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International audience; In strongly anisotropic materials the orientation-dependent fracture surface energy is a non-convex function of the crack angle. In this context, the classical Griffith model becomes ill-posed and requires a regularization. We revisit the crack kinking problem in materials with strongly anisotropic surface energies by using a variational phase-field model. The model includes in the energy functional a quadratic term on the second gradient of the phase-field. This term has a regularizing effect, energetically penalizing the crack curvature. We provide analytical formulas for the dependence of the surface energy on the crack direction and develop an open-source finite-element solver for the higher-order phase-field problem. Quantitative numerical experiments for the crack kinking problem show that the crack kinking directions observed in our phase-field simulations are in close agreement with the generalized maximum energy release rate criterion. Finally, we revisit a thermal quenching experiment in the case of slabs with strongly anisotropic surface energies. We show that the anisotropy can strongly affect the observed crack patterns, either by stabilizing straight cracks or by inducing zigzag crack patterns. In the case of zigzag cracks, we observe that crack kinking is always associated with an unstable propagation of a finite length add-crack in a single time-step.

International audience; Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

International audience; Gradient damage models used in phase-field approaches to brittle fracture are characterised by material softening and instabilities. We present novel numerical techniques for the bifurcation and stability analysis along quasi-static evolution paths as well as practical tools to select stable evolutions. Our approach stems from the variational approach to fracture and the theory of rate-independent irreversible processes whereby a quasi-static evolution is formulated in terms of incremental energy minimisation under unilateral constraints. Focussing on the discrete setting obtained with finite elements techniques, we discuss the links between bifurcation criteria for an evolution and stability of equilibrium states. Key concepts are presented through the analytical solution of a two-degreesof-freedom model featuring a continuum family of bifurcation branches. We introduce numerical methods to i) assess (second-order) stability conditions for time-discrete evolutions subject to damage irreversibility, and ii) to select possible stable evolutions based on an energetic criterion. Our approach is based on the solution of a coupled eigenvalue problem which accounts for the time-discrete irreversibility constraint on damage. Several numerical examples illustrate that this approach allows us to filter out unstable solutions provided by standard (first-order) minimisation algorithms as well as to effectively compute stable evolution paths. We demonstrate our purpose on a multifissuration problem featuring complex fracture patterns, to show how the eigenvalue analysis enables to compute and retrieve morphological properties of emerging cracks.

A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von Karman shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment.

International audience; The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (i) definition of the energy; (ii) formulation of the damage evolution problem. The total energy of the body is defined in terms of the state variables which are the displacement field and the damage field in the case of quasi-brittle materials. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem is then based on the concepts of irreversibility, stability and energy balance. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Moreover, the variational formulation leads to a natural numerical method based on an alternate minimization algorithm. Several numerical examples illustrate the ability of this approach to account for all the process of fracture including a 3D thermal shock problem where the crack evolution is very complex.

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6× faster on a majority of the test cases considered.

International audience; In a seminal paper published in 1951, Taylor studied the interactions between a viscous fluid and an immersed flat sheet which is subjected to a travelling wave of transversal displacement. The net reaction of the fluid over the sheet turned out to be a force in the direction of the wave phase-speed. This effect is a key mechanism for the swimming of microorganisms in viscous fluids. Here, we study the interaction between a viscous fluid and a special class of non-linear morphing shells. We consider pre-stressed shells showing a one-dimensional set of neutrally-stable equilibria with almost cylindrical configurations. Their shape can be effectively controlled through embedded active materials, generating a large-amplitude shape-wave associated to precession of the axis of maximal curvature. We show that this shape-wave constitutes the rotational analogue of a Taylor's sheet, where the translational swimming velocity is replaced by an angular velocity. Despite the net force acting on the shell vanishes, the resultant torque does not. A similar mechanism can be used to maneuver in viscous fluids.

International audience; Local damage models with softening needs localization limiters to preserve the mathematical and physical consistency. In this paper we compare the properties of strain-gradient and damage-gradient regularizations. Gradient-damage models introduce a quadratic dependency of the dissipated energy on the gradient of the damage field and are nowadays extensively used as phase-field approximation of brittle fracture. Their key feature is to provide a smeared approximation of a crack as a band of localised damage with a finite energy dissipation per unit of surface, that can be identified with the fracture toughness of the Griffith model. Strain gradient models introduce a quadratic dependence of the elastic energy on the gradient of the strain field. A similar term can be physically interpreted as the presence in the material of linear, but nonlocal, stiffnesses, that can be eventually be affected by damage. Despite this attractive interpretation, we have found that strain-gradient regularized models can hardly be used to approximate brittle fracture, because smeared cracks with non-vanishing and finite dissipated energies are hardly obtained. Our analysis is based on variational models and focuses on the one-dimensional traction problem.

Phase-field models, sometimes referred to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith’ criterion. For a one-dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter be identified with the material’s internal or characteristic length. In this article, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U- and V-notches to show that the nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects. Our main claim, supported by validation and verification in a broad range of materials and geometries, is that crack nucleation can be accurately predicted by minimization of a nonlinear energy in variational phase field models, and does not require the introduction of ad-hoc criteria.

International audience; 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.

