Introduction to Geometrically Nonlinear Continuum Dislocation Theory

The mechanical behavior of crystals, especially metallic ones, strongly depends on the intrinsic defect structure. For a broad class of metals, the motion of dislocations carries the plastic distortion, and new macroscopic properties emerge from the collective self-organization of dislocations [1‐6]. As a typical feature, homogeneous plastic distortion may become instable and localize in smaller domains of the material [7]. Understanding these localization phenomena is of great importance for many reasons: On the one hand, localized plastic distortion may be critical for the application of mechanical components. On the other hand, controlling and exploiting this self-organized (micro)structure formation (e. g. resulting in subgrain structures) may lead to desirable material properties. Moreover, considering localization and laminate formation, there are striking similarities between the microscopic occurrence in metallic materials and the macroscopic occurrence in geological materials (e. g. chevron folds as a collective buckling phenomenon of layered sedimentary rocks) [8, p. 34].

This chapter presents the basics of geometrically nonlinear kinematics for continuously dislocated crystals. It involves the multiplicative split of the deformation and the slip-system-based decomposition of the velocity gradient. Eventually, appropriate measures for the crystal’s strain and geometrically necessary dislocation densities are derived.

This chapter presents formulations for the free energy attributed to elastic strains and geometrically necessary dislocations of a cubic (primitive) crystal. Proposing a consistent thermodynamical framework, field equations are derived, starting from the Clausius-Planck inequality. Thereby, energy dissipation due to dislocation motion is introduced in a thermodynamically consistent way. Finally, different flow rules for the slip systems are presented and discussed with regard to the thermodynamic consequences.

The general geometrically nonlinear continuum dislocation theory on the basis of [1] presented in the previous chapter includes four important special cases, which are briefly described in this chapter. This also involves a rigorous treatment of the important case of single slip, and the geometrical linearization. Thereby, some subtle but significant differences to geometrically linear continuum dislocation theory are discovered.

In this chapter a variational formulation of the geometrically nonlinear continuum dislocation theory is developed based on the principle of virtual power [1]. Inserting special virtual motions, the balance equations of linear and angular momentum are derived. For the case of a continuously dislocated, plane, cubic primitive single crystal with two active slip systems, the governing integral equations are transformed to matrix notation.

This chapter presents a comprehensive methodology for the numerical solution of the variational formulation of the geometrically nonlinear continuum dislocation theory by means of the finite element method. For the special case of a continuously dislocated, plane, cubic primitive single crystal with two active slip systems, the discretized integral equations for one finite element are transformed to matrix notation.

This chapter presents numerical solutions of the initial boundary value problem for a continuously dislocated single crystal under plane shear deformation. To this end, the in-house finite element simulation code explained in the preview chapter is adopted. The simulation results are discussed in the light of the theory of complex, pattern forming systems. Additionally, the convergence behavior of the FE solution is examined and numerical challenges are highlighted.

This chapter first summarizes the content of the book and discusses key results. Subsequently, a multitude of promising future branches of research is outlined. On the one hand, this concerns possible theoretical extensions of the model and the numerical studies. On the other hand, proper experiments are suggested for the validation of the numerical results. Thereby, the focus is on cubic minerals.