laboratoire de physique statistique
laboratoire de physique statistique




Morphology and dynamics of a crack front propagating in a model disordered material - Chopin, J. and Boudaoud, A. and Adda-Bedia, M.

Abstract : We present an experiment on the morphology and dynamics of a crack front propagating at the interface between an elastomer and a glass slide patterned with a prescribed distribution of defects. Regimes of high and low pinning strength are explored by changing the fracture energy contrast of the defects. We first analyze the roughness of crack fronts by measuring their typical amplitude in real and Fourier space. Irrespective of the pinning regime, no well defined self-affine behavior is found which may be explained by the emergence of an intermediate lengthscale between the defect size and the sample size. Then, we show that the dynamics at high fracture energy contrast results in rapid jumps alternating with periods of arrest. The distributions of speeds, displacements and waiting times are found to have an exponential decay which is directly related to the distribution of distances between defects along the direction of propagation. (C) 2014 Elsevier Ltd. All rights reserved.
Pattern formation in fiber-reinforced tubular tissues: Folding and segmentation during epithelial growth - Ciarletta, P. and Ben Amar, M.

Abstract : Constrained growth processes in living materials result in a complex distribution of residual strains, which in certain geometries may induce a bifurcation in the elastic stability. In this work, we investigate the combined effects of growth and material anisotropy in the epithelial pattern formation of tubular tissues. In order to represent the structural organization of most organs, we adopt a strain energy density which accounts for the presence of a nonlinear reinforcement made of cross-ply fibers distributed inside a ground matrix. Using a canonical transformation in mixed polar coordinates, we transform the nonlinear elastic boundary value problem into a variational formulation, performing a straightforward derivation of the Euler-Lagrange equations for perturbations in circumferential and longitudinal directions. The corresponding curves of marginal stability are obtained numerically: the results demonstrate that both the three-dimensional distribution of residual strains and the mechanical properties of fiber reinforcements within the tissue are fundamental to determine the emergence of a specific instability pattern. In particular, different proportions of axial and circumferential residual strains can model the epithelial formation of mucosal folds in the esophagus and of plicae circulares in the small intestine. The theoretical predictions are compared with morphological data for embryonic intestinal tissues, suggesting that the volumetric growth of the epithelium can also drive the early stages of villi morphogenesis. (C) 2011 Elsevier Ltd. All rights reserved.
Buckling condensation in constrained growth - Dervaux, Julien and Ben Amar, Martine

Abstract : The multiple complexities inherent to living objects have motivated the search for abiotic substitutes, able to mimic some of their relevant physical properties. Hydrogels provide a highly monitorable counterpart and have thus found many applications in medicine and bioengineering. Recently, it has been recognized that their ability to swell could be used to unravel some of the universal physical processes at work during biological growth. However, it is yet unknown how the microscopic distinctions between swelling and biological growth affect macroscopic changes (shape, stresses) induced by volume variations. To answer this question, we focus on a clinically motivated example of growth. Some solid tumors such as melanoma or glioblastoma undergo a shape transition during their evolution. This bifurcation appears when growth is confined at the periphery of the tumor and is concomitant with the transition from the avascular to the vascular stage of the tumor evolution. To model this phenomenon, we consider in this paper the deformation of an elastic ring enclosing a core of different stiffness. When the volume of the outer ring increases, the system develops a periodic instability. We consider two possible descriptions of the volume variation process: either by imposing a homogeneous volumetric strain (biological growth) or through migration of solvent molecules inside a solid network (swelling). For thin rings, both theories are in qualitative agreement. When the interior is soft, we predict the emergence of a large wavelength buckling. Upon increasing the stiffness of the inner disc, the wavelength of the instability decreases until a condensation of the buckles occurs at the free boundary. This short wavelength pattern is independent of the stiffness of the disc and is only limited by the presence of surface tension. For thicker rings, two scenarios emerge. When a volumetric strain is prescribed, compressive stresses accumulate in the vicinity of the core and the deformation localizes itself at the boundary between the disc and the ring. On the other hand, swelling being an instance of stress-modulated growth, elastic stretches near the core saturate and the instability occurs primarily at the free boundary. Besides its implications for the mechanical stability of avascular tumors, this work provides important results concerning layered tissues growth and the role of hydrogels as biological tissues substitutes. (C) 2010 Elsevier Ltd. All rights reserved.
Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints - Ben Amar, Martine and Ciarletta, Pasquale