We study fracture and delamination of a thin stiff film bonded on a rigid substrate through a thin compliant bonding layer. Starting from the three-dimensional system, upon a scaling hypothesis, we provide an asymptotic analysis of the three-dimensional variational fracture problem as the thickness goes to zero, using Gamma-convergence. We deduce a two-dimensional limit model consisting of a brittle membrane on a brittle elastic foundation. The fracture sets are naturally discriminated between transverse cracks in the film (curves in 2D) and debonded surfaces (two-dimensional planar regions). We introduce the vectorial plane-elasticity case, applying the rigorous results established for scalar displacement fields, in order to numerically investigate the typical cracking scenarios encountered in applications. To this end, we formulate a reduced-dimension, rate-independent, irreversible evolution law for transverse fracture and debonding of thin film systems. Finally, we propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional. We provide an illustration of the capabilities of the formulation exploring complex crack patterns in one and two dimensions, showing a qualitative comparison with geometrically involved real life examples.

International audience; Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [4] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elasto-damage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracturèfracturè a la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approaches in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, owning a continuous manifold of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fiber-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully non-linear inextensible uniform-curvature shell model. We report on the fabrication, identification, and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a "gear-less motor", which is, in reality, a precession of the axis of principal curvature., Comment: 20 pages, 9 figures

Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identify the orientation of collagen fibres using inexpensive and relatively simple techniques. The method uses established histological techniques and an algorithm implemented in the MATLAB image processing toolbox. It takes an average of 15 s to evaluate one image, compared to several hours if assessed visually. The technique was applied to histological sections of human skin with different Langer line orientations and a definite correlation between the orientation of Langer lines and the preferred orientation of collagen fibres in the dermis was observed. The structural parameters of the Gasser-Ogden-Holzapfel (GOH) model were all successfully evaluated. It is expected that the results of this study will assist those wishing to model skin, and that the algorithm described will be of benefit to those who wish to evaluate the collagen dispersion of other soft tissues., to appear Annals of Biomedical Engineering

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Composite shells show a rich multistable behaviour of interest for the design of shape-changing (morphing) structures. Previous studies have investigated how the initial shape determines the shell stability properties. For uniform initial curvatures and orthotropic material behaviour, not more than two stable equilibria have been reported. In this paper, we prove that untwisted, uniformly curved, thin orthotropic shells can have up to three stable equilibrium configurations. Cases of tristability are first documented using a numerical stability analysis of an extensible shallow shell model. Including mid-plane extension shows that the shells must be sufficiently curved in relation to their thickness to be multistable. Thus, an inextensible model allows us to perform an analytical stability analysis. Focusing on untwisted initial configurations, we illustrate with simple analytical results how the material parameters of the shell control the dependence of its multistable behaviour on the initial curvatures. In particular, we show that when the bending stiffness matrix approaches a degeneracy condition, the shell exhibits three stable equilibria for a wide range of initial curvatures.

International audience; A shell can have multiple stable equilibria either if its initial curvature is suciently high or if a suitably strong pre-stress is applied. Under the hypotheses of a thin and shallow shell, we derive closed form results for the critical values of curvatures and pre-stresses leading to bistability and trista-bility. These analytical expressions allow to easily provide guidelines to build shells with dierent stability properties.

Bistable structures, such as buckled beams or plates, are characterized by a two-well potential. Their nonlinear properties are currently exploited in actuators design (e.g. MEMS micropumps, switches, memory cells) to produce relatively high displacements and forces with low actuation energies. We investigate the use of distributed multiparameter actuation to control the buckling and postbuckling behavior of a three-layer piezoelectric beam pinned at either end. A two-parameter bending actuation controls the transversal motion, whilst an axial actuation and a beam end-shortening modulate the tangent bending stiffness. The postbuckling behavior is studied by reducing to a 2 dof system a nonlinear extensible elastica model. When the bending actuation is spatially symmetric, the postbuckling phenomena are analogue to those obtained for a transversal midspan force, being characterized by a snap-through instability. The use of a two-parameter actuation opens new transition scenarios, where it is possible to get true quasi-static transitions between the two specular equilibria of the buckled beam, without any instability phenomenon. The efficiencies of these different transition paths are discussed in terms of energetic requirements and stability properties. A numerical example shows the technical feasibility of the proposed actuation technique.

Reduced-order modal models of linear piezoelectric structures are useful in vibration control and health monitoring. We study experimental identification of the fundamental parameters of these modal models. We propose two identification techniques for estimating piezoelectric modal couplings and piezoelectric modal capacitances. Both methods are easily implementable and rely on elementary vibration tests. We show the application of these methods to a sample structure hosting multiple transducers.

This paper analyzes different numerical methods for modal analysis of stepped piezoelectric beams modeled by the Euler–Bernoulli beam theory. Results from standard numerical approaches, that rely on the discretization of the stepped beam (assumed modes and finite-element methods), are compared with the solution of the exact transcendental eigenvalue problem for the infinite dimensional system. An accurate and manageable novel method, that enriches the assumed modes basis functions with special jump functions, is presented. Numerical results are compared with experimental data and the accuracy of the adopted beam model is validated.

