Locally-exact homogenization of unidirectional

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Effects of coated and hollow reinforcement on homogenized moduli and stress fields in unidirectional compos- ites are ... ferent forms that depend on the fiber/matrix system and hence ... [11,33,34,6,7,3], with a recent focus on the incorporation of ... Section 5. 2. Locally-exact homogenization theory for periodic arrays. 2.1.
Materials and Design 93 (2016) 514–528

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Materials and Design journal homepage: www.elsevier.com/locate/matdes

Locally-exact homogenization of unidirectional composites with coated or hollow reinforcement Guannan Wang, Marek-Jerzy Pindera Civil Engineering Department, University of Virginia, Charlottesville, VA 22904-4742, USA

a r t i c l e

i n f o

Article history: Received 17 November 2015 Received in revised form 28 December 2015 Accepted 29 December 2015 Available online 2 January 2016 Keywords: Micromechanics Homogenization Periodic microstructures Elasticity-based solution Coatings

a b s t r a c t Effects of coated and hollow reinforcement on homogenized moduli and stress fields in unidirectional composites are investigated using elasticity-based homogenization theory for periodic materials with hexagonal and tetragonal symmetries extended to accommodate coatings. The theory employs Fourier series representations for fiber, coating and matrix displacement fields in the cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence in the presence of thick or very thin coatings with relatively few harmonic terms. The solution's stability facilitates rapid identification of coatings' impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and coating thickness. Elastic response of unidirectional composites reinforced by hollow fibers may be obtained without specializing the theory's framework by appropriately adjusting input parameters. This is illustrated using alumina nanotubes as reinforcement, revealing new results of interest in the design of multifunctional porous materials. Equally important, the theory's analytical framework requires minimal effort in constructing input data file that defines the unit cell problem, facilitating use by researchers with little mechanics exposure. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Interfaces play a key role in stress transfer between fiber and matrix phases of a fiber-reinforced composite, which is at the core of reinforcement principles in the mechanics of composite materials. They take different forms that depend on the fiber/matrix system and hence different names have been used to describe them. Examples include regions with variable properties, or interphases, due to altered chemical bond structure of the matrix phase adjacent to the fiber's surface in polymeric matrix composites; fabrication-induced reaction zones with degraded properties in metal matrix composites reinforced by ceramic fibers; as well as coatings that promote fiber/matrix adhesion, control fracture toughness or reduce residual stresses. The effect of interfaces or interphases on the homogenized and local response of unidirectional composites has been investigated by many researchers using different modeling approaches within various micromechanics or homogenization theories, including distinct interfacial layers with properties different from those of the adjacent fiber or matrix, and spring and cohesive zone models. For very thin interface/interphase regions the latter two models offer an attractive alternative to finite-thickness interfacial layers especially when variational techniques requiring detailed geometric discretization are employed. Review of the early approaches based on simple geometric models of unidirectional composites such as the CCA (composite cylinder assemblage), Mori-Tanaka and GSC (generalized self-consistent) models was

http://dx.doi.org/10.1016/j.matdes.2015.12.168 0264-1275/© 2016 Elsevier Ltd. All rights reserved.

provided by [15]. A more recent discussion of the various approaches may be found in [8]. The simple geometric models based on a single fiber embedded in the matrix phase, which may in turn be embedded in the homogenized medium of sought properties, yield estimates of homogenized moduli in the presence of interphases or coatings with uniform or variable (so-called graded) properties, but do not provide accurate estimates of stress fields that account for adjacent fiber interaction. To gauge the effect of coatings or interphases on the homogenized moduli without sacrificing local stress field accuracy critical in failure analysis, the finite-element approach has been, and continues to be, employed by a number of investigators, cf. [17,31,1,26,21,35,27]. In the presence of thin coatings, however, detailed mesh discretization is required for converged stress fields. Alternative approaches to the homogenization of unidirectional composites include elasticity-based solutions for periodic microstructures, [23,19], and finite-volume techniques, cf. [24,5,4,29]. Interest in elasticity-based methods has revived within the past 15 years in light of advances in computational technology, as well as due to the potential advantages offered by these techniques, cf. [11,33,34,6,7,3], with a recent focus on the incorporation of interphase and spring models into elasticity-based solutions, [22,25, 12]. Optimization of interfacial properties will profit from the use of analytical techniques in the solution of unit cell problems due to the significantly smaller design variable space, more efficient specification of objective functions and implementation of more efficient search procedures. Another benefit is the efficient reconstruction of local fields from

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

homogenized-based results within a multi-scale analysis of local failure modes, [18], and material development which relies on rapid answers to what/if questions. Herein, the elasticity based locally-exact homogenization theory proposed by [7] for rectangular and square periodic microstructures, and [32] for hexagonal arrays with transversely isotropic phases, is further extended to enable rapid calculation of homogenized moduli and stress fields in unidirectional composites with coated fibers. This enhanced capability may also be used for the analysis of unidirectional composites reinforced by hollow fibers, such as alumina nanotubes being developed for microelectronic, optical and potentially structural applications, [9]. The theory differs from other elasticity-based solutions of the local unit cell problem such as the eigenstrain expansion technique[3], the equivalent homogeneity method [22], or the eigenfunction expansion technique [25], in the manner of periodic boundary conditions implementation. While the eigenstrain and eigenfunction expansion techniques employ doubly-periodic displacement field representations that satisfy periodicity conditions a priori, the present theory employs a balanced variational principle in enforcing periodicity conditions along the boundary of a unit cell. This variational principle produces rapid convergence of the displacement field in cylindrical coordinates which satisfies both the Navier's equations and interfacial continuity conditions in the interior of the unit cell representative of rectangular, square or hexagonal periodic arrays of transversely isotropic inclusions. As a result, converged homogenized moduli and local stress fields alike are obtained with relatively few terms in the displacement field representation, in contrast with the eigenstrain expansion technique which requires substantially greater number of terms to obtain converged stress fields [3]. The extended locally-exact homogenization theory that accommodates coatings exhibits convergence of both homogenized moduli and local stress fields which is just as rapid in a wide range of coating thickness and coating/fiber/matrix modulus contrast. While the effect of coating on homogenized moduli has been documented by the above-mentioned elasticity-based unit cell solutions, no data is available on local stress fields which underpin stress transfer mechanisms and ensuing coating-driven modulus alteration. Section 2 describes the locally-exact homogenization theory's extension which is validated in Section 3. In Section 4 we investigate the combined effects of coating thickness and elastic properties on the homogenized moduli and local stress fields, as well as the effect of emerging nanotube reinforcement of evolving interest in the nanotechnology areas. Specifically, we generate homogenized moduli of unidirectionally-reinforced composites with coated fibers in a wide range of inclusion volume fractions and coating/matrix modulus contrast, and illustrate new and unexpected results when nanotube reinforcement is employed. Conclusions are presented in Section 5. 2. Locally-exact homogenization theory for periodic arrays 2.1. Unit cell solution overview We investigate the elastic response of periodic materials with continuous coated reinforcement along the x1 axis, characterized by repeating unit cells representative of either hexagonal or rectangular microstructure, Fig. 1. The fibers are not centered to highlight the solution's insensitivity to fiber placement[7]. Loading the entire hexagonal or rectangular array of coated inclusions by uniform homogenized strain components ε ij is equivalent to loading a single unit cell by the same components. The solution for the ensuing local stress fields is formulated in terms of displacements in the fiber, coating and matrix phases subject to interfacial continuity conditions and periodic boundary conditions on unit cell surface displacements and tractions.

