The paper is devoted to the study of quasi-static brittle crack evolution. We work under the following assumptions: a linear elastic body, with or without initial cracks inside, evolves in a quasi-static manner under an imposed path of boundary displacements. During its evolution cracks with unprescribed geometry may appear and/or grow.
This is an alternative version of the published article [Bu3], containing an improved Mumford-Shah model based on the functional K2, a generalization of the integral J of Rice. Further information at http://irmi.epfl.ch/cag/buliga_bfrac.html or http://www.imar.ro/~marius/ .
We present a simple example of toughening mechanism in the homogenization of composites with soft inclusions, produced by crack deflection at microscopic level. We show that the mechanism is connected to the irreversibility of the crack process. Because of that it cannot be detected through the standard homogenization tool of the $Γ$-convergence.
Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introd…
The purpose of this paper is to fill the gap between the classical treatment of brittle fracture mechanics and the new idea of considering the crack evolution as a free discontinuity problem. Griffith and Irwin criterions of crack propagation are studied and transformed in order to be no longer dependent on any prescription of the geometry of the crack during its evolution. The inequality contained in theorem 6.1. represents the link between generalized Irwin and Griffith criterions of brittle crack propagation. The physical meaning of this inequality is explained in the last section. Further …
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy…
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of a brittle fracture proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energ…
This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a $Γ$-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simpli…
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the $L^2$-distance between the approximate solutions at two consecutive times. We study the continuous-time v…
In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.
By a combination of geometrical and configurational analysis we study the properties of absolute minimal and equilibrium states of general Mumford-Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results concern the mathematical relations between physical quantities as energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge.
In this paper we prove a $\mathcal C^{1,α}$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a $\mathcal C^{1,α}$ curve outside a set of zero Hausdorff measure.
The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\infty$.
We show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. The crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate.
This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a maximal-rank constraint on the subgradients, which can be approximated by $C^{1}$-local immersions on regular subdomains of the cracked set, and (ii) the construction of recovery sequences by means of suitable $W^{1,\infty}$ diffeomorphisms mapping the regular subdomains onto the fractured configuration.
Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the $Γ$-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to …
Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients, which in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of …