In this paper a Euler–Bernoulli-like model of layered piezoelectric beams is presented. It describes more accurately than the others already presented in the literature both transverse (Poisson and piezoelectrically induced) cross-sectional deformations and through-the-thickness variations of the electric field and electric displacement. A deductive approach based on a mixed variational formulation is adopted and distributions of deformation, stress, electric field and electric displacement are simultaneously prescribed. The attention is focused on the choice of the most fitting assumptions to recover complex 3D cross-sectional field distributions. In particular, transverse interactions between different layers are taken into account by enforcing specific conditions on transverse stress through the Lagrange multipliers method. The estimate of electromechanical beam constitutive coefficients is discussed and comparison with standard modelling approaches, which assume either vanishing transverse stresses or vanishing transverse strains, is emphasized. For a sandwich piezoelectric beam and for a two-layer beam, expressions of the beam constitutive coefficients are provided and the main features of the proposed model are highlighted by presenting the through-the-thickness distribution of the 3D state fields associated to beam-axis deformations and applied voltage. As a main peculiarity, the proposed beam model is able to coherently estimate the equivalent piezoelectric capacitance also when the thickness of elastic and piezoelectric layers is comparable.

International audience; In this paper models of layered piezoelectric beams are discussed. The attention is focused on the analysis of the assumptions on transversal stress and strain distribution and their influence on the deduction of the beam constitutive equations from a three dimensional description. A model accounting for non trivial transversal interactions between different layers is deduced from a mixed variational formulation where non-local conditions on transversal stress are enforced by Lagrange multipliers method. The fully coupled electromechanical nature of the system is described. For a sandwich piezoelectric beam, analytical expressions of the beam constitutive coefficients are provided and comparisons to standard modelling approaches are presented. Finally, the fundamental features of the proposed model are highlighted by presenting the through-the-thickness distribution of the 3D state fields.

This paper presents a gradient approach for the quasi-static macroscopic modeling of superelasticity in softening shape memory alloys bars. The model is assumed to be rate-independent and to depend on a single internal variable. Regularization of the model is achieved through the free energy by assuming a quadratic dependance with respect to the gradient of the internal variable. The quasi-static evolution is then formulated in terms of two physical principles: a stability criterion which consists in selecting the local minima of the total energy of the system and an energy balance condition. Both homogeneous and non-homogeneous evolutions are investigated analytically for a family of material parameters. Non-homogeneous evolutions can be divided into three stages: the localized martensite nucleation followed by the propagation of the localized phase transformation front and finally the annihilation of the austenite phase. For each stage, the local phase field profile as well as the global stress–strain response are derived in closed-form. Due to the presence of an internal length related to the regularization, size effects are inherent with such non-local model. We show that for sufficiently long bars, snap-backs occur at the onset of localized phase transformation, leading to a time discontinuity in the quasi-static evolution.

These notes give a short introduction to the methods for the study of stability of elastic structures. We consider only the finite-dimensional case, where the state of the system is represented by a discrete set of variables. The core of the exposition focuses on the illustration of energetic methods where equilibrium and stability are found by studying the point of stationarity and minima of a scalar function of the state variables. After three introductory sections presenting the links between stability and energy minimization (Section 2), potential energy (Section 3) and discretization methods (Section 4), we detail the mathematical methods required to minimize a function of n variables (Section 5-8). We include the theory and recipes to deal with equality and inequality constraints, providing several examples of applications to simple structures. We then show how to classify regular and singular points (bifurcations) in force-displacement diagrams (Section 9) and give a fully worked example with several degrees of freedom (Section 10). Section 11 presents, through an example, the dynamical theory of stability including Floquet theory for systems with periodic solutions. Finally, Section 12 shows how energetic methods can be applied to the study of material instabilities, by considering the case of springs with irreversible damage.

International audience; This paper studies the initiation of cracks in the thermal shock problem through the variational analysis of the quasi-static evolution of a gradient damage model. We consider a two-dimensional semi-infinite slab with an imposed temperature drop on its free surface. The damage model is formulated in the framework of the variational theory of rate-independent processes based on the principles of irreversibility, stability and energy balance. In the case of a sufficiently severe shock, we show that damage immediately occurs and that its evolution follows first a fundamental branch without localization. Then it bifurcates into another branch in which damage localization will take place to finally generate cracks. The determination of the time and mode of that bifurcation allows us to explain the periodic distribution of the so-initiated cracks and to calculate the crack spacing in terms of the material and loading parameters. Numerical investigations complete and quantify the analytical results.

International audience; We study the genesis and the selective propagation of complex crack networks induced by thermal shock or drying of brittle materials. We use a quasi-static gradient damage model to perform large scale numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion and depends only on the fracture toughness, while crack morphogenesis is driven by the material's internal length. Our numerical simulations feature networks of parallel cracks and selective arrest in two dimensions and hexagonal columnar joints in three dimensions, without any hypotheses on cracks geometry and are in good agreement with available experimental results.

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.

Under the effect of surface tension, a blob of liquid adopts a spherical shape when immersed in another fluid. We demonstrate experimentally that soft, centimeter-size elastic solids can exhibit a similar behavior: when immersed into a liquid, a gel having a low elastic modulus undergoes large, reversible deformations. We analyze three fundamental types of deformations of a slender elastic solid driven by surface stress, depending on the shape of its cross section: a circular elastic cylinder shortens in the longitudinal direction and stretches transversally; the sharp edges of a square based prism get rounded off as its cross sections tend to become circular; and a slender, triangular based prism bends. These experimental results are compared to analysis and nonlinear simulations of neo-Hookean solids deformed by surface tension and are found to be in good agreement with each other.

