arXiv subjects

Mokhtar Adda-Bedia

Publications and source records attributed to Mokhtar Adda-Bedia.

At least 19 recordsLinked to original sources

arXiv2026

Crack-Tip Opening as a Probe for Length-Scale Separation in Geometrically Nonlinear Solids

Soft elastic solids are highly deformable materials where fracture is driven by the complex coupling of geometric and material nonlinearities. While geometric nonlinearity (GNL) arises kinematically from the intrinsic capacity of solids to undergo large deformations, material nonlinearity stems from the constitutive behavior unique to each class of materials. Because GNL is a universal feature of all highly deformable solids, establishing its standalone impact is a prerequisite for understanding nonlinear fracture. Here, we focus on brittle soft solids to study the role of GNL alone on the near-tip fields of a static crack under mode I plane-strain conditions, providing a canonical baseline for integrating material nonlinearities in future investigations. By utilizing a compressible St. Venant-Kirchhoff material model, we analyze crack behavior under large deformations in the absence of material nonlinearity. We propose a robust postprocessing methodology based on the crack-tip opening displacement (CTOD) profile and derive asymptotic analytical solutions. Our results reveal a distinct near-tip region where the CTOD departs from classical linear elastic predictions, transitioning into a nonlinear regime dictated by Poisson's ratio. Using a matched-asymptotics approach, we define a physical nonlinear length scale $\lambda_\mathrm{nl}$ that bounds this region and scales quadratically with the far-field stress intensity factor $K_I$. We show that GNL acts as an intrinsic strain-stiffening mechanism sufficient to trigger energy partitioning, effectively shielding the crack tip and imparting an apparent toughening. Ultimately, we conclude that the geometrically nonlinear material model serves as a foundational framework for the broader study of nonlinear elastic fracture mechanics.

cond-mat.soft↗

arXiv2026

Self-focusing of helicity drives finite-time singularities in inviscid flows

This paper deals with the longstanding quest of the possible existence of finite-time singularities in the equations governing the dynamics of inviscid fluids, namely, Euler equations. Here, two contributions are brought for the case of perfect fluids with finite initial energy. First, a self-similar velocity field inspired by Leray Ansatz is proposed which allows for a separation of variables that transforms the original partial differential Euler equations to a nonlinear system of ordinary differential equations. This system can be solved semi-analytically and allows a continuum set of solutions parametrised by a self-similar exponent, $\nu$. Second, we use the conservation laws of Euler equations to select the possible finite-time singular solutions and the related self-similar exponents. We find that the helicity is the driving mechanism of the blow-up through a self-focusing mechanism. The flow near the singularity separates into two phases. A first phase is within a tubular region that shrinks as a power-law $(t_c-t)^\nu$, with $t_c$ the blow-up time, where the helicity is focused. This region is separated by a sharp interface from an outer region where the vorticity, and thus helicity, is identically zero. We found that the finite-time singularity may be either point-like or line-like depending on the dynamics of the tubular region along its axis of symmetry. Incidentally for a point-like singularity we recover the Leray scaling $\nu=1/2$ paving the way to a generalisation of this approach for the Navier-Stokes equations. Finally, we conjecture that if the helicity vanishes initially, no finite-time singularity would be possible, since in this case the singularity occurs at infinite time from the initial condition.

physics.flu-dyn↗

arXiv2026

Laddering of a knitted fabric: a topology-induced failure

Laddering is the propagation of a topological defect in an everyday-life material: weft knitted fabrics, following a broken thread or a dropped stitch. What is a minor frustration when damaging a pair of tights is a more serious issue for industrial-scale production, but might inspire new solutions to limit and mitigate damage to architected materials. In this work, laddering is investigated in a pre-stressed model knit through experiments and Discrete Element Rod simulations. The control parameter is the initial tension applied on the fabric. A force threshold due to the stitch's natural curvature is evidenced. It controls both the propagation onset and arrest, as tension is relaxed by the thread length freed by ladder growth, and enables damage prediction at moderate tension. Furthermore, we uncovered that the laddering velocity is of the order of the velocity of bending waves and exhibits an unexpected linear scaling with the fabric tension, that arises from a complex combination of elastic and friction forces. Finally, we discuss the implications of our results from the perspective of damage control and mitigation.

