A Concise Review of Gradient Models in Mechanics and Physics

TLDR: In this article, a weakly nonlocal or gradient model for elasticity, diffusion and plasticity is proposed, with a possible connection to the strong force of elementary particle physics.

Abstract: The various mathematical models developed over the years to interpret the behavior of materials and corresponding processes they undergo, were based on observations and experiments made at that time. Classical laws for solids (Hooke) and fluids (Navier-Stokes) formed the basis of current technology. The discovery of new phenomena with the aid of novel developed experimental probes have led to various modifications of these laws, especially at small scales. The emergence of nanotechnology is ultimately connected with the design of novel tools for observation and measurements, as well as the development of new methods and approaches for quantification and understanding. The paper first reviews the author’s previously developed weakly nonlocal or gradient models for elasticity, diffusion and plasticity. It then proposes a similar extension for fluids and electrodynamics. Finally, it suggests a gradient modification of Newton’s law of gravity, with a possible connection to the strong force of elementary particle physics.