International audience; Differential growth of thin elastic bodies furnishes a surprisingly simple explanation of the complex and intriguing shapes of many biological systems, such as plant leaves and organs. Similarly, inelastic strains induced by thermal effects or active materials in layered plates are extensively used to control the curvature of thin engineering structures. Such behaviour inspires us to distinguish and to compare two possible modes of differential growth not normally compared to each other, in order to reveal the full range of out-of-plane shapes of an initially flat disk. The first growth mode, frequently employed by engineers, is characterized by direct bending strains through the thickness, and the second mode, mainly apparent in biological systems, is driven by extensional strains of the middle surface. When each mode is considered separately, it is shown that buckling is common to both modes, leading to bistable shapes: growth from bending strains results in a double-curvature limit at buckling, followed by almost developable deformation in which the Gaussian curvature at buckling is conserved; during extensional growth, out-of-plane distortions occur only when the buckling condition is reached, and the Gaussian curvature continues to increase. When both growth modes are present, it is shown that, generally, larger displacements are obtained under in-plane growth when the disk is relatively thick and growth strains are small, and vice versa. It is also shown that shapes can be mono-, bi-, tri- or neutrally stable, depending on the growth strain levels and the material properties: furthermore, it is shown that certain combinations of growth modes result in a free, or natural, response in which the doubly-curved shape of disk exactly matches the imposed strains. Such diverse behaviour, in general, may help to realise more effective actuation schemes for engineering structures.

International audience; Westudymulti-fissurationanddebondingphenomenaofathinfilmbondedtoastiffsubstrateusing the variational approach to fracture mechanics. We consider a reduced one-dimensional membrane model where the loading is introduced through uniform inelastic (e.g., thermal) strains in the film or imposed dis- placements of the substrate. Fracture phenomena are accounted for by adopting a Griffith model for debonding and transverse fracture. On the basis of energy minimization arguments, we recover the key qualitative prop- erties of the experimental evidences, like the periodicity of transverse cracks and the peripheral debonding of each regular segment. Phase diagrams relate the maximum number of transverse cracks that may be created before debonding takes place, as a function of the material properties and the sample's geometry. The theo- retical results are illustrated with numerical simulations obtained through a finite element discretization and a regularized variational formulation of the Ambrosio-Tortorelli type, which is suited to further extensions in two-dimensional settings.

International audience; The small-amplitude in-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible, shearable, planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there are no qualitative changes in the mode frequencies, other frequencies experience rapid variations after the buckling threshold, the thinner the rod, the more abrupt the variations. Eventually, a mismatch for half of the frequencies at buckling arises between the zero thickness limit of the extensible model and the inextensible model.

In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In this article, we first formulate the quasi-static evolution problem for a general class of such damage models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. We construct homogeneous as well as localized damage solutions in a closed form and illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress—displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant material parameters, such as limit stress and internal length. Finally, the 1D analytical results are compared with the numerical solution of the evolution problem in a 2D setting.

The aim of this work is two-fold: to design devices for passive electric damping of structural vibrations by distributed piezoelectric transducers and electric networks, and to experimentally validate the effectiveness of such a damping concept. Two different electric networks are employed, namely a purely resistive network and an inductive-resistive one. The presented devices can be considered as distributed versions of the well-known resistive and resonant shunt of a single piezoelectric transducer. The technicalfeasibility and damping effectiveness of the proposed novel devices are assessed through the construction of an experimental prototype. Experimental results are shown to be in very good agreement with theoretical predictions. It is proved that the presented technique allows for a substantial reduction in the inductances used when compared with those required by the single resonant shunted transducer. In particular, it is shown that the required inductance decreases when the number of piezoelectric elements is increased. The electric networks are optimized in order to reduce forced vibrations close to the first resonance frequency. Nevertheless, the damping effectiveness for higher modes is experimentally proved. As well as specific results, fundamental theoretical and experimental considerations for passive distributed vibration control are provided., Comment: 10 pages

We propose two identification techniques for estimating the piezoelectric couplings and the piezoelectric capacitances of reduced order modal models of linear piezoelectric structures. The two methods are easily implementable and demand few input data, which can be obtained both with experimental testing and numerical models (e.g. finite elements). We apply these methods to a sample structure hosting multiple transducers. We discuss in details the proper definition and identification of the inherent piezoelectric capacitances, whose meaning is often misunderstood.

In this paper, we examine and compare four different techniques for modal analysis of stepped piezoelectric beams. The first technique is based on the solution of the exact transcendental eigenvalue problem, formulated in terms of the dynamic stiffness matrix. The other three techniques are based on the Galerkin method for obtaining a finite-dimensional version of the system. Besides the classical assumed modes method and finite-element method, we propose a novel enhanced version of the assumed modes method, which introduces special jump functions to enrich the standard basis functions.

International audience; In this paper a coupled Euler–Bernoulli model of laminated piezoelectric beams is proposed. It is characterized by accounting for the influence of 3D distribution of mechanical stresses and strains through corrected electromechanical constitutive equations. In particular, the hypothesis of vanishing transverse (width direction) normal stress typical of standard beam models is weakened by imposing vanishing stress resultants. This integral condition is enforced by adopting a mixed variational principle and Lagrange multiplier method. Explicit expressions for the beam constitutive coefficients are given and the sandwich and bimorph piezoelectric benders are studied in details. The model is assessed through comparisons with standard models and 3D finite element results, showing an important enhancement of standard beam theories.

