes the components of the tensor H. In these cases, the deformation gradient should be applied as the generalized strain definition and the numerical algorithm introduced in Section 3.2 should be combined with the elastic damageable material model introduced in the present section.

33

4. TRAPEZE AND POYNTING EFFECTS This chapter demonstrates the capabilities of VABS to solve the trapeze and Poynting effects. Within the small strain region, for pretiwsted beams, Section 4.1 presents the analytical solutions. Jiang, Yu, and Hodges have published these solutions in Journal of Applied Mechanics entitled “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted here by permission of the American Society of Mechanical Engineers. For prismatic beams, Section 4.2 releases the small strain assumption and provides the general numerical results. Jiang and Yu have published these results in AIAA Journal entitled “Nonlinear variational asymptotic sectional analysis of hyperelastic beams” [82] reprinted here by permission of the American Institute of Aeronautics and Astronautics, Inc. Interested readers are suggested to refer to these two journal articles.

4.1

Analytical Modeling In this section, we focus on pretwisted beams undergoing elongation and torsion.

Denote k1 as the pretwist, and k2 , k3 as the initial curvatures. As a result, κα = kα = 0. Consequently, Eqs. (2.13)-(2.15) reduce to G 1 = (1 + γ + w1′ ) B 1 + [w2′ − (x3 + w3 )(κ1 + k1 )] B 2 + [w3′

+ (x2 + w2 )(κ1 + k1 )] B 3

(4.1)

G 2 = w1,2 B 1 + (1 + w2,2 )B 2 + w3,2 B 3

(4.2)

G 3 = w1,3 B 1 + w2,3 B 2 + (1 + w3,3 )B 3

(4.3)

In light of Eq. (2.29), wi′ in Eq. (4.1) can be neglected because their contributions to the strain energy are much smaller than those of other terms. Neglecting these

34 terms, one finds that components of the deformation gradient in Eq. (2.16) may be expressed as ⎡ ⎢ 1 + γ − (x2 w1,3 − x3 w1,2 )k1 w1,2 w1,3 ⎢ ⎢ [F ] = ⎢⎢ −(x3 + w3 )κ1 − (x2 w2,3 − x3 w2,2 + w3 )k1 1 + w2,2 w2,3 ⎢ ⎢ ⎢ (x2 + w2 )κ1 − (x2 w3,3 − x3 w3,2 − w2 )k1 w3,2 1 + w3,3 ⎣ 4.1.1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.4)

Isotropic Hyperelastic Material Models

The simplest form for a nonlinear constitutive relation is a linear relation between nonlinear stress and strain measures. For example, the St-V/K model is usually valid in the region of small strain [16, 21] for thin-walled structures. It has the expression for 3D strain energy density W of the form 1 W G = λ(Eii )2 + µEij Eji 2

(4.5)

where the superscript G stands for the Green strain, and λ and µ are Lamé constants. At this point, uniaxial stress was widely assumed in the literature [44, 47, 49, 135], so that S22 = S33 = S23 = 0, S11 = EE11 , S12 = 2GE12 , and S13 = 2GE13 , where E is the Young’s modulus and G the shear modulus. Then, the energy function in Eq. (4.5) reduced to that widely applied in other works 1 E 2 2 2 ) W P = Sij Eij = E11 + 2G (E12 + E13 2 2

(4.6)

where the superscript P stands for published model. However, the uniaxial stress assumption is only approximate for nonlinear analysis because the S22 , S33 , and S23 do not vanish due to the contribution of nonlinear terms. In addition, in finite elasticity, there exists a theory that determines and classifies the energy function axiomatically on the basis of various invariant principles. For example, a widely used hyperelastic material model has W in the form of a polynomial, that is, n m 1 W = ∑ cij (I¯1 − 3)i (I¯2 − 3)j + ∑ (J − 1)2k i,j=0 k=1 dk

(4.7)

35 Here J is the Jacobian of the deformation gradient tensor F , I¯1 = J −2/3 I1 , and I¯2 = J −4/3 I2 , with I1 and I2 as the first two invariants of C . The simplest form of such stress-strain relationship, which can be shown to be a natural extension of that adopted in the study of small elastic deformations, is described as 1 W N H = c10 (I¯1 − 3) + (J − 1)2 d1

(4.8)

which is actually the situation if c11 = c01 = 0, n = 1, and m = 1 in Eq. (4.7). The superscript N H stands for Neo-Hookean (N-H) type of hyperelastic material. The coefficients c01 and d1 can be related to the equivalent Lamé parameters by c10 =

µ 2

1 K 1 2 = = (λ + µ) d1 2 2 3

(4.9)

where K is the bulk modulus. It has been demonstrated that the HASE model is in good agreement with experiment for a wide class of materials, especially under moderately large deformation. The HASE model of isotropic elasticity takes the form of W ln = µ (˜ e21 + e˜22 + e˜23 ) +

λ 2 (˜ e1 + e˜2 + e˜3 ) 2

(4.10)

in which the principal logarithmic strain e˜i can be calculated by e˜i = 1/2 ln λC i where λC i is the principal value of the deformation tensor C . Anand [72] and Degener et al [135] verify that this type of material model could provide excellent agreement with experimental data of rubber cylinder and is advantageous compared with linear stress-strain relations. Integrating the strain energy density over the cross section, we obtain twice of the strain energy per unit length along the beam axis (the 1D energy) as 2Π = ⟨2W ⟩ = 2ΠO(ε2 ) + 2ΠO(ε3 ) + 2ΠO(k1 ε2 ) + O (ε4 ) + O (k1 ε3 ) + O (k12 ε2 )

(4.11)

where the doubly underlined terms are of higher order and are assumed to be negligible for the purpose to obtain an analytical solution.

36 4.1.2

Asymptotic Analysis with Green Strain Energy

By using the St-V/K model, the single-underlined terms in Eq. (4.11) are one order higher than the non-underlined terms which correspond to the zeroth-order energy [17]: 2ΠO(ε2 ) = (λ + 2µ)⟨Γ211 + Γ222 + Γ233 ⟩ + 2λ⟨Γ11 Γ22 + Γ11 Γ33 + Γ22 Γ33 ⟩ + 4µ⟨Γ212

+ Γ213

+ Γ223 ⟩

(4.12)

in which

⎡ ⎤ 1 1 ⎢ ⎥ γ 2 (w1,2 − x3 κ1 ) 2 (w1,3 + x2 κ1 ) ⎥ ⎢ ⎢ ⎥ 1 [Γ] = ⎢⎢ (4.13) w2,2 (w3,2 + w2,3 ) ⎥⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎢ symmetry ⎥ w3,3 ⎦ ⎣ In addition, the first of the single-underlined terms in Eq. (4.11) takes the form of 2ΠG O(ε3 ) = (λ + 2µ)⟨Υ1111 + Υ2222 + Υ3333 ⟩ + 2λ⟨Υ1122 + Υ1133 + Υ2233 ⟩ + 4µ⟨Υ1212 + Υ1313 + Υ2323 ⟩

(4.14)

in which Υ1111 =γ [γ 2 + (x22 + x23 ) κ21 ] 2 2 2 ) + w3,2 + w2,2 Υ2222 =w2,2 (w1,2 2 2 2 ) Υ3333 =w3,3 (w1,3 + w2,3 + w3,3 2 2 2 ) + w3,2 + w2,2 2Υ1122 =w2,2 [γ 2 + (x22 + x23 ) κ21 ] + γ (w1,2 2 2 2 ) 2Υ1133 =w3,3 [γ 2 + (x22 + x23 ) κ21 ] + γ (w1,3 + w2,3 + w3,3

(4.15)

2 2 2 2 2 2 ) + w2,2 (w1,3 ) 2Υ2233 =w3,3 (w1,2 + w2,2 + w3,2 + w2,3 + w3,3

2Υ1212 = (w1,2 − x3 κ1 ) [γw1,2 − κ1 (w3 − x2 w3,2 + x3 w2,2 )] 2Υ1313 = (w1,3 + x2 κ1 ) [γw1,3 + κ1 (w2 − x3 w2,3 + x2 w3,3 )] 2Υ2323 = (w2,3 + w3,2 ) (w1,2 w1,3 + w2,2 w2,3 + w3,2 w3,3 ) The nonlinear contribution to the elastic strain energy, including the effect of pretwist in Eq. (4.11), is 2ΠO(k1 ε2 ) = k1 (λ + 2µ)⟨Φ1111 ⟩ + 2k1 λ⟨Φ1122 + Φ1133 ⟩ + 4k1 µ⟨Φ1212 + Φ1313 ⟩

(4.16)

37 in which Φ1111 = 2γ (x3 w1,2 − x2 w1,3 ) Φ1122 = w2,2 (x3 w1,2 − x2 w1,3 ) Φ1133 = w3,3 (x3 w1,2 − x2 w1,3 )

(4.17)

2Φ1212 = (w1,2 − x3 κ1 ) (x3 w2,2 − x2 w2,3 − w3 ) 2Φ1313 = (w1,3 + x2 κ1 ) (w2 + x3 w3,2 − x2 w3,3 ) 4.1.3

Asymptotic Analysis with Neo-Hookean Model

Following the same procedure, asymptotically expanding the small terms O(ε) of 2⟨W N H ⟩, the corresponding 2ΠO(ε2 ) and 2ΠO(k1 ε2 ) calculated by using N-H is of the same format with those calculated by St-V/K model. Note that Eq. (4.12) is calculated by ⟨2W ⟩ = ⟨Γij Cijkl Γkl ⟩

(4.18)

where the elasticity tensor Cijkl can be expressed in terms of λ and µ regarding the material isotropy. However, the third-order 1D energy is different: H 2ΠN O(ε3 ) = 2λ⟨∆eew + ∆eww + ∆www ⟩ + µ⟨Ωeee + Ωeew + Ωeww + Ωwww ⟩

in which the terms associated with µ are 2 14 Ωeee = − [ γ 3 + γκ21 (x22 + x23 )] 3 9 10 14 Ωeew = − γ 2 (w2,2 + w3,3 ) − γκ1 (x2 w1,3 − x3 w1,2 ) 9 3 2 2 + κ1 [2 (x2 w2 + x3 w3 ) − (x22 + x23 ) (w2,2 + w3,3 )] 3 2 2 2 2 2 2 ) − 3 (w1,2 Ωeww = γ [26w2,2 w3,3 + 7 (w2,2 + w3,3 + w1,3 + (w2,3 + w3,2 ) )] 9 2 + κ1 [w1,3 (3w2 + 3x3 w3,2 − x2 (2w2,2 + 5w3,3 ))] 3 2 + κ1 [w1,2 (3w3 + 3x2 w2,3 − x3 (5w2,2 + 2w3,3 ))] 3 2 2 2 2 2 Ωwww = − (w2,2 + w3,3 ) (w1,2 + w1,3 + w2,3 + w3,2 + 5w2,3 w3,2 ) 3 14 2 2 ) − 5 (w2,2 + w3,3 )] + (w2,2 + w3,3 ) [2 (w2,2 + w3,3 27

(4.19)

(4.20)

38 and the terms associated with λ are ∆eew =γ 2 (w2,2 + w3,3 ) + γκ1 (x2 w1,3 − x3 w1,2 ) 2 2 ∆eww =γ (w2,2 + w3,3 + 3w2,2 w3,3 − w2,3 w3,2 ) − κ1 (w2,2 + w3,3 ) (x2 w1,3 − x3 w1,2 ) (4.21)

∆www = (w2,2 + w3,3 ) (w2,2 w3,3 − w2,3 w3,2 ) 4.1.4

Asymptotic Analysis with Hencky Strain Energy

In order to apply VAM to 2⟨W ln ⟩, we need to obtain the principal values of C asymptotically. The principal value λC i can be obtained by √ ⎛ τ ⎞ 1 1 ] λC 1 = I1 − 2 Q cos [ arccos √ 3 3 ⎝ Q3 ⎠ C λC 2 , λ3

√ ⎞ ⎛ τ 1 1 = I1 − 2 Q cos [ arccos √ ± 2π ] 3 3 ⎠ ⎝ Q3

(4.22)

where I1 = 3 + I1(ε) + I1(ε2 ) + I1(k1 ε) + ... 1 2 (I − 3I2 ) = Q(ε2 ) + Q(k1 ε2 ) + Q(ε3 ) + ... 9 1 1 (−2I13 + 9I1 I2 − 27I3 ) = τ(ε3 ) + τ(ε4 ) + ... τ= 54 Q=

(4.23)

Consequently, the kept 1D strain energy terms are λ µ 2 2Πln O(ε2 ) = ( + ) ⟨I1(ε) ⟩ + 3µ⟨Q(ε2 ) ⟩ 4 6 λ µ 2Πln O(k1 ε2 ) = ( + ) ⟨I1(ε) I1(k1 ε) ⟩ + 3µ⟨Q(k1 ε2 ) ⟩ 2 3 λ µ 1 3 λ 2Πln O(ε3 ) = ( + ) ⟨I1(ε) I1(ε2 ) − I1(ε) ⟩ − 3 ( + µ) ⟨I1(ε) Q(ε2 ) ⟩ 2 3 6 2

(4.24)

+ 3µ⟨Q(ε3 ) + τ(ε3 ) ⟩ where 2Πln and 2Πln are same with Eqs. (4.12) and (4.16), respectively. HowO(k1 ε2 ) O(ε2 ) ever, the third-order 1D energy is different: 2Πln O(ε3 ) = λ⟨Θeee + Θeew + Θeww + Θwww ⟩ + µ⟨Ξeee + Ξeew + Ξeww + Ξwww ⟩

(4.25)

39 in which the terms associated with λ are Θeee = − γ 3 Θeew = − γ 2 (w2,2 + w3,3 ) − 2γκ1 (x2 w1,3 − x3 w1,2 ) 2 2 Θeww = − γ (w2,2 + w3,3 + 2w2,3 w3,2 ) − 2κ1 (w2,2 + w3,3 ) (x2 w1,3 − x3 w1,2 )

(4.26)

2 2 Θwww = − (w2,2 + w3,3 ) (w2,2 + w3,3 + 2w2,3 w3,2 )

and the terms associated with µ are Ξeee = − 2γ 3 − γκ21 (x22 + x23 ) Ξeew = 4γκ1 (x3 w1,2 − x2 w1,3 ) + κ21 [2 (x2 w2 + x3 w3 ) − (x22 w3,3 + x23 w2,2 ) + x2 x3 (w2,3 + w3,2 )] 2 2 ) Ξeww = − γ (w1,2 + w1,3

(4.27)

− κ1 w1,2 (2w3 − 4x3 w2,2 + x2 (3w2,3 + w3,2 )) + κ1 w1,3 (2w2 − 4x2 w3,3 + x3 (w2,3 + 3w3,2 )) 3 3 ) − w1,3 (w1,2 w3,2 + w1,3 w3,3 ) Ξwww = − 2 (w2,2 + w3,3 2 2 − w1,2 (w1,2 w2,2 + w1,3 w2,3 ) − (w2,2 + w3,3 ) (w2,3 + w3,2 + 4w2,3 w3,2 )

4.1.5

Approximation of Warping Functions

The warping functions that minimize 2ΠO(ε2 ) in Eq. (4.12) are given by Yu and Hodges [17] as w1 = wˆ1 = ψ(x2 , x3 )κ1

(4.28)

w2 = wˆ2 = −x2 νγ

(4.29)

w3 = wˆ3 = −x3 νγ

(4.30)

with ψ(x2 , x3 ) denoting the Saint-Venant warping. For notational convenience, we introduce the following A = ⟨1⟩,

J0 = ⟨x22 + x23 ⟩,

2 2 J1 = −⟨ψ,2 + ψ,3 ⟩,

JS = J0 + J1

(4.31)

where JS is the Saint-Venant torsion stiffness. According to the VAM, one can show that the warping function we have obtained is sufficient to obtain the strain energy

40 asymptotically correct through orders ε3 and k1 ε2 . A detailed proof can be found in the following section.

4.1.6

Perturbation of Warping Functions

In this section, we show that the warping functions in Eqs. (4.28), (4.29), and (4.30) are sufficient to obtain the strain energy asymptotically correct up to the order of ε3 and k1 ε2 . Now we consider the nonlinear term and the initial twist term in the 1D strain energy density in Eq. (4.11). To this end, the warping functions are perturbed a small unknown variable vi = o (wî ) as wi = wî + vi

(4.32)

Plugging Eq. (4.32) into Eq. (4.12), we have the products of strains in Eqs. (4.12) rewritten as Γ211 = γ 2

(4.33)

2 2 Γ222 = wˆ2,2 + 2wˆ2,2 v2,2 + v2,2

(4.34)

2 2 Γ233 = wˆ3,3 + 2wˆ3,3 v3,3 + v3,3

(4.35)

Γ11 Γ22 = Γ11 wˆ2,2 + Γ11 v2,2

(4.36)

Γ11 Γ33 = Γ11 wˆ3,3 + Γ11 v3,3

(4.37)

Γ22 Γ33 = wˆ2,2 wˆ3,3 + wˆ2,2 v3,3 + wˆ3,3 v2,2 + v2,2 v3,3

(4.38)

2 2 4Γ212 = wˆ1,2 + x23 κ21 − 2x3 κ1 wˆ1,2 + 2wˆ1,2 v1,2 − 2x3 κ1 v1,2 + v1,2

(4.39)

2 2 4Γ213 = wˆ1,3 + x22 κ21 + 2x2 κ1 wˆ1,3 + 2wˆ1,3 v1,3 + 2x2 κ1 v1,3 + v1,3

(4.40)

2 2 4Γ223 = wˆ2,3 + wˆ3,2 + 2wˆ2,3 wˆ3,2 + 2 (wˆ2,3 + wˆ3,2 ) (v2,3 + v3,2 ) 2 2 + v2,3 + v3,2 + 2v2,3 v3,2

(4.41)

in which, the singly underlined terms should be kept due to their unknown asymptotic role with respect to ε3 or k1 ε2 . However, the doubly underlined terms should be dropped due to their determined higher order than the singly underlined ones. It is

41 easy to see when we substitute the Eq. (4.32) into Eqs. (4.15), (4.21), (4.20), (4.26), (4.27), and (4.17), then neglect the terms higher than the kept asymptotic order ε3 and k1 ε2 , the corresponding Π(ε3 ) and Π(k1 ε2 ) will not have vi,j terms contained in their expressions. Then, taking the variation of this updated energy functional with respect to the perturbing terms of the warping functions results in the following governing equations wˆ1,22 + wˆ1,33 = 0

(4.42)

2 (1 − ν) wˆ2,22 + (1 − 2ν) wˆ2,33 + wˆ3,23 = 0

(4.43)

2 (1 − ν) wˆ3,33 + (1 − 2ν) wˆ3,22 + wˆ2,23 = 0

(4.44)

and the associated boundary conditions n3 (wˆ1,3 + x2 κ1 ) + n2 (wˆ1,2 − x3 κ1 ) = 0 2n2 n3 (wˆ2,3 + wˆ3,2 ) + [ν (γ + wˆ3,3 ) + (1 − ν) wˆ2,2 ] = 0 1 − 2ν 2n3 n2 (wˆ2,3 + wˆ3,2 ) + [ν (γ + wˆ2,2 ) + (1 − ν) wˆ3,3 ] = 0 1 − 2ν

(4.45) (4.46) (4.47)

where nα is the direction cosine of the outward-directed normal with respect to xα . Eqs. (4.42) – (4.47) are used to obtain the classical approximation of the warping functions. Thus, the classical approximation for the warping is sufficient to obtain the 1D strain energy asymptotically correct up through order ε3 and k1 ε2 .

