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Review on nonlocal continuum mechanics: Physics, material applicability, and mathematics

M. Shaat's picture

The classical continuum mechanics assumes that a material is a composition of an infinite number of particles each of which is a point that can only move and interact with its nearest neighbors. This classical mechanics has limited applications where it fails to describe the discrete structure of the material or to reveal many of the microscopic phenomena, e.g., micro-deformation and micro-dislocation. This observation motivated the need for a general point of view that instills the fact that the material particle is a volume element that would deform and rotate, and the material is generally a multiscale material. In addition, the particle's equilibrium should not be considered in isolation from its nonlocal interactions with other particles of the material. Material models with these features are the nonlocal microcontinuum theories. 

Whereas review articles and books on microcontinuum theories and nonlocal mechanics would be found in the literature, no review that extensively deals with the fundamentals of nonlocal mechanics from the physics, material, and mathematical points of view has been presented so far. There is a current scientific debate on the benefits of applying nonlocal theories to various fields of mechanics. This is due to a lack of understanding of the physics behind these theories. In addition, questions on the applicability of nonlocal mechanics for various materials are not answered yet. Furthermore, mathematicians revealed paradoxes and complications of finding solutions of nonlocal field problems. In this review, we shed light on these folders. We give extensive interpretations on the physics of nonlocal mechanics of particles and elastic continua, and the applicability of nonlocal mechanics to multiscale materials and single-scale materials is interpreted. In addition, the existing complications of solving nonlocal field problems, and the various methods and approaches to overcome these complications are collected and discussed from the physical and material points of view. Furthermore, we define the open forums that would be considered in future studies on nonlocal mechanics.


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