On the stability of nonconservative continuous systems under kinematic constraints

Lerbet, Jean; Challamel, Noël; Nicot, François; Darve, Félix

doi:10.1002/zamm.201600203

Cited by 5 publications

(11 citation statements)

References 42 publications

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“…For instance, Ingerle [1969] found a dimensionless divergence buckling load p = 20.19 in the presence of a specific constraint applied to the end of the column (the application point of the follower load), whereas the free Beck column admits a flutter instability value of 20.05, as calculated by Beck [1952] (see also [El Naschie 1976;1977] for this result). However, consider that any kinematic constraint reduces this value to π 2 as mentioned above as was observed by Ingerle [2013] and as has been definitively proved by Lerbet et al [2017].…”

Section: The Kissmentioning

confidence: 63%

“…Thompson [1982] noted the paradoxical possibility of destabilizing a nonconservative column with an additional constraint whereas Ingerle [2018] computed approximate loadings that may destabilize nonconservative columns by investigating some special constrained systems. More accurately, for the continuous Beck column, the dimensionless divergence stability value for any kinematic constraint has been calculated in [Lerbet et al 2017] and is equal to π 2 . This value had been empirically obtained by Ingerle [2013] from a discrete approach using numerical arguments.…”

Section: The Kissmentioning

confidence: 99%

“…During our investigations, it appeared that a better way to systematically tackle constrained systems consists of using so-called compressions of operators. It provides objects not only appropriate to physical systems (for example a matrix of size r for an r degree of freedom mechanical system) but also applicable to various other situations such as constrained continuous media involving infinitedimensional spaces (see [Lerbet et al 2017] for its use leading to the complete solution of the constrained Beck column). The formal definition of the compression of a linear map reads:…”

Section: The Kinematic Structural Stability (Kiss)mentioning

confidence: 99%

“…This result extends to Hilbert spaces with compressions to closed spaces. It has been used in [Lerbet et al 2017].…”

Section: The Solution Of These Issues Is Contained Inmentioning

confidence: 99%

See 3 more Smart Citations

Second-order work criterion and divergence criterion: a full equivalence for kinematically constrained systems

Lerbet

Challamel

Nicot

et al. 2020

Math. Mech. Compl. Sys.

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This paper presents stability results for rate-independent mechanical systems, associated with general tangent stiffness matrices including symmetric and nonsymmetric ones. Conservative and nonconservative as well as associate and nonassociate elastoplastic systems are concerned by such a theoretical study. Hill's stability criterion, also called the second-order work criterion, is here revisited in terms of kinematically constrained systems. For piecewise rate-independent mechanical systems (which may cover inelastic and elastic evolution processes), such a criterion is also a divergence Lyapunov stability criterion for any kinematic autonomous constraints. This result is here extended for systems with nonsymmetric tangent matrices. By virtue of a new type of variational formulation on the possible kinematic constraints, and thanks to the concept of kinematical structural stability (KISS), both criteria, Hill's stability criterion and the divergence stability criterion under kinematic constraints, are shown to be equivalent.

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Section: The Kissmentioning

confidence: 63%

Section: The Kissmentioning

confidence: 99%

Section: The Kinematic Structural Stability (Kiss)mentioning

confidence: 99%

“…This result extends to Hilbert spaces with compressions to closed spaces. It has been used in [Lerbet et al 2017].…”

Section: The Solution Of These Issues Is Contained Inmentioning

confidence: 99%

See 2 more Smart Citations

Second-order work criterion and divergence criterion: a full equivalence for kinematically constrained systems

Lerbet

Challamel

Nicot

et al. 2020

Math. Mech. Compl. Sys.

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show abstract

“…All these works have been carried out within the framework of nonconservative linear discrete mechanics and dealt with the finite dimensional “punctual” issue which involved only tools of usual linear algebra. More recently performed the extension to continuous systems and involved then infinite dimension Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

Coordinate free nonlinear incremental discrete mechanics

et al. 2018

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In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual linear case to a full non linear framework. Kinematic Structural Stability (KISS) refers to the stability of conservative or non-conservative elastic systems under kinematic constraints. We revisite and sometimes introduce some new mechanical and geometrically-related concepts, using the systematic language of vector bundles: transversality-stability, tangent stiffness tensor, loading paths and KISS. We bring an intrinsic geometrical meaning to these concepts and we extend the main results to a general nonlinear framework. The lack of connection on the configuration manifold in order to make derivatives of sections is bypassed thanks to the key rule of the nil section of the cotangent bundle. Two mechanical examples illustrate the interest of the new concepts, namely a discrete nonconservative elastic system under nonconservative loading (Ziegler's column) and, in more details, a four-sphere granular system with nonconservative interactions called diamond pattern. K E Y W O R D Scoordinate free approach, fiber bundle, kinematic structural stability, transversality

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Divergence instability of kinematically constrained Hencky chains: Analytic results and asymptotic behavior

et al. 2021

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This paper investigates the stability of any 𝑛 degree-of-freedom Hencky chain subjected to a full follower force at its tip and under kinematic constraints. We specifically solve this discrete problem and focus on its asymptotic behavior when 𝑛 tends towards infinity and compare it with its analogous continuous formulation, also referred to as the kinematically constrained Beck's column. The divergence instability load of this non-conservative discrete system under general kinematic constraints is ruled by the so-called second-order work criterion, which involves the symmetric part of the stiffness matrix. The application of the second-order work criterion to the discrete repetitive system allows to find the exact divergence load 𝑝 𝑛 under kinematic constraints, whatever the size of the discrete system. The critical value 𝑝 𝑛 is obtained as the smallest positive root of explicit transcendent equations for both the generic (𝑎 𝑛 ≠ 0) and the singular (𝑎 𝑛 = 0) cases, these two cases being illustrated by different values (respectively 2𝐶 and 𝐶) of the stiffness of the elastic torsion spring at the basis of the discrete system. The corresponding destabilizing kinematic constraint is also calculated. Asymptotic derivations show the convergence of the dimensionless buckling load towards the continuous one, equal to 𝜋 2 , and give the corresponding destabilizing kinematic constraint of the Beck's column.

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On the stability of nonconservative continuous systems under kinematic constraints

Cited by 5 publications

References 42 publications

Second-order work criterion and divergence criterion: a full equivalence for kinematically constrained systems

Second-order work criterion and divergence criterion: a full equivalence for kinematically constrained systems

Coordinate free nonlinear incremental discrete mechanics

Divergence instability of kinematically constrained Hencky chains: Analytic results and asymptotic behavior

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