Stress localization and strength optimization of frame material with periodic microstructure

Stress localization for an elastic material with periodic microstructure is applied to the optimal design of rigid-jointed cellular material. The discretized form of localization was published some 20 years ago but was primarily used for strain localization (and stiffness homogenization). The stress localization aspect did not receive much attention in the past and even more so for materials modeled as rigid- or pin-jointed lattices. With the present minimization of the maximum local stress under a macroscopic stress state efficient stress localization was a major concern and it has therefore received due consideration in this paper. A ground-structure approach was used for the optimal design where in addition to topological variables for the unit cell geometric variables determine the coordinates of two internal nodes and the overall shape of the representative volume element. In a first instance the material was subjected to a uniaxial and a pure shear macroscopic state of stress of arbitrary orientation. Three regular shapes emerged from the optimization: a square crossed by diagonals, an equilateral triangle and the Kagomé. The three figures, which exhibit predominant axial modes with no significant bending, have very similar strengths. In a second example the ground-structure was subjected to a transversely isotropic macroscopic tensile stress and to a transversely isotropic macroscopic compressive stress. The design space had a plethora of optimal configurations of same maximum strength. It is shown that these optimal designs, although anisotropic, expand isotropically under isotropic stresses and reach the Hashin-Shtrikman upper bounds for the bulk modulus of isotropic materials in the low density range. In the compressive case, slender elements were eliminated by the Euler buckling constraint and bulkier optimal patterns were obtained.