The interest of gradient models is then discussed in the context of damage mechanics and crack simulation. The phenomenon of strain localization in a time-dependent or timeindependent process of damage is explored as a convenient numerical method to simulate the propagation of cracks, in relation with some recent works of the literature, cf. Bourdin & Marigo [3], Lorentz & al [5], Henry & al [12]. | Vietnam Journal of Mechanics, VAST, Vol. 33, No. 4 (2011), pp. 293 – 301 STANDARD GRADIENT MODELS AND CRACK SIMULATION Nguyen Quoc Son, Nguyen Truong Giang Laboratoire de Mecanique des Solides Ecole Polytechnique, 91128 Palaiseau, France Abstract. The standard gradient models have been intensively studied in the literature, cf. Fremond (1985) or Gurtin (1991) for various applications in plasticity, damage mechanics and phase change analysis. The governing equations for a solid have been introduced essentially from an extended version of the virtual equation. It is shown here first that these equations can also be derived from the formalism of energy and dissipation potentials and appear as a generalized Biot equation for the solid. In this spirit, the governing equations for higher gradient models can be straightforwardly given. The interest of gradient models is then discussed in the context of damage mechanics and crack simulation. The phenomenon of strain localization in a time-dependent or timeindependent process of damage is explored as a convenient numerical method to simulate the propagation of cracks, in relation with some recent works of the literature, cf. Bourdin & Marigo [3], Lorentz & al [5], Henry & al [12]. Keywords: Gradient and higher gradient models, damage mechanics, standard viscoplasticity, localization and crack propagation. 1. INTRODUCTION The introduction of the gradients of the state variables such as the strain, the internal parameter and even the temperature in Solid Mechanics has been much discussed in the literature since the pioneering works of Mindlin and Toupin in second-gradient elasticity. Especially, in the two last decades, standard gradient theories has been considered in many papers, cf. for example [4], [6], [8] , [9], [18], for the modeling of phase change and of solids with microstructures. In particular, in Frémond or Gurtin’s approach, the governing equations have been originally derived from an additional virtual work
