Anisotropy and Asymmetry of the Elastic Tensor of Lattice Materials

Abstract

Lattice materials formed by hinged springs or linear elastic bonds may exhibit diverse anisotropy and asymmetry features of the overall elastic behavior depending on their unit cell configuration. The recently developed singum model transfers the force-displacement relationship of the springs in the lattice to the stress-strain relationship in the continuum particle, and provides the analytical form of tangential elasticity. When a pre-stress exists in the lattice, the stiffness tensor significantly changes due to the effect of the configurational stress; existing methods like the lattice spring method, relying on a scalar energy equivalence, are insufficient in such situations. Instead, a tensorial homogenization method with the new definition of singum stress and strain, should be preferred. Different lattice structures lead to different symmetries of the stiffness tensors, which are demonstrated by five lattices. When all bonds exhibit the same length, regular hexagonal, honeycomb, and auxetic lattices demonstrate that the stiffness changes from an isotropic to anisotropic, from symmetric to asymmetric tensor. When the central symmetry of the unit cell is not satisfied, the primitive cell will contain more than one singums and the Cauchy–Born rule fails by the loss of equilibrium of the single singum. A secondary stress is induced to balance the singums. Displacement gradient \(d_{ij}=u_{j,i}\) is proposed to replace strain in the constitutive law for the general case because \(d_{12}\) and \(d_{21}\) can produce different stress states. Although the hexagonal and honeycomb lattices may exhibit isotropic behavior, for general auxetic lattices, an anisotropic and asymmetric elastic tensor is obtained with the loss of both minor and major symmetry, which is also demonstrated in a square lattice with unbalanced central symmetry and a chiral lattice. The modeling procedure and results can be generalized to three dimensions and other lattices with the anisotropic and asymmetric stiffness.