Computation of High-Gradient Phenomena in Solid Mechanics

本研究的目的是开发和实现有效的数值方法来解决模拟固体非线性力学的偏微分方程。首席研究员将使用数值方法研究控制非线性弹性、弹塑性和粘塑性材料大运动的准线性偏微分方程。控制这类物质运动的方程通常是双曲守恒定律系统，非常适合于数值研究。特别是，PI将利用基于有限差分近似的有效数值方案，并受到气体动力学数值方法的启发，以研究弹塑性材料的反平面和径向运动，目标是表征冲击的形成。更多的努力将指向粘塑性材料中剪切带形成的研究。首席研究员将研究应变梯度正则化对二维非线性粘塑性剪切带发展和扩展的影响。这将包括研究的问题，其中解决方案表现出狭窄的剪切带，跨应变发生快速变化。PI将开发自适应伪谱方法来解决在塑性变形期间发生快速变化的空间区域。橡胶、钢或混凝土等材料的性能可以用复杂的偏微分方程组来描述，而这些偏微分方程组通常很难分析。然而，在某些特殊情况下，这些方程可以简化到既可以进行理论分析又可以进行数值计算的程度。例子包括一个物质块在固定平面和运动平面之间的运动，以及一个实心物质球的压缩。作为该项目的重点，首席研究员将研究这些类型运动中冲击形成和永久塑性变形的本质。第二个研究领域是剪切带的形成。这些是当韧性固体经历大变形时形成的局部强烈剪切区域。冲击和剪切带对应于可能发生材料破坏的区域，因此它们的准确测定非常重要。首席研究员将利用方程式的特殊形式来应用为气体动力学应用而开发的高性能数值方法。

The aim of this research is the development and implementationof effective numerical methods for the solution of partial differential equations which model the nonlinear mechanics of solids. The principal investigator will use numerical methods to study quasilinearpartial differential equations governing large motions of nonlinearly elastic, elastoplastic and viscoplastic materials. The equations governing the motions of such materials are generally systems of hyperbolic conservation laws which are ideally suited for numerical study. In particular, the PI will utilize an efficient numerical scheme based on finite-difference approximations and inspired by numerical methods from gas dynamics in order to study antiplane and radial motions of elastoplastic materials, with the goal of characterizing the formation of shocks. Additional effort will be directed toward the study of the formation of shear bands in viscoplastic materials. The principal investigator will study the effects of strain-gradient regularization on the development and propagation of shear bands in two-dimensional nonlinear viscoplasticity. This will include studies of problems where the solutions exhibit narrow shear bands, across which rapid variations in strain occur. The PI will develop adaptive pseudo-spectral methods to resolve spatial regions where rapid variations occur during plastic deformation.The behavior of materials such as rubber, steel, or concrete can bedescribed by complicated systems of partial differential equations which arein general difficult to analyze. However, there are certain special situations in which these equations simplify to a point where they can be bothanalyzed theoretically and computed numerically. Examples include the motion of a material block between a fixed and a moving plane andthe compression of a solid ball of material. As a focus of this project, the principal investigator will investigate the nature of shock formation and of permanent plastic deformation on the formation and propagation of shocks for these types of motions. A second area of research is the formation of shear bands. These are localized regions of intense shear that form when ductile solids undergo large deformations. Shocks and shear bands correspond to possible regions in which material failure occurs,and their accurate determination is therefore very important. The principalinvestigator will take advantage of the special form of the equations toapply high-performance numerical methods that have been developed for applications in gas dynamics.