Research On Elastic Contact Problems By Boundary Integral Equation

In mechanical structures,the contact behavior always cause stress concentration,which increase the risk for wear and fatigue fracture.However,the classical theoretical analysis method cannot be used in practical complex contact problems.Nowadays,CAE analysis has become an indisp ensable means to study the complex contact problems.As a numerical method of CAE analysis,the boundary integral equation method(BIEM)converts the partial differential equations describing the behavior of unknown internal and boundary into integral equa tions only related to the boundary values.In the elasticity problem,only the boundary needs to be discretized,and there is no need to generate domain grid.In addition,in BIEM the traction and displacement are both unknowns of the equation,so the cont act constraint can be expressed directly by these unknowns.At the same time,since the traction and displacement are unknown,the traction and displacement are at the same precision,and the stress precision is high.When the BIEM is used to solve the ela stic contact problem,only the boundary information is involved,and the intra-domain grid is not needed.Therefore,when updating the mesh,only the boundary mesh needs to be updated.As for the stress concentration in the contact zone,the BIEM has high stress accuracy and is more likely to capture this feature.Therefore,the BIEM can be used to study the elastic contact problem in a simple and effective manner.The main contributions of this dissertation are as follows:(1)The mathematical expression corresponding to the three constraint methods are given by theoretical analysis,when the BIEM is used to solve the contact problem.When the BIEM is employed to solve the contact problem,there are three commonly used constraint methods: node-to-node method,node-to-point method and integral weak form constraint method,but the naming is mainly based on the characteristics imposed by the contact constraint.According to the basic conditions of contact constraints: the gap is zero in contact and the equilibrium should be satisfied,and based on the idea of weighted residual method,it is derived that the node-to-node method and the node-to-point method are both contact constraint collocation method.The method of integral weak form constraint corresponds to t he contact constraint Galerkin method.In addition,the contact-constrained Galerkin method is extended to three-dimensional frictionless contact,and the implementation process is given in this dissertation.Numerical examples demonstrate the validity of the proposed method.(2)To treat the problem of weak pressure discontinuity at the contact boundary,the dual-layer interpolation method and mesh updating method are adopted.The transition of the contact pressure from the contact zone to the non-contact zone is not continuous.Due to the discontinuity,the accuracy of the integration is difficult to ensure,which affects the calculation accuracy.In this dissertation,for general contact problems with an a priori unknown contact boundary,the problem of weak pressure discontinuity is treated.Firstly,a method to detect the contact boundary is proposed.Then the element near the contact boundary is updated,and the ability to approximate both continuous and discontinuous fields in dual-layer interpolation method is used to treat the problem of weak pressure discontinuity at the boundary.In addition,the pressure oscillation near the contact boundary is also treated.(3)The mesh updating method is used to avoid the problem of mesh encryption in the whole contact area in the moving or rolling contact problem,and the equivalence of the boundary integral equation before or after moving or rolling is derived.There are a large number of moving or rolling contact problems in the engineering.Accurate calculation of the mechanical properties at various moments in moving or rolling contact requires mesh encryption in the entire contact area.To avoid this problem and improve the computational efficiency,this dissertation adopts the mesh updating method,by which the mesh encryption is only needed at the potential contact zone at each moment.In addition,since the space coordinates change after moving or rolling,the boundary integral equation needs to be recalculated at each moment.To avoid this problem,in this dissertation,the equivalent of the boundary integral equation before or after moving or rolling is derived by using the rotating coordinate system.Therefore,the integration data of the previous moment can be utilized in the latter moment,so the problem of recalculating boundary integral is avoided.This method is successfully applied to the two-dimensional moving or rolling contact problem.(4)A new explanation and a strategy are given for the problem that a sudden jump in contact pressure is often observed near the singularity.In the boundary element method(BEM)computations for contact problems,a jump in pressure often occurs near the singular point.The traditional explanation of this phenomenon contradicts some facts,and the proposed solution is not very successful.In this dissertation,we find that pressure jump often occur in linear and quadratic element rather than in constant element.According to this special phenomenon,this dissertation compares the differences between constant element,linear element and quadratic element interpolation.Based on this comparison,a new explanation is proposed.The new explanation not only answers the question why the pressure jump often occurs in linear and quadratic element,rather than in constant element,but also answers the question why mesh refinement does not treat the pressure jump.At the same time,based on this explanation,a strategy to treat the pressure jump is given.Finally,the explanation and the strategy are verified by two-dimensional and three-dimensional contact numerical examples.(5)Sliding direction prediction technique is adopted to determine the sliding direction,and the convergence of the 3D friction contact is addressed.When the direct method is used to solve the 3D friction contact problem in BEM,the convergence is often difficult to guarantee.The convergence depends mainly on how to determine the sliding direction,but there is still no uniform and effective way to determine the sliding direction.In this dissertation,the sliding direction prediction technique is introduced to give a way to determine the sliding direction.The sliding direction is obtained by using the sliding direction prediction technique.If sliding occurs during the subsequent iteration,the sliding direction always takes the direction obtained in the prediction technique.In addition,the frictional energy consumption condition is used to judge the contact point entering the sliding state,so it avoids the problem that sliding point is always slides no matter how it changes in the future.Finally,numerical examples demonstrate the validity and the feasibility of the prediction technique.