The present paper was conducted to calculate Young's modulus and Negative Poisson's ratio (NPR) of a warp and woof (WAW) 3D auxetic structure in three perpendicular directions using an analytical method under uniaxial loading and considering the elastic–plastic behavior of materials. Due to the periodic pattern of the auxetic structure, a unit cell was chosen as a representative volume element (RVE) and its mechanical properties were assessed by developing analytical formulations in the elastic region. To this end, the principle of virtual work along with the concept of lumped plasticity (plastic hinges) were utilized. For validating the proposed analytical model, the 3D finite element (FE) simulations were used. Fairly good correlations were found between the present analytical results and the results of FE simulations. Subsequently, the effects of parameters, such as the initial re-entrant (IRE) angle, the cell thickness, and the auxetic cell aspect ratio on the mechanical properties were investigated. The obtained results revealed that with the increase in the IRE angle, NPR increases in the vertical direction and decreases in two other horizontal directions; it also increases the cell stiffness in three perpendicular directions. The thickness of cell walls does not alter NPR, yet relatively increases the cell stiffness. Furthermore, it was found that the cell aspect ratio has an important role in the mechanical properties of WAW auxetic structures.

本文采用单轴载荷下的分析方法并考虑材料的弹塑性特性，计算了三个垂直方向上经向和纬向（WAW）3D膨胀结构的杨氏模量和负泊松比（NPR）。由于膨胀结构的周期性模式，因此选择了一个晶胞作为代表体积元素（RVE），并通过在弹性区域中开发分析配方来评估其机械性能。为此，利用了虚拟工作的原理以及集总可塑性（塑料铰链）的概念。为了验证所提出的分析模型，使用了3D有限元（FE）仿真。在当前的分析结果和有限元模拟的结果之间发现了相当好的相关性。随后，研究了初始折返角（IRE）角，泡孔厚度和膨胀泡孔长径比等参数对机械性能的影响。所得结果表明，随着IRE角的增加，NPR在垂直方向上增加而在另外两个水平方向上减少。它还在三个垂直方向上增加了单元刚度。孔壁的厚度不会改变NPR，但会相对增加孔的刚度。此外，已经发现，细胞纵横比在WAW膨胀结构的机械性能中具有重要作用。所得结果表明，随着IRE角的增加，NPR在垂直方向上增加而在另外两个水平方向上减少。它还在三个垂直方向上增加了单元刚度。孔壁的厚度不会改变NPR，但会相对增加孔的刚度。此外，已经发现，细胞纵横比在WAW膨胀结构的机械性能中具有重要作用。所得结果表明，随着IRE角的增加，NPR在垂直方向上增加而在另外两个水平方向上减少。它还在三个垂直方向上增加了单元刚度。孔壁的厚度不会改变NPR，但会相对增加孔的刚度。此外，已经发现，细胞纵横比在WAW膨胀结构的机械性能中具有重要作用。