Explicit conditions on geometry and elastic moduli of stiffest anisotropic tetrakis and tetrakis-like lattices

Over recent years there has been growing interest in the design and fabrication of lattice metamaterials exhibiting a variety of 'extreme' behaviors not found in natural materials. These may include: exceptional strength-And stiffness-To-weight ratios; excellent strain recoverability; very soft and/or very stiff deformation modes; auxetic behavior; phononic band-gaps; sound control ability; negative effective mass density; negative effective stiffness; negative effective refraction index; superlens behavior; and/or localized confined waves, to name some examples. We derive general conditions for the design of two-dimensional stiffest elastic networks with tetrakis-like (or 'Union Jack'-like) topology. Upon generalizing recent results for tetrakis structures composed of two different rod geometries (length and cross-sectional area), we derive the elasticity tensor of a lattice with generalized tetrakis architecture and anisotropic response. In addition, we derive optimality conditions for the achievement of stiffest networks formed by three different kinds of rods in the unit cell.