Stability of Discrete Non-conservative Systems 1785482866, 9781785482861

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Stability of Discrete Non-conservative Systems

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Discrete Granular Mechanics Set coordinated by Félix Darve

Stability of Discrete Non-conservative Systems

Jean Lerbet Noël Challamel François Nicot Félix Darve

First published 2020 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd 27-37 St George’s Road London SW19 4EU UK

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www.iste.co.uk

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Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. For information on all our publications visit our website at http://store.elsevier.com/ © ISTE Press Ltd 2020 The rights of Jean Lerbet, Noël Challamel, François Nicot and Félix Darve to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-286-1 Printed and bound in the UK and US

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. On Stability of Discrete and Asymptotically Continuous Systems . . . . . . . . . . . . . . . . . . . . .

1

1.1. Stability – general concepts . . . . . . . . . . . . . 1.2. Buckling of discrete conservative systems – Hencky’s column under conservative axial load . . . . 1.3. Buckling of discrete non-conservative systems – Hencky’s column under follower axial load . . . . . . 1.4. Stability under kinematic constraints . . . . . . . . 1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 1.6. References . . . . . . . . . . . . . . . . . . . . . . .

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28 37 49 50

Chapter 2. Second-order Work Criterion and Stability in the Small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.1. Some generalities in continuum mechanics for material systems . . . . . . . . . . . . . . . . . . . . . 2.2. The link between the kinetic energy and the second-order work . . . . . . . . . . . . . . . . . . . . 2.3. The second-order work in Eulerian formalism . . . 2.4. The second-order work on the material point scale 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

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60 65 67 68

Chapter 3. Mixed Perturbations and Second-order Work Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. From a quasi-static to a dynamical regime . . . . . . . . . . . . . . . . .

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3.2.1. Existence of multiple equilibrium configurations . . 3.2.2. Stability of equilibrium configurations . . . . . . . . 3.2.3. Spectral analysis of tensors K and Ks . . . . . . . . . 3.3. The case of discrete systems . . . . . . . . . . . . . . . . . 3.3.1. General framework . . . . . . . . . . . . . . . . . . . . 3.3.2. The constrained system . . . . . . . . . . . . . . . . . 3.4. Application to the generalized Ziegler column problem 3.4.1. The generalized Ziegler column problem . . . . . . . 3.4.2. Instability problem with constraints . . . . . . . . . . 3.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . 3.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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70 74 79 80 80 82 88 88 94 99 100

Chapter 4. Divergence Kinematic Structural Stability . . . . . . . . . .

103

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2. KISS issue . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The paradigmatic case of the 2 dof Ziegler system 4.4. Algebraic approach . . . . . . . . . . . . . . . . . . . 4.4.1. Schur’s complement formula . . . . . . . . . . . 4.4.2. Case of one constraint: r = 1 . . . . . . . . . . . 4.4.3. Case of any set of constraints: 1≤ r ≤ n –1 . . . 4.5. Variational approach . . . . . . . . . . . . . . . . . . 4.5.1. Variational and minimizing formulations . . . . 4.5.2. Constraints and quadratic forms a, b, q: elimination of Lagrange multipliers . . . . . . . . . . . 4.5.3. Divergence KISS issue. . . . . . . . . . . . . . . 4.6. Geometric approach . . . . . . . . . . . . . . . . . . . 4.7. Coming back to Lyapunov’s and Hill’s stabilities . 4.8. References . . . . . . . . . . . . . . . . . . . . . . . .

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103 105 109 112 112 113 113 114 114

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116 118 119 121 122

Chapter 5. Flutter Kinematic Structural Stability . . . . . . . . . . . . . .

125

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Flutter stability and flutter KISS formulations . . . . . . 5.2. Grassmannian and Stiefel manifolds . . . . . . . . . . . . . . 5.3. Case n = 3 and m = 2 . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Geometric considerations and preliminary calculations 5.3.2. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . 5.3.3. Necessary and sufficient conditions: calculations in 2,3 (ℝ) . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Necessary and sufficient conditions: calculations in 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5. Summary of the results . . . . . . . . . . . . . . . . . . .

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125 126 128 131 132 137

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143 148

Contents

5.4. Partial flutter KISS: examples . 5.4.1. Mechanical consequences . 5.4.2. Examples . . . . . . . . . . . 5.4.3. M = I3 . . . . . . . . . . . . . 5.4.4. Uniform mass distribution. 5.5. References . . . . . . . . . . . .

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148 148 149 152 154 155

Chapter 6. Geometric Degree of Non-conservativity . . . . . . . . . . .

157

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Modeling and calculation of the GDNC: examples 6.2.1. Calculation of the GDNC . . . . . . . . . . . . . 6.2.2. Examples . . . . . . . . . . . . . . . . . . . . . . . 6.3. Calculation of . . . . . . . . . . . . . . . . . . . . 6.3.1. The GDNC and the exterior calculus . . . . . . 6.3.2. Set of solutions . . . . . . . . . . . . . . . . . . . 6.3.3. An example . . . . . . . . . . . . . . . . . . . . . 6.4. Extension to the nonlinear framework . . . . . . . . 6.4.1. Nonlinear issue and notations . . . . . . . . . . . 6.4.2. Link with the linear framework. . . . . . . . . . 6.5. Solution of the nonlinear problem . . . . . . . . . . 6.5.1. The solution . . . . . . . . . . . . . . . . . . . . . 6.5.2. Effectiveness of the solution . . . . . . . . . . . 6.5.3. Set of solutions . . . . . . . . . . . . . . . . . . . 6.5.4. The example . . . . . . . . . . . . . . . . . . . . . 6.6. Duality KISS-GDNC . . . . . . . . . . . . . . . . . . 6.7. References . . . . . . . . . . . . . . . . . . . . . . . .

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199

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Chapter 7. Buckling of Granular Systems with Shear Interactions: Discrete versus Continuum Approaches . . . . . . . . . . . . . . .

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157 158 158 161 173 174 176 177 181 182 184 185 185 187 188 188 194 195

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7.1. Introduction – instabilities of granular systems . 7.2. Shear granular system – a discrete approach . . 7.3. Buckling of granular system – exact solution . . 7.4. Shear granular system – a continuous approach 7.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . 7.6. References . . . . . . . . . . . . . . . . . . . . . .

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199 201 206 211 218 218

Chapter 8. Continuous Divergence KISS . . . . . . . . . . . . . . . . . . .

223

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Strong formulation of divergence stability of the Beck column and its Ziegler system counterpart . . . . . . . . . . .

223 224 224

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8.2.2. Usual weak formulation of divergence stability of the Beck column and its Ziegler system counterpart . . . . . 8.2.3. Results in a finite dimension space and position of the problem . . . . . . . . . . . . . . . . . . . . . . 8.3. Fundamental Spaces – topological aspects. Variational formulation of the problem . . . . . . . . . . . . . . 8.3.1. Vector spaces and operator A(P): usual aspects . . . . 8.3.2. Kinematic constraints . . . . . . . . . . . . . . . . . . . 8.3.3. An alternative formulation involving only the Hilbert space equipped with : solution of issue 1 . . . . . . 8.3.4. Variational formulation for the KISS problem . . . . . 8.3.5. Geometric solution of the KISS issue: compression of an operator . . . . . . . . . . . . . . . . . . . . 8.4. Solution for the kernel of the operators As(P) and Ãs(P) 8.4.1. Calculation of As(P) . . . . . . . . . . . . . . . . . . . 8.4.2. Calculation of Ãs(P) . . . . . . . . . . . . . . . . . . . 8.4.3. Calculation of the critical load P2* . . . . . . . . . . . . 8.4.4. Calculation of ker As(P2*) = ker Ãs(P2*) . . . . . . . . . 8.4.5. Calculation of the optimal destabilizing constraint . . 8.5. P1* = P2* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Stability issues . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1. Equivalence between different forms of kinematic constraints . . . . . . . . . . . . . . . . . . . . . . 8.7.2. Eigenvalues problem for Ãs(P) . . . . . . . . . . . . . 8.7.3. Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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252 253 257 258

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

Introduction

I.1. Classes of instabilities Many natural phenomena and human domains of activity are concerned with bifurcations. Roughly speaking, bifurcations can be defined by the fact that an arbitrarily small variation of any parameter, influencing the generic state of a given system, is suddenly and drastically changing this state (Thompson and Bishop 1994). Instability is a specific type of bifurcation, often related to mechanical systems. Many definitions – specific or more general – have been proposed for instability. However, a general consensus might agree on the fact that an equilibrium state of a mechanical system is called unstable if there exists at least one mechanical perturbation, arbitrarily small, leading to a large variation of at least one variable characterizing this equilibrium state (Thompson and Hunt 1973). For rate-independent mechanical systems as concerned with this book, this change in the state of the system is instantaneous. It can be convenient to call this class of instabilities “instability in the small” in the sense of the evolution of the system around its current state. For example, a sand sample will fail instantaneously at a stress peak if the experimentalist tries to pursue the stress loading. However, if this is the asymptotic mechanical behavior of the system which is drastically modified, “instabilities in the large” are concerned, in the sense of the evolution of the system far away from its current state. It is also necessary to distinguish between “static instabilities”, which are reached by static loading conditions, and “dynamic instabilities” which necessitate dynamic loading conditions (such as periodic forcing loading) to

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Stability of Discrete Non-conservative Systems

induce an unstable state. Therefore, some static loading paths, on a rateindependent metallic or geomaterial sample for example, can lead to instabilities at the stress peaks. Let us note that generally, even if the unstable state is met under static conditions, some bifurcated branches after the unstable point can develop in a dynamic regime. Therefore, even if a fully general stability analysis must consider both kinds of instabilities for static and dynamic perturbations for any material, rate-independent systems loaded in quasi-static conditions are subjected to static instabilities, which may include divergence or flutter-type instabilities. Another point is important to be mentioned. The instability can be induced by the constitutive behavior of some components of the mechanical system. If so, the instability is characterized as “material instability”. For example, softening mechanical properties lead to material instabilities. However, if some geometric characteristics of the system are the cause of the unstable behavior, these are “geometric instabilities”. An elastic elongated column will buckle by geometric instability, while its intrinsic material hyperelastic behavior is perfectly materially stable. Finally, the experiments also lead one to distinguish between “divergence instability” and “flutter instability”. A divergence instability is characterized by suddenly monotonously growing displacements of some elements of the system, while in the case of flutter instabilities, these displacements grow cyclically. The most known example of flutter instability is given by the wings of a plane, which can crash by growing beats. Both of these kinds of instabilities are considered in this book. I.2. Non-conservative plasticity

elasticity

and

non-associate

elasto-

This book will be concerned only with rate-independent materials and systems, which basically means non-viscous materials whose behavior does not depend on any physical time. Of course, it still depends on the chronology of events. Therefore, the loading rates do not have any influence on their mechanical behavior. Only chronologies can be defined in an objective intrinsic manner and determine the behavior of these materials. The two main classes of rate-independent behaviors are elasticity and elastoplasticity or elasto-damage. Elasticity ignores any internal dissipation and is characterized by a constitutive linear or nonlinear biunivocal function

Introduction

xi

linking stress and strain tensors. Therefore, by differentiation, there exists an incrementally linear constitutive relation, which means that the relation between stress rate and strain rate (equation I.1) is fully linear – even if the elastic constitutive tensor can depend or not on some state variables. dσij = Mijkl (hpq) dεkl,

[I.1]

where hpq are the tensorial state variables. However, plasticity and damage are characterized by rate-independent dissipative mechanisms. If the strains are partly reversible by elasticity and partly irreversible by plasticity, the behavior is said to be elasto-plastic. If so, the behavior is path-dependent, which means that the mechanical response depends on the whole loading history. Thus, the stress–strain relation can no more be expressed by a function but it is a functional involving the whole loading history. Moreover, this constitutive functional is no longer differentiable, since – roughly speaking – the incremental behavior is different in loading and unloading. Therefore, any stress–strain state is singular for this functional (Owen and Williams 1969). Thus, there is no longer any linear relation between stress and strain rates. The incremental constitutive relation is essentially nonlinear, which is called the “incremental nonlinearity”, an intrinsic property of any elasto-plastic relation (Darve 1990): dσij = Mijkl (dεmn, hpq) dεkl,

[I.2]

where hpq are the internal tensorial variables. Usually the instabilities of both these elastic and elasto-plastic media or systems are tackled by different theories and analyzed in different books. This is the first main purpose of this book: to propose a unified theory of stability for both of these constitutive behaviors. As it has just been recalled, in elasticity, there is a linear relation between stress and strain rates. Thus, an elastic tensor links both these tensors. This tensor is constant in linear elasticity, and it depends on some state variables (e.g. the stress state) in nonlinear elasticity. Moreover, if an elastic

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potential can be defined, the constitutive law is called “hyperelastic” and the elastic tensor is symmetric (the so-called “major symmetry”): Mijkl (hpq) = Mklij (hpq)

[I.3]

This can be referred to as Green elasticity (Green and Naghdi 1971; Rajagopal, 2011); otherwise, the elastic law is called Cauchy elasticity, which can be classified as “hypoelastic” and the symmetry is lost (Truesdell and Noll 1965). Because of this absence of any elastic potential, hypoelasticity can be considered as a “non-conservative” law: Mijkl (hpq) ≠ Mklij (hpq)

[I.4]

As mentioned by Green and Naghdi (1971), hypoelasticity can lead to some possible infinite dissipation processes for closed cycles of deformation. All these notions have been defined in a continuous mechanics framework. However, from a historical point of view, many discrete elastic systems have been considered in parallel with the development of continuous mechanics such as, for example, the well-known Ziegler column (Ziegler 1952). In these cases, a stiffness elastic tensor links forces, rotations and displacements, respectively, applied to the structure – as a loading – or measured on the structure – as a response. The stiffness tensor is generally non-constant and depends not only on some current forces, rotations and displacements but also on some geometrical parameters. This is the reason why this operator is called “stiffness tensor” rather than “constitutive tensor”, because it describes the behavior of a structure and not that of a material point. As well as in constitutive hypoelasticity, for specific loading conditions, the stiffness tensor of discrete structures can be non-symmetric. The most known example of such non-symmetries corresponds to the case of follower forces applied to discrete systems. This kind of elastic system is thus also called “non-conservative”. Other examples will be shown in this book in relation to discrete granular systems. To conclude, in hypoelasticity – from a constitutive point of view – like in discrete non-conservative elastic systems – from a structural point of view – the constitutive tensor and the stiffness tensor, respectively, may be non-symmetric. This book will show why this non-symmetry plays such an important role in elastic stability analyses. Let us note that non-symmetric operators have a lot more various bifurcations than symmetric operators according to linear algebra.

Introduction

xiii

With respect to elasto-plasticity, first of all, its main constitutive features are recalled now. Based on the rate-independent condition, the function relating the incremental stress to the incremental strain is homogeneous of degree one. Applying Euler’s identity for homogeneous functions, it is possible to demonstrate the existence of an elasto-plastic tensor, depending on the incremental loading direction and on some internal state variables and memory parameters (Darve 1990): dσij = Mijkl (umn, hpq) dεkl,

[I.5]

with umn = dεmn / || dε ||, while hpq are the internal tensorial variables. To establish the expression of this elasto-plastic tensor Mijkl, three surfaces have to be introduced in the six-dimensional stress space: 1) the failure surface, which can be defined more rigorously as an intrinsic plastic limit condition, which cannot be trespassed by any stress state applied to the material: F(σij) = 0; 2) the yield surface, which is the boundary of the elastic domain. This yield surface is also called the “loading surface” since the elasto-plastic theory shows that the loading–unloading criterion has to be defined with respect to this surface: f(σij, akl) = 0, where akl are the tensorial hardening variables; 3) the plastic potential which gives the direction of the plastic strain increment, when the plastic strains appear, i.e. when the stress state reaches the yield surface: g(σij, akl) = 0. When the plastic potential g( ) coincides with the yield surface f( ), the elasto-plastic law is called “associate”. If not, the plastic behavior is considered as “non-associate”.

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For many materials (as metals in a first approximation), the plastic behavior is invariant with respect to the mean pressure, which leads to failure criteria like von Mises or Tresca, which are invariant with respect to any translation parallel to the hydrostatic axis. On the same line, the plastic volume variations can be considered as negligible according to the experiments. Therefore, the volumic plastic incremental strain vectors are perpendicular to the hydrostatic axis and are thus perpendicular to a yield surface like von Mises or Tresca. Eventually, for these materials, yield surface f( ) and plastic potential g( ) can be assumed to be identical. Therefore, these materials are associate. A basic consequence is that, according to elasto-plasticity theory, the related elasto-plastic tensor is symmetric: Mijkl (umn, hpq) = Mkjij(umn, hpq)

[I.6]

Now let us consider materials whose plastic dissipation is governed by a coulombian friction like geomaterials (soils, rocks, concretes). For these materials, the plastic limit surface has a conical structure, originating from the stress space origin if they are non-cohesive or with an apex on the hydrostatic axis if they are cohesive-frictional. The yield surfaces have the same conical structure. Therefore, if their plastic potential coincides with their yield surface, the plastic volumic strains would unreasonably be large according to the experiments. The most known example of coulombian materials is certainly the granular non-cohesive media like sands. While their friction angle φ is usually between 30° and 38°, their dilatancy angle ψ is between 0° and 18°. According to the definitions of both these angles, an associate behavior would have implied: φ = ψ, which is not at all verified by the experimental results. Therefore, these materials are essentially non-associate and their related elasto-plastic tensor is fully non-symmetric: Mijkl (umn, hpq) ≠ Mkjij(umn, hpq)

[I.7]

Since the theory of instability, which is proposed in this book, is essentially linked to the non-symmetry of the constitutive or stiffness operator, the idea emerges that this theory can unify the instabilities found for hypoelastic materials, non-conservative discrete elastic systems and non-associate elasto-plastic behaviors and structures.

Introduction

xv

I.3. Second-order work criterion Indeed, for all these media and mechanical systems characterized by a non-symmetric operator, some “paradoxical” instabilities and bifurcations have been observed. These instabilities have been called “paradoxical” essentially because they appear before reaching the classically established limit states and limit surfaces. Indeed, for symmetric matrices, all the mechanical bifurcations correspond to the singularities of M and thus to the vanishing of its determinant: det (M) = 0.

[I.8]

Therefore, if we consider this classical theory, all the material instabilities will appear on the limit states given by det (M) = 0. Consequently, the instabilities observed for strictly positive values of the determinant of non-symmetric tensor M are called “paradoxical”. However, there exists an instability criterion, proposed by Rodney Hill in the 1950s (Hill 1958, 1959), which is independent of both the symmetric or non-symmetric features of M (Mijkl can be equal or not to Mklij) and even to the incrementally linear or nonlinear character of M (Mijkl can depend or not to the loading direction): this is the so-called “second-order work criterion”. A material equilibrium state is reputed as stable in the sense of Hill, if for any dσ and dε, linked by the material constitutive relation, their scalar product is strictly positive: dσij dεji > 0

[I.9]

On the contrary, if there exists at least one loading direction where this scalar product is vanishing or taking negative values, this is a sufficient condition for this direction to be reputed unstable.

xvi

Stability of Discrete Non-conservative Systems

For rate-independent materials or systems, we have shown in section 1.2 a tensor linking dσ and dε always exists: dσ = M dε or dε = M–1 dσ

[I.10]

Therefore, it becomes: d2W = dσ: dε = dεt: M :dε = dσt : M–1 : dσ

[I.11]

According to linear algebra, the quadratic form of a skew-symmetric matrix is identically equal to 0. Therefore, if we decompose M into its symmetric Ms and skew-symmetric Ma parts: M = M s + Ma

[I.12]

the following is obtained: d2W = dσ : dε = dεt: Ms :dε = dσt : (M–1)s : dσ

[I.13]

Knowing that a necessary condition for a quadratic form to take negative values is that the determinant of its matrix is negative, we obtain a first necessary condition for instability: det Ms ≤ 0

[I.14]

Therefore, in hyperelasticity, in conservative elastic systems and in associate elasto-plasticity, there is no difference between the classical bifurcations or the plastic limit condition and the second-order work criterion, since M is symmetric. All the material bifurcations take place on the plastic limit surface. However, in hypoelasticity, for non-conservative elastic systems and in non-associate elasto-plasticity, there exists a bifurcation domain in the stress space between a lower boundary surface given by: det Ms = 0 and its upper limit surface given by: det M = 0 (recalling that det Ms is vanishing before det M).

Introduction

xvii

Eventually, let us note for consistency that det (M–1)s is vanishing like det Ms since: det Ms = det (M–1)s / (det M–1)2. = (det M)2 det (M–1)s

[I.15]

Of course, when M is symmetric, the above relation gives: det M–1 = 1/det M

[I.16]

Therefore, a first consequence is the existence of a whole domain of bifurcations, instabilities and losses of uniqueness in non-conservative elasticity and non-associate elasto-plasticity (Darve et al. 2004, Wan et al. 2017). Now, considering a stress–strain state inside this bifurcation domain, a question has to be asked: in which loading directions from this stress–strain state can the instability manifest and develop itself? According to linear algebra, each symmetric matrix, whose determinant takes negative values, has a so-called “isotropic cone” gathering the directions where its related quadratic form is negative. Therefore, a second consequence is that, inside the bifurcation domain, there exists some “instability cones” gathering the incremental loadings whose directions can lead to material instabilities (Darve and Laouafa 2000). Finally, a third condition for effective instabilities is that a proper loading or control parameter has to be considered. This last condition corresponds to the notion of “kinematic constraints” in non-conservative elasticity (Challamel et al. 2010, Lerbet et al. 2017) and of “mixed loadings” in non-associate elasto-plasticity (Darve and Roguiez 1998), both notions which will be shown in this book as basically equivalent. This will lead to a new unified view of these paradoxical instabilities observed both for nonconservative elastic systems and for non-associate elasto-plastic materials. Let us note here that a mixed loading means that the loading path is defined partly by strains and partly by stresses or even by linear combinations of strains (the so-called “generalized strains”) and linear combinations of stresses (“generalized stresses”). In all events, the loading path is

xviii

Stability of Discrete Non-conservative Systems

thermodynamically admissible if the contracted tensorial product of the generalized strains by the generalized stresses is equal to the energy: σij εji. Therefore, the theory is perfectly consistent with these new variables, as the second-order work given by the scalar product of the generalized stresses by the generalized strains remains equal to: dσij dεji. Besides, the notion of a limit state also has to be defined in the generalized strains – generalized stresses space as the limit point (if it exists) on the loading path, which cannot be trespassed by any additional value of the control parameter describing the motion of the loading point along the loading path. This limit state may be linked to a maximum or a minimum value of one generalized strain or one generalized stress, or may be reached asymptotically (Darve et al. 2004; Wan et al. 2017). A generalized failure state corresponds to such a generalized limit state, and the failure will be called “effective” if the bifurcation takes places effectively via a sudden transition from a static regime to a dynamic one through a burst of kinetic energy. I.4. Recent new developments in solid mechanics instability At this step of this Introduction, three questions (at least) remain open: – Is there any link between the second-order work criterion and the bursts of kinetic energy which are usually observed in practice when material failures occur? – What kind of material instabilities (i.e. instabilities by divergence or by flutter) can be described by the second-order work criterion? – Is there any link between this criterion and Lyapunov stability definition (Lyapunov 1892)? As regards the first question, it will be shown in Chapter 2 that indeed, when the second-order work is negative, if the loading is pursued in the same direction with a proper loading parameter, the second derivative of kinetic energy is strictly positive leading to a sudden burst of kinetic energy. For that, a relation relating this second derivative of kinetic energy to the difference between a so-called “external second-order work” and the material “internal” second-order work will be established (Nicot

Introduction

xix

et al. 2011, Nicot et al. 2012). For quasi-static loading conditions corresponding to a fully equilibrated loading path, external and internal second-order works are equal and the quasi-staticity is maintained by a zero value of the kinetic energy of the system. However, if the internal second order is vanishing (or a fortiori taking negative values) due to the material or the system intrinsic behavior, and if the external second-order work is maintained strictly positive by the loading conditions, the second derivative of kinetic energy will suddenly become strictly positive, leading to these observed bursts of kinetic energy. The material body or the mechanical system is not able to absorb the energy coming from outside, and it fails by suddenly transforming the supplementary available energy to kinetic energy. The profound mechanical significance of the vanishing of the internal second-order work and its link to failure emerges here (Nicot et al. 2011, Nicot et al. 2012). By letting aside the geometric instabilities, another question emerges: what kind of instabilities is the second-order work criterion describing? This question will find its proper response in Chapters 3 and 4 with the notion of KInematic Structural Stability (KISS) (Lerbet et al. 2016a). Thanks to the use of variational methods, it will be shown that the second-order work criterion is the general criterion for all divergence instabilities, for rateindependent materials and mechanical systems. By considering the virtual power principle, the second-order work appears as an inner product defined on a properly built Riemannian manifold. To obtain the vanishing values of the second-order work, it can be viewed, in this framework, as a minimization problem under constraints corresponding to the so-called “kinematic constraints” (Lerbet et al. 2017). Thus, the field of application of the second-order work criterion will be clearly established as being equivalent to all the divergence instabilities of rate-independent systems, while the case of flutter instabilities has been considered in Lerbet et al. 2016b. Since the beginning of the previous century, Lyapunov’s theory of stability (Lyapunov 1892) probably represents the main basis of any stability theory. Indeed, Lyapunov has developed his theory having initially in mind to see if some initial infinitesimal perturbations could ultimately change the course of the planets around the Sun in a finite manner.

xx

Stability of Discrete Non-conservative Systems

For rate-independent mechanical systems, Lyapunov’s definition of stability means that any mechanical perturbation applied to a given equilibrium state of the system must lead to a “small” change of this state (i.e. of arbitrarily small amplitude). This is once more the application of variational methods that will lead to establishing an equivalence between Lyapunov’s definition of stability and the second-order work criterion for rate-independent materials and mechanical systems subjected to quasi-static loading conditions (Lerbet et al. 2016a, 2016b, 2017). Therefore, a complete consistent view of divergence instability for rate-independent systems is established in this book. I.5. Main outlines of this book The first originality of this book is the fact that this intricate question of instability is tackled by different tools, issued either from tensorial linear algebra or from variational methods, or even from geometric considerations. Therefore, the eight chapters are passing from one method to another, but all converge to grasp the capacities of the second-order work criterion to describe the instability of rate-independent materials and systems and particularly to clarify how kinematic constraints can destabilize such systems. More generally, let us note here the fundamental interest of the geometrization of constitutive theory as a way to obtain perfectly intrinsic descriptions of constitutive relations in an elegant manner without the mathematical burden of tensorial writing developed sometimes in arbitrary frames (Lerbet et al. 2018). The second originality – though it is no less important – is to emphasize a single unified framework for elastic non-conservative systems and for elasto-plastic non-associate ones, their common point being the nonsymmetry of their tangent stiffness operator. Indeed, the new methods, introduced in this book, to tackle this non-symmetry of operators is widely applicable in solid and fluid mechanics, in magnetohydrodynamics, etc. (see some examples in Kirillov 2013). The questions of bifurcation, stability and uniqueness linked to non-symmetric operators find here a new theoretical framework to be developed in.

