A pressurized blister test model for the interface adhesion of dissimilar elastic–plastic materials

A pressurized blister test model for the interface adhesion of dissimilar elastic–plastic materials

Materials Science and Engineering A 487 (2008) 228–234 A pressurized blister test model for the interface adhesion of dissimilar elastic–plastic mate...

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Materials Science and Engineering A 487 (2008) 228–234

A pressurized blister test model for the interface adhesion of dissimilar elastic–plastic materials L.M. Jiang a,b , Y.C. Zhou a,b,∗ , Y.G. Liao a,b , C.Q. Sun a,c a

Key Laboratory of Low Dimensional Materials and Application Technology of Ministry of Education, Xiangtan University, Hunan 411105, China b Faculty of Materials, Optoelectronics and Physics, Xiangtan University, Hunan 411105, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore Received 1 August 2007; received in revised form 3 October 2007; accepted 3 October 2007

Abstract A concept of the interface adhesion energy, Γ 0 , has been proposed to characterize the quality of interface adhesion of the elastic–plastic materials system. A geometrically nonlinear finite element analysis of a blister test of an elastic–plastic film bonded to an elastic–plastic substrate has been conducted. The fracture process ahead of the crack tip at the interface between the thin film and the substrate is represented in terms of a built-in cohesive model, for which interface adhesion energy and the peak stress required for separation are basic parameters. Effects on the critical pressure of various influencing parameters are studied. The plastic work in the substrate contributes significantly to the critical pressure. Agreement between modeling prediction and measurement evidences the validity of the proposed concept and the method of extracting the interface adhesion energy. © 2007 Elsevier B.V. All rights reserved. Keywords: Quality of interface adhesion; Interface adhesion energy; Blister test; Elastic–plastic material; Cohesive element

1. Introduction With the increasing application of surface coatings and thinfilm systems, coating failure due to the mismatch of mechanical and thermal properties between coating and substrate is of great importance. Previous studies have demonstrated that the most important intrinsic factor affecting the life of coating materials is the quality of interface adhesion. Therefore, characterizing of the interface adhesion quality is the key in practical application. As we know, for linear elastic systems, there are two common used quantities to characterize interface adhesion strength: one is ˆ at which interface debond under uniaxial tenthe peak stress, σ, sion that is normal to the interface [1,2]; another is the interface fracture toughness, Γ ss , defined as the total work of fracture per unit area of fractured interface at a steady state of crack growth [3,4]. The former is the direct indication of the strength of adhesion between thin film and substrate and is widely accepted for

∗ Corresponding author at: Key Laboratory of Low Dimensional Materials and Application Technology of Ministry of Education, Xiangtan University, Hunan 411105, China. Tel.: +86 732 8293586; fax: +86 732 8292468. E-mail address: [email protected] (Y.C. Zhou).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.10.014

materials scientists who only considered the applied mechanical loading as the key factor. However, the scientists in mechanics field think that coatings failure is not only dependent on the applied mechanical loading but also on the flaws and defects sited at the interface. Hence, they suggest the interface fracture toughness, Γ ss , for which the mechanical and geometry factor are both considered, to characterize the quality of interface adhesion. Obviously, the latter is more comprehensive than the former. Although many experimental techniques have been developed in past decades, the pressurized blister test is still one of the few methods that can deliver quantitative and meaningful estimation on interface fracture toughness. Since the first proposal by Dannenberg [5], the blister test method has been applied to a variety of adhering systems. This testing method consists of applying a pressure through a hole in a substrate to a thin film bonded to it causing a delamination, as shown in Fig. 1. With a theoretical model, the interface fracture toughness can be evaluated from the height of the blister and the critical pressure applied during the crack growing. However, for systems that contain elastic–plastic materials, it is hard to evaluate the interface fracture toughness using this method. For linear elastic systems the interface fracture tough-

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Fig. 1. Schematic of the pressurized blister test. Fig. 2. Specification of traction separation law.

