Dynamic Fracture Research Of The Plane Elastic Continuum With Cracks Based On Finite Element Method With Generalized Degrees Of Freedom

Under the action of dynamic load,the structural response of engineering structure with cracks is a function of time.Compared with the static load,the dynamic load is more likely to cause crack propagation and cause structural instability,and the theoretical derivation and solution are more complex because of the inertial effect of load.Therefore,it is of great significance to carry out a high-precision and efficient numerical algorithm for structural fracture behavior under dynamic load.Based on the finite element method with generalized degrees of freedom,the Williams element with generalized degrees of freedom(W element)is derived for the stress intensity factors(SIFs)analysis of planar elastic structures with cracks,assuming the nodes displacement of the radial discrete element in the singular region of the crack tip satisfy the rigid body rotation theory.Because the displacement field of W element still contains the parameters related to SIFs,it can be obtained directly,which overcomes the defect that the traditional singular element needs linear extrapolation solution every time,which leads to artificial error.In this paper,the advantages of W element are fully used to study the dynamic fracture of planar elastic bodies with cracks(1)The 1/4 singular element calculation scheme for SIFs analysis of plane elastic body with cracks under dynamic load is derived,and the corresponding Fortran program is compiled.Firstly,based on the construction of conventional eight node isoparametric element and quarter singular element,the global stiffness matrix,mass matrix and load array of the model are derived according to the dynamic finite element theory,and the global control equation is established.Then,the displacement vector of the nodes is obtained by solving the global control equation with Newmark integration method,and the SIFs at the crack tip is obtained by linear fitting with displacement extrapolation method.Through the verification and analysis of classical examples,the correctness of singular element solution is proved,which lays a theoretical foundation for the follow-up research work.However,when solving SIFs by singular element,linear extrapolation is required in each load cycle.This process not only introduces artificial secondary error,but also easily causes error accumulation and affects the calculation accuracy.(2)Based on the theory and foundation of singular element,combined with finite element method with generalized degrees of freedom,the displacement vector of element is transformed into generalized parameter vector directly related to SIFs by introducing Williams series displacement field in singular region,so that directly solve global control equation and obtain high-precision SIFs at crack tip.However,since the velocity and acceleration vectors of the virtual nodes in the inner diameter of the singular region can not be expressed explicitly,it is assumed that the nodes displacement approximately satisfy the rigid body rotation theory.According to the proportional relationship between the coordinates of each node,the displacement,velocity and acceleration of the internal virtual node are calculated based on the real node outside the singular area,and then the approximate solution of the load vector of the element in the singular region is obtained.On this basis,the three important parameters of W element and the recommended values of singular region size are studied.The results show that when the three important parameters of W element are taken as m=12,α=0.95,n=400,and the size of singular region taken as (?),the calculation results have high accuracy.(3)In order to further study the universality of W element in dynamic fracture analysis,different crack parameters and dynamic load types are set up for analysis,that is,the peak value of SIFs curve is analyzed by changing the length,inclination angle and position of crack.At the same time,the step load,linear load and sinusoidal load are used to calculate in turn.The results are in good agreement with ANSYS,which proved that the W element can be applied to dynamic fracture analysis of plane elastic body with crack under different dynamic loads.On this basis,the influence of conventional grid division on SIFs is studied,and the influence of material properties on the peak value and lag time of SIFs curve is analyzed according to different values of elastic modulus,density and poisson ratio.The example analysis shows that W element has good convergence and stability;increasing elastic modulus and density of structure under dynamic load will shorten and extend SIFs at crack tip respectively The lag time of the curve and the change of poisson ratio will change the action order of stress wave on each crack tip,and then affect the variation law of SIFs at the crack tip.