cond-mat.soft↗

arXiv2025

Curved crack paths are predicted by elastic-charges

Predicting crack trajectories in brittle solids remains an open challenge in fracture mechanics due to the non-local nature of crack propagation and the way cracks modify their surrounding medium. Here, we develop a framework for analytically predicting crack trajectories, similar to predicting the motion of charged particles in external fields within Newtonian mechanics. We demonstrate that a crack can be described as a distribution of elastic charges, and within the framework of Linear Elastic Fracture Mechanics (LEFM), its interaction with the background stress can be approximated by a singular geometric charge at the crack tip. The cracks motion is then predicted as the propagation of this singular charge within the unperturbed stress field. We apply our approach to study crack trajectories near defects and validate it through experiments on flat elastomer sheets containing an edge dislocation. The experimental results show excellent agreement with theoretical predictions, including the convergence of curved crack trajectories toward a single focal point. We discuss future extension of our theory to the motion of multiple interacting cracks. Our findings highlight the potential of the elastic-charges approach to significantly advance classical fracture mechanics by enabling analytical solutions to problems traditionally requiring numerical methods.

cond-mat.soft↗

arXiv2024

Dual Role for Heterogeneity in Dynamic Fracture

We approach the problem of heterogeneous dynamic fracture by considering spatiotemporal perturbations to planar crack fronts. Front propagation is governed by local energy balance between the elastic energy per unit area available to fracture, G, and the dissipation in creating new surfaces. G is known analytically as a perturbation series in the crack front fluctuation. For dissipation that monotonically increases with the crack speed, we derive an equation of motion for crack fronts that is second-order accurate. In the linear order, heterogeneity does not change the net speed of fracture. In the second order, nonlinear interactions of the front and the heterogeneous landscape populate an intermediate-scale fluctuation spectrum. We find that, when dissipation weakly grows with velocity, nonlinearities globally amplify dissipation and reduce the crack speed. Strong velocity dependence, however, mitigates toughening effects and may facilitate fracture.

cond-mat.mtrl-sci↗

arXiv2024

A comprehensive study of nonlinear perturbations in the dynamics of planar crack fronts

The interaction of crack fronts with asperities is central to the criteria of fracture in heterogeneous materials and for predicting fracture surface formation. It is known how dynamic crack fronts respond to small, 1st-order, perturbations. However, large and localized disturbances to crack motion induce dynamic and geometric nonlinear effects beyond the existing linear theories. Because the determination of the 3D elastic fields surrounding perturbed crack fronts is a necessary step towards any theoretical study of crack front dynamics, we develop a 2nd-order perturbation theory for the asymptotic fields of planar crack fronts. Based on previous work, we consider two models of fracture. In the so-called scalar elastic model, which is analogous to anti-plane (Mode III) fracture, the stress and displacement fields are obtained through matched asymptotic expansions. A self-consistent expansion is used to resolve the fields near tensile (Mode I) crack fronts. Both methods can be extended to higher perturbation orders. The main results of this work are the explicit 2nd-order expressions of the local dynamic energy-release-rates for perturbations of straight fronts. These general formulae recover the known energy-release-rates of curved quasi-static fronts and of straight dynamic fronts.

cond-mat.soft↗

arXiv2023

Transonic and supershear crack propagation driven by geometric nonlinearities

Linear elastic fracture mechanics theory predicts that the speed of crack growth is limited by the Rayleigh wave speed. Although many experimental observations and numerical simulations have supported this prediction, some exceptions have raised questions about its validity. The underlying reasons for these discrepancies and the precise limiting speed of dynamic cracks remain unknown. Here, we demonstrate that tensile (mode~I) cracks can exceed the Rayleigh wave speed and propagate at supershear speeds. We show that taking into account geometric non-linearities, inherent in most materials, is sufficient to enable such propagation modes. These geometric non-linearities modify the crack-tip singularity, resulting in different crack-tip opening displacements, cohesive zone behavior, and energy flows towards the crack tip.

cond-mat.soft↗

arXiv2023

Crack tip kinematics reveal the cohesive zone structure in brittle hydrogel fracture

When brittle hydrogels fail, several mechanisms conspire to alter the state of stress near the tip of a crack, and it is challenging to identify which mechanism is dominant. In the fracture of brittle solids, a sufficient far-field stress results in the complete loss of structural strength as the material `unzips' at the tip of a crack, where stresses are concentrated. Direct studies of the so-called small-scale yielding zone, where deformation is large, are sparing. Using hydrogels as a model brittle solid, we probe the small-scale yielding region with a combination of microscopy methods that resolve the kinematics of the deformation. A zone over which most of the energy is dissipated through the loss of cohesion is identified in the immediate surroundings of the crack tip. With direct measurements, we determine the scale and structure of this zone, and identify how the specific loss mechanisms in this hydrogel material might generalize for brittle material failure.

