欧拉描述下求粘弹塑性材料残余应力和自由边界的有限元方法

The finite element method has been implemented to model history-dependent manufacturing problems such as forming, welding, rolling and extrusion for nearly 30 years. Lagrangian formulations have been algorithmically easier to implement and have been proven accurate and robust for many applications. However Lagrangian formulations may become computationally demanding in three-dimensional models when several thousand time increments are needed such as in the case of laser or arc welding of large structures. Moreove, they can lead to mesh distortion when the process exhibits very large deformation such as the case of friction stir welding. In Eulerian formulation, material can be treated as a fluid flow through a fix region in space with manufacturing tools, such as a die. Because the Eulerian mesh does not deform with material, Eulerian formulation can solve large deformation problem nicely. However, calculating the material evolution and tracking the free surface are difficult for Eulerian formulation. In this research, a velocity based Eulerian elasto-visco-plastic formulation using Anand's rate dependent plasticity model is proposed to solve the material evolution for history-dependent material. A 3D stabilized Petrov-Galerkin finite element formulation using four primary fields (velocity, deformation gradient, visco-plastic deformation gradient and internal variable ) will be developed in this work to predict residual stress. To track the material free surface, level set method is used to describe the discontinuity on the Eulerian material boundary. The mass conservation equation is coupled with the four fields formulation to model the evolution of the free surfaces. The evolution of the free surface can be used to predict the distorsion of the structure. In this work, an Eulerian Finite Element algorithm will be proposed to solve elasto-visco-plastic material, and a Finite Element Code will be developed in the Linux platform. With the implementation of this FEM code, the manufacture procedures with large deformation (such as Friction Stir Welding) can be completely solved. It provides an accurate and efficient way for modeling the manufacture procedures.

材料成型和加工过程通常都有非常大的几何变形，基于拉格朗日描述的有限元方法在模拟大几何变形过程中存在严重的网格畸变，已成为材料成型和加工过程数值模拟的瓶颈。欧拉描述目前主要用于流体材料的模拟，然而由于欧拉描述可以很好地解决大变形问题，因此本项目拟推导新型的基于欧拉描述的粘弹塑性材料成型和加工过程的有限元方法。着重解决欧拉描述下求解粘弹塑性材料的残余应力和自由边界两个问题。本项目拟在欧拉描述下应用速度、变形梯度、塑性变形梯度和内部变量的四场耦合Petrov-Galerkin方程求解材料的残余应力；应用eXtended FEM（XFEM）方法和Level set函数引入材料边界上的不连续性，以求解材料的自由边界。这一研究的完成，将解决忽略加工工具变形时材料成型和加工的数值模拟问题，为制造业提供一个快速准确的有限元分析的新方法。

本研究推导了欧拉坐标系下，热力耦合的粘弹塑性，和粘塑性材料的稳定的有限元方程。粘弹塑性方程由温度，速度、变形梯度、塑性变形梯度和内部变量的五场耦合的Petrov-Galerkin方程来求解材料的基于变形历史的残余应力和应变。粘塑性材料方程，由温度，速度和压力三场耦合的稳定的Petrov-Galerkin方程来求解粘塑性变形。基于本项目推导的有限元方法，本研究编写了完全自主的OnExtrude有限元求解器。该求解器用Gmsh为前处理，Paraview为后处理。粘弹塑性方法已经应用于求解铝型材挤压成型，粘弹塑性模拟已经用于焊接的模拟。OnExtrude有限元求解器提供了OpenMP和MPI两种并行计算算法，可以根据用户服务器的具体情况，提供快速准确的有限元求解。OnExtrude有限元求解器在计算精确度和速度上都已经超过了国际现有的商业软件，例如Altair公司的HyperExturde。通过该项目的实施，为我国的制造业提供了一个快速准确的有限元新工具。

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