Large Plastic Deformation of Crystalline Aggregates

This chapter attempts to show the connection between the atomistic processes that determine the stress-strain curves of metals and the phenomenological description proposed for such curves

The aim of this lecture is to review some significant aspects of the dislocation modelling of the large deformation plasticity of single crystals and crystalline aggregates, by making use of an internal-variable approach.

For single-crystal plasticity, the most important internal variables are the dislocation densities on various glide planes. Their evolution is governed by balance equations involving production and annihilation rates. Dislocation interactions determine in a basically anisotropic way the slip rates and the evolution of the critical shear stresses.

Recently, dislocation-based models of continuum plasticity have been employed for the simulation of inhomogeneously deformed crystalline aggregates. Such simulations may help understanding the influence of the crystallographic mismatch across grain boundaries and of the difference in size between neighbouring grains on the heterogeneity of plastic deformation and possibly on strain localization and damage.

One of the most striking features of the microstructural organization inside the grains is that dislocations evolve towards some steady-state microstructures, provided that a sufficient amount of monotonous deformation is allowed for along the same strain path. Reversed deformation and changes in the strain path generally tend to the modification or dissolution of preformed microstructures and the formation of new ones that correspond to the last deformation mode. The lecture will focus on the attempts to model such processes and their contribution to plastic anisotropy, by means of internal variables associated to the strength and polarity of dislocation structures.

This paper is devoted to the description of the general relationships between micro- and macroscales in non-linear mechanics. After a thermodynamical presentation of these relations, we point out some particular cases of non-linearities, especially the case of polycrystalline aggregates in finite strain. In the case of the single crystal, the energy is well defined in the frame of the crystal lattice, the deformation of which is essentially reversible. The plastic deformation preserves the orientation and the structure of the lattice. In the case of the polycrystal, the constitutive law has the same form as the single-crystal orne, the evolution of a triad of vectors is necessary to describe the evolution of the microstructure and to ensure uniqueness of the decomposition of the deformation gradient in a reversible and a plastic part.

The problem of the evolution of the internal state of a single crystal and of a polycrystal is investigated, including the symmetry of the rate boundary-value problem.

In polycrystalline metals the major cause of the anisotropic plastic response is crystallographic texture resulting from the reorientation of the crystal lattices of grains during deformation. There have been considerable recent advances in the understanding of anisotropy due to crystallographic texturing, and a reasonably successful, physically-based elasto-viscoplasticity theory for the deformation of face- and bodycentered-cubic polycrystals deforming by crystallographic slip is now at hand. The constitutive equations in the theory are reviewed, and the implementation of these equations in a finite element program is described. The theory is able to predict the macroscopic anisotropic stress-strain response, shape changes and the evolution of crystallographic texture in complex deformation modes. Also, it is beginning to be applied to the analysis of deformation-processing problems. Applications to (i) the prediction of earing defects during quasi-static cup-drawing of an f.c.c. aluminum alloy, and (ii) the ovalization of pre-textured b.c.c. tantalum cylinders during dynamic Taylor cylinder-impact experiments are described.

This lecture describes some self-consistent approaches that are used to model the plastic deformation of strain-rate dependent and of rate-independent polycrystalline materials. A discussion is presented of the grain-matrix interaction law, which in most of the approaches appears to be too stiff. Large strain deformations are considered, and the evolution of the crystallographic texture and of the morphology of grains is accounted for to model the development of the overall anisotropy of the polycrystal.

A methodology is outlined for incorporating a representation of the behavior of metals based on polycrystal plasticity into a finite element formulation for their deformations. The review begins with a summary of the finite element formulation for equations governing the viscoplastic flow of metals. The anisotropic material response is specified by the averaged response of representative aggregates of crystals residing within the elements. The aggregate behavior is based on models for crystallographic slip in single crystals and hypotheses for computing the average response of crystals comprising the aggregate. Approaches are presented for representing orientations of individual crystals, for describing distributions of crystal orientations, and for updating the distributions with deformation. The methodology is applied to the rolling of face-centered cubic and hexagonal close-packed metals.