Finite Element Implementation Of Boned-based Peridynamic Theory And Its Numerical Simulation In Damage Process Of Brittle Materials

Classical solid mechanics can be troublesome in dealing with discontinuous damage problems because its theory involves displacement differential equations which are not defined at discontinuities.The emerging peridynamic method is based on the long-range nonlocal interaction,and use the spatial integral equation of displacement to describe the mechanical behavior of the brittle material.It does not involve the spatial differentiation of the displacement,which is still defined at the discontinuity.The peridynamic theory has both the advantages of molecular dynamics and meshless methods,and is better suited for simulating cracks,damage and other discontinuous.However,since peridynamic numerical methods consider non-local interaction and dynamic relaxation numerical solution method,there are some disadvantages such as low calculation accuracy,skin effect and fuzzy boundary,and these can be solved with numerical simulation methods based on traditional continuum mechanics,such as the finite element method.Therefore,it is very advantageous to solve governing equations of peridynamic using the FEM’s theoretical framework and combine peridynamic method with traditional finite element method,and then apply this coupling method to simulate the failure of brittle materials.The application of peridynamic in practical engineering problems and the study of failure mechanism of brittle materials can be greatly promoted.Based on the principle of minimum potential energy,this paper studies the finite element implementation of peridynamic by Taylor expansion of the total potential energy of the system and the peridynamic method is combined with the traditional finite element method in finite element software ABAQUS.So that peridynamic method is more easily applied in different science fields and practical engineering analysis.Finally,an example is given to demonstrate that this method can be used to simulate the one-and two-dimensional non-fracture problems accurately,and the brittle crack propagation problem of two-dimensional plate is also studied.The research results are obtained as follows:(1)In this paper,the surface effects and the difficulties of applying boundary conditions in the peridyanmic method are discussed in detail.The causes of the boundary effects and the advantages and disadvantages of the commonly used boundary effects correction methods are analyzed.(2)Based on the principle of minimum potential energy and the idea of atomic-scale finite element method which accounts for the multi-body interactions among atoms and the basic theory of bond-based peridynamic method,a rigorous derivation of the element stiffness matrix and the non-equilibrium force vector of one-and two-dimensional cases in the finite element implementation of peridynamic method accounting for the nonlocal interactions among material points are presented.And this kind of matrix element stiffness matrix and the non-equilibrium force vector can be applied to static and dynamic problems in various dimensions.(3)The peridynamic element is developed by the user-defined element program provided by commercial finite element software ABAQUS which may be the best suite of non-linear finite element analysis.Meanwhile,all issues about computational efficiency,skin effect and fuzzy boundary are avoid.(4)The coupling method of the peridynamic and the finite element method can directly use irregular shape meshes,not just the traditional square rule grid generated by ABAQUS software.Two ways to calculate the area of the irregular quadrilateral mesh and the influence of the calculation method of its area on the numerical accuracy is discussed.When the traditional finite element is used as the boundary,the width of the finite element region should be the same as the horizon radius of material points in peridynamic.(5)By coupling of peridynamic element with the traditional finite element,the crack initiation,expansion and failure of the brittle material are simulated in the ABAQUS,and the position and path of the crack are accurately predicted.This realizes the transition from the finite element calculation of the continuum to discontinuous failure calculations spontaneously.