Designing reliable structural components that undergo plastic deformations requires incorporating the damage mechanism into models. Ductile crack initiation results from the accumulation of plastic deformations induced by different loading conditions. However, an idealized neat distinction between the plastic and the elastic domains cannot explain structural failure caused by cyclic loading below the yield stress. In this paper, we combine damage mechanics and unconventional elastoplasticity theories by coupling the ductile damage and the subloading surface model constitutive equations. This plasticity model is particularly suitable for fatigue or cyclic loading problems because it can capture the accumulation of irreversible contributions, meaning that it provides a more precise model of damage evolution and prediction of failure.


The inclusion of impurities, such as carbides or sulfides, in ductile metals can generate microvoids during a generic loading process. The voids tend to grow and coalesce, leading to the formation of a crack, which propagates fast inside the matrix, and then to material failure.

The importance of the correct description of this process has been a challenge, and many models, including micro and macro models, have been developed. For elastoplasticity and ductile damage, we can identify two main approaches (Bonora et al., 2005; De Souza et al., 2008): Gurson's void growth model (Gurson, 1977) and Lemaitre's model (Lemaitre, 1985).

The main idea in Gurson's void growth model is that porous metals behave as ideal metals formed from an elastoplastic matrix. The metals follow the von Mises constitutive law, containing microvoids with simple geometrical shapes. The effect of the ductile damage is considered by modifying the yield condition to express the joint contribution of the matrix and the void growth by progressive shrinkage of the plastic surface. Subsequently, Needleman and Tvergaard (1984), Koplik and Needleman (1987), Ohata and Toyoda (2003), and Ohata et al. (2010) have proposed an extension of the initial formulation to include the acceleration of the failure process that is observed because of the coalescence of the voids and macrocrack formation.

Gurson's void growth model, also known as continuum damage mechanics (CDM), treats the damage variable as an internal constitutive variable that is responsible for the mechanical degradation induced by an irreversible microscale process. The free energy becomes a function of the strain elastic tensor and the plastic hardening variables (isotropic and kinematic) and of an isotropic damage variable, D. The pioneering works of Lemaitre (1985a; 1985b) and Kachanov (1986) described the evolution CDM with a more complex model, considering an anisotropic damage variable (Murakami and Ohno, 1981; Lemaitre and Chaboche, 1990), or special damage evolution law during cyclic loading (Lemaitre, 1996; Andrade Pires et al., 2003, Pirondi et al., 2006).

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