Romain Lagrange

Aeroelastic instability of cantilevered flexible plates in uniform flow

We address the flutter instability of a flexible plate immersed in an axial flow. This instability is similar to flag flutter and results from the competition between destabilizing pressure forces and stabilizing bending stiffness. In previous experimental studies, the plates have always appeared much more stable than the predictions of two-dimensional models. This discrepancy is discussed and clarified in this paper by examining experimentally and theoretically the effect of the plate aspect ratio on the instability threshold. We show that the two-dimensional limit cannot be achieved experimentally because hysteretical behaviour and three-dimensional effects appear for plates of large aspect ratio. The nature of the instability bifurcation (sub- or supercritical) is also discussed in the light of recent numerical results.

Curvature-induced symmetry breaking determines elastic surface patterns

Symmetry-breaking transitions associated with the buckling and folding of curved multilayered surfaces-which are common to a wide range of systems and processes such as embryogenesis, tissue differentiation and structure formation in heterogeneous thin films or on planetary surfaces-have been characterized experimentally. Yet owing to the nonlinearity of the underlying stretching and bending forces, the transitions cannot be reliably predicted by current theoretical models. Here, we report a generalized Swift-Hohenberg theory that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. By testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions for the critical curves separating labyrinth, hybrid and hexagonal phases. Furthermore, a comparison to earlier experiments suggests that the theory is universally applicable to macroscopic and microscopic systems. Our approach builds on general differential-geometry principles and can thus be extended to arbitrarily shaped surfaces.

A rotating fluid cylinder subject to weak precession

In this paper, we report experimental and theoretical results on the flow inside a precessing and rotating cylinder. Particle image velocimetry measurements have revealed the instantaneous structure of the flow and confirmed that it is the sum of forced inertial (Kelvin) modes, as predicted by the classical linear inviscid theory. But this theory predicts also that the amplitude of a mode diverges when its natural frequency equals the precession frequency. A viscous and weakly nonlinear theory has therefore been developed at the resonance. This theory has been compared to experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude scales as the square root of the Reynolds number owing to the presence of Ekman layers on the cylinder walls. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle to the power one-third for a given resonance. The nonlinear theory also predicts the forcing of a geostrophic (axisymmetric) mode which has been observed and measured in the experiments. These results allow the flow inside a precessing cylinder to be fully characterized in all regimes as long as there is no instability.

Precessional instability of a fluid cylinder

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

Wrinkling crystallography on spherical surfaces

Significance Curved crystals cannot comprise hexagons alone; additional defects are required by both topology and energetics that depend on the system size. These constraints are present in systems as diverse as virus capsules, soccer balls, and geodesic domes. In this paper, we study the structure of defects of the crystalline dimpled patterns that self-organize through curved wrinkling on a thin elastic shell bound to a compliant substrate. The dimples are treated as point-like packing units, even if the shell is a continuum. Our results provide quantitative evidence that our macroscopic wrinkling system can be mapped into and described within the framework of curved crystallography, albeit with some important differences attributed to the far-from-equilibrium nature of our patterns. We present the results of an experimental investigation on the crystallography of the dimpled patterns obtained through wrinkling of a curved elastic system. Our macroscopic samples comprise a thin hemispherical shell bound to an equally curved compliant substrate. Under compression, a crystalline pattern of dimples self-organizes on the surface of the shell. Stresses are relaxed by both out-of-surface buckling and the emergence of defects in the quasi-hexagonal pattern. Three-dimensional scanning is used to digitize the topography. Regarding the dimples as point-like packing units produces spherical Voronoi tessellations with cells that are polydisperse and distorted, away from their regular shapes. We analyze the structure of crystalline defects, as a function of system size. Disclinations are observed and, above a threshold value, dislocations proliferate rapidly with system size. Our samples exhibit striking similarities with other curved crystals of charged particles and colloids. Differences are also found and attributed to the far-from-equilibrium nature of our patterns due to the random and initially frozen material imperfections which act as nucleation points, the presence of a physical boundary which represents an additional source of stress, and the inability of dimples to rearrange during crystallization. Even if we do not have access to the exact form of the interdimple interaction, our experiments suggest a broader generality of previous results of curved crystallography and their robustness on the details of the interaction potential. Furthermore, our findings open the door to future studies on curved crystals far from equilibrium.

Curvature-Controlled Defect Localization in Elastic Surface Crystals

We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal, and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling, or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test bed for exploring the effects of geometrically induced forces on assemblies of topological charges.

From wrinkling to global buckling of a ring on a curved substrate

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler–Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.

Compression-induced stiffness in the buckling of a one fiber composite

We study the buckling of a one fiber composite whose matrix stiffness is slightly dependent on the compressive force. We show that the equilibrium curves of the system exhibit a limit load when the induced stiffness parameter gets bigger than a threshold. This limit load increases when the stiffness parameter is increasing and it is related to a possible localized path in the post-buckling domain. Such a change in the maximum load may be very desirable from a structural stand point.

Triadic instability of a non-resonant precessing fluid cylinder

Abstract Flows forced by a precessional motion can exhibit instabilities of crucial importance, whether they concern the fuel of a flying object or the liquid core of a telluric planet. So far, stability analyses of these flows have focused on the special case of a resonant forcing. Here, we address the instability of the flow inside a precessing cylinder in the general case. We first show that the base flow forced by the cylinder precession is a superposition of a vertical or horizontal shear flow and of an infinite sum of forced modes. We then perform a linear stability analysis of this base flow by considering its triadic resonance with two free Kelvin modes. Finally, we derive the amplitude equations of the free Kelvin modes and obtain an expression of the instability threshold and growth rate.

Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.