Incompatibility and a simple gradient theory of plasticity

In the continuum theory, at finite strains the crystal lattice is assumed to distort only elastically during plastic flow, while generally the elastic distortion itself is not compatible with a single-valued displacement field. Lattice incompatibility is characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in nonlocal theories of plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations. As a result, the incremental boundary value problem for rate-independent and rate-dependent behaviors has a classical structure and rather straightforward modifications of standard finite element programs can be utilized. Two examples are presented in this paper: one for size-scale effects in the torsion of thin wires in the setting of an isotropic J2 flow theory and the other for hardening of microstructures containing small particles embedded in a single crystal matrix.

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