Research On Non Euclidean Model For Incompatible Deformation Of Materials

Fatigue,damage,fracture and failure of structures are the common problems in structural analysis encountered and not well solved in the fields of civil engineering,machinery,transportation,hydraulic engineering,aviation,aerospace and so on.The stress and strain of materials are determined based on the classical continuum mechanics and thermodynamics.One of the basic assumptions of classical elastoplastic theory is Saint-Venant deformation compatibility principle.The deformation compatibility equation of Saint-Venant must be satisfied in the process of deformation.The compatibility of material deformation is determined by the continuity of material,which assumes that there is no dislocations,disclinations and other defects in material.However,almost all engineering structures work with defects.The classical continuum mechanics is no longer applicable when the dislocations,disclinations and other incompatible deformations occur.Therefore,it is particularly important to study the incompatible deformation of materials with defects.In the 1980 s,academician Chen Zongji pointed out that: the residual stress inside the object has not been paid enough attention,and its source and nature are not clear;when the boundary of the object is free and there is no additional applied load,there is a self-balanced closed stress inside the object.For solid mechanics,only the research on the compatible deformation of materials has been carried out in the past,but the research on the incompatible deformation of materials will open up a new prospect.At present,the problems to be solved concerning the elimination of stress concentration and description of residual stress are:(1)how to eliminate the stress singularity in the cylindrical coordinate system and how to describe the distribution of the residual stresses due to the incompatible deformation in the cylindrical coordinate system.(2)How to eliminate the stress singularity in the spherical symmetric coordinate system and how to eliminate the stress concentration at the central point.(3)How to establish a two-dimensional non Euclidean geometric model to describe the distribution of stress and strain in one-dimensional bar under strain localization.In this thesis,the stress distribution is studied in the framework of Riemann model in continuous medium.The expression of material defect parameters representing the degree of deformation incompatibility is obtained.The phenomenological parameters of the expression are determined by experimental data.The results show that the non Euclidean geometry model describing the residual stress of an object in a cylindrical coordinate system is consistent with the experimental data.Residual stresses is actually the product of the incompatible deformations of the object.We also study the problem of strain localization.The elastic-plastic material with structural defects is actually a non-Riemannian manifold rather than a continuum in Euclidean space.Therefore,it is reasonable to use non Euclidean theory to solve the problem of strain localization.The two-dimensional non Euclidean geometric model constructed in this paper can well describe the stress at any point of one-dimensional rod.From a macroscopic point of view,it can well explain the balance of forces.The results show that the theoretical model is in good agreement with the experimental data.For the first time,Aifantis used gradient elasticity theory with characteristic internal length to solve the stress concentration effect of hollow specimen.However,how to eliminate the stress singularity in spherically symmetric objects has not been solved.In this thesis,the problem of classical stress distribution in spherically symmetric coordinate system is solved,and the non Euclidean geometric mathematical model of stress distribution in spherically symmetric coordinate system is established.The problem of stress concentration at the edge of an infinite small hole in spherically symmetric coordinate system is eliminated by the method of superposition of general solutions of differential equations with special solutions.The results are consistent with those obtained by Aifantis in solving the fourth-order differential equations.Both show that when the small hole is infinite,the radial stress and circumferential stress are equal to zero.