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Dislocation microstructures and the effective behavior of single crystals. (English) Zbl 1064.74144

Summary: 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.

MSC:

74Q15 Effective constitutive equations in solid mechanics
74E15 Crystalline structure
74C99 Plastic materials, materials of stress-rate and internal-variable type
74A60 Micromechanical theories
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