The objective of this thesis is to develop an efficient multi-scale finite element framework to capture the macroscopic localization due to the micro--buckling of cell walls and the size effect phenomena arising in structures made of cellular materials.
Under the compression loading, the buckling phenomenon (so-called micro-buckling) of the slender components (cell walls, cell faces) of cellular solids can occur. Even if the tangent operator of the material of which the micro--structure is made, is still elliptic, the presence of the micro-buckling can lead to the loss of ellipticity of the resulting homogenized tangent operator. In that case, localization bands are formed and propagate in the macroscopic structure. Moreover, when considering a cellular structure whose dimensions are close to the cell size, the size effect phenomenon cannot be neglected since deformations are characterized by a strain gradient.
On the one hand, a classical multi-scale computational homogenization scheme (so-called first-order scheme) looses accuracy with the apparition of the macroscopic localization or the high strain gradient arising in cellular materials because the underlying assumption of the local action principle, in which the stress state on a macroscopic material point depends only on the strain state at that point, is no--longer suitable. On the other hand, the second--order multi-scale computational homogenization scheme proposed by Kouznetsova exhibits a good ability to capture such phenomena. Thus this second--order scheme is improved in this thesis with the following novelties so that it can be used for cellular materials.
First, at the microscopic scale, the periodic boundary condition is used because of its efficiency. As the meshes generated from cellular materials exhibit a large void part on the boundaries and are not conforming in general, the classical enforcement based on the matching nodes cannot be applied. A new method based on the polynomial interpolation without the requirement of the matching mesh condition on opposite boundaries of the representative volume element (RVE) is developed.
Next, in order to solve the underlying macroscopic Mindlin strain gradient continuum of this second--order scheme by the displacement--based finite element framework, the presence of high order terms (related to the higher stress and strain) leads to many complications in the numerical treatment. Indeed, the resolution requires the continuities not only of the displacement field but also of its first derivatives. This work uses the discontinuous Galerkin (DG) method to weakly impose these continuities. This proposed second--order DG--based FE2 scheme appears to be easily integrated into conventional parallel finite element codes.
Finally, the proposed second--order DG--based FE2 scheme is used to model cellular materials. As the instability phenomena are considered at both scales, the path following technique is adopted to solve both the macroscopic and microscopic problems. The micro--buckling leading to the macroscopic localization and the size effect phenomena can be captured within the proposed framework.