Nonlinear extension of the 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 the 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, the Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for the nonlinear stress function. Our 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.

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 band aid onto a 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' 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 sheets but fails qualitatively to describe the curvature-induced delamination of ultrathin films, whose mechanics is governed by their propensity to buckle and delaminate partially, under minute levels of compression. Combining mechanical and geometrical considerations, we introduce a minimal model for curvature-induced delamination accounting for the two buckling motifs that underlie partial delamination: 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 experiments. Beyond gaining control on the use of ultrathin adhesives in cutting-edge technologies such as stretchable electronics, our analysis is a significant step toward quantifying the multiscale morphology that emerges upon imposing geometrical and mechanical constraints on highly bendable solid objects.

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-1 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.

The onset of the frictional motion of dissimilar materials.

Frictional motion between contacting bodies is governed by propagating rupture fronts that are essentially earthquakes. These fronts break the contacts composing the interface separating the bodies to enable their relative motion. The most general type of frictional motion takes place when the two bodies are not identical. Within these so-called bimaterial interfaces, the onset of frictional motion is often mediated by highly localized rupture fronts, called slip pulses. Here, we show how this unique rupture mode develops, evolves, and changes the character of the interface's behavior. Bimaterial slip pulses initiate as "subshear" cracks (slower than shear waves) that transition to developed slip pulses where normal stresses almost vanish at their leading edge. The observed slip pulses propagate solely within a narrow range of "transonic" velocities, bounded between the shear wave velocity of the softer material and a limiting velocity. We derive analytic solutions for both subshear cracks and the leading edge of slip pulses. These solutions both provide an excellent description of our experimental measurements and quantitatively explain slip pulses' limiting velocities. We furthermore find that frictional coupling between local normal stress variations and frictional resistance actually promotes the interface separation that is critical for slip-pulse localization. These results provide a full picture of slip-pulse formation and structure that is important for our fundamental understanding of both earthquake motion and the most general types of frictional processes.

Emergence of scale-free smectic rivers and critical depinning in emulsions driven through disorder.

During the past 60 min, oil companies have extracted 6 trillion liters of oil from the ground, thereby giving a striking illustration of the impact of multiphase flows on the world economy. From a fundamental perspective, we largely understand the dynamics of interfaces separating immiscible fluids driven through heterogeneous environments. In stark contrast, the basic mechanisms ruling the transport of fragmented fluids, such as foams and emulsions, remain elusive with studies mostly limited to isolated droplets and bubbles. Here, we demonstrate that the mobilization of emulsion driven through model disordered media is a critical plastic depinning transition. To elucidate this collective dynamics, we track the trajectories of hundreds of thousands of microfluidic droplets advected through random lattices of pinning sites. Their dynamics reveals that macroscopic mobilization only requires the coordinated motion of small groups of particles and does not involve any large-scale avalanches. Criticality arises from the interplay between contact and hydrodynamic interaction, which channel seemingly erratic depinning events along smectic river networks correlated over system spanning scales. Beyond the specifics of emulsion transport, we close our article discussing the similarities and profound differences with the plastic depinning transitions of driven flux lines in high-T c superconductors, charged colloids, and grain transport in eroded sand beds.

Cloaking by coating: how effectively does a thin, stiff coating hide a soft substrate?

From human tissue to fruits, many soft materials are coated by a thin layer of a stiffer material. While the primary role of such a coating is often to protect the softer material, the thin, stiff coating also has an important effect on the mechanical behaviour of the composite material, making it appear significantly stiffer than the underlying material. We study this cloaking effect of a coating for the particular case of indentation tests, which measure the 'firmness' of the composite solid: we use a combination of theory and experiment to characterize the firmness quantitatively. We find that the indenter size plays a key role in determining the effectiveness of cloaking: small indenters feel a mixture of the material properties of the coating and of the substrate, while large indenters sense largely the unadulterated substrate.

Experimental Evidence of Thermal-Like Behavior in Dense Granular Suspensions.