cond-mat.mtrl-sci↗

arXiv2022

Substrate Curvature Curbs the Coffee Ring Effect

"When the liquid phase of a particle-laden droplet evaporates, a ring of solute is typically formed - what has become known as the "coffee ring effect". A key focus of recent work has been the suppression of the coffee-ring effect to leave behind more spatially-uniform particle coatings instead. Efforts to suppress the coffee ring effect often focus on physical effects such as Marangoni flows. Here we focus on a purely geometric effect - the effect of substrate curvature - by evaporating suspension droplets on spherical surfaces of different radius of curvature. We find that stains formed on more highly curved substrates are more uniform. To understand this, we propose a model of the evaporation-induced flow, combined with a detailed calculation of how curvature modifies the local evaporation rate. This model shows that evaporation in the centre of the droplet is enhanced, leading to increased concentration of solute there and a reduction in the propensity for ring-formation.

physics.flu-dyn↗

arXiv2022

Kaluza-Klein Dimensional Reduction From Elasticity Theory of Crumpled Paper

During the last century, two independent theories using the concept of dimensional reduction have been developed independently. The first, known as Föppl-von Kàrmàn theory, uses Riemannian geometry and continuum mechanics to study the shaping of thin elastic structures which could become as complex as crumpled paper. The second one, known as Kaluza-Klein theory, uses Minkowskian geometry and general relativity to unify fundamental interactions and gravity under the same formalism. Here we draw a parallel between these two theories in an attempt to use concepts from elasticity theory of plates to recover the Einstein-Maxwell equations. We argue that Kaluza-Klein theory belongs to the same conceptual group of theories as three-dimensional elasticity, which upon dimensional reduction leads to the Föppl-von Kàrmàn theory of two-dimensional elastic plates. We exploit this analogy to develop an alternative Kaluza-Klein formalism in the framework of elasticity theory in which the gravitational and electromagnetic fields are respectively associated with stretching-like and bending-like deformations. We show that our approach of dimensional reduction allows us to retrieve the Lagrangian densities of both gravitational, electromagnetic and Dirac spinors fields as well as the Lagrangian densities of mass and charge sources.

gr-qc↗

arXiv2022

Nonlinear extension of Kolosov-Muskhelishvili stress function formalism

The method of stress-function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for nonlinear stress function. The new formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.

cond-mat.mtrl-sci↗

arXiv2022

Delamination from an adhesive sphere: Curvature-induced dewetting versus buckling

Everyday experience confirms the tendency of adhesive films to detach from spheroidal regions of rigid substrates -- what is a petty frustration when placing a sticky bandage onto an elbow or knee is a more serious matter in the coating and painting industries. Irrespective of their resistance to bending, a key driver of such phenomena is Gauss' \textit{Theorema Egregium}, which implies that naturally flat sheets cannot conform to doubly-curved surfaces without developing a strain whose magnitude grows sharply with the curved area. Previous attempts to characterize the onset of curvature-induced delamination, and the complex patterns it gives rise to, assumed a dewetting-like mechanism in which the propensity of two materials to form contact through interfacial energy is modified by an elastic energy penalty. We show that this approach may characterize moderately bendable adhesive sheets, but fails qualitatively to describe the curvature-induced delamination of ultrathin films, whose mechanics is governed by their propensity to buckle under minute levels of compression. Combining mechanical and geometrical considerations, we introduce a minimal model for curvature-induced delamination that accounts for two elementary buckling motifs, shallow "rucks" and localized "folds". We predict nontrivial scaling rules for the onset of curvature-induced delamination and various features of the emerging patterns, which compare well with experimental observations. Beyond gaining control on the use of ultrathin adhesives in cutting edge technologies such as stretchable electronics, our analysis is a significant step towards quantifying the multiscale morphological complexity that emerges upon imposing geometrical and mechanical constraints on highly bendable solid objects.

cond-mat.soft↗

arXiv2022

How hidden 3D structure within crack fronts reveals energy balance

Griffith's energetic criterion, or `energy balance', has for a century formed the basis for fracture mechanics; the energy flowing into a crack front is precisely balanced by the dissipation (fracture energy) at the front. If the crack front structure is not properly accounted for, energy balance will either appear to fail or lead to unrealistic results. Here, we study the influence of the secondary structure of low-speed crack propagation in hydrogels under tensile loading conditions. We first show that these cracks are bistable; either simple (cracks having no secondary structure) or faceted crack states (formed by steps propagating along crack fronts) can be generated under identical loading conditions. The selection of either crack state is determined by the form of the initial `seed' crack; perfect seed cracks generate simple cracks while a small local mode~III component generates crack fronts having multiple steps. Step coarsening eventually leads to single steps that propagate along crack fronts. As they evolve, steps locally change the instantaneous structure and motion of the crack front, breaking transverse translational invariance. In contrast to simple cracks, faceted cracks can, therefore, no longer be considered as existing in a quasi-2D system. For both simple and faceted cracks we simultaneously measure the energy flux and local dissipation along these crack fronts over velocities, $v$, spanning $0<v<0.2c_R$ ($c_R$ is the Rayleigh wave speed). We find that, in the presence of secondary structure within the crack front, the implementation of energy balance must be generalized for 3D systems; faceted cracks reveal energy balance, only when we account for the local dynamic dissipation at each point along the crack front.