ness is a constant under the plane strain condition and it is equal to the work of separation per unit area Γ 0 . There have been many theoretical models [3,4] for this purpose. But for elastic–plastic systems, owing to the plastic dissipation, the energy dissipation for the interface debonding Γ ss is no longer equal to Γ 0 . It is however the sum of the two quantities: the energy consumed by interface separation in the fracture process zone Γ 0 , and the energy dissipated by inelastic deformation in the film and substrate Γ p . That is, Γ ss = Γ 0 + Γ p . As the plastic dissipation always changes with the crack growth and the geometry properties of the layers, Γ ss is not a constant. Thus, it is not appropriate to use Γ ss to characterize the quality of interface adhesion for elastic–plastic system. The interface adhesion energy Γ 0 , also referred to the intrinsic interface property [6], is independent of the layer geometry or the plastic dissipation in the layers. It reflects the interface adhesion strength directly. In this paper, the interface adhesion energy, Γ 0 , is proposed to characterize the quality of interface adhesion of the elastic–plastic materials system. Because of the nonlinear properties of the elastic–plastic system, it is hard to obtain an analysis formula to calculate the Γ 0 value. A finite element analysis could be appropriate in this case. The purpose of this work is to gain an empirical equation to evaluate the Γ0 by a combination of modeling and experiments. Although several computational studies have been made [7,8], they were restricted to interfaces between the elastic materials or the elastic–plastic film and the rigid substrate. However, in many cases, an elastic–plastic thin film is bonded to an elastic–plastic substrate. So in this study, we employ a built-in cohesive zone model to characterize the properties of an interface between dissimilar elastic–plastic materials under plane strain condition. We follow the concept of Needleman [9] who suggested a traction–separation boundary condition to specify the fracture process along a plane of crack growth. Since the dominant fracture process can be measured in microns, the traction–separation relationship can be regarded as a phenomenological characterization of the zone of separation along the interface rather than atomic debonding. The primary parameters specifying the traction–separation law of the fracture process are the work of ˆ We invesseparation per unit area, Γ 0 , and the peak traction, σ. tigate the effects on the critical pressure at which separation initiated of the following parameters: interface adhesion paramˆ geometry and material properties of the layers. eters Γ 0 and σ, The parametric study provides some guidelines on our experiments and proposing the empirical equation evaluating Γ 0 . It also suggests a method to evaluate Γ 0 by fitting model prediction to the experimental data.

2. Formulation The finite analysis to be carried out here is similar to the numerical studies of the crack growth at an interface in Refs. [7,9–12]. These studies were based on an interface potential that specifies a traction–separation relation similar to the dependence of interatomic forces on interatomic separation. The traction–separation law is shown in Fig. 2. Here, δn and δt denote the normal and tangential components of relative displacement of crack faces across the interface in the zone where the fracture process is occurring. When δcn and δct are critical values of these displacement components and a single non-dimensional separation measure is defined as    2 δn 2 δt + (1) λ= c δn δct The tractions drop to zero when λ = 1. With σ(λ) displayed in Fig. 2, the interface potential from which the tractions are derived is defined as  λ c φ(δn, δt ) = δn σ(λ ) dλ (2) 0

The normal and tangential components of the traction acting on the interface in the fracture process zone are given by Tn =

∂φ σ(λ) = δn ; ∂δn λδcn

Tt =

∂φ σ(λ) δt δtn = ∂δt λ δct δct

(3)

The traction law under a purely normal separation (δt = 0), is Tn = σ(λ) where λ = δn /δcn . Under a purely tangential displacement (δn = 0), Tn = (δcn /δct )σ(λ) with λ = δt /δct . The work of separation per unit area of interface is given by Eq. (2) with λ = 1. In all these studies, the traction–separation law was implemented into an interface element in the UEL user subroutine in the finite element code ABAQUS. However, ABAQUS has developed cohesive element in these two years, though it does not include the trapezoidal traction–separation law. It has been found by Tvergaard [10,11,13] that the details of the shape of the separation law are relatively unimportant. In this paper, we use a built-in cohesive element in ABAQUS which makes our work much easier compared with previous studies. Considering an essentially triangular traction separation law so that ˆ cn Γ0 = 21 σδ

(4)

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which shows that the Γ 0 and the σˆ are most important parameters charactering the fracture process in this model. Other features of the traction separation law such as the relative peak in the shear traction to normal traction as specified by δcn /δct , is taken to be unity for the same reason [12]. The materials both for thin film and substrate considered here are elastic–plastic and the uniaxial tension is specified by  σ/E σ ≤ σy ε= (5) 1/n (σy /E)(σ/σy ) σ > σy Here, σ y is the initial yield stress and n is the power hardening exponent, while E and ν are Young’s modulus and Poisson’s ratio, respectively. The tensile behavior is generalized to multiaxial stress states assuming isotropic hardening and using the Mises yield surface. For thin-film the parameters are denoted as σ y1 , n1 , ν1 , E1 , while for substrate they are denoted as σ y2 , n2 , ν 2 , E2 . A reference length that will be used to normalize the length scale is defined by −1  Γ0 2 1 − ν22 1 − ν12 + (6) R0 = 2 2 3π(1 − β ) E1 E2 σy1 where β is the second Dundurs’ elastic mismatch parameter [13], β=

