Element-Based Peridynamic Theory And Fracture Problem Study

Fracture of materials and structures has always b een an important research topic in engineering field.Most theoretical models and numerical methods encounter singularity problem when dealing with discontinuous problem such as cracks,damage and fracture.Peridynamic theory establishes integral equations in the neighborhood of material points,it has natural advantages in dealing with discontinuity problem.However,classical peridynamics has many insufficient,such as cannot distinguish between volume deformation and shape change,Poisson’s ratio can only take a fixed value,does not contain the concept of stress and strain,cannot represent volume incompressibility when dealing with plastic deformation,the material properties should not change continuously with the angle when describing composite,and it has instability problem.In addition,peridynamics belongs to non-local theory,and its computational efficiency is much lower than that of local theory.Because of the disadvantages of classical peridynamic theory,this study proposed a new element-based peridynamic theoretical analysis method.And extended it to elastoplastic problem,heat conduction problem,coupled thermomechanical problem and composite laminates.At the same time,peridynamic theory and local theory are coupled to improve the computational efficiency.The proposed element-based peridynamic theory in the study covers the disadvantages of classical peridynamic theory largely,enriched peridynamic theory and it can further promote the theoretical research and engineering application of peridynamic theory to various defects including cracks.Firstly,some basic concepts of element-based peridynamics are introduced,including element composition rule,element stiffness density matrix,strain energy density,micro-modulus coefficient and surface correction coefficient.The static and dynamic element-based peridynamic model are derived by principle of minimum potential energy and Euler-Lagrange equation.The methods of applying initial and boundary conditions and characterizing cracks are given.Secondly,equilibrium equation and motion equation of element-based peridynamic model for elastic problem are derived using principle of minimum potential energy and Euler-Lagrange equation.The equilibrium equation in incremental form of element-based peridynamic model for elastoplastic problem is derived by principle of minimum potential energy.And the solutions of equilibrium equation and motion equation are clarified.For the elastic problem,the critical strain energy density was determined as the criterion for judging the failure of the material by considering that the interface energy needed to form a new crack was equal to the strain energy released by the material.For the elastoplastic problem,the strength value is used as the criterion to judge material failure.Thirdly,the steady-state and transient heat conduction equations of element-based peridynamic are derived by using principle of minimum potential energy and Euler-Lagrange equation.The element-based peridynamic model of coupled thermomechanical problem is proposed by combining the element-based peridynamic model of elastic problem with the heat conduct ion model.The solutions of heat conduction model and coupled thermomechanical model are given.The critical strain energy density is used as the failure cri terion to judge the material failure for the coupled thermomechanical problem.Then,the deformation and failure of composite lamina and composite laminate are analyzed by using element-based peridynamic theory.The Hashin failure criterion was introduced into element-based peridynamics as a criterion to judge the in-plane failure of composite lamina,and the effectiveness of the calculation method was verified by some examples.Finally,the peridynamic model is coupled with local model(discretized by finite element method)to improve the computational efficiency.The coupling scheme is applied to elastic problems,elastoplastic problems,heat conduction problems and coupled thermomechanical problems.The advantages of element-based peridynamic theory proposed in this study are as follows: there is no limitation of Poisson’s ratio,including the concepts of non-local stress and non-local strain;the material parameters can change continuously with the angle and contains four independent material parameters when dealing with composite material;the volume deformation and the shape deformation can be distinguished;the volume incompressibility can be represented when describe plastic deformation;and there is no instability problem in element-based peridynamic theory.