Abstract : The purpose of this work is to provide a theoretical analysis of the mechanical behavior of the growth of soft materials under geometrical constraints. In particular, we focus on the swelling of a gel layer clamped to a substrate, which is still the subject of many experimental tests. Because the constrained swelling process induces compressive stresses, all these experiments exhibit surface instabilities, which ultimately lead to cusp formation. Our model is based on fixing a neo-Hookean constitutive energy together with the incompressibility requirement for a volumetric, homogeneous mass addition. Our approach is developed mostly, but not uniquely, in the plane strain configuration. We show how the standard equilibrium equations from continuum mechanics have a similarity with the two-dimensional Stokes flows, and we use a nonlinear stream function for the exact treatment of the incompressibility constraint. A free energy approach allows the extension both to arbitrary hyperelastic strain energies and to additional interactions, such as surface energies. We find that, at constant volumetric growth, the threshold for a wavy instability is completely governed by the amount of growth. Nevertheless, the determination of the wavelength at threshold, which scales with the initial thickness of the gel layer, requires the coupling with a surface effect. Our findings, which are valid in proximity of the threshold, are compared to experimental results. The proposed treatment can be extended to weakly nonlinearities within the aim of the theory of bifurcations. (C) 2010 Elsevier Ltd. All rights reserved.
Laws of crack motion and phase-field models of fracture - Hakim, Vincent and Karma, Alain

Abstract : Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed. (c) 2008 Elsevier Ltd. All rights reserved.
Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Foppl-von Karman limit - Dervaux, Julien and Ciarletta, Pasquale and Ben Amar, Martine

Abstract : The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and physical factors at different temporal and spatial scales. This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface. The results of this theory demonstrate that the corresponding equilibrium equations are of the Foppl-von Karman type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation. (C) 2008 Published by Elsevier Ltd.
Stress field at a sliding frictional contact: Experiments and calculations - Scheibert, J. and Prevost, A. and Debregeas, G. and Katzav, E. and Adda-Bedia, M.

Abstract : A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons' friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient mu(d). but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14\% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressure-independence of mu(d) is briefly discussed. (c) 2009 Elsevier Ltd. All rights reserved.
Buckling of a stiff film bound to a compliant substrate - Part III: Herringbone solutions at large buckling parameter - Audoly, Basile and Boudaoud, Arezki

Abstract : We study the buckling of a compressed thin elastic film bonded to a compliant substrate. An asymptotic solution of the equations for a plate on an elastic foundation is obtained in the limit of large residual stress in the film. In this limit, the film's shape is given by a popular origami folding, the Miura-ori, and is composed of parallelograms connected by dihedral folds. This asymptotic solution corresponds to the herringbone patterns reported previously in experiments: the crests and valleys of the pattern define a set of parallel, sawtooth-like curves. The kink angle obtained when observing these crests and valleys from above are shown to be right angles under equi-biaxial loading, in agreement with the experiments. The absolute minimum of energy corresponds to a pattern with very slender parallelograms; in the experiments, the wavelength is instead selected by the history of applied load. (c) 2008 Elsevier Ltd. All rights reserved.
Buckling of a stiff film bound to a compliant substrate - Part II: A global scenario for the formation of herringbone pattern - Audoly, Basile and Boudaoud, Arezki

Abstract : We study the buckling of a thin compressed elastic film bonded to a compliant substrate. We focus on a family of buckling patterns, such that the film profile is generated by two functions of a single variable. This family includes the unbuckled configuration, the classical primary mode made of straight stripes, as well the pattern with undulating stripes obtained by a secondary instability investigated in the first companion paper, and the herringbone pattern studied in last companion paper. A simplified buckling model relevant for the analysis of these patterns is introduced. It is solved analytically for moderate or for large residual compressive stress in the film. Numerical simulations are presented, based on an efficient implementation. Overall, the analysis provides a global picture for the formation of herringbone patterns under increasing residual stress. The film shape is shown to converge at large load to a developable shape with ridges. The wavelength of the pattern, selected in a first place by the primary buckling bifurcation, is frozen during the subsequent increase of loading. (c) 2008 Elsevier Ltd. All rights reserved.
Buckling of a stiff film bound to a compliant substrate - Part I: Formulation, linear stability of cylindrical patterns, secondary bifurcations - Audoly, Basile and Boudaoud, Arezki

Abstract : The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become unstable with respect to undulating stripes. These results account for many of the previously published experimental or numerical results on this geometry. (c) 2008 Elsevier Ltd. All rights reserved.