4.1.7

Transverse Stresses and Strains

In this section, we examine the uniaxial stress assumption which is widely taken in the literature to modle the trapeze effects [47–49]. We take the prismatic beam with Green strains to prove that the transverse stresses are only ineffective to the ΠO(ε2 ) so they should not be neglected when the nonlinear behavior is considered. Under the uniaxial stress assumption made by published theories, the strain energy taken by the published theories is 1 W P = S11 E11 + S12 E12 + S13 E13 2

(4.48)

42 However, without this assumption we have 1 1 W = S11 E11 + S12 E12 + S13 E13 + (S22 E22 + S33 E33 ) + S23 E23 2 2

(4.49)

where the underlined terms are dropped in Eq. (4.48). And the nonlinear transverse strains resulted from the warping functions Eqs. (4.28), (4.29), and (4.30) are 1 E22 = −(νγ) + [(νγ)2 + (Ψ,2 κ1 )2 ] 2 1 E33 = −(νγ) + [(νγ)2 + (Ψ,3 κ1 )2 ] 2 1 2 E23 = [κ1 ψ,2 ψ,3 ] 2

(4.50) (4.51) (4.52)

The transverse stresses from the St-V/K model are O(ε2 ) such as S22 =

E 2 2 [γ 2 ν(1 + ν) + (x22 + x23 )κ21 ν + κ21 νψ,3 ] + κ21 (1 − ν)ψ,2 2(1 + ν)(1 − 2ν)

(4.53)

+ O(ε ) 3

S33 =

E 2 2 [γ 2 ν(1 + ν) + (x22 + x23 )κ21 ν + κ21 νψ,2 ] + κ21 (1 − ν)ψ,3 2(1 + ν)(1 − 2ν)

(4.54)

+ O(ε ) 3

S23 =

E (κ2 ψ,2 ψ,3 ) 2(1 + ν) 1

(4.55)

When these stresses are multiplied by their conjugated strains, S22 E22 and S33 E33 will result in a significant part of ΠO(ε3 ) , which is neglected in Eq.(4.48) by uniaxial stress assumption. In fact only S23 E23 can be reasonably neglected because it is O(ε4 ).

4.1.8

Strain-Dependent Beam Stiffness

For simplicity, we now choose a centroidal coordinate system with cross-sectional coordinates aligned with principal bending directions. Plugging the solutions of warping functions into the energy functional in Eq. (4.11), one obtains 2Π = [γ

κ1 ] (S + k1 T + γA) [γ

κ1 ]T

(4.56)

43 where S is the stiffness matrix for the classical beam model, and T and A are responsible for a variety of nonlinear effects in beams. For a general beam made of the isotropic material, we have ⎡ ⎢ S11 0 ⎢ S =⎢ ⎢ ⎢ 0 S22 ⎣

⎤ ⎡ ⎤ ⎥ ⎢ EA 0 ⎥ ⎥ ⎢ ⎥ ⎥=⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ 0 GJS ⎥ ⎦ ⎣ ⎦

(4.57)

The stiffness matrix associated with k1 is ⎡ ⎢ 0 T12 ⎢ T =⎢ ⎢ ⎢ T12 0 ⎣

⎤ ⎡ ⎤ ⎥ ⎢ 0 −EJ1 ⎥⎥ ⎥ ⎢ ⎥=⎢ ⎥ ⎥ ⎢ ⎥ 0 ⎥ ⎥ ⎢ −EJ1 ⎦ ⎣ ⎦

(4.58)

The stiffness matrix associated with γ is ⎡ ⎢ A11 0 ⎢ A=⎢ ⎢ ⎢ 0 A22 ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.59)

and varies depending on the various energy functions. When the St-V/K model is considered, A11 = AG 11 = EA A22 = AG 22 = E [J0 −

(4.60) 2ν (J0 + J1 )] 1+ν

(4.61)

For the N-H, 2 EA (−7 − 14ν + 20ν 2 ) 27 4 = E [J0 − (J0 + J1 )] 3

H A11 = AN 11 =

(4.62)

H A22 = AN 22

(4.63)

While for the HASE model, A11 = Aln 11 = −EA A22 = Aln 22 = E [J0 −

(4.64) 3 (J0 + J1 )] 2

(4.65)

In Eqs. (4.61), (4.63), and (4.65) the underlined terms show the differences between trapeze terms from the various energy functions versus the well-known trapeze effect

44 term EJ0 . In the following section, we will show the limitations of EJ0 and the St-V/K model. Furthermore, Eqs. (4.62) – (4.65) will be demonstrated to play an important role in determining whether the beam possesses the trapeze effect, the Poynting effect, or no nonlinear extension-torsion behavior at all, according to the sectional thickness-to-width ratio.

4.1.9

Trapeze Effect: Stiffening of Torsional Rigidity

Assume that the beam is made of Aluminum 6061 with Young’s modulus E = 68.9 GPa, and Poisson’s ratio ν = 0.33. For an elliptical section with semi-axes a and b in the direction of x2 and x3 , respectively, and r = b/a as the sectional thickness-to-width ratio, the Saint-Venant warping is found to be ψ(x2 , x3 ) =

(b2 − a2 )x2 x3 a2 + b 2

(4.66)

For a rectangular section of width 2a in x2 -direction and height 2b in x3 -direction, the Saint-Venant warping can be expressed as 2n+1 πx 32b2 ∞ (−1)n sinh ( 2 b 2 ) 2n + 1 πx3 ψ(x2 , x3 ) = −x2 x3 + 3 × ∑ ) sin ( 3 2n+1 πa π 2 b ) n=0 (2n + 1) cosh ( 2 b

(4.67)

As shown in Tables 4.1 and 4.2, results from analytical formulae AG 22 presented in Eq. (4.61) agree very well with those from the trapeze effect functionality in previous VABS version, which provides a 2D finite element solution with the St-V/K model, and has been extensively validated, for example, in [16], for thin-walled beams. It is also worth noting that the expressions for A22 , obtained by different strain energy functions, agree with each other when r is small. However, the differences increase as r increases. The generalized axial force and the twisting moment can be obtained by ∂Π 3 1 = EAγ + A11 γ 2 + A22 κ21 − EJ1 k1 κ1 ∂γ 2 2 ∂Π M1 = = (GJS + A22 γ) κ1 − EJ1 k1 γ ∂κ1 F1 =

(4.68) (4.69)

45 Table 4.1. A22 (×106 ) (elliptical section: a = 0.23 m). r

VABS in [16]

AG 22

H AN 22

Aln 22

EJ0

0.04

6.0478

6.0478

6.0154

6.0090

6.0670

0.08

12.039

12.039

11.781

11.730

12.192

0.12

17.922

17.922

17.058

16.886

18.434

0.16

23.649

23.649

21.624

21.221

24.850

0.20

29.186

29.186

25.285

24.509

31.498

Table 4.2. A22 (×107 ) (rectangular section: a = 0.23 m). r

VABS in [16]

AG 22

H AN 22

Aln 22

EJ0

0.04

1.0268

1.0268

1.0214

1.0203

1.0300

0.08

2.0450

2.0450

2.0031

1.9948

2.0698

0.12

3.0479

3.0479

2.9104

2.8830

3.1294

0.16

4.0307

4.0307

3.7136

3.6505

4.2186

0.20

4.9905

4.9905

4.3887

4.2688

5.3473

Now, let us take a look at the trapeze effect. Consider applying a couple of constant extensional forces with reverse direction on the ends of the beam. As γ ≪ 1, we neglect the second term in Eq. (4.68) and solve for γ as γ=

A22 2 F1 J1 + k1 κ1 − κ EA A 2EA 1

(4.70)

Substituting Eq. (4.70) into Eq. (4.69), expanding in Taylor series with respect to both κ1 and k1 around zero, and keeping the leading terms yields M1 (k1 , κ1 ) = −

F1 F1 J1 k1 + (GJS + A22 ) κ1 A EA

(4.71)

46 For a slender beam undergoing both extension and torsion, an explanation of the stiffening phenomenon of the trapeze effect can be formulated by calculating the effective torsional rigidity: Re =

F1 ∂M1 (k1 , κ1 ) = GJS + A22 ∂κ1 EA

(4.72)

By using different material models introduced in the previous section, F1 2ν F1 J0 − (J0 + J1 ) A 1+ν A 4 F1 F1 (J0 + J1 ) ReN H = GJS + J0 − A 3A F1 3 F1 Reln = GJS + J0 − (J0 + J1 ) A 2A ReG = GJS +

(4.73) (4.74) (4.75)

Note that the underlined term was missing from previously published formulae for trapeze effects known to the authors [27–30, 34, 47, 49]. Normalizing A22 to the traditional term (i.e., EJ0 ), we define AG 2ν JS 22 =1− EJ0 1 + ν J0 NH A 4 JS η N H = 22 = 1 − EJ0 3 J0 ln A 3 JS η ln = 22 = 1 − EJ0 2 J0 ηG =

(4.76)

which are plotted for an elliptical section in Figure 4.1 versus r and Poisson’s ratio ν. The traditional theory tends to overestimate the stiffening of the torsional rigidity caused by axial force. Only for very thin beams, when the Saint-Venant torsion stiffness JS can be neglected, the present models will reduce to the published theory. Furthermore, according to η N H and η ln , when the sectional thickness ratio increases, the beam tends to behave in a sort of reverse-trapeze effect, which is actually the positive Poynting effect.

47

St-Venant/Kirchhoff

x3 b

x2

a Hencky Strain Energy Elliptical Section

Neo-Hookean

Figure 4.1. Plot of Eq. (4.76) for an elliptical section,taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers.

4.1.10

Trapeze Effect: Untwisting of Pretwisted Beam

The result from the different strain energy functions in our analytical formulae is unique, that is, T12 = −EJ1 . As shown in Tables 4.3 and 4.4, it agrees very well with that from the VABS code developed in [39]. Rosen’s formulae for approximating the thin symmetrical cross sections within the first order of the pretwist, according to [49], can be expressed in terms of the present nomenclature as 1 F1R = EAγ + EJ0 κ21 + EJR k1 κ1 2

(4.77)

M1R = (GJS + EJ0 γ) κ1 + EJR k1 γ

(4.78)

where the subscript R stands for Rosen and the JR is defined as JR = ⟨x22 − x23 ⟩. To compare the extension-twist coupling due to pretwist, we define ξR =

JR , J0

ξ=

−J1 J0

(4.79)

48 as it is shown in Figure 4.2 for a rectangular section. From Figure 4.2, it can be seen that Rosen’s estimation is valid for the extremely thin beam, that is, with a very small r. However, it tends to underestimate the coupling stiffness for an initially twisted beam when the sectional thickness-to-width ratio is around 0.2 (a typical value for helicopter rotor blades). It is worth noting that this observation agrees with what has been pointed out by [47]. Table 4.3. Nonlinear stiffness induced by pretwist k1 (elliptical section: a = 0.23 m, k1 = 0.013963 rad/m ). r

VABS in [39] k1 T12 (×105 )

0.04

0.84173

0.84173

0.08

1.65937

1.65936

0.12

2.42981

2.42981

0.16

3.13195

3.13196

0.20

3.74748

3.74747

Table 4.4. Nonlinear stiffness induced by pretwist k1 (rectangular section: a = 0.23 m, k1 =0.008727 rad/m ). r

VABS in [39] k1 T12 (×105 )

0.04

0.89321

0.89321

0.08

1.76262

1.76262

0.12

2.58757

2.58757

0.16

3.35091

3.35091

0.20

4.03897

4.03897

49

Figure 4.2. Plot of Eq. (4.79) for a rectangular section, taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers.

For a pretwisted beam which is undergoing only extension, we have M1 (k1 , κ1 ) = 0. The explanation of the untwisting phenomenon in the trapeze effect can be concluded by rewriting Eq. (4.71) as (GJS +

F1 F1 A22 ) κ1 = J1 k1 EA A

(4.80)

To compare the results with Hodges’ theory, we follow [47] to define the torsion per unit pretwist for a slender beam as ζ = −∂κ1 /∂k1 so that (F1 /EA)J1 (G/E)JS + (F1 /EA)J0 (F1 /EA)J1 ζG = − (G/E)JS + (F1 /EA) [J0 − 2JS ν/(1 + ν)] (F1 /EA)J1 ζNH = − (G/E)JS + (F1 /EA) [J0 − 4JS /3] (F1 /EA)J1 ζ ln = − (G/E)JS + (F1 /EA) [J0 − 3JS /2]

ζH = −

(4.81)

where subscript H stands for Hodges’ theory. Setting F1 /EA = 0.005, results from Eq. (4.81) for an elliptical section are plotted in Figure 4.3 (a) and Figure 4.3 (b)

50 for two values of G/E = 0.4 and 0.025, respectively. The larger value is typical for isotropic materials whereas the smaller value could be for some composite materials. For the larger value of G/E Hodges’ theory agrees well with the present theory. For the smaller value of G/E, however, the difference between these two results is greater. Moreover, Hodges’ theory tends to slightly underestimate the untwist due to axial load of an initially twisted beam. It is interesting to note here that if we use the uniaxial stress assumption used in previous works, the present theory can reproduce the well-accepted analytical formulae in the literature. However, the uniaxial stress hypothesis is only valid for materially linear models.

(a) G/E = 0.4.

(b) G/E = 0.025.

Figure 4.3. Plot of Eq. (4.81) for an elliptical section (F1 /EA = 0.005), taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers.

4.1.11

Poynting Effect: Rubber Tube

Under the situation of F1 = k1 = 0, for a cylindrical section with outer and inner radii a1 and a0 , respectively, and furthermore J1 = 0, Eq. (4.70) reduces to γ=−

Aln A22 2 J0 2 1 2 2 κ1 = − 22 κ21 = κ = κ (a + a20 ) 2EA 2EA 4A 1 8 1 1

(4.82)

51 which is of the well-known form for the positive Poynting effect of a cylinder in [74] using the Hencky strain energy. Here it can be concluded that, for an extremely thin-walled section with Saint-Venant torsion stiffness JS ≈ 0

(4.83)

NH ln AG 22 ≈ A22 ≈ A22 ≈ EJ0

(4.84)

J1 = 0

(4.85)

then

For a circular section,

yields E H AN 22 = − J0 3 however AG 22 =

E Aln 22 = − J0 2

1−ν EJ0 1+ν

(4.86)

(4.87)

This implies that the St-V/K model cannot be used to predict the positive Poynting effect as AG 22 is positive while we need a negative value for A22 here. Both the Hencky strain energy and neo-Hooken material models can be used to explain both the trapeze effects and positive Poynting effect. H ln Another interesting conclusion can be made by setting AN 22 = A22 = 0, and the

nonlinear torsion-extension behavior in Eq. (4.68) and Eq. (4.69) for a prismatic beam will vanish. This corresponds to a specific aspect ratio. For example, the decoupling √ thickness ratio for an elliptic section will be rN H = 1/ 3 ≈ 0.577 if one uses a neo√ √ Hookean model, and rln = 2 − 3 ≈ 0.518 for the Hencky strain energy model. This conclusion will be further validated in the following example using 3D FEA.

4.1.12

Decoupling Thickness Ratio: Comparison with 3D FEA

The axial displacement is reported to be important to the dynamic behavior of the rotating beams [136]. In this section the displacement calculated by our model is

52 compared with results obtained from 3D FEA using ANSYS MAPDL, in which the 3D beam body is meshed by 20 nodes quadratic hexahedral elements. One end of the beam is clamped while the twisting angle is applied at the free end. To avoid the boundary effects and because of the first constraint in Eq. (2.3) the axial displacement data is measured from the node at the centroid of the middle-length-located section where the beam reference line is passing through. As we are comparing the numerical values, the γ 2 term associated with A11 in Eq. (4.68) can be kept without difficulty. NH ln Note that the AG 11 has a different sign from A11 and A11 . By setting F1 = k1 = 0 in

Eq. (4.68), we obtain the general equation to predict the axial displacement u1 of the beam under pure torsion as l

u1 = ∫ γdx1 = 0

√ l [ (EA)2 − 3A11 A22 κ21 − EA] 3A11

(4.88)

The displacement of the middle point of a rubber beam with an elliptic-section calculated by Eq. (4.88) with equivalent µ = 1.5 MPa and K = 0.5 GPa is shown in Figure 4.4 (a) with respect to the thickness ratio r. The 3D FEA is computed using the dimension with the semi-axes a = 0.23 m and the length l = 20a. The twisting angle at the free end is 45 degree. Another example shown in Figure 4.4 (b) uses the same geometry, but the material properties are E = 68.9 GPa and ν = 0.33 for aluminum. The twisting angle at the free end is 0.5 degree. The displacement uP1 is computed by using the well-known published values AP11 = EA and AP22 = EJ0 from the strain energy function Eq. (4.6), which is not better than prediction using the StV/K model. From these comparisons we observe that our analytical solution agrees well with 3D nonlinear FEA if the same material model is used. We can also observe that the coupling vanishes at a specific aspect ratio as pointed in the previous section. Finally, different nonlinear material models may exhibit very different nonlinear behavior. This discloses a point worthy of noting about experimental characterization of nonlinear materials using beam-like coupons and suggests that we not only need to determine the material constants but also the functional form of the material model.

53

(a) Rubber beam with µ = 1.5 MPa and K = 0.5 (b) Aluminum beam with E = 68.9 GPa and ν = GPa (κ1 = 0.171 rad/m).

0.33 (κ1 = 0.038 rad/m).

Figure 4.4. u1 of the middle position of an elliptic-section beam, taken from Jiang, Yu, and Hodges, “Analytical modeling of trapeze and Poynting effects of initially twisted beams” [81] reprinted by permission of the American Society of Mechanical Engineers.

4.1.13

Negative Poynting Effect

Consider a cylinder with sectional radius a. Motivated by [69], we firstly check the Mooney-Rivlin (M-R) model expressed by W MR =

µ K µ ¯ φ(I1 − 3) + (1 − φ)(I¯2 − 3) + (J − 1)2 2 2 2

(4.89)

And when φ = 1 it reduces to the N-H model. The ΠO(ε2 ) results form Eq. (4.89) is the same as Eq. (4.12), and obviously the warping solution is the same as Eqs. (4.28) – (4.29). Carrying out the same approach introduced in this paper and correcting the 1D strain energy density up to the ΠO(3 ) , we can obtain the nonlinear stiffness accounting for the Poynting effect as R AM 22 =

Ea4 π EJ0 (φ − 2) = (φ − 2) 6 3

(4.90)

and the sign of Poynting effect depends on φ. This result is in agreement with Ref. [69] and also validated by 3D FEA.

54 Motivated by [78], we secondly consider the 3D strain energy density of the form WM =

p + 2m 3 λ + 2µ 2 j1 − 2µj2 + j1 − 2µj1 j2 + nj3 2 3

(4.91)

where p, m, n are the third-order elastic constants, and ji are the invariants of Green strain E. Following the same approach, the nonlinear stiffness terms are AM 11 =

a2 π [3nλ2 (λ + µ) + 2µ(9λ3 + λµ(24λ + 21µ) 4(λ + µ)3 + m(3λ + 2µ)2 + 2µ2 (p + 3µ))]

AM 22 =J0

(4.92)

nλ + 4µ(m + λ + 2µ) 4(λ + µ)

M plug these AM 11 and A22 into Eq. (4.88) and expand the Taylor series with respect to

κ1 , we can obtain the axial displacement as u1 = −

M 2 (nλ + 4µ(m + λ + 2µ)) a2 (nλ + 4µ(m + λ + 2µ)) 2 κ1 l = − 1 6 2 3 l 16µ(3λ + 2µ) 4a π µ (3λ + 2µ)

(4.93)

which is same with the Eq. (2.41) in [78]. And both positive and negative Poynting effects can be predicted. Note the ΠO(ε2 ) resulting form Eqs. (4.89) and (4.91) is in the same expression as Eq. (4.12).