Introduction

xxi

Chapter 1 presents a stability analysis of discrete columns (composed of a finite number n of elements) by the second-order work criterion expressed through discrete variables. The asymptotic continuous case, obtained when n tends towards infinite, is obtained by an elegant upscaling from discrete to continuous. A basic question is treated in Chapter 2: why is the second-order work criterion able to describe the unstable states by a material divergence instability? Indeed, it is shown that, when the “external second-order work” exceeds the value of the “internal second-order work”, a burst of kinetic energy emerges, expressing the instability of the behavior. This relation between second-order work and kinetic energy is applied in Chapter 3 to discrete systems with kinematic constraints. Therefore, it is shown that the instabilities in continuous non-associate elasto-plasticity and discrete nonconservative elasticity can be treated in the single framework of secondorder work criterion. Chapters 4 and 5 are devoted to the introduction and application of the KInematic Structural Stability (KISS) concept. Based on variational methods and a geometric interpretation, the full equivalence between Lyapunov divergence instability and the second-order work criterion is established in Chapter 4 with an application to discrete column systems. Therefore, the mathematical analysis of both these instability criteria shows for the first time their essential interlink. A more complex link between flutter instability and the second-order work criterion is established in Chapter 5 with an application to Ziegler’s system. A new concept is introduced in Chapter 6: the geometric degree of nonconservativity (GDNC), which can be viewed as dual to KISS. Indeed, while KISS only depends on the symmetric part of the tangent stiffness operator, the GDNC characterizes its skew-symmetric part by an intrinsic scalar, which expresses the “distance” of the system to elastic conservativity or elasto-plasticity associativity. Chapter 7 considers the buckling of a granular elastic column lying on an elastic support. The system of equations characterizing this column appears to be equivalent to the finite difference description of a shear-deformable Bresse–Timoshenko column on a Winkler foundation, thus bridging the gap between the grain scale to the continuous scale.

xxii

Stability of Discrete Non-conservative Systems

Chapter 8 generalizes the KISS concept to a continuous mechanics framework by taking, as an example, Beck’s column subjected to follower forces such that this is a non-conservative elastic structure. The difficulty to pass from a finite number of degrees of freedom for a discrete system to an infinite number for a continuous system is circumvented such that the destabilizing kinematic constraint is exhibited. I.6. References Challamel, N., Nicot, F., Lerbet, J. and Darve, F. (2010). Stability of nonconservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. Darve F. (1990). The expression of rheological laws in incremental form and the main classes of constitutive equations. In Geomaterials, Constitutive Equations and Modelling, Darve, F. (ed.). Elsevier, Essex, UK. Darve, F. and Laouafa, F. (2000). Instabilities in granular materials and application to landslides. Int. J. Mechanics of Cohesive-Frictional Materials, 5(8), 627–652. Darve, F. and Roguiez, X. (1998). Homogeneous bifurcation in soils. In Localisation and Bifurcation Theory for Soils and Rocks, Toshihisa, A., Oka, F., Yashima A. (eds), Balkema, Boca Raton, FL, USA. Darve, F., Servant, G., Laouafa, F., Khoa, H.D.V. (2004). Failure in geomaterials, continuous and discrete analyses. Comp. Meth. in Appl. Mech. and Eng., 193(27–29), 3057–3085. Green, A.E. and Naghdi, P.M. (1971). On thermodynamics, rate of work and energy. Arch. For Ration. Mech. Anal., 40, 37–49. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. of Mech. and Phys. of Solids, 6, 236–249. Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. J. of Mech. and Phys. of Solids, 7, 209–225. Kirillov, O.N. (2013). Nonconservative Stability Problems of Modern Physics. Walter de Gruyter , Berlin, Germany. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2016a). Kinematical structural stability. Discrete and Continuous Dynamical Systems – Series S, 9(2), 529–536.

Introduction

xxiii

Lerbet, J., Hello, G., Challamel, N., Nicot, F., and Darve, F. (2016b). 3-dimensional flutter kinematic structural stability. Nonlinear Analysis, 29, 19–37. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2017). On the stability of nonconservative continuous systems under kinematic constraints. J. of Appl. Math. and Mech. (ZAMM), 1–20. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2018). Coordinate free nonlinear incremental discrete mechanics. J. of Appl. Math. and Mech. (ZAMM), 98(10), 1813–1833. Lyapunov, A.M. (1992). Problème général de la stabilité du mouvement. Annales de la Faculté des sciences de Toulouse : Mathématiques, 9, 203–274. Nicot, F., Daouadji, A., Laouafa, F., and Darve, F. (2011). Second order work, kinetic energy and diffuse failure in granular materials. Granular Matter, 13(1), 19–28. Nicot, F., Sibille, L., and Darve, F. (2012). Failure in rate-independent granular materials as a bifurcation toward a dynamic regime. Int. Journal of Plasticity, 29(1), 136–154. Owen, D.R. and Williams, W.O. (1969). On the time derivatives of equilibrated response functions. ARMA, 33(4), 288–306. Rajagopal, K.R. (2011). Conspectus of concepts of elasticity. Mathematics and Mechanics of Solids, 16(5), 536–562. Thompson, J.M.T. and Bishop, S.R. (1994). Nonlinearity and Chaos in Engineering Dynamics, Wiley, Hoboken, NJ, USA. Thompson, J.M.T. and Hunt, G.W. (1973). A General Theory of Elastic Stability, Wiley, Hoboken, NJ, USA. Truesdell, C. and Noll, W. (1965). The non-linear field theories of mechanics. In Handbuch Phys. III, Truesdell, C., Noll, W. (eds), Springer, Berlin, Germany. Wan, R., Nicot, F., and Darve, F. (2017). Failure in Geomaterials, a Contemporary Treatise. ISTE Press, London, and Elsevier, Oxford. Ziegler, H. (1952). Die stabilitaetskriterien der elastomechanik. Ing. Arch., 20, 49–56.

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1 On Stability of Discrete and Asymptotically Continuous Systems

1.1. Stability – general concepts This book is concerned with some stability problems of discrete and continuous systems. The continuous systems investigated in this chapter may be asymptotically achieved from asymptotic arguments from some repetitive chain elements. Conservative and non-conservative elastic problems will be mainly considered and can be seen as a preliminary investigation in the field of stability of inelastic systems. Euler (1744) is generally considered to be among the pioneers to have investigated the stability of continuous conservative systems (Timoshenko 1983). Euler calculated the buckling load of a clamped-free column under conservative axial loading and rigorously characterized the post-buckling state, which is known as the elastica problem. These works still remain reference works for computing the post-buckling behavior of slender elements. A stability criterion has been formulated by Lagrange (1788) for discrete systems from the total potential energy of the system. This criterion established for a finite number of degrees of freedom has been generalized by Dirichlet in the 19th Century (Dirichlet 1846). Dirichlet already formulated the stability notion in rigorous words from the evolution of the system with respect to any perturbation of the equilibrium positions: “the equilibrium is stable if, in displacing the points of the system from their equilibrium positions by an infinitesimal amount and giving each one a small

2

Stability of Discrete Non-conservative Systems

initial velocity, the displacements of different points relative to the equilibrium positions of the system, remain throughout the course of the motion, contained between certain small prescribed limits” (see also Knops and Wilkes 1973 for the historical background on stability definitions). Lagrange (1788) proved that if the potential energy is a minimum at a position of equilibrium, then this equilibrium position is stable. If the potential energy is a maximum, the equilibrium position is shown to be unstable. Lagrange (1788) has shown these results by linearizing the equations of motion about the equilibrium position. This procedure was criticized by Dirichlet (1846) who pointed out the unnecessary justification of the linearization process followed by Lagrange. Dirichlet (1846) then rigorously prescribed the bounds that should be given for the initial conditions in order for the perturbed motion to remain in the neighborhood of the studied equilibrium position. The so-called Lagrange–Dirichlet theorem based on energy arguments is valid for conservative systems. A more general definition of the stability concept was given at the end of the 19th Century to the beginning of the 20th Century by Lyapunov (1907, 1949) for general dynamical systems, thus including both conservative and non-conservative systems. Lyapunov (1949) relates the stability concept to the property of uniform continuity of the perturbed motion to the initial conditions. This approach more generally includes the energy criterion of Lagrange–Dirichlet for conservative systems. This stability definition due to Lyapunov inherently contains inertia terms and is related to the associated dynamical systems. It is quite natural that the dynamical instabilities, which may be associated with periodic attractors, are investigated through this more general stability definition. Among these systems, non-conservative systems should be preferentially investigated through dynamical methods for a rigorous link with Lyapunov stability. Stability of non-conservative elastic systems is a branch of mechanics, which has emerged essentially in the 1950s and in the 1960s, with the major contributions of Bolotin (1963) and Ziegler (1968). Non-conservative elastic systems can become instable differently by divergence or by flutter. Both are instability phenomena covered through the larger concept of Lyapunov stability. We will not discuss in this chapter the technical difficulties related to the continuous nature of some physical systems and the inherent dependence of the stability notion to the metrics of continuous systems (see the works of Movchan 1956; Koiter 1963; Knops 1982 for one-dimensional or three-dimensional elasticity problems). This chapter is mainly concerned with discrete systems; continuous systems are viewed as asymptotic discrete systems.

On Stability of Discrete and Asymptotically Continuous Systems

3

In this book, we will mainly focus on mechanical systems ruled by the following differential equation of motion for discrete systems:

 + F ( X ) = 0 MX

[1.1]

which is a discrete nonlinear autonomous dynamical system. The dot refers to the time differentiation. We assume that X, the position vector of the system, depends on n generalized coordinates xi ( i = 1, 2..., n ) and that a static equilibrium state is characterized by xi = 0 ( X = 0 ) . X can also be

seen as a perturbation of the equilibrium. We will also restrict the study to smooth nonlinearities so that F is differentiable (non-smooth linearities may be explored as well, especially for inelastic processes – see Awrejcewicz and Lamarque (2003) for the treatment of non-smooth dynamical systems). Moreover, F is assumed to be velocity-independent or rate-independent. Although this assumption is not necessary when characterizing the stability of some trivial equilibrium position via dynamics methods, we will assume at least that damping can be neglected, which is a reasonable assumption for the concerned applications. The nonlinearities covered by such a function may come from some geometrical or material sources. For the concerned mechanical application, the mass matrix M is assumed to be a symmetric definite positive matrix. A rigorous definition of stability given by Lyapunov (1907, 1949) expresses the uniform continuity of the motion with respect to  . the initial conditions X0 , X 0

(

)

∀ε > 0, ∃δ > 0 / X0 < δ and  0 if only if Y ≠ 0 , H ( Y ) ≤ 0∀ Y [1.6] The stability theorem of Lagrange–Dirichlet valid for Hamiltonian systems may be proven from direct application of the second theorem of Lyapunov, where the total energy is chosen as a function of Lyapunov. For conservative systems, it is possible to derive the force vector from a potential Π ( X ) :

F ( X ) = grad Π ( X )

[1.7]

The kinetic energy of the system is given by:  = 1X  TM X  T X 2

( )

[1.8]

so that it is possible to introduce the total energy (Hamiltonian) H = T + Π:

(

)

 = H X, X

1 T  + Π ( X) X MX 2

[1.9]

It is not difficult to show in this case that the total energy is conserved:

(

)

 =X  T M X  + F ( X )  = 0 H X, X  

[1.10]

If the potential energy Π ( X ) is definite positive, the total energy H is also positive definite, and H is therefore a Lyapunov function, indicating that the equilibrium position is stable in the sense of Lyapunov (from application of the direct method of Lyapunov). The Lagrange–Dirichlet stability theorem is directly derived from the second method of Lyapunov applied to

6

Stability of Discrete Non-conservative Systems

conservative systems: an equilibrium position is stable if the potential energy has a complete relative minimum at that point. In the more general case, it is not possible to derive the generalized force from a potential (there is no potential associated with the generalized force) and the system is then classified as a non-conservative system. The alternative method, also labeled as the first method of Lyapunov, is a method based on the linearization near the equilibrium, which consists of examining the solution of the linearized dynamical system directly. This requires the determination of the spectrum of the operator associated with the linearized system. The nonlinear dynamic system may be linearized around an equilibrium position ( Y = 0 ):

 =AY Y

[1.11]

which can be equivalently formulated with the linearized stiffness matrix:

 + K X = 0 MX

[1.12]

In the general case including non-conservative loading, the stiffness matrix is not necessarily symmetric (this stiffness matrix is symmetric only in the case of conservative loading). This matrix can be formulated from the initial nonlinear dynamic system: K=

∂F ( X = 0) ∂X

[1.13]

It is also possible to express the linearized operator in equation [1.11] from the mass and the stiffness matrix through skew symmetric and symmetric matrix:

1  0 1   M −1K 0   0 A= =   −1 1   −M K 0   −1 0   0

[1.14]

The eigenvalues μ of A follows from

det ( A − μ 1) = 0

[1.15]

On Stability of Discrete and Asymptotically Continuous Systems

7

or equivalently,

(

)

det K + μ 2 M = 0

[1.16]

Eigenvalues μ are real or conjugated complex roots since A is a real matrix. The first method of Lyapunov gives stability or instability information (about the initial nonlinear dynamic system) with respect to the values of the eigenvalues of the linearized system: If Re ( μi ) < 0 for all i, the considered equilibrium is asymptotically stable. If ∃i / Re ( μi ) > 0 , the considered equilibrium is unstable.

( )

If Re ( μi ) ≤ 0 for all i, with at least one index j such as Re μ j = 0, no conclusion is available from the linearized theorem.

[1.17]

Mainly two types of instabilities may appear when varying a structural parameter (typically a loading parameter): static instability (or divergence type instability) or dynamic instability (flutter). Divergence instability (or non-oscillatory instability) arises when at least one eigenvalue becomes real positive after passing through zero where the Jacobian A is singular. Among different scenarios of divergence instability, a real negative eigenvalue crosses the origin of the complex μ-plane and becomes positive. However, in some mathematical problems, a positive real eigenvalue may emerge through other mechanisms (see Huseyin 1975, Kuznetsov 2004). The transition between stability and divergence may be also accompanied by the merging of a pair of pure imaginary eigenvalues into a double-zero eigenvalue with the Jordan block that then split into two real eigenvalues of different sign. Thus, the boundary between the divergence and stability is characterized by the double-zero eigenvalues μ (nullity of a vibration frequency). In the present case, since the characteristic equation [1.16] is given by a real polynomial with respect to μ2, the divergence boundary is determined by the vanishing free term of the characteristic polynomial, which yields μ2 = 0. According to the Faddeev–Leverrier algorithm, the free term of the characteristic polynomial is exactly the vanishing of the determinant of the

8

Stability of Discrete Non-conservative Systems

stiffness matrix. The divergence instability criterion is then written (for both conservative and non-conservative systems):

det ( K ) = 0

[1.18]

A complex eigenvalue with the positive real part yields dynamical instability (flutter or oscillatory instability). Generically, a pair of complex conjugate eigenvalues crosses the imaginary axis, resulting in a conjugate pair with positive real part. Some alternative mechanisms may also appear for some physical systems. For instance, a transition between stability and flutter in circulatory systems may happen for circulatory systems through the confusion of two pure imaginary eigenvalues into one double pure imaginary eigenvalue (confusion of two close frequencies). Dynamic instability (or flutter instability) may be associated with the branching of limit cycles from the critical state. For conservative systems, the eigenvalues of equation [1.16] are always real or imaginary. This indicates that an equilibrium state can only become unstable through static instability for conservative systems (flutter cannot arise in such systems). Equation [1.18] is also valid for conservative systems, which, for symmetric stiffness matrix, means that the system is no longer definite positive at the stability boundary. For conservative systems, the instability criterion equation [1.18] can also be written for the symmetric part of the stiffness matrix:

( )

det K S = det ( K ) = 0

[1.19]

As a synthesis, for conservative systems, the Lagrange–Dirichlet stability criterion is obtained from application of Lyapunov’s direct method, whereas the instability associated with the loss of definite positiveness is shown from the application of the first method of Lyapunov. For non-conservative systems, the divergence instability criterion is shown from application of the first method of Lyapunov (singular condition of the stiffness matrix equation [1.18]). For non-conservative systems (linear or nonlinear systems), the nonconservative source of the system may come from the loading (non-potential loading or loads which do not derive from a potential) such as some follower

On Stability of Discrete and Asymptotically Continuous Systems

9

loads (circulatory systems) or from the constitutive law (non-hyperelastic law or Cauchy elasticity). We do not discuss the eventual deficiencies of Cauchy elasticity (as opposed to Green elasticity or hyperelasticity – see Green and Naghdi 1971). In the following, only non-conservative and conservative loading systems are explored. The next parts of this chapter are based on generalized Hencky models of dimension n loaded by conservative and non-conservative loading.

Figure 1.1. Hencky chain model using concentrated masses and concentrated stiffness (see also Wang et al. 2016)

In this chapter, stability of discrete repetitive systems as introduced by Hencky (1920) will be considered with conservative and non-conservative loading. The Hencky bar-chain model (see Hencky 1920 or, more recently, Wang et al. 2019) is a kind of lattice beam composed of rigid links of equal spacing a, connected by rotational springs of stiffness C. First, we will investigate the buckling problem of a Hencky column composed of n=2 and n=3 elements loaded at its tip by a vertical load P. It is also assumed that the rotational stiffness at the basis of the discrete column is equal to C1 = 2C, as also assumed by Hencky (1920) for modeling discrete clamped boundary

10

Stability of Discrete Non-conservative Systems

conditions. Figure 1.2 shows this clamped-free Hencky model for n=2. The internal work of this two-degree-of-freedom system is given by: 1 1 1 2 2 Wint (θ1 ,θ 2 ) = C1θ12 + C (θ 2 − θ1 ) = Cθ12 + C (θ 2 − θ1 ) 2 2 2

[1.20]

The work done by the dead load is expressed through its linearized form: Wext (θ1 ,θ 2 ) = Pa ( 2 − cos θ1 − cos θ 2 ) ≈

1 Pa θ12 + θ 22 2

(

)

[1.21]

The total potential energy Π is expanded in quadratic format: 1 Pa 2 2 Π (θ1 ,θ 2 ) = Wint − Wext = Cθ12 + C (θ 2 − θ1 ) − θ1 + θ 22 2 2

(

)

[1.22]

Figure 1.2. Two-degree-of-freedom divergence system (Hencky system under vertical conservative loading) – n=2

The stiffness matrix K is built from the Hessian of the total potential energy π :

K ij =

∂ 2Π ∂θi ∂θ j

−C   3C − Pa  K =  C − Pa   −C

[1.23]

Applying the divergence criterion for this conservative system det K = 0 gives the second-order polynomial equation:

On Stability of Discrete and Asymptotically Continuous Systems

a2 P2 − 4CaP + 2C 2 = 0

11

[1.24]

It is convenient to introduce the dimensionless buckling load

β = n2

Pa C

[1.25]

so that the buckling loads from equation [1.24] are calculated as:

β1 = 8 − 4 2 and β 2 = 8 + 4 2

[1.26]

For Hencky column with n=2, the fundamental solution θ1 = θ2 = 0 becomes instable when the dimensionless load is larger than the fundamental buckling load β > β1 . It can be shown that the fundamental branch is stable for β ≤ β1 (see also Alfutov 2000 for an analysis of a similar two-degree-offreedom conservative system). In fact, for β < β1 , when the applied force is lower than the fundamental buckling load, the total potential energy of the system in the initial equilibrium state (fundamental solution) is a minimum (the stiffness matrix is definite positive) which means that the initial equilibrium state is stable according to the Lagrange–Dirichlet theorem. For β > β1 , the stiffness matrix has one negative eigenvalue, and the equilibrium is unstable according to the Lyapunov stability theorem. For β = β1 , the nonlinear potential function needs to be expanded at a higher order: Pa 2 1 2 Π (θ1 ,θ 2 ) = Cθ12 + C (θ 2 − θ1 ) − θ1 + θ 22 2 2 Pa 4 + θ1 + θ 24 + Ο θ16 + Ο θ 26 24

(

(

)

( )

)

( )

[1.27]

The nonlinear potential energy can also be re-expressed as:

Π (θ1 ,θ 2 ) =

 12 − β C (θ1 ,θ 2 )  8  −4

+C

β 96

(θ

4 1

)

−4   θ1    4 − β  θ 2 

( )

( )

[1.28]

+ θ 24 + Ο θ16 + Ο θ 26

which is definite positive for β = β1 . Then, according to the Lagrange– Dirichlet theorem, the trivial equilibrium is also stable for β = β1 .

12

Stability of Discrete Non-conservative Systems

As another illustrative example, Hencky column with n=3 (see Figure 1.3) will be analyzed next.

Figure 1.3. Three-degree-of-freedom divergence system (Hencky system under vertical conservative loading) – n=3

The stiffness matrix for this three-degree-of-freedom system now writes: −C 0   3C − Pa   K =  −C 2C − Pa −C   0 −C C − Pa  

[1.29]

Applying the divergence criterion for this conservative system det K = 0 gives the cubic equation:

−a3 P3 + 6Ca2 P2 − 9C 2 aP + 2C3 = 0

[1.30]

On Stability of Discrete and Asymptotically Continuous Systems

13

Cardano’s method can be applied to solve this cubic equation, which admits three real roots: P=

2C   π 5π 9π   1 + cos  , ,   a   6 6 6 

[1.31]

and the buckling loads are then calculated as:  π   β1 = 18  1 − cos 6  = 18 − 9 3    n = 3   β 2 = 18   β 3 = 18 1 + cos π  = 18 + 9 3 6  

[1.32]

It will be shown in the next part that it is possible to derive a single formula which covers all these generic cases by solving a linear difference eigenvalue problem.

Figure 1.4. Two-degree-of-freedom flutter system (Ziegler’s type system) – n=2

Instability of non-conservative Ziegler’s type systems will now be investigated. The flutter load of the two-degree-of-freedom Ziegler’s type system (see Figure 1.4) can be analytically obtained. The internal virtual work of this two-degree-of-freedom system is expressed as:

14

Stability of Discrete Non-conservative Systems

δ Wint = 2Cθ1δθ1 + C (θ1 − θ 2 ) (δθ1 − δθ 2 )

[1.33]

The virtual work done by the follower force, in its linearized form, is:

δ Wext = Pa (θ1δθ1 + θ 2δθ 2 ) − Paθ 2 (δθ1 + δθ 2 ) = Pa (θ1 − θ 2 ) δθ1 [1.34] The work done by the fictitious inertial force on a virtual displacement system is: δW f = −

ma 2  θ1δθ1 + θ1δθ 2 + θ2δθ1 + θ2δθ 2 − ma 2θ1δθ1 2

(

)

[1.35]

The total virtual work δ W = δ Wint − δ Wext − δ W f is zero for or any virtual displacement system, thus leading to the vibration equation:

 + K Θ = 0 MΘ Pa  3−  K =C C   −1

with

M=

Pa   θ1  C  and Θ =    θ2  1 

−1 +

ma 2  3 1  , 2  1 1

[1.36]

Note that K is not symmetric as it also involves the effect of the non-conservative follower force. The circular frequency of the forced vibration is calculated from the determinant equation:

(

)

det K − ω 2 M = 0

[1.37]

which can be equivalently presented using the dimensionless parameters:

3Ω 2 4 32 Ω2 −1− 32

3−

β

Ω2 =

−

ω 2 L4 EI

−1 +

μ = n4

β

−

4 Ω2 1− 32

ω 2a2m C

Ω2 32

= 0 with β =

and n = 2

PL2 Pa = n2 , EI C

[1.38]

On Stability of Discrete and Asymptotically Continuous Systems

15

The quartic frequency equation is obtained from the expansion of this determinant: Ω 4 + 8Ω 2 ( β − 16 ) + 322 = 0

[1.39]

The flutter frequency corresponds to the vanishing of the discriminant: Δ = 82 ( β − 16 ) − 4 × 322 = 0  β = 8 2

[1.40]

,2 = 8 is found for the two-degree-of-freedom The flutter value β disc flut

Ziegler’s type system n=2. The flutter frequency is then obtained from equation [1.39] as: ,2 Ω disc flut = 4 2 ≈ 5.657

[1.41]

The flutter load of the three-degree-of-freedom follower load system (see Figure 1.5) can also be analytically obtained. Following the reasoning already detailed for the two-degree-of-freedom system, and again considering that the total virtual work δ W = δ Wint − δ Wext − δ W f is zero for or any virtual displacement system, leads to the vibration equation:

 + K Θ = 0 MΘ Pa  3− C  K = C  −1    0  

with

−1 Pa C −1

2−

 5 3 1 ma 2   M= 3 3 1 ,  2    1 1 1

Pa  C   θ1   Pa    and Θ =  θ 2  −1 + C  θ    3 1   

[1.42]

16

Stability of Discrete Non-conservative Systems

Figure 1.5. Three-degree-of-freedom flutter system (Ziegler’s type system) – n=3

The circular frequency of the forced vibration is calculated from the determinant equation [1.37], which can be equivalently presented using the dimensionless parameters:

 β  3 − 9   det  −1    0  

β = n2

−1 2−

β

9 −1

    9 5 3 1     β  Ω2   −1 + − 3 3 1  = 0 9  162     1 1 1  1     

β

ω 2a2m ω 2a2m Pa Pa = 81 , Ω2 = n 4 and n = 3 =9 C C C C

with

[1.43]

On Stability of Discrete and Asymptotically Continuous Systems

17

( )

Now using the dimensionless variables x = Ω 2 2n 4 and p = β n2 , the determinant can be reformulated as: −1 − 3 x p−x   3 − p − 5x   det  −1 − 3 x 2 − p − 3 x −1 + p − x  = 0  −x −1 − x 1 − x  

[1.44]

which is equivalently written by:

(

)

P ( x ) = 2 + −33 + 20 p − 3 p 2 x + ( 28 − 8 p ) x 2 − 4 x 3

[1.45]

This cubic equation can be solved using Cardano’s method. Put 1 1 y = x − ( 7 − 2 p ) , namely x = y + ( 7 − 2 p ) , then x is a root of P ( x ) = 0 3 3 if and only if y is a root of Q ( y ) = 0 , with Q ( y ) defined by:

(

)

Q ( y ) = 719 − 498 p + 123 p 2 − 10 p 3 + 873 − 468 p + 63 p 2 y − 108 y 3

(

= −108 y 3 + vy + u with u =

)

719 − 498 p + 123 p 2 − 10 p3 873 − 468 p + 63 p 2 and v = −108 −108

[1.46]

One recognizes the canonical form of the initial cubic equation (see Cardano’s method). This cubic equation has multiple roots when the discriminant of this cubic equation vanishes, i.e. when:

Δ = 4v3 + 27u 2 = 0 

9 p 6 − 192 p5 + 1702 p 4 − 7984 p3 + 20725 p 2 − 27840 p + 14656 = 0

[1.47]

This sixth-order polynomial equation can be numerically solved for the flutter load p = β n 2 . The first positive root of Δ ( p ) = 0 is p f ≈ 1.3780 , yielding the flutter load of the discrete model with n=3 cells: ,3 β disc = 9 × p f ≈ 12.4023 flut

[1.48]

18

Stability of Discrete Non-conservative Systems ,3 When Δ ( p ) = 0 , the corresponding value of the flutter frequency Ω disc flut

is the double root of the cubic equation given by:

(

2 3 3u −3 719 − 498 p f + 123 p f − 10 p f yf = − = 2v 2 873 − 468 p f + 63 p 2f



(

)

)

2 3 1 −213 + 206 p f − 61 p f + 6 p f xf = − 2 97 − 52 p f + 7 p 2f

[1.49]

One numerically finds x f ≈ 0.378693 , which yields the flutter frequency of the discrete model with n=3 cells: ,3 Ωdisc flut = 9 2 x f ≈ 7.833

[1.50]

1.2. Buckling of discrete conservative systems – Hencky’s column under conservative axial load

This part aims to contribute to a better understanding of the transition from a discrete to a continuous rod modeling in the presence of geometrical nonlinearities. Both discrete and continuous models are formulated into a geometrically exact framework. The buckling and post-buckling behaviors of the elastic lattice system are referred to herein as the discrete elastica problem in a geometrically exact framework. The investigation of buckling and post-buckling behaviors of inextensible elastic columns (linearized problem and geometrically exact formulation) dates back to the mid-1700s (Euler 1744). However, the discrete elastic bar-chain problem was only analyzed in the early 1900s. A discrete formulation of Euler’s problem was first studied by Hencky in 1920 (Hencky 1920) who considered an elastic bar-chain composed of rigid links connected by rotational springs. Hencky (1920) showed that this system may asymptotically converge towards the Euler one for an infinite number of links. Note that Hencky did not provide the analytical solutions of the buckling load for an arbitrary number of links n, but he presented solutions

On Stability of Discrete and Asymptotically Continuous Systems

19

only for n=2, 3 or 4. This problem has been reconsidered by Wang (1951, 1953) who gave the buckling solution for any number of links n. In fact, Wang (1951, 1953) solved a linear second-order difference equation (see, for instance, Goldberg (1958) or Elaydi (2005) for a general overview of difference equations) and analytically obtained the buckling load associated with the corresponding boundary value problem. Silverman (1951) also mentioned the mathematical analogy between Hencky’s system and the central finite difference formulation of Euler’s problem. This buckling problem was reconsidered by Seide (1975) in terms of the finite difference method (which may be regarded as being equivalent to the algebraic equations of the Hencky system) and compared to other numerical methods for general boundary conditions. This strong connection between the difference equations involved in lattice mechanics and the finite difference formulation of continuous mechanics problems is also discussed by Maugin (1999). Going back to Hencky’s model and due to the interest of the mechanics community in understanding the discretization properties of nonlinear systems, the post-buckling behavior of the Hencky bar-chain model (which can be also labeled as the discrete elastica) was reconsidered in the 1980s independently by El Naschie et al. (1989) and Gáspár and Domokos (1989). Gáspár and Domokos (1989), Domokos (1993) and Domokos and Holmes (1993) pointed out the very rich structure inherent to the discrete property of this nonlinear structural system, and the possible spatial chaotic behavior of Hencky bar-chain to be far away from the first initial bifurcated branch. Domokos and Holmes (1993) also showed the mathematical equivalence of the finite difference formulation of the Euler problem and the Hencky system in the nonlinear range. It is worth mentioning that the spatial chaotic behavior of discrete chains has also been observed for non-conservative systems by Kocsis and Károlyi (2006). More recently, Wang et al. (2013) or Challamel et al. (2014a, 2014b) have shown the possibility of linking the linearized Hencky system to nonlocal beam mechanics. It is also possible to generalize these results in the nonlinear range, where nonlocal beam mechanics may also be founded from discrete beam interactions, including axial, bending and shear interactions (Challamel et al., 2015a, Kocsis and Challamel 2018). Consider a Hencky’s bar-chain with pinned–pinned ends as shown in Figure 1.6.

