Mokhtar Adda-Bedia

Capillary wrinkling of elastic membranes

We present a physically-based model for the deformation of a floating elastic membrane caused by the presence of a liquid drop. Starting from the equations of membrane theory modified to account for surface energies, we show that the presence of a liquid drop causes an azimuthal compression over a finite region. This explains the origin of the wrinkling of such membranes observed recently (Huang et al., Science, 2007, 317, 650) and suggests a single parameter that determines the extent of the wrinkled region. While experimental data supports the importance of this single parameter, our theory under-predicts the extent of the wrinkled region observed experimentally. We suggest that this discrepancy is likely to be due to the wrinkling observed here being far from the threshold.

An elastica problem: instabilities of an elastic arch

Abstract We analyze the modes of instability of an elastic homogeneous arch that is loaded at its center. We study first the dynamical linear stability of the symmetric elastic arch with respect to small perturbations. We also perform a constrained minimization of the static energy of the system. The numerical resolution of both the dynamical and the static problems allows the determination of the phase diagram corresponding to the behavior of the arch. For this simple elastica problem, the phase diagram shows a very rich structure that is in agreement with the experimental results.

Thermal fracture as a framework for quasi-static crack propagation

We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith criterion. We then argue that the calculations of the stress intensity factors can be combined with the standard crack propagation criteria to obtain the evolution equation for the crack tip within any loading configuration. The theoretical results of the thermal crack problem agree with the numerical simulations we performed using a phase field model. Moreover, it turns out that the phase-field model allows to clarify the nature of the transition between straight and oscillatory cracks which is shown to be supercritical.

Mechanical response of a creased sheet.

We investigate the mechanics of thin sheets decorated by noninteracting creases. The system considered here consists of parallel folds connected by elastic panels. We show that the mechanical response of the creased structure is twofold, depending both on the bending deformation of the panels and the hingelike intrinsic response of the crease. We show that a characteristic length scale, defined by the ratio of bending to hinge energies, governs whether the structures response consists in angle opening or panel bending when a small load is applied. The existence of this length scale is a building block for future works on origami mechanics.

Seismic Radiation from Simple Models of Earthquakes

We review some basic features of shear wave generation and energy balance for a 2D anti plane rupture. We first study the energy balance for a flat fault, and for a fault that contains a single localized kink. We determine an exact expression for the partition between strain energy flow released from the elastic medium surrounding the fault, radiated energy flow and energy release rate. This balance depends only on the rupture speed and the residual stress intensity factor. When the fault contains a kink, the energy available for fracture is reduced so that the rupture speed is reduced. When rupture speed changes abruptly, the radiated energy flow also changes abruptly. As rupture propagates across the kink, a shear wave is emitted that has a displacement spectral content that decreases like ω^(-2) at high frequencies. We then use spectral elements to model the propagation of an antiplane crack with a slip-weakening friction law. Since the rupture front in this case has a finite length scale, the wave emitted by the kink is smoothed at very high frequencies but its general behavior is similar to that predicted by the simple sharp crack model. A model of a crack that has several kinks and wanders around a mean rupture directions, shows that kinks reduce the rupture speed along the average rupture direction of the fault. Contrary to flat fault models, a fault with kinks produces high frequency waves that are emitted every time the rupture front turns at a kink. Finally, we discuss the applicability of the present results to a 3D rupture model.

Seismic Radiation from a Kink on an Antiplane Fault

We study radiation and energy balance for an antiplane fault containing a kink. A semi-infinite crack with a sharp rupture front propagates along the flat portion of a kinked crack. At time t � 0, the crack reaches the kink located at the origin of coordinates and continues propagating beyond the kink at a different speed. We com- pute the exact solution for this problem using a Chaplygin transformation, a variation of the well-known Cagniard-de Hoop method. We first establish an integral equation for the computation of stress intensity factor after the kink and then we solve numeri- cally for the velocity and stress field around the crack. We find that the propagation of the crack across the kink produces a sharp change in energy balance that in turn pro- duces a circular SH wave centered at the kink that we call the kink wave. Across the wavefront of this wave there is a sudden jump in particle velocity and radial stress. At the same time, a local stress concentration appears on the external side of the kink. We establish an exact energy balance for this problem in terms of energy rates per unit crack advance. Radiated energy is shown to maintain the balance between elastic en- ergy released by the bulk and energy used to make the crack advance. The kink wave- front is the boundary between a field dominated by the initial flat portion of the crack and a region dominated by radiation from the kink.

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.

Roughness of tensile crack fronts in heterogenous materials

The dynamics of planar crack fronts in heterogeneous media is studied using a recently proposed stochastic equation of motion that takes into account nonlinear effects. The analysis is carried for a moving front in the quasi-static regime using the Self Consistent Expansion. A continuous dynamical phase transition between a flat phase and a dynamically rough phase, with a roughness exponent ζ = 1/2, is found. The rough phase becomes possible due to the destabilization of the linear modes by the nonlinear terms. Taking into account the irreversibility of the crack propagation, we infer that the roughness exponent found in experiments might become history dependent, and so our result gives a lower bound for ζ.

Elastic theory of origami-based metamaterials.

Origami offers the possibility for new metamaterials whose overall mechanical properties can be programed by acting locally on each crease. Starting from a thin plate and having knowledge about the properties of the material and the folding procedure, one would like to determine the shape taken by the structure at rest and its mechanical response. In this article, we introduce a vector deformation field acting on the imprinted network of creases that allows us to express the geometrical constraints of rigid origami structures in a simple and systematic way. This formalism is then used to write a general covariant expression of the elastic energy of n-creases meeting at a single vertex. Computations of the equilibrium states are then carried out explicitly in two special cases: the generalized waterbomb base and the Miura-Ori. For the waterbomb, we show a generic bistability for any number of creases. For the Miura folding, however, we uncover a phase transition from monostable to bistable states that explains the efficient deployability of this structure for a given range of geometrical and mechanical parameters. Moreover, the analysis shows that geometric frustration induces residual stresses in origami structures that should be taken into account in determining their mechanical response. This formalism can be extended to a general crease network, ordered or otherwise, and so opens new perspectives for the mechanics and the physics of origami-based metamaterials.

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.