4.2

Numerical Analysis

4.2.1

Material Models and Their Tensorial Derivatives

The anisotropic St-V/K model is defined by the energy function W 1 W = Eij Cijkl Ekl 2

(4.94)

where Cijkl denotes the components of the fourth-order tensor named the second elasticity tensor C which refers to the undeformed configuration. Note that Eq. (3.48) is equal to Eq. (4.94) when the damage is neglected from the problem. Besides the first elasticity tensor A, C also plays an important role in describing hyperelasticity. Its components are defined as Cijkl ≡

∂Sij ∂ 2W ≡ ∂Ekl ∂Eij ∂Ekl

(4.95)

55 where Sij denotes the components of the second Piola-Kirchhoff stress (PK2). The components of PK1 of this material model are then obtained by Pab = Fai Cibkl Ekl

(4.96)

Further the tensorial derivative of Pab with respect to Fcd gives Aabcd = δac Cdbij Eij + Fai Fcj Cibjd

(4.97)

The St-V/K model succeeds in predictions of geometrically nonlinear behaviors with small strain assumptions especially for beams with thin cross sections [11, 16, 30, 34, 38, 47, 48, 50, 52, 65]. However, its lack of practice on large strain problems has been already widely known [137]. It is pointed out by Petersen [44] that the St-V/K model is not suitable for slim beams which can be highly twisted while the strains are small. In addition, Degener et al. [135] prove that the prediction by assuming the St-V/K model is poor for finite deformation by experiments on rubber cylinders under combined extension and torsion. Compared with the St-V/K model, the HASE model is preferred [71, 72] in modeling nonlinear structural behaviors and has been applied in many commercial FEA softwares such as ANSYS [138] and Abaqus [139]. The HASE model is obtained by simply replacing the Green strain tensor components Eij in Eq. (4.94) with Hencky strains Hij , that is, 1 W = Hij Cijkl Hkl 2

(4.98)

The Hencky strain tensor is obtained by taking diagonal logarithmic functions of spectral decomposed form of the Green deformation tensor C as 1 1 1 C C H = ( ln λC 1 ) M1 + ( ln λ2 ) M2 + ( ln λ2 ) M2 2 2 2

(4.99)

and the principal directions of the principal values λC i of tensor C is expressed by Ni = Ni(1) b1 + Ni(2) b2 + Ni(3) b3

(4.100)

56 The defined eigenvalue-bases Mi are formed as dyadics of the principal vectors as M1 = N1 N1 , M2 = N2 N2 , and M3 = N3 N3 . In order to use Cijkl , we need to express the Hencky strain tensor using bi as T ⎡ ⎧ ⎫ 2 ⎪ ⎪ ⎢ b Ni(1) Ni(1) × Ni(2) Ni(1) × Ni(3) ⎪ ⎪ 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ 2 Mi = ⎨ b2 ⎬ ⎢⎢ Ni(1) × Ni(2) Ni(2) Ni(2) × Ni(3) ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ 2 ⎪ ⎪ ⎢ Ni(1) × Ni(3) Ni(2) × Ni(3) ⎪ Ni(3) ⎩ b3 ⎪ ⎭ ⎣

⎤⎧ ⎫ ⎥⎪ b1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎥⎨ b ⎪ ⎥⎪ 2 ⎬ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ b ⎭ ⎦⎩ 3 ⎪

(4.101)

i = 1, 2, 3 (no sum) Then the components of P and A are respectively Pab = Faq Sbq = 2Faq Aabcd = 4Faq Fcp (

∂Hij ∂W = 2Faq ( ) Cijkl Hkl ∂Cbq ∂Cbq

∂Hij ∂Hkl ∂ 2 Hij ∂Hij + Hkl ) Cijkl + 2δac ( ) Cijkl Hkl ∂Cbq ∂Cdp ∂Cbq ∂Cdp ∂Cbd

(4.102)

(4.103)

The strain energy for rubber-like material is commonly built in terms of the invariants of Cauchy-Green tensor C and/or principal stretches [140]. For example, the polynomial form shown in the following Eq. (4.7) is a general expression of this kind. If n = 1, c11 = 0, and m = 1, it reduces to the M-R model [62, 64, 138], that is, W = W1 (I1 , J ) + W2 (I2 , J ) + W3 (J )

(4.104)

with W1 = c10 (I¯1 − 3)

W2 = c01 (I¯2 − 3)

W3 =

1 (J − 1)2 d1

(4.105)

If c01 = 0 in Eq. (4.105), it further reduces to the N-H model [134], that is, W = W1 + W3

(4.106)

Because N-H and M-R models behave linearly in twisting test [62, 72, 141], Yeoh [142] proposes a high-order energy function as 1 1 1 W = c10 (I¯1 −3)+c20 (I¯1 −3)2 +c30 (I¯1 −3)3 + (J −1)2 + (J −1)4 + (J −1)6 (4.107) d1 d2 d3

57 To use N-H, M-R and Yeoh material models, PaB and AaBcD can be obtained for Wi in Eq. (4.104) as Pab = Pab + Pab + Pab (1)

(2)

(3)

Aabcd = Aabcd + Aabcd + Aabcd (1)

(2)

(3)

(4.108) (4.109)

where ∂Wi ∂Ii ∂Wi ∂J + i = 1, 2 (no sum) ∂Ii ∂Fab ∂J ∂Fab ∂W3 ∂J (3) Pab = ∂J ∂Fab ∂ 2 Wi ∂Ii ∂J ∂Ii ∂J ∂ 2 Wi ∂J ∂J (i) ( + )+ Aabcd = ∂Ii ∂J ∂Fab ∂Fcd ∂Fcd ∂Fab ∂J 2 ∂Fab ∂Fcd ∂ 2 Wi ∂Ii ∂Ii ∂Wi ∂ 2 Ii ∂Wi ∂ 2 J + + + ∂Ii 2 ∂Fab ∂Fcd ∂Ii ∂Fab ∂Fcd ∂J ∂Fab ∂Fcd Pab = (i)

(4.110) (4.111)

(4.112)

i = 1, 2 (no sum) Aabcd = (3)

∂ 2 W3 ∂J ∂J ∂W3 ∂ 2 J + ∂J 2 ∂Fab ∂Fcd ∂J ∂Fab ∂Fcd

(4.113)

with the following derivative recipes ∂I1 = 2Fab ∂Fab

(4.114)

∂J −1 = J Fab ∂Fab

(4.115)

∂I2 = 2 (I1 Fab − Fak Cbk ) ∂Fab ∂ 2 I1 = 2δac δbd ∂Fab ∂Fcd ∂ 2J −1 −1 −1 ) = J (Fdc Fba − Fbc−1 Fda ∂Fab ∂Fcd ∂ 2 I2 = 2 (2Fab Fcd + δac (I1 δbd − Cbd ) − δbd Fak Fck − Fad Fcb ) ∂Fab ∂Fcd 4.2.2

(4.116) (4.117) (4.118) (4.119)

Small Strain Trapeze Effect Model

The validity of the code developed herein will be shown in this section by solving nonlinear problems in a wide realm of applications to beam-like structures. For

58 convenience, we name the theory and companion code developed using the present nonlinear theory and algorithms herein by “VABS (Model of Hyperelasticity)”, and those from [16] as “VABS (Popescu and Hodges)”. We also label the results obtained by performing 3D finite element analysis as “3D FEA (Model of Hyperelasticity)”. VABS (Popescu and Hodges) deals the sectional nonlinearity using the St-V/K model under small strain assumptions. By using the VAM, the zero-order warping solution is taken to correct W up to O(ε3 ). As a result, the 1D strain energy density is expressed by 1 1 Π = ⟨Eij Dijkl Ekl ⟩ = T ( S + γA + κ1 B + κ2 C + κ3 D) 2 2

(4.120)

where S, A, B, C, and D are 4 × 4 symmetric stiffness matrices. Then the sectional stress resultants can be obtained as F1 =

∂Π ∂γ

M1 =

∂Π ∂κ1

M2 =

∂Π ∂κ2

M3 =

∂Π ∂κ3

(4.121)

Note if Γij is used as the 3D strain definition, the higher-ordered terms ΠO(ε3 ) = T (γA + κ1 B + κ2 C + κ3 D)

(4.122)

in Eq. (4.120) can be reasonably neglected. Under such circumstance the 1D strain energy density is 1 Π = T S 2

(4.123)

where the classical stiffness matrix S is obtained by ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ S = ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂F1 ∂γ

∂F1 ∂κ1

∂F1 ∂κ2

∂F1 ∂κ3

∂M1 ∂γ

∂M1 ∂κ1

∂M1 ∂κ2

∂M1 ∂κ3

∂M2 ∂γ

∂M2 ∂κ1

∂M2 ∂κ2

∂M2 ∂κ3

∂M3 ∂γ

∂M3 ∂κ1

∂M3 ∂κ2

∂M3 ∂κ3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.124)

which indicates that with the linear strain definition the nontrivial components in Eq. (4.124) can be reproduced by the tangent coefficients Tij from Eq. (3.21). This point will be illustrated in the following examples followed by cases involving finite deformation.

59 A rigorous nonlinear model should be able to reproduce the results predicted by the linear strain model. To this end we first use a few sections including a spring steel strip, a heterogeneous rotor blade, and a composite pipe to investigate the trapeze effect, a widely known nonlinear beam behavior. A rigorous analytical explanation to the trapeze effect was given in previous sections in this chapter under small strain assumption using VAM. The following sections present the more general numerical solutions without small strain assumptions.

4.2.3

Trapeze Effect: Spring Steel Strip

Consider a spring steel strip (see Figure 4.5). The material is assumed to be isotropic with the Young’s modulus E = 207.126 GPa and Poisson’s ratio ν = 0.27. The non-zero derivatives of stress resultants with respect to their corresponding strains are shown in Table 4.5, in which the Tij are obtained by setting the beam strains to 10−9 which verifies that the differences among VABS (Popescu and Hodges), VABS (St-V/K), and VABS (HASE) can be ignored at such small values of strain.

Figure 4.5. Sectional geometry of the spring steel strip.

3D FEA is conducted in ANSYS according to the geometry in Figure 4.5 and is regarded as a control group. The beam is cantilevered, straight and made of uniform cross sections. The beam length is l = 100 mm. Nodes at the free end are kinematically coupled to a master node with MPC184 elements and a constant tip rotation angle of 1 degree is applied at this master node. Simultaneously, an axial displacement is also applied at the master node so that the axial strain γ varies from 0.0 to 0.005.

60

Table 4.5. Derivatives of stress resultants with respect to beam strains of the strip. Nontrivial derivatives

VABS (Popescu and Hodges)

VABS (St-V/K)

VABS (HASE)

(×105 N)

S11 = 4.1425

T11 = 4.1425

T11 = 4.1425

(×10 Nm /rad)

S22 = 2.1473

T22 = 2.1473

T22 = 2.1473

(×10−3 Nm2 /rad)

S33 = 1.3808

T33 = 1.3803

T33 = 1.3803

(×10 Nm /rad)

S44 = 3.4521

T44 = 3.4521

T44 = 3.4521

∂F1 ∂γ ∂M1 ∂κ1 ∂M2 ∂κ2 ∂M3 ∂κ3

−3

0

2

2

For simplicity, the master node is always located at the extension center of the cross section. The reaction forces and moments are measured from the master node and compared with the results from the present theory. To avoid boundary effects, the beam is made sufficiently slender. Under uniform twist and axial strain, we can express the twisting angle as θ1 (x1 ) = κ1 x1

(4.125)

and the beam tip axial deformation as ∆l = γl

(4.126)

Substituting Eq. (4.126) and x1 = l + ∆l into Eq. (4.125) gives κ1 =

θ1tip (1 + γ)l

(4.127)

In 3D FEA, ∆l and θ1tip are applied boundary conditions at the master node along with the kinematic coupling of the free end. In VABS, corresponding γ and κ1 are non-zero inputs. The increasing of reaction twisting moment M1 with respect to the axial strain γ is shown in Figure 4.6. It is also interesting to investigate the effects of the nonlinear strain definition. Here we take the simple tension as an example. Analytic analysis in Eq. (4.68) provides 3 F1 = EAγ + A11 γ 2 2

(4.128)

61 0.0035

VABS (Popescu and Hodges) 0.0030

3D FEA (St-V/K) VABS (St-V/K) 3D FEA (HASE) VABS (HASE)

0.0020

M

1

(Nm)

0.0025

0.0015

0.0010

0.0005

0.0000 0.000

0.001

0.002

0.003

0.004

0.005

Figure 4.6. Trapeze effect of the spring steel strip.

When the St-V/K model is applied for the isotropic beam we have A11 = S11 = EA

(4.129)

A11 = −S11 = −EA

(4.130)

while for the HASE model

Define a parameter β to indicate the normalized difference between T11 and EA as β=

T11 −1 EA

(4.131)

As shown in Figure 4.7, the range of γ for the agreement between the predictions by using the linear and the nonlinear strain definitions is below 10−4 . In addition, the predictions of VABS (St-V/K) and VABS (HASE) depart from each other at the same level of the axial strain that they depart from EA. Beginning with γ = 10−4 , the St-V/K model starts to predict a higher value than EA while the HASE model shows a different trend. This phenomenon is in correspondence with the analytic

62 explanation provided by the analytical solution. In light of Figure 4.7 the differences among the strain definitions can be ignored when γ < 10−4 for the simple tension. 0.03 Jiang et al. (St-V/K) VABS (St-V/K)

0.02

Jiang et al. (HASE) VABS (HASE)

0.01

0.00

-0.01

-0.02

-0.03 -9

-8

-7

-6 Log

-5

-4

-3

-2

10

Figure 4.7. Plot of Eq. (4.131) in case of the spring steel strip.

4.2.4

Trapeze Effect for a Realistic Rotor Blade

A realistic rotor blade of length l = 8.124 m is made of a uniform cross section in the form of Bell540 (modified NACA0012) airfoil with 0.8214 m chord length. A schematic of this blade section is depicted in Figure 4.8 with both the airfoil and the sectional coordinates. The origin of the sectional coordinates is located at the extension center of the section. This blade is made of aluminum as the skin and a foam as the core. Aluminum has the properties E = 72.4 GPa, ν = 0.3, and the properties for the foam are E = 2.76 GPa, ν = 0.22. The skin thicknesses are 0.012 m, 0.00832 m, and 0.006 m for the leading edge, the web, and the trailing edge, respectively.

63

Figure 4.8. Sectional layout of the rotor blade.

The nontrivial derivatives of the stress resultants with respect to beam strains are shown in Table 4.6. The VABS (St-V/K) and VABS (HASE) values are obtained by setting the beam strains to 10−9 , which shows again that the differences among the values from different models can be ignored at this strain level. Table 4.6. Derivatives of stress resultants with respect to beam strains of the rotor blade. Nontrivial derivatives


VABS (St-V/K)

VABS (HASE)

(×108 N)

S11 = 9.8146

T11 = 9.8153

T11 = 9.8146

(×10 Nm /rad)

S22 = 8.3183

T22 = 8.3222

T22 = 8.3183

(×105 Nm2 /rad)

S33 = 5.9308

T33 = 5.9308

T33 = 5.9308

(×10 Nm /rad)

S44 = 5.2145

T44 = 5.2153

T44 = 5.2145

∂F1 ∂γ ∂M1 ∂κ1 ∂M2 ∂κ2 ∂M3 ∂κ3

5

7

2

2

3D FEA is conducted to study the trapeze effect of this blade. The mesh of the free end of the 3D model in ANSYS is shown in Figure 4.9. The validation methodology applied to the example of the spring steel strip is also applied to this blade. A constant tip rotation angle θ1tip = 1 degree is set, while the axial strain γ varies from 0.0 to 0.004. The HASE model is widely used as a default by commercial FEA softwares such as ANSYS and Abaqus when the geometrically nonlinear option is triggered and the constant elastic moduli are provided at the same time. In light

64 of this fact, we compare in this section the results among VABS (HASE), VABS (Popescu and Hodges) and 3D FEA (HASE) by ANSYS, as shown in Table 4.7. It is worth noticing that VABS (Popescu and Hodges) is based on the St-V/K model instead of the HASE model. The 3D FEA (HASE) is regarded as a control group. And the absolute percent differences (ABS DIFF%) of the VABS results with respect to the control group are listed in the square brackets. Very good agreement has been obtained between VABS and 3D FEA. The difference between VABS and 3D FEA decreases along with increased γ which means the additional twisting moment predicted by VABS is relatively larger than the predication of 3D FEA. This difference is possibly due to that the constraints applied to the ends of the beam in 3D FEA prevent the occurrence of the trapeze effect in these local boundary regions. The difference between VABS (Popescu and Hodges) and 3D FEA is also acceptable which indicates that the St-V/K model does have practical values for small cross-sectional aspect ratio.

Figure 4.9. 3D mesh of the free end of the blade in ANSYS.

65

Table 4.7. Twisting moment (kNm) [ABS DIFF%] to preserve a tip rotation of 1 degree under various levels of axial strains for the rotor blade.

4.2.5

γ

3D FEA (HASE)

VABS (HASE)


0

1.8006328

1.7683337 [1.7937637]

1.7674867 [1.8524577]

0.001

1.9091152

1.8840782 [1.3114434]

1.8757257 [1.7599532]

0.002

2.0157375

1.9989144 [0.8345892]

1.9837486 [1.5974216]

0.003

2.1215331

2.1128469 [0.4094294]

2.0915563 [1.4229745]

0.004

2.2266129

2.2258825 [0.0328011]

2.1991490 [1.2430137]

Trapeze Effect: Composite Pipe

We progress to deal with a fiber reinforced composite pipe.

A section of a

multi-layer composite pipe with the geometry and the layout information is shown in Figure 4.10. Each layer is made of orthotropic material having properties with longitudinal Young’s modulus E1 = 20.59 × 106 psi, transverse Young’s modulus E2 = E3 = 1.42 × 106 psi, shear moduli G12 = G13 = G23 = 8.7 × 105 psi, and Poisson’s ratio ν12 = ν13 = ν23 = 0.42. θ3 = - 45 degree

y3 θ3 = 90 degree y2

x3 θ3 = 0 degree θ3 = - 45 degree

1’’

Sectional Coordinates

1’’

1’’

0.4’’

y2

y3

y2

y3

y2 y3

θ3 = 45 degree

x2

0.3’’

θ3=90 degree

θ3 = 0 degree

y3 Material Coordinates

θ3 = 45 degree

y2

θ3

y1

Figure 4.10. Geometry and layout of the anisotropic pipe section.

66 The nontrivial derivatives of the stress resultants to the beam strains predicted by VABS (Popescu and Hodges), VABS (St-V/K), and VABS (HASE) are listed in Table 4.8, in which the values calculated by VABS (St-V/K) and VABS (HASE) are obtained by setting the beam strains to 10−9 . The agreement is excellent at this strain level as expected. In Table 4.8, the nonzero

∂F1 ∂κ1

represents the coupling of the

extensional and the torsional behaviors due to the material anisotropy. For small γ and κ1 , the twisting moment can be calculated as M1 = S12 γ + S22 κ1

(4.132)

Table 4.8. Derivatives of stress resultants with respect to beam strains of the composite pipe. Nontrivial derivatives


VABS (St-V/K)

VABS (HASE)

(×10 lbf)

S11 = 1.0389

T11 = 1.0389

T11 = 1.0389

(×104 lbf-in/rad)

S12 = 9.8357

T12 = 9.8357

T12 = 9.8357

(×10 lbf-in /rad)

S22 = 6.8706

T22 = 6.8706

T22 = 6.8706

(×106 lbf-in2 /rad)

S33 = 1.8823

T33 = 1.8823

T33 = 1.8823

(×10 lbf-in /rad)

S44 = 5.3815

T44 = 5.3815

T44 = 5.3815

∂F1 ∂γ ∂F1 ∂κ1 ∂M1 ∂κ1 ∂M2 ∂κ2 ∂M3 ∂κ3

7

5

6

2

2

3D FEA is conducted in Abaqus and the beam is cantilevered. Same validation methodology introduced previously is also applied to this beam. Since the material anisotropy does not couple the extension and the bending nor the twisting and the bending, Eq.(4.125) is still applicable. To apply the axial strain γ varying from 0.0 to 0.007, the corresponding displacement is imposed on the reference point at the free end, where the rotation angle θ1tip = 5 degree is also simultaneously applied. The beam length is set to 50 times the width along x2 . The increasing of the reaction twisting moment M1 to preserve the constant tip rotation angle θ1 is plotted in Figure 4.11. It can be seen that both of the linearly elastic model and VABS (Popescu and Hodges) predict higher values compared to the results by using the HASE model, which agrees with the behavior observed in Figure 4.7.