1 μ1 (1 − 2ν2 ) − μ2 (1 − 2ν1 ) 2 μ1 (1 − ν2 ) + μ2 (1 − ν1 )

(7)

This reference value R0 scales with the extent of the plastic zone size when Γ ss ∼ = Γ 0. Dimensional analysis reveals that the critical pressure Pc , at which separation is initiated, is a function of three dimensionless groups: crack advance A and layer geometry; interface adhesion parameters; film and substrate material properties. In a dimensionless form, the functional relationship can be written as, Pc a A t σy1 σy2 E2 n2 σˆ =F , , , , , , , , σy1 R0 R0 R0 E1 σy1 E1 n1 σy2  E1 Γ0 (8) 2 t (1 − ν12 )σy1 where t and a are respectively thin film thickness and the radius of the central hole in the substrate as shown in Fig. 1. Γ 0 is the interface adhesion energy defined in Eq. (4) and A is the amount of crack growth as shown in Fig. 1. It is noticed that for every crack propagation increment is a and after crack propagation n times the amount of crack growth is A = n a. 3. Numerical processing A geometrically nonlinear finite element analysis (FEA) was conducted using the commercial general FEA package ABAQUS. Because of symmetry, only half of the film and substrate is modeled as shown in Fig. 3(a). A uniform pressure is applied to the de-bonded strip. Cohesive element in ABAQUS

Fig. 3. (a) Computational model for the blister test; (b) mesh of the crack tip zone.

is used to characterize the properties of an interface of dissimilar elastic–plastic materials under plane strain condition. Biased meshes were used in front of the initial crack tip to model the process of crack growth. The smallest element size is denoted by Δ0 as shown in Fig. 3(b). Due to the fact that the reference length R0 defined by (6) scales with the extent of the plastic zone size when Γ ss ∼ = Γ 0 , the ratio R0 /Δ0 gives indication of how well the mesh is able to resolve the stress and strain fields around the crack tip. Based on the same method of evaluation of the ratio R0 /Δ0 proposed by Tvergaard and Hutchinsion [10] R0 /Δ0 = 17.8 for our study gives a reasonable resolution of the near-tip fields and the fracture process zone. 4. Results and discussion A study of the critical pressure Pc dependence on the influencing parameters displayed in Eq. (8) has been conducted. Generally, the yield stress and Young’s modulus for metal or alloy is about 0.03–1.1 GPa and 40–210 GPa, respectively. Therefore, the ratio for σ y /E is about 0.001–0.03. Based on many scientists’ research the typical values of σ y /E is 0.003 [6,10]. Other typical parameters are: ν1 = ν2 = 1/3, n2 = n1 = 0.1, t/R0 = 1.67, δc = 0.02R0 [7,10]. The appropriate values for Γ 0 and R0 can be readily calculated from (4) and (6). Before presenting the effect of various dimensionless parameters, we first modeled the blister test in ref. [12] in which the author also modeled using the same finite element code ABAQUS. The difference of the finite model between ours and [12] is that: spring elements in the ABAQUS finite code were used to simulate the traction separation laws in the directions normal and tangential to the interface in ref. [12], while in the present paper, built-in cohesive elements are used. The comparison of experiment and finite element solution in ref. [12] and finite element result obtained in this paper is shown in Fig. 4. Experiment in ref. [12] indicates that the interface is

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Fig. 4. Comparison of experiment measurement and finite element prediction.

