In this work we present a three-dimensional extension of pantographic structures and describe its properties after homogenization of the unit cell. Here we rely on a description involving only the first gradient of displacement, as the semi-auxetic property is effectively described by first-order stiffness terms. For a homogenization technique, discrete asymptotic expansion is used. The material shows two positive ($0\le {\nu }_{\mathrm{yx}},{\nu }_{\mathrm{yz}}\le 1$) and one negative Poisson’s ratios ($-1\ge {\nu }_{\mathrm{xz}}\ge 0$). If, on the other hand, we assume inextensible Bernoulli beams and perfect pivots, we find a vanishing stiffness matrix, suggesting a purely higher gradient material.

中文翻译：

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缩放几何的 3D 扩展以获得具有半拉胀特性的超材料

在这项工作中，我们提出了缩放结构的三维扩展，并描述了晶胞均质化后的特性。在这里，我们依赖于仅涉及位移的第一梯度的描述，因为半拉胀特性可以通过一阶刚度项有效地描述。对于均质化技术，使用离散渐近展开。该材料显示两个正（$0\le {\nu }_{\mathrm{yx}},{\nu }_{\mathrm{yz}}\le 1$) 和一个负泊松比 ($-1\ge {\nu }_{\mathrm{xz}}\ge 0$）。另一方面，如果我们假设不可扩展的伯努利梁和完美的枢轴，我们会发现一个消失的刚度矩阵，这表明是一种纯粹更高梯度的材料。