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Mar 15, 2007 - M. Y. Zang (B). College of Automotive Engineering,. South China University of Technology,. Guangzhou 510640, People's Republic of China.
Comput Mech (2007) 41:73–83 DOI 10.1007/s00466-007-0170-1

O R I G I NA L PA P E R

Investigation of impact fracture behavior of automobile laminated glass by 3D discrete element method M. Y. Zang · Z. Lei · S. F. Wang

Received: 11 October 2006 / Accepted: 9 February 2007 / Published online: 15 March 2007 © Springer Verlag 2007

Abstract A three-dimensional discrete element model of laminated glass plane is presented and a 3D numerical analysis code, which can simulate the impact fracture behavior of automobile laminated glass, is developed. The impact process of a single glass plane and a laminated glass plane are calculated in the elastic range by the code. Comparing its results with those calculated by the commercial FEM code LS-DYNA in the same condition, the validity of the 3D laminated glass model and the 3D discrete element method are proved. Furthermore, the impact fracture process of a single glass plane and a laminated glass plane are simulated respectively. The entire failure processes in detail are presented. By comparing the impact force and reduction of kinetic energy of impact body between those two models, the numerical method is applied to demonstrate the advantage of laminated glass in passenger’s safety.

M. Y. Zang (B) College of Automotive Engineering, South China University of Technology, Guangzhou 510640, People’s Republic of China e-mail: [email protected] Z. Lei State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, People’s Republic of China e-mail: [email protected] S. F. Wang College of Chemical and Energy Engineering, South China University of Technology, Guangzhou 510640, People’s Republic of China e-mail: [email protected]

Keywords Computational mechanics · 3D discrete element method · Laminated glass · Impact fracture behavior · Failure process 1 Introduction Automobile laminated glass, which is pressed from two pieces of glass plate and one piece of polyvinyl butyral (PVB) film under high temperature, is one of important parts of automobile. It is shown that the mechanical property of automobile laminated glass is very important for passenger’s safety from many traffic accidents. Researchers did many experimental studies for the mechanical property of automobile glass to produce more safety glass. However, because the complicacy of the structure of laminated glass, any variation in the manufacturing process and materials can induce the significant effects of the safety of the laminated glass [3], specially, safety factors related to glass failure during impact processes can neither be gained nor be trusted from only a limited number of experimental investigations [2]. Recently, many scholars have tried to solve these problems by using numerical methods [5, 8, 14]. Contrary to the experimental result, numerical method allows easy and comprehensive studying of all mechanical parameters. More important, some casual and interferential factors can be eliminated in numerical method, and simple design changes can easily be accomplished with another test when a suitable, validated numerical model is available [2]. Due to the essence of the dynamic damage and failure processes of laminated glass are the materials transferring from continuum to non-continuum, the discrete element method (DEM), which based on non-continuum mechanics, is considered one of the effective numerical methods.

74

The DEM is a numerical method which proposed firstly by Cundall et al. [4] at the beginning of 1970s, it was used to solve the non-continuous medium problems, such as geology engineering problems, as an initial stage, then the application area of DEM is expanding. After nearly more than 30 year’s development, there are many successful examples [6, 12, 13] in the field of continuum mechanics. Contray to FEM, the DEM, in which the nodal points are the centroid of elements, can simplify simulate the transition process from continuum to noncontinuum by changing the joint types between elements, without the need for specialized elements or remeshing lattices as well as ensuring mass conservation. In the middle of last century, Oda, Zang et al. used DEM to simulate the impact problems of laminated glass for the first time [10, 11, 15]. They established a 2D discrete element model of laminated glass beam and got many useful research findings. But as we know, the capability of 2D model is limited; there are 3D problems for most real conditions. In this paper, a 3D discrete element model of laminated glass plane is presented based on Oda and Zang’s 2D model. In this paper, the 3D DEM is used to study the impact fracture problem of laminated glass. Firstly, a 3D discrete element model of laminated glass plane is established. The glass and the PVB of laminated glass plane are discretized to the assembling of uniform rigid spherical elements. The joint types between elements can be classified into two kinds: contact models and connective models. There are two different kinds of theories to calculate forces between elements. The Mohr–Coulomb failure criterion is used to judge the failure of materials. The physical properties of the PVB, which are greatly related to strain velocity, are obtained from experiments. Secondly, a 3D numerical analysis code, which contains a pre–post process system and a computational program, is developed. Thirdly, the impact process in elastic range of a single glass plane and a laminated glass plane are calculated by using this code. The results are compared with those calculated by LS-DYNA [7] in the same condition. As a result, the impact fracture process of a single glass plane and a laminated glass plane are simulated, the advantages of laminated glass as automobile glass are discussed in passenger’s safety.

