Abstract

The equivalent mechanical properties of pyramidal lattice materials were established by theoretical analysis and numerical simulation. According to the deformation of pyramidal unit cells and multi-layered pyramidal materials under the action of vertical, horizontal, and shear forces, the equivalent elastic modulus, equivalent shear modulus, and equivalent Poisson’s ratios of pyramidal lattice materials in three main directions were derived. The influence of the unit cell parameters on the equivalent elastic modulus, shear modulus, and Poisson’s ratios were analyzed at constant relative density of the material. With the increase of the inclination angle of the strut, the equivalent elastic modulus in the z-direction(E_z) increases, and the equivalent elastic modulus in the x-direction(E_x) and y-direction(E_y) first increases and then decreases; the equivalent shear modulus in the x-y plane (G_xy) decreases, while the equivalent shear modulus in the x-z plane (G_xz) and y-z plane (G_yz) first increases and then decreases; the equivalent Poisson’s ratios ν_zx, ν_zy increases, while ν_xy, ν_yx, ν_xz, ν_yz decreases. The size of the unit cell did not affect the equivalent elastic properties of the material, and the axial and flexural deformations of the struts are considered in displacement calculation. The solutions of analytical and numerical of the equivalent mechanical parameters were very close, and the results of the equivalent model agree well with the exact numerical model results, which proves the rationality of the homogenized equivalent method and can realize the cross-scale characterization and analysis of the pyramidal lattice materials.

• Highlights

• The equivalent mechanical expression of multi-layered pyramidal lattice material

• Relationship between equivalent mechanical parameter and struts inclination angle

• Solution of the equivalent stiffness matrix of multi-layered pyramidal materials

• Numerical calculation of the exact finite element and the equivalent models

摘要

通过理论分析和数值模拟建立了金字塔晶格材料的等效力学性能。根据金字塔晶胞和多层金字塔材料在垂直力、水平力和剪力作用下的变形，推导出金字塔晶格材料三个主要方向的等效弹性模量、等效剪切模量和等效泊松比。在材料相对密度恒定的情况下，分析了晶胞参数对等效弹性模量、剪切模量和泊松比的影响。随着支柱倾角的增大，z方向的等效弹性模量（E _z)增大，x方向( E _x )和y方向( E _y )的等效弹性模量先增大后减小；x - y平面的等效剪切模量（ G _xy）减小，而x - z平面（G _xz）和y - z平面（G _yz）的等效剪切模量先增大后减小；等效泊松比ν _zx , ν _zy增加，而ν _xy、ν _yx、ν _xz、ν _yz减小。单元格的大小不影响材料的等效弹性特性，并且在位移计算中考虑了支柱的轴向和弯曲变形。等效力学参数的解析解和数值解非常接近，等效模型结果与精确数值模型结果吻合较好，证明了均质化等效方法的合理性，可以实现跨尺度表征分析金字塔晶格材料。

• 强调

• 多层金字塔晶格材料的等效力学表达式

• 等效力学参数与支柱倾角的关系

• 多层金字塔材料等效刚度矩阵的求解

• 精确有限元及等效模型的数值计算