arXiv subjects

Basile Audoly

Publications and source records attributed to Basile Audoly.

At least 19 recordsLinked to original sources

arXiv2025

Classifying soft elastic lattices using higher-order homogenization

We propose a methodology for the homogenization of periodic elastic lattices that covers the case of unstable lattices, having affine (macroscopic) or periodic (microscopic) mechanisms. The singular cell problems that are encountered when a periodic mechanism is present are naturally solved by treating the amplitude $\theta(X)$ of the mechanism as an enrichment variable. We use asymptotic second-order homogenization to derive an effective energy capturing both the strain-gradient effect $\nabla \varepsilon$ relevant to affine mechanisms, and the $\nabla \theta$ regularization relevant to periodic mechanisms, if any is present. The proposed approach is illustrated with a selection of lattices displaying a variety of effective behaviors. It follows a unified pattern that leads to a classification of these effective behaviors.

cond-mat.soft↗

arXiv2024

Fixing non-positive energies in higher-order homogenization

Energy functionals produced by second-order homogenization of periodic elastic structures commonly feature negative gradient moduli. We show that this undesirable property is caused by the truncation of the energy expansion in powers of the small scale separation parameter. By revisiting Cholesky's LDLT decomposition, we propose an alternative truncation method that restores positivity while preserving the order of accuracy. We illustrate this method on a variety of periodic structures, both continuous and discrete, and derive compact analytical expressions of the homogenized energy that are positive and accurate to second order. The method can also cure the energy functionals produced by second-order dimension reduction, which suffer similar non-positivity issues. It naturally extends beyond second order.

cond-mat.soft↗

arXiv2024

Asymptotic, second-order homogenization of linear elastic beam networks

We propose a general approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The network may be nearly (rather than exactly) periodic: its elastic and geometric properties are allowed to vary slowly in space, in addition to being periodic at the scale of the unit cell. The reference configuration may be prestressed. A homogenized strain energy depending on both the macroscopic strain $\boldsymbol{\varepsilon}$ and its gradient $\nabla \boldsymbol{\varepsilon}$ is obtained by means of a two-scale expansion. The homogenized energy is asymptotically exact two orders beyond that obtained by classical homogenization. The homogenization method is implemented in a symbolic calculation language and applied to various types of networks, such as a 2D honeycomb, a 2D Kagome lattice, a 3D truss and a 1D pantograph. It is validated by comparing the predictions of the microscopic displacement to that obtained by full, discrete simulations. This second-order method remains highly accurate even when the strain gradient effects are significant, such as near the lips of a crack tip or in regions where a gradient of pre-strain is imposed.

cond-mat.soft↗

arXiv2023

Twisting instabilities in elastic ribbons with inhomogeneous pre-stress: a macroscopic analog of thermodynamic phase transition

We study elastic ribbons subject to large, tensile pre-stress confined to a central region within the cross-section. These ribbons can buckle spontaneously to form helical shapes, featuring regions of alternating chirality (phases) that are separated by so-called perversions (phase boundaries). This instability cannot be described by classical rod theory, which incorporates pre-stress through effective natural curvature and twist; these are both zero due to the mirror symmetry of the pre-stress. Using dimension reduction, we derive a one-dimensional (1D) 'rod-like' model from a plate theory, which accounts for inhomogeneous pre-stress as well as finite rotations. The 1D model successfully captures the qualitative features of torsional buckling under a prescribed end-to-end displacement and rotation, including the co-existence of buckled phases possessing opposite twist, and is in good quantitative agreement with the results of numerical (finite-element) simulations and model experiments on elastomeric samples. Our model system provides a macroscopic analog of phase separation and pressure-volume-temperature state diagrams, as described by the classical thermodynamic theory of phase transitions.

cond-mat.soft↗

arXiv2021

Simple deformation measures for Discrete elastic rods and ribbons

The Discrete elastic rod method (Bergou et al., 2008) is a numerical method for simulating slender elastic bodies. It works by representing the center-line as a polygonal chain, attaching two perpendicular directors to each segment, and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulas for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy of the strain gradients are handled naturally.

cond-mat.soft↗

arXiv2021

Effective continuum models for the buckling of non-periodic architected sheets that display quasi-mechanism behaviors