20

Stability of Discrete Non-conservative Systems

Figure 1.6. Simply supported Hencky’s chain: n rigid links are connected by hinges and rotational springs. For a color version of this figure, see ww.iste.co.uk/lerbet/stability.zip

The column, composed of n repetitive cells of size denoted by a, is axially loaded by a concentrated force denoted by P. The discrete column of length L is modeled by some finite rigid segments connected by elastic rotational springs of stiffness C=EI/a, where EI is the bending rigidity of the local Euler–Bernoulli column asymptotically obtained for an infinite number n of rigid links. In other words, the total length of the structure L is equal to L=n×a, the number of rigid segments multiplied by the size of each segment. The discrete version of the local elastica can be obtained from the following system of nonlinear difference equations: M i = EI

θi +1 − θi a

and

M i − M i −1 + P sin θi = 0 a

[1.51]

Here, M i is the bending moment in the rotational spring at hinge i, and

θi is the angle of the ith link from the line of action of compressive force P. In other words, θi is the rotation angle of the segment i connecting the (i − 1)th and the ith nodes. As pointed out by Domokos and Holmes (1993), these difference equations [1.51] are similar to the forward and the backward finite difference equations of the continuous elastic problem, where the step size is equal to the length of the rigid link a. In this concept, the differential equation system of the axially compressed, hinged–hinged elastica, M = EI × dθ / ds and dM / ds + P sin θ , are discretized using forward and backward differences,

On Stability of Discrete and Asymptotically Continuous Systems

21

respectively. This yields a semi-implicit Euler method, which defines an area-preserving map. The nonlinear second-order difference equation is obtained from both equations [1.51]:

EI

θi +1 − 2θi + θi −1 a2

+ P sin θi = 0

[1.52]

This nonlinear difference equation is reformulated in a dimensionless form:

θi +1 − 2θi + θi −1 = −

β n2

sin θi

[1.53]

by using the dimensionless load β = PL2 EI . The nonlinear difference equation can be equivalently reformulated with the following relations:

θi +1 = θ i +

κˆi n

and κˆi +1 = κˆi −

β n

sin θi +1

[1.54]

with the dimensionless curvature κˆi defined by κˆi = Lκ i and the curvature

κi = M i EI . The boundary conditions of the hinge–hinge column are obtained from the vanishing of the bending moments at both ends, i.e. M 0 = 0 and M n = 0:

θ1 = θ0 and θn+1 = θn

[1.55]

Note that an equivalent system may be obtained by discretizing the differential equations of the local elastica with central differences: M i = EI

θi +1 2 − θi −1 2 a

and

M i +1 2 − M i −1 2 a

+ P sin θi = 0

[1.56]

leading to the same nonlinear difference equation:

θi +1 − 2θi + θi −1 = −

β n2

sin θi

[1.57]

22

Stability of Discrete Non-conservative Systems

The equivalence between the two discretizations is obtained from the shift relationship, originating from the definition of the bending moment Mi at node i:

θi = θi +1 2

[1.58]

It implicitly means that the new rotation variable θi is associated with the rotation measured at the center of the rigid element. The equivalent boundary conditions for the discrete elastica are now written as:

θ−1 2 = θ1 2 and θn −1 2 = θn +1 2

[1.59]

The same buckling load is obtained in both cases, but with different buckling modes. The analytical solution for the fundamental buckling load issued of the linearization process has been calculated by Wang (1951, 1953) – see also Domokos and Holmes (1993) – or more recently by Challamel et al. (2015a). The reasoning is repeated here for completeness. The linearization of equation [1.53] (or equivalently the one of equation [1.57]) for computing the buckling load gives: β  − 2 θi + θi −1 = 0 2 n 

θi +1 + 

[1.60]

The solution of this difference boundary value problem is derived by Goldberg (1958). Goldberg (1958) also mentioned that this second-order difference boundary value problem arises in the mathematical theory of scale analysis. We reproduce the main reasoning of the mathematical proof for the investigation of this discrete eigenvalue problem. The characteristic equation is obtained by replacing θi = Aγ i in equation [1.60], which leads to

γ+

1

γ

=2−

β

[1.61]

n2

Equation [1.61] is symmetrical with respect to interchanging γ and 1 γ . This equation admits the following two solutions for 1 −  

γ 1,2 = cos φ ± j sin φ with φ = arccos 1 −

β 

β 2n 2

< 1:

 and j = −1 2n 2 

[1.62]

On Stability of Discrete and Asymptotically Continuous Systems

23

The solution of the linear second-order difference equation [1.60] can be expressed with the real basis as:

θi = A cos (φ i ) + B sin (φ i )

[1.63]

Introduction of the solution expressed by equation [1.63] into the two boundary conditions equation [1.55] leads to the following buckling mode:

θi = θ 0

sin (φ i ) − sin (φ ( i − 1) ) sin φ

φ  cos  φ i −  2  = θ0 φ cos 2

[1.64]

Now, by using the same reasoning for θi = A cos (φ i ) + B sin (φ i ) and considering the boundary conditions of equation [1.59] necessarily give:

θi = θ0 cos (φ i )

[1.65]

It is worth noting that the difference between the two modes relies on a shift of φ / 2 inside the trigonometric function (see the equivalence relationship equation [1.58]). In both cases, the kth buckling load of the pinned–pinned Hencky chain is given by the following load formulae: kπ β =1− 2 n 2n  kπ   β k = 4n 2 sin 2    2n 

sin (φ n ) = 0  φ n = kπ

 cos

[1.66]

which is consistent with the results of Wang (1951, 1953). Hence, there are n − 1 buckling loads of a Hencky chain of n links, contrary to the infinitely many buckling loads of the continuous local elastica. The fundamental buckling load converges towards the Euler value (continuous simply supported elastica problem) for a sufficiently large number of links: π  lim β1 = lim 4n 2 sin 2   = π 2 n →∞  2n 

n →∞

[1.67]

24

Stability of Discrete Non-conservative Systems

As highlighted in Figure 1.7, the fundamental buckling load of Hencky’s model is lower than the one of the asymptotic continuous model, which is equivalent to say that the finite difference method gives a lower-bound approximation of the continuous Euler–Bernoulli beam model. Indeed, an asymptotic expansion of the nonlinear buckling load function shows that:

π2 π  2 = − π 1   2  2n   12n

β1 = 4n 2 sin 2 

β1

  1   + Ο 4  n  

[1.68]

12

β1= π2

10

β1 = 4n 2 sin 2 ( π/2n)

8

6

4

2

0 2

3

4

5

6

7

8

9

10

11

12

13 n

14

Figure 1.7. Dimensionless fundamental buckling load β1 versus the number of links n

The solutions of the discrete boundary value problem, defined by equation [1.54] and the boundary conditions θ1 = θ0 and θn+1 = θn (equation [1.55]), are determined by using the shooting method (see, for instance, Kocsis (2013)). The solutions are visualized in Figure 1.8 in bifurcation diagrams, 2D plots spanned by the initial angle θ0 and the dimensionless load parameter β . The bifurcation diagram of Hencky chains is shown with various number of links n in the domain β ∈ [ 0;200] and θ0 ∈ [ 0; π ] (see also Challamel

et al. 2015a). The resolution of the discretization along both θ0 and β is 10,000 for the shooting method. As pointed out by Gáspár and Domokos

On Stability of Discrete and Asymptotically Continuous Systems

25

(1989), Domokos (1993) and Domokos and Holmes (1993), the discrete system possesses a very rich structure inherent to the discrete property of the structural system, and the possible spatial chaotic behavior of Hencky bar-chain is far away from the first initial bifurcated branch, for sufficiently large n numbers (number of rigid links). The discrete system possesses a multiplicity of solutions appearing from the primary branches in saddle node bifurcation, a property which is not observed for the continuous elastica system. These solutions are classified as parasitic solutions. Also described by Domokos and Holmes (1993) is that the number of parasitic solutions increases with the number of links n.

Figure 1.8. Bifurcation diagram of Hencky chains for various numbers of links n in the domain β ∈0;200 and θ0 ∈0;π  . The value of n is plotted on the top of the figures. The resolution of the discretization along both θ0 and β is 10,000 for the shooting method (after Challamel et al. 2015a)

26

Stability of Discrete Non-conservative Systems

The Hencky chain model under clamped-free boundary conditions can be considered as well, as shown in Figure 1.9.

Figure 1.9. Clamped-free Hencky’s chain with a rotational stiffness at the clamped section equal to C1=2C and C=EI/a

The second-order linear difference equation [1.53] is still valid, but with the new boundary conditions:

θ1 = −θ0 and θn+1 = θn−1

[1.69]

The first condition is an anti-symmetrical boundary condition which is equivalent to considering an equivalent stiffness at the clamped node, twice the internal rotational stiffness C1=2C (as also recommended by Hencky (1920)): M 0 = EI

θ1 − θ 0 a

= 2 EI

θ1 a

= C1θ1

[1.70]

On Stability of Discrete and Asymptotically Continuous Systems

27

The second symmetrical boundary condition at the end node expressed the vanishing of the end moment: M n = EI

θ n +1 − θ n a

=0

[1.71]

The boundary conditions can be formulated with the shift variables:

θ−1 2 = −θ1 2 and θn −1 2 = θn +1 2

[1.72]

By using the same reasoning for θi = A cos (φ i ) + B sin (φ i ) and considering the boundary conditions of equation [1.72], it necessarily gives:

θi = B sin (φ i )  cos (φ n ) = 0

[1.73]

The buckling load is then calculated as:

φ n = ( 2k − 1)

π 2

π    β k = 4n 2 sin 2 ( 2k − 1)  4n  

[1.74]

which is consistent with the results obtained by Seide (1975) for the clamped-free column analyzed with the centered finite difference formulation. The results obtained in the first part are found again in the particular case n=2 (see also equation [1.76]):

π  n = 2  β1,2 = 16sin 2 ( 2k − 1)  = 8 ± 4 2 8 

[1.75]

For n=3, we also find the results derived in the first part of the chapter (see equation [1.32]):

(

n = 3  β1,2,3 = 18;18 ± 9 3

)

[1.76]

28

Stability of Discrete Non-conservative Systems

The fundamental buckling load for each system parameterized by the number of links n is then equal for n=1, n=2 and n=3:  2π   n = 1  β1 = 4sin  4  = 2     π   n = 2  β1 = 8 1 − cos  = 8 − 4 2 4    π   n = 3  β1 = 18 1 − cos  = 18 − 9 3 6  

[1.77]

The fundamental buckling load converges towards the Euler value (continuous clamped-free elastica problem) for a sufficiently large number of links: 2

π  π lim β1 = lim 4n 2 sin 2   = n →∞ n →∞  4n  4

[1.78]

The asymptotic analysis of the post-bifurcation branches of the discrete elastica is developed in Challamel et al. (2015a), who also approximated the Hencky system with a non-local Euler–Bernoulli rod model. The Hencky system is essentially based on concentrated microstructures ruled by some finite difference formulation of the equivalent continuous system. It is worth mentioning that distributed microstructure could be considered instead of concentrated microstructure, leading to an equivalent finite-element elastica as shown by Kocsis and Challamel (2016). 1.3. Buckling of discrete non-conservative systems – Hencky’s column under follower axial load

The aim of this chapter is to generalize previous results mainly devoted to conservative systems to non-conservative elastic systems, such as circulatory systems. The structural paradigm of Beck’s system composed of a cantilever elastic column loaded by a concentrated follower load has been studied in detail (see Beck 1952, Bolotin 1963, Ziegler 1968, Leipholz 1970, Carr

On Stability of Discrete and Asymptotically Continuous Systems

29

and Malhardeen 1979, Elishakoff 2005). The discrete version of this structural problem can be understood as Ziegler’s column, at least for the two-degree-of-freedom system (Ziegler 1952). In other words, there is a kind of continuous transition from the two-degree-of-freedom system (Ziegler’s column – Ziegler 1952) to the continuous one (Beck’s column is asymptotically found for an infinite degree of freedom; Beck 1952), using an n-degree-of-freedom Hencky bar-chain model under follower load. This chapter is also related to the effect of discretization in the characterization of the stability domain of non-conservative elastic systems. Moreover, as discussed in Challamel et al. (2015b), there is a close connection between lattice mechanics and finite difference methods. Hencky’s system can be considered as the physical representation of the finite difference method applied to a continuous beam problem (Silverman 1951). Leipholz (1962), Sugiyama et al. (1971), Sugiyama and Kawagoe (1975), El Naschie and Al-Athel (1979a) and El-Naschie and Al-Athel (1979b) investigated non-conservative problems with distributed axial forces by the finite-difference method (or the discrete element method). El Naschie and Al-Athel (1979a) and El-Naschie and Al-Athel (1979b) used a discrete element method (equivalent to the Hencky bar-chain system) under distributed axial follower load with a modified clamped boundary condition based on a clamped rotational stiffness larger than two times the internal rotational stiffness. This approach can be shown to be equivalent to the finite difference method, except eventually for the modeling of the clamped section (which should be analyzed with some specific elastic connection at the basis of the column). More specifically with respect to the present chapter devoted to Beck’s column, Sugiyama et al. (1971) also specifically investigated Beck’s column by the equivalent finite difference method. El Naschie and Al-Athel (1979b) studied a massless Beck’s column with a concentrated tip mass, using Hencky’s system. More recently, the specific effect of discretization of columns in the presence of follower loads has been numerically handled by Gasparini et al. (1995) without any theoretical correspondence with nonlocal theories. Luongo and D’Annibale (2014) investigated the destabilizing role of damping in discrete and continuous systems. The main results in this part are also developed in Challamel et al. (2016).

30

Stability of Discrete Non-conservative Systems

The discrete Beck’s problem is shown in Figure 1.10.

Figure 1.10. Discrete Beck’s problem

This microstructured elastic system comprises n cells, as it is a discretized beam of length L composed of n repetitive cells of size a. In other words, the total length of the structure L is equal to L=n×a, i.e. the number of repetitive cells multiplied by the size of each cell. The cells are connected by frictionless hinges of mass m and coupled by elastic rotational springs of stiffness C. The correspondence between the discrete and the continuous systems yields C=EI/a for the spring stiffness and m=μa for the mass, where EI is the bending stiffness and μ is the mass per unit length of the continuum. The end inertial mass is half of that in the middle in order to maintain the total mass conservation, nm=μL. The end of the beam is attached to the ground and equipped with a rotational spring of stiffness C1=2C (as recommended by Hencky (1920) in case of conservative loading). The other (free) end is loaded by a follower force denoted by P. The stability of the discrete system under the non-conservative follower force is studied with the dynamic method under the assumption of small displacements. Note that the equilibrium method cannot capture dynamic loss of stability of

On Stability of Discrete and Asymptotically Continuous Systems

31

non-conservative systems, as it was demonstrated on similar discrete models by Kocsis et al. (2006) and Kocsis (2013). In order to obtain the governing equations of the system, D’Alembert’s principle, along with the principle of virtual displacements, is used (see, for instance, Gantmacher (1975)). The virtual work done by the moments in the spring on a virtual displacement system is given by: w1 (t ) δ w1 a a  wi +1 (t ) − 2 wi (t ) + wi −1 (t )   δ wi +1 − 2δ wi + δ wi −1     a a   

δ Wint = C1 n

+C

 i =1

[1.79]

 Here, wi (t ) = w ( x = ia, t ) is the vertical translation of node i as the function of time, δ wi is an arbitrary virtual displacement of node i and C1=2C is the rotational spring stiffness at the clamped end. The virtual work done by the follower force is:  wi (t ) − wi −1 (t )  δ wi − δ wi −1   w (t ) − wn −1 (t )  − P n    δ wn [1.80] a a a     i =1  n

δ Wext = Pa  

The work done by the fictitious inertial force on a virtual displacement system is: n −1

i (t )δ wi − δ W f = − m w i =1

m n (t )δ wn w 2

[1.81]

The total virtual work δ W = δ Wint − δ Wext − δ W f is zero for or any virtual displacement system. It yields: wi + 2 (t ) − 4 wi +1 (t ) + 6 wi (t ) − 4 wi −1 (t ) + wi − 2 (t ) a4 w (t ) − 2 wi (t ) + wi −1 (t ) i (t ) = 0 + P i +1 + μw a2 EI

[1.82]

32

Stability of Discrete Non-conservative Systems

for i = 1… n − 1, with the following boundary conditions prescribed at the clamped end:

w0 = 0 and w−1 = w1

[1.83]

The far-end boundary conditions of this non–self-adjoined problem are obtained from the virtual work theorem for node i = n :

wn+1 − 2wn + wn−1 = 0 and wn +1 − 3 ( wn − wn −1 ) − wn − 2 +

Ω wn = 0 2n 4

[1.84]

Note that the last condition (free-end boundary condition) reduces to the one obtained by Leckie and Lindberg (1963) for a free vibration problem with the finite difference method. We can immediately recognize that the considered difference equations are the finite difference version of the differential equations of Beck’s column. Hence, the discrete, microstructured system is mathematically equivalent to the finite difference format of the so-called “local” continuum (see the remark of Silverman (1951) and the debate about the physical interest of Hencky’s bar-chain compared to the “abstract” numerical-based finite difference method). The discrete Beck’s problem, under investigation, is nothing other than the finite difference formulation of the Beck’s continuous problem governed by:

   = 0 EI δ 04 w + Pδ 02 w + μ w

[1.85]

where wi + 2 (t ) − 4 wi +1 (t ) + 6 wi (t ) − 4 wi −1 (t ) + wi − 2 (t ) a4  w (t ) − 2 wi (t ) + wi −1 (t ) δ 02 w = i +1 a2



δ 04 w =

and

Here, δ 0 is the first-order central difference. This linear finite difference equation can be solved exactly. First, the nodal displacement is written in the form of

wi (t ) = wi ⋅ e jω ⋅t

[1.86]

On Stability of Discrete and Asymptotically Continuous Systems

33

with j being the imaginary unit, ω being the vibration frequency of the beam and wi being the nodal displacement amplitude. It is substituted in equation [1.85], leading to the time-independent difference equation: Hi +

β n

Gi − 2

Ω2 wi = 0 ( i ∈ [1, n − 1] ) n4

[1.87]

where

Hi = wi +2 − 4wi +1 + 6wi − 4wi −1 + wi −2 Gi = wi +1 − 2wi + wi −1 and β and Ω are given by:

β=

PL2 ω 2 L4 μ and Ω2 = EI EI

[1.88]

Next, the linear fourth-order difference equation is exactly solved, following the procedure described for the conservative buckling problem. The discrete displacement field of the microstructured beam model can be assumed as

wi = Bγ i

[1.89]

where B is a constant. Therefore, equation [1.87] may be written as

Γ 2 − (4 − A1 )Γ + 4 − 2 A1 + A2 = 0

[1.90]

where

Γ =γ +

1

γ

, A1 =

β n

2

, A2 = −

Ω . n4

By solving equation [1.90], we obtain Γ1,2 = 2 −

β 2n 2



β 2 + 4Ω 2n 2

[1.91]

34

Stability of Discrete Non-conservative Systems

Therefore γ 1,2 = 1 −

1  β 1  β + β 2 + 4Ω   j 1 − 1 − 2 − 2  4n 2  4 n 4 n 

γ 3,4 = 1 −

1  β 1  β − β 2 + 4Ω   1 − 2 + 2  4n 2  4n  4n

 

β 2 + 4Ω   

2

2

β 2 + 4Ω  − 1

[1.92a] [1.92b]

where j = −1 is the imaginary unit. From equation [1.92], we can assume that cos φ = 1 −

β 4n

cosh ϑ = 1 −

2

−

β 4n 2

1 4n 2 +

1 4n 2

β 2 + 4Ω β 2 + 4Ω

[1.93a] [1.93b]

Therefore

γ 1,2 = cos φ  j ⋅ sin φ

[1.94a]

γ 3,4 = cosh ϑ  sinh ϑ

[1.94b]

In view of equation [1.94], the general solution for wi can be represented as

wi = A1 cos ( iφ ) + A2 sin ( iφ ) + A3 cosh ( iϑ ) + A4 sinh ( iϑ )

[1.95]

In view of the boundary condition equations [1.83] and [1.84], the load– frequency relationship can be obtained from the following characteristic equation: 1 0 1 0 0 sin φ 0 sinh ϑ =0 cos(nφ ) ( cos φ − 1) sin(nφ ) ( cos φ − 1) cosh(nϑ ) ( cosh ϑ − 1) sinh(nϑ ) ( cosh ϑ − 1) F1 F2 F3 F4

[1.96]

On Stability of Discrete and Asymptotically Continuous Systems

35

where  Ω  F1 =  4 − 2(cos φ − 1) 2  cos(nφ ) − [ 2(cos φ − 1)] sin(nφ )sin φ  2n   Ω  F2 =  4 − 2(cos φ − 1)2  sin(nφ ) + [ 2(cos φ − 1)] cos(nφ )sin φ  2n  F3 = ( 4 − 2cosh ϑ ) cosh(nϑ ) cosh ϑ − ( 2 − 2cosh ϑ ) sinh(nϑ )sinh ϑ Ω   −  2 − 4  cosh(nϑ ) n  2  F4 = ( 4 − 2cosh ϑ ) sinh( nϑ ) cosh ϑ − ( 2 − 2cosh ϑ ) cosh(nϑ )sinh ϑ Ω  −2− 4 2n 

  sinh(nϑ ) 

,n disc ,n The flutter load, β disc flut , and flutter frequency, Ω flut , can be computed

from equation [1.96] for a given number of cells n. The roots of equation [1.96] for zero load β = 0 yields the natural vibration frequencies of the microstructured model. The flutter load for the two-degree-of-freedom Ziegler-type system, n=2, ,2 is β disc = 8. This value can be numerically obtained from the load– flut frequency relationship equation [1.96], or analytically obtained, as detailed in the first part of this chapter – see equation [1.40]. A closed-form equation ,3 of the flutter load can also be obtained for n=3, leading to β disc ≈ 12.4023, flut as also shown in the first part of this chapter – see equation [1.48]. As shown in Figure 1.12, the non-dimensional flutter load parameter increases with the number of cells n. This property is valid for both conservative and nonconservative loadings. For larger values of n, the flutter load and the flutter frequency parameters can be obtained numerically, by using the shooting method. Figure 1.11 shows the load–frequency curve, f ( β , Ω) = 0, for different values of n cells in the domain of β ∈ [0, 25], Ω ∈ [0, 25]. The flutter load and ,4 ,4 flutter frequency parameters are: β disc = 15.0834 and Ω disc flut flut = 9.011, ,4 respectively, for n=4. The first natural frequency is Ω disc free = 3.342.

36

Stability of Discrete Non-conservative Systems

The same curve is plotted for n=10 cells. The flutter load and flutter ,10 frequency parameters in this case are: β disc = 19.1175 and flut ,10 ,10 Ω disc = 10.653, respectively. The first natural frequency is Ωdisc = 3.487. flut free

The case of n=100 cells almost corresponds to the so-called local Beck’s column (without scale effects). The flutter load and flutter frequency ,100 ,100 parameters in this case are: β disc = 20.0413 and Ω disc = 11.012, flut flut ,100 = 3.516. All of these respectively. The first natural frequency is Ω disc free

values tend to Beck’s solution as n increases. As classically observed for these lattice-type systems (see Challamel et al. 2014b), the microstructure effect tends to soften the lattice system when compared to the local continuous one. This softening effect is also confirmed here for the flutter load dependency on the number of cells n.

Figure 1.11. Comparison of load–frequency curves of the microstructured model for different values of n, n=4 (cyan dashed line), n=10 (blue dash–dot line) and n=100 (red solid line). The flutter load–flutter frequency pair is denoted by a box. The horizontal dashed line shows the flutter load level of the local continuum Beck = 20.051 ). For a color version of this figure, see ww.iste.co. (Beck’s solution; βflut uk/lerbet/stability.zip

On Stability of Discrete and Asymptotically Continuous Systems

37

Figure 1.12. Flutter load of the discrete model versus the number of cells n. Beck = 20.051 . The dashed line shows the flutter load level of Beck’s column; βflut For a color version of this figure, see ww.iste.co.uk/lerbet/stability.zip

1.4. Stability under kinematic constraints

For undamped discrete systems, the linearized equations of motion can be formulated from the second-order differential equation:

 + K X = 0 MX

[1.97]

In this equation, the stiffness matrix is not necessarily symmetric, whereas the mass matrix is generally symmetric and definite positive. In a conservative system, the matrix K is symmetric. Stability (in the sense of Lyapunov) of the equilibrium can be investigated by means of the Lagrange– Dirichlet criterion. The positive definitiveness of the stiffness matrix K is easily checked from Sylvester’s criterion (La Salle and Lefschetz 1961). The loss of positive definiteness is reached when the determinant of one of the submatrices of Sylvester’s criterion vanishes. In the general case (conservative or non-conservative systems), the stability domain can be checked from the application of the Routh–Hurwitz criteria. In case of divergence instabilities, the boundary between stability and instability is

38

Stability of Discrete Non-conservative Systems

generally given by the singularity condition (see, for instance, Gajewski and Zyczkowski (1988) or Leipholz (1987)):

det ( K ) = 0

[1.98]

In 1958, Hill defined a stability criterion for elastoplastic systems which obey the normality rule for the evolution of the plastic strain rate. This criterion restricted to the present elastic eventually non-conservative discrete systems would be formulated as:

 , F . X  ≥ 0 with F = K X ∀X

[1.99]

This criterion can be equivalently written:

∀ X , XT K X = XT K S X ≥ 0

[1.100]

where K S is the symmetric part of K. The new stability criterion, when applied to non-conservative systems then writes:

( )

det K S = 0

[1.101]

The justification of the stability concept induced by equation [1.101] related to the Lyapunov stability of constrained systems, at least for nonconservative systems, is elaborated in Challamel et al. (2010). Equation [1.101] can also be interpreted as the lower bound of the divergence stability domain for the free system (Huseyin and Leipholz 1973). Challamel et al. (2010) proved that this so-called second-order work criterion formulated for non-conservative systems is the divergence stability criterion for any kinematically constrained system with one kinematic constraint. The result has been generalized by Lerbet et al. (2012) for n kinematic constraints and appears to be identical to the one induced in the presence of one kinematic constraint. This last result shows that the second-order work criterion has a wide spectrum of application. Challamel et al. (2010) applied this criterion for a two-degree-of-freedom Ziegler model with partial follower load. Lerbet et al. (2012) applied this criterion to a three-degree-of-freedom Ziegler system under follower load. Lerbet et al. (2013) reconsidered the Ziegler system and showed the possible destabilization effect of additional constraints. Lerbet et al. (2017) generalized the results for continuous systems including the analysis of the constrained Beck’s column.

