From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models

Abstract

. From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Continuum Mechanics and Thermodynamics, Springer Verlag, 2013, 25 (2-4), pp.147-171. 10.1007/s00161-011-0228-3. hal-00647860To be submitted to Continuum Mechanics and Thermodynamics Abstract We propose a construction method of non homogeneous solutions for the traction problem of an elastic damaging bar. This bar has a softening behavior which obeys a gradient damaged model. The method is applicable for a wide range of brittle materials. For sufficiently long bars, we show that localization arises on sets whose length is proportional to the material internal length and with a profile which is also a material characteristic. From its onset until the rupture, the damage profile is obtained either in a closed form or after a simple numerical integration depending on the model. Thus, the proposed method provides definitions for the critical stress and fracture energy that can be compared with experimental results. We finally discuss some features of the global behavior of the bar such as the possibility of a snap-back at the onset of damage. We point out the sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples.

A convergence study of phase-field models for brittle fracture

“…2 For a sufficient large bar, this results in the localization of damage in one single zone which corresponds to the lowest energy and is associated to a snap-back as shown by (Pham and Marigo, 2013), which the authors interpret as the nucleation of a single crack. cf.…”

“…For a discussion of the uniqueness and the stability of the homogeneous response see also (Pham et al, 2011;Pham and Marigo, 2013).…”

A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations

“…A major step forward was the removal of the ill-posedness of the boundary value problem that is caused by smearing out the decohesion over a finite domain. Regularisation can be obtained by means of non-local approaches (Pijaudier-Cabot and Bažant, 1987), and more effectively in a finite element context, by gradient approaches (Peerlings et al, 1996), see also de Borst et al (2004) and Pham and Marigo (2013).…”

“…• g ′ (1) = 0, since the energy must converge to a finite value for the fully broken state, see Braides (1998); Pham et al (2011);Pham and Marigo (2013) for a general discussion. For the degradation function g(d) use has often been made of the quadratic function (Miehe et al, 2010b):…”

“…Furthermore, a smaller length scale parameter ℓ results in a more pronounced snapback behaviour. Pham et al (2011);Borden et al (2012) and Pham and Marigo (2013) also state that the length scale parameter ℓ in the model by Bourdin et al (2000) can be interpreted as a material parameter since it influences the critical stress. This makes it less straightforward to decide how ℓ should be handled in phase-field models for brittle fracture.…”