The Riemann Problem Of Elastoplastic Solids

In elastic-plastic fluid mechanics, impact physical, detonation physics, and many other fields of engineering application, people pay special attention to the problem of the deformation of the solid and the propagation of related stress waves. Numerical simulation of deformation process is an important method to study the properties of material under high pressure and intermediate pres-sure as well as laws of physics. In this article we study the numerical methods of stress wave propagation in linear elastic ideal plastic material and linear e-lastic hardening material.The main point is the numerical methods of Riemann problem.All content can be divided into four parts. The first part is about the problem of one-dimensional elastic-plastic bar. We solver the Riemann problem for one-dimensional decreasing hardening and linear hardening bar, and give the accurate Riemann solution. Then we give the numerical solution with Godunov scheme and random choice method(RCM) which was proposed by Glimm.At last,we verify the correctness of this method using two classic example heading loading problem and catching unloading problems. The second part discuss the one-dimensional strain plane wave of the hardening materials and ideal plastic material. Due to containing the lateral normal stress which is used for horizontal restraint, the problem can be regarded as a degradation of two-dimensional prob-lem. Its Riemann solution can be directly applied to the numerical solver of two-dimensional problems. The third part takes the two-dimensional anti-plane shear problem an example to study the problem of Riemann solver of two-dimensional solid. Different from most studies which pay attention to the steady state crack problem, we still consider the two-dimensional stress waves and therefore still treat them as hyperbolic conservation rate equations. The numerical method of this problem is cell center Godunov scheme. We turn the problem of the bound-ary flux solver to solving an one dimensional Riemann problem by finding an rotation invariant. Due to the constitutive relation no longer meeting the ideal plastic condition, the loading paths of stress should be carefully chosen. As the fundament of the Riemann of high pressure state multi-dimensional elastic-plastic problem which will be study in the future,the last part of the article studies the numerical methods of euler equation of fluid mechanics under cylindrical coordi-nate system. We use the numerical two-dimensional Riemann solver, which can inhibit nonphysical numerical phenomenon, to propose the area weighted numer-ical scheme, which can keep symmetry on the framework of arbitrary Lagrangian Eulerian (ALE). We compare the difference of three schemes the control volume scheme,the area weighted scheme and the plane plus source scheme.