debonded when the pressure reached 0.78 MPa and the critical central deflection was 0.9 mm. The solid curve in Fig. 4, corresponding to finite element results obtained by Shirani, illustrates that the critical pressure is 0.815 MPa, and the central deflection is about 0.8 mm. However, the critical pressure and central deflection evaluated by the model in this paper is 0.781 MPa and 1.0 mm, respectively. The difference between our numerical results and the experiment is about 10%. Thus, we have shown that our finite element model can well predict the solution for an elastic–plastic thin film bonded to an elastic–plastic substrate. In order to verify the validity of our finite element model further, large finite element calculations are carried out in cases where the substrate does not yield plastically, while the film is elastic–plastic. The analyses focus on the effect of elastic modulus mismatch on the critical pressure, critical central deflection and the product of pressure with central deflection, Pc ω0 , which is proved to be proportional to the energy release rate in ref. [7], where ω0 is the critical central deflection. In all these analyses the interface ˆ y1 = 3.0 [10]. The values of E2 /E1 to adhesion parameter is σ/σ be considered here are 1, 2, 6, 1000 [10]. The calculations results are displayed in Fig. 5. The general trend is that the critical pressure, central deflection, and Pc ω0 increases when the E2 /E1 value increases. This is because the plastic zone and plastic dissipation increases with E2 /E1 . It also can be seen that the critical pressure is decreasing with increasing crack length, the central deflection is increasing with crack length, while the Pc ω0 increases with crack growth, and almost ceases to increase after a small amount of crack growth, implying that steady-state conditions were almost attained. The above observations are consistent with the experiment results in refs. [12,14] and calculation results in refs. [8,15,16]. ˆ y1 is studNext, the dependence of critical pressure Pc on σ/σ ied. The results are shown in Fig. 6. The critical pressure is ˆ y1 increases. Fig. 6 also shows that the critical increasing as σ/σ pressure ceases increasing when E2 /E1 = 10.

Fig. 5. (a) Pc /σ y1 vs. A/R0 ; (b) ω0 /R0 vs. A/R0 ; (c) Pc ω0 /Γ 0 vs. A/R0 for several E2 /E1 .

The calculation results above further prove that our finite element model can well predict the solution for an elastic–plastic film bonded on an elastic–plastic substrate. So with this finite element model, we study the dependence of the critical pressure on the various influencing parameters given by Eq. (8) in the rest of the paper.

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ˆ y1 . Fig. 6. Steady state Pc /σ y1 vs. E2 /E1 for several σ/σ

tations. The elastic property of the substrate is represented by ˆ y1 E2 /E1 = 6. Here, it is noticed that the choice of parameter σ/σ will be argued in the Section 4.3 in this paper. From Fig. 7(a), it is found that the critical pressure is decreasing with crack advancing. These results are close to those found for the elastic substrate. Fig. 7(a) also illustrates, for a given value of A/R0 , that the critical pressure decreases when the value of σ y2 /σ y1 is increased. The critical pressure has an obviously increase for σ y2 /σ y1 = 0.15, compared to the results for an elastic substrate (σ y2 /σ y1 = ∞). The influence of plasticity in the substrate is further illustrated in Fig. 7(b) by the curve of crack steady state critical pressure vs. σ y2 /σ y1 . It shows that when σ y2 /σ y1 increases to unity, the critical pressure tend to be a constant and independent of σ y2 /σ y1 . This is an interesting phenomenon and it is not found in the references as we know. The phenomenon happens because when the initial yield stress of the substrate is more than that of the film, the crack initiates before plastic zone forms in the substrate, and the substrate behaviors like an elastic material. Fig. 8 shows the equivalent plastic strain zones of substrate near crack tip for several values of σ y2 /σ y1 . As shown in Fig. 8, there is nearly no plastic deformation when σ y2 /σ y1 = 1. Therefore, in the calculations following, σ y2 /σ y1 is always set to be less than unity. 2 t and n /n 4.2. Effects of E1 Γ0 /(1 − ν12 )σy1 2 1

The parameter n describes the strain-hardening characteristics of the elastic–plastic layer, with n = 0 corresponding to an elastic–perfectly plastic material, and n = 1 corresponding to a linear elastic material. The dependence of steady-state critical pressure Pc on the value of n2 /n1 is shown in Fig. 9(a) for a number of different cases of interface adhesion energy. The fixed dimensionless parameters are as follows: n1 = 0.5, ˆ y1 = 0.5. The general trend that σ y2 /σ y1 = 0.25, E2 /E1 = 1, σ/σ the critical pressure Pc decreases with increasing value of n2 /n1 can readily be seen. Strain hardening increases the traction ahead of the tip and makes it easier to attain the peak stress σˆ [10]. Curve in Fig. 9(b), for the case where n2 /n1 = 0.2, shows an approximately linear relationship of the critical pressure Pc with the interface adhesion energy. The same finding was reported in Ref. [7]. ˆ y2 and E2 /E1 4.3. Effects of σ/σ

Fig. 7. (a) Pc /σ y1 vs. A/R0 for several σ y2 /σ y1 ; (b) steady state Pc /σ y1 vs. σ y2 /σ y1 .