515

Fig. 1. Repeating unit cells of hexagonal (a) and rectangular (b) arrays of coated fibers.

where (xo, xo + d) ∈ S, S is the unit cell boundary, d is a characteristic distance that defines the unit cell array microstructure, and ti = σjinj from Cauchy's relations, with nj denoting the jth component of the unit normal to the boundary. The solution for the displacement fields is carried out within the homogenization theory's framework wherein the global coordinates x = (x1, x2, x3) describe the average response of the entire periodic array, and the local coordinates y = (y1, y2, y3) describe the interior unit cell response, [2,28]. Accordingly, a two-scale displacement field representation is employed in the individual phases as follows. ðkÞ

ðkÞ

ui ðx; yÞ ¼ ε ij x j þ u0 i ðyÞ

ð2Þ

where the fluctuating displacement components ui′ caused by the material's heterogeneity are functions of the local coordinates (y2, y3) given the unidirectional constraint along the x1 direction by continuous reinforcement, and the superscripts k = f, c, m denote fiber, coating and matrix phases, respectively. The above displacement field generates the local strains. ðkÞ

ðkÞ

ε ij ðyÞ ¼ εij þ ε0 ij ðyÞ

ð3Þ

from which local stresses follow, with continuous reinforcement yieldui ðxo þ dÞ ¼ ui ðxo Þ þ ε ij d j ;

t i ðxo þ dÞ ¼ −t i ðxo Þ

ð1Þ

ðkÞ

ing the constraint ε11 ¼ ε11 .

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G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

The boundary–value problem is solved exactly in the interior of the unit cell using cylindrical coordinate system wherein the problem is separable, facilitating exact solution. The inseparable exterior problem involves satisfaction of the periodic boundary conditions in the Cartesian coordinate system and is efficiently solved using the balanced variational principle proposed by [7]. Solution of the exterior problem entails minimization of the functional Z Z Z 1 H D−P ¼ σ ij εij dV− ti uoi dS− toi ui dS ð4Þ 2 v

Su

St

in the equilibrium equations in the cylindrical coordinate system yields the Navier's equations for the three unknown fluctuating displacements u'z, u'r and u'θ in each phase 2

2

∂ u0z 1 ∂u0z 1 ∂ u0z þ 2 2 ¼0 þ r ∂r r ∂θ ∂r 2 ! 2 0 2 0 2 0 0 0 ∂ ur 1 ∂ur ur μ T ∂ ur kT ∂ uθ 2ðkT þ 2μ T Þ ∂u0θ − ¼ 0: ðkT þ μ T Þ þ ¼ þ þ r ∂r 2 ∂θ2 r2 r 2 ∂θ2 r 2 ∂r∂θ ∂r 2 ∂θ ! 2 0 2 2 ∂ uθ 1 ∂u0θ u0 ðkT þ μ T Þ ∂ u0θ kT ∂ u0r 2ðkT þ 2μ T Þ ∂u0r − θ2 þ ¼0 þ þ þ μT r ∂r∂θ2 r ∂r r2 r2 ∂r 2 ∂θ rθ ∂θ2

ð8Þ

where t = to and u = uo are periodic traction and displacement constraints imposed on S t and S u , respectively. The first variation of HD − P, and the fact that the interior solution satisfies the stress equilibrium equations a priori, yields the variational principle in the final form. Z Z     δui ti −toi dS þ δti ui −uoi dS ¼ 0: ð5Þ

2.2.1. Displacement field solution The solution for the fluctuating out-of-plane and in-plane displacements uz′ and ur′, uθ′, respectively, in the fiber, coating and matrix phases is obtained in the form

ST

u0z

Su

ðkÞ

∞ h    i X ðkÞ ðkÞ ðkÞ ðkÞ a ζ n Hn1 þ ζ −n Hn3 cosnθ þ ζ n H n2 þ ζ −n H n4 sinnθ

ðr; θÞ ¼

n¼1

The displacement and traction components on the six and four surfaces of hexagonal and rectangular unit cells, respectively are related through the periodicity conditions (Eq. (1)).

ðkÞ u0r ðr; θÞ ðkÞ u0θ ðr; θÞ

ðkÞ

ðkÞ

¼ F 01 aζ þ F 02 aζ −1 þ ∞ X 4 X

¼

aβnj ζ

pnj

h

∞ X 4 X

h i ðkÞ ðkÞ aζ pnj F nj cosnθ þ Gnj sinnθ

n¼2 j¼1 ðkÞ F nj

ðkÞ

sinnθ−Gnj cosnθ

i

n¼2 j¼1

ð9Þ

2.2. The interior problem The solution of the interior problem for unit cells with coated fibers follows the solution procedure described by [32]. Hence only the main results are stated here for transversely isotropic phases possessing plane of isotropy perpendicular to the reinforcement direction x1, with appropriate modifications that include the coating. The generalized Hooke's law for such materials takes the uncoupled form, 2

3 2 σ zz C 11 6 7 6 4 σ rr 5 ¼ 4 C 12 σ θθ 2

σ rθ

6 4 σ zr σ zθ

C 12 3

C 12 C 22

32 3 εzz C 12 76 7 C 23 54 εrr 5

C 23

C 22

εθθ

2

1 ðC 22 −C 23 Þ 2 7 6 6 5¼4 0 0

0 C 66 0

32 3 2εrθ 76 7 7 0 54 2εzr 5 2εzθ C 66

ð6Þ

where rigid body terms associated with n = 0 term in the expression for (k) (k) u′(k) z , and n =1 terms in the expressions for u′r , and u′θ have been ex(k) (k) (k) cluded by fixing the fiber at the origin. Hnj , Fnj , Gnj (j = 1, 2, 3, 4) are unknown coefficients, ζ = r/a is the nondimensionalized radial coordinate with respect to the fiber radius a, and the four eigenvalues pnj are pn1 = n + 1, pn2 = n − 1, pn3 = − (n + 1), and pn4 = − (n − 1) with the corresponding eigenvectors β(k) nj given by

ðkÞ βnj

0

   ðkÞ ðkÞ ðkÞ 1−p2nj þ μ T n2 kT þ μ T   : ¼ ðkÞ ðkÞ ðkÞ n kT pnj −kT −2μ T

To ensure that the displacements remain bounded in the fiber, we ) (f ) set H(f n3 =Hn4 = 0 for n ≥ 1 in the case of the out-of-plane displacement,

where the elements Cij may be expressed in terms of five engineering moduli: C11 = EA + 4kTν2A, C12 = 2kTνA, C22 = kT + μT, C23 = kT − μT and C66 = μA, where EA, μA and νA are the axial Young's and shear moduli and Poisson's ratio, and kT, and μT are the transverse plane strain bulk and shear moduli. Use of the Hooke's law and the strain–displacement relations

