The variational approach to fracture: formulation, general properties and examples

Abstract : The lecture 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 [2] 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 [5], whereas they contain also the plastic strain field in the case of ductile materials [1]. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem then based on the concepts of irreversibility, stability and energy balance, as well inquasi-static as in dynamic [4]. 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 will 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 [3].

References
[1] R. Alessi, J.-J. Marigo, and S. Vidoli. Gradient damage models coupled with plasticity: variational formulation and main properties. Mechanics of Materials, 80:351–367, 2015.
[2] B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. J. Elasticity, 91(1–3):5–148, 2008.
[3] B. Bourdin, J.-J. Marigo, C. Maurini, and P. Sicsic. Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys. Rev. Lett., 112(1):014301, 2014.
[4] T. Li, J. J. Marigo, D. Guilbaud, and S. Potapov. Numerical investigation of dynamic brittle fracture via gradient damage models. Advanced Modeling and Simulation in Engineering Sciences, DOI: 10.1186/s40323-016-0080-x:3–26, 2016.
[5] K. Pham, H. Amor, J.-J. Marigo, and C. Maurini. Gradient damage models and their use to approximate brittle fracture. International Journal of Damage Mechanics, 20(4):618–652, 2011

Bio : Jean-Jacques Marigo obtained his PhD in Mechanics from the University Pierre and Marie Curie (Paris, France). He is currently a full-time professor in the mechanics department at the Ecole Polytechnique (Palaiseau, Île-de-France, France). He has received the Prix Paul Doistau-Emile Blutet in 2002 from the l'Académie des Sciences (France) and the Tullio Levi Civita International Prize in Mechanics and Applied Mathematics in 2011 by the Tullio Levi Civita Foundation (Italy). Since 2009, Prof. Marigo is an associated editor of the Journal of Elasticity.