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

摘要

For a composite containing one isotropic elastic material, with positive Lamemoduli, and void, with the elastic material occupying a prescribed volumefraction $f$, 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 boundsalso provide sharp bounds on the possible $({{ {\sigma}}}^0,{{{\epsilon}}}^0)$-pairs that can coexist in such composites, and thus solve theweak $G$-closure problem for $3d$-printed materials. The materials we use toachieve the extremal $(\sigma^0,\epsilon^0)$-pairs are denoted as near optimalpentamodes. We also consider two-phase composites containing this isotropicelasticity material and a rigid phase with the elastic material occupying aprescribed volume fraction $f$, and with the composite being subject to anaverage strain, $\epsilon^0$. For such composites, Allaire and Kohn provided asharp lower bound $\widetilde{W}_f({{ {\epsilon}}}^0)$ on the minimum elasticenergy $\sigma^0:\epsilon^0$. We show that these bounds also provide sharpbounds on the possible $({{ {\sigma}}}^0,{{ {\epsilon}}}^0)$-pairs that cancoexist in such composites of the elastic and rigid phases, and thus solve theweak $G$-closure problem in this case too. The materials we use to achievethese extremal $({{ {\sigma}}}^0,{{ {\epsilon}}}^0)$-pairs are denoted as nearoptimal unimodes.