The geometric nature of plasticity laws
The classical constitutive equations of plasticity for small strains are rewritten in terms of differential geometry. The system state is defined as a point of a differentiable manifold. The state variables are the local coordinates of this point in a chart. The internal stresses are the components of a covariant vector of the cotangent bundle to internal state manifold and the elastic domain is a convex part of cotangent vector space. The plastic yield criteria, such as von Mises condition, define a Riemannian structure over the manifold. The metric element is identified with the internal dissipation element. Constitutive laws link the covariant derivatives of the thermodynamic stress with the state variable. Hardening modulus splits up in two parts, kinematic hardening and metric hardening. The last is defined by Christoffel connection coefficients. Applied to von Mises isotropic yield condition, the metric hardening is identified with isotropic hardening. Other more involved yield conditions are also considered. The Drucker's postulate requires the introduction of parallel transport of the internal stress covector.
von Mises, Baltov-Sawczuk and Tresca models
scientific article; zbMATH DE number 3907883