On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials
On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials
复制标题
关于二维和三维印刷材料可能的有效弹性张量
DOI：
--
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
D. Harutyunyan
中科院分区：
文献类型：
--
作者：
G. Milton;M. Briane;D. Harutyunyan
由两种材料组成的复合材料可能有效弹性张量集合$GU_f$，其中弹性张量$\BC_1>0$和$\BC_2=0$组成集合$U=\{\BC_1,\BC_2\}$，并按$f$和$1-f$的比例混合。具有张量$\BC_2=0$的材料对应于一个空的材料。（由于技术原因$\BC_2$实际上是非零的，我们取极限$\BC_2\to 0$）。具体来说，回顾$GU_f$完全通过涉及一组施加应变的能量总和的最小值和涉及一组施加应力的互补能量来表征，我们提供了在适当限制下在许多情况下实现最小值的微观几何的描述。在这些情况下，最小值的计算被简化为一个有限维的最小化问题，可以用数值方法来完成。每个微几何结构都由适当方向上的墙壁组合组成，其中墙壁中的材料是适当的$p$ -模态材料，这很容易适应$p\leq 5$独立的施加应变，但在正交空间中支持任何应力。因此，这种材料很容易沿着墙壁向某些方向滑动。墙外的区域包含“互补的Avellaneda材料”，这是一种分层层压板，可以最大限度地减少互补能量的总和。
The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor $\BC_2=0$ corresponds to a material which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero and we take the limit $\BC_2\to 0$). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.