In this work, we construct an effective continuum model for architected sheets that are composed of bulky tiles connected by slender elastic joints. Due to their mesostructure, these sheets feature quasi-mechanisms -- low-energy local kinematic modes that are strongly favored over other deformations. In sheets with non-uniform mesostructure, kinematic incompatibilities arise between neighboring regions, causing out-of-plane buckling. The effective continuum model is based on a geometric analysis of the sheets' unit cells and their energetically favorable modes of deformation. Its major feature is the construction of a strain energy that penalizes deviations from these preferred modes of deformation. The effect of non-periodicity is entirely described through the use of spatially varying geometric parameters in the model. Our simulations capture the out-of-plane buckling that occurs in non-periodic specimens and show good agreement with experiments. While we only consider one class of quasi-mechanisms, our modeling approach could be applied to a diverse set of shape-morphing systems that are of interest to the mechanics community.

cond-mat.mtrl-sci↗

arXiv2020

Asymptotic derivation of high-order rod models from non-linear 3D elasticity

We propose a method for deriving equivalent one-dimensional models for slender non-linear structures. The approach is designed to be broadly applicable, and can handle in principle finite strains, finite rotations, arbitrary cross-sections shapes, inhomogeneous elastic properties across the cross-section, arbitrary elastic constitutive laws (possibly with low symmetry) and arbitrary distributions of pre-strain, including finite pre-strain. It is based on a kinematic parameterization of the actual configuration that makes use of a center-line, a frame of directors, and local degrees of freedom capturing the detailed shape of cross-sections. A relaxation method is applied that holds the framed center-line fixed while relaxing the local degrees of freedom; it is asymptotically valid when the macroscopic strain and the properties of the rod vary slowly in the longitudinal direction. The outcome is a one-dimensional strain energy depending on the apparent stretching, bending and twisting strain of the framed center-line; the dependence on the strain gradients is also captured, yielding an equivalent rod model that is asymptotically exact to higher order. The method is presented in a fully non-linear setting and it is verified against linear and weakly non-linear solutions available from the literature.

cond-mat.soft↗

arXiv2020

What is the stiffness of a bent book?

We study the bending of a book-like system, comprising a stack of elastic plates coupled through friction. The behavior of this layered system is rich and nontrivial, with a non-additive enhancement of the apparent stiffness and a significant hysteretic response. A dimension reduction procedure is employed to develop a centerline-based theory describing the stack as a non-linear planar rod with internal shear. We consider the coupling between the nonlinear geometry and the elasticity of the stacked plates, treating the interlayer friction perturbatively. This model yields predictions for the stack's mechanical response in three-point bending that are in excellent agreement with our experiments. Remarkably, we find that the energy dissipated during deformation can be rationalized over three orders of magnitude, including the regimes of a thick stack with large deflection. This robust dissipative mechanism could be harnessed to design new classes of low-cost and efficient damping devices.

cond-mat.soft↗

arXiv2020

A one-dimensional model for elasto-capillary necking

We derive a non-linear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from 3d finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurately, including well into the localized regime, and that it has a vastly larger domain of validity than 1d model proposed so far.

cond-mat.soft↗

arXiv2019

Asymptotically exact strain-gradient models for nonlinear slender elastic structures: a systematic derivation method

We propose a general method for deriving one-dimensional models for nonlinear structures. It captures the contribution to the strain energy arising not only from the macroscopic elastic strain as in classical structural models, but also from the strain gradient. As an illustration, we derive one-dimensional strain-gradient models for a hyper-elastic cylinder that necks, an axisymmetric membrane that produces bulges, and a two-dimensional block of elastic material subject to bending and stretching. The method offers three key advantages. First, it is nonlinear and accounts for large deformations of the cross-section, which makes it well suited for the analysis of localization in slender structures. Second, it does not require any a priori assumption on the form of the elastic solution in the cross-section, i.e., it is Ansatz-free. Thirdly, it produces one-dimensional models that are asymptotically exact when the macroscopic strain varies on a much larger length scale than the cross-section diameter.