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elements are governed by the second Newton’s law. For element i: d2 ui = Fi dt2 dωi Ii = Mi dt

mi

2.1 Equation of motion In the present work, the object of study is divided into a set of rigid sphere elements, there are one or more forces between two neighboring element pairs, the motion of

(2)

where ui is the displacement of the center of mass of element i; ωi is the rotational velocity of element i; mi and Ii are the mass and inertia moment of element i, respectively; Fi and Mi present the total external force and centroidal moment, which act on the element i, respectively. The differential Eqs. (1) and (2) can be solved by explicit finite difference method. Assuming that the displacement ui (t) , velocity vi (t), rotational velocity ωi (t), acceleration ai (t) and rotary acceleration β i (t) of element i in time t are known, respectively. The following Eqs. (3) and (4) can be obtained from Eqs. (1) and (2): Fi (t) mi Mi (t) β i (t) = Ii ai (t) =

(3) (4)

in time t + ∆t: vi (t + ∆t) = vi (t) + ai (t) ∆t

(5)

ωi (t + ∆t) = ωi (t) + β i (t) ∆t

(6)

ui (t + ∆t) = ui (t) + vi (t + ∆t) ∆t

(7)

ei (t + ∆t) = ei (t) + wi (t + ∆t) × ei (t) ∆t ei (t + ∆t) ei (t + ∆t) = |ei (t + ∆t) |

(8)

where ei , which is bound to the element i, is a unit vector. In the present work, ei is used to record the rotation of the element i. It will be used to establish the local coordinates between elements i and j in Sect. 2.2.1. ∆t presents the time step, which is a basic parameter for difference methods of dynamic problems. Though, the small ∆t makes the poor efficiency of calculation, the large ∆t may cause instability of analysis. Based on the idea that in a period of time step ∆t, the impact wave doesn’t pass on to the next element, the suitable ∆t can be obtained from formula (9). ∆t
cAc − Fn,ij tan(φ)

(47)

the shear failure takes place. Once the tension or shear failure happens, the join type of interelement changes from connect to contact. In the contact models, tension force can’t work between the pair of elements. The failure criterion is shown in Fig. 3b. The compression failure between the pair of elements is not a true damage, it is that the material has had a plasticity compression deformation and the join type does not change. In this paper, if a compression failure occurs, the normal force is set: Fn = −σc Ac .

3 Effect of strain velocity on Young’s modulus of PVB Comparing to glass, the Young’s modulus of PVB is greatly related to strain velocity. Generally, it is predicted that the tensile strength also changes corresponding to strain velocity. In order to find the relation, tensile experiments of PVB were carried out by three strain velocities (0.425E-3, 0.425, 156.5 (1/s)) [11], and the results are shown in Fig. 4. Assuming the experiment result of 0.425E-3(1/s)(Young’s modulus is 5.88 MPa, tensile strength is 18.62 MPa) as the static value, the

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(MPa)

150

0.176

30

0.142

25

Young’s modulus of film:5.88+29.21| ˙|

impact force(KN)

200

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Tensile strength of film:18.62+12.85| ˙| Young’s modulus of film:Experimental values Tensile strength of film:Experimental values

100

20 15 10 5

50

0

0 −4 10

DEM FEM

10

−3

10

−2

10

−1

10

0

10

1

10

2

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3

10

0

50

100

150

200

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time (us)

4

Fig. 5 Comparison of the impact force on the impact body by single glass model

Strain velocity (1/s)

Fig. 4 Variations of Young’s modulus and tensile strength of PVB on strain velocity

Young’s modulus and tensile strength corresponding to strain velocity can be calculated approximately as follows: E = 5.88 + 29.21|˙ |0.176 σt = 18.62 + 12.85|˙ |

0.142

(48) (49)

Based on these calculations, we can get the elastic coefficient and strength between the two elements of PVB for each ∆t.

