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Modeling fatigue-driving delamination using a thick level set interface model

Latifi, Mohammad; van der Meer, Frans; Sluys, Lambertus J.

Publication date 2016

Document Version Final published version Published in

Proceedings of the 17th European Conference on Composite Materials

Citation (APA)

Latifi, M., van der Meer, F., & Sluijs, B. (2016). Modeling fatigue-driving delamination using a thick level set interface model. In K. Drechsler (Ed.), Proceedings of the 17th European Conference on Composite Materials: Munich, Germany (pp. 1-5). KIT.

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MODELING FATIGUE-DRIVEN DELAMINATION USING A THICK

LEVEL SET INTERFACE MODEL

M. Latifi1, F.P. van der Meer, L.J. Sluys

Delft University of Technology, Faculty of Civil Engineering and Geosciences, Delft, Netherlands

1Email: m.latifi@tudelft.nl

Keywords: Delamination, Composite, Fatigue, Thick level set, Interface elements

Abstract

This paper presents a new discontinuous damage model for simulating fatigue-driven delamination in composites. Fatigue models commonly describe crack growth using the Paris law which provides the link between the energy released due to delamination and the crack growth rate. A core issue in implementing the Paris law is to accurately compute the energy release rate. In common cohesive fatigue models the energy release rate is extracted locally from the cohesive law where improving the accuracy of computed energy release rate needs extra treatments. The new fatigue model proposed in this paper provides an accurate non-local method for extracting the energy release rate based on the thick level set approach. This model also profits from the kinematic capabilities of interface elements for modeling discontinuities along the interface. The model is validated by comparing the model predictions under different fracture modes with theoretical and experimental data which shows good agreement in all cases.

1. Introduction

Delamination is a common mode of failure in laminated composites under quasi-static and fatigue load-ing conditions. To ensure the safety of composite structures at the service work predictload-ing their fatigue behavior is necessary. To predict delamination the cohesive zone method has been widely applied by means of interface elements. This method was used for delamination modeling under quasi-static load-ing and later extended to fatigue analysis [1–5]. For this extension, the energy released due to delami-nation is extracted from the cohesive law, and the computed energy is used to obtain the crack growth rate according to the Paris law. However, limitations arise in application of this method because of inac-curacy in local extraction of energy release rate [6], and the need for estimating the length of cohesive zone ahead of the crack tip based on geometry and loading conditions [7] . Kawashita and Hallett [7] improved the method by developing a crack tip tracking algorithm which eliminates the dependency of the method on the length of the cohesive zone. This algorithm was applied to orthogonal meshes, and recently Xu and Wang [8] proposed a new crack tip tracking algorithm which is applicable to both or-thogonal and non-oror-thogonal meshes; however, convergence problems restrict the applicability of this model. In earlier work [6] on fatigue analysis we have proposed an alternative for cohesive zone method in the framework of fracture mechanics. This 3D level set model was able to predict fatigue delamination growth rate and its front shape accurately. However, this model is only applicable for simulating single delamination in thin structures. In this paper following the thick level set (TLS) approach [9, 10] a new method for delamination modeling under fatigue is proposed and validated for different fracture modes. The TLS method enables a non-local computation of energy which provides a better match with the Paris law. In this method a damaged band with a predefined length is located between sane and fully damaged material. The non-locality in computing energy appears by integrating local values of energy across the

M. Latifi, F.P. van der Meer and L.J. Sluys

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Damaged band Cracked Uncracked

φ

1

Figure 1. Damaged zone between cracked and uncracked material

damaged band. In this paper a combination of the thick level set method and interface elements is used to develop a new discontinuous method for modeling fatigue driven delamination in composite.

2. Method

To define the damage parameter, a length scale (lc) is considered which determines the size of the

dam-aged zone between the delaminated and non-delaminated parts of the interface (see Fig. 1). The damage is defined as a function of distance to the crack front which rises from 0 to 1 as a distance of lc. The crack

front location is determined using a level set function where the iso-zero of this function is the damage front. This definition of damage is used in the constitutive relation of interface elements. The level set function and front location are updated based on the computed crack growth rate with Paris law:

da dN = C ∆G Gc !m (1) where da/dN is the crack growth rate, Gcis the fracture energy, and∆G is the cyclic variation of energy

release rate. The material parameters C and m must be determined experimentally. An accurate compu-tation of energy release rate is the main issue in implementing the Paris law. The TLS method provides an accurate non-local method for computation of the energy release rate by integrating local values of the driving force inside the damaged band. In the proposed discontinuous version of the TLS the local driving force is defined as a function of displacement jump[11].

