Modified Gradient Elasticity And Its Damage Theory-Finite Element Implementations And Its Applications

The failure of the material is century difficult problem in the development of mechanics,gradient theory is the necessary way to solve the material failure mechanism. Because the assumption of homogeneity can not be strictly satisfied, classical continuum theories fail to predict the failure phenomena which depends on micro-structure, such as strain or damage localization, size effects, and mesh-size independence of numerical calculations. In order to describe the mechanical behavior more reasonably, we attempt to propose a modified gradient elasticity theory and its damage theory by introducing strain gradient, an internal length scale vector and damage in the continuum equations.By defining an internal length scale vector which is equal to the ratio of strain and strain gradient and assuming that the strain energy density depends on both the strain tensor and the strain gradient tensor, a modified gradient elasticity(MGE) theory is derived. Due to the strain gradient term which is introduced in the continuum equations, the MGE model can describe the strength and deformation behavior of materials when the strain gradient is great;the MGE can be simplified to classical elasticity theory when the internal characteristic length vector vanishes. Based on the principle of virtual work, the corresponding variational principle, boundary condition for finite deformations and finite element formulation of the MGE for statics are derived. A Fortran90 code for the numerical modeling is given. The results of a numerical example demonstrate that the MGE can be simplified to classical elasticity theory when the internal characteristic length vector vanishes and the MGE code is correct.The bi-material strip with shear and tensile boundary layers and the singularity at crack tip is numerically calculated by the MGE. It is found that the thickness of shear and tensile boundary layers strongly depends on the internal characteristic length, in addition, the singularity at crack tip the mesh-dependence can be avoided.Finally, we present a modified gradient elasticity theory and its gradient elasticity damage theory by introducing the damage variable in the continuum equations of the MGE.The MGE damage model and its finite element formulation is established. A Fortran90 code is given to simulate damage localization phenomena of uniaxial compression of geo-material.It is found that MGE damage theory can eliminate the mesh-dependence and simulate theinitiation of geo-materials with localized damage and the development of the failure process;the width of localized damage shear bands depends on the internal length scale vector.