67 1100 VABS (Linearly Elastic Model) 1000

900

VABS (Popescu and Hodges) VABS (HASE)

800

700

M

1

(lbf-in)

3D FEA (HASE)

600

500

400

300 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Figure 4.11. Twisting moment (lbf-in) to preserve a tip rotation of 5 degree under various levels of axial strains for the composite pipe.

4.2.6

Poynting Effect: Circular Cross Section

The present theory and the companion VABS code will be used to study the Poynting effect, another widely known nonlinear beam behavior. Both of the positive and the negative Poynting effects have been analytically explained in section 4.1. The positive Poynting effect is much more common in engineering structures so we focus on this phenomenon in this section. In this section, the experimental results by Rivlin and Saunders [62] will be compared with the numerical solution from the present theory and 3D FEA in cooperation with multiple typical hyperelastic models. The sectional dimension of the specimen is shown in Figure 4.12. For simple tension test D = 3/8 in. while for simple torsion test D = 1 in. The length is 5 times D in 3D FEA and the same validation methodology introduced previously is also applied to all of the 3D models in this section. We use the shear modulus set up by Anand [72] and Rivlin and Saunders [62] for their cylinder samples made of the vulcanized rubber, i.e., µ = 382.5 KPa. The other ma-

68 terial constants are calculated by assuming the Poisson’s ratio ν = 0.4999, as shown in Table 4.9.

x3 D

x2

Figure 4.12. Geometry of the cylinder section.

Table 4.9. Material constants for cylinder depicted in Figure 4.12. Matrial Constants

M-R

N-H

HASE

ν

0.4999

0.4999

0.4999

µ

382.5 (KPa)

382.5 (KPa)

382.5 (KPa)

E

No Need

No Need

2µ(1 + ν)

K

2µ(1+ν) 3(1−2ν)

2µ(1+ν) 3(1−2ν)

No Need

c10

0.4375µ

0.5µ

No Need

c01

0.0625µ

No Need

No Need

d1

2K −1

2K −1

No Need

Rivlin and Saunders [62] provide data of the tensile force F1 as a function of γ, which are plotted together with the numerical values calculated by both VABS and 3D FEA in Figure 4.13. Axial strain γ increases from 0 to 0.4341. A conclusion by Anand [72] states that the mismatch between the predictions by the HASE model and the experimental data will occur when the principal stretches are larger than

69 0.3. In correspondence with this insight, Figure 4.13 shows that when γ > 0.3, the experimental data agree more with the curves predicted by using N-H and M-R. 25.0 22.5

EXPERIMENTAL Rivlin and Saunders

20.0 17.5

12.5

NUMERICAL

F

1

(N)

15.0

10.0

VABS (N-H) 3D FEA (N-H)

7.5

VABS (M-R) 5.0

3D FEA (M-R) VABS (HASE)

2.5

3D FEA (HASE)

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Figure 4.13. Data comparison of the axial tensile force for various values of the axial strain for the beam with the cylindrical section.

Rivlin and Saunders [62] also give the experimental data of M1 as a function of κ1 for γ = 0 in the simple torsion test. These data are plotted together with the numerical results calculated by VABS and 3D FEA in Figure 4.14 with κ1 ranging from 0 to 45 rad/m. The curves are seemingly linear, which motivates Yeoh [142] to develop the cubic model which will be investigated later in this section. The difference between present code and 3D FEA is caused by ignoring the boundary effects on the two ends of the 3D model. This difference is getting bigger when the twist is huge. However, the maximum difference is merely 1.49% for the results presented in Figure 4.14. The Poynting effects predicted by VABS and 3D FEA with typical hyperelastic models are shown in Figure 4.15 together with the experimental data measured by Rivlin and Saunders [63]. Indeed in order to produce simple torsion without axial displacement, it can be seen that a normal compression force must be applied to the end sections of a cylinder in addition to the twisting moment. In correspondence with the prediction

70 by Anand [72], at a relatively huge twist value the experimental data varies from the HASE curve to the M-R curve. 0.75 NUMERICAL VABS (N-H) 0.60

3D FEA (N-H)

3D FEA (M-R)

0.45

VABS (HASE) 3D FEA (HASE)

M

1

(Nm)

VABS (M-R)

0.30

0.15 EXPERIMENTAL Rivlin and Saunders 0.00 0

5

10

15

20 1

25

30

35

40

45

(rad/m)

Figure 4.14. Data comparison of the axial twisting moment for various values of the twist for the beam with the cylindrical section.

EXPERIMENTAL

0

Rivlin and Saunders

-10 NUMERICAL

1

(N)

-5

F

VABS (N-H) 3D FEA (N-H)

-15

VABS (M-R) 3D FEA (M-R) -20

VABS (HASE) 3D FEA (HASE)

-25 0

5

10

15

20

25 1

30

35

40

45

(rad/m)

Figure 4.15. Data comparison of the Poynting effect for the beam with the cylindrical section.

71 4.2.7

Poynting Effect: Tubular Cross Section

Some soft rubbers can be largely stretched and behaves relatively more nonlinearly, and for some cases the twisting behaviors are not as linear as those predicted by N-H and M-R models. There is a common belief that more number of terms in Eq.(4.7) would lead to a better fit with nonlinear constitutive relation, for this reason Yeoh [142] proposed a cubic model. A tube made of such type of material is analyzed by both VABS and 3D FEA. The sectional geometry is depicted in Figure 4.16, with the inner radius Di = 0.0071 m and the outer radius Do = 0.011 m. The length is built as L = 1.1 m in 3D FEA and the same validation methodology introduced previously is also applied to all of the 3D models in this section.

x3

x2

Figure 4.16. Geometry of the tube section.

Two sets of material constants of Yeoh’s hyperelastic model are examined in the simulations of the simple torsion. Material #1 has the constants c10 = 511 KPa, c20 = −63.8 KPa, c30 = 11.6 KPa, d1 = d2 = d3 = 1.95697 GPa−1 . For material #2, c10 = 511 KPa, c20 = −6380 KPa, c30 = 1160 KPa, d1 = d2 = d3 = 1.95697 GPa−1 . The data of simple torsion test are plotted in Figure 4.17 showing agreement between VABS (Yeoh) and 3D FEA (Yeoh). Additionally, due to larger c20 and c30 in

72 material #2, the corresponding curves from using this material shows the inflection point at a lower strain value compared to the curves predicted by using material #1. The variation of the tangent coefficient T22 with respect to κ1 is shown in Figure 4.18. Although this value is not directly available from 3D FEA, investigating the curve of the tangent in Figure 4.17 shows the relation between Figure 4.17 and Figure 4.18. The curves of the Poynting effects are shown in Figure 4.19. The tangent coefficients T12 conjugating the twist κ1 to the axial compression force F1 are shown in Figure 4.20 with respect to κ1 . This parameter directly explains the Poynting effect. In addition, as shown in Figure 4.21, the simple tension is also simulated with material #1 which provides two inflection points. The tangent coefficient T11 obtained by

∂F1 ∂γ

is plotted

in Figure 4.22 with respect to γ. It can be seen that the tangent extension stiffness predicted by using Yeoh model firstly decreases when γ is smaller than 0.75 and then increases after γ = 0.75. This observation agrees the curve in Figure 4.21. In these simulations, VABS agrees with 3D FEA on predicting nonlinear phenomena by holding a beam-like coupon. VABS also provides the tangent beam stiffness which is not directly available from 3D FEA. 0.150 VABS (Yeoh, Material #1) 3D FEA (Yeoh, Material #1)

0.125

VABS (Yeoh, Material #2) 3D FEA (Yeoh, Material #2)

0.075

M

1

(Nm)

0.100

0.050

0.025

0.000 0

1

2

3

4 1

5

6

7

8

(rad/m)

Figure 4.17. Data comparison of the torque for various values of the twist for the beam with the tubular section and Yeoh model.

73 0.020

0.019

0.018

(Nm /rad)

0.017

2

0.016

T

22

0.015

0.014

0.013 Yeoh Material #1

0.012

Yeoh Material #2 0.011 0

1

2

3

4 1

5

6

7

8

(rad/m)

Figure 4.18. Plot of VABS (Yeoh) prediction of the tangent coefficient of the twist in Figure 4.17 for the beam with the tubular section.

0.0

-0.1

F

1

(N)

-0.2

-0.3

-0.4 VABS (Yeoh, Material #1) 3D FEA (Yeoh, Material #1) -0.5

VABS (Yeoh, Material #2) 3D FEA (Yeoh, Material #2)

-0.6 0

1

2

3

4 1

5

6

7

8

(rad/m)

Figure 4.19. Data comparison of the Poynting effect for the beam with the tubular section and Yeoh model.

74

0.000

-0.050

-0.075

T

12

(Nm/rad)

-0.025

-0.100

-0.125

Yeoh Material #1 Yeoh Material #2

-0.150 0

1

2

3

4 1

5

6

7

8

(rad/m)

Figure 4.20. Plot of VABS (Yeoh) prediction of the tangent coefficient of the axial strain in Figure 4.19 for the beam with the tubular section.

700 VABS (Yeoh) 600

3D FEA (Yeoh)

400

F

1

(N)

500

300

200

100

0 0.00

0.25

0.50

0.75

1.00

1.25

1.50

Figure 4.21. Data comparison of the axial tensile force for various values of axial strain for the beam with the tubular section and Yeoh model #1.

75

1100 1000 900

700 600

T

11

(N)

800

500 400 300 200 100 0.00

0.25

0.50

0.75

1.00

1.25

1.50

Figure 4.22. Plot of VABS (Yeoh, Material #1) prediction of the tangent coefficient of the axial strain in Figure 4.21 for the beam with the tubular section.

76

77

5. NONLINEAR SHEAR BEHAVIOR IN COMPOSITE BEAMS Accurate predictions of physically nonlinear elastic behaviors of a material point in the structure are essential to the further analyses which are beyond the linear elasticity regime, for example, the progressive damage and the failure. In light of substantial experimental evidence of nonlinear shear stress-strain responses in composites, it is necessary to consider them in the structure-level simulations rigorously. In the present work, the Hahn-Tsai [84] nonlinear in-plane shear model is used to the validation by comparing the VABS results with those from 3D FEA. Both static and dynamic examples are given. The ±45○ -laminated coupon tensile tests are simulated. 3D local fields such as the free-edge stresses are precisely captured by the present model. Nominal stress-strain curves predicted for various composite beams with different cross-sections are compared to show the impact of the cross-sectional designs of the coupons on their performances in calibrating the material constants.

5.1

Nonlinear Beam Constitutive Relation To validate the present VABS theory, consider the third-order elastic nonlinear

shear model proposed by Hahn and Tsai [84], that is, 2Γ12 =

3 σ12 σ12 + G12 X12

(5.1)

So the tangent compliance matrix can be obtained by [S] = [H] + 3

2 σ12 [Q] X12

(5.2)

78 where ⎡ 1 ⎢ ⎢ E1 ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ [H] = ⎢ ⎢ ν12 ⎢ − E1 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ν13 ⎢ − E1 ⎣

0

0

− νE212

0

1 G12

0

0

0

0

1 G13

0

0

0

0

1 E2

0

0

0

0

1 G23

0

0

− νE232

0

⎤ − νE313 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ν32 ⎥ − E3 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 1 ⎥ E3 ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [Q] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ 0 0 0 0 0 0 ⎥⎥ ⎥ 0 1 0 0 0 0 ⎥⎥ ⎥ ⎥ 0 0 0 0 0 0 ⎥⎥ ⎥ ⎥ 0 0 0 0 0 0 ⎥ ⎥ ⎥ 0 0 0 0 0 0 ⎥⎥ ⎥ ⎥ 0 0 0 0 0 0 ⎥⎦

(5.3)

The elastic properties are E1 = 155.65 GPa, E2 = E3 = 8.977 GPa, ν12 = ν13 = 0.3197, ν23 = 0.44, G13 = 4.88 GPa, G23 = 3.117 GPa, G12 = 6.18345 GPa, and X12 = 3.3984×1025 Pa3 . The X12 value is obtained by fitting the shear stress-strain measured from the tensile tests of [45/ − 45]4s coupon composed by IM7/977-3 graphite/epoxy lamina [143]. According to Hahn and Tsai [84], the X12 value is reported to be 13.8764 × 1025 Pa3 for Morganite II/4617 system. The geometry and mesh of the cross section of a [45/ − 45]s square-sectioned laminate beam is shown in Figure 5.1 and is analyzed by VABS. The section is meshed with 1296 eight-node quadrilateral elements, containing 4033 nodes.

45 deg

- 45 deg

0.005 m

- 45 deg

45 deg 0.02 m

Figure 5.1. Cross-sectional geometry, layout, and mesh.

79 The corresponding 3D model is analyzed by using the commercial FEA software Abaqus [139]. The material is applied by programming user-defined material subroutine (UMAT). The length of the beam is 0.2 m, ten times its cross-sectional width. All nodes on one of the beam end section are totally fixed. The nodes on the other end section are completely kinematically constrained to a reference point located at the geometric center of the section. The displacement boundary condition is applied at this reference point with Ux = 0.015 × 0.2 m. The reaction force is measured from the reference point. The 3D laminate body is analyzed using a global-local approach by partitioning the structure into three parts including one small part in the middle and two large parts on the sides, as shown in Figure 5.2. The three parts have the same mesh division in Y-Z plane as the cross-sectional mesh shown in Figure 5.1. In Z direction, the large parts are each meshed with 179 divisions, and the small part is meshed with 4 divisions. The large parts are each meshed by 231,987 20-node C3D20 elements containing 970,991 nodes. The small part is meshed by 5,184 20node brick elements (C3D20) containing 25,641 nodes, where the nodal coordinates and displacements are reported from the two cross sections located at X f = 0.10056 m and X b = 0.09944 m, respectively.

Z

X

Y

Figure 5.2. Partition of the 3D [45/ − 45]s square-section composite beam.

To validate the VABS predictions, it is necessary to make sure the axial strains applied on the 3D FEA model and VABS sectional model are the same. The axial

80 strain input into VABS is computed from the 3D FEA model. The averaged final axial displacements from these two sets of nodes at X f = 0.10056 m and X b = 0.09944 are Uxf = 0.0015087 m and Uxb = 0.0014914 m, respectively. Consequently, the final axial strain of the effective 1D beam can be computed as γ=

Uxf − Uxb = 0.01556 Xf − Xb

(5.4)

This strain is input into VABS to obtain the nonlinear beam sectional effective forcestrain constitutive curve. This curve is compared with the reaction force-strain data reported from the Abaqus 3D FEA in Figure 5.3. It is worth to note that the intrinsic feature of VABS enables it to obtain the good accuracy comparable to 3D FEA within the tremendously reduced computational time cost. The computation aspects of VABS and 3D FEA are compared in Table 5.1. 7E4

VABS 6E4

3D FEA

Axial force (N)

5E4

4E4

3E4

2E4

1E4

0 0.000

0.004

0.008

0.012

Axial strain (m/m)

Figure 5.3. Comparison of extensional constitutive data.

0.016

81

Table 5.1. Comparison of computation aspects. Computation Aspect

3D FEA

Nonlinear VABS

Time

19 hr 39 min 41 sec

3 min 43 sec

Increments

4

20

Multi-Processor

40

1

Multi-GPU

4

0

A good fitting equation to the VABS numerical prediction of the axial beam constitutive law shown in Figure 5.3 could be γ = c1 F1 + c2 F13

(5.5)

where c1 and c2 are constants fitted from Figure 5.3. Solving Eq. (5.5) for F1 results in

√ 2/3 s1 + s2 (s3 γ + s4 s5 + s6 γ 2 ) F1 = √ 1/3 (s3 γ + s4 s5 + s6 γ 2 )

(5.6)

where s1 , s2 , s3 , s4 , s5 , and s6 are constants in terms of c1 and c2 . From Eq. (5.6) the beam tangent stiffness can be computed as √ √ 2/3 (s4 s6 γ + s3 s5 + s6 γ 2 ) [−s1 + s2 (s3 γ + s4 s5 + s6 γ 2 ) ] ∂F1 T11 = = √ √ 1/3 ∂γ 3 s5 + s6 γ 2 (s3 γ + s4 s5 + s6 γ 2 )

(5.7)

This analytical expression agrees the numerical results of T11 predicted by VABS, as shown in Figure 5.4. To validate the T11 from VABS with the result of 3D FEA, we examine the extension mode natural frequency ω of the beam under the axial pre-strain. Under the circumstance of pinned-pinned boundary conditions, ω can be computed by π ω= l

√

T11 ρA

(5.8)

where l is the beam length. The natural frequencies under various axial pre-strains are normalized by the natural frequency without any pre-strain. These normalized

82 values are plotted in Figure 5.5 with respect to the corresponding pre-strains. It can be seen that the natural frequency is reduced by the introduction of the pre-strain and the nonlinearity.

Figure 5.4. Tangent beam stiffness predicted by VABS as a function of axial strain.

Figure 5.5. Reduction of the natural frequency due to pre-strain and shear nonlinearity.

83 To examine the bending behavior of the beam, in the 3D FEA, the nodes on the two end sections of the beam are both kinematically constrained to the reference points located at the geometric centers of the two sections, respectively. The first reference point at x = 0 is fixed in Ux and Uz and is applied with a rotation −θy . The second reference point at x = l (l is the overall beam length) is fixed only in Uz and is applied with a rotation θy . The reaction bending moment M2 is measured from the second reference point. The bending curvature of the 3D beam is approximately estimated by κ2 = 2θy /l. This bending curvature is input into VABS to obtain the nonlinear beam bending constitutive curve. This curve is compared with the reaction bending moment and curvature data reported from the 3D FEA, as shown in Figure 5.6.

Figure 5.6. Comparison of bending constitutive data.

84 To examine the twisting behavior of the beam, the layup is switched to [0/90]s while the geometry and the mesh are kept unchanged. In the 3D FEA, one reference point at x = 0 is overall fixed in displacements and rotations. The other reference point at x = l is applied with a rotation θx . The twisting moment M1 is measured from the reference point at x = l. The twisting curvature of the 3D beam is approximately estimated by κ1 = θx /l. This twisting curvature is input into VABS to obtain the nonlinear beam twisting constitutive curve. This curve is compared with the reaction twisting moment and curvature data reported from the 3D FEA, as shown in Figure 5.7.

Figure 5.7. Comparison of twisting constitutive data.

The discrepancies between VABS and 3D FEA in Figures 5.6 and 5.7 are mainly caused by the boundary effects due to the kinematical couplings on the end sections. The approximations of the bending and the twisting curvatures, κ1 and κ2 , in these

85 examples may also induce the differences. However, the agreement is good considering that the computational cost is dramatically reduced.

5.2

Three-Dimensional Stresses From 3D FEA of simple tension, coordinates and stress components (values are

measured in lamina coordinate) on the integration points located at the beam axial coordinate X = 0.09975 m are reported. The contour plots of the sectional stress fields are compared to the field results obtained from VABS for the final incremental step, as shown in Figures 5.8, 5.9, 5.10, 5.11, 5.12, and 5.13 for σ11 , σ12 , σ13 , σ22 , σ23 , and σ33 , respectively. Note that in 3D FEA model the reference coordinate originates from the left-bottom corner of the section. And in VABS the reference coordinate originates from the sectional geometric center. Good agreement can be found between VABS and 3D FEA. From Figures 5.10, 5.12, and 5.13, it can be seen that highly concentrated stress field occurs at the interfaces between two dissimilar layers along the free edges. The stress fields are localized within the boundary region and exhibit steep gradients with a rapidly decaying behavior towards the inner laminate region. This is the so-called free-edge effect [144, 145]. To further investigate this phenomenon, σ13 , σ23 , and σ33 data along the free-edges and the interface boundaries are plotted in Figures 5.14, 5.15, and 5.16, respectively. The agreement between VABS and 3D FEA data is excellent. For comparison, the coordinates of the integration point data from Abaqus 3D FEA are transformed into the beam sectional triad. Consider the bending deformation. The stresses predicted by VABS and 3D FEA are contour-plotted and compared in Figures 5.17, 5.18, 5.19, 5.20, 5.21, and 5.22. For the twisting example, nontrivial stresses under twisting are plotted in Figures 5.17, 5.19, and 5.21. The kinematical couplings on the beam end sections and the approximations of κ1 and κ2 in 3D FEA cause the differences. However, good agreement between VABS and 3D FEA can be seen from the plots.