On Stability of Discrete and Asymptotically Continuous Systems

39

We reproduce the proof detailed in Challamel et al. (2010) in this chapter. We would like to investigate the properties of such a dynamical system in the presence of an additional kinematics constraint, given by the holonomic constraint:

CT X = 0

[1.102]

This constraint does not affect the trivial equilibrium position X = 0. The Lagrange multiplier λ can be introduced for the constrained system as:

 + K X + λC = 0 MX

[1.103]

The new dynamical constrained system can be characterized by a system of dimension n + 1 :

M 0 K  ( C ) =   ( C ) =   Y  + K  ( C ) Y = 0 with M M , K T T 0 0 C  X Y=  λ 

C  and 0 [1.104]

Assuming that no flutter instabilities prevail in this constrained system, the loss of stability (divergence instability) is given by the singularity condition dealing with the matrix of dimension n + 1 :

(

)

 (C) = 0 det K

[1.105]

Therefore, the stability criterion of the constrained system is naturally affected by the choice of the given structural parameter C. According to the implicit function theorem, the stability criterion given by equation [1.105] locally gives the buckling load p as a function of the kinematics parameters parameterized by the vector C: K ( p) C CT

0



p = p (C)

[1.106]

40

Stability of Discrete Non-conservative Systems

For det ( K ) ≠ 0 by Schur’s determinant identity (Schur 1917), we get

(

(

)

)

 ( C ) = det K det −CT K −1C = −CT K −1C det K det K

[1.107]

which yields an expression for the divergence boundary

CT K −1C = 0

[1.108]

The minimization of p with respect to C can then be computed directly from the implicit function via:

(

)

 (C) ∂ det K ∂p =0  =0 ∂C ∂C

[1.109]

Such a derivative can be analytically achieved from the expansion of the stability criterion detailed in equation [1.107]:

(

 (C) ∂ det K ∂C

) = − det K ∂ C K T

∂C

C 

−1

( )

= −2det K  K −1 

S

C = 0 

[1.110]

 det K S = 0 Note that det K ≠ 0 was specifically assumed at the beginning of the reasoning. Equation [1.110] is based on the remarkable property:

( )

det K −1

S

=

det K S

( det K )2

[1.111]

Therefore, the optimization process of the buckling load with respect to the kinematic constraints leads to the second-order work criterion given by equation [1.101], assuming that only divergence instabilities are studied. In other words, when considering the minimization problem, the second-order work criterion is associated with the lower bound of the critical load of all the constrained systems. The critical constraint C* (related to a critical instability direction) has some particular features that we will investigate.

(K ) −1

S

C* = 0

[1.112]

On Stability of Discrete and Asymptotically Continuous Systems

41

In the next section, we apply the described procedure to the generalized Ziegler’s pendulum with partial follower force and investigate its critical divergence load as a function of kinematics constraints. The Ziegler column, loaded by a partial follower load (Hermann and Bungay 1964, Leipholz 1987) can be considered as an interesting structural system, because instability by divergence and instability by flutter may both appear, depending on the structural parameters. This undamped structural system is sometimes called a generalized Ziegler column. It is a pinned column with a sub-tangential or super-tangential buckling load P (sub-tangential for γ < 1 , super-tangential for γ > 1 ). This is a two-degreeof-freedom system with a state vector ΘT = (θ1 ,θ 2 ) , where θi is the rotation in each spring (see Figure 1.13). The equilibrium position is given by (θ1 ,θ 2 ) = ( 0,0 ) . The stiffness of each spring is denoted by C.

Figure 1.13. Ziegler’s model under partial follower load-free system

The elastic potential V of this system can be written as: 1 1 2 V = Cθ12 + C (θ 2 − θ1 ) 2 2

[1.113]

42

Stability of Discrete Non-conservative Systems

The virtual work of external forces is given by:  u = 2a − a cos θ1 − a cos θ 2 [1.114]  v = a sin θ1 + a sin θ 2

δ W = P cos ( γθ 2 ) δ u − P sin ( γθ 2 ) δ v with 

This variation can also be presented in the condensed form:

δ W = Pa sin (θ1 − γθ 2 ) δθ1 + Pa sin (θ 2 − γθ 2 ) δθ 2

[1.115]

Clearly, this system is a conservative system only when γ = 0. The stiffness matrix K is obtained from the linearized equations around the equilibrium position (θ1 ,θ 2 ) = ( 0,0 ) :  θ1   θ2 

δ V − δ W = ΘT K δ Θ with Θ = 

[1.116]

The stiffness matrix is then written as (see also Leipholz (1987)):

−C + Paγ  γ p −1   2C − Pa 2− p K =  or K = C   with C − Pa (1 − γ )   −C  −1 1 − (1 − γ ) p  Pa [1.117] p= C where p is the loading parameter and γ is the parameter that characterizes the orientation of the follower load (see Figure 1.13). γ = 0 corresponds to the conservative case and γ = 1 to the academic case of Ziegler’s column. The mass matrix is given, for instance, by Leipholz (1987):  3 1 M = ma 2    1 1

[1.118]

The dynamics stability of this non-conservative system is treated by Hermann and Bungay (1964), including flutter instabilities. The flutter load is calculated as:

p=

8−γ ±

(8 − γ )2 − 41(1 + (1 − γ )2 ) 2 2 1 + (1 − γ )   

[1.119]

On Stability of Discrete and Asymptotically Continuous Systems

43

The “static” criterion can be applied to characterize the instability boundary by divergence of the free system (equation [1.98]): det ( K ) = 0 ⇔

(1 − γ ) p 2 − 3 p (1 − γ ) + 1 = 0

[1.120]

The linearized constraints are written for the structural model considered in this part as:

c1θ1 + c2θ2 = 0

[1.121]

The physical meaning of this kinematic constraint can be discussed. Let us consider that point Q (parameterized by the distance l from the second spring) moves along a vertical axis. Figure 1.14 shows a basic example of such a constraint. The constraint on the lateral displacement of point Q can be expressed by:

l sin θ1 + a sin θ2 = 0

[1.122]

leading to the constrained linearized kinematics equation:

c1θ1 + c2θ2 = 0 with

c2 l = c1 a

Figure 1.14. Ziegler’s model under partial follower load-free system with kinematics constraint

[1.123]

44

Stability of Discrete Non-conservative Systems

The instability boundary of the constrained system necessarily arises by divergence (single-degree-of-freedom system), leading to the instability load, obtained from equation [1.105]:

(

)

 (C) = 0 ⇔ det K

p ( c1 , c2 ) =

c12 + 2c1c2 + 2c2 2 c12 (1 − γ ) + γ c1c2 + c2 2

[1.124]

In Figure 1.15, the divergence boundary of the free system is plotted and compared to the instability boundary of the constrained system.

Figure 1.15. Stability domain of the free Ziegler model loaded by a partial follower load; destabilizing effect of additional constraint

It is observed that there is a region γ ∈ [1 / 2;3 / 4] where the system with

an infinite stiffness for one spring, leading to θ1 = θ2 , has a lower instability load than the initial one, even in the divergence transition area γ ∈ [1 / 2;5 / 9] . For this non-conservative system, an increase in stiffness may destabilize the system, even if only divergence instabilities are considered. This is typically a particular feature of the non-conservative

On Stability of Discrete and Asymptotically Continuous Systems

45

nature of the system. Indeed, in case of conservative systems, an increase of stiffness generally leads to an increase of the buckling load (for the natural vibrations, see Rayleigh’s theorem – Tarnai 2004). Note that this property, even in case of conservative systems, is no more guaranteed if the equilibrium position depends on the loading range. We keep in mind that in some cases an increase in stiffness in a structure may also decrease the buckling load, even for conservative systems (Parnes 1977, Tarnai 1980, Panovko and Sorokin 1993). In other words, a particular kinematics constraint can stabilize or destabilize a non-conservative system (we give some specific examples in this chapter). The second-order work criterion equation [1.101] leads to the lower bound of the parameterized instability load:

 γ2  det K S = 0 ⇔ 1 − γ −  p 2 − 3 p (1 − γ ) + 1 = 0 4  

( )

[1.125]

Application of equation [1.109] to this two-degree-of-freedom system leads to the coupled system: K11 ∂ K 21 ∂c1 c1

K12 K 22 c2

c1 K11 ∂ c2 = 0 and K 21 ∂c2 0 c1

K12 K 22 c2

c1 c2 = 0 0

[1.126]

It is easy to develop these two systems in a linear system of two equations with two unknowns ( c1 , c2 ) :

 − ( K12 + K 21 ) c2 + 2 K 22 c1 = 0   2 K11c2 − ( K12 + K 21 ) c1 = 0

[1.127]

Obviously, equation [1.101] is found again from equation [1.127], and more specifically the second-order work criterion for a two-dimensional system: 2

 K + K 21  K11 K 22 −  12 = 0 ⇔ det K S = 0  2  

( )

[1.128]

46

Stability of Discrete Non-conservative Systems

For the present problem, the critical parameter C* is then computed from equation [1.127] as: c 2 + 2c1c2 + 2c2 2 c2 γ p−2 with p ( c1 , c2 ) = 2 1 = c1 2 ( 2 − p ) c1 (1 − γ ) + γ c1c2 + c2 2

[1.129]

This equation is equivalent to the second-order polynomial equation for the critical set of parameters: 2

c  c 2 ( γ − 1)  2  + 2 (1 − 2γ ) 2 + ( 2 − 3γ ) = 0 c1  c1 

[1.130]

whose solution is given for the minimization problem by: *

2γ − 1 +  c2    =  c1 

( 2γ − 1)2 − ( 2γ − 2 )( 2 − 3γ ) ≤0 2 ( γ − 1)

[1.131]

The other solution leads to the maximization problem: *

2γ − 1 −  c2    =  c1 

( 2γ − 1)2 − ( 2γ − 2 )( 2 − 3γ ) ≥0 2 ( γ − 1)

[1.132]

The critical parameter C* depends on the non-conservative structural parameter γ for this problem. This result shows the key role played by critical kinematics constraint and the possible destabilizing phenomenon for non-conservative systems. For conservative problems

(γ = 0 ) ,

these critical parameters are

calculated as: *

*

 c2  c  1− 5 1+ 5 ≈ −0.618 and  2  = ≈ 1.618   = 2 2  c1 min  c1 max

[1.133]

On Stability of Discrete and Asymptotically Continuous Systems

47

Note of course that these values are the exact ones of the buckling modes of the free system (without additional constraints). This optimization process is clearly shown in Figure 1.16 for the conservative case. The second-order work criterion corresponds to the minimization and the maximization problem. Figure 1.16 shows that the stability boundary of the constrained system is above the one of the free system, and is tangent to the boundary of the free system at the critical point corresponding to the minimization problem. It has to be outlined that an additional constraint cannot destabilize the system for the conservative problem. This property is not true for the non-conservative problem, where an additional constraint can destabilize the structural system (see Figure 1.17). There is clearly a destabilizing zone around the critical point (associated with the second-order work criterion) where the boundary of the constrained system is below the one of the free system.

p

3

2.5

2

~ det( K ( C ))=0

1.5

det( K )=0 1

0.5

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

c2 c1

2

Figure 1.16. Effect of an additional constraint on the buckling load; conservative system γ = 0; additional constraint cannot destabilize the conservative system

48

Stability of Discrete Non-conservative Systems

3

p 2.5

~ det( K ( C ))=0

2

det( K )=0

1.5

1

0.5

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

c2 c1

2

Figure 1.17. Effect of an additional constraint on the buckling load; non-conservative system γ = 0.5; additional constraint can destabilize the non-conservative system

For Ziegler’s column under follower load (γ = 1), the second-order work criterion gives a dimensionless divergence stability value p = 2 for any kinematic constraints (see, for instance, Challamel et al. 2010 – equation [1.125] with γ = 1). This value has also been found by Absi and Lerbet (2004) and Ingerle (2013) from direct application of equation [1.199] (see also Ingerle 2018). For the continuous Beck’s column, the dimensionless divergence stability value for any kinematic constraints has been calculated by Lerbet et al. (2017) and is equal to β = π2. This value has been empirically obtained by Ingerle (2013) from a discrete approach using numerical arguments (from a Hencky-type system). It is worth mentioning that the stability limit under kinematic constraints is the generalization of the one under some specific constraint especially applied to the boundaries of the system. For instance, Ingerle (1969) found a dimensionless divergence buckling load β = 20.19 in the presence of a specific stability constraint applied to the end of the column (point of application of the follower load), whereas the free Beck column has a flutter instability value of 20.05, as calculated by Beck (1952) (see also El Naschie 1976, 1977 for this result). However, considering any kinematic constraints reduces this value to β = π2 (as observed by Ingerle 2013, and as proven by Lerbet et al. 2017).

On Stability of Discrete and Asymptotically Continuous Systems

49

As a conclusion of this part, it is now widely accepted that the secondorder work criterion, also called Hill’s criterion, has a strong basis related to Lyapunov stability, in terms of kinematically constrained systems. In order to deeply investigate the paradoxical destabilization phenomenon of additional kinematics constraints in non-conservative systems, the concept of kinematic structural stability (KISS) has been recently introduced. The possible destabilization of non-conservative systems by additional constraints has been studied and illustrated from Ziegler’s discrete model or Beck’s continuous column. It is worth mentioning that the possible destabilization of the flutter-type system by additional constraint was experimentally noted by Benjamin (1961) and then specifically studied by Thompson (1981) for structural problems in the presence of fluid flow. For the fluid–structure interaction problem investigated by Benjamin (1961) and Thompson (1981), the additional kinematic constraint changes the instability mode from flutter to divergence. In this chapter, we explored a similar phenomenon where additional constraints may destabilize a nonconservative system, which loses its stability by divergence without kinematic constraints. 1.5. Conclusion

This chapter presented some stability results for discrete columns composed of a finite number n of rigid links, which asymptotically converge towards the continuous elastica problem when n tends towards infinite. The strong link between the difference equations of the discrete column (or lattice column) and the finite difference formulation of the asymptotic continuous column are also shown. Both conservative and non-conservative loadings are considered. Divergence or flutter loads are calculated for both discrete models for various boundary conditions, including the clamped-free boundary condition. The effect of additional kinematic constraint is also studied for such discrete structural elements. The destabilizing phenomenon by additional constraint is explained for the non-conservative systems, a property already commented on in the past by Benjamin et al. (1961) and by Thompson (1982) for some fluid–structure interaction problems. This destabilizing effect of additional constraints may also be explained by some singular geometrical arguments as developed by Kirillov et al. (2014). Some similar mathematical arguments may be used to explain the destabilizing effect of additional damping for non-conservative systems (see, for instance,

50

Stability of Discrete Non-conservative Systems

Ziegler 1952, Bottema 1955, Bottema 1956, Hermann and Jong 1965, Seyranian and Mailybaev 2003, Kirillov and Seyranian 2005). 1.6. References Absi, E. and Lerbet, J. (2004). Instability of elastic bodies. Mech. Res. Comm., 31(1), 39–44. Alfutov, N.A. (2000). Stability of Elastic Structures. Springer-Verlag, Berlin. Awrejcewicz, J. and Lamarque, C.H. (2003). Bifurcations and Chaos in Nonsmooth Mechanical Systems. World Scientific, Singapore. Bazant, Z.P. and Cedolin, L. (2003). Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. Dover Publications, New York. Beck, M. (1952). Die Knicklast des einseitig eingespannten tangential gedrückten Stabes. Z. Angew. Math. Phys., 3, 225–228. Benjamin, T.B. (1961). Dynamics of a system of articulated pipes conveying fluid I Theory II – Experiments. Proc. Royal Soc. A., 261, 457–486. Bolotin, V.V. (1963). Nonconservative Problems of the Theory of Elastic Stability. New York, Pergamon Press. Bottema, O. (1955). On the stability of the equilibrium of a linear mechanical system. Journal of Appl. Math. Phys. (ZAMP), 6, 97–104. Bottema, O. (1956). The Routh-Hurwitz condition for the biquadratic equation. Indagationes Mathematicae, 18, 403–406. Carr, J. and Malhardeen, Z.M. (1979). Beck’s problem. SIAM J. Appl. Math., 37, 261–262. Challamel, N., Nicot, F., Lerbet, J. and Darve, F. (2010). Stability of nonconservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. Challamel, N., Lerbet, J., Wang, C.M. and Zhang, Z. (2014a). Analytical length scale calibration of nonlocal continuum from a microstructured buckling model, Z. Angew. Math. Mech., 94(5), 402–413. Challamel, N., Wang, C.M. and Elishakoff, I. (2014b). Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis. Eur. J. Mech. A/Solids, 44, 125–135.

On Stability of Discrete and Asymptotically Continuous Systems

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Challamel, N., Kocsis, A. and Wang, C.M. (2015a). Discrete and nonlocal elastica. Int. J. Non-linear Mech., 77, 128–140. Challamel, N., Picandet, V., Collet, B., Michelitsch, T., Elishakoff, I. and Wang, C.M. (2015b). Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua. Eur. J. Mech. A/Solids, 53, 107– 120. Challamel, N., Kocsis, A., Wang, C.M. and Lerbet, J. (2016). From Ziegler to Beck’s column: A nonlocal approach. Archive of Applied Mechanics, 86(6), 1095–1118. Como, M. and Grimaldi, A. (1995). Theory of Stability of Continuous Elastic Structures. CRC Press, Boca Raton, FL. Dirichlet, G.L. (1846). Ueber die Stabilität des Gleichgewichts. J. Reine Angew. Math., 32, 85–88. Domokos, G. (1993). Qualitative convergence in the discrete approximation of the Euler problem. Mech. Struct. Mach., 21(4), 529–543. Domokos, G. and Holmes, P. (1993). Euler’s problem, Euler’s method, and the standard map; or, the discrete charm of buckling. J. Nonlinear Science, 3, 109–151. El Naschie, M.S. (1976). Post-critical behaviour of the Beck problem. J. Sound Vibration, 48(3), 341–344. El Naschie, M.S. (1977). Some remarks on the Beck problem. AIAA Journal, 15, 1200–1201. El Naschie, M.S. and Al-Athel, S. (1979a). Remarks on the stability of flexible rods under follower forces. J. Sound Vibration, 64, 462–465. El Naschie, M.S. and Al-Athel, S. (1979b). On certain finite-element like methods for non-conservative sets. Solid Mechanics Archives, 4(3), 173–182. El Naschie, M.S., Wu, C.W. and Wifi, A.S. (1988). A simple discrete element method for the initial postbuckling of elastic structures. Int. J. Num. Meth. Eng. 26, 2049–2060. Elaydi, S. (2005). An Introduction to Difference Equations. Springer, Berlin. Elishakoff, I. (2005). Controversy associated with the so-called “follower forces”: Critical overview. Applied Mechanics Reviews, 58(1–6), 117–142.

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Euler, L. (1744). Methodus inveniendi lineas curvas maxima minimive proprietate gaudentes Additamentum I, De Curvis Elasticis, Lausanne and Geneva. Gantmacher, F. (1975). Lectures in Analytical Mechanics. MIR Publishers, Moscow. Gáspár, Z. and Domokos, G. (1989). Global investigation of discrete models of the Euler buckling problem. Acta Technica Acad. Sci. Hung., 102, 227–238. Gasparini, A.M., Saetta, A.V. and Vitaliani, R.V. (1995). On the stability and instability regions of non-conservative continuous system under partially follower forces. Computer Methods in Applied Mechanics and Engineering, 124, 63–78. Goldberg, S. (1958). Introduction to Difference Equations with Illustrative Examples From Economics, Psychology and Sociology. Dover Publications, New York. Green, A.E. and Naghdi, P.M. (1971). On thermodynamics, rate of work and energy. Arch. Rat. Mech. Anal., 40, 37–49. Hermann, G. and Bungay, R.W. (1964). On the stability of elastic systems subjected to nonconservative forces. J. Appl. Mech., 86, 435–440. Herrmann, G. and Jong, I.C. (1965). On the destabilizing effect of damping in nonconservative elastic systems. ASME J. Appl. Mech., 32(3), 592–597. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids, 6, 236–249. Huseyin, K. (1975). Nonlinear Theory of Elastic Stability. Noordhoff International Publishing, Leiden. Huseyin, K. and Leipholz, H. (1973). Divergence instability of multiple-parameter circulatory systems. Quarterly Applied Mathematics, July, 185–197. Ingerle, K. (1969). Die Verwendung des statischen Stabilitätskriteriums für Probleme der Elastostatik bei Fehlennicht-trivialer Gleichgewichtslagen. Forschung im Ingenieurwesen, 35(6), 196–197. Ingerle, K. (2013). Stability of massless non-conservative elastic systems. J. Sound Vibration, 4529–4540. Ingerle, K. (2018). Non-conservative Systems – New Static and Dynamic Stability Criteria, CRC Press, Boca Raton.

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Kirillov, O.N. and Seyranian, A.P. (2005). Stabilization and destabilization of a circulatory system by small velocity-dependent forces. J. Sound Vib., 283(3–5), 781–800. Kirillov, O.N., Challamel, N., Darve, F., Lerbet, J., Nicot, F. and Prunier, F. (2014). Singular divergence instability thresholds of kinematically constrained circulatory systems. Physics Letters A, 378, 147–152. Knops, R.J. (1982). Instability and the Ill-posed Cauchy Problem in Elasticity. In Mechanics of Solids, Hopkins, H.G. and Sewell, M.J. (eds). The Rodney Hill 60th Anniversary Volume, Pergamon Press, New York. Knops, R.J. and Wilkes, E.W. (1973). Theory of Elastic Stability. In Handbuch der Physik, von Flügge, H.S. (ed.). Springer-Verlag, Berlin. Kocsis, A. (2013). An equilibrium method for the global computation of critical configurations of elastic linkages. Computer and Structures, 121, 50–63. Kocsis, A. and Challamel, N. (2016). On the post-buckling of distributed microstructured system: The finite element elastica. Int. J. Mech. Sc., 114, 12– 20. Kocsis, A. and Challamel, N. (2018). On the foundation of a generalized nonlocal extensible shear beam model from discrete interactions. In Generalized Models and Non-classical Approaches in Complex Materials, Altenbach, H., Pouget, J., Rousseau, M., Collet, B. and Michelitsch, T. (eds). Springer, Berlin. Kocsis, A. and Károlyi, G. (2006). Conservative spatial chaos of buckled elastic linkages. Chaos, 16(033111), 1–7. Kuznetsov, Y.A. (2004). Elements of Applied Bifurcation Theory, 3rd Edition, Springer, Berlin. La Salle, J. and Lefschetz, S. (1961). Stability by Liapunov’s Direct Method with Applications. Academic Press, New York. Leckie, F.A. and Lindberg, G.M. (1963). The effect of lumped parameters on beam frequencies. Aeronaut. Quart., 14(234), 224–240. Leipholz, H. (1962). Die Knicklast des einseitig eingespannten Stabes mit gleichmässig verteilter, tangentialer Längsbelastung. Z. Angew. Math. Mech., 13, 581–589. Leipholz, H. (1970). Stability Theory, Academic Press, London.

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Lerbet, J., Aldowaji, M., Challamel, N., Nicot, F., Prunier, F. and Darve, F. (2012). P-positive definite matrices and stability of nonconservative systems. Zeitschrift für Angewandte Mathematik und Mechanik, 92(5), 409–422. Lerbet, J., Kirillov, O., Aldowaji, M., Challamel, N., Nicot, F. and Darve, F. (2013). Additional constraints may soften a non-conservative structural system: Buckling and vibration analysis. Int. J. Solids Structures, 50(2), 363–370. Lerbet, J., Challamel, N., Nicot, F. and Darve, F. (2017). On the stability of nonconservative continuous systems under kinematic constraints. Zeitschrift für Angewandte Mathematik und Mechanik, 97(9), 1100–1119. Love, A.E. (1927). A Treatise on the Mathematical Theory of Elasticity, Dover Publications, 4th edition, New York. Luongo, A. and D’Annibale, F. (2014). On the destabilizing effect of damping on discrete and continuous circulatory systems. J. Sound Vibration, 333(24), 6723– 6741. Lyapunov, A.M. (1907). Problème général de la stabilité des mouvements. Annales de la Faculté des Sciences de Toulouse, 9, 203–274. Lyapunov, A.M. (1949). Problème général de la stabilité du mouvement. Annales de la Faculté des sciences de Toulouse : Mathématiques, Princeton University Press, NJ. Maugin, G.A. (1999). Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford. Nguyen, Q.S. (1995). Stabilité des structures élastiques, Springer-Verlag, Berlin. Nguyen, Q.S. (2000). Stabilité et mécanique non linéaire. Hermes, Paris. Panovko, Y.G. and Sorokin, S.V. (1993). Paradoxical influence of an increase of rigidity on buckling loads and natural frequencies for elastic systems. Mech. Res. Comm., 20(1), 9–14. Parnes, R. (1977). A paradoxical case in a stability analysis. American Institute of Aeronautics and Astronautics Journal, 15, 1533–1534. Schur, I. (1917). Potenzreihen, die im Innern des Einheitskreises Beschrankt sind [I], Journal fur die reine und angewandte Mathematik, 147, 205–232. Seide, P. (1975). Accuracy of some numerical methods for column buckling. J. Eng. Mech., 101(5), 549–560. Seyranian, A.P. and Mailybaev, A.A. (2003). Multiparameter Stability Theory With Mechanical Applications. World Scientific Publishing, Singapore.

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Silverman, I.K. (1951). Discussion on the paper of “Salvadori M.G., Numerical computation of buckling loads by finite differences, Trans. ASCE, 116, 590–636, 1951”, Trans. ASCE, 116, 625–626. Simitses, G.J. and Hodges, D.H. (2006). Fundamentals of Structural Stability. Elsevier, Burlington. Sugiyama, Y. and Kawagoe, H. (1975). Vibration and stability of elastic columns under the combined action of uniformly distributed vertical and tangential forces. J. Sound Vibration, 38(3), 341–355. Sugiyama, Y., Fujiwara, N. and Sekiya, T. (1971). Studies on nonconservative problems of instabilities of columns by means of analog computer [in Japanese]. Transactions of the Japan Society of Mechanical Engineers, 37(297), 931–940. Tarnai, T. (1980). Destabilizing effect of additional restraint on elastic bar structures. Int. J. Mech. Sc., 22, 379–390. Tarnai, T. (1995). The Southwell and the Dunkerley theorems. In Summation Theorems in Structural Stability, Tarnai, T. (ed.). Springer-Verlag, New York. Tarnai, T. (2004). Paradoxical behaviour of vibrating systems challenging Rayleigh’s theorem. 21st International Congress of Theoretical and Applied Mechanics, Warsaw. Thompson, J.M.T. (1982). Paradoxical mechanics under fluid flow. Nature, 5853(296), 135–137. Timoshenko, S.P. (1983). History of Strength of Material. Dover Publications, New York. Timoshenko, S.P. and Gere, J.M. (1961). Theory of Elastic Stability, 2nd edition. MacGraw Hill, New York. Vlassov, B.Z. (1962). Pièces longues en voiles minces. Eyrolles, Paris. Wang, C.T. (1951). Discussion on the paper of “Salvadori M.G., Numerical computation of buckling loads by finite differences, Trans. ASCE, 116, 590–636, 1951”, Trans. ASCE, 116, 629–631. Wang, C.T. (1953). Applied Elasticity, McGraw-Hill, New York. Wang, C.M., Wang, C.Y. and Reddy, J.N. (2005). Exact Solutions for Buckling of Structural Members, CRC Press, Boca Raton. Wang, C.M., Zhang, Z., Challamel, N. and Duan, W.H. (2013). Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model. J. Phys. D: Appl. Phys., 46, 345501.

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Wang, C.M., Zhang, H., Challamel, N., and Xiang, Y. (2016). Buckling of nonlocal columns with allowance for selfweight. J. Eng. Mech., 04016037, 1–9. Wang, C.M., Zhang, H., Challamel, N. and Pan, W. (2019). Hencky-Bar-Chain/Net Models for Structural Analysis. World Scientific, Singapore. Ziegler, H. (1952). Die Stabilitätskriterien der Elastomechanik. Ingenieur-Archiv, 20, 49–56. Ziegler, H. (1968). Principles of Structural Stability. Blaisdell Publishing Company, New York.

