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An overview of the modelling of fracture by gradient damage models

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Abstract

The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (1) definition of the energy; (2) formulation of the damage evolution problem. The total energy of the body is defined in terms of the state variables which are the displacement field and the damage field in the case of quasi-brittle materials. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem is then based on the concepts of irreversibility, stability and energy balance. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Moreover, the variational formulation leads to a natural numerical method based on an alternate minimization algorithm. Several numerical examples illustrate the ability of this approach to account for all the process of fracture including a 3D thermal shock problem where the crack evolution is very complex.

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Notes

  1. In the thermal shock problem treated in the last section, the pre-strain is the thermal strain induced by the given temperature field which depends on time. Therefore, we will assume that \(\varvec{\varepsilon }^0\) is a given time-dependent field.

  2. The continuity of \(\alpha '\) at \(x_{0}\pm D\) is obtained as a first order optimality condition on \({\mathcal{P}}(u,\alpha )\).

  3. cracks being surfaces in dimension 3 and lines in dimension 2.

  4. Denoting by \((r,\theta )\) the polar coordinates and \((e_1, e_2)\) the Cartesian unit vectors,

    $${\bar{{\mathbf{U}}}} = \frac{K_I}{2\mu }\sqrt{\frac{r}{2\pi }}\left( \frac{3-\nu }{1+\nu }-\cos \theta \right) \left( \cos {({\theta }/{2})} e_1 + \sin ({\theta }/{2}) e_2\right),$$
    (86)

    where \(\mu\) is the shear modulus, and \(L_c\) is the length of the preexisting crack.

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Authors and Affiliations

  1. Laboratoire de Mécanique des Solides, Ecole Polytechnique, 91128, Palaiseau, France

    Jean-Jacques Marigo

  2. Institut Jean Le Rond d’Alembert, Sorbonne Universités, UPMC, Univ Paris 06, CNRS, UMR 7190, Paris, France

    Corrado Maurini

  3. IMSIA, ENSTA ParisTech, CNRS, CEA, EDF, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762, Palaiseau Cedex, France

    Kim Pham

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  1. Jean-Jacques Marigo
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Correspondence to Jean-Jacques Marigo.

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Marigo, JJ., Maurini, C. & Pham, K. An overview of the modelling of fracture by gradient damage models. Meccanica 51, 3107–3128 (2016). https://doi.org/10.1007/s11012-016-0538-4

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