86

(a) 3D FEA.

(b) VABS.

Figure 5.8. Comparison of stress σ11 contour plots at γ = 0.01556.

87

(a) 3D FEA.

(b) VABS.


88

(a) 3D FEA.

(b) VABS.


89

(a) 3D FEA.

(b) VABS.


90

(a) 3D FEA.

(b) VABS.


91

(a) 3D FEA.

(b) VABS.


92

8.0E7 VABS 6.0E7

3D FEA

4.0E7

13

(Pa)

2.0E7

0.0

-2.0E7

-4.0E7

-6.0E7

-8.0E7 -0.010

-0.005

0.000

x

3

0.005

0.010

= Z - 0.01 (m)

(a) x2 = 0.00994, Y = 0.01994. 8.0E7 VABS 6.0E7

3D FEA

4.0E7

13

(Pa)

2.0E7

0.0

-2.0E7

-4.0E7

-6.0E7

-8.0E7 -0.010

-0.005

0.000

x

2

0.005

0.010

= Y - 0.01 (m)

(b) x3 = 0.00506, Z = 0.01506.

Figure 5.14. Comparison of stress σ13 curve plots at γ = 0.01556.

93

1.0E8 VABS

8.0E7

3D FEA 6.0E7

4.0E7

23

(Pa)

2.0E7

0.0

-2.0E7

-4.0E7

-6.0E7

-8.0E7

-1.0E8 -0.010

-0.005

0.000

x

3

0.005

0.010

= Z - 0.01 (m)

(a) x2 = 0.00994, Y = 0.01994. 1.0E8 VABS

8.0E7

3D FEA 6.0E7

4.0E7

23

(Pa)

2.0E7

0.0

-2.0E7

-4.0E7

-6.0E7

-8.0E7

-1.0E8 -0.010

-0.005

0.000

x

2

0.005

0.010

= Y - 0.01 (m)

(b) x3 = 0.00506, Z = 0.01506.


94

3.0E7 VABS 3D FEA

2.0E7

1.0E7

33

(Pa)

0.0

-1.0E7

-2.0E7

-3.0E7

-4.0E7 -0.010

-0.005

0.000

x

3

0.005

0.010

= Z - 0.01 (m)

(a) x2 = 0.00994, Y = 0.01994. 3.0E7 VABS 3D FEA

2.0E7

0.0

33

(Pa)

1.0E7

-1.0E7

-2.0E7

-3.0E7

-4.0E7 -0.010

-0.005

0.000

x

2

0.005

0.010

= Y - 0.01 (m)

(b) x3 = 0.00506, Z = 0.01506.


95

(a) 3D FEA.

(b) VABS.

Figure 5.17. Comparison of stress σ11 contour plots at κ2 = 2.0 rad/m.

96

(a) 3D FEA.

(b) VABS.


97

(a) 3D FEA.

(b) VABS.


98

(a) 3D FEA.

(b) VABS.


99

(a) 3D FEA.

(b) VABS.


100

(a) 3D FEA.

(b) VABS.


101

(a) 3D FEA.

(b) VABS.


102

(a) 3D FEA.

(b) VABS.


103

(a) 3D FEA.

(b) VABS.


104 5.3

Coupon Performance Mechanical properties of a material can be measured from loaded coupons which

are in fact structures. Then structural models are used to decipher the material properties from the coupon measurements. For example, to calibrate the in-plane shear response of a polymer matrix composite, slender coupons made by ±45○ laminae are loaded with force F1 in the beam axial direction. The in-plane shear stress is calculated by σ12 =

F1 2A

where A is the beam cross-sectional area. In-plane responses

under such a loading are measured from the beam coupon, and the in-plane shear strain is obtained by Γ12 =

Γx −Γy 2 .

This methodology is created based on the Classical

Laminated Plate Theory (CLPT). Based on the plane stress assumption, the material constitutive law is reduced to ⎤⎡ ⎡ ⎤ ⎡ ⎤ ν21 ⎥ ⎢ σ11 ⎥ ⎢ Γ11 ⎥ ⎢ E1 0 ⎥⎢ ⎢ ⎥ ⎢ 1 − E2 ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎢ Γ ⎥ = ⎢ − ν12 ⎥ (5.9) 0 σ ⎥ ⎢ 22 ⎥ ⎢ 22 ⎥ ⎢ E1 E2 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ σ12 ⎥ ⎢ 1 ⎥⎢ ⎢ 2Γ12 ⎥ ⎢ 0 ⎥ 0 G12 + X12 ⎦ ⎣ σ12 ⎦ ⎣ ⎦ ⎣ The ±45○ beam coupon is recommended by literatures [146, 147] due to its simplicity, reproducibility, and economy. Several limitations are applied on the coupon design in order to keep the acceptable accuracy. Petit [148] pointed out that small positive or tensile strains exist in addition to the relatively large shear strains in the principal direction of the laminae. Consequently, tensile stress should exist in the longitudinal and transverse directions of the laminae, which will be proved with examples in the present work. Rosen [149] considered the free edge effect in the choice of the specimen width. Hahn [150] highlighted that the layers should be arranged as homogeneous as possible to obtain the acceptable results. He also claimed that σ12 =

F1 2A

was valid

only if the shear and extension was not coupled. The present beam model is used to investigate such a material calibration method. The first goal is to exam the assumptions and the limitations. And the second goal is to find out if the assumptions made in the calibration are valid when the calibrated material properties are utilized to predict the structural behaviors. The crosssectional schematics of the virtual beam coupons are listed in the following

105 Thick Square: The geometry of Figure 5.1 with layup [±4537 ]s is considered.

There are 148 plies and the single ply thickness is 1.351315 × 10−4 m. This example represents an inhomogeneous layup. Thick Circular: A tubular cross section of Figure 5.26 with layup [±4537 ]s is

considered. There are 148 plies and the single ply thickness is 1.351315×10−4 m. This example represents an inhomogeneous layup and also the shear-extension coupling. Thin Square: A very thin layup [±45]s is considered. The single ply thickness

is 1.33 × 10−4 m. The width of the cross section is 0.02 m. Thin Circular: A very thin tube with layup [±45]s is considered. The single ply

thickness is 1.33 × 10−4 m. The outmost radius is 0.075 m. ASTM D3518: A real coupon cross-sectional geometry with layup [(+45/−45)4 ]s

is considered. There are 16 plies and the single ply thickness is 1.32775 × 10−4 m. The width of the cross section is 0.02465 m. CLPT: The above five examples are recreated in terms of using shell elements

(S4R) in Abaqus and the UMAT subroutine is used to apply the in-plane nonlinear shear constitutive law in Eq. (5.9). Extensional displacement boundary conditions are applied on these virtual coupons. The predicted constitutive laws are plotted in Figure 5.27. All of the five models based on CLPT predict the same result. In light of this fact, instead of plotting multiple overlapped curves, only one curve representing the five cases is plotted in Figure 5.27 and marked by the legend CLPT. VABS - Thin Square, VABS - Thin Circular, and VABS - ASTM D3518 agree with CLPT, while VABS - Thick Square and VABS Thick Circular coupons deviate from CLPT. It is worth to notice that in the VABS - Thick Circular case, there is a coupling between the extension and the twisting. In light of this, a twisting moment is also predicted under the axially extensional displacement boundary conditions. This coupling behavior reduces to a negligible value

106 along with the increasing of the radius-thickness ratio in the VABS - Thin Circular case. The 3D stress fields of the ASTM D3518 virtual coupon predicted by VABS are plotted in Figure 5.28. Firstly, the plane stress assumption is valid only if the layer is arranged as homogeneous as possible, and the layer width is large enough compared to the layer thickness. In this situation, the free-edge stress will be confined in a relatively small cross-sectional area so that its effects on the global force-displacement law can be reasonably neglected. As shown in Figure 5.29, the in-plane stress components (from the location centered in the cross section) predicted from VABS - ASTM D3518 are compared to the in-plane stress values predicted by CLPT. The agreement is good. The values of VABS in Figure 5.29 are obtained by averaging the values at the integration points located at x2 = ±4.63E-6. Under the in-plane shear nonlinearity, σ11 and σ22 are also in nonlinear relations with respect to the beam axial strain γ. Secondly, the plane stress assumption made during the calibration is not adaptable in the structural simulation when the layer is not arranged as homogeneous as possible, or the layer width is not large enough compared to the layer thickness, for example, the case of VABS - Thick Square. Although the plane stress assumption is successful in some specific scenarios, the out-of-plane stresses will become critical when the damage and the delamination are considered.

x3 x2

45 deg, 0.005 m -45 deg, 0.005 m -45 deg, 0.005 m 45 deg, 0.005 m

Figure 5.26. Cross-sectional geometry and layout of the tube section.

107

Figure 5.27. Comparison of performances of virtual beam coupons.

108

(a) σ11 .

(b) σ22 .

(c) σ12 .

(d) σ23 .

(e) σ13 .

(f) σ33 .

Figure 5.28. Local stress fields on the cross-sectional domain of ASTM D3518 virtual coupon at γ = 0.01556.

109

(a) σ11 .

(b) σ22 .

(c) σ12 .

Figure 5.29. Comparison of the in-plane stresses of ASTM D3518 virtual coupon predicted by VABS and CLPT.

110

111

6. COMPOSITE BEAM DAMAGE MODEL In this chapter, the present theory is implemented to capture the high-fidelity elasticto-damaged beam constitutive behaviors by the cross-sectional analyses. The mesh objectivity feature of VABS theory on the prediction using damage softening constitutive law is investigated analytically and validated numerically. Various continuum damage models in literature are taken into the present beam model from which the predicted beam constitutive law and local fields are compared to the data results from 3D solid element analysis, 2D plate element analysis, and experiments. Good agreements are observed from the validations.

6.1

Isotropic Damage Following the example of 1D isotropic damage model proposed by Einav et al [151]

the damage evolution law can be expressed as d=1−

¯ Γ Π ∣γ∣

¯ ∣γ∣ ≥ Γ

(6.1)

¯ is the damage initiation strain, and Π where d is the isotropic damage variable, Γ ¯ = 0.0025. is a monotonic function of d with Π = 1 at d = 0. For simplicity, we set Γ Three cases of Π = 1, Π = Exp(d), and Π = Exp(−d) are substituted into Eq. (6.1) accounting for damage perfect, hardening, and softening constitutive laws, respectively. These three damage evolution laws are programmed into the user material subroutine (UMAT) in Abaqus [139] 3D FEA. The kinematic constraint and boundary conditions applied in the 3D FEA are shown in Figure 6.1. In 3D FEA, the length of the beam is 0.25 m, five times its crosssectional width. The longitudinal edges are meshed by 60 divisions, and the transverse edges are meshed by 12 divisions. The sectional geometry and mesh division of

112 the sectional sides are used correspondingly in the VABS analysis. To ensure pure tension, all nodes on one of the beam end section located at X = Xmin are subjected to boundary condition UX = 0. The node located at the undeformed coordinate (Xmin , Ymin , Zmin ) is also subjected to boundary condition UY = UZ = 0. Another node located at the undeformed coordinate (Xmin , Ymax , Zmin ) is also subjected to boundary condition UZ = 0. The nodes on the other end section located at X = Xmax are kinematically constrained in terms of UX to a reference point located at the geometric center of the section. The displacement boundary condition is applied at this reference point with UX = 0.01 × 0.25 m. This boundary condition ensures a uniform distributed warping along the beam reference line and eliminate rigid body motions. Consequently, γ = Γ11 = 0.01. The reaction force is measured also from the reference point. To validate the VABS prediction, it is necessary to make sure the axial strains applied on the 3D FEA model and VABS sectional model are the same. The axial strain input into VABS is from the 3D FEA.

UY(Xmin, Ymin, Zmin) = UZ(Xmin, Ymin, Zmin) = 0 UX(Xmax) = 0.0025 m

Z

Y

X

UX(Xmin) = 0

UZ(Xmin, Ymax, Zmin) = 0

Figure 6.1. Geometry and boundary conditions in 3D FEA of isotropic elastic damageable beam in tension.

In VABS, the sectional is meshed with 144 eight-node quadrilateral elements, containing 481 nodes. This beam tensile constitutive curves are compared in Figure 6.2.

113

Figure 6.2. Comparison of beam tensile constitutive curves with isotropic damage.

The excellent agreement found in Figure 6.2 can be explained by analytically solving the warping functions. Assume isotropic damage of an isotropic material Ed = (1 − d)E

(6.2)

where Ed and E are damaged and undamaged Young’s moduli, respectively. The damaged elasticity tensor can be expressed as Cdijkl =

(1 − d)E (1 − d)Eν δij δkl + (δik δjl + δil δjk ) (1 + ν)(1 − 2ν) 2(1 + ν)

(6.3)

In light of the constitutive law in Eq. (3.43) and the simple tension condition of κi = 0, the virtual work principle results in δU = ⟨σij δΓij ⟩ = ⟨Γkl Cdijkl δΓij ⟩ = 0

(6.4)

114 Since tensor Cdijkl possesses major symmetry, and because the damage evolution law in Eq. (6.1) is not a functional of the unknown warping function, we can pull the variation and the damage variable in Eq. 6.4 out of the cross-sectional integral as 2 2 2 0 = (1 − d)δ ⟨Eγ 2 + G [w1,2 + w1,3 + (w3,2 + w2,3 ) ]⟩

⎧ ⎪ ⎪ Eν ⎪ νγ + w2,2 + (1 − d)δ ⟨ ⎨ (1 + ν)(1 − 2ν) ⎪ ⎪ ⎪ ⎩ νγ + w3,3

T⎡ ⎫ ⎪ ⎪ ν ⎪ ⎢⎢ 1 − ν ⎬ ⎢ ⎢ ⎪ ⎪ 1−ν ⎪ ⎭ ⎢⎣ ν

⎤⎧ ⎥⎪ ⎪ νγ + w2,2 ⎥⎪ ⎥⎨ ⎥⎪ ⎪ νγ + w3,3 ⎥⎪ ⎦⎩

⎫ ⎪ ⎪ ⎪ ⎬⟩ ⎪ ⎪ ⎪ ⎭

(6.5)

According to Yu and Hodges [17], the following solutions will satisfy the variational statement in Eq. (6.5) w1 = 0

w2 = −νx2 γ

w3 = −νx3 γ

(6.6)

Finally, with A denoting the cross-sectional area, F1 can be expressed in terms of d and γ as F1 =

1 ∂⟨W ⟩ = (1 − d)EAγ 2 ∂γ

(6.7)

which agrees with the curves plotted in Figure 6.2.

6.2

Mesh Objectivity When the material exhibits strain-softening behavior, the stress-strain constitu-

tive formulation results in strong mesh dependency in the conventional FEA. To investigate the mesh objectivity of the present VABS methodology, four different mesh schemes, as shown in Figure 6.3, are taken into consideration. The sections are meshed by eight-node quadratic elements. Red elements are showing the constituent with the damage initiation strains 0.5% lower that the damage initiation strains of the gray elements. Similar numerical experiments can be found in [152] for 3D FEA and in [111] for 2D plate FEA. In these conventional FEA, it is found that when the positive definiteness of the material tangent stiffness is lost, the failure will be localized within a single band of elements. As a result, the energy dissipated is proportional to the volume of the failed element rather than the area of the fractured surface

115 and, consequently, it tends to zero with the mesh refinement. This phenomenon is physically inadmissible.

(a) 3 by 3 elements.

(b) 6 by 6 elements.

(c) 15 by 15 elements.

(d) 30 by 30 elements.

Figure 6.3. Four different VABS mesh schemes in the mesh objectivity study.

Due to the strain softening damage evolution, damage growth is driven by 3D strains which are computed from generalized beam strain and warping solutions. Because the damage results in an updated elasticity and heterogeneity of the section, the warping solutions are affected by the damage state. So it is of interest to investigate the impact of the warping on the mesh dependency of damage solution.

116 Firstly, assume that the damage evolves by Γ11 , that is, d=1−

¯ 11 Γ Exp(−d) ∣Γ11 ∣

¯ 11 ∣Γ11 ∣ ≥ Γ

(6.8)

¯ 11 is the damage initiation strain associated with Γ11 . When the absolute where Γ ¯ 11 , the damage will be initiated and computed by Eq. (6.8). In value of Γ11 reaches Γ case of simple tension, κi = 0, Eq. (6.8) becomes d=1−

¯ 11 Γ Exp(−d) ∣γ∣

¯ 11 ∣γ∣ ≥ Γ

(6.9)

In case of pure bending, γ = κ1 = κ3 = 0, Eq. (6.8) becomes d=1−

¯ 11 Γ Exp(−d) ∣(x3 + w3 )κ2 ∣

¯ 11 ∣(x3 + w3 )κ2 ∣ ≥ Γ

(6.10)

Secondly, assume that the damage evolves by transverse strain Γ22 or Γ33 , that is, ¯T ¯T Γ Γ Exp(−d) = 1 − Exp(−d) ∣Γ22 ∣ ∣w2,2 ∣ ¯T ¯T Γ Γ Exp(−d) = 1 − Exp(−d) d=1− ∣Γ33 ∣ ∣w3,3 ∣ d=1−

¯T ∣w2,2 ∣ ≥ Γ

(6.11)

¯T ∣w3,3 ∣ ≥ Γ

(6.12)

¯ T is the damage initiation strain associated with the for Γ22 and Γ33 , respectively. Γ ¯ T , the transverse strains. When the absolute value of the transverse strain reaches Γ damage will be initiated and computed by Eq. (6.11) or Eq. (6.12). Thirdly, because twisting results in Γ12 and Γ13 , assume that the damage evolution is driven in the form of d=1−

¯L Γ Exp(−d) Γeq

¯L Γeq ≥ Γ

(6.13)

where the equivalent shear strain Γeq is defined by √ √ Γeq = 2 Γ212 + Γ213 = (w1,2 − w3 κ1 − x3 κ1 )2 + (w1,3 + w2 κ1 + x2 κ1 )2

(6.14)

¯ L is the damage initiation strain associated with Γeq . When the value of Γeq reaches Γ ¯ T , the damage will be initiated and computed by Eq. (6.13). The elastic constants Γ and the maximum elastic strains for the constituent in color gray are listed in Table 6.1.

