Near optimal pentamodes as a tool for guiding stress while minimizing compliance in 3d-printed materials: A complete solution to the weak G-closure problem for 3d-printed materials

For a composite containing one isotropic elastic material, with positive Lame moduli, and void, with the elastic material occupying a prescribed volume fraction f, and with the composite being subject to an average stress, σ0, Gibiansky, Cherkaev, and Allaire provided a sharp lower bound Wf(σ0) on the minimum compliance energy σ0:ϵ0, in which ϵ0 is the average strain. Here we show these bounds also provide sharp bounds on the possible (σ0,ϵ0)-pairs that can coexist in such composites, and thus solve the weak G-closure problem for 3d-printed materials. The materials we use to achieve the extremal (σ0,ϵ0)-pairs are denoted as near optimal pentamodes. We also consider two-phase composites containing this isotropic elasticity material and a rigid phase with the elastic material occupying a prescribed volume fraction f, and with the composite being subject to an average strain, ϵ0. For such composites, Allaire and Kohn provided a sharp lower bound W˜f(ϵ0) on the minimum elastic energy σ0:ϵ0. We show that these bounds also provide sharp bounds on the possible (σ0,ϵ0)-pairs that can coexist in such composites of the elastic and rigid phases, and thus solve the weak G-closure problem in this case too. The materials we use to achieve these extremal (σ0,ϵ0)-pairs are denoted as near optimal unimodes.