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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。
A homogeneous field theory-based analytic homogenization method developed in granular materials was applied to two-dimensional periodic grid materials. Based on the principle of scale separation and statistical uniform characterization, a two-scale uniform model of grid materials was constructed, including thin The constitutive relations of the bar elements, the relationship between the micromechanical displacement and the macroscopic strain, and the micromechanical expression of the stress. The equivalent elastic parameters of two kinds of two-dimensional periodic grids including elastic modulus, Poisson’s ratio and shear Meso - mechanics expressions of shear modulus of elasticity. The results show that when the equilateral triangle structure is equivalent to an isotropic continuum, the elastic parameter expression is consistent with that obtained by other methods in the literature. When the square structure is homogenized into an orthotropic continuum, the principal plane elastic modulus The axial stiffness, Poisson’s ratio and shear modulus of elasticity of rod element are determined by the Poisson’s ratio and the shear stiffness of the rod element, respectively, which are in accordance with the mechanical properties of the square grid material. For the square structure in the non-main plane Matrix, taking the direction that the included angle between the coordinate axis and the material spindle is 45 ° as an example, the results of this method and the coordinate transformation method are consistent. The above results verify the effectiveness of the method developed in this paper.