A fully second‐order homogenization formulation for the multi‐scale modeling of heterogeneous materials

Abstract

A homogenization‐based multi‐scale model at finite strains is proposed linking the micro‐scale to the macro‐scale level through a second‐gradient constitutive theory. It is suitable for modeling localization and size effects in multi‐phase materials. The macroscopic deformation gradient and second gradient are enforced on a microstructural representative volume element (RVE). The coupling between the scales is defined by means of kinematical insertion and homogenization operators, carefully postulated to ensure kinematical conservation in the scale transition, according to the method of multi‐scale virtual power. From the solution of the micro‐scale equilibrium, the macroscopic stress tensors are derived based on the principle of multi‐scale virtual power. The kinematical constraints at the RVE level are enforced with Lagrange multipliers. A mixed finite element approach is adopted for the numerical solution of the micro and macro second gradient equilibrium problems. It conveniently leads to a formulation where the Lagrange multipliers are the homogenized stresses, and the macroscopic consistent tangent operators for FE2 simulations can be directly derived. The Newton–Raphson scheme is employed to solve the non‐linear systems of equations. The numerical results demonstrate the predictive capability of the proposed model. A systematic analysis of the influence of RVE length and the micro‐constituent size is performed.

Efficient strain‐gradient mixed elements using shared degrees of freedom for the discretised fields

“…To use the more common $$$ {C}^0 $$$ interpolations, alternative mixed formulations have been developed, using an additional ‘displacement gradient’ field. This is constrained to (approximately) match the gradient of the displacement field using either Lagrange multipliers 19–22 or penalty methods 13,15,18,23 …”

“…9,17,18 To use the more common C 0 interpolations, alternative mixed formulations have been developed, using an additional 'displacement gradient' field. This is constrained to (approximately) match the gradient of the displacement field using either Lagrange multipliers [19][20][21][22] or penalty methods. 13,15,18,23 In C 1 elements, all degrees of freedom contribute to the interpolation of the displacement field 24 ; however, the interpolations are of higher polynomial order.…”