  In this paper we study quantitative stochastic homogenization of a
nonlinearly elastic composite material with a laminate microstructure. We prove
that for deformations close to the set of rotations the homogenized stored
energy function $W_{\rm hom}$ is $C^3$ and that $W_{\rm hom}$, the
stress-tensor $DW_{\rm hom}$, and the tangent-moduli $D^2W_{\rm hom}$ can be
represented with help of stochastic correctors. Furthermore, we study the error
of an approximation of these quantities via representative volume elements.
More precisely, we consider periodic RVEs obtained by periodizing the
distribution of the random material. For materials with a fast decay of
correlations on scales larger than a unit scale, we establish error estimates
on the random and systematic error of the RVE with optimal scaling in the size
of the RVE and with a multiplicative random constant that has exponential
moments.
