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 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 length scale, 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.
© Springer-Verlag Berlin Heidelberg 2005. Communicated by the Editors. This revised version was published in April 2005. The volume number has now been inserted into the citation line. This work was partly carried out during MO's stay at the Max Planck Institute for Mathematics in the Sciences of Leipzig, Germany, under the auspices of the Humboldt Foundation. MO gratefully acknowledges the financial support provided by the Foundation and the hospitality extended by the Institute. The work of SC was partially supported by Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1095 Analysis, Modeling and Simulation of Multiscale Problems. We are grateful to Stefan Müller for useful discussions and encouragement.