  In this paper, we study a hyperelastic composite material with a periodic
microstructure and a prestrain close to a stress-free joint. We consider two
limits associated with linearization and homogenization. Unlike previous
studies that focus on composites with a stress-free reference configuration,
the minimizers of the elastic energy functional in the prestrained case are not
explicitly known. Consequently, it is initially unclear at which deformation to
perform the linearization. Our main result shows that both the consecutive and
simultaneous limits converge to a single homogenized model of linearized
elasticity. This model features a homogenized prestrain and provides
first-order information about the minimizers of the original nonlinear model.
We find that the homogenization of the material and the homogenization of the
prestrain are generally coupled and cannot be considered separately.
Additionally, we establish an asymptotic quadratic expansion of the homogenized
stored energy function and present a detailed analysis of the effective model
for laminate composite materials. A key analytical contribution of our paper is
a new mixed-growth version of the geometric rigidity estimate for Jones
domains. The proof of this result relies on the construction of an extension
operator for Jones domains adapted to geometric rigidity.
