An Overview and Advances of Nonlocal Elasticity Theories in the Background of Contemporary Materials

The principle of local action plays an important role in the development of constitutive relations of classical continuum mechanics, and the theory of simple materials that is derived from this principle has been widely applied. However, with the development of science and technology, various new materials which have microstructures are emerging. It has been theoretically and experimentally demonstrated that nonlocal theories may better depict the macroscopic behavior of these materials. In this article, we briefly review some traditional nonlocal elasticity theories, including the Eringen theory, the Kunin theory, the Mindlin theory, and the Willis theory that is developed for composite materials and is characterized by temporal and spatial nonlocality, the newly developed spatiotemporal nonlocal elastodynamic theory, and the peridynamic theory. The spatiotemporal nonlocal theory reflects the inherent nonlocal characteristics of the macroscopic properties of composite materials, and the spatially nonlocal peridynamic theory facilitates processing problems with discontinuities. Finally, we point out some issues that may deserve consideration in the development of nonlocal theories.