A configurational force for adaptive re-meshing of gradient-enhanced poromechanics problems with history-dependent variables

Abstract

We introduce a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multiphysical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models. To resolve sharp gradients of both displacement and pore pressure, we introduce an energy-estimate-free re-meshing criterion by extending the configurational force theory to consider the energy dissipation due to the fluid diffusion and the gradient-dependent plastic flow. To establish new equilibria after remeshing, the local tensorial history-dependent variables at the integration points are first decomposed into spectral forms. Then, the principal values and directions are projected onto smooth fields interpolated by the basis function of the finite element space via the Lie-algebra mapping. Our numerical results indicate that this Lie algebra operator in general leads to a new trial state closer to the equilibrium than the ones obtained from the tensor component mapping approach. A new configurational force for dissipative fluid-infiltrating porous materials that exhibit gradient-dependent plastic flow is introduced such that the remeshing may accommodate the need to resolve the sharp pressure gradient as well as the strain localization. The predicted responses are found to be not influenced by the mesh size due to the micromorphic regularization, while the adaptive meshing enables us to capture the width of deformation bands without the necessity of employing fine mesh everywhere in the domain. c

Phase field modeling of coupled crystal plasticity and deformation twinning in polycrystals with monolithic and splitting solvers

“…The phase field‐dependent Schmid tensor of slip system is constructed through the regular Schmid tensor as: where the reorientation tensor associated with twinning is defined through the Lie‐algebra based interpolation: 11,38,39 …”

“…For completeness, the definitions of SO(3) and so(3) are given below (cf. References 11,38,39): where GL(3) is the set of 3‐by‐3 invertible tensor and gl(3) is the corresponding Lie algebra of GL(3), which is the set for 3‐by‐3 tensor. When the twinning phase field equals to one, the reorientation matrix represents the rotation operation 180° around the axial vector : …”

“…Note that in the current simulation, although the [001] axis of all the grains is aligned with the z axis, the [001] axis of the twinning interfacial region is not aligned with the z axis and the twinning shear strain is out of the x − y plane. This is due to the Lie algebra interpolation, which guarantees that any interpolation between two rotation tensors in the SO (3) group remains in the SO (3) 11,38,39 . As a result, the crystal orientation is rotated smoothly from the parent region to the twinning region.…”

“…A physics‐based twin nucleation 8 and propagation 9 model is proposed, where twin propagation is achieved by checking the status of the neighboring element around the twin nucleus at the end of each time step. Since the twinning region is formed by the element sets, this approach may require specific treatment to circumvent the mesh bias 10,11 . Other numerical methods, for example, adaptive mesh 12 and embedded weak discontinuity, 13,14 are also applied to explicitly model the evolution of deformation twinning and its interaction with surrounding grains.…”

An Open-Source Fenics Implementation of a Phase Field Fracture Model for Micropolar Continua

“…Since the size effect is stemmed from the microstructures and topological features, the classical phenomenological constitutive laws, which rely on the evolution of local internal variables to represent history, may not be sufficient (Fleck et al, 1994;Mota et al, 2013;Miehe et al, 2013;Lin et al, 2015). Furthermore, the size effect also makes the first-order homogenization not suitable, because the corresponding effective medium does not capture the highorder kinematics (Bryant and Sun, 2019;Na et al, 2019;Hu and Oskay, 2019). From the numerical perspective, rate-independent constitutive laws formulated at material points may also lead to spurious mesh-dependence when softening and/or damage leads to strain localization (Bazant et al, 1984;Song et al, 2008;Na and Sun, 2016).…”