Nonlocal elasticity and related variational principles

Abstract The Eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied. The model is refined by assuming an attenuation function depending on the `geodetical distance' between material particles, such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented. A suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model. The nonlocal elasticity boundary-value problem for infinitesimal displacements and quasi-static loads is addressed and the conditions for the solution uniqueness are established. Three variational principles, nonlocal counterparts of classical ones, i.e. the total potential energy, the complementary energy, and the mixed Hu–Washizu principles, are provided. The former is used to derive a nonlocal-type FEM (NL-FEM) in which the (symmetric) global stiffness matrix reflects the nonlocality features of the problem. An alternative standard-FEM-based solution method is also provided, which consists in an iterative procedure of the type local prediction/nonlocal correction, in which the nonlocality is simulated by an imposed-like correction strain. The potentialities of these analysis methods are pointed out, their numerical implementations being the object of an ongoing research work.

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