University of Essen & T.U. Darmstadt

We are concerned with the derivation of the $\Gamma$-limit to a three-dimensional geometrically exact

Cosserat model as the relative thickness $h>0$ of a flat domain tends to zero. The Cosserat bulk model involves

already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear.

It is shown that the $\Gamma$-limit based on a natural scaling assumption

consists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with $h$,

augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with $h$.

No specific bending term appears in the dimensional homogenization process. The formulation

exhibits an internal length scale $L_c$ which survives the homogenization process.

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A major technical difficulty, which we encounter in applying the $\Gamma$-convergence arguments,

is to establish equi-coercivity of the sequence of

functionals as the relative thickness $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption.

While the three-dimensional problem is well-posed for the Cosserat couple modulus $\mu_c\ge 0$, equi-coercivity forces us

to assume a strictly positive Cosserat couple modulus $\mu_c>0$. The $\Gamma$-limit model determines the

midsurface deformation $m\in H^{1,2}(\omega,\R^3)$. For the case of zero Cosserat couple modulus $\mu_c=0$

we obtain an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$, without equi-coercivity which is then strenghtened to a $\Gamma$-convergence result for zero Cosserat couple modulus. The classical linear

Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for $\mu_c=0$

apart from a stabilizing shear energy term.

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.