一种基于平均场理论的格栅材料均匀化方法

A homogenization method for the lattice material based on the average-field theory

将颗粒材料中发展的一种基于平均场理论的解析均匀化方法应用于二维周期格栅材料;依据尺度分离原理和统计均匀表征元概念构建了格栅材料的两尺度均匀化模型,包括细观杆件单元的本构关系、细观位移-宏观应变关系式以及应力的细观力学表达式;推导了两种二维周期格栅材料等效弹性参数包括弹性模量、泊松比和剪切弹性模量的细观力学表达式。结果表明:等边三角形结构等效为各向同性连续体时,弹性参数表达式与文献中其他方法所得结果一致;正方形结构均匀化为正交各向异性连续体时,主平面内弹性模量等于杆件单元轴向刚度,泊松比和剪切弹性模量分别由杆件单元的泊松比和剪切刚度决定,符合正方形格栅材料的力学特性;对于非主平面内的正方形本构矩阵,选取坐标轴与材料主轴夹角为45°的方向为例进行推导,本文方法与坐标变换方法所得结果一致。以上结果均验证了本文所发展方法的有效性。

An analytical homogenization method for granular materials based on the average-field theory is applied to the derivation of macro effective elastic properties for 2-D periodic lattice materials, including elastic modulus, Poisson ratio and shear ealstic modulus. The two-scale homogenization model is introduced on the basis of the scale separation principle and the statistically homogeneous representative volume element. The model consists of three aspects, i.e. the micro constitutive relation of the rod element, micromechanically based expression of the macro stress tensor, and the macro strains-micro displacements relaiton. Two typical 2-D periodic lattice structures are analyzed and discussed. The equilateral triangle lattice is homogenized as isotropic continuum and expressions of effective elastic parameters are consistent with the existing results in the literatures.The square lattice is homogenized as orthotropic continuum. In the material principal plane, the ealstic modulus expression is equal to the axial stiffness of the rod element, the Poisson ratio and the shear ealstic modulus is determined by the Poisson ratio and the shear stiffness of the rod element, respectively. These results accord with the mechanical characteristic of square lattice. For the constitutive behavior in other planes, taking the 45 ° direction between coordinate axis and material principal axis as an example, the constitutive matrix via the present method is consistent with the result from the coordinate transformation. These results show the validity of the presented approach.