∂u0 ε zz ¼ εzz þ z ¼ εzz ∂z ∂u0 ε rr ¼ ε rr þ r ¼ εrr þ ε0rr ∂r u0 1 ∂u0θ ε θθ ¼ εθθ þ r þ ¼ εθθ þ ε0θθ r r ∂θ  1 1 ∂u0r ∂u0θ u0θ ¼ εrθ þ ε0rθ ε rθ ¼ εrθ þ þ − 2 r ∂θ r ∂r 1 ∂u0z ε rz ¼ ε rz þ ¼ εrz þ ε0rz 2 ∂r 0 1 ∂uz ε θz ¼ εθz þ ¼ εθz þ ε0θz 2r ∂θ

(f ) (f ) ) (f ) (f) =0 and Fn3 =Fn4 =0, G(f and F02 n3 = Gn4 = 0 for n ≥2 in the case of the (k) (k) in-plane displacements. The remaining coefficients H(k) nj , Fnj and Gnj in the coating and matrix phases (k = c, m) are subsequently obtained in (f) (f) terms of the fiber coefficients H(f) nj , Fnj and Gnj from the interfacial displacement and traction continuity conditions at the fiber/coating and coating/matrix interfaces. The continuity conditions at r = a are,

ðfÞ

ða; θÞ ¼ u0z ða; θÞ; u0r

ðfÞ

ðcÞ

u0z

ðcÞ

ðfÞ

ð cÞ

ða; θÞ ¼ u0r ða; θÞ; u0θ

ðfÞ

ð cÞ

ðfÞ

ðcÞ

ða; θÞ ¼ u0θ ða; θÞ

ðfÞ

ð cÞ

σ zr ða; θÞ ¼ σ zr ða; θÞ; σ rr ða; θÞ ¼ σ rr ða; θÞ σ rθ ða; θÞ ¼ σ rθ ða; θÞ ð10Þ and at r = b, ð7Þ ð cÞ

u0z ðb; θÞ ¼ u0z ð cÞ σ zr ðb; θÞ

¼

ðmÞ

ðcÞ

ðb; θÞ; u0r ðb; θÞ ¼ u0r

ðmÞ σ zr ðb; θÞ;

ð cÞ σ rr ðb; θÞ

¼

ðmÞ

ðcÞ

ða; θÞ; u0θ ðb; θÞ ¼ u0θ

ðmÞ σ rr ðb; θÞ;

ðcÞ σ rθ ðb; θÞ

¼

ðmÞ

ðb; θÞ

ðmÞ σ rθ ðb; θÞ

ð11Þ

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

where the traction components at the two interfaces are obtained from Hooke's law (Eq. (6)), and the strain–displacement relations (Eq. (7)), ðkÞ σ zr

¼

ðkÞ 2μ T ε zr

þ

ðkÞ μT

∞ X n¼1

  ðkÞ ðkÞ n ζ n−1 H n1 −ζ −n−1 Hn3 cosnθ

ð12Þ

#   n−1 ðkÞ −n−1 ðkÞ þ ζ H n2 þ ζ H n4 sinnθ

ðkÞ pnj −1

P nj ζ



 ðkÞ ðkÞ F nj cosnθ þ Gnj sinnθ

n¼2 j¼1 ðkÞ σ rθ

n¼2 j¼1

(k) (k) (k) (k) (k) (k) (k) and P(k) nj = (kT + μT )pnj + (kT − μT )(1 + nβnj ), Rnj = μT [(pnj − 1)β(k) − n]. The periodic boundary conditions imposed on the unit cell nj are given in terms of the macroscopic strains in Cartesian coordinates, and hence the macroscopic strains in cylindrical coordinates are transformed accordingly using strain transformation equations.

2.2.2. Interfacial continuity at r = a The axial shear problem is decoupled from the transverse normal and shear problems. Hence applying the two interfacial continuity conditions on the axial displacement and axial shear stress at r = a and using the orthogonality of cosnθ and sinnθ terms, we obtain the following relations between coating and fiber coefficients associated with different-order harmonic terms for n≥ 1, ð13Þ

(f) (f) T (c) (c) (c) (c) (c) T where H(f) n = [Hn1, Hn2] , Hn = [Hn1 , Hn2 , Hn3 , Hn4 ] , and the matrices c1, c2 are given in Appendix A. The Kronecker delta term δn1 is present because the average strains are introduced only through the n = 1 terms cosθ and sinθ. The transverse normal and shear problems in the r − θ plane are coupled. Hence applying the four interfacial continuity conditions, and using orthogonality of cosnθ and sinnθ terms, we obtain the following relations between coating and fiber coefficients, for n = 0.

ð cÞ

ðfÞ

F0 ¼ b0 F 01 þ c0 ε11 þ d0 ðε 22 þ ε33 Þ

ð14Þ

(c) (c) T where F(c) 0 = [F01 , F02 ] , and the matrices b, c and d are given in Appendix A. For n ≥2

AðncÞ FðncÞ ¼ Aðnf Þ Fðnf Þ þ δn2 A0 ðε22 −ε33 Þ AðncÞ GðncÞ ¼ Aðnf Þ Gðnf Þ þ δn2 A0 2ε23

ð15Þ

(c) (c) (c) (c) (T (f) (f) (f) T (c) (c) (c) (c) where F(c) n = [Fn1 , Fn2 , Fn3 , Fn4 ] , Fn = [Fn1, Fn2] , Gn = [Gn1 , Gn2 , Gn3 , (c) T (f) (f) (f) T (c) (f) Gn4 ] , Gn = [Gn1, Gn2] , and the matrices An , An , and A0 are given in Appendix A. The Kronecker delta term δn2 is present because the average strains are introduced only through the n = 2 terms cos2θ and sin2θ.

2.2.3. Interfacial continuity at r = b The application of interfacial continuity conditions at r = b is analogous to the preceding case and produces similar sets of equations that relate the unknown coefficients in the matrix phase to their counterparts in the coating. For the axial shear we have for n ≥1, HðnmÞ

¼

c1 HðncÞ

þ δn1 c2 ½2ε12 2ε13 

T

0

ð cÞ

ð17Þ

AðnmÞ FðnmÞ ¼ AðncÞ FðncÞ þ δn2 A0 ðε22 −ε 33 Þ AðnmÞ GðnmÞ ¼ AðncÞ GðncÞ þ δn2 A0 2ε23 :

ð18Þ

2.3. The exterior problem

∞ X 4   X ðkÞ ðkÞ ðkÞ ¼ 2μ T εrθ þ Rnj ζ pnj −1 F nj sinnθ−Gnj cosnθ

HðncÞ ¼ c1 Hðnf Þ þ δn1 c2 ½2ε12 2ε 13 T

ð cÞ

¼ b0 F 01 þ b 0 F 02 þ c0 ε11 þ d0 ðε22 þ ε33 Þ

F0

and for n ≥2

    ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ σ rr ¼ 2kT ν A ε zz þ kT þ μ T εrr þ kT −μ T εθθ þ 2kT F 01 −2μ T F 02 ζ −2 þ ∞ X 4 X

while for the inplane loading we have for n = 0 ðmÞ

"

