PSI - Issue 17

Siegfried Frankl et al. / Procedia Structural Integrity 17 (2019) 51–57 Siegfried Frankl / Structural Integrity Procedia 00 (2019) 000 – 000

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There are various test methods to measure interfacial strength, such as a simple test for a single fibre described in Hampe et al. (1995) and Zhandarov et al. (2018) but also for fibre bundles, as described in Beter et al. (2019). All those tests feature highly non-uniform interfacial stress fields. Calculating the interfacial strength by dividing reached forces by interface area thus does not capture the physics of the problem. To treat the problem, a fracture mechanics approach is more realistic, by using a model that accounts for the stress fields and a delamination crack. This work proposes a finite element model for such a fibre pull-out test regarding the energy release rate of an interfacial crack and introducing a methodology for an incremental fibre-matrix delamination. The setup of the model is based on the pull-out test of Beter et al. (2019). There, a fibre bundle is pulled out of a rectangular rubber block which is held by a specimen holder. The load is applied as a monotonically increasing displacement, causing a stable crack growth with increasing applied displacement. An implicit finite element (FE) model for the test setup of the fibre bundle pull-out test from Beter et al. (2019) is developed using the commercial FE code ABAQUS (2014). The test is modelled as an axial symmetric model as shown in Fig. 1, where the geometric parameters of the model are defined. As indicated, the left end of the fibre is pulled horizontally to the left with a displacement u , causing the rubber part to contact the rigid punch, which is fixed in all directions. There can be an initial delamination between fibre bundle and rubber part with the delamination length a . Between the punch, the rubber part and the crack faces, contact is modelled with a penalty formulation. The tangential contact uses a Coulomb friction law with a friction coefficient of µ 1 and µ 2 for punch rubber and crack faces contact, respectively. The deformable bodies use four-node axisymmetric quadrilateral elements with hybrid formulation. The model uses nonlinear geometry to account for geometric and material nonlinearities. 2. Modeling 2.1. Fibre pull-out model

Fig. 1: Setup of the axial-symmetric fibre pull-out model.

The fibre bundle and the rubber part are modelled as one part with shared nodes but different material models. The fibre bundle uses homogenized isotropic material properties with linear elastic behaviour. As in Beter et al. (2019), an E-glass fibre bundle with the classification EC9- 68x5t0 of CS Interglas AG is used, a Young’s modulus is taken from the corresponding datasheet as 33 GPa. For the homogenized fibre bundle, a Poisson’s ratio of 0.3 is assumed. A hyperelastic material model with the Yeoh-form is used to describe the deformation of rubber part, which in Beter et al. (2019) consists of silicone rubber. The Yeoh parameters are fitted to a uniaxial test curve for a similar silicone rubber material from Hoffmann (2012) in ABAQUS, setting the Poisson’s ratio for the fit to 0.4. Those Yeoh parameters are given in Table 1. Table 2 shows the parameters chosen in the model to represent the geometry of the test from Beter et al. (2019). Concerning the friction coefficients and the fracture energy of the interface, typical values are assumed. In addition, the two friction coefficients were assumed to be the same. These parameters are used as values in the model, some of which will be varied in the results section.