Véronique Lazarus

From craquelures to spiral crack patterns: influence of layer thickness on the crack patterns induced by desiccation

As a film of dispersed colloidal particles consolidates by desiccation on a rigid substrate, enormous stresses develop. Most of the time, these stresses exceed the strength of the material causing crack formation. Although undesirable in most industrial cases, crack patterns exhibit a useful method to characterize paintings. Indeed, crack patterns are the signature of the particles, solvent and substrate type, the drying conditions and the film thickness. Here we focus on the influence of this last parameter on the final crack morphologies. In particular, using a colloidal dispersion with transparency properties, we observe that craquelures, delamination and spiral crack morphologies can be obtained by changing only the film thickness. An extensive description of the formation dynamics and final geometry of each pattern is presented, highlighting the key geometric parameters that may easily be used by a broad audience, including the mechanical engineering, physics, chemistry, geology and art communities.

Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part II: Comparison with experiments

Abstract Using the results presented in Part I, together with various criteria, we try to explain qualitatively and quantitatively crack front rotation and segmentation under mixed mode I+III or I+II+III conditions. We first give a qualitative explanation of segmentation in mode I+III, based on the energetic theory of fracture. However, a satisfactory quantitative interpretation would require knowledge of a parameter which theory is unable to predict and which is not provided in experimental reports. We then concentrate on crack front rotation. It is shown that a suitable energetic criterion is able to reproduce quantitatively crack front “rotation rates” observed in brittle fracture in mode I+III. In fatigue, a criterion considering the sole mode I stress intensity factor is found to be more appropriate. Finally, for fatigue in general mode I+II+III conditions, we show that the crack front rotation rate is again satisfactorily reproduced by the same criterion as before, and also that a suitable extension of the well-known principle of local symmetry allows satisfactory prediction of the mean kink angle.

Shrinkage star-shaped cracks: Explaining the transition from 90 degrees to 120 degrees

Contraction due to drying or cooling of materials yields various self-organized crack patterns. The junctions between the cracks are complex and form in some conditions, star-shaped cracks with mostly 90 degrees or 120 degrees intersection angles. Any physical explanation of the selection of the angle is lacking. Here, we report directional drying of colloids experiments in capillary tubes allowing to obtain a reversible transition between 90 degrees and 120 degrees. We show that the transition is governed by a linear elastic fracture mechanics energy minimization principle hence by only one dimensionless parameter: the ratio between the Griffith length (balance between the energy needed to create cracks and to deform the material elastically) and the cell size. We give a straightforward characterization technique to estimate Griffiths length by changing the cell geometry. As a bonus, we deduce from it the toughness of drying colloidal suspensions. We underline that the method may be applied to a broad area of materials, from suspensions (colloids, paints or mud) to engineering (ceramics, coatings) and geological materials (basalt, sediments).

Failure mechanisms and surface roughness statistics of fractured Fontainebleau sandstone.

In an effort to investigate the link between failure mechanisms and the geometry of fractures of compacted grains materials, a detailed statistical analysis of the surfaces of fractured Fontainebleau sandstones has been achieved. The roughness of samples of different widths W is shown to be self-affine with an exponent zeta=0.46+/-0.05 over a range of length scales ranging from the grain size d up to an upper cutoff length xi approximately =0.15 W. This low zeta value is in agreement with measurements on other sandstones and on sintered materials. The probability distributions pi delta z(delta h) of the variations of height over different distances delta z>d can be collapsed onto a single Gaussian distribution with a suitable normalization and do not display multiscaling features. The roughness amplitude, as characterized by the height-height correlation over fixed distances delta z, does not depend on the sample width, implying that no anomalous scaling of the type reported for other materials is present. It is suggested, in agreement with recent theoretical work, to explain these results by the occurrence of brittle fracture (instead of damage failure in materials displaying a higher value of zeta approximately =0.8 ).

Crack paths in three-dimensional elastic solids. ii: three-term expansion of the stress intensity factors—applications and perspectives

Abstract This work continues the calculation of the stress intensity factors, as a function of position s along the front of an arbitrary (kinked and curved) infinitesimal extension of some arbitrary crack on some three-dimensional body. More precisely, e denoting a small parameter which the crack extension length is proportional to, what is studied here is the third term, proportional to e fn2 = e and noted K (1) (s) e, of the expansion of these stress intensity factors at the point s of the crack front in powers of e. The novelties with respect to previous works due to Gao and Rice on the one hand and Nazarov on the other hand, are that both the original crack and its extension need not necessarily be planar, and that a kink (discontinuity of the tangent plane to the crack) can occur all along the original crack front. Two expressions of K (1) (s) are obtained; the difference is that the first one is more synthetic whereas the second one makes the influence of the kink angle (which can vary along the original crack front) more explicit. Application of some criterion then allows to obtain the apriori unknown geometric parameters of the small crack extension (length, kink angle, curvature parameters). The small scale segmentation of the crack front which is observed experimentally in the presence of mode III is disregarded here because a large scale point of view is adopted; this phenomenon will be discussed in a separate paper. It is shown how these results can be used to numerically predict crack paths over arbitrary distances in three dimensions. Simple applications to problems of configurational stability and bifurcation of the crack front are finally presented.