517

ð16Þ

To complete the solution to the unit cell problem, the unknown coefficients Ffn, Gfn, and Hfn are determined by applying the variational principle (Eq. (5)), with the surface displacements and tractions on the opposite faces of the unit cell related through the periodic boundary conditions. Use of the two-scale displacement representation given by Eq. (2) in the periodic displacement boundary conditions (Eq. (1)), reduces these periodicity conditions to constraints on the fluctuating displacement components. Hence for the hexagonal unit cell with the six surfaces S1,…S6, periodic displacement and traction boundary conditions become, u0i ðS1 Þ ¼ u0i ðS4 Þ; u0i ðS2 Þ ¼ u0i ðS5 Þ; u0i ðS3 Þ ¼ u0i ðS6 Þ t i ðS1 Þ ¼ −t i ðS4 Þ; t i ðS2 Þ ¼ −t i ðS5 Þ; ui ðS3 Þ ¼ −t i ðS6 Þ

ð19Þ

Since the out-of-plane and inplane problems are uncoupled, the coefficients Ffn, Gfn are found independently of the coefficients Hfn upon utilizing the reduced periodicity conditions in the variational principle. Implementing the reduced periodicity conditions for the inplane problem in the first variation of the functional (Eq. (5)), we obtain 3 Z

X i¼1

Si

6 Z

X i¼4

  δt 2 ðSi Þ u02 ðSi Þ−u02 ðSiþ3 Þ þ δt 3 ðSi Þ u03 ðSi Þ−u03 ðSiþ3 Þ dSþ δu02 ðSi Þ½t 2 ðSi Þ þ t 2 ðSi−3 Þ þ δu03 ðSi Þ½t 3 ðSi Þ þ t 3 ðSi−3 Þ dS ¼ 0

ð20Þ

Si

from which the system of equations for the unknown coefficients Ffn, and Gfn is obtained in the form h ^ Ff A

Gf

iT

^ε ¼B in

ð21Þ

where εin ¼ ½ε11 ; ε22 ; ε33 ; 2ε23 T and Ff = [Ff1,…, FNmaxf], Gf = [Gf1,…, GNmaxf]. Similarly, for the out-of-plane loading, the first variation of the functional becomes 3 Z X i¼1

6 Z X  δt 1 ðSi Þ u01 ðSi Þ−u01 ðSiþ3 Þ dSþ δu01 ðSi Þ½t 1 ðSi Þ þ t 1 ðSi−3 ÞdS ¼ 0

Si

i¼4

Si

ð22Þ from which the system of equations for the unknown coefficients Hfn is obtained in the form ~ f ¼ Bε ~ out AH

ð23Þ

where εout ¼ ½2ε 12 ; 2ε13 T and Hf = [Hf1,…, HNmaxf]. The elements of the ~ and B; B ~ are obtained in terms of surface integrals along matrices A; A

518

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

Table 1 Elastic fiber and matrix properties employed in the calculations. Note that the AS4 graphite fiber is transversely isotropic with E22 =E33, ν12 =ν13 and G23 = E22/2(1+ν23) and the remaining constituents isotropic. Material

EA (GPa)

ET (GPa)

GA (GPa)

GT (GPa)

νA

Graphite fiber 3501-6 epoxy Interface

214 3.5 5.25

14 3.5 5.25

7 1.3 2.059

5.83 1.3 2.059

0.25 0.35 0.275

the six sides S1,…, S6 of the unit cell. Similar results are obtained for rectangular or square unit cells with four exterior surfaces. To implement periodicity conditions in the variational principle, the fluctuating displacements in cylindrical coordinates are transformed to Cartesian coordinates using the displacement transformation equations. Similarly the fluctuating Cartesian strains, used in the calculation of stresses and tractions through Hooke's law and Cauchy's relations, respectively, are obtained from the strain transformation equations, with the cylindrical fluctuating strains determined from the strain–displacement relations (Eq. (7)). The expressions for the axial shear stress-

es and transverse normal and shear stresses in the Cartesian coordinate system along the unit cell's boundary obtained from Hooke's law are,     ðmÞ ðmÞ 0 ðmÞ 0 ðmÞ σ 12 ¼ 2μ m σ 13 ¼ 2μ m A ε 12 þ ε 12 ; A ε 13 þ ε 13     m   ðmÞ ðmÞ ðmÞ m m m σ 22 ¼ 2kT νm ε22 þ ε0 22 þ kT −μ m ε þ ε0 33 A ε 11 þ kT þ μ T T     m   ðmÞ ðmÞ ðmÞ m m σ 33 ¼ 2kT νm ε11 þ kT −μ m ε22 þ ε0 22 þ kT þ μ m ε33 þ ε0 33 T A  T ðmÞ 0 ðmÞ σ 23 ¼ 2μ m T ε 23 þ ε 23 : ð24Þ

2.4. Homogenized constitutive equations The homogenized Hooke's law is obtained by averaging local constitutive equations in each phase, σ¼

1X k V

Z

CðkÞ εðkÞ dV k ¼

X

υ CðkÞ εðkÞ k ðkÞ

ð25Þ

Fig. 2. Comparison of stress distributions in a hexagonal unit cell with a dilute fiber volume fraction as a function of the number of harmonics with the Eshelby solution, demonstrating the locally-exact solution's stability with increasing harmonic number for Ec/Em =103 and b/a=1.1.

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

where the phase volume fractions obey the relationship ∑ υðkÞ ¼ 1. k

With the knowledge of the fiber phase coefficients Ffn, Gfn and Hfn, the remaining coating and matrix phase coefficients may be calculated using Eqs. (13)–(15) and (16)–(18). Hence the average strains εðkÞ ðk ¼ f ; c; m Þ in the individual phases may be related to average macroscopic strains through the localization relations εðkÞ ¼ AðkÞ ε

ð26Þ

where A(k) are the Hill's elastic strain concentration matrices for the three phases [14]. The homogenized relationship between stress and strain averages then becomes σ¼

X

υ CðkÞ AðkÞ ε k ðkÞ

¼ C ε

ð27Þ

where C ¼ ∑ υðkÞ CðkÞ AðkÞ. In light of the phase volume fraction relationk

ship above, the homogenized stiffness matrix for the unit cell may be written,     C ¼ CðmÞ þ υð f Þ Cð f Þ −CðmÞ Að f Þ þ υðcÞ CðcÞ −CðmÞ AðcÞ :

ð28Þ

To determine the fiber and coating strain concentration matrices, the average fiber and coating strains are obtained in closed form upon integrating the local expressions over the respective phase cross sections. The resulting expressions contain only the applied average strains and the displacement coefficients associated with the n = 1 harmonic in the case of axial shear strains, and the n = 0, 2 harmonics in the case of transverse normal and shear strains, 1 ðkÞ ðkÞ ε 12 ¼ ε12 þ H 11 ; 2 ðkÞ

ε 22

1 ðkÞ ðkÞ ε13 ¼ ε 13 þ H 12 2 ðkÞ ðkÞ ðkÞ k k  3kT  F 21 þ F 22 ¼ ε22 þ F 01 þ γ ðkÞ ðkÞ 2 kT −μ T ðkÞ

