L3

One of the intrinsic methods in elasticity is to consider the Cauchy-Green tensor as the primary unknown, instead of the deformation realizing this tensor, as in the classical approach.

Then one can ask whether it is possible to recover the deformation from its Cauchy-Green tensor. From a differential geometry viewpoint, this amounts to finding an isometric immersion of a Riemannian manifold into the Euclidian space of the same dimension, say d. It is well known that this is possible, at least locally, if and only if the Riemann curvature tensor vanishes. However, the classical results assume at least a C2 regularity for the Cauchy-Green tensor (a.k.a. the metric tensor). From an elasticity theory viewpoint, weaker regularity assumptions on the data would be suitable.

We generalize this classical result under the hypothesis that the Cauchy-Green tensor is only of class W^{1,p} for some p>d.

The proof is based on a general result of PDE concerning the solvability and stability of a system of first order partial differential equations with L^p coefficients.

We discuss a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition a thermodynamically admissible flow rule for F_p is described involving as plastic gradient Curl F_p. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependenceon the plastic metric C_p=F_p^T F_p.
The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied.
Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: d_tp.τ=0 on the microscopically hard boundary Γ_D⊂∂Ω and [Curlp].τ=0 on the microscopically free boundary ∂Ω\Γ_D, where τ are the tangential vectors at the boundary ∂Ω. Moreover, I show that a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasivariational inequality with symmetric and coercive bilinear form, following the abstract framework proposed by Reddy. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity.