Mathematical and Multi-Scale Foundations of Nonlocal Modeling: Research Topic 1.2 Solid Mechanics
Abstract
Major Goals: The proposed research embarks on a systematic and coordinated analysis of nonlocal constitutive laws for fracture modeling. The proposed work will develop the mathematical foundations that will further the use of nonlocal continuum models as reliable computational tools for fracture modeling. The approach taken here is new and is based on a mathematically rigorous development of peridynamic models with convex-concave energies. These energies are well suited to nonlocal formulations of peridynamic type in both state based and bond based models. The thrust of the proposed research is to develop theoretical and mathematically rigorous connections between the new nonlocal models with convex-concave energies and classic brittle fracture models. The first goal will seek to show that smooth solutions of the nonlocal model converge to the solution of the linear elastic wave equation as the length scale of nonlocal interaction goes to zero. Here we will provide convergence rates given in terms of the length of nonlocal interaction. A second goal is to provide a stable and consistent numerical scheme for the nonlocal evolution of smooth solutions. A third goal will develop convergence rates for nonlocal evolutions to known fracture solutions. The fourth goal is to develop a stable and consistent numerical scheme for nonlocal fracture modeling. The final objective is to develop a fundamentally sound quasi-static limit of nonlocal dynamics for reliable modeling of quasi-static crack growth. In all cases the primary interest is to explore and rigorously determine the range of validity for nonlocal models contingent on the physical dimensions of the sample, length scale of nonlocal interaction, material properties and characteristics of the loading. The overall goal of this effort is to establish the theoretical foundations necessary for nonlocal models to provide reliable and predictive numerical fracture simulations for problems of brittle fracture.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 20, 2019
- Accession Number
- AD1097134
Entities
People
- Robert Lipton
Organizations
- Louisiana State University