ð29Þ

3kT ðkÞ ðkÞ kÞ ðkÞ  F ð21 ε 33 ¼ ε33 þ F 01 −γðkÞ  −F 22 ðkÞ ðkÞ 2 kT −μ T ðkÞ

3kT ðkÞ kÞ ðkÞ  Gð21 ε 23 ¼ ε23 þ γ ðkÞ  þ G22 ðkÞ ðkÞ 2 kT −μ T

519

the homogenized moduli and local stress fields as a function of the number of harmonic terms for non-dilute fiber volume fractions with different coating thicknesses. Finally, we compare our solution's predictions with those reported in the literature based on finite-element and elasticity-based calculations. 3.1. Stability of the solution To demonstrate the solution's stability and reduction to known solutions, we subject a hexagonal unit cell with the fiber volume fraction of 0.2% to uniaxial loading by σ 22 ¼ 100 MPa with the remaining stresses zero. The material system is graphite/epoxy considered by [15] with the fiber and matrix elastic moduli given in Table 1. The solutions are then generated using an increasing number of harmonics from n =2, which corresponds to the Eshelby solution as the fiber volume fraction tends to zero, to n = 16. The inplane stresses σ22(y2, y3), σ33(y2, y3) and σ23(y2, y3) are illustrated in Fig. 2 for n = 2 and 16 harmonics and compared with the exact Eshelby solution for the coating that is thousand times stiffer than the matrix. The stress distributions have been plotted in the square region in the vicinity of the fiber with the hexagonal boundaries far outside this region. The locally-exact homogenization theory solution is seen to remain very stable regardless of the number of harmonics employed (at least up to sixteen). Comparison with the Eshelby solution is very favorable. Moreover, examination of the leading coefficients Ff01, and Ff22 as a function of harmonics vis-a-vis those of the Eshelby solution indicates insignificant differences, with the higherorder coefficients practically zero. The above results have been plotted using a color map that highlights the fiber and matrix stresses at the expense of very large coating stresses owing to the large coating stiffness. Hence these stresses are compared separately with the Eshelby solution along radial paths with the largest stress gradients in Fig. 3 for the n = 16 harmonic case, illustrating the locally-exact solution's ability to accurately capture stress fields in thin interfacial layers. Setting the coating Young's modulus to a very small value prevents stress transfer into the fiber, thereby mimicking a dilute hexagonal array of non-interacting holes equivalent to an infinite plate with a hole (Fig. 4). For this problem, the maximum σ22(y2, y3) stress occurs at the top and bottom of the hole whose magnitude is three time the far-field stress, or 300 MPa. Similarly, the minimum σ33(y2, y3) stress

ðkÞ

where k = f, c, and γ(f) = 1, γ(c) = 1+ (b/a)2. Moreover, ε11 ¼ ε11 due to uniaxial reinforcement. The columns of the matrices A(f) and A(c) are generated by solving the unit cell problem for one non-zero average strain applied at a time, with the remaining average strains kept zero. The solution pro(k) (k) duces the unknown coefficients F(k) n , Gn or Hn for the applied loading, and thus the average fiber and coating strains. The elements of the strain concentration matrices occupying the column that corresponds to the applied non-zero average strain are then obtained by taking the ratios of the averaged strain in the fiber and coating phases and the average applied strain. 3. Validation We validate the developed solution by first showing that in the limit as the fiber volume fraction becomes very small, the results of [10,16] are recovered regardless of the number of terms employed in the displacement field representation in the fiber, coating and matrix phases. This also demonstrates the solution's stability with increasing number of harmonics. In the first case, we take the coating Young's modulus to be thousand times stiffer than that of the matrix, while in the second case we take it thousand times softer. In both cases the ratio of the coating to fiber outer radii is b/a = 1.1. Then, we study the convergence of

Fig. 3. Comparison of converged stress distributions (N = 16) along radial paths with largest gradients in the stiff coating of a hexagonal unit cell with a dilute fiber volume fraction with the Eshelby solution for Ec/Em =103 and b/a=1.1.

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Fig. 4. Comparison of stress distributions in a hexagonal unit cell with a dilute fiber volume fraction as a function of the number of harmonics with the Kirsch solution, demonstrating the locally-exact solution's stability with increasing harmonic number for Ec/Em =10−3 and b/a=1.1.

which is compressive occurs along the load axis at the hole boundary whose magnitude is equal to the far-field stress. These stresses are captured very well by the locally-exact theory regardless of the number of harmonics used in the displacement field representation.

3.2. Convergence study We consider hexagonal and square unit cells representative of graphite/epoxy composites with non-dilute reinforcement, and investigate convergence of the homogenized moduli and local stress fields with the number of harmonics in the displacement field representation for three interfacial layer thicknesses. The elastic moduli of the fiber and matrix phases are the same as in the preceding stability study listed in Table 1 which includes the coating Young's modulus 1.5 times that of the epoxy matrix. The non-dilute fiber volume fraction is 0.50 and the coating thicknesses yield coating/fiber radius ratios b/a of 1.01, 1.05 and 1.10, with the thinnest coating providing a demanding test of the theory's computational capability.

The homogenized engineering moduli were calculated using the homogenized compliance matrix obtained from the inverse of the homogenized stiffness matrix established in Eq. (28), S⁎ = [C⁎]−1, as described in [32]. Fig. 5 illustrates convergence behavior of the homogenized transverse Young's and shear moduli ET⁎ and GT⁎, respectively, and axial shear modulus GA⁎ for hexagonal and square unit cell architectures. Similar behavior (not shown) is observed for the homogenized Poisson's ratios νA⁎ and νT⁎. The moduli have not been normalized to highlight the effect of the coating thickness whose increase yields stiffer response given its greater Young's modulus relative to that of the matrix. The results indicate that at the considered fiber volume fraction, generally quicker convergence of the homogenized moduli is observed for the hexagonal array, with as few as 5 harmonics yielding converged transverse and axial shear moduli, and 10 yielding converged transverse Young's modulus. Included in the figure are the corresponding results in the absence of coating (b/a = 1.0), illustrating that the convergence behavior is not altered by the coating presence in the considered thickness range. Less than 1.5 s was needed to generate the full set of homogenized moduli with coated fibers as compared to less than 0.5 s without

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

521

Fig. 5. Convergence of selected homogenized moduli with the number of harmonics for a graphite/epoxy composite with the fiber volume fraction 0.50 and coating thickness as a parameter for Ec/Em =1.5.

