PSI - Issue 71

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

447

∑ ∫ 0, (3) In each element, the shape functions are denoted by P α , where e and α represent the nodes within the discretized domain. The volume of the element in its reference configuration is given by V 0,e , while J e and J A refer to the Jacobians associated with volume and surface transformations, respectively. Nodal displacements are obtained by solving ( )=∑ ∫ 0, 0, −∑ ∫ 0 0 =0 (4) which ensures equilibrium in a weak sense at chosen time points. In Eq. 4, are the components of Cauchy stress tensor such that = ( , , , ) (5) where is the deformation gradient, is the plastic deformation gradient, and and are the slip system resistances and backstresses respectively. While can be obtained from gradient of displacement field, , and can be obtained by integrating first order rate equations defined by crystal plasticity model. The details of the model and implicit integration scheme are provided in Joseph and Chakraborty (2010). Cyclic loading involves two- time scales: (i) the fine time scale τ for high -frequency oscillations, and (ii) the coarse time scale N for underlying monotonic evolution. Using this assumption, Eq. 4 can be expressed as ( , ) = ∑ ∫ 0, 0, −∑ ∫ 0 0 =0 (6) In the multi- time scale method, Haar scaling functions are used to transform FE variables. If f(N, τ) is any variable in single time response, then it can be represented in two-scales as ( ) = ( , ) = ∑ ( ) ( ) (7) where are the coefficients and evolve with cycles while ( ) are the scaling functions with the property ( )= 1 for τ ∈ [ , +1 ) and zero otherwise. are the time points within a cycle such that ∈ [0, T] where T is the time period of loading. Applying the transformation in Eq. 7 to Eq. 6 results in 0, −∑ ∫ 0 0 =0