We experimentally investigate the statistical behavior of a model two-dimensional granular system undergoing stationary sedimentation. Buoyant cylindrical particles are rotated in a liquid-filled drum, thus confined in a harmonic centripetal potential with tunable curvature, which competes with gravity to produce various stationary states: though heterogeneous, the packing fraction of the system can be tuned from fully dispersed to crystallized as the rotation rate is increased. We show that this dynamical system is in mechanical equilibrium in the confining potential and exhibits a thermal-like behavior, where the granular pressure and the packing fraction are related through an equation of state. We obtain an expression of the equation of state allowing us to probe the nature of the hydrodynamic interactions between the particles. This description is valid in the whole range of the physical parameters we investigated and reveals a buoyant energy scale that we interpret as an effective temperature. We finally discuss the behavior of our system at high packing fractions and the relevance of the equation of state to the liquid-solid phase transition.

Crackling Dynamics in the Mechanical Response of Knitted Fabrics.

Crackling noise, which occurs in a wide range of situations, is characterized by discrete events of various sizes, often correlated in the form of avalanches. We report experimental evidence that the mechanical response of a knitted fabric displays such broadly distributed events both in the force signal and in the deformation field, with statistics analogous to that of earthquakes or soft amorphous materials. A knit consists of a regular network of frictional contacts, linked by the elasticity of the yarn. When deformed, the fabric displays spatially extended avalanchelike yielding events resulting from collective interyarn contact slips. We measure the size distribution of these avalanches, at the stitch level from the analysis of nonelastic displacement fields and externally from force fluctuations. The two measurements yield consistent power law distributions reminiscent of those found in other avalanching systems. Our study shows that a knitted fabric is not only a thread-based metamaterial with highly sought after mechanical properties, but also an original, model system, with topologically protected structural order, where an intermittent, scale-invariant response emerges from minimal ingredients, and thus a significant landmark in the study of out-of-equilibrium universality.

Nonlinear Focusing in Dynamic Crack Fronts and the Microbranching Transition.

Cracks in brittle materials produce two types of generic surface structures: facets at low velocities and microbranches at higher ones. Here we observe a transition from faceting to microbranching in polyacrylamide gels that is characterized by nonlinear dynamic localization of crack fronts. To better understand this process we derive a first-principles nonlinear equation of motion for crack fronts in the context of scalar elasticity. Its solution shows that nonlinear focusing coupled to rate dependence of dissipation governs the transition to microbranching.

Non-wetting drops at liquid interfaces: from liquid marbles to Leidenfrost drops.

We consider the flotation of deformable, non-wetting drops on a liquid interface. We consider the deflection of both the liquid interface and the droplet itself in response to the buoyancy forces, density difference and the various surface tensions within the system. Our results suggest new insight into a range of phenomena in which such drops occur, including Leidenfrost droplets and floating liquid marbles. In particular, we show that the floating state of liquid marbles is very sensitive to the tension of the particle-covered interface and suggest that this sensitivity may make such experiments a useful assay of the properties of these complex interfaces.

Effect of the material properties on the crumpling of a thin sheet.

While simple at first glance, the dense packing of sheets is a complex phenomenon that depends on material parameters and the packing protocol. We study the effect of plasticity on the crumpling of sheets of different materials by performing isotropic compaction experiments on sheets of different sizes and elasto-plastic properties. First, we quantify the material properties using a dimensionless foldability index. Then, the compaction force required to crumple a sheet into a ball as well as the average number of layers inside the ball are measured. For each material, both quantities exhibit a power-law dependence on the diameter of the crumpled ball. We experimentally establish the power-law exponents and find that both depend nonlinearly on the foldability index. However the exponents that characterize the mechanical response and morphology of the crumpled materials are related linearly. A simple scaling argument explains this in terms of the buckling of the sheets, and recovers the relation between the crumpling force and the morphology of the crumpled structure. Our results suggest a new approach to tailor the mechanical response of the crumpled objects by carefully selecting their material properties.

Structure and mechanics of aegagropilae fiber network.

