Non-local models and length scale effects on metal forming processes

The classical theories of plasticity or continuum damage mechanics do not include scale effects associated to a characteristic length of the material, which is often present in deformation processes, and which results in a limitation of the localization of the plastic deformation or damage in narrow bands. The numerical models based in these theories suffer from size and orientation pathological dependence of the geometrical discretization and are unable to accommodate that limitation of the localization. This is true for both the classical finite element method as well as the novel meshless methods, for example. In fact the homogenization hypotheses usually admitted for the continuum and that are the basis of those numerical methods do not accommodate large gradients of the internal variables like the plastic deformation or damage in the localization zones and therefore second order constitutive laws are needed.

Length scale effects can be explained by micro mechanics theories but their numerical implementation is still computational very expensive. One way of bridging the gap between the classical continuum s and micro mechanics theories is the recourse to non-local theories. In the nonlocal approaches the history of the internal variables at a material point is influenced by the surrounding material points. Non-local counterparts of those variables are then defined by integral averaging techniques or by including gradients of the internal variables in the formulation. These nonlocal formulations can also be generalised to deal, for example, with micro forming processes where geometric and microstructure size effects are present.

The softening induced by the standard implementation of damage models in finite element solutions leads to mesh and orientation dependence, as the localization effects are not correctly dealt with by mesh refinement. Non-local models are a solution to solve this pathological behaviour. They include some length scale information, related with localization effects due to microstructure heterogeneity, to average an internal variable associated with dissipative process. Two types of models are usually adopted for this purpose: integral and gradient models. In this work, non-local damage models, in which the internal dissipative non-local variable adopted is the damage, are implemented. Damage definition is based on the Lemaitre’s work, which is enhanced by the introduction of a crack closure scalar parameter, allowing to treat differently the damage evolution in either traction or compression stress states. The idea is to compare the performance of some of those models and to assess the performance of some novel methods, particularly meshless methods, that some claim that may solve the numerical pathologies referred above. Comparisons are performed with non-local integral, implicit and explicit gradient methods implemented in two different frameworks, namely the Finite Element Method and the Reproducing Kernel Particle Method.