Research On The Problem Of Velocity Fields In Plastic Deformation Based On The Theory Of Rotation-rate Continuity

In the process of metal plastic deformation,the velocity field distribution in the plastic deformation region can reveal the principle of metal plastic flow,which has important theoretical guiding significance for the formulation of technological process and parameter optimization in plastic processing.The velocity fields determined by many traditional analytical methods are usually kinematically admissible fields with non-unique properties,which restrict the application of metal plasticity flow theory in solving velocity field.In view of this,this paper investigates the velocity characteristics of the theory of rotation-rate continuity based on a physcial background in crystallography.In the rigid-plastic solids which flow plastically by the mechanism of “extended slip”,the sliding crystalline plane is parallel to its maximum shear stress plane.At this time,the rotation-rate vector remains continuous in space and the velocity vector field follows the Laplace equation,which is unique with prescribed boundary conditions.On this basis,this paper derives the corresponding partial differential equations for velocity fields following the Laplace equation in different coordinate systems and deformation conditions,gives analytical solutions for the corresponding multiple plastic deformation problems(including three-dimensional problems),and discusses the differences between the velocity fields following the Laplace equation and other analytical and numerical solutions.This study establishes a quantitative mathematical relationship between microscopic crystal slip and macroscopic principle of material plastic flow.It has the advantages of a crystallographic physical background and uniqueness of solution compared to traditional analytic methods,and provides a reference for further study of the fundamental principles in plastic mechanics.Firstly,the similarities and differences between the theory of rotation-rate continuity and the slip-line theory and ideal plastic deformation theory are discussed,the Euler-Lagrange variational equations based on the theory of rotation-rate continuity are derived,the partial differential equations associated with the velocity field corresponding to the E-L equations in different coordinate systems and under different deformation conditions are given,and the general solution of the velocity field is obtained.It is shown that the partial differential equations associated with the velocity field obtained from the E-L variational equation are consistent with those obtained when the velocity vector field follows the Laplace equation and when the divergence of the strain-rate tensor is zero,and the three can be transformed into each other.Then,the differences in the velocity fields in the single curvature and double curvature sheet bulging models and their influence in predicting the configuration and strain distribution are analyzed.The sheet will bulge into a sphere of single curvature if it follows the theory of rotation-rate continuity,with a velocity field that satisfies the Laplace equation and always points in the direction of the outer normal of the instantaneous profile of the bulging specimen.If the velocity field does not satisfy this condition,the sheet will bulge into a ellipsoid with double curvature,and an incremental iterative algorithm is proposed accordingly.There is no significant difference between the two models in predicting the bulging profile,and the double curvature model can better predict the strain distribution in thickness direction of the sheet.However,the double curvature model has no explicit solution and requires a numerical iterative algorithm for calculation,while the single curvature model is simple and easy to calculate and suitable for application in engineering.Secondly,the differences between the velocity field that follows the Laplace equation and the analytical solution obtained from stress equilibrium in the plastic plane torsion of the disc are analyzed.The velocity field which obeys the Laplace equation and its corresponding displacement field are obtained based on the E-L variational equations associated with the theory of rotation-rate continuity,and the analytical solutions of the velocity fields and displacement fields are also given for different strain-hardening material models.The results show that the velocity fields satisfying the Laplace equation are asymptotic solutions to those of nonlinear strain-hardening materials obtained from the static stress equilibrium equation,and the results of the finite element analysis agree well with the theoretical calculations,the reasons for the differences in the velocity fields are given from the thermodynamic and crystallographic perspectives.After that,based on the method of fundamental solution(MFS),the general solution method when the velocity field in the Cartesian coordinate system obeys the Laplace equation is given,and it is applied to solve the generalized problem of eccentric transverse deflection of a clamped rigid-plastic thin plate under quasi-static loading of a rigid flat-ended punch(including loading combinations of arbitrary cross-sectional shapes of punch vs.plate and simultaneous loading of multiple punches).By finding the analytical solution which satisfies the Laplace equation in the doubly-connected solution domain with prescribed boundary conditions,the deflection of any point on the sheet and the p unching force required when the punch is loaded at different positions are obtained.The finite element analysis results agree well with the theoretical calculation results,validating the correctness of the model based on the theory of rotation-rate continuity.Finally,taking the analysis of the characteristics of the velocity field in the disk and ring upsetting problem as an example,the uniqueness and applicable conditions of the kinematically admissible velocity field in the plastic deformation process of the material obeying the theory of rotation-rate continuity are discussed.It is proved that if the rotation-rate vector of the disk or ring remains continuous during the upsetting process,the strain-rate tensor is divergence-free,and the radial velocity field on the lateral surface is evenly distributed,and no lateral bulging occurs.This deformation state only occurs when the anvil surface is smooth.When the anvil surface is rough,using a parabolic type of velocity field to describe the radial velocity distribution on the lateral surface can well predict the lateral bulging during the upsetting process.In the end,the influence of lateral bulging on the limit load in the process of ring upsetting is analyzed.