We produce a coordinate free presentation of some concepts usually involved in incremental mechan... more We produce a coordinate free presentation of some concepts usually involved in incremental mechanics (tangent linear stiffness matrix, stability, loading paths for example) but not always well founded. Thanks to the geometric language of vector bundles, a well defined geometrical object may be associated to each of these tools that allows us to understand some latent difficulties linked with these tools due to the absence of a natural connection and also to extend some of our recent results of linear stability to a non linear framework.
The stability of a granular column composed of a finite number of grains is investigated through ... more The stability of a granular column composed of a finite number of grains is investigated through an exact and some approximated continuum models. Shear and rotational interactions are taken into account at the rigid grain interfaces. This system can be considered as a discrete Cosserat chain with two independent degrees-of-freedom, namely the deflection and the rotation of each grain. The buckling of this discrete granular system on elastic foundation with translational and rotational stiffness (to account for some possible transversal grain interactions) is calculated whatever the number of grains. The formulation of the discrete boundary value problem is based on the exact resolution of a fourth-order linear difference equation. This solution is compared to the one of a continuous Cosserat chain asymptotically obtained for an infinite number of grains. In this last case, the asymptotic solution converges towards the one of a Bresse-Timoshenko beam under Winkler-Pasternak foundation. A more refined Cosserat continuum is built by continualization of the difference equations valid for the discrete Cosserat medium. It is shown that this more refine continuous model can be classified as a gradient elasticity Cosserat continuum, which is able to reproduce the scale effects observed for the buckling of the discrete granular system. These scale effects are related to the grain size, as compared to the structural length of the granular system. The key role played by the shear interaction in the instabilities of granular structural system is revealed, especially when the bending interaction can be neglected.
Les systèmes non conservatifs conduisantà des matrices de rigidité non symétriques sont connus po... more Les systèmes non conservatifs conduisantà des matrices de rigidité non symétriques sont connus pour présenter des comportements paradoxaux relativementà la viscosité. L'étude de la stabilité par divergence sous contraintes cinématiques a récemment débuté pour le cas d' une contrainte et nousétudions ici le cas de deux contraintes pour relier la stabilité par divergence au critère du travail du second ordre. L'exemple d'un système de Zieglerà trois degrés de liberté soumisà une force suiveuse illustre le résultat gńéral.
This paper is concerned with the dynamic stability of a microstructured elastic column loaded by ... more This paper is concerned with the dynamic stability of a microstructured elastic column loaded by circulatory forces. This nonconservative lattice (or discrete) problem is shown to be equivalent to the finite difference formulation of Beck's problem (cantilever column loaded by follower axial force). The lattice problem can be exactly solved from the resolution of a linear difference eigenvalue problem. The first part of the paper deals with the theoretical and numerical analyses of this discrete Beck's problem, with a particular emphasis on the flutter load sensitivity with respect to the discretization parameters, such as the number of links of the lattice. The second part of the paper is devoted to the elaboration of a nonlocal equivalent continuum that possesses similar mathematical or physical properties as compared to the original lattice model. A continualized nonlocal model is introduced first by expanding the difference operators present in the lattice equations in terms of differential operators. The length scale of the continualized nonlocal model is size independent. Next, Eringen's nonlocal phenomenological stress gradient is considered and applied at the beam scale in allowance for scale effects of the microstructured Beck column. The nonlocal Euler-Bernoulli beam model is able to capture the softening scale effect of the lattice model, even if the length scale of Eringen's model appears to be size dependent in this case. The continualized nonlocal continuum slightly differs from the Eringen's one, in the sense that the length scale affecting the static and the inertia terms differs in the deflection equation. A general parametric study illustrates the capability of each nonlocal model, the phenomenological and the continualized one, with respect to the reference lattice model. Nonlocal Beck's column is shown to be a transient medium from Ziegler's column (two-degree-of-freedom system) to the local continuous Beck's column (with an infinite degree of freedom).
This paper deals with both divergence and second order work criteria and the kinematical structur... more This paper deals with both divergence and second order work criteria and the kinematical structural stability called ki.s.s. In this context, kinematical structural stability means that the criterion remains valid even if the system is subjected to additional kinematic constraints. First some developments about the effect of additional kinematics constraints are presented on divergence instability. Secondly, divergence and second order work criteria are addressed. Using a variational formulation, previous results from a usual algebraic formulation using Schur's complement formula are highlighted and finally translated through the ki.s.s. concept: unconditional ki.s.s. for the second order work and the divergence of conservative systems and conditional ki.s.s. for the divergence of nonconservative elastic systems.
