周期性六手性和四手性胞状固体的均匀化
Homogenization of periodic hexa- and tetrachiral cellular solids
摘要：
用两种不同的方法研究了具有周期性六手性和四手性微结构的拉胀胞状固体的均匀化。第一种方法是基于将蜂窝状固体表示为被均匀化为微极连续体的梁晶格。第二种方法被开发来分析周期性细胞设想为一个二维域组成的可变形部分，如环，韧带和可能的嵌入式矩阵内部这些。该方法基于作者提出的第二位移梯度计算均匀化（Bacigalupo和Gambarotta，2010）。由微极均匀化得到的弹性模量表示在解析形式，从它出现明确的手征性参数，这是相对于网格的线连接的环的中心的韧带的倾斜角的依赖。对于六手性细胞，Liu等人，2012年，发现，显示拉胀性质的晶格与弹性耦合模量之间的正常和不对称的应变;一个属性，已确认在这里的四手征晶格。与六角晶格不同，四角晶格的经典本构方程的特征在于通过弹性模量的正应变和剪切应变之间的耦合，该弹性模量是手性参数的奇函数。此外，这种晶格被发现表现出显着的变化的杨氏模量和泊松比与所施加的单轴应力的方向。在等效二阶梯度连续谱中也定性地检测了等效微极连续谱的性质。
英文摘要：
The homogenization of auxetic cellular solids having periodic hexachiral and tetrachiral microstructure is dealt with two different techniques. The first approach is based on the representation of the cellular solid as a beam-lattice to be homogenized as a micropolar continuum. The second approach is developed to analyse periodic cells conceived as a two-dimensional domain consisting of deformable portions such as the ring, the ligaments and possibly an embedded matrix internally to these. This approach is based on a second displacement gradient computational homogenization proposed by the Authors (Bacigalupo and Gambarotta, 2010). The elastic moduli obtained by the micropolar homogenization are expressed in analytical form from which it appears explicitly their dependence on the parameter of chirality, which is the angle of inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. For hexachiral cells, the solution of Liu et al., 2012, is found, showing the auxetic property of the lattice together with the elastic coupling modulus between the normal and the asymmetric strains; a property that has been confirmed here for the tetrachiral lattice. Unlike the hexagonal lattice, the classical constitutive equations of the tetragonal lattice turns out to be characterized by the coupling between the normal and shear strains through an elastic modulus that is an odd function of the parameter of chirality. Moreover, this lattice is found to exhibit a remarkable variability of the Young's modulus and of the Poisson's ratio with the direction of the applied uniaxial stress. The properties of the equivalent micropolar continuum are qualitatively detected also in the equivalent second-gradient continuum.
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