Saint-Venant Principle on Problems of Nonlocal Elasticity Theory

Abstract

Simulating the modern structural materials requires introduction of a model that takes into account structural features at the micro-level. The Eringen’s nonlocal elasticity theory model could be referred to such models. This model introduction was considered in comparison with the elasticity classical model. Main feature of the nonlocal model is that it takes into consideration the long-range interactions between the continuous medium particles; classical formulation is its special case. In this case, equations are having the integral differential form, which significantly complicates obtaining the analytical solutions. In this regard, the finite elements method was applied to find a solution using the isoparametric finite elements. Here, the main balance relations are satisfied, as in the classical elasticity theory model. However, the solutions obtained are differing to a larger extent from the classical solutions, since such solutions exhibit the edge effect in vicinity of the domain free boundaries. This effect, as well as preservation of the balance of forces, are demonstrated on the example of the Saint-Venant principle feasibility at stretching the rectangular plate. Solutions obtained in the nonlocal formulation have a significant decrease in the tensile stress decrease near the free boundaries and shear stresses in the cross section