Electric vibration absorbers made of distributed piezoelectric devices for the control of beam vibrations are studied. The absorbers are obtained by interconnecting an array of piezoelectric transducers uniformly distributed on a beam with different modular electric networks. Five different topologies are considered and their damping performance is analysed and compared. Their optimal parameters are found by adopting a criterion for critical damping of k-waves: the parameters are suitably chosen to have the quickest temporal vibration decay for a single wave number k. The analysis is based on homogenized models of the modular piezo-electromechanical systems, i.e. they are regarded as continuous systems by assuming that the number of modules per unit length is high enough with respect to the considered wave numbers. Calling k -absorbers the corresponding optimal absorbers, we show that: (i) k-waves are damped in k-absorbers with an optimal decay time which is independent of the absorber interconnecting topology, while it depends only on the piezoelectric coupling coefficient; (ii) the efficiency of k-absorbers depends significantly on the absorber interconnecting topology for k different from k; (iii) one of the proposed absorbers (which is made of a fourth-order electric transmission line with a second-order electric dissipation) equally performs for all the wave numbers and accomplishes an effective multi-modal damping for the mechanically forced response; (iv) the optimal values of the electric parameters differently depend on the number n of used circuit modules for different interconnecting topologies and, in particular, the optimal inductance per module needed in a fourth-order electric transmission line is proportional 1/ n 3 .

Bistable structures, such as buckled beams, are characterized by a two-well potential. Their nonlinear properties are currently exploited in actuators to produce relatively high displacements and forces with low actuation energies. We investigate the use of distributed multiparameter actuation to control the buckling and postbuckling behaviour of a three-layer piezoelectric beam pinned at either end. A two-parameter bending actuation controls the transversal motion, whilst an axial actuation modulates the tangent bending stiffness. The postbuckling behaviour is studied by reducing to a 2 dof system a nonlinear extensible elastica model. When the bending actuation is spatially symmetric, the postbuckling phenomena are characterized by a snapthrough instability. The use of a two-parameter actuation opens new transition scenarios, where it is possible to get quasi-static transitions between the two equilibria of the buckled beam, without any instability phenomenon.

The work carried out during this thesis focused on the study of the behavior of a bistable buckled beam subjected to several types of actuations. The static and dynamic aspects of switching mecanism are analyzed. The kinematic model used is the Elastica model. This latter is adapted to take into account deformations, displacements and rotations of significant amplitudes. In addition, the extensible nature of the beam is considered. The static part is devoted to the study of bifurcation diagrams for three types of actuations: ponctual force, piezoelectric elements and Laplace forces. For each of these technologies, an optimization procedure is implemented. At the end of the static part, experimental tests are carried out. They made it possible to validate the model in the case of actuation via Laplace forces. In the dynamic part a new model is developed to simulate the dynamic behavior of the beam. This model taking into account three modes of buckling. We excited the beam via a sinusoidal point force. The influence of several parameters on the response of the beam is discussed. In particular, we established that bringing the beam into resonance reduces the switching force. In addition, the analysis of the diagrams of the Poincaré sections allows us to understand the origins of the chaotic behavior of the beam. Experimental tests were carried out at the end of the dynamic part in order to validate the model. The results issued from these studies are used to design a new technology for actuating Braille devices. This technology is based on bistable beams. The possibility of actuating these structures via electromagnetic forces and via piezoelectric elements is discussed.; Le travail effectué pendant cette thèse a porté sur l'étude de comportement d'une poutre flambée bistable soumise à plusieurs types d'actionnements. Les aspects statique et dynamique du comportement du bistable ont été analysés de manière détaillée. Le modèle cinématique exploité est le modèle Elastica extensible. La partie statique a été consacrée à l'étude des diagrammes de bifurcation pour trois types d'actionnements : force ponctuelle, éléments piézoélectriques et forces de Laplace. Pour chacune de ces technologies, une procédure d'optimisation a été mise en œuvre. À la fin de la partie statique, des essais expérimentaux ont été réalisés. Ils ont permis de valider le modèle dans le cas de l'actionnement via les forces de Laplace. Dans la partie dynamique, un nouveau modèle permettant de simuler le comportement dynamique de la poutre a été développé. Cette approche prenant en compte trois modes de flambage a mis en évidence le phénomène de croisement de modes. Nous avons excité la poutre via une force ponctuelle sinusoïdale. L'influence de plusieurs paramètres sur la réponse de la poutre a été discutée. De plus, l'analyse des diagrammes de sections de Poincaré nous a permis de comprendre les origines du comportement chaotique. Des essais expérimentaux ont été réalisés à la fin de la partie dynamique. Les résultats des études menées ont été exploités pour la conception d'une nouvelle technologie d'actionnement des dispositifs Braille fondée sur les poutres bistables. La possibilité d'actionner ces structures via les forces électromagnétiques et via les éléments piézoélectriques a été discutée.

