Numerical Simulation of the Mode I Fracture of Angle-ply Composites Using the Exponential Cohesive Zone Model

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1 Numerical Simulation of the Mode I Fracture of Angle-ply Composites Using the Exponential Cohesive Zone Model Numerical Simulation of the Mode I Fracture of Angle-ply Composites Using the Exponential Cohesive Zone Model Guowei Zhu a, Peng Qu b, Jiaqi Nie c, Yunli Guo d, and Yuxi Jia e * Key Laboratory for Liquid-Solid Structural Evolution & Processing of Materials (Ministry of Education), Shandong University, Jinan , China Summary The cohesive zone model (CZM), which is based on the elasto-plastic fracture mechanics, is nowadays widely used in the numerical simulation of the composites ductile crack and progressive delamination. For the sake of tracking the angle-ply composites delamination evolution accurately, the exponential CZM is applied. At first the standard double cantilever beam (DCB) specimen is modeled using the exponential interface damage law to verify the feasibility of it. The simulated results are in good agreement with the analytical results and corrected beam theory. Then in order to investigate how the ply angle influences the mode I fracture, five laminates with different ply orientations, /30 12 /45 12 /60 12 and, are studied. The main results are obtained as follows. The delamination evolutions corresponding to different ply angles are quite non-uniform. The unidirectional laminate exhibits the stable, continuous crack growth along the specimen, so does the specimen. Whereas the other three specimens show very uneven fracture surfaces. When the plies change from to, the stiffness along the specimen is gradually reduced correspondingly, so are the distance of the delamination propagation and the peak load, while the corresponding loading displacement increases slowly. The research provides the numerical simulation method to evaluate the multidirectional laminated composites fracture toughness, and then to optimize the composite performance by the design of ply angles. Keywords: Angle-ply composites, Exponential CZM, Delamination evolution, Numerical simulation 1. Introduction One of the most common failure modes in continuous fiber-reinforced laminated composites is the delamination 1. Traditionally, the double cantilever beam (DCB) specimen is used to test the mode I delamination, and get the critical strain energy release rate, G IC, which is used to rank the delamination fracture resistance of different materials. In the standard DCB test, the delamination is initiated between two plies with the same fiber orientation, and the delamination front is constrained to move along the fiber direction. However, in typical multidirectional laminates, the delamination may occur between plies with different orientations. The shape of the delamination front and the direction Smithers Rapra Technology, 2014 of the delamination growth will not be in such a manner as the unidirectional composites. Therefore, it is important to predict the delamination growth of the multidirectional composites, and then evaluate the value of fracture toughness as a function of the fiber orientation 2. For this reason, some researchers attempted to investigate the composites where the delaminations have been inserted between plies with different orientations. For example, Ghasemnejad et al. 3 studied the effect of fiber orientation and stacking sequence on G IC experimentally. Chai 4 measured the interlaminar toughness between +q and -q plies. However, the experimental tests of the delamination for all possible laminates a yhzuiai2006@yahoo.cn, b qproc@163.com, c jiaqi0204@126.com, d yunlinihao@126. com, e jia_yuxi@sdu.edu.cn configurations are expensive and timeconsuming work 5. Moreover, it is hard to capture the accurate delamination evolution of multidirectional composites. With the development of computer software and hardware, the methods such as J-integral and virtual crack closure technique (VCCT) are widely used. The crack growth in DCB has been investigated in Ref. 6 and the progressive crack growth under mixed-mode loading has been addressed in Ref. 7. But VCCT is based on the assumption of self-similar crack growth, which is often not the case in multidirectional composites. Recently, the cohesive zone model is widely used in describing the composites failure process, especially the delamination. It brings in the cohesive elements between composite layers. The behavior of the cohesive elements is controlled by the relationship between the displacement jump and the traction across the interface. Cotterell et al. 8 investigated Polymers & Polymer Composites, Vol. 22, No. 1,