physics.app-ph↗

arXiv2019

Photo-Motile Structures

Actuation remains a signifcant challenge in soft robotics. Actuation by light has important advantages: objects can be actuated from a distance, distinct frequencies can be used to actuate and control distinct modes with minimal interference and signifcant power can be transmitted over long distances through corrosion-free, lightweight fiber optic cables. Photo-chemical processes that directly convert photons to configurational changes are particularly attractive for actuation. Various researchers have demonstrated light-induced actuation with liquid crystal elastomers combined with azobenzene photochromes. We present a simple modeling framework and a series of examples that studies actuation by light. Of particular interest is the generation of cyclic or periodic motion under steady illumination. We show that this emerges as a result of a coupling between light absorption and deformation. As the structure absorbs light and deforms, the conditions of illumination change, and this in turn changes the nature of further deformation. This coupling can be exploited in either closed structures or with structural instabilities to generate cyclic motion.

cond-mat.soft↗

arXiv2018

Shape-morphing architected sheets with non-periodic cut patterns

We investigate the out-of-plane shape morphing capability of single-material elastic sheets with architected cut patterns that result in arrays of tiles connected by flexible hinges. We demonstrate that a non-periodic cut pattern can cause a sheet to buckle into three-dimensional shapes, such as domes or patterns of wrinkles, when pulled at specific boundary points. These global buckling modes are observed in experiments and rationalized by an in-plane kinematic analysis that highlights the role of the geometric frustration arising from non-periodicity. The study focuses on elastic sheets, and is later extended to elastic-plastic materials to achieve shape retention. Our work illustrates a scalable route towards the fabrication of three-dimensional objects with nonzero Gaussian curvature from initially-flat sheets.

cond-mat.soft↗

arXiv2018

A diffuse interface model for the analysis of propagating bulges in cylindrical balloons

With the aim to characterize the formation and propagation of bulges in cylindrical rubber balloons, we carry out an expansion of the non-linear axisymmetric membrane model assuming slow axial variations. We obtain a diffuse interface model similar to that introduced by van der Waals in the context of liquid-vapor phase transitions. This provides a quantitative basis to the well-known analogy between propagating bulges and phase transitions. The diffuse interface model is amenable to numerical as well as analytical solutions, including linear and non-linear bifurcation analyses. Comparisons to the original membrane model reveal that the diffuse interface model captures the bulging phenomenon very accurately, even for well-localized phase boundaries.

physics.class-ph↗

arXiv2017

Buckling of an elastic ridge: competition between wrinkles and creases

We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $\phi$ below a critical value $\phi^\star\approx 90^\circ$ experiments reveal an extended sinusoidal mode, while for $\phi$ above $\phi^\star$ we observe a series of creases progressively invading the lateral faces starting from the ridge. A numerical linear stability analysis is set up using the finite-element method and correctly predicts the sinusoidal mode for $\phi \leq \phi^\star$, as well as the associated critical strain $\epsilon_{\mathrm{c}}(\phi)$. The experimental transition at $\phi^\star$ is found to occur when this critical strain $\epsilon_{\mathrm{c}}(\phi)$ attains the value $\epsilon_{\mathrm{c}}(\phi^\star) = 0.44$ corresponding to the threshold of the sub-critical surface creasing instability. Previous analyses have focused on elastic crease patterns appearing on planar surfaces, where the role of scale-invariance has been emphasized; our analysis of the elastic ridge provides a different perspective, and reveals that scale-invariance is not a sufficient condition for localization.

physics.class-ph↗

arXiv2016

The surprising dynamics of a chain on a pulley: Lift-off and snapping

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. Here, we consider the dynamics of the chain when one of the weights is removed and, thus, one end is pulled with constant acceleration. This simple change has dramatic consequences for the ensuing motion: at a finite time, the chain `lifts off' from the pulley and the free end subsequently accelerates faster than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack, or snap, of a whip. We combine experiments, numerical simulations, and theoretical arguments to explain key aspects of this dynamical problem.

cond-mat.soft↗

arXiv2014

Liquid ropes: a geometrical model for thin viscous jets instabilities

Thin viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitch-like patterns including meanders, W-patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasi-static geometrical model which captures the patterns, consisting of three coupled ODEs for the radial deflection, the orientation and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the fall height.

physics.flu-dyn↗

arXiv2014

"Wunderlich, meet Kirchhoff": A general and unified description of elastic ribbons and thin rods

The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich's model) and the limit of small width (Sadowksky's model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Fr\'enet-Serret's frame, the present work unifies the description of thin ribbons and thin rods.

cond-mat.soft↗

arXiv2013

A non-linear rod model for folded elastic strips

We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin- walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.

cond-mat.soft↗