M

P P

c

Mixed mode I/II Mode II

Mode I

M

M

ȡM

Figure 2. Loading conditions in three different fracture modes

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3. Results and discussion

The proposed model has been validated under mode I, mode II and mixed-mode loading cases. For this purpose the DCB, 4ENF and mixed-mode test cases (see Fig. 2) have been simulated and the obtained results are compared with experimental data as well as theoretical Paris law. The boundary conditions for these simulations are presented in Fig. 2 which provide a constant crack growth rate under cyclic loading. In these simulations a load ratio (R) equal to 0.1 is considered, and a load envelope strategy is used which means that instead of the complete cyclic load a constant load equal to the maximum value of the cyclic boundary conditions is applied. The simulated specimen in all simulations is 150 mm long, 20 mm wide with two arms of 1.5 mm thick with an initial crack of 35 mm. The material properties used for the simulations are related to carbon/epoxy HTA/6376C obtained from [1, 12]. 3D wedge elements connected with 6-node interface elements are used. Furthermore, a conforming 2D mesh is defined on the interface for definition of the level set field. A value of lc = 5 mm was considered and the smallest

size of the finite elements in the 2D mesh at the interface was 0.618 mm. Figures 3, 4 and 5 compare the results obtained from the thick level set model with experimental data and the Paris relation. In these figures, the energy release rate along the horizontal axis is computed from the applied load with beam theory. The crack growth rate along the vertical axis is the averaged crack growth rate measured in the simulation at the side of the specimen. An excellent agreement is observed between the results which demonstrates the accuracy of obtained crack growth rate and computed energy release rate using the thick level set interface model. It is illustrated in figure 5 that the thick level set interface approach does not only allow accurate computation of the energy release rate, but also accurate decomposition of this quantity into the three pure-mode contributions.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10−6 10−5 10−4 10−3 10−2 10−1

Normalized energy release rate GI/GIc

Crack growth rate [mm/cycle]

Experimental results [12] Paris relation Thick level set model

Figure 3. Comparison of crack growth rate from thick level set model with experimental data and Paris relation for DCB tests

4. Conclusion

An interface thick level set model is introduced and validated in this paper. A new definition of damage is used in the constitutive relation of interface elements. Unlike common fatigue cohesive models which

M. Latifi, F.P. van der Meer and L.J. Sluys

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10−6 10−5 10−4 10−3 10−2 10−1

Normalized energy release rate G

II/GIIc

Crack growth trate [mm/cycle]

Experimental results [12] Paris relation Thick level set model

Figure 4. Comparison of crack growth rate from thick level set model with experimental data and Paris relation for 4ENF tests

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10−6 10−5 10−4 10−3 10−2 10−1 100 101

Normalized energy release rate G/G c

Crack growth rate [mm/cycle]

Experimental results [12] Paris relation Thick level set model

Figure 5. Comparison of crack growth rate from thick level set model with experimental data and Paris relation for mixed-mode tests

extract the energy release rate from the cohesive law, the energy is computed non-locally based on the thick level set approach. The comparison between propagation rates in mode I, mode II and mixed-mode simulations with the Paris law demonstrates the accuracy of the proposed model.

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References

[1] P. Robinson, U. Galvanetto, D. Tumino, G. Bellucci, and D. Violeau. Numerical simulation of fatigue-driven delamination using interface elements. International Journal for Numerical Methods in Engineering, 63(13):1824–1848, aug 2005.

[2] J. Munoz, U. Galvanetto, and P. Robinson. On the numerical simulation of fatigue driven delami-nation with interface elements. Interdelami-national Journal of Fatigue, 28(10):1136–1146, oct 2006. [3] A. Turon, J. Costa, P. P. Camanho, and C. G. D´avila. Simulation of delamination in composites

under high-cycle fatigue. Composites Part A: Applied Science and Manufacturing, 38(11):2270– 2282, nov 2007.

[4] P. W. Harper and S. R. Hallett. A fatigue degradation law for cohesive interface elements Develop-ment and application to composite materials. International Journal of Fatigue, 32(11):1774–1787, nov 2010.

[5] P. Naghipour, M. Bartsch, and H. Voggenreiter. Simulation and experimental validation of mixed mode delamination in multidirectional CF/PEEK laminates under fatigue loading. International Journal of Solids and Structures, 48(6):1070–1081, mar 2011.

[6] M. Latifi, F. P. van der Meer, and L. J. Sluys. A level set model for simulating fatigue-driven delamination in composites. International Journal of Fatigue, 80:434–442, nov 2015.

[7] L. F. Kawashita and S. R. Hallett. A crack tip tracking algorithm for cohesive interface element analysis of fatigue delamination propagation in composite materials. International Journal of Solids and Structures, 49(21):2898–2913, oct 2012.

[8] C. Wang and X. Xu. Cohesive element analysis of fatigue delamination propagation in composite materials with improved crack tip tracking algorism. Composite Structures, 134:176–184, 2015. [9] N. Mo¨es, C. Stolz, P. E. Bernard, and N. Chevaugeon. A level set based model for damage

growth: The thick level set approach. International Journal for Numerical Methods in Engineering, 86(3):358–380, apr 2011.

[10] F. P. van der Meer and L. J. Sluys. The Thick Level Set method : sliding deformations and damage initiation. Computer Methods in Applied Mechanics and Engineering, 2014.

[11] M. Latifi, F. P. van der Meer, and L. J. Sluys. An interface thick level set model for simulating delamination in composites. Submitted to: International Journal of Numerical Methods in Engi-neering, 2016.

[12] L. E. Asp, A. Sj¨ogren, and E. S. Greenhalgh. Delamination Growth and Thresholds in a Car-bon/Epoxy Composite Under Fatigue Loading. Journal of Composites Technology and Research, 23(2):55–68, 2001.

M. Latifi, F.P. van der Meer and L.J. Sluys

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