2 Second-order Work Criterion and Stability in the Small

2.1. Some generalities in continuum mechanics for material systems Failure, as discussed in the introduction, can be related to a transition (a bifurcation) from a quasi-static regime towards a dynamical regime. This chapter aims to translate this definition within the solid mechanics formalism, by investigating in which conditions the kinetic energy of a non-viscous material system, in equilibrium at a given time, may increase. For this purpose, a system made up of a volume Vo of a given material initially in a configuration Co (reference configuration) is considered. After a loading history, the system is in a strained configuration C and occupies a volume V , in equilibrium under a prescribed external loading (Figure 2.1). Each material point of the volume Vo is transformed into a material point of the volume V . During this transformation, the matter is likely to undergo a rigid body motion, together with a pure strain composed of stretch and spin deformations. If large strains take place, both the initial Co and current C configurations cannot be confused. It will be assumed hereafter that the physicochemical properties of the constituents of the system are not altered and that no matter is added to or removed from the system. In addition, only isothermal transformations will

58

Stability of Discrete Non-conservative Systems

be considered, so that the system can only exchange mechanical energy with the exterior. We introduce the transformation χ relating each material point x of the current configuration C to the corresponding material point X of the initial configuration Co . The continuity of the matter ensures that χ is bijective. Using the change in variable x = χ ( X ) , any field f ( x ) of the current positions x can be transformed into a field f ( X ) = f ( x ) of the initial

positions X . When no confusion is possible, the notation ~ is omitted.

Figure 2.1. Transformation of a material system. Initial and current (strained) configurations. All material points are displaced, inducing a deformation of any geometrical body like vectors, surfaces and volumes

As χ is bijective, the Jacobian J of the tangent linear transformation F , with Fij = ∂xi / ∂X j , is strictly positive. F is a function of the positions

Second-order Work Criterion

59

X. The displacement field u(X) of material points x between both initial and current configurations is defined by the relation x = χ ( X ) = X + u ( x ) = X + u ( X ) .

 Thus, Fij = δ ij + ∂ui / ∂X j , and Fij = ∂ui / ∂X j . Recall: Fundamentals in continuum mechanics Adopting a Eulerian description, the mechanical equilibrium of the volume V implies that, at any material point of the volume V, both displacements and stress tensor fields must fulfill the following governing equations:

∀M ∈ V

∂σ ij ∂x j

=0

(Cauchy equation)

∀ M ∈ Γσ

σ ij n j = fi Γσ

∀M ∈ Γ ε

ui = fi Γε

where σ is the Cauchy stress tensor expressed at each material point M of coordinates xi with respect to a suitable frame. Moreover, the strain rate tensor D is defined as follows:

Dij =

1  ∂ui ∂u j  + 2  ∂x j ∂xi  

Both the strain rate tensor D and a suitable objective time derivative σ of the Cauchy stress σ are related through a constitutive relation that models the rateindependent constitutive behavior of the materials. Governing and constitutive equations can also be expressed through a Lagrangian formalism.

60

Stability of Discrete Non-conservative Systems

2.2. The link between the kinetic energy and the second-order work

The kinetic energy of the system, in the equilibrium configuration C at time t , is given by the following energy conservation equation expressed under rate form:

∂ ( ui ) E c ( t ) = σ ij n j ui dS − σ ij dV x ∂ j Γ V





[2.1]

where E c represents the kinetic energy rate of the system. It is convenient to express the integrals in equation [2.1] with respect to the initial configuration by using the change in variables x = χ (X ) . Recalling that dV = J dVo , and

( )

using Nanson’s formula, n dS = J F −1

t

N dSo , which relates the current

surface vector n dS to the corresponding surface vector N dS o in the initial configuration, Equation [2.1] is written as: E c ( t ) =

 J σ ( χ ( X )) F ij

−1 kj

Γo

− σ ij ( χ ( X ) )



Vo

N k ui ( χ ( X ) ) dSo

∂ui ( χ ( X ) ) ∂X k J dVo ∂X k ∂x j

[2.2]

which yields:

E c ( t ) =





Π ij N j ui dSo − Π ij

Γo

Vo

( ) dV

∂ ui

∂X j

o

[2.3]

with:

( )

Π = J σ F -1

t

[2.4]

where Π denotes the Piola–Kirchhoff stress tensor of the first kind and Γ o is the boundary of Vo . As Π is built from the current forces acting in the

Second-order Work Criterion

61

current configuration C with respect to the initial configuration Co , this is a semi-Lagrangian tensor. It is worth noting that this tensor can be related to Hill’s nominal tensor s H , since Π = s H t . Recall: Derivation of Nanson’s formula Let dA = N dS o and da = n dS , and let us consider any vector dL in the reference configuration, transformed in vector dl in the current configuration. We have dVo = dA ⋅ dL and dV = da ⋅ dl , with dl = F dL . Thus, as dV = J dVo :

da ⋅ dL = Ft da ⋅ dL = J dA ⋅ dL This relation must be verified for any vector dL , which in turn imposes that:

Ft da = J dA Or

( )

n dS = J F −1

t

N dSo

Thus, Nanson’s formula is established.

The advantage of the formulation given in equation [2.3] is that all integrals are given with respect to a fixed domain. Thus, differentiation of equation [2.3] simply gives: Ec ( t ) =

 Π

ij

Γo

( )

N j ui dSo +

Π

ij

N j ui dSo

Γo

( )

~ ~   ij ∂ ui dVo − Π ij ∂ ui dVo − Π  ∂X j ∂X j Vo Vo

[2.5]

62

Stability of Discrete Non-conservative Systems

Recall: The Green formula For any tensors A and B, the following transformation equation (the Green formula) holds:

 Aij

D

∂ (Aij ) ∂(bi ) dV =  Aij bi n j dS −  bi dV ∂x j ∂x j ∂D D

where n is the current outward normal to the boundary ∂D of the domain of interest D . If A does not depend on time, then the above relation reduces in:

A

ij

D

∂ (bi ) dV =  Aij bi n j dS ∂x j ∂D

As this relation is verified for any tensor A , the following equation (the second Green formula) can finally be obtained:

∂ (bi ) dV =  bi n j dS ∂x j D ∂D



It is worth noting that both operators ∂. / ∂X j (spatial differentiation) and

δ . / δ t (particulate time derivative) commute in the Lagrangian description, which is not the case in the Eulerian description due to the existence of a convective term. Moreover, in essence, the time differentiation of the Piola–Kirchhoff stress tensor is objective. Taking advantage of the standard Green formula:

 Π ij

Vo

( )

∂ u~i  N dS − ∂ (Π ij ) u~  dV dVo =  Π ij u~ i j o i o  ∂X j ∂X j Γo Vo

Second-order Work Criterion

63

and recalling that ∀M ∈ Vo , ∂Π ij / ∂X j = 0 (Cauchy equation in Lagrangian formulation), it follows that:

Ec ( t ) =





 f j ui dSo − Π ij

Γo

Vo

( ) dV

∂ ui

∂X j

[2.6]

o

where f = Π n denotes the current stress distribution applied to the initial (reference) configuration. Furthermore, for any time increment Δt , the second-order Taylor expansion of the kinetic energy reads:

( Δt ) E t + o Δt 3 Ec ( t + Δt ) = Ec ( t ) + Δt E c ( t ) + ( ) ( ∀ Δt ) c( ) 2 2

[2.7]

As the system is in an equilibrium state at time t , E c ( t ) = 0 . Equation [2.7] therefore reads:

E ( t + Δt ) − Ec ( t ) Ec ( t ) = 2 c + o ( Δt ) ( Δt )2

[2.8]

Combining both equations [2.6] and [2.8] and ignoring o ( Δt ) terms finally yields:

Ec (t + Δt ) − Ec (t ) =

( ) dV 

 ∂ u~i  u~ dS − Π  f j i o V ij  ∂X j 2  Γo  o

(Δt )2 

 

o

 

[2.9]

Equation [2.9] explicitly introduces the so-called (internal) second-order work that is expressed through a semi-Lagrangian formalism as:

 F dV W2int = Π ij ij o



Vo

[2.10]

64

Stability of Discrete Non-conservative Systems

~

with Fij =

( )

∂( xi ) ∂ u~i = . ∂X j ∂X j

The terminology of internal second-order work is justified by the fact that expression [2.10] introduces internal variables. As a standard, the usual international terminology in the literature is “second-order work”.

 f

The first term

j

ui dSo of the right-hand side of equation [2.9]

Γo

involves both stresses and displacements acting on the boundary of the volume. This boundary second-order term is therefore an external secondorder work, and will be denoted thereafter W2ext . Thus, equation [2.9] indicates that the increase in the kinetic energy over the small time range [t , t + Δt ] equals the difference between the external second-order work W2ext and the (internal) second-order work W2int :

Ec ( t + Δt ) − Ec ( t )

2 Δt ) ( =

2

(W

ext 2

− W2int

)

[2.11]

In conclusion, the kinetic energy increases over the small time range [t , t + Δt ] if W2ext is larger than W2int . A meaningful situation corresponds to the case where W2ext is nil. The increase in kinetic energy is then related to the vanishing of the internal second-order work. The occurrence of such situations will be investigated in the following sections. Equation [2.11] holds true over a quasi-static transformation between two equilibrium states at times t and t + Δt . In that case, equation [2.11] reads W2ext − W2int = 0 , i.e.:

( ) dV

 ~   ∂ ui Π  ij  ∂X j Vo 

 

o

=

 f

Γo

j

u~i dSo =

 Π

∂Vo

ij

N j u~i dSo

[2.12]

Second-order Work Criterion

65

Beyond its nice symmetry, we can learn more from equation [2.12]. Indeed, a consequence of this equation is that the internal second-order work, along a quasi-static transformation, can be entirely determined from the boundary variables, without requiring any information on both strain and stress fields within the volume of interest. This is greatly advantageous, particularly for large-scale boundary value problems, where the boundary conditions are generally well known (often imposed), whereas both strain and stress fields may be difficult to evaluate. The investigation of laboratory tests will also benefit from this important property. 2.3. The second-order work in Eulerian formalism

The internal second-order work can also be expressed through a Eulerian formalism. Differentiation of equation [2.4] yields:

δ ~  ~t (Jσ ) = Π F + Π F~ t δt

[2.13]

Thus, by combining equations [2.4] and [2.13]:

( )

( )

t  =  J σ + J σ − Jσ F −1 F t  F −1 Π  

t

[2.14]

Finally, equation [2.10] can be expressed as:

W2int =

~ ~ ~ ~ ~ ~ ~  (J σ + J σ − Jσ (F ) F ) (F ) : F dV −1 t

t

−1 t

[2.15]

o

Vo

Recalling

that

(

)

for

(

any

)

two-order

tensors

A,

B

and

C,

A : ( B C) = A C : B = B A : C , thus equation [2.15] can be rewritten as: t

t

( )

W2int =  J σ + J σ − Jσ F −1  V



o

t

  F t  : F F −1 dVo 

[2.16]

66

Stability of Discrete Non-conservative Systems

It is worth noting that:

( )

∂ u~ i ∂ (u i ) ∂x k ~ ~ = = Lik Fkj Fij = ∂X j ∂x k ∂X j

[2.17]

which gives:  L = F F −1

[2.18]

where L = ∂ ( u ) / ∂x is the velocity gradient tensor, a function of current

positions x . Then, using the change in variables X = χ −1 ( x ) and recalling that dV = J dVo , the integral of equation [2.16] can be expressed with respect to the current configuration, leading to the Eulerian expression of the internal second-order work:

  J W2int =  σ + σ − σ Lt  : L dV J  V



[2.19]

In contrast to the Piola–Kirchhoff stress tensor, the time derivative σ = δ σ / δ t of the Cauchy stress tensor σ is not objective, in the sense that this derivative depends on the observer. What is intrinsic for the matter is not the stress evolution with respect to a fixed (Galilean) frame, but the stress evolution with respect to the matter itself, which may evolve too. A different objective time derivative of the Cauchy stress exists as, for instance, the  Zaremba–Jaumann formula, σ = σ + σ ω − ω σ , where ω = L − Lt / 2

(

)

denotes the spin rate tensor. Thus, equation [2.19] writes:  J   W2int = σ : L dV + σ :  L − ω L + L ω − L2  dV J  V V





[2.20]

Second-order Work Criterion

67

   Taking advantage of the symmetry of σ , σ : L = σ : Lt = σ : ( L + Lt ) / 2,

which yields:  J   W2int = σ : D dV + σ :  L − ω L + L ω − L2  dV J  V V





[2.21]

As a consequence, the internal second-order work can be expressed either  with a Eulerian formulation (equation [2.21]), where σ and D are linked by the constitutive relation g h , or with a semi-Lagrangian formulation

~  and F are linked by a constitutive relation Gh . (equation [2.10]), where Π

It is worth emphasizing that the internal second-order work Eulerain expression, as indicated in equation [2.21], is composed of two terms:

a

material



 σ : D dV ,

term

and

a

complementary

term

V

 σ : (J / J L − ω L + L ω − L ) dV 2

that takes geometry changes into

V

account. In many situations, especially when large strains take place, this last term cannot be neglected. Thus, the internal second-order work, in the Eulerian description, does not reduce into a single term



 σ : D dV .

V

2.4. The second-order work on the material point scale

The equations derived in this section, giving the expression of the internal second-order work in its relation with the kinetic energy, can be simplified when reduced to a single material point of volume dVo . In that case, integral notations vanish, and we readily obtain:

dEc ( t + Δt ) − dEc ( t )

2 Δt ) ( =

2

( Π

ij

N j ui dSo − dW2int

)

[2.22]

with dEc ( t ) = ρo ui ui dVo / 2, where ρ o is the initial density of the matter at the material point considered.

68

Stability of Discrete Non-conservative Systems

The internal second-order work for the material point of volume dVo is expressed in a Lagrangian formulation as:

( ) dV

 ~   ∂ ui dW2int = Π ij   ∂X j

 

o

[2.23]

In a Eulerian formulation, we obtain:

 J   dW2int = σ : D dVo + σ :  L − ω L + L ω − L2  dVo J 

[2.24]

When geometry changes can be omitted, equation [2.24] merely simplifies to:

 dW2int = σ : D dVo

[2.25]

2.5. Conclusion

In this chapter, basic concepts related to the second-order work were developed to highlight the role of kinetic energy in the process of failure in material systems. Failure can also be interpreted as a loss of constitutive uniqueness in a material response, leading to the classical concept of generalized limit states involving mixed stress–strain loading parameters. More interesting when such limit states are reached, the generalized stress (or strain) necessarily reaches an extremum. This physically translates into an imbalance between external loading and internal stresses, thereby resulting in an increase in kinetic energy. As a final conclusion, a material system becomes unstable (failure can develop) when the external second-order work, involving the external loading exerted on the system boundary, cannot be balanced by the internal second-order work involving the material properties of this system. This result has opened on a very active field in geomechanics, where failure can be interpreted from the spectral properties of the constitutive operator that depend on the loading conditions. Various engineering applications such as landslide analysis or geotechnical structure failure have been intensively developed.

3 Mixed Perturbations and Second-order Work Criterion

3.1. Introduction Structure stability is one of the key issues that deserve attention for both engineering and academic purposes. In particular, the stability of elastic structures has been abundantly investigated since the 1960s, extending to non-conservative loading (Bolotin, 1963; Ziegler, 1968; Nguyen, 2000). Inspired by the work in soil mechanics, the influence of additional kinematic constraints on the stability of non-conservative elastic systems constitutes a novel field that has been insufficiently considered until now (Challamel et al., 2009 and 2010). In soil mechanics, an equilibrium configuration of a given soil body subjected to a given external loading is stable if an incremental displacement field exists, associated with a strictly positive value of the second time derivative of the kinetic energy. Formally, as established in Chapter 2, the following basic equation holds: Ec = W2ext − W2int , where W2ext is the external second-order work term involving both incremental displacements and incremental external forces applied to the boundary of the system, and W2int is referred to as the secondorder work involving both incremental stress and strain fields acting within the soil body (Hill, 1958). Existence of stress–strain incremental fields that ensure that the quantity − W2int is strictly positive has been broadly discussed in soil mechanics, especially when the boundary term is nil. This has shed light on the basic role played by the second-order work and leads to an investigation of the

W2ext

70

Stability of Discrete Non-conservative Systems

vanishing of the second-order work (see, for example, Bigoni and Hueckel, 1991; Petryk, 1993; Bazant and Cedolin, 2003; Darve et al., 2004; Nicot and Darve, 2007; Nicot et al., 2007 and 2009; Prunier et al., 2009a and 2009b; Sibille et al., 2007a and 2007b). This approach was applied to a variety of engineering purposes, such as the analysis of landslides occurring along very gentle slopes (Lignon et al., 2009). This chapter aims to extend this approach to the context of structural systems with a finite number of freedoms, subjected to non-conservative loadings. Application of a non-conservative load makes the tangent stiffness matrix non-symmetric, even though the constitutive behavior of the structure is reversible (elastic). As the second-order work is a quadratic form associated with the tangent stiffness matrix, this matrix’s loss of symmetry was shown to be a basic ingredient in the occurrence of instability in soil mechanics (Nicot et al., 2010). Section 2 develops a theoretical framework to derive the general second = W ext − W int in the context of structural systems. Then, the order relation E c 2 2 case of discrete systems is developed, and the incidence of the spectral properties of the symmetric part of the tangent stiffness matrix on the occurrence of instability is thoroughly considered. Finally, section 3 revisits the well-known Ziegler column in the context of this approach. 3.2. From a quasi-static to a dynamical regime 3.2.1. Existence of multiple equilibrium configurations Let a material system be considered, subjected to an external loading characterized by a distribution of M forces F p , with p = 1,  M . Any coordinate of F p is denoted Fqp , with q = 1,3 . In what follows, for the

sake of simplicity, Fk will denote any term Fqp , with k = 1,  3M . Any geometrical configuration Cx of the system is assumed to be described by a finite set of N kinematical parameters xi , with i = 1,  N . Terms xi stand indifferently as positions or angles. By extension, x will denote indifferently the vector composed of the N coordinates xi or the associated configuration.

Mixed Perturbations and Second-order Work Criterion

71

Equilibrium configurations are determined using the virtual work theorem, according to the following variational equation:

δ Wext − δ Wint = 0

[3.1]

which must be verified for any admissible variational field δ xi , where Wint and Wext denote, respectively, the internal and external works. Let C x* be an equilibrium configuration associated with the family of coordinates xi* . The question of the existence of other equilibrium configuration Cx in a neighboring of C x* may arise: x* − x 0 , and the immediate change in the kinetic energy of the system is governed by equation [3.23]. DEFINITION 3.1.– The equilibrium configuration is reputed time-locally stable (or stable in the small, as discussed in the Introduction) at time t * , if for any velocity disturbance, Δt > 0 exists such that the kinetic energy decreases over the small time range t * , t * + Δt  . Otherwise, the equilibrium configuration is reputed time-locally unstable at time t * . Herein, the notion of local (temporal) stability at time t * contrasts with that of Lyapunov stability (as introduced in Chapter 1) or the notion of asymptotic stability, in that only the time range t * , t * + Δt  is considered from an equilibrium configuration for a specific class of perturbation.

Mixed Perturbations and Second-order Work Criterion

79

3.2.3. Spectral analysis of tensors K and Ks

According to equation [3.23], the modifications in the system require that a velocity field x exist ensuring that W2int is negative. In that case, the equilibrium configuration C x* is reputed locally unstable. On the contrary, if

W2int is positive regardless of the velocity field x , then the equilibrium configuration C x* is reputed stable with respect to a velocity disturbance class. The sign of the second-order work W2int is closely related to the spectral properties of the symmetric part of the tangent stiffness matrix K s . If K s is definitely positive, then for any velocity field x , W2 is always positive. No increase in the system’s kinetic energy is therefore possible from the equilibrium configuration C x* if velocity disturbances are applied. The equilibrium configuration C x* is reputed locally stable. If the symmetric part K s of the stiffness matrix admits at least one strictly negative eigenvalue, then a velocity field x exists, which ensures that W2int is negative. In this case, the system’s kinetic energy may increase, depending on a certain velocity disturbance. The equilibrium configuration C x* is reputed locally unstable. The transition between the two situations (locally stable or unstable equilibrium states) corresponds to the existence of a nil eigenvalue for K s , all other eigenvalues being strictly positive. In that case, for velocity fields x belonging to the kernel of K s , the second-order work is nil, and the kinetic energy remains constant. Returning to equation [3.9], it was established that other equilibrium configurations Cx exist if the matrix K is singular, i.e. if K admits at least one nil eigenvalue. Taking advantage of the Bromwich theorem (Ishaq, 1955; Iordache and Willam, 1998), stating that the smallest eigenvalue of the symmetric part A s of any square matrix A is lower than any real part of the eigenvalues of A (the inequality is strict when A is non-symmetric), it follows that for non-conservative systems, the determinant of K s always vanishes before that of K . In that case, when the determinant of K is zero,

80

Stability of Discrete Non-conservative Systems

the equilibrium configuration C x*

is not unique: all the positions

*

x = x + Δx , where Δx is the solution of equation [3.9], are equilibrium states. Moreover, at least one eigenvalue of K s is strictly negative. The equilibrium configurations x* + Δx are therefore locally unstable.

{

}

3.3. The case of discrete systems 3.3.1. General framework

In this section, a discrete material system made up of a finite number N p of discrete parts is considered. This context corresponds also to the discretized model of a continuum system (using, for example, a finite element method), as that analyzed in the previous section. A geometrical configuration Cx of the system is defined by the geometrical position of each part (or node). Interaction forces f int exist between connective parts or nodes that may be subjected to an external loading f ext (composed of forces and torques). The external loading may evolve over time. Any geometrical configuration Cx of the system is assumed to be described by a finite set of N kinematical parameters xi , with i = 1, N . Terms xi stand indifferently as positions or angles coordinates. By extension, X will denote the vector composed of the N coordinates xi . Starting from an equilibrium configuration at time to , the evolution of the system under the prescribed loading can be expressed by the differential equations:

 + f int = f ext MX

[3.24]

where M is the inertial matrix and f int (resp. f ext ) corresponds to the internal (resp. external) forces. The tensor M is assumed not to depend on time. Differentiating equation [3.24] gives:

M  X + f int = f ext

[3.25]

Mixed Perturbations and Second-order Work Criterion

81

The internal forces f int (between connective discrete parts or adjoining nodes) embed the constitutive behavior of the system into account. We assume in this chapter that a constitutive relation between internal forces and displacements holds in dynamical regime. Even though such relations are usually developed in a quasi-static regime, the extension to the dynamical regime (beyond the scope of this book) will be admitted. For rateindependent systems, this relation can be expressed under the rate form as:

 f int = K X

[3.26]

where K is the tangent stiffness matrix. In the general context of rateindependent incrementally nonlinear behaviors (Darve, 1990), K depends  / X  and on history variables, on the direction of the loading u = X hardening parameters, etc. Finally, the evolution of the system can be described by the following nonlinear differential equation:

 = f ext M  X+K X

[3.27]

In the general case of a non-conservative system, the stiffness matrix K is non-symmetric, whereas M is symmetric and definite positive. The Euclidean norm of f ext is supposed to remain bounded over time. At the initial time to , the system lies in a configuration C xo that can correspond to an equilibrium state or not. The stability of the equilibrium solution C xo is related to the properties of the solutions of equation [3.27]. Different classes of perturbations can be considered: perturbation of the initial positions or velocities, of the external forces and of the mass or stiffness properties. When f ext and K do not depend on time, some classical results can be briefly reviewed: – when K is diagonalizable with all eigenvalues being real and strictly positive, the solution of equation [3.27] remains bonded regardless of the initial conditions; – when K admits at least one negative eigenvalue, the solution of equation [3.27] diverges exponentially, regardless of the initial conditions. This situation corresponds to a divergence instability.

82

Stability of Discrete Non-conservative Systems

These results hold in the absence of any additional constraint. In some cases, additional kinematic constraints can be assigned to the system, changing therefore the stability features of the system. In particular, the following counter-intuitive result holds: a system lying in a stable equilibrium configuration (the non-symmetric tensor K is diagonalizable with all eigenvalues being real and strictly positive) can be destabilized when specific kinematical constraints are prescribed (Challamel et al., 2009; Nicot et al., 2011). 3.3.2. The constrained system

The system considered in the previous section is now subjected to an additional set of p kinematic constraints. These constraints are modeled by linear relations affecting the N kinematic variables ( x1 , xN ) as follows:

Ci1 x1 +  + CiN xN = 0 i = 1, N − p where C is an Ci =

t

(( N − p ) × N )

( Ci1 ,, CiN )

[3.28]

matrix. t C is composed of N − p vectors

of R N , each vector Ci is normal to the hyperplane

( H i ) associated with each kinematic constraint. For

any

velocity

 = ( x ,, x ) , X 1 N

the

N−p

constraints

C1i x1 +  + CNi xN = 0 apply. The constrained system can be described by introducing a set of N − p Lagrangian parameters λi , possibly depending on time, as follows:

 + t C λ = f ext M  X+K X

[3.29a]

 =0 CX

[3.29b]

(

)

where λ = λ1 ,, λN − p .

Mixed Perturbations and Second-order Work Criterion

83

The set of N − p vectors Ci can be completed in a base of R N , with a set of p orthonormalized vectors B j =

t

(B

j1 ,  B jN

).

The vectors B j

belong to the orthogonal subspace of C : C B j = 0 . The set of p vectors B j constitutes the ( N × p ) matrix B and stands as an orthonormal basis of the subspace defined by the kinematic constraints in equation [3.29b]. Equations  is prescribed to [3.28] and [3.29b] mean that any kinematic vector X or X belong to the subspace generated by the p orthonormalized vectors Bi : p

X=

α

i

Bi = B α

[3.30]

i =1

with α =

t

(α , α ) . 1

p

Thus, from equation [3.29a], it can be derived: t

B M B  α + t B K B α + t B t C λ = t B f ext

[3.31]

As C B = 0 , it follows that:

M B  α + K B α = f B

[3.32]

with f B = t B f ext , and where both M B = t B M B and K B = t B K B are two

( p × p)

matrices. From a geometrical point of view, matrices M B and K B

can be regarded as the so-called compressions of M and K on the intersection of the N − p hyperplanes ( H i ). The concept of compression will be discussed in detail in the following chapters. As a result, the N constrained dynamic problem reduces formally to a p -free dynamic problem. Applying holonomic constraints formally reduces the dimension of the problem. All the results derived for the free problem related to its stability properties can therefore be applied to the constrained dynamic problem.

