MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Dislocation Microstructures and the Effective Behavior of Single Crystals

Sergio Conti and Michael Ortiz


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 restrict attention throughout to linearized kinematics and deformation theory of plasticity, which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well-defined variational structure analogous to elasticity.

We first study a scale-invariant (local) problem. We show that, by developing microstructures in the form of sequential laminates of finite depth, crystals can beat the single-slip constraint, i.e., the macroscopic (relaxed) constitutive behavior is indistinguishable from multislip ideal plasticity.

In a second step, we include dislocation line energies, and hence a lengthscale, into the model. Different regimes lead to several possible types of microstructure patterns. We present constructions which achieve the various optimal scaling laws, and discuss the relation with experimentally known scalings, such as the Hall-Petch law.


Related publications

2005 Repository Open Access
Sergio Conti and Michael Ortiz

Dislocation microstructures and the effective behavior of single crystals

In: Archive for rational mechanics and analysis, 176 (2005) 1, pp. 103-147