The strain gradient elasticity via nonlocal considerations

Strain gradient elasticity and nonlocal elasticity are two enhanced elastic theories intensively used over the last fifty years to explain static and dynamic phenomena that classical elasticity fails to do. The nonlocal elastic theory has a clear differentiation from the classical case by considering stresses at a point of the continuum as an integral of classical stresses defined in the treated elastic body. On the other hand, strain gradient elasticity is characterized as a non-classical theory because considers both potential and kinetic energy densities as not only functions of strains and velocities, respectively but also functions of their gradients. Although the two considerations seem to be completely different from each other, it is a common belief that strain gradient elasticity has a lot in common with nonlocal elasticity. The goal of the present work is to derive all the strain gradient elastic theories appearing so far in the literature via the nonlocal definitions of the potential and kinetic energy densities and Hamilton’s principle. Such a consideration manifests the nonlocal nature of the SGE theories, correlates their internal length scale parameters with the nonlocal horizon of each material point, and proposes nonlocality in both elastic and inertia behavior of the continuum. For the sake of simplicity and brevity, only one-dimensional wave propagation phenomena in infinite domain are considered. However, since the Hamilton’s principle is employed, the definition of the corresponding boundary conditions for finite domains is accomplished simultaneously with the definition of the equation of motion.