117

Table 6.1. Material constants in mesh objectivity study. E (MPa)

ν

¯ 11 Γ

¯T Γ

¯L Γ

30000

0.2

0.0025

0.0005

0.004

In the tensile analysis, we have κi = 0. We can further assume that the axial strain γ does not result in out-of-cross-sectional warping w1 on the cross sections which are shown in Figure 6.3. Without κi and w1 we can formulate the virtual work principle as ⟨σ11 δΓ11 + σ22 δΓ22 + σ33 δΓ33 ⟩ = 0

(6.15)

which results in the corresponding governing equations as (1 − d) (2(1 − ν)w2,22 + (1 − 2ν)w2,33 + w3,23 ) − 2d,2 (ν(γ + w3,3 ) + (1 − ν)w2,2 ) − d,3 (1 − 2ν) (w2,3 + w3,2 ) = 0 (1 − d) (2(1 − ν)w3,33 + (1 − 2ν)w3,22 + w2,23 ) − d,2 (1 − 2ν) (w2,3 + w3,2 ) − 2d,3 (ν(γ + w2,2 ) + (1 − ν)w3,3 ) = 0

(6.16)

(6.17)

and the corresponding boundary conditions are the same those in the elasticity solution provided by Yu and Hodges [17]. The elastic solution, wα = −νxα γ, satisfy Eqs. (6.16) and (6.17). So the damage can be individually determined by damage evolutions law. VABS seeks this solution by discretizing the warping field with finite elements. The prediction of VABS is shown in Figure 6.4, the softening part of the beam tensile constitutive curve does not depend on the definition of the strain which is driving the damage, neither the mesh of the cross section. In this case, the damage does not localize in the red area in Figure 6.3. From our analytical solution, it also can be seen that the damage evolution laws Eq. (6.11) and (6.12) have the same effects on the beam tensile constitutive law. In the bending analysis, we have γ = κ1 = κ3 = 0. Further, assume that the bending curvature κ2 does not result in out-of-cross-sectional warping w1 on the cross sections

118 which are shown in Figure 6.3. According to Eq. (6.15), we can formulate the virtual work principle as (1 − d) (2(1 − ν)w2,22 + (1 − 2ν)w2,33 + w3,23 ) − 2d,2 (ν(x3 κ2 + w3,3 ) + (1 − ν)w2,2 ) − d,3 (1 − 2ν) (w2,3 + w3,2 ) = 0 (1 − d) (2(1 − ν)w3,33 + (1 − 2ν)w3,22 + w2,23 ) − d,2 (1 − 2ν) (w2,3 + w3,2 ) − 2d,3 (ν(x3 κ2 + w2,2 ) + (1 − ν)w3,3 ) = 0

(6.18)

(6.19)

and the corresponding boundary conditions are the same those in the elasticity solution provided by Yu and Hodges [17]. Under pure bending κ2 , the exact elastic solution of warping functions are w1 = 0 ⟨x2 x3 ⟩ − x2 x3 ) κ2 A ⟨x22 ⟩ − ⟨x23 ⟩ ν 2 2 ) κ2 w3 = (x2 − x3 + 2 A w2 = ν (

(6.20)

which also solves the above governing equations and satisfies the corresponding boundary conditions. The strains are computed from the warping functions as Γ2,2 = Γ3,3 = w2,2 = w3,3 = −νx3 κ2 . These strains can be used for computing the cross-sectional damage field. The solution by VABS agrees with the analytical investigation. As shown in Figures 6.5 and 6.6, the softening part of the beam bending constitutive curve does not depend on the definition of strain driving the damage. This explains why the damage initiated at the same curvature value. In light of this, on the cross-sectional domain, the first damage initiated is homogeneous in x2 direction, but heterogeneous in x3 direction. This damage field also agrees with the analytical solutions of the strain and damage. Consequently, the cross-sectional damage field does not localize in the red area in Figure 6.3. However, because the warping solution depends on the mesh and also affects the 3D strain solution, the softening beam constitutive relation converges at 225-element mesh. The cross-sectional damage contour is shown by Figure 6.7 which indicates the damage initiates at the upper and lower boundary of the cross section and then spreads into the inner sectional area during damage evolution.

119

Figure 6.4. Beam tensile constitutive law under isotropic damage driven by Eqs. (6.8), (6.11), and (6.12).

Figure 6.5. Beam bending constitutive laws with isotropic damage driven by Eq. (6.8).

120

Figure 6.6. Beam bending constitutive laws with isotropic damage driven by Eqs. (6.11) and (6.12).

Figure 6.7. Damage contour at κ2 = 0.8 rad/m with damage driven by Eq. (6.8).

121 In the twisting analysis, we have γ = κ2 = κ3 = 0. It has been shown in [17] that in the infinitesimal strain theory that w3 and w2 has very little impact on twisting. Neglecting these we have the equivalent strain as Γeq =

√

(w1,2 − x3 κ1 )2 + (w1,3 + x2 κ1 )2

(6.21)

Then the virtual work principle ⟨σij δΓij ⟩ = 0 gives ⟨(1 − d)(Γ12 δ(w1,2 − x3 κ1 ) + Γ13 δ(w1,3 + x2 κ1 ))⟩ = 0

(6.22)

which results in the governing equations (1 − d)(w1,22 + w1,33 ) − (Γ12 d,2 + Γ13 d,3 ) = 0

(6.23)

and the corresponding boundary conditions are the same those in the elasticity solution provided by Yu and Hodges [17]. Eq. (6.23) indicates two points. First, if the sectional damage field is uniform through the section, the governing equation reduces to the elasticity solution, which has been observed in the tensile example. Second, if the elastic warping of damaged section is not affected by damage field, that is, w1,22 + w1,33 = 0, we should have Γ12 d,2 + Γ13 d,3 = 0

(6.24)

which is corresponding to the situation of a circular section with isotropic damage material. For an isotropic elastic circular section, we have w1 = 0 2Γ12 = −x3 κ1

2Γ13 = x2 κ1

(6.25)

Damage will results in heterogeneity in the radius dimension. However, this radial heterogeneity do not affect the solution of warping, w1 = w1,22 + w1,33 = 0. In light of Eqs. (6.13) and (6.21), Γ12 d,2 + Γ13 d,3 =

⎞ x2 x3 κ31 x2 x3 κ31 ∂d ⎛ √ − √ =0 ∂Γeq ⎝ 4 (x22 + x33 )κ2 4 (x22 + x33 )κ2 ⎠

(6.26)

To validate this point, a tube section with the outer and inner radius as 0.3 m and 0.22 m, respectively. The mesh of this section is shown in Figure 6.8. This discretized

122 field is input into VABS from which the predicted damage field at κ1 = 0.019 rad/m is shown in Figure 6.9. Note that the damage on the outer and inner surfaces of the cross section are in 50% difference, however, no unrealistic strain localization is found.

Figure 6.8. Mesh of the tube section.

Figure 6.9. Damage field of the tube section at twist curvature κ1 = 0.019 rad/m.

123 When the twisting of a rectangular-sectioned beam is under analysis, it is not necessary to introduce artificial weaker cross-sectional area in Figure 6.3. As shown in Figure 6.10, in the elasticity region, the maximum equivalent shear strain Γeq are located at some small cross-sectional areas around four points located at (0.0, −0.015), (0.0, 0.015), (−0.015, 0.0), and (0.015, 0.0), respectively. The damage is initiated in these small areas. Consequently, in this pure twisting case, the more elements are used in the cross-sectional mesh, the more accurate are the predictions of the local strain and damage solutions. This solution agrees with the analytical solution provided by Yu and Hodges [17].

Figure 6.10. Cross-sectional contour of elastic equivalent shear strain.

Regarding that refining the mesh will result in a better representation of the crosssectional field to solve the governing equation, we consider mesh the cross section with 5 × 5 (25 elements), 15 × 15 (225 elements), 31 × 31 (961 elements), 61 × 61 (3724 elements), and 91 × 91 (8281 elements) quadratic 8-node elements. Figure 6.11 shows

124 the softening part of the beam twisting constitutive curves predicted by different mesh refinement. The values in the damaged configuration are normalized to the maximum elastic values. Different from the conventional analysis using 3D FEA, the area under the softening constitutive curve does not converge to zero along with the refinement of mesh.

Figure 6.11. Twisting constitutive law with damage driven by Eq. (6.13).

The warping field of the damaged configuration predicted by 91 × 91 eight-noded quadratic elements is shown in Figure 6.12. The shear strain 2Γ12 and 2Γ13 are plotted in Figures 6.14, and 6.15 respectively. The cross-sectional damage resulting in this warping field is plotted in Figure 6.13. Along with the mesh refinement, the damaged area on the cross section approaches to being similar to a single crack-like line, which is a quasi-fracture behavior. The twisting moment and curvature associated with this damage configuration are 199.668 Nm and 0.1978 rad/m, respectively.

125

Figure 6.12. Warping of the damaged cross section.

Figure 6.13. Damage of the cross-section.

126

Figure 6.14. Shear strain 2Γ12 on damaged cross section.

Figure 6.15. Shear strain 2Γ13 on damaged cross section.

127 According to Griffith theory of fracture [153], the energy released from the elastic body due to the generation of new cracks is G=−

∂U 1 lc (M1d κd1 − M1e κe1 ) 1 M1e κe1 − M1d κd1 =− = ∂Ac 2 ad lc 2 ad

(6.27)

where U is the elastic energy in the body; M1e and κe are the maximum elastic twisting moment and curvature, respectively; M1d and κd are the first damaged twisting moment and curvature, respectively; lc is the length of the beam coupon under twisting, which is known as the characteristic length of beam element; ad is the length of the crack observed from the cross-sectional damage field. The convergence of G is plotted in Figure 6.16. Instead converging to zero, the energy release rate is stabilized at a constant value when the mesh is refined. 900

700

Energy release rate

G

(N/m)

800

600

500

400

300

200 0

1660

3320

4980

6640

8300

Number of elements

Figure 6.16. Convergence of the energy release rate with respect to the mesh refinement.

We also consider an irregular refinement of mesh as shown in Figure 6.17 where 6node triangular elements are used to discretize the field. The cross-sectional damage predicted by this mesh plotted in Figure 6.18 agrees with the result predicted by

128 91 × 91 8-node quadratic elements. The twisting moment and curvature associated with this damage configuration are 202.822 Nm and 0.1978 rad/m, respectively.

Figure 6.17. Irregular mesh refinement.

d

0.015

0.995 0.871

0.010

0.746 0.005

0.000

0.498

x

3

(m)

0.622

0.373 -0.005

0.249 -0.010

0.124 0.000

-0.015 -0.015 -0.010 -0.005 0.000

x

2

0.005

0.010

0.015

(m)

Figure 6.18. Damage of the cross-section predicted by the mesh in Figure 6.17.

129 In this section, it has been shown that the damage prediction by VABS is not sensitive to mesh. To further investigate that if VABS is sensitive to the damage model applied, a regularization parameter r can be used to modify the softening degree of the damage model, that is, d=1−

¯L Γ Exp(−rd) Γeq

When r = 0, Eq. (6.28) reduces to the case of d = 1 −

(6.28) ¯L Γ Γeq .

And when r is bigger, it

will result in more severe damage softening. As shown in Figure 6.19, the twisting moment at κ1 = 0.1978 rad/m predicted by 91 × 91 eight-noded quadratic elements is slightly reduced due when the regulation parameter is small. However, as shown in Figure 6.20, the twisting moment is dramatically dropped when the regulation parameter is larger than 0.7. In sum, VABS prediction is not sensitive to the mesh but to the damage model utilized in the prediction.

Figure 6.19. Twisting moments on damage configuration versus small softening regulation parameters.

130

Figure 6.20. Twisting moments on damage configuration versus large softening regulation parameters.

6.3

Comparison with 3D FEA In this section, the damage behavior for the fiber-reinforced epoxy lamina is de-

scribed using the model proposed by Linde et al [154]. For the sake of simplicity and space, we only consider the matrix damage which is the first damage mode in FRP [96]. The damaged stiffness matrix is computed by ⎡ ⎢ C11 0 0 (1 − dm )C14 0 C16 ⎢ ⎢ ⎢ 0 (1 − dm )C22 0 0 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 C33 0 0 0 ⎢ [Cd ] = ⎢ ⎢ 0 0 (1 − dm )C44 0 (1 − dm )C46 ⎢ (1 − dm )C41 ⎢ ⎢ ⎢ 0 0 0 0 C55 0 ⎢ ⎢ ⎢ ⎢ C61 0 0 (1 − dm )C64 0 C66 ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6.29) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

131 which assumes the matrix damage only affects the in-plane transverse stiffness and the other terms coupled with the in-plane transverse stiffness. Without distinguishing between tension and compression, dm is defined as the internal variable denoting the damage in the matrix. The damage initiation criteria Fm is expressed in terms of 3D strains: ¿ 2 Á Á Γ222 1 1 2Γ12 À + ( − ) Γ + ( Fm = Á ) 22 F,c 2ΓF12 ΓF,t ΓF,t ΓF,c 22 Γ22 22 22

(6.30)

F,C F where ΓF,T 22 , Γ22 , and Γ12 denote the maximum elastic strains in extension, compres-

sion, and in-plane shear modes of matrix failure, respectively. Once Eq. (6.30) is satisfied, the evolution of transverse damage dm is governed by dm = 1 −

Lc 1 2 Exp ( c C44 (1 − Fm )(ΓF,T 22 ) ) Fm Gm

(6.31)

where Lc is the characteristic length which is based on the element geometry and formulation: for a first-order solid element, it is a typical length of a line across an element; for a second-order solid element, it is half of the same typical length; for a beam element, it is defined as the length along the element axis; for a shell element, it is a characteristic length in the reference surface. To validate the present VABS model, the characteristic length Lc is approximately estimated from the geometry of the solid elements in the 3D FEA model, that is, Lc = Ve

1/3

(6.32)

where Ve is the volume of the 3D solid element. Gcm in Eq. (6.31) denotes the fracture energy controlling the evolution of dm . The usage of the fracture energy and the characteristic length help minimize the strain localization and the mesh sensitivity of the numerical results from conventional FEA. Although it has been shown in the previous section that unrealistic strain localization and the mesh sensitivity are not observed from VABS, it is possible that these numerical issues will exist in the 1D beam analysis.

132

Table 6.2. Elastic constants considered in the validation with 3D FEA. E1 (MPa)

E2 = E3 (MPa)

G12 = G13 (MPa)

G23 (MPa)

ν12 = ν13

ν23

55000

9500

5500

3000

0.33

0.45

Table 6.3. Constants of CDM model in the validation with 3D FEA. ΓF,t 22

ΓF,c 22

ΓF12

0.00399

0.01196

0.00455

Gcm (N/mm) Lc (mm) 1.0

0.5138

The constants of orthotropic elasticity and CDM model in validation with 3D FEA are listed in Tables 6.2 and 6.3, respectively. The example considered here is a 64-layer laminates, each layer is 25 mm in width and 0.125 mm in thickness. In VABS, the sectional domain is meshed with 1536 8-node quadrilateral elements, containing 4785 nodes. The corresponding 3D model is analyzed in Abaqus. The material is applied to Abaqus using UMAT. The length of the beam is 250 mm, ten times of its cross-sectional width. In light of the SaintVenant principle, the laminate is partitioned into three parts including one small part in the middle and two large parts on the sides, as illustrated in Figure 6.21. The large parts are each meshed by 182,784 20-node brick elements (C3D20) containing 767,575 nodes. The small part is meshed by 3072 20-node C3D20 elements containing 17,605 nodes. In the case of examining the beam tensile behavior, the composite layup is [0/903 ]8s . In the 3D FEA, the nodes on the two end sections of the beam are both kinematically constrained in Ux to the reference points located at the geometric centers of the two sections, respectively. The first reference point at x = 0 is fixed in Ux , Uy , and Uz . The second reference point at x = l (l is the overall beam length) is fixed in Uy , and Uz and is applied with a displacement Ux = 0.015 × l. The extensional force F1 is mea-

133 sured from the second reference point. The axial strain of the 3D beam is estimated by γ = Ux /l. This γ value is input into the present VABS algorithm to obtain the beam tensile constitutive curve. This curve is compared with the force and strain data reported from the 3D FEA, as shown in Figure 6.22. It can be seen that the agreements between VABS and 3D FEA are excellent. The contour plot of the matrix damage variable dm in the small part of the 3D FEA model at the final incremental step is shown in Figure 6.23. From this part, the Y and Z coordinates and dm on the integration points located at the beam axial coordinate X = 124.075733 mm are reported and plotted in Figure 6.24 (a). The contour plot of this sectional dm field is compared to the result obtained from VABS which is shown in Figure 6.24 (b). Note that in 3D FEA model the coordinate originates from the left-bottom corner of the section, while in VABS the coordinate originates from the sectional geometric center. In Figure 6.24, the maximum difference between VABS and 3D FEA is 3.6 % of the maximum value predicted by 3D FEA. This discrepancy is possibly caused by the different values of Lc used in VABS and 3D FEA.

Figure 6.21. Partition and mesh of the 3D composite beam.

134

Figure 6.22. Comparison of beam tensile constitutive law of [0/903 ]8s coupon predicted by VABS and 3D FEA.

Z X

Y

Figure 6.23. Contour of damage dm on the small part of [0/903 ]8s coupon at γ = 0.015 in 3D FEA.

135

(a) 3D FEA X = 124.075733 mm.

(b) VABS.

Figure 6.24. Comparison of matrix damage dm contour plots of [0/903 ]8s coupon at γ = 0.015.

136 The bending behavior of the [0/45/90/ − 45]8s laminate is also examined. In the 3D FEA, the nodes on the two end sections of the beam are both completely kinematically constrained to the reference points located at the geometric centers of the two sections, respectively. The first reference point at x = 0 is fixed in Ux and Uz and is applied with a rotation −θy . The second reference point at x = l is fixed only in Uz and is applied with a rotation θy . The reaction bending moment M2 is measured from the second reference point. The contour plot of Uz obtained from 3D FEA results from such boundary condition is shown in Figure 6.25. The bending curvature of the 3D beam is approximately estimated by κ2 = 2θy /l which is input into VABS. The reaction bending moment and curvature data predicted by VABS and 3D FEA are compared in Figure 6.26. The contour plot of the matrix damage dm in the small part of the 3D FEA model at the final incremental step is shown in Figure 6.27. From this small part, Y and Z coordinates and dm on the integration points located at the beam axial coordinate X = 124.075733 mm are reported and plotted in Figure 6.28 (a). The contour plot of this sectional dm is compared to the result obtained from VABS which is shown in Figure 6.28 (b). The maximum difference between VABS and 3D FEA is 4.4 % of the maximum value predicted by 3D FEA. This discrepancy is possibly caused by the different values of Lc used in VABS and 3D FEA and the boundary effects in the 3D FEA. The computational aspects of this bending example are summarized in Table 5.1. U, U3 +2.831e−02 +2.595e−02 +2.359e−02 +2.123e−02 +1.887e−02 +1.652e−02 +1.416e−02 +1.180e−02 +9.437e−03 +7.078e−03 +4.719e−03 +2.359e−03 +0.000e+00

Z X

Y

Figure 6.25. Contour plot of Uz of [0/45/90/ − 45]8s coupon under bending predicted by 3D FEA.

137

Figure 6.26. Comparison of beam bending constitutive law of [0/45/90/ − 45]8s coupon predicted by VABS and 3D FEA.

Z X

Y

Figure 6.27. Contour of damage dm on the small part of [0/45/90/ − 45]8s coupon at κ2 = 3.6 rad/m.

138

(a) 3D FEA X = 124.075733 mm.

(b) VABS.

Figure 6.28. Comparison of matrix damage dm contour plots of [0/45/90/ − 45]8s coupon at κ2 = 3.6 rad/m.

139

Table 6.4. Computation aspects of bending examples with CDM.

