We consider a family of non-convex integral functionals
$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$
where
W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the
Γ-limits corresponding to linearization (
h → 0) and homogenization (
\({\varepsilon\rightarrow 0}\)) commute, provided
W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of
W.