This paper is focused on the buckling and the vibration analyses of microstructured structural el... more This paper is focused on the buckling and the vibration analyses of microstructured structural elements, i.e., elements composed of repetitive structural cells. The relationship between the discrete and the equivalent nonlocal continuum is specifically addressed from a numerical and a theoretical point of view. The microstructured beam considered herein is modeled by some repetitive cells composed of finite rigid segments and elastic rotational springs. The microstructure may come from the discreteness of the matter for small-scale structures (such as for nanotechnology applications), but it can also be related to some larger scales as for civil engineering applications. The buckling and vibration results of the discrete system are numerically obtained from a discrete-element code, whereas the nonlocal-based results for the equivalent continuum can be analytically performed. It is shown that Eringen's nonlocal elasticity coupled to the Euler-Bernoulli beam theory is relevant to capture the main-scale phenomena of such a microstructured continuum. The small-scale coefficient of the equivalent nonlocal continuum is identified from the specific microstructure features, namely, the length of each cell. However, the length scale calibration depends on the type of analysis, namely, static versus dynamic analysis. A perfect agreement is found for the microstructured beam with simply supported boundary conditions. The specific identification of the equivalent stiffness for modeling the equivalent clamped continuum is also discussed. The equivalent stiffness of the discrete system appears to be dependent on the static-dynamic analyses, but also on the boundary conditions applied to the overall system. Satisfactory results are also obtained for the comparison between the discrete and the equivalent continuum for other type of boundary conditions.
International Journal of Non-Linear Mechanics, 2012
This paper is focused on the geometrically exact elastic stability analysis of two interacting ki... more This paper is focused on the geometrically exact elastic stability analysis of two interacting kinematically constrained, flexible columns. Possible applications are to partially composite or sandwich columns. A partially composite column composed of two inextensible elastically connected sub-columns is considered. Each sub-column is modeled by the Euler-Bernoulli beam theory and connected to each other via a linear constitutive law for the interlayer slip. The paper discusses the validity of parallel and translational kinematics beam assumptions with respect to the interlayer constraint. Buckling and postbuckling behavior of this structural system are studied for cantilever columns (clamped-free boundary conditions). A variational formulation is presented in order to derive relevant boundary conditions in a geometrically exact framework. The exact post-buckling behavior of this partially composite beam-column is investigated analytically and numerically. The Euler elastica problem is obtained in the case of noncomposite action. The ''partially composite elastica'' is then treated analytically and numerically, for various values of the interaction connection parameter. An asymptotic expansion is performed to evaluate the symmetrical pitchfork bifurcation, and comparisons are made with some exact numerical results based on the numerical treatment of the non-linear boundary value problem. A boundary layer phenomenon, similar to that also observed for the linearized bending analysis of partially composite beams, is observed for large values of the connection parameter. This boundary layer phenomenon is investigated with a straightforward asymptotic expansion, that also is valid for large rotations. Finally, the paper analyses the effect of some additional imperfection eccentricities in the loading mode, that lead to some pre-bending phenomena.
In this paper, we revisit the capability of numerical approaches such as finite difference method... more In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pad e approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.
In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on sim... more In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen's model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen's nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.
The effect of additional kinematic constraints on eigenfrequencies of non conservative systems pr... more The effect of additional kinematic constraints on eigenfrequencies of non conservative systems presenting a non symmetric stiffness matrix is investigated with the use of the second order work criterion. It is shown that there are always additional constraints that may soften structural systems, from both buckling and vibration points of view. The steps for building such constraints are given, consequences on stability are discussed and several illustrating examples are presented.