84

Stability of Discrete Non-conservative Systems

The basic question is to know whether the matrix K B possesses singular eigen properties, i.e. whether the matrix K B is diagonalizable with at least one negative eigenvalue. As in the context of the continuum mechanics, the stability of the system can be analyzed through the properties of the scalar term of the kinetic energy (Nicot et al., 2012b). Considering that the system lies in an equilibrium configuration at time to , Ec ( to ) = 0 . If the kinetic energy can be  ( t ) > 0 ), then shown to be convex over a given time range [t , t + Δt [ ( E o

∀t ∈ ]to , to + Δt [ both Ec ( t )

o

c

and Ec ( t ) are strictly positive. The kinetic

energy of the system diverges over this time range. Experimentally, this divergence corresponds to an outburst in kinetic energy, which can be regarded as an indication of (at least time-local) instability. The kinetic energy of the system reads:

Ec =

1 t  XM X 2

[3.33]

Accounting for kinematic constraints, equations [3.30] and [3.33] give:

Ec =

1 t t 1 α B M B α = t α M B α 2 2

[3.34]

A second-order differentiation of equation [3.34] yields:

 Μ B α  Ec = t α M B  α + tα

[3.35]

By combining equations [3.32] and [3.35], it can be obtained:

 M B α  + t α f B − t α K B α Ec = t α

[3.36]

Therefore, the same formalism as that derived for continuous systems is obtained for discrete systems. Indeed, the second-order time derivative of the kinetic energy is composed of three terms:

 M X  = t α  M B α  is an inertial term, and is always – the first term I 2 = t X positive (or nil);

Mixed Perturbations and Second-order Work Criterion

85

 f = t α f B involves the external loading – the second term W2ext = t X applied to the structure, and is referred to as the external second-order work;  Ks X  = ( B α ) K s ( B α ) involves the – the third term W2int = t X constitutive behavior of the structure and as thus the internal forces developed. It corresponds to the internal second-order work computed along the direction B α . t

As a result and omitting the initial term I 2 , the basic equation [3.19] inferred for continuous systems is recovered: the second-order time derivative of the kinetic energy equals the sum of a strictly positive inertial term with the difference between both external and internal second-order works. Thereafter, the analysis is restricted to the case where the external loading remains constant over time. Thus, Ec is strictly positive when the term

( B α ) K s ( B α ) is negative. Discussing the conditions t that make the term ( B α ) K s ( B α ) negative, regardless of α is now the W2int = t α K B α =

t

problem in hand. DEFINITION 3.2.– The cone gathering all vectors x that ensure to the quadratic form associated with K s being negative ( K ijs xi x j < 0 ) is denoted the negative isotropic cone Ι − of K s . LEMMA 3.1.– If {vi } is a family of n independent unit eigenvectors of K s associated

with negative eigenvalues μi , then the subspace defined by this family is included within the negative isotropic cone Ι − of K s . PROOF 3.1.– n

Let x =

a

i

i =1

v i , then:

86

Stability of Discrete Non-conservative Systems

t

x Ks x =

a

i

a j t vi K s v j =

i, j

a

i

a j μ j t vi v j

[3.37]

i, j

where μi is the eigenvalue associated with the eigenvector v i . As K s is symmetric, all eigenvectors associated with different eigenvalues are orthogonal. Thus, equation [3.37] reads: t

x Ks x =

n

a

2

i

μi

[3.38]

i =1

which yields that

t

x K s x < 0 . The vector x belongs therefore to the

negative isotropic cone Ι − of K s , which establishes the lemma. As a result, if K s admits p negative eigenvalues, the vectors Bi being the associated eigenvectors, the lemma states that any vector B α belongs to the negative isotropic cone Ι − of K s , regardless of the vector α . The term W2int = t α K B α is negative, and Ec takes strictly positive values. It is worth emphasizing that the role played by the kinematic constraints is fundamental. If no kinematic constraint is assigned to the system, the second-order time derivative of the kinetic energy of the system writes:

 M X  − t X  Ks X  Ec = t X

[3.39]

where the vector α of R p in equation [3.35] is replaced with the vector X of R N .

 M X  is always positive, whereas the second term The first term I 2 = t X  Ks X  can take positive or negative values. Even though K s W2int = t X  to admits negative eigenvalues, no prescription is assigned to the vector X belong to the associated negative isotropic cone of K s . The negativeness of  Ks X  is not guaranteed, and E is thereby not necessarily W2int = t X c positive.

Mixed Perturbations and Second-order Work Criterion

87

The kinematic constraints for discrete systems can be regarded as the counterpart of the control (or loading) variables. For both continuous and discrete systems, the notion of instability from an equilibrium state can be related to the occurrence of a sudden increase in kinetic energy. Moreover, for both systems, it is notable that the increase in kinetic energy is formally governed by the same equation, involving external and internal second-order works. When the external second-order work is nil, the increase in kinetic energy depends on the vanishing of the internal second-order work, i.e. to the spectral properties of the symmetric part K s of the stiffness matrix. When K s admits at least one negative eigenvalue, additional kinematic conditions should be assigned (kinematic constraints for discrete systems, reducing the degree of freedom of the system; introduction of specific control parameters for continuous systems) so that the direction of the velocities (boundary velocities for continuous systems, nodes or discrete bodies velocities for discrete systems) induced by some (even infinitesimal) perturbations ensures the internal second-order work to take negative values. In both discrete and continuous situations, the stability of nonlinear (elastoplastic) systems can be only investigated over a given time range, from a given time to . The constitutive properties are known at this time to , but are not predictable beyond this point because of the path dependency. Hence, developing a more general theory stating for the stability properties of elastoplastic systems over any time range seems compromised in a general manner. For these systems, asymptotic stability can be analyzed only by restricting the loading conditions to a given “tensorial zone” (as defined in Darve, 1990) in the case of homogeneous stress–strain fields or to the “linear comparison solid” (Raniecki and Bruhns, 1981) for general boundary value problems. It is notable that the local stability properties are directly related to K s and not to K . Taking advantage of the Bromwich theorem (Ishaq, 1955; Iordache and Willam, 1998), stating that the smallest eigenvalue of the symmetric part A s of any square matrix A is lower than any real part of the eigenvalues of A (the inequality is strict when A is non-symmetric), it follows that for non-conservative systems ( K is non-symmetric), K s can admit negative eigenvalues even though all eigenvalues of K are strictly positive (the first vanishing of one eigenvalue of K corresponds to the standard plastic limit condition). As discussed in many papers (see, for example, Piccolroaz et al., 2006; Bigoni and Noselli, 2011; Nicot et al.,

88

Stability of Discrete Non-conservative Systems

2014), divergence and flutter instabilities can occur before the plastic limit condition is met. In the next section, the theoretical results derived in section 2 are applied to a meaningful academic case, namely the generalized Ziegler column problem. 3.4. Application to the generalized Ziegler column problem 3.4.1. The generalized Ziegler column problem

An illustrating example is given with the non-conservative generalized Ziegler column, loaded by a partial follower load (Hermann and Bungay, 1964; Leipholz, 1987). The structure is composed of two bars, AB and BC, of length L – articulated on extremities A and B with a torque bending k . A force F is imposed at extremity C, with an angle α θ2 with the vertical direction y . The rotation of each bar is described with angles θ1 and θ2 (Figure 3.1). Henceforth, the range of α is restricted to [ 0,1] .

Figure 3.1. Generalized Ziegler model subjected to partial follower load

Mixed Perturbations and Second-order Work Criterion

89

Given a force F with an inclination α θ2 , the question of the existence of other geometrical configurations in the vicinity of the trivial solution θ1 = 0 and θ2 = 0 , corresponding to the structure equilibrium, arises. This geometrical configuration is defined with angles θ1 and θ2 . Assuming that both angles θ1 and θ2 are small with respect to 1, equation [3.9] yields:

α p −1 2− p   −1 1 − (1 − α ) where p =

  θ1   0  =  p   θ 2   0 

α p −1 2− p FL . K = k  −1 1 − (1 − α )

[3.40]

  is the stiffness matrix. The p

existence of another solution ( θ1* , θ2* ) different from the trivial solution (0, 0) requires that the determinant of K vanish, i.e.:

(1 − α ) p 2 − 3 (1 − α ) p + 1 = 0

[3.41]

For a given value of α , equation [3.41] admits p-solutions – if and only 5 if α ≤ . In that case, the two positive solutions p1 and p2 are: 9 p1 =

If

3 9 1 3 9 1 and p2 = + − − − 2 4 1−α 2 4 1−α

[3.42]

5 < α < 1 , det K never vanishes and is strictly positive. 9

5 , the two solutions correspond to the configurations defined 9 by the relation: When α
0 and χ ( a ) is of the sign of K11 = 2 − p ; as

p < p1s < 2 , it follows that χ is positive, and regardless of the value of a , the vector

t

( a, − 1)

does not belong to Ι − .

When p1s < p < p2s , det K s < 0 . As K11 = 2 − p , then: – If p1s < p < 2 , the vector – If

2 < p < p2s ,

the

t

vector

a ∈ ]−∞; a2 [ ∪ ]a1; + ∞[ with a1 =

( a, − 1)

belongs to Ι − for a ∈ ]a1 , a2 [ t

( a, − 1)

belongs

to

Ι−

for

K12 + K 21 − 2 − det K s K + K 21 + 2 − det K s and a2 = 12 . 2 K11 2 K11

It is worth noting that a1 ⎯⎯⎯ → −∞ and a1 ⎯⎯⎯ → +∞ , whereas p → 2− p → 2+ → a2 ⎯⎯⎯ p→2

2 K 22 2α − 1 . = K12 + K 21 α − 1

When p > p2s , det K s > 0 ; the discriminant of χ ( a ) is therefore

negative, and χ ( a ) is of the sign of K11 = 2 − p . Recalling that p2s ≥ 2 , it follows that χ is strictly negative. Regardless of the value of a , the vector t

( a, − 1)

belongs to Ι − .

As a result, the value of a ensuring that the vector −

t

( a, − 1)

belongs to

Ι strongly depends on both parameters p and α . The critical loading for the constrained Ziegler column corresponds to the first value vanishing

Mixed Perturbations and Second-order Work Criterion

99

det K s , i.e. to p = p1s . Under the corresponding loading force, the system is at a bifurcation point: the state

(θ

* 1

= 0, θ 2* = 0

)

corresponds to an

undifferentiated equilibrium, the transition between a stable state and a timelocally unstable state. The related kinematical constraint is obtained for a=

K12 + K 21 = 2 K11

3 (1 − α ) − 10α 2 − 14α + 5 K 22 α p1s − 2 s = p . , with = 1 K11 2 2 − p1s  α2  2 1 − α −  4  

(

)

3.5. Concluding remarks

This chapter has investigated the occurrence of instability for structural systems subjected to non-conservative loadings. Extending the approach developed in soil mechanics, a general equation relating both the secondorder time derivative of the kinetic energy and the second-order work was recovered. This equation specifies in which conditions, under a constant loading, the system’s kinetic energy may increase under proper velocity disturbances. As in soil mechanics, the second-order work is shown to play a basic role. As a quadratic form, the spectral properties of the symmetric part of the associated (tangent stiffness) matrix entirely determine the existence of velocity disturbances directing a nil or negative value of the second-order work. When the symmetric part of the tangent stiffness matrix admits at least one negative eigenvalue, velocity disturbances directing a nil or negative value of the second-order work exist. Then, kinematically-constrained discrete structures were examined. The structure is subjected to a set of constraints, as linear combinations of the kinematic variables. As a result, these constraints reduce the kinematic degrees of freedom. Each constraint is described by the equation of a hyperplane. In cases where the symmetric part of the tangent stiffness matrix admits one negative eigenvalue, it is shown that the structure is more prone to destabilize when the associated eigen subspace is included within the hyperplane describing each constraint. The critical case is obtained when the intersection of all the hyperplanes corresponds to the eigen subspace associated with a negative eigenvalue. Then, any velocity disturbance

100

Stability of Discrete Non-conservative Systems

applied to the structure will entail an increase in the kinetic energy of the structure. These results were applied to the case of the generalized Ziegler column, loaded by a partial follower force. It is shown how specific kinematic constraints have to be chosen in order to minimize the loading force leading to an instability situation. 3.6. References Bažant, Z. and Cedolin, L. (2003). Stability of Structures, New York. Dover Publications. Bigoni, D. and Hueckel, T. (1991). Uniqueness and localization, I. Associative and non-associative elastoplasticity. International Journal of Solids and Structures, 28,(2), 197–213. Bigoni, D. and Noselli, G. (2011). Experimental evidence of flutter and divergence instabilities induced by dry friction. Journal of the Mechanics and Physics of Solids, 59, 2208–2226. Bolotin, V.V. (1963). Nonconservative Problems of the Theory of Elastic Stability. New York, Pergamon Press. Challamel, N., Nicot, F., Lerbet, J., and Darve, F. (2009). On the stability of nonconservative elastic systems under mixed perturbations. EJECE, 13(3), 347–367. Challamel, N., Nicot, F., Lerbet, J., and Darve, F. (2010). Stability of nonconservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. Darve, F. (1990). The expression of rheological laws in incremental form and the main classes of constitutive equations. In Geomaterials Constitutive Equations and Modelling, Darve, F. (ed.), Abingdon Taylor and Francis Books, 123–148. Darve, F., Servant, G., Laouafa, F., and Khoa, H.D.V. (2004). Failure in geomaterials, continuous and discrete analyses. Computer Methods Applied Mechanics and Engineering, 193, 3057–3085. Hermann, G. and Bungay, R.W. (1964). On the stability of elastic systems subjected to nonconservative forces. Journal of Applied Mechanics, 86, 435–440. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids, 6, 236–249.

Mixed Perturbations and Second-order Work Criterion

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Iordache, M.M. and Willam, K.J. (1998). Localized failure analysis in elastoplastic Cosserat continua. Computer Methods in Applied Mechanics and Engineering, 151(3–4), 559–586. Ishaq, M. (1955). Sur les spectres des matrices. Séminaire Dubreuil, Algèbre et théorie des nombres. 9, 1–14. Leipholz, H. (1987). Stability Theory, New York. John Wiley & Sons, Inc. Nguyen, Q.S. (2000). Stabilité et mécanique non linéaire. Hermès, Paris. Nicot, F. and Darve, F. (2007). A micro-mechanical investigation of bifurcation in granular materials. International Journal of Solids and Structures, 44, 6630– 6652. Nicot, F., Challamel, N., Lerbet, J., and Darve, F. (2010). Mixed loading conditions, revisiting the question of stability in geomechanics. International Journal of Numerical and Analytical Methods in Geomechanics, in press. Nicot, F., Daouadji, A., Laouafa, F., and Darve, F. (2011). Second order work, kinetic energy and diffuse failure in granular materials. Granular Matter, 13(1), 19–28. Nicot, F., Darve, F., and Khoa, H.D.V. (2007). Bifurcation and second-order work in geomaterials. International Journal of Numerical and Analytical Methods in Geomechanics, 31, 1007–1032. Nicot, F., Lerbet, J., Darve, F. (2014). On the divergence and flutter instabilities of some constrained two-degree of freedom systems. Journal of Engineering Mechanics, 140(1), 47–52. Nicot, F., Sibille, L., and Darve, F. (2009). Bifurcation in granular materials: An attempt at a unified framework. International Journal Solids and Structures, 46, 3938–3947. Petryk, H. (1993). Theory of bifurcation and instability in time-independent plasticity. In Bifurcation and Stability of Dissipative Systems, CISM Courses and Lecturers, Nguyen, Q.S. (ed.), 327, New York Springer, 95–152. Piccolroaz, A., Bigoni, D., and Willis, J.R. (2006). A dynamical interpretation of flutter instability in a continuous medium. Journal of the Mechanics and Physics of Solids, 54, 2391–2417. Prunier, F, Nicot, F., Darve, F., Laouafa, F., and Lignon, S. (2009a). 3D multiscale bifurcation analysis of granular media. Journal of Engineering Mechanics (ASCE), 135(6), 1–17.

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Prunier, F., Laouafa, F., Lignon, S., and Darve, F. (2009b). Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analyses. International Journal for Numerical and Analytical Methods in Geomechanics, 33(9), 1169–1202. Sibille, L., Donzé, F., Nicot, F., Chareyre, B., and Darve, F. (2007b). Bifurcation detection and catastrophic failure. Acta Geotecnica, 3(1), 14–24. Sibille, L., Nicot, F., Donze, F., and Darve, F. (2007a). Material instability in granular assemblies from fundamentally different models. International Journal for Numerical and Analytical Methods in Geomechanics, 31, 457–481. Ziegler, H. (1968) Principles of Structural Stability. Massachusetts, Blaisdell Publishing Company.

4 Divergence Kinematic Structural Stability

4.1. Introduction As it has been accurately recalled in the previous chapters, we handle two approaches of the stability of mechanical systems and these two approaches lead to two different stability criteria. The first one is the most usual and refers to Lyapunov’s stability. In the linear and static framework, it leads to the so-called Euler criterion of instability that reads: det(K) = 0

[4.1]

On the other hand, the mixed perturbation of an equilibrium referring to Hill’s approach of the stability leads to the so-called second-order work criterion of the instability of the involved equilibrium that reads: det(Ks ) = 0

[4.2]

when, as usual, monotone loading paths are considered. More generally, the second-order work instability criterion refers to the statement: K(p) is no longer positive definite. In fact, when p ≥ 0 is the increasing load parameter so that K = K(p), [4.1] leads to a critical load pdiv , whereas [4.2] leads to a critical load psow . Obviously, in practice, the load can be a multi-parameter p = (p1 , . . . , pm ) load. However, any load path s → p(s) = (p1 (s), . . . , pm (s)) allows us to consider the issue parametrized by only one increasing parameter s. We call this parameter s, the load parameter. Moreover, in order to get analytic and

104

Stability of Discrete Non-conservative Systems

explicit results, for the rest of this chapter, we only deal with a static, linear and discrete framework. Especially, neither dissipation nor inertial effects are taken into account. This very simple framework allows us to provide one of the keys of the link between the two different criteria recalled above. In fact, there is nothing essential to prevent the same link in the most general framework of rate-independent mechanical systems. We however have to stress that the extension to a general smooth nonlinear framework of the results presented in this chapter involves tools of differential geometry that exceed the frame of this book. For rate-independent non-smooth systems, a definitive presentation of the extension is still a critical issue. A usual result of linear algebra claimed that det Ks ≤ det K so that, considering increasing loads from a conservative state (purely elastic) of the mechanical system, we deduce that psow ≤ pdiv : as soon as the mechanical system looses its Lyapunov’s stability, it has already lost its Hill’s stability. The main question, which will be solved at the end of this chapter, addresses to the reciprocal sense and will lead to an equivalence between both approaches that obviously coincide for conservative systems because, in these cases, K is symmetric and K = Ks . To do it, we need to focus on a specific property of non-conservative systems, which may be viewed as a “paradox” for the one who is used to deal with conservative systems. We are then referring to the possibility to destabilize a mechanical system only by adding kinematic constraints. On the contrary, Lyapunov’s stability for any kinematically constrained system leads to the concept of Kinematic Structural Stability (KISS or KISS) for the mechanical system. This is the central issue of this chapter, which is mainly devoted to the KISS for divergence instability, whereas the next chapter will deal with the KISS for flutter instability. The main result about divergence KISS will lead to a full equivalence between the above-mentioned two approaches of stability as long as these two criteria are required for the system and for any kinematically constrained system. It should then close the 60-year-old apparent antagonism between these two criteria. This chapter introduces one of the key concepts to investigate some characteristic features of non-conservative rate-independent systems. It is then especially central in this book. We start by recalling some elements about the stability of discrete constrained or unconstrained systems that have been introduced in Chapter 1. Then, the KISS concept is introduced and the KISS issue regarding the divergence stability is tackled thanks to different

Divergence Kinematic Structural Stability

105

approaches. The first one, as usual in mechanics, uses Lagrange multipliers and the Schur formula. Unfortunately this approach has various disadvantages such as the impossibility to extend it to continuous systems. In a second time, we propose a variational formulation which leads to the geometric concept of the compression of operators. We believe that this second approach, less usual in mechanics, is the most appropriate to tackle constrained problems. It allows us to propose a condensed proof of the most significant result of this chapter, giving a full equivalence between Hill and Lyapunov divergence stabilities. It will be also intensively used in the following chapters. 4.2. KISS issue Let Σ be a discrete holonomic mechanical system with q = (q1 , . . . , qn ), a Lagrangian coordinate system of its configuration, space M. We identify points of M with their coordinates, which is without real significance because we investigate the stability problem around an equilibrium whose coordinates are q ∗ = (q1∗ , . . . , qn∗ ). Our considerations then remain local. As claimed in the Introduction section of this chapter, we deal with linearized motions with respect to q ∗ . Geometrically speaking, that means that the motion equations, that usually lie on M, are approximated by their linearized equations on the tangent space Tq∗ M whose current tangent vector  n  ∂ xk , and we identify X ∈ Tq∗ M is called X. We write X = ∂qk q=q∗ k=1 ⎛ ⎞ x1 ⎜ .. ⎟ with the column vector of its coordinates X = ⎝ . ⎠. Thus, referring to xn Chapter 1, the linear dynamics of Σ about the equilibrium q ∗ is governed by ¨ + K(p)X = 0 MX

[4.3]

whereas the static one is K(p)X = 0 without forgetting that M = M(q ∗ ) and K = K(q ∗ ).

[4.4]

106

Stability of Discrete Non-conservative Systems

The Lyapunov stability of q ∗ , when dynamics is neglected, then means that the only solution of [4.4] is X = 0. The Lyapunov instability may then be viewed as the loss of unicity of the equation K(p)U = V: it is usually referred to as Lyapunov instability by loss of unicity. In the static linear framework, the Lyapunov stability is then equivalent to the invertibility of the stiffness matrix K(p). From now on, and without explicit mention of the contrary, we suppose that K(p) is invertible or equivalently that p < pdiv . We now consider that Σ may be constrained by additional kinematic constraints. Any holonomic constraint may be viewed in the coordinate system q = (q1 , . . . , qn ) as a relation g(q) = 0 so that any set of constraints C is defined in the coordinate system q = (q1 , . . . , qn ) by a map f : q → f (q) = (f1 (q), . . . , fr (q)) of C ∞ functions. The system, constrained by the family C, is called ΣC , and its configuration space is called MC so that: q ∈ MC ⇐⇒ f (q) = 0 = (0, . . . , 0)

[4.5]

so that MC is a sub-manifold of M. The dimension of MC is n − r. The main hypothesis of these investigations is that q ∗ is again an equilibrium configuration of MC . Then, Tq∗ MC is an n − r-dimensional vector subspace of Tq∗ M whose equations read: X ∈ Tq∗ MC ⇐⇒ df (q ∗ )X = 0 = (0, . . . , 0)

[4.6]

where df (q) = (df1 (q), . . . , dfr (q)) is the derivative of f at q. For all i = 1, . . . , r, ∗

dfi (q )(X) =

n  k=1

xk

∂fi ∗ (q ) ∂qk

In the linear framework we are here concerned in, the constraints relations [4.5] will then be involved only through their following linear counterparts X ∈ T MC ⇐⇒

n 

q∗

k=1

xk

∂fi ∗ (q ) = 0 ∀i = 1, . . . , r ∂qk

[4.7]

Divergence Kinematic Structural Stability ∂fi ∗ ⎞ ∂q1 (q )

107

⎛

⎞ ci1 ⎟ ⎜ . ⎟ ⎜ .. If Ci = ⎝ ⎠ = ⎝ .. ⎠ ∈ Mn1 (R), then a set of r additional . ∂fi ∗ cin ∂qn (q ) linear constraints C is a family C = {C1 , . . . , Cr } of r independent vectors of Mn1 (R). Equivalently, C = (C1 . . . Cr ) = (cij )1≤i≤n,1≤j≤r ∈ Mnr (R) built with column vector Ck as the kth column of C is a r full-rank matrix because the constraints are independent. The set of r full-rank matrices of Mnr (R) is denoted Gnr (R). In this linear framework, the dependency in the family of constraints C is equivalent to a dependency in the corresponding matrix C, and these two notations are here indifferently used. ⎛

In this framework, pdiv,C denotes the divergence Lyapunov stability threshold of ΣC and the so-called KISS issue questions the stability of ΣC for all C ∈ Gnr (R) and all 1 ≥ r ≥ n − 1. r = n because obviously if r = n, there is no possible motion and the stability issue is then irrelevant. The set of all the possible additional kinematic constraints is then Gnr (R). This chapter deals with divergence stability, but at this C= 1≤r≤n−1

stage, we can formally consider any kind of instability. In the context of this book, it means divergence instability and flutter instability for a Lyapunov approach of the linear stability, as described in Chapter 1, plus Hill instability, as described in Chapters 2 and 3. For each kind of instability, the critical load is denoted pcri , which leads to the respective three values pdiv , pf l and psow for the corresponding kind of instability. The corresponding thresholds of instabilities for the constrained system ΣC constrained by C ∈ C then read pdiv,C , pf l,C and psow,C . We then put the following: D EFINITION 4.1.–

– the KISS is universal if

pcri ≤ pcri,C ∀C ∈ C

[4.8]

which means that the stability of Σ implies the stability of any constrained system ΣC ; – the KISS is conditional if it exists a maximal value pcri,c < pcri pcri,c ≤ pcri,C ∀C ∈ C

[4.9]

which means that pcri,c is the maximal value of the load parameter, ensuring the stability of Σ and of any constrained system ΣC and that pcri,c < pcri .

108

Stability of Discrete Non-conservative Systems

Conversely, it means that there is a family of constraint(s) C ∗ defined by ∈ C such that, for p = pcri,c = pcri,C ∗ , the constrained system ΣC ∗ becomes unstable (for the involved criterion), whereas Σ remains stable since pcri,c = pcri,C ∗ < pcri . This is the paradox of the existence of a “destabilizing kinematic constraint”; C∗

– the KISS is partial if it exists pcri,p such that pcri,p ≤ pcri,C ∀C ∈ C

[4.10]

which means that pcri,p is the maximal value of the load parameter ensuring the stability of any constrained system ΣC . Contrary to the conditional KISS, the condition pcri,c < pcri is not assumed, which means that the system Σ itself can be unstable or not when p < pcri,p . Obviously, the following implications hold: universal KISS =⇒ conditional KISS =⇒ partial KISS The rest of this chapter is now devoted to investigate the question of KISS for the divergence-type stability. In the next chapter, these definitions will be used for flutter-type stability KISS analysis. To sum up, the present chapter is then devoted to investigate the nature of the divergence KISS for any mechanical system Σ and any equilibrium position q ∗ under the assumption that additional kinematic constraints do not remove the involved equilibrium position q ∗ . Since Rayleigh’s works, it is well-known that, for a conservative system Σ, divergence instability is the unique mode of instability of a given equilibrium q ∗ . Moreover, it is also well-known that additional kinematic constraints make the fundamental eigenfrequency higher so that these constraints prevent the occurrence of divergence. With the language of KISS, that means that, for conservative systems, the (divergence) KISS is universal. This fact is doubtless the reason why the KISS concept has never emerged until these last years. As a direct result, the non-universal KISS also implies the occurrence of the “paradox” of destabilizing of a mechanical system by additional kinematic constraints. This few-referenced paradox is then a signature of the non-conservativity of a mechanical system and may be compared with the

Divergence Kinematic Structural Stability

109

other but full-referenced “paradox” of destabilizing by additional damping (Bolotin (1963) or Kirillov and Verhulst (2010), for example). To the best of our knowledge, the unique reference to this paradox may be found in Thompson’s paper (Thompson 1982), where the paradox is only mentioned and not investigated. The divergence KISS issue and solution will be tackled by three distincts, but equivalent ways in our framework. The last one, however, will be the most appropriate for the extensions to flutter instability or to infinite dimensional mechanical systems, namely for continuous systems. We successively present these approaches, and we conclude this chapter by coming back to our first question about the comparison between Lyapunov’s and Hill’s instabilities, and by giving a complete solution and a full equivalence between these both criteria thanks to a variational formulation on all C ∈ C. From a mechanical point of view, that may be considered as the main result of this chapter and constitutes a deep result. Before tackling the general problem, we start our investigations by showing what happens in the simplest possible case where n = 2 and r = 1. To illustrate this case, we use the two-degree-of-freedom Ziegler system and we observe what happens when the system is subjected to one additional kinematic constraint. 4.3. The paradigmatic case of the 2 dof Ziegler system Let us then consider as in Figure 4.1 a holonomic mechanical system Σ made up of two bars S1 = OA and S2 = OB of the same length . Each joint at O and A is elastic with same rigidity k. The chosen coordinate system is q = (θ, φ). The non-conservative action on Σ is a complete follower force P acting on the extremity B of S2 . The corresponding dimensionless load parameter is p = Pk , where P > 0 in compression is the unique investigated case. Straightforward calculations first show that the only equilibrium configuration   2 − p −1 + p ∗ ∗ so is q = (0, 0). At q , the stiffness matrix reads K(p) = −1 1 that det(K(p)) = 1 for all p > 0 and then pdiv = +∞. That means that, in the linear framework, no load P can destabilize Σ by divergence. Thus, the divergence KISS of the Ziegler column cannot be partial and is either universal or conditional.

