Fluid model of crystal plasticity - mathematical properties and computer simulations

Looking at severe plastic deformation experiments, it seems that crystalline materials at yield behave as a special kind of anisotropic, compressible, highly viscous fluid. In the presented approach the plastic behaviour of crystalline solids is treated as a highly viscous material flow through an adjustable crystal lattice.The main purpose of this dissertation is to investigate model of a plastic flow of a highly viscous fluid that describes the ultra-fine structure formation induced by severe plastic deformation. The idea behind is to apply and further develop methods known from fluid mechanics to modelling crystal solids.We first provide the thermodynamic description of a fluid like - Eulerian model of crystal plasticity. The model derivation is based on the application of the Gibbs potential to obtain a rate type stress strain constitutive relation. The result is compared to the approaches of traditional plasticity.The second part of this thesis is devoted to the mathematical analysis of a simplified problems originating from the visco-elastic model derived to describe flows of crystal plastic materials. Even after simplifications, neglecting the plastic effects, the problem seems to be difficult to establish local-in-time existence of a smooth solution and the existence of a weak solution. We propose a regularisation of the stress evolution equation and prove the global-in-time existence of a weak solutions, in the case of two dimensional bounded domain, for general initial data. The results, obtained by the Galerkin method, hold true for the periodic case. We then give a thorough description of the numerical methods used in the dissertation. Necessary tools in order to discretize the fluid model of crystal plasticity both in space and time, such that: triangulation of a domain, proper finite dimensional spaces and time discretization schemes, are recalled. The discrete system resulting from the weak formulation of the considered system is solved by means of the Newton’s method.Further the results of numerical simulations are reported. Performed simulations aim to justify that the presented approach is capable of capturing large strains in typical experimental settings. We give a detailed description of performed numerical simulations. Two different deformation settings were considered: uniaxial compression and channel extrusion. In the case of a simple compression we deal with a free boundary problem. For this reason we employ the Arbitrary Lagrangian Eulerian (ALE) approach in the sense that we use the Eulerian formulation on a moving mesh which captures the free boundary. As a test example we analyse a plane strain compression in a channel die and an uniaxial compression of a pillar-shaped sample. The second case focuses on 2-turn equal channel angular pressing (ECAP) in two dimensions. We take advantage of the Eulerian formulation and obtain high strains in a single 2-turn ECAP pass.