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,andAllaireprovidedasharplowerbound$W_f({{ {\sigma}}}^0)$ontheminimumcomplianceenergy$\sigma^0:\epsilon^0$, in which ${ϵ}^0$ is the average strain. Here we show these bounds also provide sharp bounds on the possible $({{ {\sigma}}}^0,{{ {\epsilon}}}^0)$-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 $({\sigma }^0,{ϵ}^0)$-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, ${ϵ}^0$. For such composites, Allaire and Kohn provided a sharp lower bound ${\tilde{W}}_f({ϵ}^0)$ on the minimum elastic energy ${\sigma }^0:{ϵ}^0$. We show that these bounds also provide sharp bounds on the possible $({\sigma }^0,{ϵ}^0)$-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 $({\sigma }^0,{ϵ}^0)$-pairs are denoted as near optimal unimodes.