Analysis Of Serveral Problems In Nonlocal Continuum Mechanics

Nonlocal continuum mechanics is a generalization of classical continuum theory. In this theory, the influences of scale effects and long-range interactions among microstructures within materials on the macroscopic mechanical properties are considered. Therefore, many problems that are difficult to be explained by classical continuum theory can be investigated effectively by the nonlocal continuum mechanics. These problems include the strain localization, crack tip stress singularity and wave dispersion etc. The nonlocal continuum mechanics provides a new and valuable approach to solve the macro-micro correlation problems in mechanics.The main objective of this thesis is to investigate some controvertible and unsolved problems existing in the nonlocal continuum mechanics, including the definition of nonlocal plane strain/stress problem in the nonlocal elastic theory and analysis for the accuracy of solutions; the simulation for the strain localization behavior based on the nonlocal damage model; the evolution of strain localization zone; nonlocal cohesive model at the crack tip and the implementation of nonlocal constitutive model in the finite element (FE) software ABAQUS. The whole thesis is divided into 7 parts. Chapter 1 begins with a brief introduction of the background and development of the nonlocal continuum mechanics. Chapters from 2 to 6 constitute the main body of this thesis. The problems mentioned above are investigated in these chapters. The concluding remarks are given in Chapter 8. The main academic contributions of this thesis are listed as following:(1) The plane strain and stress problems are defined in the framework of the nonlocal elasticity. The governing equations of the two problems are respectively derived from 3-dimensional theory. The accuracy of solutions to the two plane problems in the nonlocal elasticity is analyzed in detail. It is found that in the nonlocal plane strain problem, the governing equations reduce to a group of two-dimensional equations accurately, and corresponding solutions are also accurate compared with 3-dimensional nonlocal elasticity. On the other hand, the governing equations in the nonlocal plane stress problem are only a group of approximate equations due to the nonlocal effects in the direction of thickness. Through analysis for the Fourier transform of the strain compatibility equations, we find that the solutions to the nonlocal plane stress problem violate the strain compatibility conditions in 3-dimensional nonlocal elasticity. So the solutions of the nonlocal plane stress problem are also approximate relative to 3-dimensional theory.(2) Through taking strain and damage as independent nonlocal variables, a thermodynamics -consistent integral-type nonlocal brittle-elastic damage model is established. The nonlocal elastic damage constitutive equation and the damage evolution equation are deduced. On the basis of these results, we simulate the strain localization phenomenon in a bar subjected to uni-axial tension. The results show that the strain will concentrate to the central part of the bar in the damage process, and the stress-strain relation in the localization zone exhibits an obvious softening feature. In addition, the influences of different nonlocal kernel functions on the strain locaization solutions are analyzed. A rational proposal on how to select the nonlocal kernel function is given.(3) The uniaxial tension tests of duralumin and the low carbon steel specimens are carried out so as to investigate the change of the strain localization zone with loading. By comparing the deformation patterns of specimens at the beginning and at the end of damage, it can be inferred that the strain localization zone decreases with the load increasing. This conclusion is also verified by the existing experimental data obtained by ESPI method. In order to explain this conclusion in theory, correlation between characteristic length within material and damage is firstly investigated. The variability of the characteristic length is found, and the nonlocal kernel with variable equation is determined. These theoretical results can rationally show the decrease of the strain localization zone with loading.(4) A new cohesive crack model is constructed based on the equivalence between the cohesive stress and the surface-induced traction (nonlocal surface residual) occurring in the nonlocal stress boundary condition. By means of the rational mechanics approach, a new cohesive stress law is derived logically. In this new cohesive law, the dependence of the cohesive stress on the surface energy density and curvature of crack surface is determined. Applying this new cohesive crack model, we investigate the stress field ahead of the crack-tip in the brittle fracture problem. The results show that the stress singularity at the crack tip is removed, the maximum stress occurs within the cohesive zone away from the crack tip, and the cohesive stress vs crack opening displacement (COD) curve exhibits significant softening characteristics in the cohesive zone. Moreover, the nonlocal interaction radius is introduced to act as the actual crack tip opening displacement (CTOD), which gives a simple way to calculate the size of cohesive zone.(5) On the finite element software ABAQUS, the VUMAT subroutine for the gradient-type nonlocal elastic constitutive model is implemented. Based on this subroutine, we calculate two examples in the nonlocal elasticity. By comparison between the nonlocal solution and the classic elastic solution, it can be found that a boundary effect induced by non-locality appears in the nonlocal solution, while it can not be revealed by the classical elastic theory. In addition, a VUMAT subroutine for the nonlocal gradient-dependent plasticity with kinetic hardening is implemented and testified by some simple examples.