2 Guowei Zhu, Peng Qu, Jiaqi Nie, Yunli Guo, and Yuxi Jia the delamination of composites using the trapezoidal model. Mi et al. 9 studied the progressive delamination using the interface elements. Sebaey et al. 10 used cohesive elements to simulate the onset and growth of the delamination of multidirectional composites. Goyal et al. 11 modeled the mixed-mode fracture using an exponential constitutive law. Alfano et al. 12 analyzed the delamination of composites using bilinear relationship. Among these CZMs, the exponential CZM is widely used because of some advantages over other CZMs 13. Namely, the tractions and their derivatives are continuous, which is attractive from a numerical viewpoint. This paper is concerned with the prediction of the mode I interlaminar fracture toughness of angle-ply composites using the exponential CZM. Aiming to provide a detailed analysis tool of the multidirectional composites failure mechanism, the delamination evolution corresponding to different ply angles, namely /30 12 /45 12 /60 12 and, is studied, which has not been reported as yet. 2. Mechanical Model 2.1 Geometric Model and Boundary Conditions The commercial software ANSYS is used to generate a 3D mechanical model (as shown in Figure 1) representative for the mode I DCB test. The size of the entire model is mm 2 with a thickness of 3 mm, and the initial crack length a=30 mm. Since the thickness of the interface is thin enough to be considered negligible, compared with the overall geometric dimensions, its thickness is taken as exactly zero 12. The cohesive elements are inserted at the initially non-delaminated parts of the interface between the 12th layer and the 13th layer. In the numerical analysis, the element type inter205/ solid46 is used to simulate the mode I delamination propagation of the multidirectional composites. The loading and boundary conditions for the DCB test are depicted in Figure 1. The load is applied by the displacement method: a Y-direction displacement (V=10mm) is applied to the upper edge (X=0, Y=1.5), for which U=W=0. For the lower edge (X=0, Y=-1.5), displacements in the X, Y and Z directions are full constrained to prevent the rigid body motion 10. Figure 1. Finite element model of the DCB test 2.2 Exponential CZM The cohesive zone model used in this paper is the exponential relation (see Figure 1) proposed by Xu and Needleman 14. It relates cohesive surface tractions, s, to displacement jumps, d. As the cohesive surfaces separate, the traction initially increases to a maximum value, s max, and then falls to zero as a complete separation occurs. For the mode I fracture, the governing equation is given by: ( ) =eσ max δ n 1 1+ Δ n φ δ δ n exp Δ n δ n exp Δ 2 t 2 δ t where f(d) is the surface potential. D and D n denote the normal and tangential t displacement jump, respectively. d n is the normal separation across the interface where the maximum normal traction s is attained at D =0. d max is the shear t t separation where the maximum shear traction t is attained at D =d / 2. max t t The normal traction at the interface can be obtained by differentiating Eq. (1) and is given by: T n = eσ max Δ n exp - Δ n exp - Δ 2 t 2 Δ t Δ n The mode I separation potential is the area under the normal traction separation curve, which is related to the interfacial normal strength by: f n =es max d n. (1) (2) 26 Polymers & Polymer Composites, Vol. 22, No. 1, 2014

3 Numerical Simulation of the Mode I Fracture of Angle-ply Composites Using the Exponential Cohesive Zone Model 3. Results and Discussion 3.1 Verification of Exponential CZM For the mode I fracture, a typical DCB test has been analyzed under displacement control conditions. The geometry is described in Figure 1, and the material parameters are listed in Table 1. The traction-displacement jump curves obtained from the simulation and Eq. (2) are plotted in Figure 2. As expected, they are perfectly matched each other, confirming the applicability of the exponential CZM. Furthermore, the load response to the tip displacement is plotted in Figure 3. It can be seen that the simulated results are in good agreement with the results from the corrected beam theory 15, except for the difference of peak load. The maximum error is less than 5.88%, thus the exponential CZM can capture the mode I delamination evolution of composites with reasonable accuracy. 3.2 Effect of Ply Angle Figure 4 illustrates the normal traction evolution along the interface, using the mode I DCB test of composites. When the tip displacement initially increases from 0.2 mm to 1.2 mm, the cohesive element adjacent to the pre-crack tip rapidly reaches s max, then comes into the softening region of the traction-displacement response (Figure 2). As the tip displacement increases further (2.2 mm), the element adjacent to the pre-crack tip is out of work completely since the traction becomes zero, and the delamination starts to propagate, which coincides with the result shown in Figure 3. At the same time, the normal traction of more and more cohesive elements increases up to s max, but they are located at the positions ahead of the pre-crack tip. Hereafter, the normal traction of the interface goes forward keeping this figure. The result indicates that the delamination front follows the maximum normal traction. Since the shape of the delamination front can be directly obtained from the simulations by studying the distribution of the normal traction, five different laminates are numerically modeled to investigate the effect of ply orientations on the mode I delamination propagation. As shown in Figure 5, the delamination fronts of different angle-ply composites are quite nonuniform. It can be seen that they are nearly straight lines perpendicular to the X-direction for the and composites. Figure 2. Comparison between the simulated results and the analytical results from Eq. (2) Figure 3. Comparison between the simulated results and the analytical results from the corrected beam theory Table 1. Material properties for simulation of the mode I delamination 12 Ply E 11 = [GPa E 22 = E 33 = 9 [GPa v 12 = v 13 = 0.24 v 23 = 0.46 G 12 = G 13 = 5.2 [GPa Interfacial property G IC =0.28 [N mm -1 s max = 15 [MPa Note: E is the Young s modulus, n is the Poisson s ratio, G is the shear modulus, and G IC is the critical energy release rate Polymers & Polymer Composites, Vol. 22, No. 1,