Time-periodic systems are an important niche in dynamical systems, analysing these systems profoundly opens up new possibilities for engineering. The goal of this thesis is to provide a comprehensive guide on computing Floquet forms and application of modal analysis of structures in time-periodic elastic state. The method is illustrated by fundamental examples. Modal analysis consists of computing the mode shape and the natural frequency of the equations of motion, this method is limited to linear time-invariant structures. Linear time-periodic systems have attracted attention over the past decades. Floquet theory has been applied to analyse stability. Using the same theory it is possible to compute periodic eigenvectors, known as Floquet forms, which are time-periodically equivalent to modes. Thus allowing the generalisation of modal analysis to linear time-periodic structures. Floquet forms are computed as eigensolutions of Hill’s matrix. The eigenspectrum requires treatment to find linearly independent Floquet forms. By using the archetypical case of a Ziegler column eigenvector sorting is shown to be the most efficient method to sort out Floquet forms. Projecting the physical equations of motion on Floquet forms results in a system of uncoupled equations with time independent coefficients. Truncating the number of Floquet forms results in a reduced order model. Computing Floquet forms is simplified by first projecting the equations of motion on classic modes. A finite element model of a periodically prestressed beam shows convergence of the solution with increasing number of Floquet forms. Thus proving the concept of time-periodic modal analysis.; Les systèmes périodiques sont une niche importante dans l’analyse des systèmes dynamiques. L'analyse modale ouvre de nouvelles opportunités pour l’ingénierie et la recherche. Le but de cette thèse est de montrer de manière exhaustive comment calculer une base modale pour des systèmes périodiques et comment appliquer ça pour réduire les équations de mouvement. Pour montrer la méthode marche des cas fondamentaux sont analysés.

The work carried out during this thesis focused on the study of the behavior of a bistable buckled beam subjected to several types of actuations. The static and dynamic aspects of switching mecanism are analyzed. The kinematic model used is the Elastica model. This latter is adapted to take into account deformations, displacements and rotations of significant amplitudes. In addition, the extensible nature of the beam is considered. The static part is devoted to the study of bifurcation diagrams for three types of actuations: ponctual force, piezoelectric elements and Laplace forces. For each of these technologies, an optimization procedure is implemented. At the end of the static part, experimental tests are carried out. They made it possible to validate the model in the case of actuation via Laplace forces. In the dynamic part a new model is developed to simulate the dynamic behavior of the beam. This model taking into account three modes of buckling. We excited the beam via a sinusoidal point force. The influence of several parameters on the response of the beam is discussed. In particular, we established that bringing the beam into resonance reduces the switching force. In addition, the analysis of the diagrams of the Poincaré sections allows us to understand the origins of the chaotic behavior of the beam. Experimental tests were carried out at the end of the dynamic part in order to validate the model. The results issued from these studies are used to design a new technology for actuating Braille devices. This technology is based on bistable beams. The possibility of actuating these structures via electromagnetic forces and via piezoelectric elements is discussed.; Le travail effectué pendant cette thèse a porté sur l'étude de comportement d'une poutre flambée bistable soumise à plusieurs types d'actionnements. Les aspects statique et dynamique du comportement du bistable ont été analysés de manière détaillée. Le modèle cinématique exploité est le modèle Elastica extensible. La partie statique a été consacrée à l'étude des diagrammes de bifurcation pour trois types d'actionnements : force ponctuelle, éléments piézoélectriques et forces de Laplace. Pour chacune de ces technologies, une procédure d'optimisation a été mise en œuvre. À la fin de la partie statique, des essais expérimentaux ont été réalisés. Ils ont permis de valider le modèle dans le cas de l'actionnement via les forces de Laplace. Dans la partie dynamique, un nouveau modèle permettant de simuler le comportement dynamique de la poutre a été développé. Cette approche prenant en compte trois modes de flambage a mis en évidence le phénomène de croisement de modes. Nous avons excité la poutre via une force ponctuelle sinusoïdale. L'influence de plusieurs paramètres sur la réponse de la poutre a été discutée. De plus, l'analyse des diagrammes de sections de Poincaré nous a permis de comprendre les origines du comportement chaotique. Des essais expérimentaux ont été réalisés à la fin de la partie dynamique. Les résultats des études menées ont été exploités pour la conception d'une nouvelle technologie d'actionnement des dispositifs Braille fondée sur les poutres bistables. La possibilité d'actionner ces structures via les forces électromagnétiques et via les éléments piézoélectriques a été discutée.

The work carried out during this thesis focused on the study of the behavior of a bistable buckled beam subjected to several types of actuations. The static and dynamic aspects of switching mecanism are analyzed. The kinematic model used is the Elastica model. This latter is adapted to take into account deformations, displacements and rotations of significant amplitudes. In addition, the extensible nature of the beam is considered.The static part is devoted to the study of bifurcation diagrams for three types of actuations: ponctual force, piezoelectric elements and Laplace forces. For each of these technologies, an optimization procedure is implemented. At the end of the static part, experimental tests are carried out. They made it possible to validate the model in the case of actuation via Laplace forces.In the dynamic part a new model is developed to simulate the dynamic behavior of the beam. This model taking into account three modes of buckling. We excited the beam via a sinusoidal point force. The influence of several parameters on the response of the beam is discussed. In particular, we established that bringing the beam into resonance reduces the switching force. In addition, the analysis of the diagrams of the Poincaré sections allows us to understand the origins of the chaotic behavior of the beam. Experimental tests were carried out at the end of the dynamic part in order to validate the model.The results issued from these studies are used to design a new technology for actuating Braille devices. This technology is based on bistable beams. The possibility of actuating these structures via electromagnetic forces and via piezoelectric elements is discussed.; Le travail effectué pendant cette thèse a porté sur l'étude de comportement d'une poutre flambée bistable soumise à plusieurs types d'actionnements. Les aspects statique et dynamique du comportement du bistable ont été analysés de manière détaillée. Le modèle cinématique exploité est le modèle Elastica extensibleLa partie statique a été consacrée à l'étude des diagrammes de bifurcation pour trois types d'actionnements : force ponctuelle, éléments piézoélectriques et forces de Laplace. Pour chacune de ces technologies, une procédure d'optimisation a été mise en œuvre. À la fin de la partie statique, des essais expérimentaux ont été réalisés. Ils ont permis de valider le modèle dans le cas de l'actionnement via les forces de Laplace.Dans la partie dynamique, un nouveau modèle permettant de simuler le comportement dynamique de la poutre a été développé. Cette approche prenant en compte trois modes de flambage a mis en évidence le phénomène de croisement de modes. Nous avons excité la poutre via une force ponctuelle sinusoïdale. L'influence de plusieurs paramètres sur la réponse de la poutre a été discutée. De plus, l'analyse des diagrammes de sections de Poincaré nous a permis de comprendre les origines du comportement chaotique. Des essais expérimentaux ont été réalisés à la fin de la partie dynamique.Les résultats des études menées ont été exploités pour la conception d'une nouvelle technologie d'actionnement des dispositifs Braille fondée sur les poutres bistables. La possibilité d'actionner ces structures via les forces électromagnétiques et via les éléments piézoélectriques a été discutée.