the coating using 10 harmonics and a PC platform running Windows 7 Ultimate 64-bit operating system with 16 GB RAM and Intel(R) Core(TM) i5-3320M CPU @ 2.6 GHz. Fig. 6 compares the converged stress fields σ22(y2, y3), σ23(y2, y3) and σ33(y2, y3) for hexagonal arrays of uncoated and coated fibers with the smallest and largest coating thickness. Similar stress fields (not shown) are obtained for the square array, with the higher transverse stress σ22(y2, y3) in the fiber yielding higher transverse Young's modulus than the hexagonal array for all coating thicknesses (see Fig. 5). These stress fields were generated using 12 harmonics under unidirectional loading by the homogenized stress σ 22 ≠0 at the applied strain of ε22 ¼ 0:01. Unidirectional loading was achieved by adjusting the homogenized strains in Eq. (27) to obtain σ 22 as the only nonzero homogenized stress. When the coating stiffness is higher than that of the matrix, increasing coating thickness produces greater stress transfer into the fiber for both hexagonal and square geometries for this loading, yielding increasingly greater moduli seen in the

convergence study of Fig. 5. Comparison of the stress distributions in both arrays without and with the thinnest coating for the given coating/matrix modulus contrast indicates very little difference, suggesting little effect on the homogenized moduli as observed in Fig. 5 and further illustrated in Section 4. The locally-exact elasticity solution is sensitive enough to accurately capture the small differences in stress distributions in the presence of very thin coatings, highlighting the method's ruggedness. The rapid convergence of both the homogenized moduli and local stress fields sets our method apart from other elasticitybased solutions such as the eigenstrain expansion approach, cf. [3], which require substantially greater number of harmonics for converged stress fields. 3.3. Comparison with published results The results of an extensive investigation of the impact of interphases or coatings on homogenized moduli of unidirectionally-reinforced

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Fig. 6. Converged stress distributions in a hexagonal unit cell of a graphite/epoxy composite with the fiber volume fraction 0.50 and two coating thicknesses with modulus contrast Ec/Em = 1.5 subjected to the unidirectional loading σ 22 ≠0 at the applied strain ε 22 ¼ 0:01.

composites have been reported by [17]. The authors employed the finite-element method to generate the full set of homogenized moduli of transversely isotropic composites based on a hexagonal array of coated fibers with different coating moduli. The numerical solution was implemented following elements of the 0th-order homogenization theory, including displacement decomposition into average and fluctuating components within generalized plane strain framework, and periodic boundary conditions applied on the faces of a hexagonal unit cell representative of the transversely isotropic composite. Hence the numerical results of [17] may be compared directly with the present results based on the same geometry and solution methodology applied within elasticity, rather than variational, framework. The results used in the comparison were generated for a unidirectional composite with the fiber volume fraction of 0.50 comprised of isotropic fibers embedded in an isotropic matrix with the elastic moduli: Ef = 84 GPa, νf =0.22 and Em =4 GPa, νm =0.34. These moduli are representative of a glass/epoxy composite. Four values of the interphase Young's modulus were used, namely 4, 6, 8 and 12 GPa with the Poisson's ratio

fixed at νc =0.34. The fiber radius was 8.5 μm and the coating thickness 1.0 μm with the unit cell dimensions adjusted accordingly. Table 2 presents comparison of the finite-element calculations and the present theory for three of the five homogenized moduli normalized by the corresponding matrix moduli, ET⁎/Em GT⁎/Gm and GA⁎/Gm. These moduli are more sensitive to changes in the coating's stiffness than the axial Young's modulus EA⁎ and axial and transverse Poisson's ratios Table 2 Comparison of selected homogenized moduli predicted by the locally-exact homogenization theory with the PMH model of [17] for coated fiber unidirectional composite with different coating moduli. Coat Ec

E⁎T/Em

G⁎T/Gm

GA⁎/Gm

(GPa)

PMH

Present

PMH

Present

PMH

Present

4 6 8 12

2.6887 2.9112 3.0425 3.1916

2.6636 2.9255 3.0841 3.2677

2.5495 2.7751 2.9108 3.0667

2.5195 2.7840 2.9474 3.1393

2.7126 2.9367 3.0654 3.2083

2.6362 2.9618 3.1181 3.248

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528 Table 3 Comparison of the normalized homogenized transverse shear modulus G⁎T/Gm predicted by the locally-exact homogenization theory with the three-phase square array calculations of [25] for unidirectional composite with a very thin fiber coating. vf

LEHT Three-phase model

10

20

30

40

50

60

70

75

1.139 1.142

1.287 1.293

1.455 1.461

1.662 1.671

1.939 1.948

2.353 2.360

3.109 3.129

3.852 3.817

νA⁎ and νT⁎. Overall, the results in Table 2 are seen to agree to one significant digit, whereas the axial and transverse Poisson's ratios differ only in the third decimal place. We note that when Ec = 4 GPa, that is when the matrix and coating moduli are the same, we recover the results for the uncoated fiber composite. For this case, the moduli produced by the present method have been shown by [32] to be accurate to 4 significant digits at the fiber volume fraction of 0.50 upon comparison with the results of [13], often employed as a gold standard, which were

523

generated using an integral equation method. In the present case, the discrepancy is likely due to a relatively coarse discretization of both the matrix and the coating, with the coating discretized into a doublelayer triangular element mesh as a compromise between efficiency and accuracy. The recent finite-element results of [27] suggest that a larger number of elements is needed for fully converged results. Our results for the transverse Young's modulus ET⁎ also compare favorably with those of [22] (reported in Table 1 of this reference in a different form) which were generated using the authors' elasticity-based equivalent inhomogeneity method under plain strain constraint. We also compare our results with those of [25] based on the complex potential representation of displacement and stress fields in an infinite series form for a square array of isotropic fibers coated by a very thin interphase layer such that b −a =0.001. The fiber, coating and matrix moduli for this system are: Ef = 24 GPa, νf = 0.20; Ec = 3.03 GPa, νc = 0.50; and Em = 2.7 GPa, νc = 0.35. Table 3 presents comparison of the normalized homogenized transverse shear modulus GT⁎/Gm in a wide fiber volume range predicted by the two analytical methods,

Fig. 7. Homogenized moduli as a function of fiber volume fraction for a graphite/epoxy composite with hexagonal and square architectures and modulus contrast Ec/Em = 1.5, demonstrating the effect of coating thickness.

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showing very good agreement to within graphical resolution accuracy of the Sevostianov et al. data for this computationally demanding case. 4. Numerical results Having validated the locally-exact homogenization theory, we first conduct a parametric study to demonstrate the effects of coating thickness and stiffness on the homogenized moduli of unidirectional composites in a wide fiber volume fraction range. The previously considered graphite/epoxy composite represented by both hexagonal and square arrays is employed towards this end. Then we investigate the efficiency of reinforcing epoxy by alumina nanotubes with different wall thickness in a hexagonal array under development for a wide range of applications. 4.1. A parametric study with coated fibers We illustrate the effects of coating's thickness and stiffness for three of the five homogenized moduli which exhibit the greatest sensitivity to these parameter variations, namely ET⁎, νT⁎ and GA⁎, calculated as a function of the fiber volume fraction for the graphite/epoxy material system. Fig. 7 illustrates the effect of coating thickness represented by the ratios b/a =1.0, 1.01, and 1.10, see Fig. 1, on the three homogenized moduli for the coating/epoxy modulus contrast Ec/Em = 1.5. The results were generated in the fiber volume interval [0.05, 0.75] using increments of 0.05, normalized by the corresponding matrix moduli in the case of the transverse Young's and axial shear moduli. The fiber volume fraction was incremented by changing the unit cell dimensions in order to keep

