A variational approach to higher-order homogenization

It is convenient to describe materials possessing a microstructure using homogeneous effective properties. When the ratio η between the size of the microstructure and the size of the overall structure is very small, effective models can be obtained by applying a homogenization procedure. Asymptotic homogenization is particularly well-suited to treat periodic microstructures. This approach relies on an expansion of the mechanical fields in powers of the scale ratio η. At leading order, classical homogenization is recovered and a standard Cauchy medium is obtained. Next orders yield additional corrections, that depend on the successive strain gradients. By contrast with phenomenological approaches, asymptotic homogenization allows to rigorously identify the microscopic solution accounting for the effective behavior. Besides, the higher-order corrections permit to improve the accuracy of the model by capturing size-effects. These effects typically occur when the scale ratio η is not sufficiently small or when the microscopic constituents have contrasted stiffness properties. Although accurate in the bulk region, the higher-order corrections produced by homogenization still miss significant boundary effects, which can strongly degrade the quality of the predictions. Besides, in these higher-order models, the order of the equilibrium equation is increased and new boundary conditions are required. In continuum mechanics, boundary conditions are usually derived by minimizing the energy of the system. For higher-order models such an energy is missing, which makes the identification of appropriate boundary conditions particularly difficult. Deriving such an energy present several challenges. First, the strain-gradient stiffness appearing in these models can be non-positive. A raw truncation of the bulk energy therefore yields a non-positive strain energy density. This property is highly undesirable as it leads to ill-posed boundary-value problems. Secondly, in order to guarantee the accuracy of the model, one must account for the energy generated by the boundary layers. This produces boundary contributions to the energy, that can be non-positive as well. In this work, we tackle the limitations mentioned above and uncover such an energy in 1D. We proceed in two steps. First, we combine asymptotic homogenization in the bulk and a rigorous boundary layer analysis in order to derive a formal approximation of the microscopic energy, that we call a pseudo-energy. This pseudo-energy contains a bulk term and a boundary term that are accurate up to order η^2 included. Besides, we show that the higher-order forces entering the boundary term satisfy some compatibility conditions which guarantee that the pseudo-energy can be made stationary order by order (perturbatively). As an example, we treat a 1D spring network with next-nearest neighbor interaction. We demonstrate that the predictions of this perturbative model are asymptotically exact at order η^2. Next, we propose a new procedure that transforms such a pseudo-energy into a truncated energy, restoring positivity. Two key ingredients are involved in this procedure. First, we restore positivity on the boundaries by transferring the boundary elasticity into a ghost term which, by construction, can be ignored. Next, we restore positivity in the bulk by applying a truncation method inspired from Cholesky decomposition. As a result, the obtained effective energy is positive and produces a well-posed boundary-value problem variationally. The energy is of the strain-gradient type, and is formulated in terms of a generalized strain. We further show on the example of the spring network, that the predictions of the model coincide with that of the perturbative approach. Finally, as an extension, we derive the pseudo-energy of a continuous laminate. In that case, the effective stiffnesses on boundary need to be computed numerically. Keywords : linear elasticity, higher-order homogenization, variational approach, boundary effects, positive energy, asymptotic analysis