6.4

Computational Aspects

3D FEA

VABS

Time Elapsed

18 hr 39 min 41 sec

4 min 33 sec

Increments

5

20

Multi-Processor

44

1

Comparison with 2D FEA The damage model applied in this section is modified according to the Hashin

damage model based on the plane stress assumption. For details of this model please refer to by Lapczyk and Hurtado [111]. We need to modify the Hashin damage model to be compatible with the VABS theory. The damaged configuration experiences Γij and σij . Assume there exits an imaginary undamaged configuration which experiences ˆ ij and σ Γ îj . We further hypothesize that the beam idealizations of the damaged and undamaged 3D configuration experience the same beam strains γ, κ1 , κ2 , and κ3 . The constitutive laws in the damaged and undamaged configurations are respectively by Γij = Sdijkl σkl

(6.33)

˜ = Sijkl σ Γij ˜kl

(6.34)

where the damaged compliance matrix is computed by ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ d d −1 [S ] = [C ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 E1 (1−df )

0

0

− νE212

0

0

1 G12 (1−ds )

0

0

0

0

0

1 G13

0

0

− νE121

0

0

1 E2 (1−dm )

0

0

0

0

0

1 G23

− νE131

0

0

− νE232

0

⎤ − νE313 ⎥⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ − νE323 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 1 ⎥ E3 ⎦

(6.35)

140 while the undamaged compliance matrix [S] is obtained by setting df = ds = dm = 0. The utilization of the set of scalar damage variables, for example, df , dm , and ds in Eq. (6.35), results in the material symmetry of the composite ply even in the damaged configuration. For another example of such damage modeling methodology, please refer to Maim´ı et al. [109, 110]. The damage variable dm is associated with matrix cracking. ds is a damage variable influenced by fiber and matrix cracks. The closure of cracks under compressing strain, also known as the unilateral effect, is taken into account by defining two matrix damage variables dmt and dmc tension and compression, respectively. To determine the active damage mode, the matrix damage is formulated as dm = dmt

⟪Γ22 ⟫ ⟪−Γ22 ⟫ + dmc ∣Γ22 ∣ ∣Γ22 ∣

(6.36)

where the Macauley operator ⟪ ⟫ is defined as ⟪x⟫ =

x + ∣x∣ 2

(6.37)

We further adapt the assumption that the shear damage is not independent and can be expressed as ds = 1 − (1 − dmt )(1 − dmc )

(6.38)

Due to the limit of space and regarding that the composite beam experiences buckling in compression, only matrix tensile damage mode is introduced herein as an example. Damage initiation is determined based on Hashin’s failure criteria in which the failure surface is expressed in the elastic stress space. Herein we directly adapt the model proposed in Hashin and Rotem [155]. The initiation criteria has the following form for the matrix in tension: Matrix tension (˜ σ22 ≥ 0) ∶ Fmt = (

σ ˜22 2 σ ˜12 2 ) +( ) =1 YT SL

(6.39)

In the above equation, YT denotes the tensile strength in the matrix direction, and SL denotes the longitudinal in-plane shear strength. Although it has been shown that the strain localization and the mesh dependency are not observed from VABS, it is possible that these numerical issues will exist in

141 1D beam analysis. To conquer this behavior Lapczyk and Hurtado [111] use the crack band model proposed by Baˇzant and Oh [156]. In their theory, the fracture is modeled as a smeared crack band. The strain strength limit Γf is not kept constant but is adjusted in such a way that the fracture energy (also called the energy release rate) Gc is preserved by 1 0 f Gc = σ ˜ δ 2

(6.40)

where δ f denotes the equivalent displacement δ at which the material is fully damaged, and σ ˜ 0 denotes the equivalent stress σ ˜ at which a damage initiation criteria is satisfied. The equivalent displacement at damage initiation corresponding to σ ˜ 0 is denoted by δ f . A general definition of equivalent displacement reads δ = ΓLc

(6.41)

where Lc is the characteristic length of the 2D plat/shell element. Herein, for a beam model, Lc is properly defined as the length of the beam element. For validation purpose, the characteristic length is kept to be the same as the characteristic length obtained from the integration point of S4R element used in the Abaqus 2D FEA model which is uniformly meshed. According to Lapczyk and Hurtado [111] the Lc of S4R element can be approximately computed by Lc =

√

AS

(6.42)

where AS is the elemental area. Barbero et al. [157] proposed a method of calibrating the model in [111] from experimental date of beam coupon tensile tests without considering the effect of Lc on the damage evolution law. The equivalent stress associated with the matrix tensile mode is formulated as σ ˜mt =

˜ 22 ⟫ + σ ˜ 12 ) ⟪˜ σ22 ⟫⟪Γ ˜12 (2Γ √ ˜ 22 ⟫2 + (2Γ ˜ 12 )2 ⟪Γ

(6.43)

The evolution of the matrix tensile damage is governed by the equivalent displacement associated with the matrix tensile mode. dmt evolves according to dmt =

f 0 δmt (δmt − δmt ) f 0 δmt (δmt − δmt )

(6.44)

142 where the corresponding equivalent displacements are computed by √ ˜ 22 ⟫2 + (2Γ ˜ 12 )2 δmt = Lc ⟪Γ

δmt 0 δmt =√ Fmt

f δmt =

√

Fmt

2Gcmt σ ˜mt

(6.45)

The glass fiber-reinforced epoxy layers are used to perform the numerical validations. The orthotropic elastic constants are listed in Table 6.5. The material constants for the modified Hashin’s damage initiation and evolution laws are listed in Table 6.6. Table 6.5. Elastic properties of glass fiber-reinforced epoxy lamina. E1 (MPa)

E2 = E3 (MPa)

G12 = G13 (MPa)

G23 (MPa)

ν12 = ν13

ν23

55000

9500

5500

3298.6

0.33

0.44

Table 6.6. Constants of the modified Hashin damage initiation criterion evolution laws in matrix tensile damage mode. YT (MPa) SL (MPa) Gcmt (N/mm) Lc (mm) 2000

50

1.0

2.5416

Slender laminate coupons with length of 250 mm will be used in the validation. Figure 6.29 illustrates the cross-sectional geometry of the coupons. The laminate schematics and cross-sectional geometries are listed in Table 6.7. The prediction of tensile constitutive curves of [0/45/90/ − 45]2s coupon by VABS and by Abaqus S4R elements are compared in Figure 6.30. Small discrepancy between the two predictions can be observed beyond the damage initiation points. This acceptable difference is mainly caused by the difference of the assumptions on which the two modeled are formulated. VABS is built upon the 3D continuum while S4R is formulated based on conventional shell assumptions in which only in-plane stresses (σ11 , σ12 , σ22 ) are involved. The matrix tensile damage dmt predicted by VABS is plotted in Figure 6.31.

143 Even though the damage only considers the in-plane mode, because VABS utilizes the information of all of the components in the stress and strain tensors, more damage can be found along the free edges (x2 ≈ ±0.013 m) in the layers with 90 and ±45 layup angels. This phenomenon is also known as the free-edge effect [145, 158].

x3

a

x2

b Figure 6.29. Cross-sectional schematic of laminate beam coupons.

Table 6.7. Layup and geometry of the beam coupons. Layup Scheme

a [Ply Thickness] (mm)

b (mm)

Loading Type

[0/45/90/ − 45]2s

2.080 [0.103]

25.416

Tension

[30/60/90/ − 60/ − 30]2s

2.660 [0.133]

25.420

Tension

[60/0/ − 60]3s

2.412 [0.134]

25.396

Tension

[0/45/90/ − 45]8s

8.000 [0.125]

25.000

Bending

The prediction of tensile constitutive curves of [30/60/90/ − 60/ − 30]2s coupon by VABS and by S4R element in Abaqus are compared in Figure 6.32. The matrix tensile damage dmt at γ = 0.007 predicted by VABS is plotted in Figure 6.33 in the form of 3D surface due to the severe free edge effect at the boundary around x2 ≈ ±0.013 m. Especially, within the layers of ±30 layup angels, dmt on the boundary is considerable while the inner part is not damaged at all.

144

Figure 6.30. Comparison of tensile constitutive curves of [0/45/90/ − 45]2s coupon.

Figure 6.31. Matrix tensile damage dmt contour of tensile simulation of [0/45/90/ − 45]2s coupon at axial strain γ = 0.01 by VABS.

145 The predictions of tensile constitutive curves of [60/0/ − 60]3s coupon by VABS and Hashin damage model with S4R element in Abaqus are plotted in Figure 6.34. The matrix tensile damage dmt at γ = 0.007 is plotted in Figure 6.35 in the form of 3D surface due to the severe free edge effect at the boundary around x2 ≈ ±0.013 m within the layers of ±60 layup angels. The predictions of bending constitutive curves of [0/45/90/ − 45]8s coupon by VABS and S4R element in Abaqus are plotted in Figure 6.36. The matrix tensile damage dmt predicted by VABS is shown in Figure 6.37. It can be seen that the dmt is also found in the sectional area where the compressive stress is dominant. This is because the matrix tensile damage model also can be triggered by in-plane shear in the layers of ±45 degrees. In some layers, the dmt is not uniformly distributed due to out of plane stress fields, which is different from the CLPT.

Figure 6.32. Comparison of tensile constitutive curves of [30/60/90/ − 60/ − 30]2s coupon.

146

Figure 6.33. Matrix tensile damage dmt contour of tensile simulation of [30/60/90/ − 60/ − 30]2s coupon at axial strain γ = 0.007 by VABS.

Figure 6.34. Comparison of tensile constitutive curves of [60/0/ − 60]3s coupon.

147

Figure 6.35. Matrix tensile damage dmt contour of [60/0/ − 60]3s coupon at axial strain γ = 0.006 by VABS.

100

80

S4R element (Hashin damage)

Bending moment

M

2

(Nm)

VABS (modified Hashin damage)

60

40

20

0 0.0

0.5

1.0

1.5

2.0

Bending curvature

2.5

2

3.0

3.5

(rad/m)

Figure 6.36. Comparison of bending constitutive curves of [0/45/90/ − 45]8s coupon.

148

Figure 6.37. Matrix tensile damage dmt contour of [0/45/90/ − 45]8s coupon by VABS at curvature κ2 = 3.6 rad/m.

6.5

Comparison with Experimental Data The damage model proposed by Hou [159] based on Chang-Chang failure crite-

ria [86] is modified to fit the algorithm of the VABS theory. The damaged compliance matrix is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [S] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 E1 (1−d11 )

0

0

− νE212

0

− νE313

0

1 G12 (1−d12 )

0

0

0

0

0

0

1 G13 (1−d13 )

0

0

0

− νE121

0

0

1 E2 (1−d22 )

0

− νE323

0

0

0

0

1 G23 (1−d23 )

0

− νE131

0

0

− νE232

0

1 E3 (1−d33 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6.46)

In Eq. (6.46), dij are scalar damage variables which ensure the symmetry of the composite ply even in the damaged configuration. The damage variable d11 is associated

149 with fiber breakage, whereas d22 and d33 are the damage variables associated with matrix cracking and delamination, respectively. The closure of cracks under compressing strain is taken into account by defining four damage variables, d1t , d1c , d2t , and d2c , for fiber tension, fiber compression, matrix tension, and matrix compression, respectively. To determine the active damage mode, the 3D normal damage variables are formulated as ⟪Γ11 ⟫ ⟪−Γ11 ⟫ + d1c ∣Γ11 ∣ ∣Γ11 ∣ ⟪Γ22 ⟫ ⟪−Γ22 ⟫ d22 = d2t + d2c ∣Γ22 ∣ ∣Γ22 ∣ ⟪Γ33 ⟫ d33 = d3t ∣Γ33 ∣

d11 = d1t

(6.47) (6.48) (6.49)

where d3t denotes the delamination damage mode. It is noticed that the throughthickness compression damage mode is neglected in such a model due to the lack of experimental evidence. We further adapt the assumption that the shear damage mode is not independent and can be expressed as a function of the other variables. In light of this assumption, d12 is a damage variable influenced by fiber and in-plane matrix cracks; d13 is a damage variable influenced by fiber and delamination; d23 is a damage variable influenced by in-plane matrix cracks and delamination. Consequently, these 3D shear damage variables are computed by d12 = 1 − (1 − d1t )(1 − d1c )(1 − d2t )(1 − d2c )

(6.50)

d13 = 1 − (1 − d1t )(1 − d1c )(1 − d3t )

(6.51)

d23 = 1 − (1 − d2t )(1 − d2c )(1 − d3t )

(6.52)

Due to the limit of space and for simplicity, we only consider the tensile modes of three-dimensional damage. The in-plane damage initiation is determined based on Chang-Chang’s failure criteria [86]. The delamination criterion proposed by Brewer and Lagace [160] is applied as well. The initiation criteria have the following form for fiber tension, in-plane matrix tension, and delamination, as follows: Fiber tension (˜ σ11 ≥ 0) ∶ F1F = (

σ ˜11 2 ) =1 XT

(6.53)

150 Fiber tension and shear (˜ σ11 ≥ 0) ∶ F1S = (

σ ˜12 2 σ ˜11 2 ) +( ) =1 XT SF

σ ˜22 2 σ ˜12 2 σ ˜23 2 Matrix tension (˜ σ22 ≥ 0) ∶ F2t = ( ) +( ) +( ) =1 YT S12 S23M σ ˜33 2 σ ˜13 2 σ ˜23 2 Delamination (˜ σ33 ≥ 0) ∶ F3t = ( ) +( ) +( ) =1 ZT S13 S23L

(6.54) (6.55) (6.56)

In the above equations, XT denotes the tensile strength in the fiber direction; YT denotes the tensile strength in the matrix direction; ZT denotes the tensile strength in the through-thickness direction; SF denotes the shear strength involving fiber failure; S12 denotes the shear strength in the fiber and transverse plane; S13 denotes the shear strength in the fiber and through-thickness plane; S23M denotes the shear strength for matrix cracking in the transverse and through-thickness plane; S23L denotes the shear strength for delamination in the transverse and through-thickness plane. Similarly to the modified Hashin damage model, to conquer the numerical issues induced by the strain localization and the mesh dependency in the 1D beam analysis, the crack band method is applied. The equivalent stresses associated with the damage evolution laws in fiber tension, matrix tension, and delamination are defined as follows: ˜ 11 ⟫ ⟪˜ σ11 ⟫⟪Γ σ ˜1F = √ ˜ 11 ⟫2 ⟪Γ ˜ 11 ⟫ + σ ˜ 12 ) ⟪˜ σ11 ⟫⟪Γ ˜12 (2Γ σ ˜1S = √ ˜ 11 ⟫2 + (2Γ ˜ 12 )2 ⟪Γ ˜ 22 ⟫ + σ ˜ 12 ) + σ ˜ 23 ) ⟪˜ σ22 ⟫⟪Γ ˜12 (2Γ ˜23 (2Γ σ ˜2t = √ ˜ 22 ⟫2 + (2Γ ˜ 12 )2 + (2Γ ˜ 23 )2 ⟪Γ ˜ 33 ⟫ + σ ˜ 13 ) + σ ˜ 23 ) ⟪˜ σ33 ⟫⟪Γ ˜13 (2Γ ˜23 (2Γ σ ˜3t = √ ˜ 33 ⟫2 + (2Γ ˜ 13 )2 + (2Γ ˜ 23 )2 ⟪Γ

(6.57)

(6.58)

(6.59)

(6.60)

The evolution of fiber tension damage d1t is formulated by including the fiber normal damage d1F and fiber shear damage d1S as d1t = 1 − (1 − d1F )(1 − d1S )

(6.61)

151 where the evolution of fiber normal damage d1F is f 0 δ1F (δ1F − δ1F ) d1F = f 0 δ1F (δ1F − δ1F )

(6.62)

where the corresponding equivalent displacements are computed by δ1F = Lc

√

δ1F 0 δ1F =√ F1F

˜ 11 ⟫2 ⟪Γ

f δ1F =

√

F1F

2Gc1F σ ˜1F

(6.63)

The evolution of fiber shear damage d1S is d1S =

f 0 δ1S (δ1S − δ1S ) f 0 δ1S (δ1S − δ1S )

(6.64)

where the corresponding equivalent displacements are computed by δ1S = Lc

√

˜ 11 ⟫2 + (2Γ ˜ 12 )2 ⟪Γ

δ1S 0 =√ δ1S F1S

f = δ1S

√

F1S

2Gc1S σ ˜1S

(6.65)

The evolution of matrix tension damage d2t is d2t =

f 0 δ2t (δ2t − δ2t ) f 0 δ2t (δ2t − δ2t )

(6.66)

where the corresponding equivalent displacements are computed by δ2t = Lc

√

˜ 22 ⟫2 + (2Γ ˜ 12 )2 + (2Γ ˜ 23 )2 ⟪Γ

δ2t 0 δ2t =√ F2t

f δ2t =

√

F2t

2Gc2t σ ˜2t

(6.67)

The evolution of delamination damage d3t is d3t =

f 0 δ3t (δ3t − δ3t ) f 0 δ3t (δ3t − δ3t )

(6.68)

where the corresponding equivalent displacements are computed by δ3t = Lc

√

˜ 33 ⟫2 + (2Γ ˜ 13 )2 + (2Γ ˜ 23 )2 ⟪Γ

δ3t 0 δ3t =√ F3t

f δ3t =

√

F3t

2Gc3t σ ˜3t

(6.69)

The carbon fiber-reinforced epoxy layers are used to perform the numerical validations. The orthotropic elastic constants are listed in Table 6.8 according to [143] except that ν23 is assumed. The normal damage initiation thresholds are listed in Tables 6.9 according to [143] and [88]. The shear damage initiation thresholds are listed in Tables 6.10 according to [159] except that SF is modified according to the

152 present damage model. The calibrated fracture energies and the characteristic length are listed in Table 6.11. A more rigorous way to obtain these material constants is to directly calibrate them from the coupon test data when it is available. Experimental results in [143] are compared with the VABS prediction using the damage model in this section. Figure 6.29 illustrates the cross-sectional geometry of

Table 6.8. Elastic properties of carbon fiber-reinforced epoxy lamina. E1 (MPa) E2 = E3 (MPa) G12 = G13 (MPa) G23 (MPa) ν12 = ν13 159000

8770

4880

3117

0.3197

ν23 0.44

Table 6.9. Material properties of initial normal damage thresholds of carbon fiber-reinforced epoxy lamina. XT (MPa) YT (MPa) ZT (MPa) 2300

80

80

Table 6.10. Material properties of initial shear damage thresholds of carbon fiber-reinforced epoxy lamina. SF (MPa) S12 (MPa) S13 (MPa) S23L (MPa) S23M (MPa) 100

64

86

64

64

Table 6.11. Material properties of damage evolution of carbon fiberreinforced epoxy lamina. Gc1F (N/mm) Gc1S (N/mm) Gc2t (N/mm) Gc3t (N/mm) Lc (mm) 42.0

20.7

6.0

6.0

2.5

153 the coupons. The laminates scheme and geometries of [30/60/90/ − 60/ − 30]2s and [0/45/90/ − 45]2s coupons are listed in Table 6.7. The predictions of beam tensile constitutive curves of [30/60/90/ − 60/ − 30]2s coupon by VABS are compared with experimental data in Figure 6.38. The 3D surface contours of fiber directional damage d11 , in-plane matrix directional damage d22 , and delamiantion damage d33 at the beam axial strain γ = 0.012 are plotted in Figures 6.39, 6.40, and 6.41, respectively. The prediction of the tensile constitutive curve of [0/45/90/−45]2s coupon by VABS is compared with experimental data in Figure 6.42. The predicted ultimate strength is relatively lower than the experimental data. It is because of the coupling of in-plane shear stress in the matrix damage initiation criteria. The contours of matrix directional damage d22 and delamination damage d33 at the beam axial strain γ = 0.012 are plotted in Figure 6.43 and Figure 6.44, respectively. In both laminates, highly concentrated damage fields are also observed on the vertical free edge boundaries of the coupon laminates.

Figure 6.38. Comparison of tensile constitutive curves of [30/60/90/ − 60/ − 30]2s carbon/epoxy coupon.

154

Figure 6.39. Damage d11 contour of [30/60/90/ − 60/ − 30]2s carbon/epoxy coupon by VABS at γ = 0.012.


155


Figure 6.42. Comparison of tensile constitutive curves of [0/45/90/ − 45]2s carbon/epoxy coupon.

156

Figure 6.43. Damage d22 contour of tensile simulation of [0/45/90/ − 45]2s carbon/epoxy coupon by VABS at γ = 0.012.

Figure 6.44. Damage d33 contour of tensile simulation of [0/45/90/ − 45]2s carbon/epoxy coupon by VABS at γ = 0.012.

157

7. LARGE BENDING NONLINEARITIES This chapter demonstrates the capabilities of VABS to predict the nonlinearities induced by large bending. Section 7.1 presents the modeling of extension-bending coupling effect. Jiang, Tian, and Yu have published this model in 57th AIAA/ASCE/AHS/ ASC structures, structural dynamics, and materials conference paper entitled “Nonlinear modelling of axially deformable elastica based on hyperelasticity” [161] reprinted here by permission of the American Institute of Aeronautics and Astronautics, Inc. Interested readers are suggested to refer to this article. Section 7.2 provides the VABS solution to the Brazier effect.

