PSI - Issue 28

Ping Zhang et al. / Procedia Structural Integrity 28 (2020) 1176–1183 P. Zhang et al. / Structural Integrity Procedia 00 (2019) 000–000

1177

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1. Introduction It is acknowledged that short cracks exhibit a propagation behaviour different from long cracks, showing tortuous slip-controlled crack path and fluctuations in crack growth rate. In single crystals, short cracks depend more on the crystallographic orientation and temperature (Ma et al., 2008). Based on the finite element method (FEM), numerous techniques have been adopted to simulate the crack initiation and propagation, such as the Virtual Crack Closure Technique (VCCT) and the Cohesive Zone Model (CZM). However, simulated crack paths are highly sensitive to the mesh and need to be predefined. Also, crack surfaces need to be aligned with element edges, thus a remeshing is required. The Extended Finite Element Method (XFEM) (Moës et al., 1999) can overcome these disadvantages. By introducing enrichment functions into the standard FEM to describe arbitrary discontinuous structures, cracks can be modelled in a mesh-independent way, without the introduction of predefined paths or remeshing. To evaluate the driving force of crack growth, various macroscopic fracture criteria have been developed to simulate crack growth in metals and proved to work well for long cracks. However, the lack of crystallographic mechanism prevents them from capturing the behaviour of early-stage cracks. Hence, crystal plasticity (CP) theory has been embedded in numerical models to introduce microscopic fracture criteria. As the cumulative plastic strain has been related to crack growth by experimental studies (Carroll et al., 2013), it was employed to evaluate the crack growth (Zhao and Tong, 2008). Combined with the remeshing technique, it was also used to recover the crack path in a polycrystalline Ni-based superalloy (Lin et al., 2011). In this case, the slip trace corresponding to the maximum cumulative slip was set as the crack path, however, the intragranular deflection of crack growth was not considered. In another study, a mechanistic framework based on XFEM and CP was developed to investigate the crack growth in HCP, BCC and FCC polycrystals (Wilson et al., 2019). Dislocation configurational stored energy was proposed as the damage criterion and the crack extended along crystallographic traces. This approach can capture the alternating crack path and the variations in propagation rate. However, experimentally observed crack deflections within grains for FCC crystals (Zhang et al., 2019) were not addressed in this work. So far, there is no existing work combining the XFEM and CP that can successfully capture the tortuous crack path and fluctuating growth rate for short cracks in FCC single crystals, despite the XFEM’s high accuracy at a relatively low computational expense. In this study, we propose a computational approach that combines slip-based CP and XFEM. The CP model is calibrated against low cycle fatigue experiments of [111] orientation at 24 °C and 650 °C and then used with the XFEM. Based on a crystallographic damage criterion, the crack path and slip behaviours around the crack tip were investigated in [111] and [001] orientations at both temperatures. The crack growth process is also discussed by tracking the activities of slip systems. Lastly, the numerically obtained crack growth rates are compared with experimental data. 2. Methodology 2.1. Crystal plasticity (CP) model The model is based on the slip-based and rate-dependent classical CP theory (Peirce et al., 1982), and takes into account the effect of kinematic hardening. The plastic shear rate of each slip system takes the form

n

0               





 

sign

,

(1)

  

 

g



  is the resolved shear stress, g  represents the critical shear stress

where 0   denotes the reference shear strain rate,

of the th  slip system for isotropic hardening and n is the rate sensitivity exponent. The initial values of g  are assumed to be the same value 0 g for all slip systems. The back stress for kinematic hardening,   , can be obtained as