A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left( \nabla _{\alpha}u_\varepsilon\big| \frac{1}{\varepsilon}\nabla _3 u_\varepsilon\right) $ bounded in $L^p (\Omega; \mathbb{R}^9 ), \ 1 < p < +\infty .$ Here it is shown that, up to a subsequence, $u_\varepsilon$ may be decomposed as $w_\varepsilon+ z_\varepsilon,$ where $z_\varepsilon$ carries all the concentration effects, i.e.$\left\{ \left| \left( \nabla _{\alpha }w_\varepsilon| \frac{1}{\varepsilon }\nabla _3 w_\varepsilon\right) \right| ^{p} \right\} $ is equi-integrable, and $w_\varepsilon$ captures the oscillatory behavior, i.e.$z_\varepsilon\to 0$ in measure. In addition, if $\{ u_\varepsilon\} $ is a recovering sequence then $z_\varepsilon= z_\varepsilon(x_\alpha )$ nearby $\partial \Omega.$

Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.