Computation of High-Gradient Phenomena in Solid Mechanics

本研究的目的是发展和实施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。 主要研究者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的大运动的准线性偏微分方程。 控制这些材料运动的方程通常是双曲守恒律系统，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，其目标是表征激波的形成。另外的努力将针对粘塑性材料中剪切带的形成的研究。 主要研究者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传播的影响。 这将包括研究的问题，其中的解决方案表现出狭窄的剪切带，在应变发生快速变化。 PI将开发自适应伪谱方法来解决塑性变形过程中发生快速变化的空间区域。橡胶、钢或混凝土等材料的行为可以用复杂的偏微分方程系统来描述，这些系统通常很难分析。 然而，在某些特殊情况下，这些方程可以简化到理论分析和数值计算的程度。 例子包括材料块在固定和移动平面之间的运动以及材料的固体球的压缩。 作为该项目的重点，首席研究员将研究冲击形成的性质和永久塑性变形对这些类型运动的冲击形成和传播的影响。 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。 激波和剪切带对应于材料破坏可能发生的区域，因此它们的精确确定是非常重要的。 主要研究者将利用方程的特殊形式来应用高性能的数值方法，这些方法是为气体动力学的应用而开发的。

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.