Abstracts

Jan Kratochvil, Prague.
"Variational formulation of microstructure formation based on the continuum theory of dislocations"
Deformation microstructure controls plastic, fatigue and fracture properties of ductile materials. There are two types of microstructure of the different physical nature: dipolar and lamellar.
Typically, a dipolar microstructure is observed as a quasi regular pattern of high and low dislocation density regions. The high density regions (tangles, veins, walls) serve as a storage and annihilation facilities for left-overs of glide dislocations. The left-overs are stored predominantly in the form of dipolar dislocation loops. The dipolar microstructure occur typically in easy cross-slip materials at early stages of single slip deformation. Additionally, dipolar high dislocation density regions may be formed as a substructure within a lamellar structure (e.g. a ladder dipolar wall structure in a lamella of a persistent slip bands).
Lamellar structures consisting of lamellae of differing slip occur at the later stages of deformation. The reason for the slip differences is a tendency to decrease locally the number of active slip systems and that way to minimize the energy of the deformation process. In the case of single slip the plastic strain is concentrated into shear band or kink band lamellae. In the case of double or multi slip the plastic strain may occur in several systems of lamellae simultaneously forming a pattern of misoriented cells. In that case the boundary between cells are formed by geometrically necessary dislocations.
The proposed variational formulation is based on crystal plasticity coupled to the dislocation motion through the Orowan equation. The dislocation population is divided in glide dislocations, which carry plastic deformation, and dipolar dislocation loops which hinder the glide dislocation motion causing strain hardening. The loop density is governed by a balance equation, the equations for glide dislocations are derived from the continuum theory of dislocations. The dislocation self-force and the interaction between glide dislocations represent non-local effects that introduce intrinsic length scales into the model.
As long as the energy functional is positive, lamellae do not form, however, a homogeneous loop distribution can be instable and a dipolar structure may occur. If the functional becomes non-positive, the lamellae of differing slip may be formed. The non-positivity may arise either due to the non-positivity of some hardening coefficients or in the case of double or multi slip due to anisotropy of the hardening matrix.
Three types of microstructure are analyzed: a vein (tangle) dipolar structure in metal crystals deformed by single slip, a lamellar structure in the case of single slip formed by shear or kink bands, and a lamellar structure in the case of double slip formed by misoriented cells.

Klaus Hackl, Bochum.
"Relaxed energies and evolution equations for microstructures"
Whereas for elastic potentials the formation of microstrucutres can be very well descibed using the concept of relaxation, the treatment of inelastic materials still poses fundamental difficulties. This is mainly due to the fact that it is not clear how to update microstuctures when proceeding from one time-increment to the following one. In this presentation we would like to propose one possible solution of this problem.
We start from volume-averaged energies depending on distributions (Young-measures) of internal variables and compatible distributions (Gradient Young-measures) of deformation-gradients. Minimization with respect to the latter yields a relaxed energy which is approximately (cross-) quasiconvex ensuring the corresponding favorable properties such as existence of minimizers for the corresponding boundary-value problem.
Legendre-transform of this relaxed energy with respect to the distributions of internal variables introduced above then leads to evolution equations which possess a formal similarity to those of plasticity theory, apart from the fact that numerical complications are caused by the requirement on the density-distributions to be nonnegative everywhere.
Nevertheless it is possible to construct efficient integration-algorithms. Numerical examples involving shape-memory-alloys will be presented.

Sergio Conti, Leipzig.
"Dislocation microstructures and the effective behavior of single Crystals"
We consider single-crystal plasticity in the limiting case of infinite latent hardening, which signifies that the crystal must deform in single slip at all material points. This requirement introduces a nonconvex constraint, and thereby induces the formation of fine-scale structures.
We show by means of concrete examples how the mathematical tools of the calculus of variations permit to investigate both the macroscopic material behavior and the pattern formation on a microscopic scale.

Erik van der Giessen, Groningen.
"Nonlocal crystal plasticity: a statistical-mechanics based formulation versus discrete dislocation plasticity"
Conventional continuum mechanics models of plastic deformation are size scale independent. In contrast, there is considerable experimental evidence that plastic flow in crystalline materials is size dependent over length scales on the order of tens of microns and smaller. Various so-called nonlocal plasticity theories have been proposed that incorporate a size dependence, e.g. Aifantis (1984), Fleck and Hutchinson (1997), Acharya and Bassani (2000), Gurtin (2002), but they differ strongly in origin and mathematical structure. Although dislocation-based arguments have sometimes been used as a motivation, the theories mentioned above are phenomenological and have not been quantitatively derived from considerations of the behavior of dislocations. Therefore, the material length scale that enters in such theories needs to be fitted to experimental results, e.g., Fleck et al. (1994) and Fleck and Hutchinson (1997), or results of numerical discrete dislocation simulations, e.g. Bassani et al. (2001), Bittencourt et al. (2002).
By contrast, I will present a different new nonlocal plasticity theory that combines a standard crystal plasticity model with a two-dimensional statistical-mechanics description of the collective behavior of dislocations due to Groma (1997). The theory involves two coupled transport equations for two dislocation density fields: one is the total dislocation density and the other is a net-Burgers vector density. The length scale that enters this theory during the derivation is not a fixed length but one that varies with the distribution of the net-Burgers vector density.
After presentation of the theory for single slip, I will discuss the fit to dislocations simulations of the same boundary-value problem, namely that of the shear of a composite. The resulting solution will be contrasted with solutions using the phenomenological theories of Acharya and Bassani (2000) and Gurtin (2002). With the parameters from this fit, bending is analyzed, and compared with discrete dislocation plasticity also. Both calculations emphasize the importance of dislocation availability, a notion that is often ignored but can control the response of small systems. Finally, I will discuss some issues in the generalization of the theory to multiple slip.

Francesco Maggi, Florence.
"A new approach to counterexamples to estimates: Korn's inequality and geometric rigidity"
The derivation of counterexamples to L¹ estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn's inequality, and of the corresponding geometrically nonlinear rigidity result, in L¹ . Secondly, we construct a function f:R²->R which is separately convex but whose gradient is not in BV_loc in the sense that the mixed derivative f₁₂ is not a bounded measure.