One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects

Abstract

ABSTRACTIn the present work, the close similarity that exists between Mindlin’s strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and nonstandard boundary conditions in both the cases. C0-based finite element methods (FEMs) are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long-range interactions. In this case, the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom is required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher-order approximations are compared. Lagrangian, Bèzier and B-Spline approximations are considered for the analysis. It has been shown that such higher-order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion, both the models predict size effect for one-dimensional example. Further, the higher-continuous approximation results in less computational effort for nonlocal-differential model.