Abstract

Summary and Conclusions

• The nonlocal plasticity models proposed by Eringen do not act as localization limiters. The first model does not prevent localization into a set of measure zero, and the second model does not allow any localization at all (in the one-dimensional setting).
• Models with the yield stress dependent only on the nonlocal cumulative plastic strain (basic nonlocal formulation, model of Borino et al., and Nilsson's model)  provide only a  partial regularization and are essentially equivalent to a cohesive zone model. Plastic strain is localized into a set of zero measure but, in contrast to the local formulation, the global structural response in terms of the load-displacement diagram and work spent during the failure process is captured correctly. In finite element simulations in multiple dimensions, such models are likely to exhibit mesh-induced directional bias, since the plastic yielding would localize into one layer of elements.
• Models that combine in a suitable way the effect of the local and nonlocal cumulative plastic strain on the current yield stress act as true localization limiters and lead to a nonzero size of the localized plastic zone. This is true for the Vermeer-Brinkgreve model, models motivated by ductile damage (integral-type version of the implicit gradient plasticity model due to Geers et al. and nonlocal extension of the Gurson model), and the thermodynamically motivated model proposed by Svedberg and Runesson. The model proposed by Bazant and Lin has similar properties. At the first bifurcation from a uniform strain state, all these models are essentially equivalent. The subsequent evolution of the plastic strain profile and the stress transmitted by the plastic zone depend on the specific form of the softening law.
• If the softening process needs to be simulated until the complete loss of material integrity (zero residual yield stress), the model should be selected with great care. At late stages of the softening process, certain formulations produce pathological effects such as stress locking or spatial expansion of the plastic zone. Such formulations have a limited scope and they should be combined with appropriate tools for the description of highly localized strain and of the transition to fracture. Only the Vermeer-Brinkgreve model and the ductile damage models seem to be suitable for a pure continuum-based description of the complete failure process.
• The models of Borino et al. and of Svedberg and Runesson comply with the postulate of maximum plastic dissipation (in a modified form for the entire body) and the dissipation is guaranteed to be nonnegative. An alternative formulation due to Nilsson, also motivated by thermodynamic considerations, is not really consistent. The evolution laws cannot be derived from a maximum dissipation postulate and, in some particular cases (e.g., in the presence of compressive stresses within the interaction distance from the plastic zone yielding under tension), the model can give negative dissipation.
• Nonlocal extension of the postulate of maximum plastic dissipation leads to theoretically appealing models with a symmetric structure. These models require a double application of the nonlocal averaging operator, which complicates their numerical implementation. Also, at present there does not seem to be any model of this type that would act as a true localization limiter and at the same time could describe the complete failure process without any locking effects.
• For nonlocal models  formulated ad hoc, without any recourse to thermodynamics, it is not trivial to check their thermodynamic admissibility. For the basic nonlocal model, it is possible to prove that, if the free-energy potential is assumed in the same form as for the thermodynamically motivated model due to Borino et al., the dissipation is always nonnegative  (but the evolution law for the softening variable is of course not associated). For models with the softening law contaning both local and nonlocal terms, the construction of a suitable free-energy potential is more tricky and will be the subject of further research.
• If the nonlocal weight function is scaled in the proximity of boundaries, as is routinely done, the solution with a plastic region localized at the boundary usually dissipates much less energy than if the plastic region localizes inside the body. Since the solution that would actually occur is that with the steepest descent of the   post-peak branch of the load-displacement diagram (Bazant and Cedolin, 1991), the boundary acts as a weak layer that attracts localization. Whether this is physically realistic depends on the actual structure of the material in the boundary layer. For the thermodynamically motivated models with double nonlocal averaging, the largest plastic strain does not develop directly on the boundary but at a finite distance from it (for the Svedberg-Runesson model this is true if the nonlocal hardening modulus exceeds the magnitude of the local softening modulus by at least 25%).

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