Microstructure evolution in crystal plasticity: strain path effects and dislocation slip patterning

Abstract

Deformed polycrystalline metals tend to develop heterogeneous plastic deformation fields, as the amount of plastic strain varies spatially, depending on local grain orientation, geometry and defects. While grain boundaries are natural places triggering plastic slip accumulation and geometrically necessary dislocations that accommodate the gradients of the inhomogeneous plastic strain, the deformation localizes within grains revealing dislocation cell structures or micro slip bands (e.g. clear band formation in irradiated materials). Across grains, macroscopic plastic slip bands (Luders bands, etc.) exist as well. These intergranular and intragranular deformation patterns are stated to be inherent minimizers of the free energy (including the microstructurally trapped plastic energy). These microstructures may macroscopically manifest themselves through softening of the material or through plastic anisotropy in hardening under strain path changes. These effects are crucial with respect to the mechanics of the materials under consideration and should be taken into account in the constitutive modeling. In this thesis, the computational modeling of microstructure evolution (with softening or plastic anisotropy) is covered in different crystal plasticity frameworks. The scope is basically two-fold. First, in the chapters 2 and 3 the plastic anisotropy of Body Centered Cubic crystals is studied from the onset of deformation due to an intrinsic orientation dependence from non-planar dislocation core structures, to the anisotropy upon a strain path change owing to resulting dislocation cell formation. In this part of the thesis, after developing a proper BCC crystal plasticity framework taking into account the intrinsic anisotropy, a composite cell model was established where the evolution of dislocation cells was modeled under monotonic and non-proportional loading histories. Here, the existence of a dislocation microstructure is introduced into the model in terms of internal variables and the evolution was described by phenomenologically based evolution equations. However, this phenomenological approach is not able to incorporate the formation stage of the microstructure. Hence, a crystal plasticity framework called for an extension in order to capture the evolution of the microstructure driven by the free energy of the material. In order to complete the missing link between the formation of the microstructure and its evolution in crystal plasticity frameworks, the second part of the thesis concentrates on the development of a non-convex rate dependent crystal plasticity model, which reveals a rate dependent dislocation microstructure formation and evolution together with macroscopic hardening-softening-plateau stress-strain responses. To this purpose, nonconvexity is treated as an intrinsic property of the plastic free energy of the material. First, this non-convex contribution is incorporated in a strain gradient crystal plasticity framework with a double-well character, which results in a computational routine partially dual to the Ginzburg-Landau type of phase field modeling approaches (with high and low slipped regions representing the different phases). In this model, both the displacement and the plastic slip fields are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equation which is rederived in a thermodynamically consistent manner. In chapter 4, the analysis is conducted in a 1D mathematical setting in order to illustrate the ability of the model to capture the patterning of plastic slip. In chapter 5 and inspired by the literature, the non-convexity originates from latent hardening in a multislip strain gradient crystal plasticity framework. Hence, the 1D approach pursued in chapter 4 is extended to a 2D plane strain setting. It is thereby assumed that in crystals exhibiting latent hardening, the energy function is non-convex and has wells corresponding to the single slip deformations. Even though the phenomenological double-well free energy function used in the 1D approach allows to track non-equilibrium states during microstructure evolution, it does not rely on a physically based expression for non-convexity, but presents a generic formulation. Therefore chapter 5 concentrates more on the physical reasons of plastic slip localization, where a slip interaction potential is analyzed and incorporated into the rate dependent strain gradient crystal plasticity framework. The non-convexity due to the slip interactions is explicitly illustrated and the possibility of deformation patterning in the material is discussed in a boundary value problem. The last part of the thesis, chapter 6, presents a discussion and conclusions.