Research On Peridynamic Lattice Model And Its Applications On Dynamic Brittle Fracture Modeling

Fracture behavior of structures has received extensive attentions and researches in the past decades.Methods based on continuum mechanics,such as finite element method and boundary element method,are widely used in the fracture modeling.A common problem of these methods is that the stress singularity caused by the spatial discontinuity at the crack tip,which also poses new challenges to the guiding criteria of crack initiation,propagation,bifurcation and other behaviors.Silling proposed a general mechanical model in the theoretical,mechanical framework,called Peridynamics(PD).Peridynamics theory is a continuous generalization of molecular dynamics equation.This model corrects the shortcomings of continuum model which can not calculate continuity and discontinuity simultaneously.In Peridynamics,the motion equation does not assume the spatial differentiability of displacement field,and allows discontinuity to appear as part of the solution.It has great advantages in dealing with discontinuous problems such as crack propagation.Because all material points within a fixed distance can interact with each other in the reference configuration,the Peridynamics is classified as a non-local method.Same as other nonlocal theories/models,combination of PD with other mature methods is necessary.Among some PD combination methods,Peridynamic lattice model(PDLM)is a model applicable to engineering structures.It can treat the ambiguity of boundary treatment,the influence of boundary effect on simulation results,the difficulty of imposing displacement boundary conditions and so on well.However,as a new combination model,there are still some issues need to improved and complemented in Peridynamics.Therefore,on the basis of the above problems,the Peridynamic lattice model is inherited and developed in this thesis to study the dynamic crack behaviour of brittle materials.The main work includes the following:Firstly,the Peridynamics lattice model based on regular lattices is improved.The bond-based PD theory is introduced,and the constitutive relation,failure criterion and numerical realization are summarized.Theoretical derivation of PDLM is given based on the regular lattice as the basic element,failure model based on critcal stretch and boundary condition treatment method are also developed.In order to make full use of the computational efficiency of lattice model,the matrix calculation process of PDLM discrete model is given.The explicit expression of the stiffness matrix of the bond in the form of "bar" is obtained.The lattice forms suitable for PDLM are compared.Secondly,a parameterized study is carried out on PDLM to investigate its capability of dynamic crack propagation in engineering structures.One and two-dimensional isotropic material are used in the PDLM simulaiton.Firstly,the elastic vibration(wave)response of one-dimensional model is analyzed by PDLM.In the vibration(wave)analysis of the bar,results obtained by PDLM is in good agreement with the theoretical solution.After parameter analysis of the lattice radius,it is found that the lattice radius has little influence on the simulation effect of the wave in the rod,but it has great influence on the total calculation time.In the two-dimensional problem,focus is put on the dynamic crack propagation problems.The parametric analysis of two-dimensional model needs to consider more parameters,including the lattice radius,the lattice rotation angle,the topological relationship of the lattice,the modeling method,etc.The above parameters are analyzed in detail,and many meaningful conclusions are obtained.Considering that the failure criterion based on critical stretch in PDLM is local,two kinds of nonlocal failure criteria in PDLM are extended: maximum principal stress criterion and strain energy criterion.Their expressions are given respectively.The flat plate with two sides constrained is subjected in horizontal/vertical directions.Three kinds of failure models are evaluated by a progressive failure example under tension.Then three kinds of failure models are applied to the dynamic crack growth model.The simulation proves the ability of the maximum principal stress and strain energy criterion in dealing with size dependence and mesh bias.Thirdly,in order to solve the problem of constant Poisson's ratio in bond-based PDLM,a volume compensation VPDLM is proposed to add the volume potential energy caused by volume change to the cell strain energy density.Firstly,the volume strain energy caused by expansion is introduced into the cell,and the relationship between bond micromodulus parameters and engineering constants is deduced,and the explicit expression and range of Poisson's ratio are obtained.Then a non-local failure model for VPDLM is presented.The relationship between critical stretch and strain energy release rate is obtained by energy equivalence.VPDLM is validated by elasticity problem and dynamic fracture damage problem respectively.The elasticity problem proves the ability of VPDLM to simulate different Poisson's ratios.The damage and fracture process of Kalthoff-Winkler test model was studied by VLPDM.The complete crack propagation process was successfully given.The crack analysis of each crack stage was given.The pure shear damage mode mechanism of K-W test was analyzed by wave analysis.At last,PDLM for fracture simulation of composite laminates is proposed.According to the directional characteristics of the lattice,the rotational direction of the lattice is taken as the direction of the fiber,and the analytical expression of the properties of PDLM bonded by the fiber and the matrix is obtained.By the equivalence of strain energy density with strain energy density in classical lamination theory,the expression of micromodulus is obtained,and the relationship between critical stretch and strain energy release rates of type I and type II is obtained.The elastic response of composite laminates under fiber orientation and tension was studied by PDLM method.The simulation results are in good agreement with the test results.The numerical analysis of laminates with circular and linear pre-cracks is carried out.Cack propagation path obtained by PDLM is in good agreement with the experimental results,which shows the capability of PDLM in modeling dynamic fracture of composites.In this paper,the foudation of the Peridynamic lattice model is constructed,and its constitutive relation,failure criterion,boundary treatment method and numerical realization process are given.The model is parameterized by one-dimensional and two-dimensional numerical examples.Based on parameterized verification,two kinds of non-local damage criteria are proposed and systematically evaluated.By the volume-compensation in strain engergy density in a cell,a general Peridynamic lattice model which can simulate different Poisson's ratios is obtained.Based on this model,a Peridynamic lattice model with arbitrary geometric mesh is developed.Finally,the Peridynamic lattice model is applied to the simulation of composite structures.The fiber bond and matrix bond parameters are obtained by homogenization method with the rotation angle of the lattice as the direction of the fiber bonds.The Peridynamic model and its extended model proposed in this paper have certain theoretical and engineering guiding significance for the dynamic brittle fracture analysis of brittle materials.