4.1. Effects of A/R0 and σ y2 /σ y1 Fig. 7(a) shows the critical pressure Pc /σ y1 vs. crack growth A/R0 for σ y2 /σ y1 = 0.15, 0.25, 0.5, ∞. The same traction ˆ y1 = 2/3 in all four compuseparation law is used with σ/σ

The ratio E2 /E1 is one measure of the elasticity mismatch between the substrate and the film. Fig. 10 shows the effects of ˆ y2 and E2 /E1 (=1, 6, 1000) on the critical pressure Pc . The σ/σ solid-line curves were computed with σ y2 /σ y1 = 0.25, while the dash curves were computed with σ y2 /σ y1 = 0.15. Consequently, ˆ y2 , the peak stress σˆ of the solid curves for a given value of σ/σ is lager than that of the dash curves and the critical pressure Pc needed for solid curves also is lager. Evidently, an increase in ˆ y2 leads to a corresponding increase in the critical pressure σ/σ Pc . It is also found that the critical pressure Pc decreases with the increasing of E2 /E1 . This is different with the results reported in ref. [13] for dissimilar elastic–plastic solids and the results in ref. [6] for elastic–plastic thin film on elastic substrate. The differ-

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ence results from the yield stress of the substrate, σ y2 . Contrarily to [13], σ y2 is always lower than σ y1 in our studies. This enhances substrate effect on the critical pressure compared with those of

ˆ y2 for several E2 /E1 and σ y2 /σ y1 . Fig. 10. Initiate state Pc /σ y1 vs. σ/σ

2 t; (b) Fig. 9. (a) Steady state Pc /σ y1 vs. n2 /n1 for several E1 Γ0 /(1 − ν12 )σy1 2 t when n /n = 0.2. steady state Pc /σ y1 vs. E1 Γ0 /(1 − ν12 )σy1 2 1

the thin film. Hence, when the E2 /E1 value increases, although the plastic zone size of the thin film is larger, the critical pressure decreases eventually for the reason that higher elastic modulus of substrate increases the traction ahead of the crack tip and makes it easer to attain the peak stress. Whilst, curves in Fig. 10 also ˆ y2 for each case. Pc illustrate that there is a critical value of σ/σ ˆ y2 approaches this critical value, increases dramatically as σ/σ since a traction ahead of a crack cannot attain the peak stress and ˆ y2 is greater than this value. separation cannot initiate when σ/σ For example, for σ y2 /σ y1 = 0.15 and E2 /E1 = 1000, the critical ˆ y2 is 4.78. For σ y2 /σ y1 = 0.25 and E2 /E1 = 1, it is 3.2. The σ/σ above observations are consistent with the finding reported in refs. [10,11,13]. ˆ y1 = Now, the argument about the choice of the parameter σ/σ ˆ y2 2/3 is given. In order to ensure separation can initiate, σ/σ must less than 5 as discussed above. In the case of σ y2 /σ y1 = 0.15, ˆ y2 could ˆ y1 < 1, the value of σ/σ 0.25, 0.5, ∞, only when σ/σ ˆ y1 in Section 4.1 be less than 5. Therefore, the parameter of σ/σ was set to be 2/3.

Fig. 8. Plastic zones of substrate near crack tip for several values of σ y2 /σ y1 : (a) σ y2 /σ y1 = 0.15; (b) σ y2 /σ y1 = 0.25; (c) σ y2 /σ y1 =0.5; (d) σ y2 /σ y1 =0.75; (e) σ y2 /σ y1 = 1; (f) σ y2 /σ y1 = ∞. The region with light blue to red color represents the equivalent plastic strain that is greater than zero and the color corresponds to the minimum and maximum equivalent plastic strain respectively.

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6. Conclusions

Fig. 11. Steady state Pc /σ y1 vs. t/R0 for several σ y2 /σ y1 .