Z displacement (mm)

0.4

DEM FEM

0.2 0 −0.2 −0.4

0

100

200

300 time (us)

400

500

Fig. 6 Comparison of the displacement on vertical direction at the center of the free side by single glass model

4 Numerical simulation Based on the theory mentioned above, a numerical analysis code is developed. The code, which can be used to simulate the impact fracture behavior of automobile laminated glass, contains a pre–post process system and a calculation program. By using the code, the following problems are simulated in this paper. 4.1 Study on impact elastic wave transmission in different structure planes using 3D DEM To test calculation accuracy of the 3D DEM, the impact process of a single glass plane and a laminated glass plane in elastic range are numerical simulated by using DEM and the results are compared with those calculated by LS-DYNA. The planes in those two examples, with around fixed, under a lateral impact body at center of plane, are considered. The radius of the impact body is 6 mm, the velocity is 3.0E4 mm/s. It is assumed that the planes and the impact body just contact at the initial time. In example 1, the plane is a single glass plane, with the size of 180 mm × 180 mm × 12.8 mm. Material parameters are: Young’s modulus Eg = 74.1 GPa, the Poisson’s ratio µg = 0.2, density ρg = 2, 500 kg/m3 ; the mass of impact body is 2.2E-2 kg. In the discrete element model, the uniform radius of elements are 1.5 mm, the total

number of elements in this model is 20,873 and the time step is ∆t = 0.1 µs. While in the finite element model, the plane is divided into uniform solid mesh, of which size is 3 mm. In example 2, the plane is a laminated glass, which is consist of two outer layers glass and one piece of interlayered PVB, with the size of 100 mm × 98.8 mm × 4.76 mm, the parameters of glass are the same as example 1. Material parameters of PVB are: Young’s modulus Ep = 100 MPa (the value of Young’s modulus in high strain velocity is appropriately considered), the Poisson’s ratio µp = 0.42, density ρp = 101 kg/m3 ; the mass of impact body is 1.85E-3 kg. In the discrete element model, PVB and glass are divided into the same size sphere elements with the radius of 0.38 mm, the total number of elements in this model is 139,767 and the time step is ∆t = 0.05 µs. While in the finite element model, the plane is divided into uniform solid mesh, of which size is 0.76 mm. To make sure the calculation is in elastic field, the strength parameters of those two examples are set large enough to ensure no failure occur during impact. In this paper, DEM and FEM are used to calculate the examples mentioned above. Comparisons have been made with those results such as the impact force on the impact body (Figs. 5, 8), the displacement on vertical direction at the center of the free side (Figs. 6, 9) and the

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Fig. 7 Comparison of the displacement distribution on vertical direction by single glass model at t = 100 µs

4 impact force(KN)

displacement distribution on vertical direction on the impact side of planes (Figs. 7, 10). From the numerical analysis, it is obvious that results of 3D DEM agree very well with the corresponding results of LS-DYNA. Thus, the accuracy of the algorithm presented in this paper is proved in elastic field.

3 2 1 0

As an application of DEM, the fracture behaviors of a single glass plane and a laminated glass plane are simulated. The results are compared from a security perspective. The single glass plane model is shown in Fig. 11a, while the laminated one is shown in Fig. 11b, with the plane size of 101 mm × 98.5 mm × 4.76 mm and 100 mm × 98.8 mm × 4.76 mm, respectively. As the same as stated above, the planes, under a lateral impact body at center of plane at the velocity of 3.0E4 mm/s, are fixed around. The radius of the impact body is 6 mm and the mass is 1 kg. In order to reduce the calculation time, the models are assumed symmetry. The single glass plane is divided into 63,438 uniform elements with the radius of 0.4036 mm. In the laminated glass model, PVB and glass are divided into the same size sphere elements with the radius of 0.38 mm. The total number of elements in this model is 71,223. Some other material parameters for analysis are shown in Table 1, where the asterisked parameters of PVB are strain rate dependence. The impact failure processes of the single glass plane and the laminated glass plane are shown in Figs. 14 and 15. To facilitate observation, Figs. 14 and 15 are the profile of the symmetrical plane of the glasses. To compare the mechanical property of those two kinds of glass, two aspects are considered, such as impact force and reduction of kinetic energy of impact body shown in Figs. 12 and 13. The reduction of kinetic energy

0

10

20

30

40

50 60 time (us)

70

80

90

100

Fig. 8 Comparison of the impact force on the impact body by laminated glass model

0.4 Z displacement (mm)

4.2 Study on impact fracture behavior of different structure planes using 3D DEM

DEM FEM

DEM FEM

0.2 0 −0.2 −0.4

0

100

200

300 time (us)