This paper investigates the macroscopic behaviour of an axially loaded discrete granular system f... more This paper investigates the macroscopic behaviour of an axially loaded discrete granular system from a stability perspective. The granular system comprises uniform grains that are elastically connected with some bending and shear interactions and confined by some elastic supports. This structural system can then be classified as a discrete repetitive system, a lattice elastic model or a Cosserat chain model. It is shown that this Cosserat chain model is exactly tantamount to the finite difference formulation of a shear-deformable Timoshenko column in interaction with a Winkler foundation. The buckling of the discrete column with pinned ends is first analytically investigated through the resolution of a finite difference equation. The solution is compared to a nonlocal approach derived by continualizing the discrete problem. The approximated Timoshenko nonlocal approach appears to be efficient with respect to the reference lattice problem and highlights some specific scale effects. T...
International Journal of Solids and Structures, 2012
This paper is an attempt to extend the approach of the second-order work criterion to the analysi... more This paper is an attempt to extend the approach of the second-order work criterion to the analysis of structural system instability. Elastic structural systems with a finite number of freedoms and subjected to a given loading are considered. It is shown that a general equation, relating the second-order time derivative of the kinetic energy to the second-order work, can be derived for kinetic perturbations. The case of constant, nonconservative loadings are then investigated, putting forward the role of the spectral properties of the symmetric part of the tangent stiffness matrix in the occurrence of instability. As an illustration, the case of the generalized Ziegler column is considered and the case of aircraft wings subjected to aeroelastic effects is investigated. In the both cases, the consequences of additional kinematic constraints are discussed.
Cet article étudie l'influence du mode de chargement sur la stabilité de systèmes élastiques disc... more Cet article étudie l'influence du mode de chargement sur la stabilité de systèmes élastiques discrets non conservatifs. La stabilité du système contraint est comparée à celle du système libre, par l'introduction de multiplicateurs de Lagrange. L'approche est illustrée avec le pendule généralisé de Ziegler. Elle est ensuite appliquée à un modèle à deux degrés de liberté représentant un sol contraint par un chargement isochore. On montre que le chargement isochore affecte sensiblement la frontière de stabilité pour le problème conservatif et pour le problème non conservatif. En dehors des instabilités par flottement, le critère de travail du second-ordre constitue une borne inférieure de la frontière de stabilité du système libre ainsi que la frontière du domaine de stabilité du système sous chargements mixtes proportionnels en déplacement. ABSTRACT. This paper shows that the loading mode strongly influences the stability of discrete nonconservative elastic systems. The stability of the constrained system is compared to the stability of the unconstrained system, through the incorporation of Lagrange multipliers. Initially, the approach is illustrated for a two-degrees-of-freedom generalized Ziegler's column. Then, it is applied to a two-degrees-of-freedom model representing a soil constrained with isochoric loading. The isochoric instability load is not necessarily greater than the instability load of the free problem. Excluding flutter instabilities, it is shown that the second-order work criterion is not only a lower bound of the stability boundary of the free system, but also the boundary of the stability domain, in presence of mixed perturbations based on proportional kinematic conditions. MOTS-CLÉS : stabilité des structures élastiques, problèmes non conservatifs, perturbations mixtes, systèmes contraints, stabilité matérielle, chargement isochore.
Static instability or divergence threshold of both potential and circulatory systems with kinemat... more Static instability or divergence threshold of both potential and circulatory systems with kinematic constraints depends singularly on the constraints' coefficients. Particularly, the critical buckling load of the kinematically constrained Ziegler's pendulum as a function of two coefficients of the constraint is given by the Plücker conoid of degree n = 2. This simple mechanical model exhibits a structural instability similar to that responsible for the Velikhov-Chandrasekhar paradox in the theory of magnetorotational instability.
The lateral-torsional stability of circular arches subjected to radial and follower distributed l... more The lateral-torsional stability of circular arches subjected to radial and follower distributed loading is treated herein. Three loading cases are studied, including the radial load with constant direction, the radial load directed towards the arch centre, and the follower radial load (hydrostatic load), as treated by Nikolai in 1918. For the three cases, the buckling loads are first obtained from a static analysis. As the case of the follower radial load (hydrostatic load) is a non-conservative problem, the dynamic approach is also used to calculate the instability load. The governing equations for out-of-plane vibrations of circular arches under radial loading are then derived, both with and without Wagner's effect. Flutter instabilities may appear for sufficiently large values of opening angle, but flutter cannot occur before divergence for the parameters of interest (civil engineering applications). Therefore, it is concluded that the static approach necessarily leads to the same result as the dynamic approach, even in the non-conservative case.
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Papers by Noël Challamel