Limited Influence Region Nonlocal Theory And Finite Element Method

Nonlocal continuum mechanics is a promising theoretical approach, because it endows the materials with microstructure information, namely the full consideration of the long-range influence between the particles, so that it can appropriately solve some problems that the classical mechanics cannot explain such as the deformation localization, the crack tip stress singularity, etc. The classical nonlocal theory includes a global field integral term, so it will have a large amount of calculation. Therefore, because the theory of nonlocal elasticity of Eringen has the characteristics of rapid attenuation of interaction between particles, the global domain will not be calculated in this thesis, but to build finite element models in a limited domain of influence so as to greatly improve the efficiency of solving equations. By comparing the finite element solution with the analytic solution thus to evaluate the impact of weighted kernel function related to nonlocal theory, the length of the range, the size of finite element unit, etc. on the result of calculation. The main contents and results in this thesis are as follows:(l)Discuss the classical one-dimensional Eringen model, analyze the characteristics of analytic solution and then study the boundary effect of one-dimensional models that is the strain field has the feature to increase rapidly in the model boundaries and quickly attenuate within a certain range.(2) Firstly, obtain the nonlocal minimum potential energy functional through variational principle. And then use finite element method to discrete the functional so as to formulate and implement the finite element equations. Finally arrived at a conclusion that the total displacement field got by nonlocal minimum potential energy principle is also smaller than the analytic solution.(3) Discuss the performance of the global stiffness matrix under nonlocal theory and reached that the global stiffness matrix of nonlocal limited influence domain has the characteristics of symmetry, sparsity, singularity, etc. According to these characteristics, half the bandwidth of the storage method is adopted in the thesis for storing the stiffness matrix, which greatly reduced the computer memory.(4) Use finite element method to solve one-dimensional limited influence domain of nonlocal Eringen models, compare the results with analytic solutions thus obtained that when the limited influence domain length is 6 times the element length meanwhile the difference between the length of finite elements and the length of characteristic scale is small, the result of the finite element solution is more accurate.(5) By comparing the different results solved by finite element method, when kernel function as exponential kernel and triangle kernel function. It reached that exponential kernel function is more appropriate to be used in one-dimensional nonlocal finite element method.Through this paper, it comes to a conclusion that finite element solution of nonlocal limited influence domain has the characteristics of efficiency and accuracy which is a promising method of numerical calculation and is worth promoting in two-dimensional and three-dimensional models.