Abstract
The elastic tensor of any triangular (2D) lattice material is given with respect to the geometry and the mechanical properties of the links between the nodes. The links can bear central forces (tensional material, for example with hinged joints), momentums (flexural materials) or a combination of the two. The symmetry class of the stiffness tensor is detailed in any case by using the invariants of Forte and Vianello. A distinction is made between the trivial cases where the elasticity symmetry group corresponds to the microstructure’s symmetry group and the non-trivial cases in the opposite case. Interesting examples of isotropic auxetic materials (with negative Poisson’s ratio) and non-trivial materials with isotropic elasticity but anisotropic fracturation (weak direction) are shown. The proposed set of equations can be used in an engineering process to create a 2D triangular lattice material of the desired elasticity.
Highlights
Trusses have been known for their mechanical performances for centuries.Recent progresses in manufacturing have made possible to generate lattice materials for which the truss microstructure is small with respect to the overall structure size
We refer to trivial cases when the symmetry groups of the lattice and the tensor are the same or at least when Hermann’s theorem can be applied and find some interesting non-trivial cases for their original properties
When no particular relation exists between the stiffness tensor components the elastic tensor belongs to the Z2 symmetry class which is called digonal
Summary
Recent progresses in manufacturing (such as 3D printers) have made possible to generate lattice materials for which the truss microstructure is small with respect to the overall structure size. We refer to trivial cases when the symmetry groups of the lattice and the tensor are the same or at least when Hermann’s theorem can be applied (for example a D3 lattice obviously leads to an isotropic stiffness tensor) and find some interesting non-trivial cases for their original properties. The flexural lattice material is shown to have a null dilatational mode (in the Kelvin sense), to belong only to the tetragonal or isotropic classes and to have Kelvin elasticity (without the full index symmetry). Special attention is payed to a nontrivial isotropic case which presents a weak direction inducing an anisotropic (orientated) fracturation process
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