PSI - Issue 71

Manan Ghosh et al. / Procedia Structural Integrity 71 (2025) 445–452

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the materials is less than their yield strengths. Also failure may happen after a larger number of cycles (>10000). Failure under HCF occurs in three stages: (i) Crack nucleation due to inhomogeneous plastic flow or grain boundary failure; (ii) micro-crack growth due to cyclic loading, and (iii) agglomeration of these cracks to form longer cracks and resulting in final failure (Suresh (1998)). Thus, it is imperative that accurate fatigue life prediction requires precise tracking of evolving deformation and damage under cyclic loading. To achieve the same, micromechanical models with explicit grain structure in the framework of CPFEM have been developed in the last few decades. The conventional approach in CPFEM involves single time scale integration and typically requires a high-resolution of timesteps for each cycle, making cyclic CPFEM simulations for HCF computationally impractical. Some approaches attempt to predict cyclic damage by extrapolating from a few cycles of simulation, but these methods are often inaccurate and inefficient, especially when stabilization occurs after large number of cycles. Also, they are largely limited to macroscopic variables and struggle to accurately capture local variable evolution (Bennett and McDowell (2003) and Goh et al. (2003)). Deformation under cyclic loading such as in HCF involves two distinct time scales. One of them corresponds to high frequency loading and is referred to as fast time scale or fine scale. The other corresponds to accumulation of damage in every cycle and occurs in the slow time or coarse scale. Multi-time scale methods are meant to decouple these responses, thereby improving the computational efficiency by integrating the slow time response with coarser time steps (cycle jumps). Various multi-time scale methods have been developed for cyclic plasticity and crystal plasticity. However, a majority of the methods assume periodicity in response, and become inapplicable to decouple non-periodic and spatially localized responses seen in cyclic CPFEM simulations (Oskay and Fish (2004) and Manchiraju et al. (2007)). This paper uses the WATMUS method (Joseph and Chakraborty (2010)) which overcomes the aforementioned drawbacks for accelerating cyclic CPFEM simulations. However, instead of Daubechies wavelet transformation, Haar scaling functions are used. While the use of former can reduce the size of the modified FEM problem, the latter is deemed more stable, since any undesirable effect of neglecting certain coefficients can be avoided. Comparison of the coarse scale method and single time scale CPFEM simulations for beta-Titanium alloys are performed to exhibit the accuracy along with the computational advantage of the method. 2. Haar Scaling Transformation Based Multi-Time Scale Method The quasi-static equilibrium condition for rate-dependent plasticity or crystal plasticity (excluding body forces) is given by = 0 ℎ = = (1) where S u and S T are the surfaces on which the essential and natural boundary conditions are applied. The weak form of the partial differential equations (Eq. 1) can be expressed as ∫ − ∫ = 0 (2) Finite element discretization and interpolation of Eq. 2 at every time point results in the form