A variational approach of homogenization of heterogeneous materials towards second gradient continua

A methodology for the construction of effective strain gradient media for heterogeneous materials is proposed, combining a variational principle in linear elasticity with the extended Hill lemma accounting for the generalized kinematics in the framework of periodic homogenization. The microscopic displacement field of the heterogeneous continuum is decomposed additively into a polynomial homogeneous part in the generalized kinematic measures of the strain gradient effective continuum and a periodic fluctuation. The homogeneous part of the displacement is evaluated in a first stage versus the macroscopic kinematic variables by introducing the average statics in conjunction with Hill lemma. Closed form expressions of the tensors of effective moduli are obtained from the knowledge of the displacement localizators, as the solutions of the classical and higher order unit cell problems. Correction factors for the strain gradient and coupling moduli are introduced, resulting in higher order moduli that do not depend on the unit cell choice and size, and are thus intrinsic to the microstructure. The strain gradient effective moduli are computed for composite materials prone to strong internal strain gradients to illustrate the proposed methodology.

Automated summary

A methodology is proposed for constructing effective strain gradient media for heterogeneous materials, combining linear elasticity variational principle and extended Hill lemma. The method decomposes the microscopic displacement field of the heterogeneous continuum into a polynomial homogeneous part in the generalized kinematic measures of the strain gradient effective continuum and a periodic fluctuation, and computes strain gradient effective moduli for composite materials.