110

Stability of Discrete Non-conservative Systems

P B φ

y

A k θ

ey O k

ex

x

Figure 4.1. 2 dof Ziegler System

We now investigate the possibility to destabilize some constrained system ΣC . Because only one constraint may be involved, the associated matrix C is an element of G21 (R). Σ is now subjected to the following different additional kinematic constraints. The threshold of divergence stability of each corresponding one degree of freedom mechanical system is easy to compute and is given without justification.   1 namely for the constraint θ = 0, we find pdiv,C1 = +∞ – For C1 = 0   0 , namely for the constraint φ = 0, we find pdiv,C2 = 2 – For C2 = 1   1 , namely for the constraint θ + 2φ = 0, we find – For C3 = 2 pdiv,C3 = 52

Divergence Kinematic Structural Stability

 – For C4 = pdiv,C4 = +∞

 −1 , namely for the constraint −θ + φ = 0, we find 1

– For Cα = C(α) = pdiv,Cα =

111

α2 ++2α+2 α+1

  α , namely for the constraint αθ + φ = 0, we find 1

Regrouping C1 and Cα for α ∈ R allows us to take into account all C ∈ C. We are then able to calculate min pdiv,C . Because obviously C∈C

min pdiv,C < pdiv = +∞, min pdiv,C = pdiv,c and the divergence KISS is in C∈C

C∈C

this case necessarily conditional. Σ Straightforward calculations then show that pdiv,c = 2: as long as p <2,  0 and any ΣC are divergence stable, and when p = 2, the constraint C∗ = 1 leads to a mechanical system ΣC ∗ that is divergence unstable. We question now the meaning of the value p = 2 for Σ.   2 − p −1 + p2 and The symmetric parts of K(p) read Ks (p) = −1 + p2 1 p2 det(Ks (p)) = 1 − . It follows that the value p = 2 is the lowest positive 4 value of the load parameter that vanishes det(Ks (p)). It is also the lowest positive value of the load parameter so that the quadratic form X → X T Ks (p)X T = X T K(p)X T is no longer positive definite or such that the isotropic cone is no longer reduced to {0}. That means that it is the critical value psow for Hill’s second-order work criterion, or in other words for Hill’s stability of q ∗ = 0, but for the system Σ without any consideration of additional kinematic constraint: pdiv,c = psow This astonishing result observed for the simple example of the 2-degree-of -freedom Ziegler system is in fact general and it is now systematically tackled.

112

Stability of Discrete Non-conservative Systems

4.4. Algebraic approach 4.4.1. Schur’s complement formula The usual way for dealing with constrained systems consists of introducing corresponding Lagrange’s multipliers. Let C be of r constraints so ⎞ ⎛a family λ1 ⎜ ⎟ that C ∈ Gnr (R). Thus, there is a vector Λ = ⎝ ... ⎠ ∈ Mr,1 (R) so that the λr quasi-static evolution of ΣC is governed by: K(p)X + CΛ = 0 [4.11] CT X = 0   K(p) C which also reads KC (p)Y = 0 with KC (p) = ∈ Mn+r (R) and CT 0   X ∈ Mn+r,1 (R). This is now the trick of Schur’s complement Y = Λ formula. Since we assume that p < p∗div , K(p) is an invertible matrix and we may do the following block matrix product:    −1    K(p) C K (p) −K−1 (p)C Inn 0 = CT 0 0 Irr CT K−1 (p) −CT K−1 (p)C From this equality, by taking the determinant of both sides: det KC (p) det K−1 (p) = (−1)p det(CT K−1 (p)C) which gives det KC (p) = (−1)p det K(p) det(CT K−1 (p)C) ˜ T K(p)C), ˜ ˜ = K−1 (p)C, then det(CT K−1 (p)C) = det(C But first, if C ˜ = K−1 (p)C is a bijective map from the set Gnr (R) onto and second, C → C itself. Thus, we conclude that: {p ∈ [0, pdiv [|KC (p) ∈ Gn+r (R)} ˜ T K(p)C ˜ ∈ Gr (R)} = {p ∈ [0, pdiv [| C

[4.12]

Divergence Kinematic Structural Stability

113

4.4.2. Case of one constraint: r = 1 For this case, the main reference is Challamel et al. (2010). ˜ is a scalar (real number). The determinant of the ˜ T K(p)C When r = 1, C corresponding 1 × 1 matrix is the scalar itself. We then deduce that pdiv,c = max{p ∈ [0, pdiv [| min det KC (p) = 0} C∈C

˜ = 0} = psow ˜ T K(p)C = max{p ∈ [0, pdiv [| min C C∈C

[4.13]

which proves, for r = 1, the claimed result above. Moreover, coming back to the KISS issue, we easily deduce the following P ROPOSITION 4.1.– For one-constrained systems, namely for systems subjected to one additional kinematic constraint C = (C1 ), the KISS is conditional and pdiv,c = psow . Moreover, this KISS criterion is effective in the following sense: for p = pdiv,c = psow , an appropriate constraint C1 may be chosen as C1 = K(psow )−1 Xsow , where Xsow = 0 is on the isotropic cone of Ks (psow ), which is then non-reduced to {0}: XTsow Ks (psow )Xsow = 0. Thanks to the increasing continuity of the loading p, Xsow is also in the kernel of Ks (psow ): Ks (psow )Xsow = 0. 4.4.3. Case of any set of constraints: 1 ≤ r ≤ n − 1 For this case, the main reference is Lerbet et al. (2012). Suppose now that the family of constraints C is such that C ∈ Gnr (R) for any 1 ≤ r ≤ n − 1. From [4.12], p∗div = max{p ∈ [0, p∗div [| (min det KC (p)) = 0} C∈C

˜ T K(p)C)) ˜ = 0} = max{p ∈ [0, p∗div [| (min det(C C∈C

[4.14]

We then are led to investigate the set of matrices K such that det(AT K(p)A)) = 0 for all A ∈ C. Since the load p is increasing from 0 and K(0) is positive definite, we put the following:

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Stability of Discrete Non-conservative Systems

D EFINITION 4.2.– For any 1 ≤ r ≤ n − 1, a matrix K ∈ Mn (R) is said to be r-positive definite if det(AT K(p)A)) > 0 for all A ∈ Gnr (R). Obviously, 1-positive definite means positive definite and the main result for us about r-positive definite matrices reads: T HEOREM 4.1.– Let 1 ≤ r ≤ n − 1. If Ks ∈ Mn (R) is positive definite, then K is r-positive definite. (see Lerbet et al. 2012) for a proof of this statement). It follows that: P ROPOSITION 4.2.– K is r-positive definite for all 1 ≤ r ≤ n − 1, if and only if Ks is definite positive. and, regarding the divergence KISS issue, we deduce the following. P ROPOSITION 4.3.– For the divergence KISS issue, the number of constraints is irrelevant and the conclusion of proposition 4.1 for 1-constraint systems remains valid for any set of constraint. To better understand this surprising result, we may also question the reciprocal point. For p = psow and for a fixed number 1 ≤ r ≤ n − 1 of constraints, is it possible to find a family of constraints C with C ∈ Gn,r (R) of destabilizing constraints? The answer is positive by choosing, for example, the first constraint C1 such that C1 ∈ K(psow )−1 (ker Ks (psow )), which is ˜ T K(p)C ˜ is not reduced to {0} by definition of psow . The first column of C T ˜ ˜ then the nil column and C K(p)C ∈ / Gr (R). [4.12] allows us to conclude that ΣC is then divergence unstable, the other constraints C2 , . . . , Cr of C being then irrelevant for the divergence stability issue. 4.5. Variational approach This section is mainly contained in Lerbet et al. (2015) . 4.5.1. Variational and minimizing formulations The previous section led to the solution for the divergence KISS issue and for its astonishing relationship with the second-order work criterion. However, the used approach – called the algebraic approach – strongly depends on the

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Schur’s formula trick, whose generalized use would be difficult to carry out, especially in an infinite dimensional framework for continuous systems. The variational approach we are dealing with in this section is a way to remove this delicateness. This will be however fully achieved only in the following section. Moreover, this variation also aims to bring new insights into the KISS issue and its solution for divergence stability by proposing other formulations of divergence stability equivalent to [4.1] or [4.4]. These developments began by focusing on the conservative case. For a conservative mechanical system (remember that K = Ks is then symmetric), the equilibrium is stable for the load p, when one of the following three and even four different equivalent formulations holds: – det K(p) = 0. It will be called an algebraic formulation of the criterion. It is the Euler criterion (4.1); – ∀X ∈ Mn1 (R) \ {0} XT K(p)X = (>)0. It will be called a variational formulation of the criterion. It is the second-order work criterion; XT K(p)X = min XT K(p)X = (>)0 with Sn1 XT X X∈Mn1 (R)\{0} X∈Sn1 (R) (R) = {X ∈ Mn1 (R) | XT X = 1} the unit sphere of Mn1 (R). It will be called a minimizing formulation; –

min

– The most famous minimizing formulation deals with Rayleigh’s quotient whose minimum is the lowest eigenfrequency ω1 (p) of the mechanical system. It reads XT K(p)X = ω1 (p) = (>)0 X∈Mn1 (R)\{0} XT MX min

More formally, a variational formulation reads: ∀X ∈ D, P r(X)

[4.15]

where D ⊂ Rn and P r(X) a logical property right that is true for the concerned X. This often looks like ∀X ∈ Dh ,

h(X) > 0

[4.16]

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where h is a real-valued function defined on D ⊃ Dh . A minimizing formulation often occurs as follows: min j(X) > 0

[4.17]

X∈Dm

where j is a real-valued function defined on D ⊃ Dm . From the above remarks, it is easy to identify what are h and j and their respective domain Dh and Dm for the divergence stability of conservative systems. The goal is then to supply such equivalent variational and minimizing formulations for the divergence stability of non-conservative systems. Let q,a and b be the three quadratic forms defined on Mn1 (R) by q(X) = XT X

[4.18]

a(X) = XT KX = XT Ks X

[4.19]

b(X) = XT KT KX

[4.20]

Then, the solution is given by the following P ROPOSITION 4.4.– Equivalent variational and minimizing formulations for the divergence stability read: – ∀X ∈ Mn1 (R) \ {0} b(X) = (>)0 –

b(X) >0 X∈Sn1 (R) q(X) min

namely h = b and j =

b q

4.5.2. Constraints and quadratic forms a, b, q: Lagrange multipliers

elimination of

Consider now a family C of r independent additional kinematic constraints C = (C1 , . . . , Cr ) with the corresponding matrix C ∈ Gnr (R) where, if

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necessary, every column Cj of C can be normalized such that CTj Cj = 1. The equations of the constrained system ΣC are given by [4.11]. If B is a subset of a vector space E, we note Vect(B) the vector subspace spanned by B and B ⊥ the subspace of E orthogonal to all the vectors of B. Thus, Vect(B ⊥ ) = (Vect(B))⊥ = B ⊥ for any subset B ⊂ E. Let T (C) = Vect {C1 , . . . , Cr } and let H(C) = T (C)⊥ be orthogonal to T (C) in Mn1 (R). Thus, dim T (C) = r and dim H(C) = n − r. Let (t1 (C), . . . , tr (C)) be an orthonormal basis of T (C) (built by the Gramm–Schmidt method from (C1 , . . . , Cr ), for example) and another one (hr+1 (C), . . . , hn (C)) of H(C) such that B(C) = (t1 (C), . . . , tm (C), hm+1 (C), . . . , hn (C)) is an orthonormal basis of Mn1 (R) and let P = P(C) ∈ On (R) be the orthogonal matrix passing from the canonical basis Bc of Mn1 (R) to B(C): P = P(C) = (t1 (C) . . . tm (C) hm+1 (C) . . . hn (C))

[4.21]

Let Z ∈ Mn1 (R) be defined by X = P(C)Z. Then, [4.11] reads, after having multiplied on the left-hand side by P(C)T = P(C)−1 :

P(C)T K(p)P(C)Z + P(C)T CΛ = 0 CT P(C)Z = 0

[4.22]

Consider now KC (p) ∈ Mn−r (R) the square submatrix of P(C)T K(p)P(C) built by removing its first r rows and its first r columns and XC ∈ Mn−r 1 (R) built by removing its r first rows of Z = P(C)T X. Decomposing the matrix P(C)T K(p)P(C) by blocks leads to:  P(C) K(p)P(C) = T

RK(p),C BK(p),C

AK(p),C KC (p)



with RK(p),C ∈ Mrr (R), AK(p),C ∈ Mr n−r (R), BK(p),C Mn−r r (R), KC (p)Mn−r (R). The most interesting block KC (p) reads ⎛

hr+1 (C)T K(p)hr+1 (C) ⎜ .. KC (p) = ⎝ . hn (C)T K(p)hr+1 (C)

... .. . ...

∈

⎞ hr+1 (C)T K(p)hn (C) ⎟ .. ⎠ [4.23] . T hn (C) K(p)hn (C)

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so that [4.22] becomes equivalent to KC (p)XC = 0

[4.24]

where Lagrange’s multiplier has then been removed. Let now aC , bC , qC be the corresponding quadratic forms of ΣC . They are then defined on Mn−r 1 (R) by qC (U) = UT U

[4.25]

aC (U) = UT KC (p)U

[4.26]

bC (U) = UT KTC (p)KC (p)U

[4.27]

The relationships between aC , bC , qC and a, b, q are given by the P ROPOSITION 4.5.– For all X ∈ H(C): q(X) = qC (XC )

[4.28]

a(X) = aC (XC )

[4.29]

b(X) = bC (XC ) + bad (XC )

[4.30]

with bad (XC ) = XTC ATK(p),C AK(p),C XC  The proof is left to the reader The fact that b(X) = bC (XC ) for all X ∈ H(C) is the key of the paradox of the existence of destabilizing kinematic constraints. We now question the divergence KISS thanks to variational and/or minimizing formulations. 4.5.3. Divergence KISS issue From the last relation [4.30], it seems obvious that divergence stability of Σ does not necessarily ensure the divergence stability of a constrained system ΣC . Indeed, b(X) = bC (XC ) + bad (XC ) > 0 ⇒ bC (XC ) > 0. We suppose now that the system Σ is divergence stable, namely that p < which means that b(X) > 0 ∀X ∈ Mn1 (R) \ {0}. Then,

p∗div

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P ROPOSITION 4.6.– bC (XC ) > 0 ∀C ∈ C ∀X ∈ H(C) \ {0} if and only if a(X) > 0 ∀X ∈ Mn1 (R) \ {0} It obviously follows that p∗div = p∗sow .  Suppose that there is X∗ ∈ Mn1 (R) \ {0} so that a(X∗ ) = 0, namely that p = p∗sow . Let C ∗ = (C1 ) be the constraint such that T (C ∗ ) = Vect(K(p)X∗ ), namely C1 is a normalized vector collinear to K(p)X∗ . Remark that K(p)X∗ = 0 because if not, then b(X∗ ) = 0. Thus, X∗ ∈ H(C ∗ ) because X∗T K(p)X∗ = 0. But it then follows that X∗C ∗ = 0 and then that KC ∗ (p)X∗C ∗ = 0 and finally that bC ∗ (X∗C ∗ ) = 0. The reciprocal sense is left to the reader, and the proof shows again that the number of constraints is irrelevant for the KISS issue. 4.6. Geometric approach The above variational formulation was an intermediate matrix formulation that finally leads to the most intrinsic formulation we now deal with and which is called the geometric formulation. The advantage of this geometric approach of the divergence KISS is its ability to be extended by a natural way to other frameworks. For example, it will be used: – to tackle the flutter KISS (see Chapter 5); – to question the dual issue through the concept of geometric degree of non-conservativity (see Chapter 6); – to extend the divergence KISS to the case of continuous mechanics, namely from finite to infinite dimension spaces (see Chapter 8). We now start from the above variational formulation with the same n  notations. If U = (ur+k )1≤k≤n−r ∈ Mn−r 1 (R), put YU = uk hk (C) ∈

H(C). Remember that, for all k, , hk (C)T h (C)

k=r+1

=

δk while

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hk (C)hT (C) = Pk (C) with Pkk (C)2 = Pkk (C) is the projection matrix on Vect(hk (C)) and Pk (C)2 = 0 if k = . Thus, n 

Pkk (C) = PH(C)

k=r+1

is the projection matrix on H(C). For all 1 ≤ i, j ≤ n − r, calculations give: (KC (p))i,j = hTr+i (C)K(p)hr+j (C) (KC (p)T )i,j = hTr+i (C)K(p)T hr+j (C) (KC (p)T KC (p))i,j =

n−r 

hTr+i (C)K(p)T hr+k (C)hr+k (C)T K(p)hr+j (C)

k=1

= hr+i (C) K(p) ( T

T

n−r 

hr+k (C)hTr+k (C))K(p)hr+j (C)

k=1

= hr+i (C) K(p) ( T

T

n 

P (C))K(p)hr+j (C)

=r+1

= hr+i (C)T K(p)T PH(C) K(p)hr+j (C) = hr+i (C)T K(p)T P2H(C) K(p)hr+j (C) = hr+i (C)T K(p)T PTH(C) PH(C) K(p)hr+j (C) = hr+i (C)T (PH(C) K(p))T (PH(C) K(p))hr+j (C) But when X ∈ HC , PH(C) X = X so that we may rewrite the forms qC , aC , bC as the restrictions to HC of some quadratic forms qc C , ac C , bc C obtained from q, a, b by replacing K(p) by PH(C) K(p). We are then led to put the following. D EFINITION 4.3.– Let (E, (. | .)) be an Euclidean space, u ∈ L(E) a linear map of E and F a subspace of E. The compression u|F of u on F is the element of u ∈ L(F ) defined by u|F = pF ◦ u ◦ iF , where iF : F → E is the canonical injection map from F to E and pF : E → F is the orthogonal projection on F .

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Extending the previous reasoning to any linear property, we may prove that P ROPOSITION 4.7.– Let Σ be a mechanical system. Suppose that a mechanical property of Σ is described in a linear framework by a linear map u of Rn . The same property for a constrained system ΣC is described by u| ⊥ , with F

C

FC the space generated by the vectors C1 , . . . , Cr of Rn defining the set of constraints C. Coming back to the divergence KISS issue for discrete mechanical systems, the problem is fully solved by the following nice result on compressions: T HEOREM 4.2.– Let u ∈ L(E) be an injective linear map of a Euclidean space E. All these compressions on the subspaces are again injective if and only if the symmetric part us of u is definite. As us looses its definiteness, we may build a compression (on an hyperplan of E, for example) such that u|F is not injective.  The proof is similar to the one of proposition [4.6] above. It is worth noting that this result is still valid for a Hilbert space E, for a continuous linear injective map u and when compressions are understood as compressions on closed subspaces of E. This framework will be of great importance in Chapter 8 when we will deal with continuous systems. 4.7. Coming back to Lyapunov’s and Hill’s stabilities From any of the above points of view (namely algebraic, variational or geometric), the divergence KISS issue has been solved. When it is interpreted in terms of stability, it leads to the full equivalence between both types of stability and both the corresponding criteria: the Lyapunov divergence criterion and the Hill second-order work criterion. It especially allows us to close an issue which is born with Hill’s papers of 1958 and 1959 (Hill 1958, Hill 1959) regarding the link between the divergence stability criterion and the second-order work criterion. This result may be written through three formulations in fact equivalent but each of them being interesting. The first one is T HEOREM 4.3.– Hill’s stability of Σ is equivalent of Lyapunov’s divergence stability of Σ and of all ΣC for C ∈ C.

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We may do remark however that Hill’s stability of Σ implies Hill’s stability of all ΣC for C ∈ C as it has already mentioned in Lerbet and Absi (2009). In other words, the KISS of Hill’s stability is universal. It follows that the above proposition takes the following fully symmetric equivalent form between Lyapunov’s and Hill’s stabilities: T HEOREM 4.4.– Hill’s stability (s.o.w. criterion) of ΣC Lyapunov’s stability of ΣC ∀C ∈ C

∀C ∈ C ⇔ (div.)

This statement may be compared with the well-known statement concerning only conservative systems or incremental associate systems: Hill’s stability (s.o.w. criterion) of Σ ⇔ Lyapunov’s divergence stability of Σ which highlights that the extension from the conservative or associate case to any non-conservative or hypoelastic or incrementally non-associate case consists of passing from a statement concerning the system Σ alone to the same statement but involving, from a variational point of view, Σ and all these constrained subsystems ΣC for all C ∈ C. To definitively conclude, let us recall that this variational form can be restricted to only subsystems ΣC constrained by only one scalar kinematic constraint, namely T HEOREM 4.5.– Hill’s stability of Σ is equivalent of Lyapunov’s divergence stability of Σ and of any ΣC ∀C ∈ Gn1 (R). 4.8. References Bolotin, V.V. (1963). Non-conservative Problems of the Theory of Elastic Stability. New York. Pergamon Press. Challamel, N., Nicot, F., Lerbet, J. and Darve, F. (2010). Stability of non-conservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids, 6, 236–249. Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. Journal of the Mechanics and Physics of Solids, 7, 209–225.

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Kirillov, O.N. and Verhulst, F. (2010). Paradoxes of dissipation-induced destabilization or who opened Withney’s umbrella? Z. Angew. Math. Mech., 90(6), 462–488. Lerbet, J., Absi E., and Rigolot, A. (2009). About the stability of nonconservative undamped elastic systems: Some new elements. International Journal of Structural Stability and Dynamics, 9(2), 357–367. Lerbet, J., Aldowaji, M., Challamel, N., Nicot, F., Prunier, F., and Darve, F. (2012). Ppositive definite matrices and stability of nonconservative systems, Z. Angew. Math. Mech., 92(5), 409–422. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2015). Variational formulation of divergence stability for constrained systems. Applied Mathematical Modeling, 39(23–24), 7469–7482. Thompson, J.M.T. (1982). ‘Paradoxical’ mechanics under fluid flow. Nature, 5853(296), 135–137.

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5 Flutter Kinematic Structural Stability

5.1. Introduction In the previous chapter, the KISS issue has been fully solved for divergence-type stability: the divergence KISS is conditional, and the threshold of conditional divergence KISS is pdiv,c = psow . In this chapter, we are now involved with the other type of instability that may occur in linear non-conservative systems: the flutter type of instability or flutter instability. Sometimes flutter instability also refers to dynamic instability, whereas divergence instability refers to static or quasi-static instability because it does not involve the inertia forces. Remark however that both are Lyapunov’s instability modes for the linear framework. The flutter instability is systematically invoked to solve the paradoxical impossibility to destabilize by divergence the 2 dof Ziegler system subjected to a complete follower force, as mentioned in the previous chapter (pdiv = +∞). The KISS issue that has been investigated for divergence instability has also been defined for any type of instability and, in this chapter, we will deal with the thresholds pf l , pf l,C with C ∈ C any set of constraints and eventually with pf l,c and pf l,p for conditional and partial flutter thresholds, as defined in the previous chapter. However, the KISS problem is much more difficult here because of the mathematical formulation of the criterion of flutter instability that is used for defining and calculating pf l . It will not lead to as nice a result as the one of divergence KISS (pdiv,c = psow ), and the mathematical tools involved to calculate pf l,p and/or pf l,c will be much less common.

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Roughly speaking, the flutter instability occurs when two eigenfrequencies collapse together, which generates an auto-resonance phenomenon in the mechanical system Σ. That means that, in order to occur, the degree of freedom (dof) of the mechanical system must be at least 2. However, we are investigating the behavior of the flutter instability for constrained systems. Thus, the dof of the constrained systems ΣC must be greater than or equal to 2, which implies that the dof of Σ is greater than or equal to 3. The minimal dimension configuration for investigating the flutter KISS is then n = 3 and r = 1. The calculations will then be expanded in this case, whilst also taking advantage that usual mathematical tools may be used in this simplest case. We will then present both of the calculations with usual tools (Sphere manifold) and the ones with less usual ones (Grassmannian and Stiefel manifolds) that can be generalized for any kind of dimension configuration. The first section is devoted to finding a formulation of flutter instability and to deduce the one for flutter KISS. To do this, we systematically use the concept of compressions, which was introduced in the previous chapter. A short second section deals with the minimal results about Grassmannian and Stiefel manifolds that are naturally involved to solve the general KISS issue. The last and third section focuses on the minimal dimension configuration case. It will lead to a partial flutter KISS, giving the way to find such a pf l,p for each system Σ. This value depends on the mass matrix, and several examples illustrate the results. 5.1.1. Flutter stability and flutter KISS formulations We use the same notations as in the previous chapter: Σ is the mechanical system, q = (q1 , . . . , qn ) is the coordinate system of M the configuration manifold of Σ, q ∗ is the equilibrium position, M is the mass matrix at q ∗ and K(p) is the stiffness matrix at q ∗ . With the same notations, the linear dynamic equations can be written as ¨ + K(p)X = 0 MX

[5.1]

where X ∈ Tq∗ M. Usually, as M is symmetric positive definite, M = S2 , then S is also symmetric positive definite because it is the square root of M. We are then led

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127

˜ to put K(p) = S−1 K(p)S−1 so that [5.1] is: ¨ + K(p)Y ˜ Y =0

[5.2]

where Y = SX. Physically, flutter instability occurs for p = pf l when the amplitude of the solution t → Y (t) of [5.2] increases indefinitely with time t. Let λ1 , . . . , λn ˜ be the eigenvalues of K(p). Looking for solutions Y(t) = Y0 est leads to the characteristic equation ˜ =0 P (s, p) = det(s2 I + K(p))

[5.3]

namely s2 = −λk for k = 1, . . . , n. For p = 0, the eigenvalues λ1 , . . . , λn are > 0 so that s = ±iωk , 1 ≤ k ≤ n. Flutter-type instability means that, for the value p = pf l , [5.3] has a double (multiple) root with a one-dimensional ˜ f l ) no longer diagonalizable, corresponding eigenspace. That makes K(p which induces a non-bounded response t → Y(t). The eigenvalues λk (p) and λ (p) that were distinct for p < pf l merge for p = pf l , whereas the dimension of the corresponding eigenspace remains equal to 1. It is not easy to characterize by a single equation the loss of diagonalizibility ˜ of K(p). As usual, we will only use the necessary condition of the existence for a double root of the characteristic equation thanks to the discriminant ΔK(p) = ˜ Δ(P (s, p)) of [5.3] that always exists and may be calculated as a resultant of s → P (s, p) and its derivative. However, because we will use the compressions ˜ of operators, we associate as usual to K(p) a linear map u(p) of E = Rn Euclidean vector space equipped with its canonical scalar product (.|.). Thus, we assume that: D EFINITION 5.1.– pf l is the lowest positive value of p solution of =0 Δu(p) = ΔK(p) ˜

[5.4]

and the flutter stability condition is Δu(p) > 0

[5.5]

If F is a subspace of E, u|F = pF ◦ u ◦ iF is, as defined in the previous chapter, the compression of u on F . The flutter KISS is then defined by the following variational and minimizing formulations:

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Stability of Discrete Non-conservative Systems

D EFINITION 5.2.– Flutter KISS means that ∀F = {0} ⊂ E Δu|

F

(p)

>0

[5.6]

where implicitly dim(F ) ≥ 2 because, if not, the degree of the characteristic polynomial is less than 1 and the discriminant is meaningless. An equivalent minimizing formulation is min

F ={0}⊂E

Δu |

F

(p)

>0

[5.7]

Let m ≤ n and Gm,n (R) be the so-called Grassmannian of all m-planes of E = Rn . If φm : Gm,n (R) → R F → Φm (F ) = Δu| (p) , then [5.7] is F equivalent to: min

min

2≤m 0

[5.8]

The flutter KISS issue is then brought back to the calculation of the minimum of a function whose argument lies on a Grassmannian manifold. Concretely, the function Φm will present a nicer form when its argument is not an m-plane F ∈ Gm,n (R) but an orthonormal basis e = (e1 , . . . , em ) of F . We will then be led to make the calculations on the so-called Stiefel manifold Sm,n (R) while checking that the results do not depend on e but only on F = Vect(e). Basic results on Grassmannian and Stiefel manifolds are now recalled through some exercises. 5.2. Grassmannian and Stiefel manifolds This section will present as a sequence of exercises the manifold structures of Sn , Gm,n (R) and Sm,n (R) and the elementary properties which will be used to solve the KISS issue. We start with the usual case of the n-dimensional sphere Sn of Rn+1 . E XERCISE 5.1.– Manifold structure of the sphere Sn of Rn+1 . 1) We choose a pole, called the “north pole”, N = (0, . . . , 0, 1) (resp. its opposite S = (0, . . . , 0, −1) = −N the “south pole”). H = {x = (x0 , . . . , xn ) ∈ Rn+1 | xn+1 = 0} is the “horizontal” equatorial subspace. We note UN = Sn \ {N } (resp. US = Sn \ {S}) and ψN : UN → H (resp.ψS :

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129

US → H) the stereographic corresponding projections. Show that the charts (UN , ψN , H) and (US , ψS , H) build an atlas of Sn . We will calculate the two −1 transition maps of this atlas ψN S = ψS ◦ψN : ψN (UN ∩US ) → ψS (UN ∩US ) −1 −1 and ψSN = ψN S = ψN ◦ ψS : ψS (US ∩ UN ) → ψN (US ∩ UN ). 2) Show that the sphere Sn = {x = (x0 , . . . , xn ) ∈ Rn+1 | x20 +. . .+x2n = 1} is an n-dimensional compact manifold. 3) Propose another atlas of Sn . The atlas with two charts is the smallest. Why? 4) Show directly that Sn is also a submanifold of Rn+1 . 5) Let a = (a0 , . . . , an ) ∈ Sn . Compute the tangent space Ta Sn of Sn at a and show that as affine subspace of Rn+1 , its direction is set up by vectors of Rn+1 orthogonal to a. E XERCISE 5.2.– Let m ≤ n and Gm,n (R) be the set of subspaces of dimension m of Rn . They are called m-planes of Rn , and Gm,n (R) is called the Grassmannian of m-planes of Rn . In this exercise, Gm,n (R) is equipped with a manifold structure. 1) What are G0,n (R), Gn,n (R), G1,n (R)? 2) Metric structure on Gm,n (R). Let V, W be two sub(vector)spaces of Rn with same dimension m, namely V, W ∈ Gm,n (R). Let d(V, W ) = |||ΠV − ΠW |||, where ΠF denotes the orthogonal projector on F . Show that d is a distance on Gm,n (R). 3) Show that Gm,n (R) is compact. 4) Manifold structure on Gm,n (R). Let V ∈ Gm,n (R) and UV = {W ∈ Gm,n (R)|W ⊕ V ⊥ = Rn }. a) Show that UV is an open set of Gm,n (R) with V ⊂ UV . b) Let W ∈ UV . Show that there is a unique linear map uV,W ∈ L(V, V ⊥ ) such that W = {x + uV,W (x)|x ∈ V }. c) Show that φV : W → uV,W is a bijective map and deduce that (UV , φV , L(V, V ⊥ )) is a chart of Gm,n (R) at V . d) Show that (UV , φV , L(V, V ⊥ ))V ∈Gm,n (R) is an atlas Gm,n (R), which is then an m(n − m)-dimensional manifold.

