Initiation of cracks with cohesive force models: a variational approach

Charlotte, Miguel; Laverne, Jérôme; Marigo, Jean‐Jacques

doi:10.1016/j.euromechsol.2006.05.002

Cited by 61 publications

(39 citation statements)

References 9 publications

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“…This is the penalty to pay in order to recover the Griffith theory via a variational approach, which frees itself from the usual constraints that a pre-existing crack must be assumed a priori. In agreement with other work by Marigo et al [11], this means that if local minima are looked for, the load required to open a crack is infinite and the sound elastic state is always locally stable. This conclusion is in agreement with the most classical and definite tests on glass filaments by Griffith [18], who experimentally showed that brittle materials without pre-existing flaws are practically unbreakable or, more precisely, that their strength approaches the theoretical value of the same order of the elastic modulus.…”

Section: Introductionsupporting

confidence: 86%

“…On the other hand, the functional (3.1) does not account for the possibility of pre-existing flaw. For the reasons mentioned at length in the Introduction, already noted by Marigo and his collaborators [11], such functional does not appear appropriate to model the phenomenon of fracture. On the other hand, the approximating functional is consistently applicable.…”

Section: The Proposed Modified Variational Modelmentioning

confidence: 96%

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The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

2009

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Recently, Francfort and Marigo (J. Mech. Phys. Solids 46, 1319-1342 have proposed a novel approach to fracture mechanics based upon the global minimization of a Griffith-like functional, composed of a bulk and a surface energy term. Later on the same authors, together with Bourdin, introduced (in J. Mech. Phys. Solids 48, 797-826, 2000) a variational approximation (in the sense of -convergence) of such functional, essentially for computational purposes.Here, we utilize this new variational approach to show how it might be altered to incorporate the idea of less brittle, "deviatoric-type fracture" and apply to materials such as confined stone. To do so, we modify the original formulation of Francfort and Marigo, in particular its approximation of Bourdin, Francfort and Marigo, to only allow for discontinuities in the deviatoric part of the strain. We apply such modified model to gain insight on the deterioration and cracking in the ashlar masonry work of the French Panthéon, which are so repetitious and particular to be a distinguishable symptom of ongoing damage. Numerical experiments are performed and the results compared to those obtained using the original Francfort-Marigo model and to actual crack patterns from the Panthéon.The modified formulation allows one to reproduce fracture paths surprisingly similar to that observed in situ, to sort out the possible causes of damage, and to confirm, with a quantitative analysis, the main structural deficiencies in the French monument. This practical example enhances the importance of this promising new theory based in the mathematical sciences.

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Section: Introductionsupporting

confidence: 86%

Section: The Proposed Modified Variational Modelmentioning

confidence: 96%

The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

2009

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“…For example, a first extension should be to consider prestresses in the composite and hence to develop the procedure in the case of an affine stress-strain relation. Another natural extension could be to consider more general and more realistic contact conditions between matrix and fibers: unilateral contact without friction or cohesive forces [9], for instance. The difficulty would be to solve non linear cell problems, and in such cases the effective behavior would no longer be described by a finite number of coefficients.…”

Section: Discussionmentioning

confidence: 99%

The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers

Berrehili

Marigo

2014

Math. Mech. Compl. Sys.

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Abstract. This paper is devoted to the analysis of the homogenized behavior of unidirectional composite materials once the fibers are debonded from (but still in contact with) the matrix. This homogenized behavior is built by an asymptotic method in the framework of the homogenization theory. The main result is that the homogenized behavior of the debonded composite is that of a generalized continuous medium with an enriched kinematics. Indeed, besides the usual macroscopic displacement field, the macroscopic kinematics contain two other scalar fields. The former one corresponds to the displacement of the matrix whereas the two latter ones correspond to the sliding and the rotation of the debonded fibers with respect to the matrix. Accordingly, new homogenized coefficients and new coupled equilibrium equations appear. This problem is addressed in a linear elastic three-dimensional setting.

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“…An important step will be to apply the MAM in the case of a cohesive crack [Barenblatt 1962;Dugdale 1960;Del Piero and Raous 2010], which automatically contains a critical stress and even a characteristic length. The goal will be to study the influence of the critical stress and characteristic length on the nucleation and the propagation of a crack in the spirit of the previous works based on the variational approach to fracture [Abdelmoula et al 2010;Bourdin et al 2008;Charlotte et al 2006;Del Piero and Truskinovsky 2009;Ferdjani et al 2007;Giacomini 2005;Jaubert and Marigo 2006;Marigo and Truskinovsky 2004].…”

Section: Discussionmentioning

confidence: 99%

Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch

Dang

Halpern

Marigo

2014

Math. Mech. Compl. Sys.

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We use matching asymptotic expansions to treat the antiplane elastic problem associated with a small defect located at the tip of a notch. In a first part, we develop the asymptotic method for any type of defect and present the sequential procedure which allows us to calculate the different terms of the inner and outer expansions at any order. This requires in particular separating in each term its singular part from its regular part. In a second part, the asymptotic method is applied to the case of a crack of variable length located at the tip of a given notch. We show that the first two nontrivial terms of the expansion of the energy release rate are sufficient to well approximate the dependence of the energy release rate on the crack length in the range of values of the length which are sufficient to treat the problem of nucleation. This problem is considered in the last part where we compare the nucleation and the propagation of a crack predicted by two different models: the classical Griffith law and the Francfort-Marigo law based on an energy minimization principle. Several numerical results illustrate the interest of the method.

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Initiation of cracks with cohesive force models: a variational approach

Cited by 61 publications

References 9 publications

The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers

Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch

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