Time-periodic systems are an important niche in dynamical systems, analysing these systems profoundly opens up new possibilities for engineering. The goal of this thesis is to provide a comprehensive guide on computing Floquet forms and application of modal analysis of structures in time-periodic elastic state. The method is illustrated by fundamental examples.Modal analysis consists of computing the mode shape and the natural frequency of the equations of motion, this method is limited to linear time-invariant structures.Linear time-periodic systems have attracted attention over the past decades. Floquet theory has been applied to analyse stability. Using the same theory it is possible to compute periodic eigenvectors, known as Floquet forms, which are time-periodically equivalent to modes. Thus allowing the generalisation of modal analysis to linear time-periodic structures.Floquet forms are computed as eigensolutions of Hill’s matrix. The eigenspectrum requires treatment to find linearly independent Floquet forms. By using the archetypical case of a Ziegler column eigenvector sorting is shown to be the most efficient method to sort out Floquet forms.Projecting the physical equations of motion on Floquet forms results in a system of uncoupled equations with time independent coefficients. Truncating the number of Floquet forms results in a reduced order model. Computing Floquet forms is simplified by first projecting the equations of motion on classic modes. A finite element model of a periodically prestressed beam shows convergence of the solution with increasing number of Floquet forms. Thus proving the concept of time-periodic modal analysis.; Systèmes périodiques sont un niche important dans l’analyse des systèmes dynamiques. Analyse modale ouvert des nouveaux opportunités pour l’ingénierie et la recherche. Le but de cette thèse est de montrer exhaustif comment calculer un base modale pour des systèmes périodiques et comment appliquer ça pour réduire les équations de mouvement. Pour montrer la méthode marche des cas fondamentaux sont analysés.

This thesis studies elastic structures including a bistable microstructure, where local instabilities introduce an equivalent macroscopic dissipation We focus on the one-dimensional case of a dimpled strip. Our dimpled strip is an unidirectional arrangement of bi-stable cells. In a first part, we study the properties of a single bi-stable cell. After presenting the main theoretical and numerical modelling tools, we propose two design concepts of bi-stable cells based obtained playing either with the initial shape or the pre-stress. The macroscopic behavior of the cell is approached from a study of equilibria and critical states by reduced models. This leads to the characterisation of the two-well non-convex energy potential of a bi-stable cell. The macroscopic behavior of the dimpled strip is obtained by studing the equilibrium of a series of bi-stable cells and selecting among all possible equilibrium paths the one passing through metastable states involving the successive transition of each of the elements. In the case where the element number becomes very large, the macroscopic behavior happens to be a continuous transition similar to a phase transformation mechanism. A phenomenological model of phase transformation is proposed in order to model in the context of generalized standard materials the transformation of the dimpled strip.; Dans cette thèse on étudie des structures élastiques ayant une microstructure bistable. Dans ces structures les phénomènes d’instabilité à l’échelle microscopique introduisent une dissipation effective à l’échelle macroscopique. On se concentre sur le cas d’une bande élastique constituée d’un arrangement unidirectionnel de cellules bi-stables compatibles entres elles. L’étude se déroule sur deux axes. Dans un premier temps on étudie la cellule bi-stable. Après avoir présenté les outils théoriques (modèles des poutres non-linéaires) et numériques (éléments finis et solveurs nonlinéaires) nécessaires à sa modélisation, on propose deux concepts de cellules bi-stables basées sur la forme initiale de la cellule, et sur le flambement. Le comportement macroscopique de la cellule est approché à partir d’une étude des équilibres et des états critiques par des modèles réduits. De cette étude on définit le potentiel d’une cellule bi-stable comme une énergie non convexe à deux puits. Le comportement macroscopique de la bande indentée est obtenu par l’équilibre d’une série de cellules bi-stables, en sélectionnant parmi toutes les branches d’équilibres possibles, le trajet d’équilibre passant par des états métastables impliquant la transition successive de chacun des éléments. Dans le cas où le nombre d’éléments devient très grand, le comportement macroscopique se trouve être une transition continue similaire aux mécanismes de transformation de phase. Un modèle phénoménologique de transformation de phase est proposé afin de modéliser dans le cadre des matériaux standards généralisés la transformation de la bande indentée.

