Keywords: non local plasticity, constitutive model, variational formulation, rate model, finite-step non local plasticity, convex analysis.

Classical theories of continuum plasticity exhibit serious drawbacks in the case of strain softening behaviour. In fact most materials show a loss of positive definiteness of the tangent stiffness operator which yields to the localization of plastic deformations in narrow bands until the occurrence of cracks appear, see for example [1].

If a localization phenomenon occurs in a body, the deformation pattern evolves from relatively smooth into a one in which shear bands of highly strained material appears whereas the remainder part of the body unloads.

In classical plasticity the size of the localization zone is unspecified. On the contrary nonlocal plasticity introduces a length scale which reflects the ability of the microstructure to transmit information to neighbouring points within a certain distance[2].

This paper concerns with a nonlocal theory which introduces, in the constitutive model, state variables defined in an average form over a finite volume of the body.

The nonlocal plasticity model is addressed in the framework provided by internal variables[3]. In particular, we introduce local internal variables, which govern kinematic hardening and isotropic softening, and a nonlocal internal variable which can be defined as the sum between a new internal variable and its spatial weighted average.

The nonlocal internal variable is added to the local variables, governing iso-tro-pic hardening, in the definition of the elastic domain.

It is shown that a theoretical analysis of the nonlocal constitutive problem can be performed by resorting to convex analysis [4] and potential theory for monotone multivalued operators [5]. The validity of the maximum dissipation theorem is assessed and the constitutive problem for the nonlocal model is provided.

The structural problem is then formulated. The nonlocal variational principle in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular two variational formulations are given: in the former the independent state variables are displacement, plastic strain and kinematic internal variables, in the latter the independent state variables are displacement, plastic multiplier, plastic strain, stress, and static and kinematic internal variables.

These two formulations show that two different approaches can be followed in order to perform a finite element approximation of the proposed nonlocal plastic model. In the former approach the expression of the free energy and of the dissipation have to be approximated. In the latter formulation the plastic multipliers explicitly appear.

Attention is then focused in the latter model and the variational principle is reformulated so that plastic multipliers explicitly appear. This is the starting point for a finite element approximation which stands as a generalization to a nonlocal context of the classic displacement method and will be analysed in a forthcoming paper.

1

Lasry, D., Belytschko, T., "Localization limiters and transient problems", Int. J. Solids Structures, 24, 581-597, 1988. doi:10.1016/0020-7683(88)90059-5

2

Bazant Z.P., Pijaudier-Cabot, G., "Nonlocal continuum damage, localization instability and convergence", J. Appl. Mech. (ASME), 55, 287-293, 1988.

3

Halphen, B., Nguyen, Q.S., "Sur les materiaux standards generalises", Jour. Mech., 14, 39-63, 1975.

4

Hiriart-Urruty, J.B., Lemarechal, C., "Convex Analysis and Minimization Algorithms I", Springer-Verlag, New York, 1993.

5

Romano, G., Rosati, L., Marotti de Sciarra, F., Bisegna, P., "A potential theory for monotone multi-valued operators", Quart. Appl. Math., LI, 4, 613-631, 1993.

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