PSI - Issue 13

Zhengkun Liu et al. / Procedia Structural Integrity 13 (2018) 787–792 Z. Liu et al. / Structural Integrity Procedia 00 (2018) 000–000

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3. Numerical example

In this section, the proposed phase-field model is used to simulate the crack propagation in polycrystalline materials and the influence of the direction of anisotropy θ on the crack path is analyzed. The material properties in the numerical simulation are defined in Table 1(Paggi et al. (2018)). The principal sti ff ness matrix C 0 in a two dimensional setting is chosen as C 0 =    165700 63900 0 63900 165700 0 0 0 79600    [ N / mm 2 ] , (9) with the corresponding bulk modulus K = 97833 . 33 [ N / mm 2 ]. The phase-field model for anisotropic fracture has been implemented as a user defined element in FEAP (Taylor and Govindjee (2017)).

Table 1. Material parameters used in the numerical simulations. Name Symbol

Unit

Value

10 − 6 2 . 04 1000

Sti ff ness resistance

-

η

G c M

N / mm

Fracture energy release rate

mm 2 / N ∙ s

Mobility factor

3.1. Crack propagation in polycrytalline materials

To demonstrate the ability of the phase-field model for anisotropic fracture to predict the failure processes in a comparable real solar-grade polycystalline silicon, crack propagation in a square polycrystalline microstructure with 10 mm × 10 mm domain consisting of an initial crack and 10 grains under uniaxial tension is performed. The details of the geometry and boundary conditions are schematically shown in Fig. 1(a). Each grain has been associated a randomly direction of anisotropy, as depicted in in Fig. 1(b). The unstructured mesh contains 50141 quadrilateral elements with an e ff ective element size h of 0.05 mm. Fig. 2(a)- (d) depict the evolution of crack phase-field at several failure stages under tensile loading. In each trans granular fracture system, the direction of crack path appears not identical. Fig. 2(c) shows that the crack runs within the grain boundary from the secondary transgranular fracture system and the crack swifts to the neighbouring grain. In this grain, crack propagates horizontally. A phenomenon for inter- / transgranular fracture can be observed for the numerical results that the direction of anisotropy influences the crack propagation in each transgranular fracture sys tem. The intergranular fracture will take place mainly when the orientation of the boundary of grains resists the crack propagation from transgranular fracture system. The same geometry and material properties of the polyctrystals are applied to another case for the verification of the influences of the direction of anisotropy in fracture processes in polycrystalline microstructure. In this case, the direction of anisotropy in a chosen grain within the same polycrystalline structure is set to 0 ◦ . A detail of the modification for the direction of anisotropy θ is shown in Fig. 3(a). It can be observed that the crack propagates nearly horizontally in the modified grain due to the direction of anisotropy θ = 0 ◦ which is illustrated in Fig. 3(b). On the other hand, the resulting crack path may be changed by using a new direction of anisotropy of a grain. The simulation results lead to the assumption that the direction of anisotropy influences the crack orientation and plays a crucial role in the evolution processes of crack phase-field.

4. Conclusion

In this contribution, we provided a brief overview of the phase-field model for crack propagation in anisotropic brittle materials. Here, the material properties of solar-grade polycrystalline silicon have been used. Next, The numer ical simulation of crack propagation in polycrystalline materials have been done and the influence of the direction of anisotropy is analyzed. The numerical results show that the phase-field method is able to capture di ff erent scenarios