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

Abstract

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 $({\sigma }^0,{\epsilon }^0)$-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 $({\sigma }^0,{\epsilon }^0)$-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 ${\tilde{W}}_f({\epsilon }^0)$ 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.

20 pages, 2 figures

References in corpus (5)

• Acoustic cloaking theory
• Special transformations for pentamode acoustic cloaking
• On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials
• A parameteric class of composites with a large achievable range of effective elastic properties
• Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

Cited by in corpus (2)

• Some open problems in the theory of composites
• Planar polycrystals with extremal bulk and shear moduli