cond-mat.soft↗

arXiv2022

Transition from viscoelastic to fracture-like peeling of pressure-sensitive adhesives

We investigate the process of the slow unrolling of a roll of typical pressure-sensitive adhesive, Scotch tape, under its own weight. Probing the peeling velocities down to nm/s resolution, which is three orders of magnitudes lower than earlier measurements, we find that the speed is still non-zero. Moreover, the velocity is correlated to the relative humidity. A humidity increase leads to water uptake, making the adhesive weaker and easier to peel. At very low humidity, the adhesive becomes so stiff that it mainly responds elastically, leading to a peeling process akin to interfacial fracture. We provide a quantitative understanding of the peeling velocity in the two regimes.

cond-mat.soft↗

arXiv2021

Curving Origami with Mechanical Frustration

We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-like origami, made from an elastic sheet decorated by a series of parallel creases crossed by a central longitudinal crease. Surprisingly, while the imprinted crease network does not exhibit a geodesic curvature, the emergent structure is characterized by an effective curvature produced by the deformed central fold. Moreover, both finite element analysis and manually made mylar origamis show a robust empirical relation between the imprinted crease network's dimensions and the apparent curvature. A detailed examination of this geometrical relation shows the existence of three typical elastic deformations, which in turn induce three distinct types of morphogenesis. We characterize the corresponding kinematics of crease network deformations and determine their phase diagram. Taking advantage of the frustration caused by the competition between crease stiffness and kinematics of crease network deformations, we provide a novel tool for designing curved origami structures constrained by strong geometrical properties.

cond-mat.soft↗

arXiv2020

Plasticity and Aging of Folded Elastic Sheets

We investigate the dissipative mechanisms exhibited by creased material sheets when subjected to mechanical loading, which comes in the form of plasticity and relaxation phenomena within the creases. After demonstrating that plasticity mostly affects the rest angle of the creases, we devise a mapping between this quantity and the macroscopic state of the system that allows us to track its reference configuration along an arbitrary loading path, resulting in a powerful monitoring and design tool for crease-based metamaterials. Furthermore, we show that complex relaxation phenomena, in particular memory effects, can give rise to a non-monotonic response at the crease level, possibly relating to the similar behavior reported for crumpled sheets. We describe our observations through a classical double-logarithmic time evolution and obtain a constitutive behavior compatible with that of the underlying material. Thus the lever effect provided by the crease allows magnified access to the material's rheology.

cond-mat.soft↗

arXiv2020

Flowing emulsions through disorder: Critical depinning and smectic rivers

During the past sixty minutes only, oil companies have extracted six trillions liters of oil from the ground, i.e. the volume of about two hundreds Olympic swimming pools. This phenomenal number gives a striking illustration of the impact of multiphase flows on the world economy and environment. From a fundamental perspective, we now clearly understand the large-scale patterns formed when liquid interfaces are driven through heterogeneous environments. In stark contrast, the displacement of fragmented fluids through disordered media remains limited to isolated droplets and bubbles. Here, we elucidate the collective dynamics of emulsions hydrodynamically driven through disordered environments. Advecting hundreds of thousands of microfluidic droplets through random lattices of pinning sites, we establish that the mobilization of confined emulsions is a critical dynamical transition. Unlike contact-line depinning, emulsion mobilization is not triggered by large-scale avalanches but merely requires the coordinated motion of small groups of particles. Criticality arises from the correlations of seemingly erratic depinning events over system-spanning scales along smectic river networks. We elucidate the microscopic origin of these self-organized flow patterns: contact interactions and hydrodynamic focusing conspire to mobilize emulsion out of disorder. We close our article commenting on the similarities (and profound differences) with the plastic depinning transitions of driven flux lines in high-$T_{\rm c}$ superconductors, and grain transport in eroded sand beds.

cond-mat.soft↗

arXiv2018

Local Mechanical Description of an Elastic Fold

To go beyond the simple model for the fold as two flexible surfaces or faces linked by a crease that behaves as an elastic hinge, we carefully shape and anneal a crease within a polymer sheet and study its mechanical response. First, we carry out an experimental study that consists on recording both the shape of the fold in various loading configurations and the associated force needed to deform it. Then, an elastic model of the fold is built upon a continuous description of both the faces and the crease as a thin sheet with a non flat reference configuration. The comparison between the model and experiments yields the local fold properties and explains the significant differences we observe between tensile and compression regimes. Furthermore, an asymptotic study of the fold deformation enables us to determine the local shape of the crease and identify the origin of its mechanical behaviour.

cond-mat.soft↗