Regularity of stresses in Prandtl-Reuss perfect plasticity

Abstract

We study the differential properties of solutions of the Prandtl-Reuss model. We prove that in dimensions n = 2, 3 the stress tensor has locally square-integrable first deriv-The result is based on discretization of time and uniform estimates of solutions of the incremental problems, which generalize the estimates in the case of Hencky perfect plasticity. Counterexamples to the regularity of displacements and plastic strains in the quasistatic case are presented.

On boundary regularity for the stress in problems of linearized elasto-plasticity

“…See also Bensoussan and Frehse (1996) for the extension of the same result to (PR) (and Demyanov (2009) for an alternative proof). On the other hand, there are only few results for regularity of σ σ σ near the boundary: while the results proved in Frehse and Málek (1999); Knees (2006);Steinhauer (2003) and Blum and Frehse (2008) suggest that at least L 2 -regularity of the tangential derivatives of σ σ σ should hold up to the boundary and that the normal derivatives belongs to some fractional Sobolev space, the result of Seregin (1996b) who constructed a sequence of approximations for which some essential quantity explodes when approaching the limit problem (H) gives a significant warning regarding the global (even fractional) regularity.…”

Quasi-Static Small-Strain Plasticity in the Limit of Vanishing Hardening and Its Numerical Approximation

“…the above references and e.g., [3]. Actually, regularity for perfect plasticity up to the boundary is a difficult problem, requiring high smoothness of the boundary, as documented in [7,13,16,29]. Besides, the hardening parameters are considered fixed.…”

“…Then one has an analog of (4.10): 13) where the last convergence uses the L 2 (Ω) convergence assumed in (5.11). Taking into account the quadratic structure of both E εh (if restricted on the finitedimensional FE subspaces) and (5.13), we have…”

A first regularity result for the Armstrong–Frederick cyclic hardening plasticity model with Cosserat effects

“…This framework is perfectly adapted for inelastic deformation processes of metals that are characterized by monotone flow rule (associated plasticity). In that case the finite difference method was very useful to prove regularity of stresses in the PrandtlReuss and Norton-Hoff models [4,5,15,18,37], because this method allows to cancel the monotone nonlinearities. Using this method D. H. Alber and S. Nesenenko [3] have shown L ∞ (H 1/3−δ )-regularity for stresses and plastic strain for coercive models of viscoplasticity with variable coefficients.…”