A Piecewise Parabolic Method For Elastic-plastic Flow In Solids

Elastic-plastic flow problems that involve high-load conditions and large defor-mations in solids,are widely applied to military and industrial areas and have great theoretical and practical significance.To numerically solve these problems,this work will be devoted to construct a monolithic model that can describe the mechanical be-havior in solid materials on one hand and to design high-precision and high-resolution numerical methods that can deal with large deformation of solid materials on the other hand.In this thesis,the Eulerian hyperbolic conservation laws are established based on the fact that the deformation gradient tensor instead of the stress tensor as primitive variables is used to describe the mechanical behavior of materials.Therefore,the most commonly used numerical methods in computational fluid dynamics can be extended to solve the elastic-plastic flow problems.Meanwhile these methods overcome some difficulties in solving nonconservative equations in the original Eulerian model.The main work includes the following aspects.First,this thesis is engaged in establishing the Euler equations coupled with an elastic model whose variable is the deformation gradient tensor and studying local Rie-mann problems of non-linear elasticity.A five-wave HLLD Riemann solver for the Euler equations is presented.Here,the robustness and accuracy of the HLL(Harten-Lax-van Leer)family of Riemann solvers are examined in conjunction with Godunov' s scheme in non-linear problems.Numerical results show that HLLD("D" denotes Dis-continuities)Riemann solver has distinct advantage in dealing with large deformation of non-linear elasticity and resolving complex multi-wave structure,compared with the HLL and HLLC("C" denotes Contact)solvers.Second,a high-precision and high-resolution piecewise parabolic method(PPM)coupled with HLL-type Riemann solvers for non-linear coupled model of multiple equa-tions is developed.Numerical results show that the technique,PPM + HLLD,provides a higher resolution solutions for single material problems than other techniques.Third,a interface tracking method(i.e.,the level set algorithm)for multi-material components is applied to track the interface between different materials.Here,the in-teraction among interfaces is realized based on HLLD Riemann solver combined with modified ghost fluid method.Both the solid/solid "stick" problem with the same ma-terial at the two sides of contact interface and the solid/solid "slip" problem with dif-ferent materials at the two sides are calculated.It is shown by the numerical results that this method composed of HLLD solver,PPM and level set algorithm can capture the material interface effectively and suppress spurious oscillations therein significantly.Finally,the hyper-elasticity model under the Eulerian framework is extended to study the plastic flow problem and numerical methods for the generated model is developed to model the elastic-plastic deformation behavior of solid materials.The numerical results of the test cases involving large deformations and high strain rates illustrate that high-order accurate PPM coupled with interface tracking techniques can realize to calculate the elastic-plastic behavior and capture the elastic-plastic waves accurately and capture multi-material interfaces accurately and keep the interfaces sharp.