This thesis studies elastic structures including a bistable microstructure, where local instabilities introduce an equivalent macroscopic dissipation We focus on the one-dimensional case of a dimpled strip. Our dimpled strip is an unidirectional arrangement of bi-stable cells. In a first part, we study the properties of a single bi-stable cell. After presenting the main theoretical and numerical modelling tools, we propose two design concepts of bi-stable cells based obtained playing either with the initial shape or the pre-stress. The macroscopic behavior of the cell is approached from a study of equilibria and critical states by reduced models. This leads to the characterisation of the two-well non-convex energy potential of a bi-stable cell. The macroscopic behavior of the dimpled strip is obtained by studing the equilibrium of a series of bi-stable cells and selecting among all possible equilibrium paths the one passing through metastable states involving the successive transition of each of the elements. In the case where the element number becomes very large, the macroscopic behavior happens to be a continuous transition similar to a phase transformation mechanism. A phenomenological model of phase transformation is proposed in order to model in the context of generalized standard materials the transformation of the dimpled strip.; Dans cette thèse on étudie des structures élastiques ayant une microstructure bistable. Dans ces structures les phénomènes d’instabilité à l’échelle microscopique introduisent une dissipation effective à l’échelle macroscopique. On se concentre sur le cas d’une bande élastique constituée d’un arrangement unidirectionnel de cellules bi-stables compatibles entres elles. L’étude se déroule sur deux axes. Dans un premier temps on étudie la cellule bi-stable. Après avoir présenté les outils théoriques (modèles des poutres non-linéaires) et numériques (éléments finis et solveurs nonlinéaires) nécessaires à sa modélisation, on propose deux concepts de cellules bi-stables basées sur la forme initiale de la cellule, et sur le flambement. Le comportement macroscopique de la cellule est approché à partir d’une étude des équilibres et des états critiques par des modèles réduits. De cette étude on définit le potentiel d’une cellule bi-stable comme une énergie non convexe à deux puits. Le comportement macroscopique de la bande indentée est obtenu par l’équilibre d’une série de cellules bi-stables, en sélectionnant parmi toutes les branches d’équilibres possibles, le trajet d’équilibre passant par des états métastables impliquant la transition successive de chacun des éléments. Dans le cas où le nombre d’éléments devient très grand, le comportement macroscopique se trouve être une transition continue similaire aux mécanismes de transformation de phase. Un modèle phénoménologique de transformation de phase est proposé afin de modéliser dans le cadre des matériaux standards généralisés la transformation de la bande indentée.

This work is essentially based on the phenomenon of multistability of thin structures as plates and shells and some associated applications. The work is divided in two parts. The first part aims to study theoretically, numerically and experimentally the multistability of thin orthotropic shallow shells with uniform curvature. We show notably that such a shell, when submitted to the combination of initial curvature and pre-stresses sufficiently high, possesses up to three stable states towards the choice of the material. First, we propose criteria to design and manufacture multistable shells up to tristability ; this work is validated by finit element simulations and experiments. After, we apply those criteria to the design and manufacture ofcylindrical multistable shells for which the energetic gap between stable states is minimal. The second part is dedicated to direct applications of bistability of thin cylindrical bistable shells with low energetic gap. We first propose an application on shape control via the use of active materials which we bond on the structure. This includes a first phase of theoretical design of both the structure and the actuation law, and a second phase of experimental demonstration. After, we study both theoretically and experimentally the non-linear dynamic properties of such structures with the aim to highlight the different modes of oscillations intrinsic to an external excitation source. Finally, we propose an application to non-linear broadband energy harvesting from vibrations based on multistable piezoelectric excited shells., Ces travaux de thèse sont basés principalement sur le phénomène de multistabilité des structures minces de type plaques et coques ainsi que quelques applications associées. Les travaux sont divisés en deux parties. La première partie a pour objet l’étude théorique, numérique et expérimentale de la multistabilité des coques minces orthotropes peu profondes à courbures uniformes. On montre notamment qu’une telle coque, lorsqu’elle est soumise à la combinaison d’une courbure initiale et d’une précontrainte suffisamment élevées, possède jusqu’à trois configurations stables vis-à-vis des propriétés matériaux. Dans un premier temps, nous proposons des critères de conception et fabrication de coques multistables allant jusqu’à la tristabilité, validés numériquement et expérimentalement. Ensuite, nous appliquons ces critères à la conception et à la fabrication de coques multistables cylindriques dont la différence de niveau énergétique entre les deux états stables est minime. Sur ce support, la deuxième partie est consacrée à des applications exploitant la bistabilité des coques cylindriques minces à faible différence énergétique. Nous effectuons tout d’abord une application au contrôle de forme via l’utilisation de matériaux actifs que l’on attache à la structure. Cela comprend une première phase théorique de conception de la structure et de la loi d’actionnement, et une seconde phase de mise en œuvre expérimentale. Ensuite, nous étudions théoriquement et expérimentalement les propriétés de dynamique non-linéaire de ce type de coques dans le but de mettre en évidence les modes d’oscillations intrinsèques à une source d’excitation externe. Enfin, nous proposons une application à la récupération d’énergie vibratoire non-linéaire de coques multistables cylindriques métalliques par voie piézoélectrique.