4 Guowei Zhu, Peng Qu, Jiaqi Nie, Yunli Guo, and Yuxi Jia But for the other three specimens, the delamination propagation shows different morphologies along the width direction. Furthermore, the distance of the delamination propagation is gradually reduced from the to composites. Except for the shape of the delamination fronts, the relation between the load and the displacement is quite different. The sensitivity of the load-displacement curves to different ply angles is illustrated in Figure 6. Figure 4. Normal traction evolution along the interface for composites It can be seen from Figure 6 that when the ply angles vary from to, the peak loads decrease from 59.35N to 23.03N, and the corresponding loading displacements increase from 2.2 mm to 5.2 mm. In the present case, the different mechanical behaviors are attributed to the change of stiffness along the specimen, which is related with the ply angles. In the DCB test, the composite is with the representative stiffness E 11 along the specimen (X-direction). When the ply angles are changed from 30 to 90, the representative stiffness reduces gradually, and the 90 -ply is with the representative stiffness E 22 along the X-direction especially. As listed in Table 1, E 11 is much larger than E 22, thus the mechanical response to the same bending deflection is different for the different ply angles. 4. Conclusions Figure 5. Distribution of the normal traction for different angle-ply composites A rational application of the exponential CZM to the capture of the delamination evolution in DCB is presented. Five types of composite laminates with different ply orientations have been used to investigate the mode I delamination propagation. The comparison between the analytical Figure 6. Load-displacement curves of different angle-ply composites 28 Polymers & Polymer Composites, Vol. 22, No. 1, 2014

5 Numerical Simulation of the Mode I Fracture of Angle-ply Composites Using the Exponential Cohesive Zone Model solutions and the simulated results shows a perfect agreement. The delamination evolutions corresponding to different ply angles are quite non-uniform. They are nearly straight lines for the and composites, while the other three specimens show different morphologies along the width direction. When the plies change from to, the stiffness along the specimen is gradually reduced correspondingly, so are the distance of the delamination propagation and the peak load, while the corresponding loading displacement increases slowly. The results will provide scientific guidance for the prediction of the delamination growth in multidirectional laminates. Acknowledgement This work was supported by the National Key Basic Research Program of China (2010CB631102, 2012CB821500), the National Natural Science Foundation of China ( ), and the Natural Science Foundation of Shandong Province (JQ201016). References 1. Prombut P., Michel L., Lachaud F., and Barrau J.J., Engineering Fracture Mechanics, 73 (2006) Robinson P. and Song D.Q., Journal of Composite Materials, 26 (1992) Ghasemnejad H., Blackman B.R.K., Hadavinia H., and Sudall B., Composite Structures, 88 (2009) Chai H., International Journal of Fracture, 43 (1990) Shokrieh M., Heidari-Rarani M., and Ayatollahi M., Aerospace Science and Technology, 15 (2011) Fleming D.C., Journal of Composite Materials, 35 (2001) De X. and Sherrill B.B.Jr., Finite Elements in Analysis and Design, 42 (2006) Amrutharaj G.S., Lam K.Y., and Cotterell B., Theoretical and Applied Fracture Mechanics, 24 (1995) Mi Y., Crisfield M.A., Davies G.A.O., and Hellweg H.B., Journal of Composite Materials, 32 (1998) Sebaey T.A., Blanco N., Lopes C.S., and Costa J., Composites Science and Technology, 71 (2011) Goyal V.K., Johnson E.R., and Davila C.G., Composite Structure, 65 (2004) Alfano G. and Crisfield M.A.. International Journal for Numerical Methods in Engineering, 50 (2001) VandenBosch M.J., Schreurs P.J.G., and Geers M.G.D., Engineering Fracture Mechanics, 73 (2006) Kulkarni M., Carnahan M.D., Kulkarni K., Qian D., and Abot J.L., Composites Part B: Engineering, 41 (2010) Harper P.W. and Hallett S.R., Engineering Fracture Mechanics, 75 (2008) Polymers & Polymer Composites, Vol. 22, No. 1,

6 Guowei Zhu, Peng Qu, Jiaqi Nie, Yunli Guo, and Yuxi Jia 30 Polymers & Polymer Composites, Vol. 22, No. 1, 2014