Plasticity-induced damage in metals : nonlocal modelling at finite strains

The plasticity models that are generally adopted to predict the response of e.g. a deforming piece of metal assume that the material behaves like a true local continuum. This implies that the evolution of a state variable in a single material point only depends on the material state of that particular point itself and is not influenced by the value of the state variables in the surrounding material. Although this is a valid assumption for homogeneous deformation and stress fields it is inappropriate to properly capture e.g. the localised deformation in case of shear bands or size effects that are observed when the impact of the underlying microstructure becomes apparent at the global scale. The proper modelling of these phenomena requires the addition of spatial or nonlocal interactions into the continuum framework, which is achieved by considering either the gradient of a state variable or a nonlocal state variable field besides the usual local fields. An internal length parameter determines the global scale at which these nonlocal effects are manifested. The objective of this research is to develop a continuum plasticity-damage model that accounts for the nonlocal interactions of the underlying micromechanical mechanisms during ductile failure up to a satisfactory extent. In the developed framework ductile failure is introduced to the plasticity model by the concept of plasticity-induced damage, i.e. with ongoing plastic deformation microvoids initiate that weaken the material’s resistance to plastic deformation. In the model this is achieved by the gradual reduction of the material strength by a damage variable that is governed by a nonlocal effective plastic strain. This implies that the elastic material properties are not affected by the ductile damage. The nonlocal field acts as an additional degree of freedom which is determined by solving an additional partial differential equation that is equivalent to the integral definition of a nonlocal quantity. In the current implementation the additional differential equation is solved in a coupled fashion with the equilibrium equation, which permits the derivation of a consistent tangent operator which is beneficial for the convergence of the numerical solution scheme. Furthermore, this framework resolves a number of issues that are encountered when introducing alternative gradient-enhanced formulations to plasticity. Both small and finite strain variants of the plasticity-induced nonlocal damage model are presented. Various finite element simulations show the capability of these frameworks to properly deal with localised deformations and vanishing stress states at complete failure. The results indicate that the softening behaviour is not sensitive to the size of the mesh discretisation, but is governed by the value of the internal length parameter instead. Although a constant length parameter is adopted the limit behaviour of the model towards complete failure reveals the convergence towards a displacement discontinuity, which is beneficial for the transition from continuum failure to discrete cracking. Within the context of large deformations the nonlocality can be defined either on the undeformed (material) configuration or the deformed (spatial) configuration, which affects the interpretation of the internal length scale and leads to less or more ductility in case of large strains. These computational findings are substantiated by the analytical assessment of a number of competing models, which represent the entire range of higher-order models that are currently available in literature. These models are evaluated for two diverse fields of application. First, their capability to display a size effect in hardening is considered, which is investigated by means of the predicted moment that is required for the pure bending of a thin sheet as a function of the internal length scale. Although the current model does predict an increased strengthening effect for a smaller length scale, the quantitative effect for the Fleck and Hutchinson models are in better agreement with experimentally observed values. Second – and more important for this particular research – the ability to undergo the transition from hardening to localised softening behaviour is investigated. This is analysed by means of a one-dimensional bifurcation problem of an initially homogeneous bar under uniform straining. The result reveals the finite nontrivial wave numbers for which the particular rate equilibriumequation is able to accommodate a harmonic incremental solution. It is concluded that only the currentmodel properly handles the transition from hardening to localised softening and shows the desired behaviour in the limit of complete failure. In order to improve the accuracy of the evolution law for the damage-governing variable the underlying micromechanical mechanisms of ductile failure in metals are considered. In particular the growth and deformation of micro voids that nucleate within the metal matrix is regarded. Homogenisation of the response of voided unit cell simulations, which consider the entire scope of the deformation modes of micro voids ranging from volumetric growth to pure isochoric elongation, reveal the impact on the global material response of both the void volume fraction and the void shape. For both these variables evolution laws are presented, where the evolution of the void shape is obtained from a kinematical micromechanical model that describes the non-confocal evolution of two ellipses, which represent the void surface and the outer boundary of the cell respectively. It is shown that these evolution laws yield a good prediction of the results obtained with the unit cell simulations for the entire range of applied global loads between volumetric and purely deviatoric modes.