基于迭代法的非线性弹性均质化研究

STUDY ON NONLINEAR ELASTIC HOMOGENIZATION WITH ITERATIVE METHOD

微观结构对复合材料的宏观力学性能具有至关重要的影响,通过合理设计复合材料微观结构可以得到期望的宏观性能.均质化方法作为一种有效的设计方法,它从微观结构的角度出发,利用均匀化的概念,实现了对复合材料宏观力学性能的预测和设计.而当考虑非线性因素,均质化的实现就非常困难.本文利用双渐近展开方法,将位移按照宏观位移和微观位移展开,推导了非线性弹性均质化方程.通过直接迭代法,对非线性弹性均质化方程进行了求解,并给出了具体的迭代方法和实现步骤.本文基于迭代步骤和非线性弹性均质化方程编写MATLAB程序,对3种典型本构关系的周期性多孔材料平面问题进行了计算,对比细致模型的应变能、最大位移和等效泊松比,对程序及迭代方法的准确性进行了验证.之后对一种三元橡胶基复合材料进行多尺度均质化,将其分为芯丝尺度和层间尺度.用线弹性的均质化方法得到了芯丝尺度的等效弹性参数,并将其作为层间尺度的材料参数.在层间尺度应用非线性弹性均质化方法对结构进行计算,得到材料的宏观等效性能,并以实验结果为基准进行评价.

Microstructure is critical to affect or change the macroscopic mechanical properties of composites, and the desired material properties can be obtained by rationally designing the composite microstructure. As an effective design method, homogenization method is used to obtain and design the macro-mechanical properties on the basis of microstructure. However, once considering the nonlinear factors, the realization of homogenization can be very difficult. Therefore, this paper focuses on the nonlinear elastic homogenization of composite materials by theoretical deduction, and solves the problem by direct iteration method. In this study, the equation of nonlinear elastic homogenization is deduced by the asymptotic expansion homogenization method. The iterative steps of direct iteration method are given to solve the nonlinear elastic homogenization equation. According to the iterative steps and the nonlinear elastic homogenization equation, the program in MATLAB language is obtained. The porous materials with three typical constitutive relations are chosen to be the study object. The program and iterative method is verified by comparing the strain energy, maximum displacement and equivalent Poisson＇s ratio with the results of detailed model. Then, the application of nonlinear elastic homogenization method is extended to three-dimensional composite materials with multi-scale periodic microstructure, a three-element rubber-based composite material. It is divided into core-scale and layer-scale and homogenized with multi-scale homogenization method. The equivalent elasticity modulus of the core-scale are obtained by linear elastic homogenization method and used as a parameter of a component in layer-scale. Then, the nonlinear elastic homogenization method is used for layer-scale. The macroscopic equivalent performance of the material is obtained and compared with experimental results. The nonlinear elastic homogenization method has certain guiding significance and reference value for the nonlinear homogenization and microstructure design of the composite material.