几种变截面点阵承受面外压缩载荷的力学行为

Mechanical Behaviors of Lattices with Non-uniform Longitudinal Cross-sections under Out-of-plane Compressive Loading

增材制造工艺的出现使得轴向变截面点阵结构设计成为可能,变截面点阵的刚度、强度和稳定性等力学行为研究具有重要的工程意义.假定杆件横截面沿轴向呈双曲线或椭圆型变化,积分推导变截面对于杆件刚度产生的影响,在平衡方程中进行参数变换,推导出变截面对于杆件失稳特征值的解析解.通过对比发现,双曲线截面杆件在刚度、稳定性方面较等截面杆并无优势.但当参数ξ=0.906时,椭圆型截面变化设计可使杆件失稳特征值提高31.8%.最终,将椭圆型截面杆件应用于实际的金字塔、Kagome点阵设计,并引入了特征值算法和弧长法以证实解析解的正确性.弧长法用于计算含几何缺陷的点阵胞元后屈曲行为.解析解、特征值算法与弧长法（缺陷0.01mm）均表明,与等截面杆件相比,椭圆型截面杆分别可以提高承载能力31.5%、31.4%和29.0%.Kagome点阵可以得到相同结果.此外,当缺陷增加到0.1mm时,杆件的弯曲会使得胞元承载能力降低,但椭圆型截面杆的优势仍然存在.研究结果可以为高性能变截面点阵的设计提供理论基础.

The appearance of additive manufacturing makes it possible to design a lattice structure with a non-uniform cross-section.Therefore,its mechanical behaviors need to be studied.Considering that the lattice is made of multi-trusses,two kinds of trusses with non-uniform cross-sections are firstly introduced.The longitudinal cross-sections of the trusses are assumed to be hyperbolic and elliptic,respectively.Their stiffnesses are calculated analytically by axial integration,and their first buckling eigenvalues are derived explicitly through introducing parameter variation into equilibrium equations.By comparing these analytical solutions,it is found that both the stiffness and the buckling eigenvalue of the truss with a hyperbolic cross-section are lower than those of the truss with a uniform cross-section.However,the elliptic cross-section can increase the buckling eigenvalue by up to 31.8% when the parameterξ=0.906.Then,the truss with an elliptic cross-section is applied in the practical pyramid lattice and Kagome lattice.The stiffness,the strength and the buckling eigenvalue are analyzed.When the relative density of the lattice is much lower,the truss with an elliptic cross-section has superior critical compressive strength.This is also shown by a special aluminum alloy lattice.Finally,the analytical solutions are validated by two kinds of numerical methods,the eigenvalue buckling method and the Riks method.The latter is used to calculate the post-buckling behaviors of pyramid lattice with geometric imperfection.For the pyramid lattice,it is proved that the elliptic cross-section can enhance the critical compressive strength by 31.5%,31.4% and29.0%,respectively,for the three solutions,i.e.,the analytical solution,the eigenvalue buckling method and the Riks method with the geometric imperfection of 0.01 mm.Similar results are also obtained for the Kagome lattice.Moreover,when the geometric imperfection increases to 0.1 mm,the bending of trusses can cause the loss of critical compressive strength,while the elliptic cross-section still maintains the superiority.The research results provide a theoretical basis for designing high-performance lattice materials.