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, sigma(0), Gibiansky, Cherkaev, and Allaire provided a sharp lower bound W-f(sigma(0)) on the minimum compliance energy sigma(0) : epsilon(0), in which epsilon(0) is the average strain. Here we show these bounds also provide sharp bounds on the possible (sigma 0, epsilon)-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 (sigma(0), epsilon(0))-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, epsilon(0). For such composites, Allaire and Kohn provided a sharp lower bound (W) over tilde (f)(epsilon(0)) on the minimum elastic energy sigma(0): epsilon(0). We show that these bounds also provide sharp bounds on the possible (sigma(0), epsilon(0))-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 (sigma(0), epsilon(0))-pairs are denoted as near optimal unimodes. (C) 2018 Elsevier Ltd. All rights reserved.