The Scale Affect In Constitutive Equation Of Periodic Mesoscale Material

The material which is discussed in classical theory is the leading order material, and the stress of some point only lies on the strain or strain's history of the point. But recent years, many examinations indicate that in some situation, the stress of some point not only lies on the strain, but also lies on the gradient strain of the point. This is the non-local affection. From physics, it is close to the essence of problem if you study the interaction of basal particle when considering material's action. So we should constitute a model between the classical model and atomistic model to resolve the problem which needs to consider the scale. In this way, we need to research a multi-scale modeling method which can obtain the macro-scopical properties from meso-scale phenomena. Accordingly, this paper use mathematical homogenization theory to research the non-local constitutive relationship of macro-scale periodic material.Chapter one set forth that people are interested in the characteristics of scale which represented by material properties, and it put forward the concept of non-local affects, consequently, it demonstrates the necessity of studying this problem. From reading a lot of literatures which are at home and abroad, I summarize the history and the present state of the higher order theory and multi-scale modeling. After the analysis of this project's background, I present the research goal, method and content.Chapter two provide a constitutive relationship of non-local affection generally. It transforms the variance speed of time to space-time based on kinematics description by Lagrange. Then an example is adopted here to discuss the variance of wave speed from three phases which are elasticity, elastoplasticity and strain localization. After the strain localization, we use the constitutive description of non-local affection to discuss the property of the wave's propagation in the elastoplastic material.Chapter three make use of the one-dimensional chain of atoms, and use mathematical homogenization theory to research the non-local equation of motion. The total energy stored in the crystal is comprised of the kinetic and potential energies, and we can utilize the Hamilton principle to acquire the balance equation. In order to simulate the model phenomena at both the continuum and discrete scales, we utilize the double scale asymptotic expansion, and introduce usual time coordinate and slow time scale. Then we use the periodicity of the unit and the Taylor expansion to obtain a series of properties. And then we use mathematical homogenization theory to obtain the equation of motion, finally, we can acquire the non-local equation of motion which is considered by scale affect from eliminating the dependence of slow time.Chapter four make use of the two-dimensional model, and it use mathematical homogenization theory to research the non-local equation of motion. For two-dimensional model, it can easily obtain the equation of motion of any atom, and write them by discrete form. Then we utilize Taylor expansion of displacement to obtain a series of properties and equation of motion by mathematical homogenization theory. In the two-dimensional model, there are not only axes spread, but also rolling, so we also study the equation of motion of the two-dimensional model when considering the rolling of two atoms.Chapter five make use of the BCC crystal, and use mathematical homogenization theory to research the non-local equation of motion. For square-hexahedron, it can easily obtain the equation of motion of any atom, and write them by discrete form. Then we utilize Taylor expansion of displacement to obtain a series of properties and equation of motion by mathematical homogenization theory, finally, we can acquire the non-local equation of motion by eliminating the dependence of slow time.Chapter six analyzes the development of meso-distortion of soil by examination data and the meso-scale parameter's affection on macro-character. Soil is an important material of engineering construction, the variance of it's inter structure decide the macro action's property of evolvement. So the studying of the kinetic variance of micro or meso-structure is important. This chapter introduces the meaning of the CT experimentation firstly, and it also introduces the method and procedure of the experimentation. Then we obtain the variance rule of each parameter by analyzing the initial data of the experimentation. Then, we select a sample of farinaceous soil to analyze, and obtain the development of meso-distortion. Finally, we discuss the meso-scale parameter's affection on macro-scale character from studying the rigid tensor of equation of motion.Chapter seven present the results in this thesis and set forth the further research.