Initiation of cracks in Griffith's theory: an argument of continuity in favor of global minimization

International audienceThe initiation of a crack in a sound body is a real issue in the setting of Griffith's theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith's criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort-Marigo's criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle $\omega$ is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to $\omega$, since no initiation occurs when $\omega>0$ while a crack initiates when $\omega=0$. In contrast, the Francfort-Marigo's criterion delivers a response which is continuous with respect to $\omega$, even though the onset of cracking is necessarily brutal when $\omega>0

Soft beams: when capillarity induces axial compression

We study the interaction of an elastic beam with a liquid drop in the case where bending and extensional effects are both present. We use a variational approach to derive equilibrium equations and constitutive relation for the beam. This relation is shown to include a term due to surface energy in addition of the classical Young's modulus term, leading to a modification of Hooke's law. At the triple point where solid, liquid, and vapor phases meet we find that the external force applied on the beam is parallel to the liquid-vapor interface. Moreover, in the case where solid-vapor and solid-liquid interface energies do not depend on the extension state of the beam, we show that the extension in the beam is continuous at the triple point and that the wetting angle satisfy the classical Young-Dupr\'e relation

The effective behavior of elastic bodies containing microcracks or microholes localized on a surface

International audienceWe propose a two-scale method to find the effective behavior of a three-dimensional linear elastic medium containing a series of microcracks or microholes located on a surface. The obtained effective behavior is that of a homogeneous body with, in place of the actual microdefects, a surface across which the displacements and the stresses suffer jump discontinuities. The transmission conditions are in general of Ventcel's type. The coefficients entering in these jump conditions are obtained by solving six elastic problems posed on an infinite representative cell. The theoretical analysis is illustrated by a few examples

From gradient damage laws to Griffith's theory of crack propagation

International audienceThis paper is devoted to the comparison of the evolution of damage governed by a gradient damage model with the evolution of a crack predicted by Griffith's law. The analysis is made in a two-dimensional setting, assuming that damage is concentrated inside thin bands whose width is proportional to the internal length of the material. Taking advantage of the variational formulation based on the three principles of irreversibility, stability and energy balance, one introduces a generalized Rice path integral which contains terms involving the gradient of damage. Assuming that the internal length of the material is small by comparison with the dimension of the body, a separation of scales is achieved. Owing to the energy balance and the stability condition, one first proves some properties of this path integral with respect to the path. Then, one shows that the evolution of the damage zone is governed by Griffith's law, the dissipated surface energy being given by the energy dissipated in the damage process zone

Morphogenesis and propagation of complex cracks induced by thermal shocks

We study the genesis and the selective propagation of complex crack networks induced by thermal shock or drying of brittle materials. We use a quasi-static gradient damage model to perform large scale numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion and depends only on the fracture toughness, while crack morphogenesis is driven by the material's internal length. Our numerical simulations feature networks of parallel cracks and selective arrest in two dimensions and hexagonal columnar joints in three dimensions, without any hypotheses on cracks geometry and are in good agreement with available experimental results

Approche variationnelle de l'endommagement : II. Les modèles à gradient

International audienceThis second part of the variational approach to damage is devoted to the construction of non-local gradient-enhanced models. That consists in extending to these regularized models the concepts introduced in the first part for local damage models. Specifically, once the gradient of damage has been inserted into the energy expression, the damage evolution problem is still based on the three physical principles of irreversibility, stability and energy balance. This new formulation is compared with that usually postulated and its merits are emphasized

Construction and analysis of localized responses for gradient damage models in a 1D setting

International audienceWe propose a method of construction of non homogeneous solutions to the problem of traction of a bar made of an elastic-damaging material whose softening behavior is regularized by a gradient damage model. We show that, for sufficiently long bars, localization arises on sets whose length is proportional to the material internal length and with a profile which is also characteristic of the material. We point out the very sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples

A non-periodic two scale asymptotic method to take account of rough topographies for 2D elastic wave propagation

International audienceWe propose a two scale asymptotic method to compute the effective effect of a free surface topography varying much faster than the minimum wavelength for 2-D P-SV elastic wave propagation. The topography variation is assumed to be non-periodic but with a deterministic description and, in this paper, the elastic body below the topography is assumed to be ho- mogeneous. Two asymptotic expansions are used, one in the boundary layer close to the free surface and one in the volume. Both expansions are matched appropriately up to the order 1 to provide an effective topography and an effective boundary condition. We show that the effective topography is not the averaged topography but it is a smooth free surface lying below the fast variations of the real topography. Moreover, the free boundary condition has to be modified to take account of the inertial effects of the fast variations of the topography above the effective topography. In other words, the wave is not propagating in the fast topography but below it and is slowed down by the weight of the fast topography. We present an iterative scheme allowing to find this effective topography for a given minimum wavelength. We do not attempt any mathematical proof of the proposed scheme, nevertheless, numerical tests show good results

Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting

International audienceConsidering a family of gradient-enhanced damage models and taking advantage of its variational formulation, we study the stability of homogeneous states in a full three-dimensional context. We show that gradient terms have a stabilizing effect, but also how those terms induce structural effects. We emphasize the great importance of the type of boundary conditions, the size and the shape of the body on the stability properties of such states