7.1

Extension-Bending Coupling

7.1.1

Beam Constitutive Law

In Chapter 4, we have shown that when the strain is relatively small and O(ε3 ) is negligible, it is possible for us to find analytical solution for the nonlinear crosssectional analysis with the 1D strain energy Π corrected to the third order of ε. Focusing on modeling the combined bending-stretching behavior for isotropic beams, we assume w1 = κ1 = 0

(7.1)

Then the components of deformation gradient tensor defined by Eq. (2.16) can be written as ⎡ ⎢ 1 + γ + (x3 + w3 )κ2 − (x2 + w2 )κ3 0 0 ⎢ ⎢ [F ] = ⎢⎢ 0 1 + w2,2 w3,2 ⎢ ⎢ ⎢ 0 w2,3 1 + w3,3 ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7.2)

158 In this section we follow the methodology of Section 4.1. We assume N-H hyperelastic model. However, it is not limited to the N-H material. Integrating the N-H hyperelastic energy function over the cross-section, we have obtained that ΠO(ε2 ) in Eq. (4.12). In light of Eq. (7.2) and the definition of Jaunmann-Biot-Cauchy strain, we have 2 2 2ΠO(ε2 ) = (λ + 2µ)⟨Γ211 + w2,2 + w3,3 ⟩ + 2λ⟨Γ11 w2,2 + Γ11 w3,3 + w2,2 w3,3 ⟩ + µ⟨Γ223 ⟩ (7.3)

in which Γ11 = γ + x3 κ2 − x2 κ3

Γ23 = w3,2 + w2,3

(7.4)

In addition, ΠO(ε3 ) takes the form of ΠO(ε3 ) = λ⟨Υλ ⟩ + µ⟨Υµ ⟩

(7.5)

in which Υλ = (Γ11 + w2,2 + w3,3 ) [Γ11 (w2,2 + w3,3 ) + w2,2 w3,3 − w2,3 w3,2 + (w3 κ2 − w2 κ3 )] (7.6) 7 Υµ = (w2,2 + w3,3 ) [Γ11 (Γ11 + w2,2 + w3,3 ) + 3w2,2 w3,3 ] 9 4 14 3 [Γ + (w2,2 + w3,3 )3 ] + Γ11 [ w2,2 w3,3 + 2(w3 κ2 − w3 κ3 )] − 3 27 11 1 − (w2,3 w3,2 )(w2,2 + w3,3 ) − (Γ11 + w2,2 + w3,3 )(w2,3 + w3,2 )2 3

(7.7)

The warping functions that minimize 2ΠO(ε2 ) in Eq. (7.3) are governed by the Euler-Lagrange equations of this functional 2 (1 − ν) w2,22 + (1 − 2ν) w2,33 + w3,23 + 2νΓ11,2 = 0

(7.8)

2 (1 − ν) w3,33 + (1 − 2ν) w3,22 + w2,23 + 2νΓ11,3 = 0

(7.9)

and the associated boundary conditions 2n2 [ν (Γ11 + w3,3 ) + (1 − ν) w2,2 ] = 0 1 − 2ν 2n3 n2 (w2,3 + w3,2 ) + [ν (Γ11 + w2,2 ) + (1 − ν) w3,3 ] = 0 1 − 2ν n3 (w2,3 + w3,2 ) +

where nα is the direction cosine of outward normal with respect to xα .

(7.10) (7.11)

159 The solutions of the warping functions are given by Yu and Hodges [17] as I23 I2 − I3 ν − x2 x3 ) νκ2 + (x22 − x23 + ) κ3 A A 2 I23 ν I − I 2 3 ̂3 = −x3 νγ + (x2 x3 − w3 = w ) νκ3 + (x22 − x23 + ) κ2 A A 2

̂2 = −x2 νγ + ( w2 = w

(7.12) (7.13)

with A = ⟨1⟩ I2 = ⟨x23 ⟩ I3 = ⟨x22 ⟩ I23 = ⟨x2 x3 ⟩

(7.14)

According to VABS we can show that the warping function we have solved is sufficient to obtain the strain energy asymptotically correct up to the order of ε3 . For simplicity, we further choose a centroidal coordinate system. The generalized axial force and the bending moment can be obtained by F1 =

∂Π ∂γ

γ E = EA [1 + (−7 − 14ν + 20ν 2 )] γ + (I2 κ22 + I3 κ23 ) (−7 − 23ν + 20ν 2 ) 9 9 ∂Π M2 = ∂κ2 2γ (−7 − 23ν + 20ν 2 )] κ2 = EI2 [1 + 9 Eκ22 + ⟨x3 [27x22 ν + x23 (−14 − 55ν + 40ν 2 )]⟩ 18 Eκ23 + ⟨x3 [9x23 ν + x22 (−14 − 73ν + 40ν 2 )]⟩ 18 Eκ2 κ3 + ⟨x2 [−9x22 ν + x23 (14 + 73ν − 40ν 2 )]⟩ 9 ∂Π M3 = ∂κ3 2γ (−7 − 23ν + 20ν 2 )] κ3 = EI3 [1 + 9 Eκ23 + ⟨x2 [−27x23 ν + x22 (14 + 55ν − 40ν 2 )]⟩ 18 Eκ22 ⟨x2 [−9x22 ν + x23 (14 + 73ν − 40ν 2 )]⟩ + 18 Eκ2 κ3 + ⟨x3 [9x23 ν + x22 (−14 − 73ν + 40ν 2 )]⟩ 9

(7.15)

(7.16)

(7.17)

160 7.1.2

One-Dimensional Beam Model

The scenario we are going to study herein is the static large bending behavior of a slender hyperelastic beam. Consequently, Eq. (2.17) reduces to l

∫0 −δUdx1 = δA

(7.18)

ˆ ) ∣x1 =l ¯ ) dx1 = (δq T Fˆ1 + δψ T M κT M ∫0 (δγF1 + δ¯ x1 =0

(7.19)

which can be expanded to l

¯ is the sectional moment resultant array which can be expressed for the where M Euler-Bernoulli beam model as ¯ = [M2 M

M3 ]

T

(7.20)

δ¯ κ is the virtual beam strain array with κ ¯ defined by κ ¯ = [κ2

κ3 ]

T

(7.21)

ˆ are the force and moments, respectively, evaluated at the ends of space Fˆ1 and M interval; δq and δψ are the virtual displacement and rotation, respectively. The FEA method and corresponding numerical recipes to solve Eq. (7.19) has been completed and can be referred to the GEBT. The advantage of GEBT is that it can capture all geometrical nonlinearities in a systematical way, which is very important for structures or mechanisms containing large deformations. Another feature of this approach is that the formulation remains the same no matter what kind of material is used. The detailed information of the global beam analysis can be found in [22] and [162].

7.1.3

Semi-Analytical Solution versus 3D FEA

A slender rubber beam with a rectangular cross section shown in Figure 7.1 is used herein as example.

161

h = 1cm

x3

x2

h = 1cm

Figure 7.1. Cross-sectional geometry of rubber beam.

With such a section, Eqs. (7.15), (7.16), and (7.17) reduce to the following expressions F1 = Eh2 [1 +

γ Eh4 2 (−7 − 14ν + 20ν 2 )] γ + (κ + κ23 ) (−7 − 23ν + 20ν 2 ) 9 108 2 Eh4 2γ (−7 − 23ν + 20ν 2 )] κ2 [1 + 12 9 Eh4 2γ (−7 − 23ν + 20ν 2 )] κ3 M3 = [1 + 12 9 M2 =

(7.22) (7.23) (7.24)

The bulk modulus of the rubber is K = 1.9123725 MPa and the shear modulus µ = 0.3825 MPa. The beam-structural constitutive relations obtained by using numerical VABS and analytical formulas in Eqs. (7.22)-(7.24) are plotted in Figure 7.2. It is noted that VABS does not restrict to the small strain yet the analytical solution rely on the assumption that the beam axial strain is small. Figure 7.2 (a) shows that F1 does not depend on the curvature very much. The analytical method can provide the accurate solution when the axial strain is under 0.15. However as shown in Figure 7.2 (b), the bending stiffness can be heavily influenced by the axial strain. And the analytical solution can hardly provide the accurate solution when the axial strain is relatively large. As a control group, 3D FEA model according to Figure 7.1 is built in ANSYS with length l = 50 × h = 50 cm with the same material model and constant values.

162

(a) F1 (γ, κ2 ).

(b) M2 (γ, κ2 ).

Figure 7.2. Comparison of numerical and analytical solutions of extension-bending coupling behavior predicted by VABS.

In ANSYS the 3D body of the cantilevered beam is meshed by 20-node quadrilateral elements with elemental volume size of 0.1 × 0.1 × 0.1cm3 . The schematic of the FEA model in ANSYS is shown in Figure 7.3. One tip of the elastica is clamped and the other is made rigid by using multiple point constraint element (MPC184). Bending moment M2 is applied to a master node on the free end. An example of large nonlinear bending deformation of the 3D elastica FEA results from the applied M2 is shown in Figure 7.4. The deformed axial length lB is obtained by summing the distances between neighboring nodes located in the center of the 3D body. Tip rotation angle θ2 are measured from the master node. Consequently, the axial strain and the curvature can be calculated from these measurements as γ = and κ2 =

θ2 lB ,

lB l

−1

respectively.

Nonlinear beam element BEAM189 in ANSYS is also used to solve this problem with the same material model and material constants to examine the difference between the present theory and the traditional beam model. Note ANSYS BEAM189 computes the beam stiffness based on the rigid cross section assumption.

163

Master Node Slave Node Clamped Node

Multiple Point Constraint Element (MPC 184)

Figure 7.3. Schematic of FE model of the cantilevered rubber beam.

Figure 7.4. Contour plot of the displacement solution of 3D FEA for M2 = 0.01 Nm.

On the other hand, to obtain the axial strain and the curvature of the deformed beam, substituting the following into Eqs. (7.15), (7.16), and (7.17) F1 = M3 = κ3 = 0

M2 = M2ANSYS

(7.25)

one can solve for the γ and κ2 by the following nonlinear system of equations γ Eh4 2 (−7 − 14ν + 20ν 2 )] γ + κ (−7 − 23ν + 20ν 2 ) 9 108 2 2γ Eh4 (−7 − 23ν + 20ν 2 )] κ2 [1 + M2ANSYS = 12 9 0 = Eh2 [1 +

(7.26)

This example is also analyzed by using Eqs. (7.22)-(7.24) and GEBT. The beam bending constitutive curves obtained by several approaches are plotted in Figure 7.5.

164 Predictions by the present theory match the result of 3D FEA. However, the traditional nonlinear beam element fails on predicting the extension-bending coupling effect.

Figure 7.5. Axial strain γ induced by bending moment M2 .

7.2

Brazier Effect The Brazier effect [112] is defined to deal with the cross-sectional in-plane oval-

ization of the long tube which led to the bending capacity limit. Harursampath and Hodges [121] obtained an asymptotically corrected analytical model for a long tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. This model releases the Kedward’s assumption on small cross-sectional displacement. However, the contribution of transverse deformation of the shell circumference was also neglected from the first-order approximation. Combining these two works with plate theory, we directly provide their analytical model on the Brazier effect in order to compare with the present beam sectional modeling approach. Considering a tubular cross section with diameter R, the beam bending constitutive law can be computed by Mα =

9(Rκα )2 (144µ + 5(Rκα )2 ) πR3 [1 − ] A′11 2(72µ + 5(Rκα )2 )2

(7.27)

165 where [A′ ] = [A]−1

µ=

A′11 D22 R2

(7.28)

with [A] and [D] the classical plate stiffness matrices. Consider a 12 in (152.4 mm) diameter long tube. A wall thickness of 0.04 in (1.016 mm) is used. To compare the predictions from Eq. (7.27) and the present model, the impact of sectional ovalization of the circular section, an isotropic material is firstly used to make the tube. The Young’s modulus is 1500 MPa, and we vary the Poisson’s ratio ν to see the difference among predictions which is shown in Figure 7.6. It can be seen that the present sectional analysis approach is more sensitive to the change of ν than the plate theory does. When ν = 0.45, a stiffening effect is also predicted by present model before the occurrence of the Brazier effect. This is one of the advantages of using 3D material constitutive model and abandoning the plane stress assumption in the plate theory. Also, the prediction of the ovalization limit is higher than the prediction of the plate theory. The deformed cross-sectional profile with Poisson’s ratio ν = 0.35 can be found in Figure 7.7.

Figure 7.6. Data comparison of predictions to the Brazier effect.

166

(a) κ2 = 0.01 rad/m.

(b) κ2 = 0.02 rad/m.

(c) κ2 = 0.03 rad/m.

(d) κ2 = 0.04 rad/m.

(e) κ2 = 0.05 rad/m.

(f) κ2 = 0.06 rad/m.

(g) κ2 = 0.07 rad/m.

(h) κ2 = 0.08 rad/m.

(i) κ2 = 0.09 rad/m.

Figure 7.7. Deformed tube sections with respect to bending curvatures κ2 .

167

8. SUMMARY In this dissertation, the theory and the code of VABS are extended to handle the finite deformation, nonlinear stress-strain law, and continuum damage. Identifying the cross section of the beam as the 2D SG, VAM is applied in light of the intrinsic slenderness feature so that VABS rigorously captures the beam constitutive relation and 3D local fields, such as strain, stress, and damage with the accuracy comparable to 3D FEA, while the computational cost is significantly reduced. Nonlinear extension-twist coupling effects for isotropic beams are modeled analytically. The deformation gradient contains the generalized nonlinear beam strains to evaluate the strain energy of the original 3D structure. Physical models of SaintVenant/Kirchhoff materials, Neo-Hookean materials, and the Hencky strain energy are considered along with the variation of the stored energy. It is consequently concluded that the stored-energy function impacts the nonlinear behavior of beam-like structures. Asymptotic analysis proves that the first approximation of the warping functions successfully corrects the strain energy into the nonlinear region. Because of the validity of the present formulation for cross-sectional warping, the theory provides improved comprehensive models for trapeze and Poynting effects for arbitrary beam cross sections. Simple examples show that the rigorous variational asymptotic method and the corresponding updated theory surpass the widely accepted analytical formulae in the literature in both generality and accuracy. It is proved that in the hyperelastic mechanics, the uniaxial stress assumption in extension-twisting coupling analysis is no longer validated because significant transverse stresses can result from the twisting of the beam. Numerically, the trapeze effect and Poynting effect of various sample beam structures are simulated successfully with the VABS code. Compression axial force is predicted by twisting a rubber cylinder. Increasing of

168 twisting moment results from adding extensional strain onto the twisted strip. These results demonstrate excellent agreement with 3D FEA. The present theory also provides an efficient high-fidelity approach for general predictions of nonlinear shear constitutive relations of composites in beams. By comparing with the predictions from 3D FEA, the nonlinear 1D constitutive relations and 3D stress fields including the free-edge effects, are proven to be rigorously captured by the present theory and the computational cost is greatly reduced. Under the in-plane shear nonlinearity, local stresses are also in nonlinear relations with the beam axial strain. Limitations of using the plane stress assumption to calibrate the nonlinear material model is examined. On one hand, the plane stress assumption is valid only if the layer is arranged as homogeneous as possible, and the layer width is large enough compared to the layer thickness. In this situation, the free-edge stress will be confined in a relatively small cross-sectional region so that its effects on the global force-displacement law can be reasonably neglected. On the other hand, the plane stress assumption made during the calibration is not adaptable in the structural simulation when the layer is not arranged as homogeneous as possible, or the layer width is not large enough compared to the layer thickness. Nevertheless, although the plane stress assumption is successful in some specific scenarios, the out-of-plane stresses will become critical when the damage is considered. Furthermore, an efficient high-fidelity damage analysis methodology is developed for composite beams based on the VABS theory. Validation from comparison with 2D FEA and experimental data proves the generality of the present approach which is capable for various scenarios of composite beams. Analytical studies and numerical results of extension, bending, and twisting examples also show that the semi-analytical feature of VABS prevents the mesh sensitivity when the damage softening constitutive law is involved in the FEA. Instead of introducing a so-called characteristic length to relate the post-peak softening response to a traction-separation law, VABS solves the governing equations of the principle of virtual work to obtain the physically admissible cross-sectional warping and damage field under the knowledge of the generalized

169 beam strain. These generalized beam strains form intrinsic constant values into the initial strain field of the cross section in light of a kinetically admissible description of deformation formulated by the geometric exact theory. These constants prevent the unrealistic and physically inadmissible strain localizations. In the example of twisting a rectangular-sectioned beam, the crack-like damage and strain localizations are found. For this case, the energy release rate is computed from the predicted beam twisting constitutive relation and the length of the localization. This value converges to a constant when the mesh is refined. By regulating the degree of softening, it is proved that VABS prediction is sensitive to the damage model taken into the prediction. This feature is hardly possessed in conventional 3D FEA. In this sense, VABS can predict better results than 3D FEA due to its semi-analytical nature. Examples also prove the capabilities of VABS to predict the nonlinearities in large bending problems. Nonlinear contributions in the analytical beam bending constitutive relations are obtained which predicts the extension-bending coupling phenomenon. The numerical and analytical solution of VABS approach are compared, and the analytical model is validated to be applicable when the axial strain is under 0.05. The results of analyzing the large bending of a rubber beam by using different approaches show that with the present theory fully intrinsic nonlinear behavior can be captured by a 1D beam model with accuracy comparable to 3D FEA. In addition, the present theory is shown to be able to predict the generalized Brazier effect. By comparing with the conventional models, it is shown that the Poisson effect plays an important role in the prediction of the Brazier effect, which is widely neglected by adapting the plane stress assumptions. When the Poisson’s ratio is large, which is the very situation for rubber, the bending rigidity tends to be even stiffened before the occurrence of the Brazier effect. The present beam theory is shown to be successful in the analyses of nonlinear elastic and progressive damage problems of a wide range of slender composite structures. It can be used to design and analyze composite beams in the construction scales, and makes obtaining the reliable prediction possible. The accuracy is com-

170 parable to and even better than 3D FEA while the computational cost is greatly reduced. Furthermore, the present theory could also serve as a high-fidelity virtual coupon in the calibration of nonlinear elastic and inelastic material properties. Based on the present work and the conclusions, the author would like to recommend the possible future work of this research. First, extending the present theory to handle nonlinear problem using 3D SG of the beam will be an interesting potential topic. Second, considering the geometrical nonlinearities has been rigorously treated in the present theory, nonlinear local buckling behaviors can be expected to be another following work. Third, enabling VABS to deal with beams made of nonlinear viscoelastic materials would contribute to the development of deployable high strain composite structures. Fourth, refining the present Euler-Bernoulli type analysis into the Timoshenko type model will further improve the fidelity. Fifth, developing a more rigorous, realistic, and practical 3D continuum damage model will assist enhancing the overall accuracy. Finally, extending GEBT code to take the nonlinear beam constitutive law predicted by VABS will enable the solution of more complex scenarios.

REFERENCES

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VITA Fang Jiang was born in Beijing, China on May 3rd, 1987. After graduating from Beijing No. 4 High School (BHSF), he continued his higher education at Beijing Institute of Technology (BIT) where he obtained his Bachelor and Master degrees in Mechanical Engineering. His master thesis focused on the kinematics and dynamics of a spatial parallel robot. He got admitted into the Ph.D. program of the Mechanical and Aerospace Engineering Department at Utah State University in Fall 2012. Since then, he has been following his advisor, Prof. Wenbin Yu, in the research of computational structural mechanics and solid mechanics. In January 2014, he transferred into the Ph.D. program of the School of Aeronautics and Astronautics at Purdue University where he continued this study in Prof. Wenbin Yu’s group. His expected date of graduation is May 2017. His expertise is finite element analysis (FEA) of composites, progressive damage and fracture, finite deformation, hyperelasticity, nonlinear beams and shells, mechanics of anisotropic heterogeneous materials, heat transfer, and thermomechanical effects.