A multi‐scale approach to condense the cyclic elastic‐plastic behaviour of the crack tip region into an extended constitutive model

Abstract

A B S T R A C TThe general idea of this paper is to model the mixed-mode cyclic elastic-plastic behaviour of the crack tip region at the global scale. It should be helpful, for instance, for predicting the effect of mixed mode overloads in fatigue. It is aimed at establishing a model reasonably precise (compared with elastic-plastic finite element (FE) computations) but condensed into a set of partial derivative equations so as to avoid huge elastic-plastic FE computations in the future. For this purpose, the kinematics of the crack tip region is characterized by a set of condensed variables. This is classical in linear elastic fracture mechanics (LEFM), the displacement field is approached by the product of spatial reference fields (u e I and u e II ) and nominal stress intensity factors (K ∞ I and K ∞ II ). Therefore, in LEFM, two condensed variables only, K ∞ I and K ∞ II , fully define the kinematics in the crack tip region. So as to generalize this approach to mixed mode cyclic elastic-plastic conditions, we define first the intensity factors (˙K I and˙K II ) of the elastic spatial reference fields (u e I and u e II ) and we introduce two additional spatial reference fields (u c I and u c II ) and their intensity factors (ρ I andρ II ) to account for plastic deformation within the crack tip region. Such an approximation is shown to be reasonably precise using FE computations. Therefore, the velocity field in the crack tip region is fully defined by only four condensed variables (˙K I ,K II ,ρ I andρ II ). Using the multi-scale approach proposed herein, evolutions of ρ I and ρ II for various mixed-mode loading conditions defined by K ∞ I and K ∞ II were generated using the finite-element method (FEM). Then, it was shown that we can model these evolutions at the global scale through a yield locus, a flow rule and a kinematics hardening rule. It is also suggested how this model could be employed for predicting the effects of mixed mode plasticity on fatigue crack growth. FEM = finite-element method LEFM = linear elastic fracture mechanics K ∞ I , K ∞ II = mode I and mode II nominal stress intensity factors u e I , u e II = mode I and mode II elastic reference displacement fields u c I , u c II = mode I and mode II complementary reference displacement fields K I ,K II = mode I and mode II pseudo-elastic intensity factors ρ I , ρ II = mode I and mode II plastic intensity factors I N T R O D U C T I O NFatigue crack growth under non-proportional mixedmode loading conditions remains an open problem. As a matter of fact, history effects in fatigue were shown to

Multi‐scale experimental study on fatigue damage behaviour and its effect on structural nonlinear response

“…This was a damage‐oriented model that accounts for the major effects of the damage, local deterioration processes and nonlinear influence on structural response. Pommier et al 30–33 . proposed their respective multi‐scale models to discuss the fatigue failure behaviour in the crack tip region or in the pre‐tensioned bridge cables.…”

“…This was a damage-oriented model that accounts for the major effects of the damage, local deterioration processes and nonlinear influence on structural response. Pommier et al [30][31][32][33] proposed their respective multi-scale models to discuss the fatigue failure behaviour in the crack tip region or in the pre-tensioned bridge cables. Recently, a reduced scale specimen of a longitudinal welded truss with pre-defect on welded joint was studied to experimentally investigate the local plastic damage evolution and its effects on the nonlinear response of the structure during static load testing process.…”

Short fatigue crack growth from a blunt notch in plate specimens

“…In this case, crack tip cyclic plastic behavior is complex. Pommier et al have recently utilized a multi-scale approach for modeling the mixed-mode cyclic elastoplastic behavior of the crack tip (Pommier et al 2009). It should be noted, however, that the notched specimen geometry used in the present study for short crack growth from a notch under mode I loading by using fourpoint bending is not suitable for study of mixed-mode crack growth.…”

Multi-scale approach for the analysis of the stress fields at a contact edge in fretting fatigue conditions with a crack analogue approach

“…Among the different approaches available in fracture mechanics one in particular can be really useful to describe the stress gradient effect in fretting-fatigue. An original model has been proposed by Pommier [7,8] aiming at describing mixedmode cyclic elastic-plastic behavior of the crack tip at the global scale. The purpose was to establish a reasonably precise model but condensed into a set of partial differential equations so as to avoid huge elastic-plastic FE computations.…”

“…The purpose was to establish a reasonably precise model but condensed into a set of partial differential equations so as to avoid huge elastic-plastic FE computations. The model proposed hereunder exploits the results obtained by Giannakopoulos [5,6] and Pommier [7,8]. A description of fretting-fatigue using nonlocal quantities has been developed.…”