Three-dimensional crack-face weight functions for the semi-infinite interface crack—I: Variation of the stress intensity factors due to some small perturbation of the crack front

The main goal of this paper and its companion is to provide the expressions of Bueckners fundamental three-dimensional crack-face weight functions for a semi-infinite interface crack in an infinite body. The method of solution avoids the calculation of the full mechanical fields in the elasticity problems implied but concentrates instead on the sole feature of interest, namely the distribution of the stress intensity factors along the crack front. As a part of the method, one must derive the expression for the variation of the stress intensity factors along the crack front arising from an infinitesimal coplanar perturbation of that front. The present Part I is devoted to this question. As an application, we examine the problem of the stability of the fundamental, straight configuration of the propagating crack front versus small in-plane perturbations.

Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.

In-plane perturbation of the tunnel-crack under shear loading I: bifurcation and stability of the straight configuration of the front

Abstract One considers a planar tunnel-crack embedded in an infinite isotropic brittle solid and loaded in mode 2+3 through some uniform shear remote loading. The crack front is slightly perturbed within the crack plane, from its rectilinear configuration. Part I of this work investigates the two following questions: Is there a wavy “bifurcated” configuration of the front for which the energy release rate is uniform along it? Will any given perturbation decay or grow during propagation? To address these problems, the distribution of the stress intensity factors (SIF) and the energy release rate along the perturbed front is derived using Bueckner–Rices weight function theory. A “critical” sinusoidal bifurcated configuration of the front is found; both its wavelength and the “phase difference” between the fore and rear parts of the front depend upon the ratio of the initial (prior to perturbation of the front) mode 2 and 3 SIF. Also, it is shown that the straight configuration of the front is stable versus perturbations with wavelength smaller than the critical one but unstable versus perturbations with wavelength larger than it. This conclusion is similar to those derived by Gao and Rice and the authors for analogous problems.

Finite size effects on crack front pinning at heterogeneous planar interfaces: Experimental, finite elements and perturbation approaches

Understanding the role played by the microstructure of materials on their macroscopic failure properties is an important challenge in solid mechanics. Indeed, when a crack propagates at a heterogeneous brittle interface, the front is trapped by tougher regions and deforms. This pinning induces non-linearities in the crack propagation problem, even within Linear Elastic Fracture Mechanics theory, and modifies the overall failure properties of the material. For example crack front pinning by tougher places could increase the fracture resistance of multilayer structures, with interesting technological applications. Analytical perturbation approaches, based on Bueckner-Rice elastic line models, focus on the crack front perturbations, hence allow for a description of these phenomena. Here, they are applied to experiments investigating the propagation of a purely interfacial crack in a simple toughness pattern: a single defect strip surrounded by homogeneous interface. We show that by taking into account the finite size of the body, quantitative agreement with experimental and finite elements results is achieved. In particular this method allows to predict the toughness contrast, i.e. the toughness difference between the single defect strip and its homogeneous surrounding medium. This opens the way to a more accurate use of the perturbation method to study more disordered heterogeneous materials, where the finite elements method is less adequate. From our results, we also propose a simple method to determine the adhesion energy of tough interfaces by measuring the crack front deformation induced by known interface patterns.

Three-dimensional crack-face weight functions for the semi-infinite interface crack—II: Integrodifferential equations on the weight functions and resolution

Abstract In Part I, we considered a small coplanar perturbation of the front of a semi-infinite interface crack and derived the expression of the resulting variation of the stress intensity factors along that front. In the present Part II, we apply this result to some special loadings, namely those which define the crack-face weight functions, and to some special perturbation, namely a rotation of the crack front about the direction normal to the crack plane; the interest of such a perturbation is that it preserves the shape of the crack so that the perturbed stress intensity factors are still expressible in terms of crack-face weight functions. The result consists in some integrodifferential equations for these functions. These equations are transformed into ordinary differential equations through Fourier transformation in the direction of the crack front. The solution is obtained in closed form to first order in the “bimaterial constant” e. Its sole non-elementary feature is the appearance of indefinite integrals of the form ∝[ ln x (x + a) ] dx .