400

500

Fig. 9 Comparison of the displacement on vertical direction at the center of the free side by laminated glass model

is defined as the kinetic energy decrement relative to initial state of the impact body. Comparing with the impact force curve and the reduction of kinetic energy of impact body, it is obviously shown that: (1) The maximum impact force of laminated glass model is much smaller than that of single glass model. In the present work, the element numbers, between which a crack occurred, were recorded. Thus the detail of crack process can be reviewed. From the information of the computational process, the failure process of laminated glass is gradually developed after impact. Shearing and tension cracks occur between glass

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81

Fig. 10 Comparison of the displacement distribution on vertical direction by laminated glass model at t = 170 µs

(a) The model of Single glass plane

(b) The model of Laminated glass plane

Fig. 11 Numerical model

Impact Force Reduction of Kinetic energy

impact force (KN)

2

4 3

1.5 2 1 1

0.5 0

0

100

200 300 time (µs)

400

Reducton of kinetic energy (J)

2.5 2.4172

0 500

Table 1 Basic constants used in calculation Name of constant

Glass

PVB

Material density ρ (kg/m3 ) Young’s modulus E (GPa) Poisson’s ratio µ Tensile strength σt (MPa) Compressive strength σc (MPa) Internal frictional angle φ Searing tear strength σs (MPa)

2500 74.1 0.20 34.6 350 0.197

101 0.006∗ 0.42 18.62∗ 18.62∗ 0.0

∗ Experimental

17.9

value of 0.425E-3(1/s) is used as static value

Fig. 12 Time variations of impact force and reduction of kinetic energy of impact body for single glass type

4

Impact Force Reduction of Kinetic energy

impact force (KN)

2

3

1.5

1.5239

2 1 A

1

0.5 0

B

0

100

200 300 time (µs)

400

Reducton of kinetic energy (J)

2.5

0 500

Fig. 13 Time variations of impact force and reduction of kinetic energy of impact body for laminated glass type

elements right under the impact point, while there is no crack in the lower glass plate. As time passes, the cracks extend in the upper glass plate. After PVB com-

presses fully, the tension crack occurs in the free side of lower glass plate and propagates up. Therefore, It can be considered that the maximum impact force is mainly depends on the thickness of one piece of laminated glass, which is only half of the single one. The result that the maximum impact force of the laminated model is 60% of that in the single one proves that assumption. Apparently, the less impact force is very helpful for personnel safety during a traffic accident. (2) PVB plays an important role for appeasing the impact. As shown in Figs. 12 and 13, the single glass almost lost the resistance ability for impact in seconds (about 100 µs, see in Fig. 12). The performances are that the impact force reduce remarkably and the reduction of kinetic energy almost has no increase. For the laminated glass, the resistance can be holded for quite some time until PVB is fractured. As shown in Fig. 13, the impact force reduces rapidly during the section AB. After the initial crack occur in the PVB at time 183 µs (point A on

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Fig. 15 Impact fracture behavior of laminated glass

Fig. 14 Impact fracture behavior of single glass

the curve), an extensive area on PVB was destroyed rapidly during 30 µs, the resistance ability for impact of laminated glass almost takes a fade until 215 µs. (3) The reduction of kinetic energies are 3.918 and 2.016 J in the model of laminated glass and single glass respectively. The reduction of kinetic energy in laminated model is almost double of that in single model. It’s showing that the PVB can absorb more energy to prevent penetration during impact. When a traffic accident happens, laminated glass can reduce the probability that the head of

passengers penetrate through front windshield or missiles come into automobile from outside.

5 Conclusions The results obtained in the present investigation can be summarized as follows: 1.

A 3D discrete element model, in which the layered setup of laminated glass is full considered, is presented. The model can expediently simulate the

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material transitional process from continuum to non-continuum. 2. A corresponding numerical analysis code is developed. 3. The accuracy of the 3D model and numerical analysis code are more validated in the elastic range by comparing with FEM. 4. The impact fracture processes of a single glass plane and a laminated glass plane are simulated by using the code. Because of the complicacy of the impact fracture of automobile glass, the present investigation is still in trial stage. As a matter of fact, many aspects need to be improved in the future work, especially for PVB model, which is very crude in this paper and how to improve it is the first imperative. At the same time, the parameters with more precise need to be obtained through experiments, such as the more accurate physical properties of PVB and the parameters of glass under high impact speed. The comparison between computational results and experimental results is also necessary in our future study.

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