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The high order homogenization techniques potentially generate the so-called infinite order homogenized equations. Since long ago, the coefficients at higher order derivatives in these equations have been calculated within various refined theories for both periodic composites and thin structures. However, it was not always clear, what is a well-posed mathematical formulation for such equations. In the present paper, we discuss two techniques for constructing a second-order homogenized equation. One of them is concerned with the projection of a weak formulation of the original problem on an ‘ansatz subspace’. The second one corresponds to the traditional two scale asymptotic expansion using the representation of a second-order corrector via the solution of the classical (leading order) homogenized equation.

Homogenized equation of second-order accuracy for conductivity of laminates