the coating thickness fixed. At this modulus contrast, measurable increases in the homogenized transverse Young's and axial shear moduli are observed for the largest ratio b/a = 1.1, with a decrease observed in the transverse Poisson's ratio. The effect of coating increases with increasing fiber volume fraction, and appears to be comparable for both arrays, accounting for the greater Young's and axial shear moduli exhibited by the square array at each fiber volume fraction. Within each array type, the coating tends to enhance the axial shear modulus GA⁎ to a somewhat greater extent than the transverse Young's modulus ET⁎ for the considered coating/epoxy modulus contrast, suggesting more efficient stress transfer mechanism into the coated fiber discussed in the sequel. Next, we choose coating thickness with the largest ratio b/a = 1.10 and investigate the effect of both fiber volume fraction and coating stiffness on the homogenized transverse Young's and axial shear moduli. The results are presented in Fig. 8 as three-dimensional carpet plots, highlighting the efficiency of generating homogenized moduli with our elasticity-based approach. These results have been generated for hexagonal arrays and normalized by the respective moduli of the matrix phase. Consistent with expectation, the effect of increasing coating's Young's modulus on the homogenized moduli increases with increasing fiber volume fraction as the coating volume fraction also increases even if the coating thickness remains fixed. Included in the figure are twodimensional projections of the carpet plots in a small range of Ec/Em ratios, namely [0.0, 0.275], wherein rapid changes in the homogenized moduli values are observed in the carpet plots. These projections illustrate the minimum values of the coating Young's modulus as a function of the fiber volume fraction which yield homogenized moduli greater

Fig. 8. Homogenized moduli as a function of fiber volume fraction and coating stiffness for a graphite/epoxy composite with hexagonal architecture and coating/fiber radius ratio b/a=1.1.

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

than the matrix modulus. This information may be useful in the design of engineered coatings. As observed in Figs. 7 and 8, the loading direction influences the effectiveness of fiber reinforcement vis-a-vis respective homogenized moduli. For both arrays, the axial shear modulus GA⁎ increases faster with fiber volume fraction relative to the matrix than the transverse Young's modulus ET⁎, with the concomitant effectiveness increase of the coating. Fig. 9 illustrates the effectiveness of stress transfer from the matrix into the fiber under uniaxial loading by transverse tension and axial shear in the presence of coating with the Ec/Em =4.0 ratio relative to the uncoated fiber when the fiber volume fraction is 0.6. The stress fields were calculated for the respective macroscopic strains of 0.01 or 1% under corresponding unidirectional loading. Under transverse tension, the coating experiences substantially larger normal stress σ22(y2, y3) than the fiber in the angular sectors away from the applied load, producing a uniform stress field throughout most of the fiber which is greater relative to the uncoated fiber. In contrast, under axial shear loading, the coating experiences smaller axial shear stress σ22(y2, y3) than the fiber. Nonetheless, the axial shear stress in the fiber is substantially enhanced by the coating's presence relative to the uncoated fiber, producing a concomitant increase in the homogenized

525

axial shear modulus GA⁎. Overall, for the chosen parameters, the coating tends to promote greater stress transfer into the fiber under axial shear loading relative to transverse normal loading.

4.2. Efficiency of nanotube reinforcement The developed solution may be specialized to composites reinforced by hollow tubes by treating the coating as a hollow fiber upon setting the elastic moduli of the solid core to very small values. This also demonstrates the method's ruggedness. Hollow fiber reinforcement of traditional composites has not been attempted on a large scale due to fabrication difficulties. Fabrication techniques developed during the past decade for nanotechnology applications, however, make possible reinforcement of different types of matrix materials by inorganic nanotubes with precisely-controlled diameters, wall thicknesses and placement, such as hexagonally-arrayed alumina nanotubes, that have potential applications in microelectronics, nanofluidics, drug delivery and optical devices, amongst others. In light of the emerging applications of these nanostructures, little data is available on their homogenized properties.

Fig. 9. Stress distributions in hexagonal unit cells of a graphite/epoxy composite with the fiber volume fraction 0.60 subjected to uniaxial loadings σ 22 ≠0 and σ 12 ≠0 at the applied strains ε22 ¼ 0:01 and ε 12 ¼ 0:01, respectively: (left) uncoated fibers; (right) coated fibers with Ec/Em =1.5 and b/a=1.1.

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Fig. 10. Homogenized moduli of an alumina nanotube-reinforced epoxy matrix as a function of the nanotube's apparent volume fraction based on the outer radius of 70 nm for different tube thicknesses.

Herein, we illustrate the extended theory's applicability by calculating homogenized moduli of an epoxy resin reinforced by atomic layer deposited (ALD) alumina nanotubes arranged in a hexagonal array. In particular, we consider alumina nanotubes 70 nm in diameter with wall thicknesses ranging from 6 nm to 15 nm, fabricated successfully by [30], and calculate homogenized moduli as a function of effective nanotube volume fraction. Only mechanical reinforcement effects are considered given that the present extension does not include surfaceenergy effects that are important at very small scales. These will be incorporated in our future work. We note, however, that [9] did not find significant effect of surface energy on homogenized plane strain and axial and transverse shear moduli of nano-porous aluminum with cylindrical porosities having radii greater than 10 nm using the CCA and GSC models. Hence continuum-level calculations remain valid for the considered size ranges, with interaction effects from neighboring nanotubes afforded by our locally-exact homogenization. The elastic moduli of the alumina nanotubes and epoxy matrix used in the present calculations were: EAl2O3 =166 GPa, νAl2O3 = 0.20 and Em = 4 GPa, νm = 0.34, with the former taken from [20]. The Young's modulus and Poisson's ratio of the solid core were Ecore = 10−3 GPa and νcore = 0.34.

These properties effectively produced a porosity in the region occupied by the solid core which plays the role of the fiber in the analytical solution. Fig. 10 presents homogenized transverse Young's and shear moduli ET⁎, and GT⁎ normalized by the respective moduli of the epoxy matrix for alumina nanotubes with three wall thicknesses as a function of the volume fraction of an equivalent solid nanocylinder of the same radius as the nanotube. The chosen mode of data display was motivated by the actual nanotube dimensions that had been successfully realized using the ALD method. The results may also be displayed as a function of the porosity volume fraction. As expected, the homogenized moduli increase with increasing alumina nanotube wall thickness for a fixed volume fraction of an equivalent solid nanocylinder. What is less expected, however, are the very small changes in the homogenized moduli over a large nanotube volume fraction relative to the epoxy matrix modulus when the nanotube wall thickness is 6 nm. In fact, while the transverse Young's modulus increases slightly, the transverse shear modulus initially decreases. In both instances the ratios ET⁎/Em and GT⁎/ Gm remain in the vicinity of unity over a large volume fraction range. This suggests that materials with enhanced functionality due to the

Fig. 11. Normal stress fields σ22(y2, y3) in the epoxy matrix (top) and alumina nanotubes of different thickness (bottom) under uniaxial loading by at the applied strain ε 22 ¼ 0:001.

