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This contribution is devoted to the formulation and numerical implementation of a ductile damage constitutive model enriched with a thermodynamically consistent nonlocal theory of integral type. In order to describe ductile deformation, the model takes finite strains into account. To model elasticity, a Hencky-like hyperelastic free energy potential coupled with nonlocal damage is adopted. The thermodynamic consistency of the model is ensured by applying the first and second thermodynamical principles in the global form and the dissipation inequality can be re-written in a local form by incorporating a nonlocal residual that accounts for energy exchanges between material points of the nonlocal medium. The thermodynamically consistent nonlocal model is compared with its associated classical formulation (in which nonlocality is merely incorporated by averaging the damage variable without resorting to thermodynamic potentials) where the thermodynamical admissibility of the classical formulation is demonstrated. Within the computational scheme, the nonlocal constitutive initial boundary value problem is discretized over pseudo-time where it is shown that well established numerical integration strategies can be straightforwardly extended to the nonlocal integral formulation. A modified NewtonRaphson solution strategy is adopted to solve the nonlinear complementarity problem and its numerical implementation, regarding the proposed nonlocal constitutive model, is presented in detail. The results of two-dimensional finite element analyses show that the model is able to eliminate the pathological mesh dependence inherently present under the softening regime if the local theory is considered.

In this contribution, a continuum-discontinuum model for ductile failure is presented. The degradation of material properties through deformation is described by a Continuum Damage Mechanics model, which uses a nonlocal integral formulation to avoid mesh dependence. In the final stage of failure, the damaged zone is replaced by a macro crack for a more realistic representation of the phenomenon. The inclusion of the discontinuity surfaces is performed by the XFEM and Level Set Method to avoid the spurious damage growth typical of this class of models.

Over the past years, the non-local method has established itself as an effective remedy to the well-known pathological mesh dependency that inherently affects softening media. The non-local method incorporates an intrinsic length into the traditional continuum theory and therefore the size of the localising zone is resolved, attenuating the unwanted effects of spurious mesh dependency. However, despite many contributions that have successfully employed the non-local theory, it is still not clear how exactly should non-locality be formulated in the general sense. Aiming to answer the question of which non-local formulations effectively lead to mesh-insensitive results, we select in this article several constitutive variables to be non-local quantities by taking both Lemaitre and Gurson–Tvergaard–Needleman models as the base for the non-local enhancement. The resulting non-local constitutive models are employed in the numerical simulation of various specimens which are subjected to different values of stress triaxiality and third invariant at the fracture zone. The goal is to find which models present the best performance in the task of providing mesh-insensitive solutions for different stress states. The results show that strain-softening mesh dependency is stronger in plane strain than in the axisymmetric case. It is also found that the variables that regularise the solution in the axisymmetric case do not necessarily eliminate mesh sensitivity in the other cases. Furthermore, the results indicate that damage should be the preferred non-local variable in the case of implicit damage models. This result is in sharp contrast with the case of explicit damage models, for which it has already been shown in the literature that damage is a bad candidate for non-local variable.

In this contribution, the implementation of a nonlocal formulation is described aiming to improve the reliability of the numerical prediction of ductile failure. The nonlocal model, which is based on a simplified version of Lemaitre's material model [1], is achieved by redefining the damage variable to be nonlocal using an integral formulation. The nonlocal constitutive problem is then discretised with respect to pseudo-time by means of a backward Euler scheme. A numerical approach similar to a fully implicit elastic predictor/return mapping algorithm [2] is adopted here with the difference that in the present algorithm all material points are integrated simultaneously. The results show that the proposed numerical strategy improves the numerical reliability of ductile failure prediction by attenuating the pathological dependency of numerical results upon mesh refinement.KEY WORDS: continuum damage mechanics, ductile failure, nonlocal formulation, mesh dependency. I TRODUCTIOWhen a metal component or structure is subject to critical loadings, ductile failure usually takes places after progressive internal degradation of the material. With the aim of predicting and describing such a phenomenon, many constitutive models and numerical strategies have been developed over the last years [1][2][3][4][5][6][7]. Within the framework of Continuum Damage Mechanics, Lemaitre [3,4] has proposed a set of thermodynamically consistent elasto-plastic equations coupled with damage to predict ductile behaviour. In addition to the damage internal variable, ˖, Lemaitre's original model also considered the evolution of both isotropic and kinematic hardening. A simplified version of the material model was later implemented numerically by disregarding the evolution of kinematic hardening [1]. The latter case is particularly suitable when load reversal is negligible or inexistent. However, if the standard local continuum theory is considered, Lemaitre's model cannot accurately predict the material stiffness reduction. In the presence of softening, the set of equilibrium equations that governs the structural problem becomes ill-posed, leading to multiple solutions. Consequently, the numerical solution can suffer from a pathological dependency of results on spatial discretisation. In a typical finite element problem, this corresponds to the size of the elements of the mesh. A possible solution to overcome the problem of illposedeness is the introduction of an internal length, ℓ, into the continuum model, where ℓ is a parameter related to the microscopic structure of the material. This modification regularises the solution and can be achieved by employing nonlocal integral or gradientenhanced formulations [5-7], among others. In this paper, the simplified version of Lemaitre's model is enhanced by redefining the damage variable to be nonlocal through an integral formulation. The constitutive problem is then numerically solved by an algorithm which performs the integration of all material points of the body simultaneously. This procedure is ...