Global regularity of the elastic fields of a power-law model on Lipschitz domains

Abstract

SUMMARYIn this paper, we study the global regularity of the displacement and stress ÿelds of a nonlinear elastic model of power-law type. It is assumed that the underlying domains are Lipschitz domains which satisfy an additional geometric condition near those points, where the type of the boundary conditions changes. The proof of the global regularity result relies on a di erence quotient technique. Finally, a global regularity result for the stress ÿelds of the elastic, perfect plastic Hencky model is derived. This model appears as a limit model of the power-law model.

Regularity of stresses in Prandtl-Reuss perfect plasticity

“…Indeed, every absolutely continuous functions can be made Lipschitz just by time reparametrization, which leads to a corresponding reparametrization of the solutions, the problem being rate-independent. In other words, we may suppose, that 9) where δh N m in understood as in (3.6) and D N m denotes the increment of the data of the problem, defined by (3.7).…”

“…The issue of boundary regularity was discussed also in [6]. To our best knowledge, the only global regularity result for the stress in the case of Hencky perfect plasticity is contained in [9], where under appropriate assumptions it is proved that σ ∈ W 1/2−δ,2 ( ) for every δ > 0.…”

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

“…Moreover, if Ω is a d-dimensional ball, we establish the estimates on L 2 -norm of σ σ σ on the boundary that are independent of approximation. Having in addition interior regularity and also regularity in tangential directions, this also indicates that at least W 1 2 −δ,2 regularity should hold that is in perfect coincidence of 2D or 3D results in Knees (2006); Blum and Frehse (2008). For simplicity we prove our results only for homogeneous Dirichlet boundary condition on ∂Ω 1 , i.e, we set…”

“…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. We discuss both types of results in a more detail.…”

“…The first global result was established by Frehse and Málek (1999) where the Norton-Hoff approximation is used and the uniform (p-independent) estimates on tangential derivatives in whole Ω in case that Ω is a two-dimensional ball and if ∂Ω 2 = ∅ and u 0 ≡ 0 are established. If d = 2, 3, Knees (2006) (Theorem 3.6, page 1373) showed (by using the Norton-Hoff approximation) that for sufficiently smooth data the solution σ σ σ belongs for all δ > 0 to the space W 1 2 −δ,2 (Ω) uniformly as p → ∞. Steinhauer (2003) showed that for d = 2 and all p ∈ (2, ∞) fixed, the solution σ σ σ of the Norton-Hoff approximation of the Hencky problem belongs to C α (Ω) 2×2 for some α > 0 (for similar result ses also Bildhauer and Fuchs (2007)).…”