G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

527

Fig. 12. Transverse shear stress fields σ23(y2, y3) in the epoxy matrix (top) and alumina nanotubes of different thickness (bottom) under uniaxial loading by at the applied strain ε23 ¼ 0 :001.

presence of thin-walled tubes of varying dimensions may be designed without altering the elastic moduli of the base material. Figs. 11 and 12 illustrate the normal and transverse shear stress fields that arise in the nanotube reinforcement and epoxy matrix under respective uniaxial loadings in the case of nanotubes with the smallest and largest wall thickness at the apparent volume fraction of 0.60. While the maximum normal and transverse shear stresses are comparable in the thick and thin walled nanotubes, the wall thickness dramatically alters the epoxy matrix stress fields, thereby producing large differences in the homogenized moduli seen in Fig. 10.

This feature makes the theory a robust and reliable tool which may be used as a standard for comparison with other approaches. Acknowledgments The lead author gratefully acknowledges the financial support of the Civil and Environmental Engineering Department at the University of Virginia through a teaching assistanship. Appendix A A.1. Interfacial continuity at r = a

5. Conclusions For axial shear loading, n≥ 1 contributions are, The extension of the locally-exact homogenization theory to accommodate fiber coatings in periodic unidirectional composites with hexagonal and square microstructures enables rapid identification of the effects of coating dimensions and elastic properties on homogenized moduli and local stress fields. By setting the elastic moduli of the reinforcement phase to very small values in the input file, the theory readily accommodates analysis of materials reinforced by thin-walled tubes, including nanotubes of dimensions at which surface effects are negligible. The generated results for alumina nanotube-reinforced epoxy illustrate the possibility of developing multifunctional materials which retain the same elastic moduli as the matrix material. Because of the simple input data construction arising from the computationally stable analytical framework, the theory may be used efficiently by specialists and non-specialists alike to rapidly calculate both homogenized moduli and local stress fields in a wide fiber volume fraction range regardless of coating thickness and fiber/coating/matrix modulus contrast. The extended theory's success is rooted in the variational principle proposed by [7] which facilitates the demonstrated rapid convergence of the unknown Fourier coefficients in the unit cell displacement field representation upon application of periodic boundary conditions even when fibers with very thin coatings are considered.

2 3 3 0 −1 0

ð f Þ

6 7 2ε12 H c1 7 0 −1 7 7 n1 þ δn1 c2 6 4 5 5 1 0 0 Hn2 2ε13 0 1 c2

2

3ðcÞ 2 H n1 c1 6 H n2 7 6 6 7 ¼ 60 4 H n3 5 4 c2 0 H n4

(f) (c) where c1 = (μ(c) A + μA )/2μA and c2 = 1 − c1. For transverse normal and shear loading, n = 0 contributions are,



F 01 F 02

ðcÞ

¼





c b01 ð f Þ d F 01 þ 01 ε11 þ 01 ðε22 þ ε33 Þ c02 b02 d02

where

b01 ¼ b02 ¼

ðfÞ ðfÞ

ðfÞ

ð cÞ

ð cÞ

; c01 ¼ ð cÞ

kT þ μ T kT þ μ T ðcÞ

ðfÞ

ð cÞ

ð cÞ

kT −kT

kT þ μ T

;

ð cÞ ð cÞ

kT νA −kT ν A 1   ; d01 ¼ − b02 ð cÞ ð cÞ 2 2 kT þ μ T

c02 ¼ −c01 ;

d02 ¼ −d01

:

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G. Wang, M.-J. Pindera / Materials and Design 93 (2016) 514–528

n ≥2 contributions are, 2

AðncÞ

1 6 βn1 ¼6 4 P n1 Rn1

1 βn2 P n2 Rn2

1 βn3 P n3 Rn3

3ðcÞ 1 βn4 7 7 ; P n4 5 Rn4

2

3ð f Þ 1 βn2 7 7 P n2 5 Rn2

1 6 βn1 ¼6 4 P n1 Rn1

Aðnf Þ

2

3 0 0  7 6 6 7 A0 ¼ 6 μ ðTf Þ −μ ðTcÞ 7: 4  5 ðfÞ ð cÞ − μ T −μ T

A.2. Interfacial continuity at r = b For axial shear loading, n ≥1 contributions are, 3ðmÞ 2 −2n 0 ðb=aÞ c2 c1 Hn1 6 6 Hn2 7 0 0 c1 7 ¼6 6 6 4 Hn3 5 4 ðb=aÞ2n c2 0 c1 2n Hn4 ðb=aÞ c2 0 0 3 2 −1 0

60 −1 7 7 2ε12 þδn1 c2 6 2 5 2ε13 4 ðb=aÞ 0 2 0 ðb=aÞ 2

32 3 ðc Þ 0 Hn1 76 −2n ðb=aÞ c2 76 Hn2 7 7 74 5 Hn3 5 0 Hn4 c1

(c) (m) where c1 = (μ(m) and c2 = 1 − c1. A + μA )/2μA For transverse normal and shear loading, n = 0 contributions are,



F 01 F 02

ðmÞ

¼

0



c b01 ðcÞ d b ðcÞ F 01 þ 01 F 02 þ 01 ε11 þ 01 ðε22 þ ε 33 Þ 0 c02 b02 d02 b02

where b01 ¼

ð cÞ

ðmÞ

ðmÞ

; ðmÞ

kT þ μ T

þ μT

kT

ðcÞ ðcÞ

c01 ¼

ðmÞ ðmÞ

ðmÞ

ðmÞ

ðmÞ

kT

þ μT

ð Þ

ð Þ

ðb=aÞ

2

c m kT ν A −kT ν A k −kT   ; d01 ¼  T  ðmÞ ðmÞ ðmÞ ðmÞ kT þ μ T 2 kT þ μ T ðmÞ

b02 ¼

ð cÞ

μ T −μ T

0

b01 ¼ −

ðcÞ

kT −kT ðmÞ

kT

2

ð cÞ

0

ðb=aÞ ; b02 ¼ ðmÞ

þ μT

2

ðmÞ

μ T þ kT ðmÞ

kT

ðmÞ

þ μT

2

c02 ¼ −c01 ðb=aÞ ;

d02 ¼ −d01 ðb=aÞ

and n ≥ 2 contributions for k =c, m are, 2

AðnkÞ

p

ðb=aÞ n1 6 β ðb=aÞpn1 6 ¼ 4 n1 p P n1 ðb=aÞ n1−1

pn1−1

Rn1 ðb=aÞ

2

p

ðb=aÞ n2 βn2 b=aÞpn2 p P n2 ðb=aÞ n2−1 p Rn2 ðb=aÞ n2−1 3

p

ðb=aÞ n3 p βn3 ðb=aÞ n3 p P n3 ðb=aÞ n3−1 p Rn3 ðb=aÞ n3−1

3ðkÞ p ðb=aÞ n4 pn4 7 βn4 ðb=aÞ 7 ; p P n4 ðb=aÞ n4−1 5 p Rn4 ðb=aÞ n4−1

0 0  7 6 6 ð cÞ 7 A0 ¼ 6 μ T −μ ðTmÞ 7: 4  5 ð cÞ ðmÞ − μ T −μ T

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