A complete description of bi-dimensional anisotropic strain-gradient elasticity

Abstract

In the present paper spaces of fifth-order tensors involved in bidimensional strain gradient elasticity are studied. As a result complete sets of matrices representing these tensors in each one of their anisotropic system are provided. This paper completes and ends some previous studies on the subject providing a complete description of the anisotropic bidimensional strain gradient elasticity. It is proved that this behavior is divided into 14 non equivalent anisotropic classes, 8 of them being isotropic for classical elasticity. The classification and matrix representations of the acoustical gyrotropic tensor are also provided, these results may find interesting applications to the study of waves propagation in dispersive micro-structured-media.

Three Decades of Auxetics Research − Materials with Negative Poisson's Ratio: A Review

“…Design and optimization of auxetic structures have been carried out using topology optimization, [145][146][147][148] homogenization, [51,75,149,150,155,159,160] and genetic algorithm. [130,131,151,152] Detailed homogenization theory is not a part of this review.…”

“…Interested readers can refer refs. [51,75,149,150,[153][154][155]159,160] In the work of Kaminakis et al, [148] topology optimization was used to design novel auxetic structures. The steps used in mathematical formulation of topology optimization [148] are as follows:…”

“…Although, the analytical beam models have been used in existing literatures but they are satisfying as a crude approach. The feedback obtained from experiments advocates for the use of either enriched continuum models [157][158][159][160] or full field finite element simulations which are able to describe the geometry properly. [51,75,[164][165][166] The recognized contributions in the field of homogenization of auxetic/periodic structures were made by Dirrenberger et al [51,75,160] It was the work of Dirrenberger only, which made the theory of homogenization popular in auxetics community as it was previously used in mechanics of composites community.…”

Pantographic metamaterials: an example of mathematically driven design and of its technological challenges

“…Henceforth, the subscript X will be omitted in ∇ X and each space derivative will be considered a material derivative. When the strain energy densityÛ (G, ∇G) is considered to be depending quadratically upon the deformation tensor G and its gradient ∇G, the following representation formula applies [49] In order to account for anisotropy of the material, we must assume invariance of the strain energy density under the action, on the Cartesian coordinate system O, ê 1 ,ê 2 labeling points of the reference configuration, of some symmetry group S of transformations, which could be any subgroup of Orth. When the symmetry group is the dihedral group D4 (orthotropic material), the representations for the matrices C 3×3 and A 6×6 read …”

Invariant-based reconstruction of bidimensional elasticity tensors

“…In such as case a functional basis is known ( Forte and Vianello, 1998 ). Another interesting extension of the present work would be to consider 2D generalized continuum model ( Auffray et al, 2009;2015 ).…”