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Elastic-plastic rate-independent materials with isotropic hardening/softening of nonlocal nature are considered in the context of small displacements and strains. A suitable thermodynamic framework is envisaged as a basis of a nonlocal associative plasticity theory in which the plastic yielding laws comply with a (nonlocal) maximum intrinsic dissipation theorem. Additionally, the rate response problem for a (continuous) set of (macroscopic) material particles, subjected to a given total strain rate field, is discussed and shown to be characterized by a minimum principle in terms of plastic coefficient. This coefficient and the relevant continuum tangent stiffness matrix are shown to admit, in the region of active plastic yielding, some specific series representations. Finally, the structural rate response problem for assigned load rates is studied in relation to the solution uniqueness, and two variational principles are provided for this boundary value problem.

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a b s t r a c tA finite element based method, theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle, is numerically implemented for solving 2D nonlocal elastic problems. The key idea of the method, known as nonlocal finite element method (NL-FEM), relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of (cross-stiffness) element's matrices able to interpret the (nonlocality) effects induced in the element itself by the other elements in the mesh. An Eringen-type nonlocal elastic model is assumed with a constitutive stress-strain law of convolutive-type which governs the nonlocal material behaviour. Computational issues, as the construction of the nonlocal element and global stiffness matrices, are treated in detail. Few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness.

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