Fiber networks encompass a wide range of natural and manmade materials. The threads or filaments from which they are formed span a wide range of length scales: from nanometers, as in biological tissues and bundles of carbon nanotubes, to millimeters, as in paper and insulation materials. The mechanical and thermal behavior of these complex structures depends on both the individual response of the constituent fibers and the density and degree of entanglement of the network. A question of paramount importance is how to control the formation of a given fiber network to optimize a desired function. The study of fiber clustering of natural flocs could be useful for improving fabrication processes, such as in the paper and textile industries. Here, we use the example of aegagropilae that are the remains of a seagrass (Posidonia oceanica) found on Mediterranean beaches. First, we characterize different aspects of their structure and mechanical response, and second, we draw conclusions on their formation process. We show that these natural aggregates are formed in open sea by random aggregation and compaction of fibers held together by friction forces. Although formed in a natural environment, thus under relatively unconstrained conditions, the geometrical and mechanical properties of the resulting fiber aggregates are quite robust. This study opens perspectives for manufacturing complex fiber network materials.

Geometry and design of origami bellows with tunable response.

Origami folded cylinders (origami bellows) have found increasingly sophisticated applications in space flight and medicine. In spite of this interest, a general understanding of the mechanics of an origami folded cylinder has been elusive. With a newly developed set of geometrical tools, we have found an analytic solution for all possible cylindrical rigid-face states of both Miura-ori and triangular tessellations. Although an idealized bellows in both of these families may have two allowed rigid-face configurations over a well-defined region, the corresponding physical device, limited by nonzero material thickness and forced to balance hinge and plate-bending energy, often cannot stably maintain a stowed configuration. We have identified the parameters that control this emergent bistability, and we have demonstrated the ability to design and fabricate bellows with tunable deployability.

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves.

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

Popping Balloons: A Case Study of Dynamical Fragmentation.

Understanding the physics of fragmentation is important in a wide range of industrial and geophysical applications. Fragmentation processes involve large strain rates and short time scales that take place during crack nucleation and propagation. Using rubber membranes, we develop an experimental analysis that enables us to track the fragmentation process in situ in both time and space. We find that bursting a highly stretched membrane yields a treelike fragmentation network that originates at a single seed crack, followed by successive crack tip-splitting events. We show that a dynamic instability drives this branching mechanism. Fragmentation occurs when the crack tip speed attains a critical velocity for which tip splitting becomes the sole available mechanism of releasing the stored elastic energy. Given the general character of the fragmentation processes, this framework should be applicable to other crack networks in brittle materials.

Dynamic stability of crack fronts: out-of-plane corrugations.

The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech. Phys. Solids 45, 591 (1997)] to study the out-of-plane stability of planar crack fronts in the framework of linear elastic fracture mechanics. We discuss a minimal scenario in which linearly unstable crack front corrugations might emerge above a critical front propagation speed. We calculate this speed as a function of Poisson's ratio and show that corrugations propagate along the crack front at nearly the Rayleigh wave speed. Finally, we hypothesize about a possible relation between such corrugations and the long-standing problem of crack branching.

Prototypical model for tensional wrinkling in thin sheets.

The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length--a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.

Statics and inertial dynamics of a ruck in a rug.

We consider the familiar problem of a bump, or ruck, in a rug. Under lateral compression, a rug bends to form a ruck--a localized region in which it is no longer in contact with the floor. We show that when the external force that created the ruck is removed, the ruck flattens out unless the initial compression is greater than a critical value, which we determine. We also study the inertial motion of a ruck that is generated when one end of the rug is moved rapidly. We show that the equations of motion admit a traveling ruck solution for which a linear combination of the tension and kinetic energy is determined by the ruck size. We confirm these findings experimentally. We end by discussing the potential implications of our work for the analogous propagation of localized slip pulses in the sliding of two bodies in contact.

In silico leaf venation networks: growth and reorganization driven by mechanical forces.

Development commonly involves an interplay between signaling, genetic expression and biophysical forces. However, the relative importance of these mechanisms during the different stages of development is unclear. Leaf venation networks provide a fitting context for the examination of these questions. In mature leaves, venation patterns are extremely diverse, yet their local structure satisfies a universal property: at junctions between veins, angles and diameters are related by a vectorial equation analogous to a force balance. Using a cell proliferation model, we reproduce in silico the salient features of venation patterns. Provided that vein cells are given different mechanical properties, tensile forces develop along the veins during growth, causing the network to deform progressively. Our results suggest that the local structure of venation networks results from a reorganization driven by mechanical forces, independently of how veins form. This conclusion is supported by recent observations of vein development in young leaves and by the good quantitative agreement between our simulations and data from mature leaves.