4.4. Effect of t/R0 and σ y1 /E1 Fig. 11 displays the steady state critical pressure Pc /σ y1 vs. film thickness for σ y2 /σ y1 = 0.25, 0.5, 0.75, ∞. The fixed dimenˆ y1 = 0.75, E2 /E1 = 1. sionless parameters are as follows: σ/σ Critical pressure increases nonlinearly with increasing thickness. These results are in good agreement with those reported in ref. [7]. The effect of the yield stress of the film σ y1 /E1 on the critical pressure has also been analyzed. It is found that the critical pressure exhibits a weak dependence on σ y1 /E1 , for the reason that σ y2 is always less than σ y1 in our studies. This makes the material properties of the substrate have a larger effect on the critical pressure compared with those of the thin film. 5. Further discussion Although the analytical formula of Eq. (8) is not obtained, the numerical result of Eq. (8) has been obtained in this paper. ˆ only the value of Γ 0 and Once one determines the value of σ, Pc in Eq. (8) is unknown. From the blister test, one can obtain the value of the steady state critical pressure Pc . Generally, σˆ could be measured by using a uniaxial tension that is normal to the interface [1,2]. Therefore, our studies suggest that if we know σˆ and Pc , the interface parameters Γ 0 could be obtained by using the model proposed in this paper for elastic–plastic thin film on elastic–plastic substrate. We are now ready to exploit the scaling relation between the critical pressure Pc and the interface adhesion energy,

Pc E1 Γ0 =f (9) 2 t σy1 (1 − ν12 )σy1 The above function can be determined by the analyses described in this paper. An example of the scaling relation in Eq. (9) for a specific peak stress σˆ is shown in Fig. 9(b). With the measured value of Pc , normalized by σ y1 , one extracts the value of Γ 0 .

In stead of the interface tensile strength and interface fracture toughness, a concept of interface adhesion energy, Γ 0 , has been proposed in this work to improve our understanding of the quality of interface adhesion of the elastic–plastic materials system. The critical pressure Pc of a blister test depends on three parameter groups: crack advance A and layer geometry; interface adhesion parameters; film and substrate material properties. The study of the effects on the critical pressure of above parameters shows that: plastic work in substrate contributes significantly to the critical pressure; for blister test, the substrate behaviors like an elastic material during crack growth when σ y2 /σ y1 > 1. It also predicts that the interface debonding will not take place when ˆ y2 is too large. More importantly, the normalized peak stress σ/σ our parameter study suggests that an uniaxial tension that is normal to the interface and a blister test are sufficient to calibrate ˆ The parametric study also the interface parameters, Γ 0 and σ. provides some guides on our experiments and proposing of the empirical equation evaluating Γ 0 in the future. Acknowledgements This work was financially supported by Key Project of National Natural Science Foundation of China (50531060), National Science Found for Distinguished Young Scholars of China (10525211) and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (076044). References [1] H. Zhou, F. Li, B. He, J. Wang, B.D. Sun, Surf. Coat. Technol. 201 (2007) 7360–7367. [2] A. Morales-Rodr´ıguez, M. Moevus, P. Reynaud, G. Fantozzi, J. Eur. Ceram. Soc. 27 (2007) 3301–3305. [3] A.N. Gent, L.H. Lewandowski, J. Appl. Polym. Sci. 33 (1987) 1567–1577. [4] Y.C. Zhou, T. Hashida, J. Eng. Mater. Technol. 125 (2003) 176–182. [5] H. Dannenberg, J. Appl. Polym. Sci. 5 (1961) 125–134. [6] P. LIU, L. Cheng, Y.W. Zhang, Acta Mater. 49 (2001) 817–825. [7] K. Hbaieb, Y.W. zhang, Mater. Sci. Eng. A Struct. 390 (2005) 385–392. [8] L. Figiel, Int. J. Fract. 139 (2006) 71–89. [9] A. Needleman, Int. J. Fract. 42 (1990) 21–40. [10] V. Tevergaard, J.W. Hutchinson, J. Mech. Phys. Solids 40 (1992) 1377–1397. [11] V. Tevergaard, J.W. Hutchinson, J. Mech. Phys. Solids 40 (1993) 1119–1135. [12] A. Shirani, K.M. Liechti, Int. J. Fract. 93 (1998) 281–314. [13] V. Tevergaard, J. Mech. Phys. Solids 49 (2001) 2689–2703. [14] N. Taheri, N. Mohammadi, Polym. Test. 19 (2000) 959–966. [15] S. Guo, K.-T. Wan, D.A. Dillard, Int. J. Solid Struct. 42 (2005) 2771– 2784. [16] C.-Y. Lee, M. Dupeux, Scripta Mater. 54 (2006) 453–457.