Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature.

Disclinations and GND tensor effects on the multislip flow rule in crystal plasticity

This paper deals with new elastoplastic models for crystalline materials with microstructural defects, such as dislocations and disclinations, which are consistent with the multislip plastic flow rule, and compatible with the free energy imbalance principle. The defect free energy function is a function of the disclination tensor and its gradient, and of the geometrically necessary dislocation (GND) tensor, via the Cartan torsion. By applying the free energy imbalance, the appropriate viscoplastic (diffusion-like) evolution equations are derived for shear plastic rates (in slip systems) and for the disclination tensor. The two sets of differential (or partial differential, i.e., non-local) equations describe the rate form of the adopted disclination–dislocation model. The first set is typical for finite deformation formalism, while the second set refers to the evolution equations with respect to the reference configuration. The dislocation appears to be a source for producing disclination defects. A pure dislocation elastoplastic model is also proposed. Multislip models with disclination within the small deformation approach are derived from the finite deformation models. The initial and boundary value problems are formulated and the incremental (rate) equilibrium equation leads to a variational equality for the velocity field, at any time, which is coupled with the rate type models for the set of variables. First, the elastic problem is solved for a certain time interval by assuming that the existing defects inside the body remain inactive. Subsequently, the variational equality is solved for the velocity field, at any time, if the slip systems are activated. Consequently, the state of the body with defects is defined by the solution of the differential-type equations, when the velocity field is known for a certain time interval. Appropriate initial conditions are necessary, including those associated with defects which became active. Finally, an update algorithm must be provided in order to compute the fields at the current moment.

Influence of Dislocations on the Deformability of Metallic Sheets

Within the constitutive framework adopted here, the plastic distortion is described by multislips in the appropriate crystallographic system, the dislocation densities $\rho^{\alpha}$ and hardening variables $\zeta^{\alpha}$ in the $\alpha-$slip system are the internal variables involved in the model. The rate type boundary value problem at time $t$ leads to an appropriate variational equality to be satisfied by the velocity field when the current state of the body is known. Numerical solutions are analyzed in a tensile problem when only two physical slip systems are activated in the single (fcc) crystal sheet. The slip directions are in the plane of the sheet, while the normals to the slip planes are spatially represented. At the initial moment the distribution of the dislocation density is localized in a central zone of the sheet and in the tensile problem no geometrical imperfection has been introduced. The plane stress state is compatible with the rate type constitutive formulation of the model. The FEM is applied for solving the variational problem in the actual configuration, together with a temporal discretization of the differential system to update the current state in the sheet. The activation condition, which is formulated in terms of Schmid's law, allows us to describe the spread of the plastically deformed zone on the sheet.

Analysis of an Antiplane Contact Problem with Adhesion for Electro-Viscoelastic Materials

We consider a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The adhesion of the contact surfaces, caused by the glue, is taken into account. The material is assumed to be electro-viscoelastic and the foundation is assumed to be electrically conductive. We derive a variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a time-dependent variational equation for the electric potential field and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators and fixed point.