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e) Let F ∈ Gm,n (R). Calculate the tangent space TF Gm,n (R) of Gm,n (GR) at F . Show that it is isomorphic to L(F, F ⊥ ). R EMARK .– The used Euclidean structure on E is actually not necessary to define the manifold structure on Gm,n (R) E XERCISE 5.3.– The aim of this exercise is to carry out a matrix approach of Gm,n (R). We will then deduce a submanifold structure on Gm,n (R). We already use that giving a (vector) subspace F of E is equivalent to giving the orthogonal projector on F . We may then identify Gm (Rn ) with the set of orthogonal projectors on m planes of E, which is then in bijection with the subset of matrices of orthogonal projections with rank m, namely we have by identification: Gm,n (R) = {A ∈ Mn (R)|A2 = A, AT = A, rank(A) = m} = {A ∈ Mn (R)|A2 = A, AT = A, tr(A) = m} where Im is the diagonal matrix whose first m coefficients are 1 and the following n − m are 0. If A ∈ Mn (R), A has the following block ˆ B C with B ∈ Mm (R). As in the previous chapter, decomposition A = E DE we put Gn,m (R) = {M ∈ Mn,m (R)|rank(M ) = m}. 1) Show that Gm,n (R) = {P −1 Im P |P ∈ On (R)}. 2) Show that V = {A| det(B) = 0} is an open set of Mm (R). Let A ∈ V . Show that rank A = m if and only if E − DB −1 C = 0 3) Let Φ : Gln,m (R) → Mn,m (R) be the map defined by Φ(M ) = M (M T M )−1 M T . Show that Φ(M ) is the only matrix of the orthogonal projection on Im(Φ(M )) = ImM . 4) Let Ψ : Mn−m,m (R) → Mn,m (R) be the map defined by Ψ(M ) = I Φ( m ). Calculate dΨ and deduce that Ψ is an immersion C ∞ . M 5) Show that Ψ is one-to-one and that Ψ(Mn−m,m (R)) = V ∩ Gm,n (R).

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6) Show that Ψ as a map into V is proper and deduce that Gm,n (R) is an m(n − m) sub-manifold C ∞ of Mn (R). 7) Show that Gm,n (R) is compact. 8) Compute the tangent space TIm Gm,n (R). E XERCISE 5.4.– The goal of this exercise is to deal with another type of manifold, one that is often easier to manipulate than the Grassmannian manifolds Gm,n (R) for 1 ≤ m ≤ n. If m ≤ n, we identify a family X = (X1 , . . . , Xm ) of m vectors of Rn with the matrix (still noted X) of Mm,n (R) whose m columns are the m vectors X1 , . . . , Xm . The set Sm,n (R) is the set of families of m orthonormal vectors of Rn . Thus, Sm,n (R) = {X ∈ Mn,m (R) | X T X = Im }. 1) What are Sn,n (R) and S1,n (R)? 2) Show that Sm,n (R) is a sub-manifold of Mm,n (R) and compute its dimension. 3) Compute its tangent space TX Sm,n (R) at any X in Sm,n (R). 4) What is the relationship between Sm,n (R) and Gm,n (R)? We can use the orthogonal group Om (R). R EMARK .– As quotient manifolds, we write Gm,n (R) = Gm,n (R)/Gm (R) = Sm,n (R)/Om (R) and Sm,n (R) = On (R)/On−m (R). We deduce that Gm,n (R) = On (R)/(Om (R) × On−m (R)). ˆ n−1 /{−1, 1}, which means that Sn−1 is a Note also that G1,n (R) = ES covering of G1,n (R). Using the orthogonal supplementary spaces, the same result holds for Gn−1,n (R). The relation Gm,n (R) = Sm,n (R)/Om (R) can also be understood in this way: Sm,n (R) is the total space of a principal fiber bundle with fiber Om (R) and base space Gm,n (R). 5.3. Case n = 3 and m = 2 From now on, we suppose that n = 3. The main part of this section concerns Lerbet et al. (2016).

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5.3.1. Geometric considerations and preliminary calculations Let F be a 2-plane of E. Then, Δu|F (p) = Tr2 (u|F (p)) − 4 det(u|F (p)) and the KISS means, by the minimizing formulation: min

F ∈G2,3 (R)

Tr2 (u|F ) − 4 det(u|F ) > 0

[5.9]

Because of the compactness of Gr2,3 (R) and the continuity of Φ : Gr2,3 (R) → R F → Φ(F ) = Tr2 (u|F ) − 4 det(u|F )

[5.10]

the minimum exists and is reached for an element Ff∗l . The corresponding constraint is then given by any vector C∗1 ∈ (Ff∗l )⊥ so that C ∗ = {C∗1 } and C∗ = C∗1 . Actually, the norm and the orientation of C∗1 are without significance and we may choose a normalized vector C∗1 = e∗3 (there are two possible choices): e∗3 ∈ (Ff∗l )⊥ ∩ S2 . Parametrizing the problem by e(F ) = e3 ∈ F ⊥ ∩ S2 instead F itself is possible only because of the specific dimension framework of this issue. It allows us to bring back the differential calculation from the Grassmannian Gr2,3 (R) to the more usual manifold sphere S2 . However, to also show how it is working in the general way, we will carry out both computations. To do it, we systematically associate to any vector e3 ∈ S2 that represent a constraint, an orthonormal basis (e1 , e2 ) in the Stiefel manifold S2,3 (R). We will complete the computations thanks to the parametrization by e3 ∈ S2 and also by (e1 , e2 ) ∈ S2,3 (R). They must lead to the same result and the fact, mentioned in the above remark, that S2 is a covering of G1,3 (R) ≈ G2,3 (R) and that G2,3 (R) = S2,3 (R)/O2 (R) means that our computations must be invariant by e3 → −e3 and e = (e1 , e2 ) → e = (e 1 , e 2 ) = R(θ)e = (R(θ)e1 , R(θ)e2 ), where R(θ) ∈ O2 (R) is any rotation about e3 . We invite the reader to systematically check it. The quotient G2,3 (R) = S2,3 (R)/O2 (R) means that there is a projection map π : S2,3 (R) → G2,3 (R). If e = (e1 , e2 ) ∈ S2,3 (R), then π(e) = Vect{e1 , e2 }, whereas if F ∈ G2,3 (R), the so-called fiber over F , denoted by π −1 (F ), is built by all orthonormal bases of F .

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133

The aim of this paragraph is to transform [5.10] in order to solve [5.9]. Indeed, the natural way to calculate Ff∗l consists of differentiating Φ to get the critical points. The derivative of Φ is however difficult to calculate because the “points” are vector spaces and, as mentioned above, we then start by transforming Φ. Let e = (e1 , e2 ) be an orthonormal basis of F = π((e1 , e2 )). We use the letter φ instead of the same capital letter Φ to refer to the function of the variables in S2,3 (R) so that Φ ◦ π = φ. φ is a lift of Φ. Two expressions of Φ that will be used are given by: L EMMA 5.1.– Φ(F ) = φ(e) = φ((e1 , e2 )) = 4(u(e1 ) | e2 )(u(e2 ) | e1 ) + ((u(e1 ) | e1 ) − (u(e2 ) | e2 ))2

[5.11]

= ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))2 − 4(ua (e1 ) | e2 )2

[5.12]

 The following transformations hold: Tr(u|F ) =

2 

(u|F (ei ) | ei ) =

i=1

2  i=1

(p ◦ u(ei ) | ei ) =

2 

(u(ei ) | ei )

[5.13]

i=1

because p is self-adjoint and det(u|F ) = (u|F (e1 ) | e1 )(u|F (e2 ) | e2 ) − (u|F (e1 ) | e2 )(u|F (e2 ) | e1 ) = (p ◦ u(e1 ) | e1 )(p ◦ u(e2 ) | e2 ) − (p ◦ u(e1 ) | e2 )(p ◦ u(e2 ) | e1 ) = (u(e1 ) | e1 )(u(e2 ) | e2 ) − (u(e1 ) | e2 )(u(e2 ) | e1 )

[5.14]

for the same reasons. Straightforward calculations then show that: Φ(F ) = φ(e) = 4(u(e1 ) | e2 )(u(e2 ) | e1 ) + ((u(e1 ) | e1 ) − (u(e2 ) | e2 ))2 which is exactly [5.11]. Here, e = (e1 , e2 ) is viewed as an element of π −1 (F ) ⊂ S2,3 (R). Direct calculations may show that this expression of

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φ(e) = φ((e1 , e2 )) fortunately does not depend on the choice of e = (e1 , e2 ) as the orthonormal basis of F , namely in the fiber over F and then justifying the notation Φ(F ). That may be directly checked at each step of the calculations, but we will not do it and we suggest that the reader does it instead. To better understand the flutter as a competition between the symmetric part us of u and its skew symmetric part ua and because the second-order work criterion involves us , we now transform the last expression by using the symmetry of us and the skew symmetry of ua . Calculations give: (u(e1 ) | e2 )(u(e2 ) | e1 ) = ((us (e1 ) | e2 ) +(ua (e1 ) | e2 ))((us (e2 ) | e1 ) + (ua (e2 ) | e1 )) = (us (e1 ) | e2 )2 − (ua (e1 ) | e2 )2 leading to: Φ(F ) = φ((e1 , e2 )) = 4((us (e1 ) | e2 )2 − (ua (e1 ) | e2 )2 ) ˆ +((us (e1 ) | e1 ) − (us (e2 ) | e2 ))2 E = ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))2 − 4(ua (e1 ) | e2 )2 which is exactly [5.12] It may be checked again that this expression of φ(e) = φ((e1 , e2 )) does not depend on the choice of e = (e1 , e2 ) but only on its equivalence class under the group action by O2 (R), which justifies the expression Φ(F ). However, among all the possible choices, there is one which is very significant. Let e3 be any of the two unit vectors of F ⊥ . Suppose that e3 ∈ / ker ua (this last case will be handled separately, and when n = 3, it is a one-dimensional vector space). Then, the orthogonal space (F ⊥ )⊥ = F is 3) spanned by (ua (e3 ), u2a (e3 )) so that we can choose e1 = ||uuaa (e (e3 )|| , e2 = e3 ∧

ua (e3 ) ||ua (e3 )|| .

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135

Define Δ1 (e3 ) = det(e3 , ua (e3 ), us (e3 )), Δk (e3 ) = det(e3 , ua (e3 ), uka (e3 )) for k ≥ 2. Straightforward calculations give: Φ(F ) = φ((e1 , e2 )) = φ(e) = h(e3 ) = ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))2 − 4(ua (e1 ) | e2 )2  2 2 ua (e3 ) | e3 ∧ ua (e3 ) = (Trus − (us (e3 ) | e3 ))2 − 4 || ua (e3 ) ||2  2 Δ2 (e3 ) = (Trus − (us (e3 ) | e3 ))2 − 4 || ua (e3 ) ||2

[5.15]

Four intrinsic quantities are involved in the issue: the three real eigenvalues of us : α1 ≤ α2 ≤ α3 with (vi )i an adapted orthonormal basis of E (us (vi ) = αi vi for all i) and −β 2 (β > 0) the unique not nil eigenvalue of the symmetric linear map u2a . Indeed, because n = 3, rank ua = 2, ker ua = ker u2a and (ker ua )⊥ = Imua = Imu2a = E−β 2 =< w1 , w2 >, with w1 and w2 two orthonormal eigenvectors of u2a so that u2a (wi ) = −β 2 wi for i = 1, 2. We then deduce, after having chosen w3 ∈ ker ua , that ua (w1 ) = βw2 , ua (w2 ) = −βw1 , ua (w3 ) = 0 ((wi )i is still an orthonormal basis of E). The quantities Δk (e3 ) will play a significant role and they can be explicitly evaluated: P ROPOSITION 5.1.–

1) For all x ∈ E and for all k ≥ 1, Δ2k+1 (x) = 0;

2) for all x ∈ S(E), Δ2k (x) = (−1)k−1 β 2(k−1) Δ2 (x); 

3) for all x ∈ S(E), Δ2k (x) = (−1)k−1 β 2(k−1) || ua (x) ||2 β 2 − || ua (x) ||2 .

 These results are obvious if x ∈ ker ua , and because Δk is a 3 homogeneous function of x, we may suppose that x ∈ S(E) \ ker ua . Moreover, the minimal polynomial of u2a is πu2a = X(X + β 2 ) meaning that u4a = −β 2 u2a which leads to the second assertion 2.

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Stability of Discrete Non-conservative Systems

Moreover, Δ3 (x) = det(x, ua (x), u3a (x)) = (x ∧ ua (x) | u3a (x)) = − || ua (x) ||2 (x ∧ ua (x) | ua (x)) +

Δ2 (x) (x ∧ ua (x) | ua (x ∧ ua (x))) || ua (x) ||2

=0+0=0 which proves the first assertion. Finally, the last assertion comes from the following calculation of Δ2 (x). ua (x) 2 Using the orthonormal basis (x, ||uuaa (x) (x)|| , x ∧ ||ua (x)|| ), ua (x) is given by: u2a (x) = (u2a (x) | x)x + +

(u2a (x) | ua (x)) ua (x) || ua (x) ||2

(u2a (x) | x ∧ ua (x)) x ∧ ua (x) || ua (x) ||2

= (u2a (x) | x)x +

Δ2 (x) x ∧ ua (x) || ua (x) ||2

= − || ua (x) ||2 x +

Δ2 (x) x ∧ ua (x) || ua (x) ||2

[5.16]

Calculating now the square of the norm of the right-hand side and the left-hand side of the last equation, we get: Δ2 (e3 ) e3 ∧ ua (e3 ) ||2 || ua (e3 ) ||2  2 Δ2 (e3 ) 2 2 4 (ua (e3 ) | ua (e3 )) = || ua (e3 ) || + || e3 ∧ ua (e3 ) ||2 + || ua (e3 ) ||2 || u2a (e3 ) ||2 = || − || ua (e3 ) ||2 e3 +

−2 || ua (e3 ) ||2

Δ2 (e3 ) (e3 | e3 ∧ ua (e3 )) || ua (e3 ) ||2

Flutter Kinematic Structural Stability

(u4a (e3 ) | e3 ) = || ua (e3 ) ||4 +

Δ22 (e3 ) || ua (e3 ) ||2

−β 2 ((u2a (e3 ) | e3 ) = || ua (e3 ) ||4 +

Δ22 (e3 ) || ua (e3 ) ||2

β 2 (ua (e3 ) | ua (e3 )) = || ua (e3 ) ||4 +

Δ22 (e3 ) || ua (e3 ) ||2

β 2 || ua (e3 ) ||2 = || ua (e3 ) ||4 +

Δ22 (e3 ) || ua (e3 ) ||2

137

leading to   Δ22 (e3 ) =|| ua (e3 ) ||4 β 2 − || ua (e3 ) ||2

[5.17]

and because Δ2 (e3 ) ≥ 0: Δ2 (e3 ) =|| ua (e3 ) ||2



β 2 − || ua (e3 ) ||2

[5.18]

which proves the last assertion. 5.3.2. Sufficient conditions First and foremost, remark that in the degenerated case where the secondorder work criterion (sow criterion) fails with a two-dimensional isotropic cone C, choosing F ⊂ C and the constraint in F ⊥ destabilizes the system and flutter KISS fails. Indeed, we then get Tr(u|F (p)) = 0 and, according to [5.12], we get Φ(F ) = −4(ua (e1 ) | e2 )2 ≤ 0. On the contrary, suppose, now to simplify the reasoning, that the sow criterion holds, meaning here that α1 > 0. Then, without calculating the minimum of Φ (or φ, h), the following sufficient flutter KISS condition holds: P ROPOSITION 5.2.– As long as α1 + α2 > 2β

[5.19]

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Stability of Discrete Non-conservative Systems

the flutter KISS holds. Moreover, if, as usual, for p = 0, the system is elastic conservative stable, then the flutter KISS is ensured on [0, ps,f l ], where ps,f l is the minimal positive root of α1 (p) + α2 (p) − 2β(p) = 0

[5.20]

 Because of the well-known results about Rayleigh’s quotient for us , α1 ≤ (us (x) | x) ≤ α3 for all unit vector x and the extrema are, respectively, reached for the eigenvectors v1 (minimum) and v3 (maximum) associated with α1 and α3 . Then α1 + α2 ≤ (us (e1 ) | e1 ) + (us (e2 ) | e2 ) = Tr(us ) − (us (e3 ) | e3 ) =

3 

αi − (us (e3 ) | e3 )

i=1

≤ α 2 + α3 with a minimum value for e3 = v3 and a maximum value for e3 = v1 . Moreover, 0 ≤| (ua (e1 ) | e2 ) |≤ β with a minimum when e3 ∈ ker u⊥ a and a maximum when e3 = w3 . Then Φ(F ) ≥ (α1 + α2 )2 − 4β 2

[5.21]

and we deduce that a flutter KISS sufficient condition can be written as [5.19], meaning that, as long as the arithmetic mean of both the lowest eigenvalues of us is greater than the square root of the not nil eigenvalue of u2a , no additional kinematic constraint may destabilize the system Σ and the flutter KISS is then ensured. Moreover, suppose as usual that, for p = 0, the system is elastic conservative stable. Then, α1 (0) > 0, α2 (0) > 0 and β(0) = 0. Thus, by continuity, the minimal positive value ps,f l root of α1 (p) + α2 (p)−2β(p) = 0 is > 0. On [0, ps,f l ], the flutter KISS holds. Suppose now that α1 + α2 ≤ 2β. Because both terms in competition are reached for e3 = v3 and for e3 = w3 and because v3 = w3 , there is no chance

Flutter Kinematic Structural Stability

139

in order that this equality should hold for a convenient constraint and the flutter KISS is still ensured. The sufficient condition [5.19] is then neither necessary nor optimal, and we now tackle the issue of necessary and sufficient flutter KISS conditions. 5.3.3. Necessary and sufficient conditions: calculations in S2,3 (R) We now start from the expression of φ given by Lemma 5.1, and we investigate the minimum of φ. We will find the extremums of φ. We then have to evaluate the function φ at the extremums to find the minimum. Extrema are critical points. We have to compute φ (e) : Te S2,3 (R) → R and to solve φ (e) = 0 where Te S2,3 (R) is the tangent space of S2,3 (R) at e. Calculation of Te S2,3 (R) has been done Exercise 5.4 above. It is summed up in: L EMMA 5.2.– For any orthonormal family e = (e1 , e2 ) ∈ S2,3 (R), Te S2,3 (R) = {(1 = se2 + t1 e3 , 2 = −se2 + t2 e3 ) | s, t1 , t2 ∈ R} and Tπ(e1 ,e2 ) G2,3 (R) = TF G2,3 (R) = π T (Te S2,3 (R)) = {(1 = t1 e3 , 2 = t2 e3 ) | t1 , t2 ∈ R} where e3 is any unit vector orthogonal to F = π((e1 , e2 )) = π(e). With the same notations, critical points of Φ are then characterized by the following: P ROPOSITION 5.3.– F = π((e1 , e2 )) is a critical point of Φ if both the following relations hold if (us (e3 ) | ua (e3 )) = 0

[5.22]

(Tr us − (e3 | us (e3 )))Δ1 (e3 ) + 2Δ2 (e3 ) = 0

[5.23]

for any normalized vector e3 such that F =< e3 >⊥ .

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Stability of Discrete Non-conservative Systems

 Usual derivative calculations give: φ (e1 , e2 )(1 , 2 ) = 2((us (e1 ) | e1 ) +(us (e2 ) | e2 ))(2(us (e1 ) | 1 ) + 2(us (e2 ) | 2 )) −8(ua (e1 ) | e2 )((ua (1 ) | e2 ) + (ua (e1 ) | 2 )) and thus Φ (π((e1 , e2 ))(t1 e3 , t2 e3 ) = 4(((us (e1 ) | e1 ) +(us (e2 ) | e2 ))(t1 (us (e1 ) | e3 ) + t2 (us (e2 ) | e3 )) −8(ua (e1 ) | e2 )(t1 (ua (e3 ) | e2 ) + t2 (ua (e1 ) | e3 )) F = π((e1 , e2 )) is a critical point of Φ if and only if Φ (π((e1 , e2 )) (t1 e3 , t2 e3 ) = 0 for all t1 , t2 ∈ R. For successively (t1 , t2 ) = (1, 0) and (t1 , t2 ) = (0, 1), it leads to: ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))(us (e1 ) | e3 ) = 2(ua (e1 ) | e2 )(ua (e3 ) | e2 )

[5.24]

((us (e1 ) | e1 ) + (us (e2 ) | e2 ))(us (e2 ) | e3 ) = 2(ua (e1 ) | e2 )(ua (e1 ) | e3 )

[5.25]

(the reader may check that these relations are really invariant by any rotation about e3 ∈ F ⊥ ). These relations are both (nonlinear) conditions on e = (e1 , e2 ) but actually on F defining the critical points of Φ. They may however be transformed by using a unit vector e3 ∈ F ⊥ . From us (ei ) =

3 

(us (ei ) | ek )ek and ua (ei ) =

k=1

3 

(ua (ei ) | ek )ek ,

k=1,k=i

we deduce (us (e1 ) | ua (e1 )) = (us (e1 ) | e2 )(ua (e1 ) | e2 ) +(us (e1 ) | e3 )(ua (e1 ) | e3 )

[5.26]

(us (e2 ) | ua (e2 )) = (us (e2 ) | e1 )(ua (e2 ) | e1 ) +(us (e2 ) | e3 )(ua (e2 ) | e3 )

[5.27]

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141

But from [5.24] and [5.25], we get (ua (e3 ) | e2 ) 2(ua (e1 ) | e2 ) (us (e1 ) | e3 ) = (= ) (us (e2 ) | e3 ) (ua (e1 ) | e3 ) (us (e1 ) | e1 ) + (us (e2 ) | e2 ) or (us (e1 ) | e3 )(ua (e1 ) | e3 ) = (ua (e3 ) | e2 )(us (e2 ) | e3 ) or still (us (e1 ) | e3 )(ua (e1 ) | e3 ) + (ua (e2 ) | e3 )(us (e2 ) | e3 ) = 0 But as ua is skew symmetric and us is symmetric, from [5.26] and [5.27] (us (e1 ) | ua (e1 )) + (us (e2 ) | ua (e2 )) = (us (e1 ) | e3 )(ua (e1 ) | e3 ) + (us (e2 ) | e3 )(ua (e2 ) | e3 ) Thus, (us (e1 ) | ua (e1 )) + (us (e2 ) | ua (e2 )) = 0 But, (us (e3 ) | ua (e3 )) = (us (e3 ) | e1 )(ua (e3 ) | e1 ) + (us (e3 ) | e2 )(ua (e3 ) | e2 ) = −(us (e1 ) | e3 )(ua (e1 ) | e3 ) − (us (e2 ) | e3 )(ua (e2 ) | e3 ) Thus, (us (e3 ) | ua (e3 )) = 0

[5.28]

which is exactly [5.22]. Because we may choose any unit vector orthogonal to e3 as e1 , we use ua (e3 ) e3 ∧ ua (e3 ) e1 = and e2 = e3 ∧ e1 = . [5.24] is then reduced to || ua (e3 ) || || ua (e3 ) || 0 = 0, and calculations give successively: (us (e2 ) | e3 ) = (e2 | us (e3 )) =

Δ1 (e3 ) det(e3 , ua (e3 ), us (e3 )) = || ua (e3 ) || || ua (e3 ) ||

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Stability of Discrete Non-conservative Systems

(ua (e1 ) | e2 ) = ( =

e3 ∧ ua (e3 ) u2a (e3 ) | ) || ua (e3 ) || || ua (e3 ) ||

det(e3 , ua (e3 ), u2a (e3 )) Δ2 (e3 ) = 2 || ua (e3 ) || || ua (e3 ) ||2

[5.29]

and (ua (e1 ) | e3 ) = −(e1 | ua (e3 )) = −(

ua (e3 ) | ua (e3 )) = − || ua (e3 ) || || ua (e3 ) ||

Thus, (ua (e1 ) | e2 )(ua (e1 ) | e3 ) = −

det(e3 , ua (e3 ), u2a (e3 )) || ua (e3 ) || || ua (e3 ) ||2

=−

det(e3 , ua (e3 ), u2a (e3 )) || ua (e3 ) ||

=−

Δ2 (e3 ) || ua (e3 ) ||

and finally, [5.25] ⇔ ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))Δ1 (e3 ) +2Δ2 (e3 ) = 0 or

[5.30]

⇔ (trus − (e3 | us (e3 )))Δ1 (e3 ) + 2Δ2 (e3 ) = 0 the last equation being exactly [5.23]. By evaluating Φ at a critical point, we get the following flutter KISS condition: P ROPOSITION 5.4.– Flutter KISS holds as long as

Δ21 (e3 ) < 1 or when || ua (e3 ) ||4

| Δ1 (e3 ) = det(e3 , ua (e3 ), us (e3 )) |⊥ is a critical point of Φ. Geometrically speaking, [5.34] means that the volume of the parallelepiped built on (e3 , ua (e3 ), us (e3 )) is lower than that of the one built on (e3 , ua (e3 ), e3 ∧ ua (e3 )) (equal to || ua (e3 ) ||2 ).  Remark that from [5.29] and [5.30], we deduce that, at a critical point F = π((e1 , e2 )) =< e3 >⊥ , Φ is given by: Φ ◦ π((e1 , e2 )) = ((us (e1 ) | e1 ) + (us (e2 ) | e2 ))2 − 4(ua (e1 ) | e2 )2  2 2Δ2 (e3 ) 2 = ((us (e1 ) | e1 ) + (us (e2 ) | e2 )) − || ua (e3 ) ||2   Δ21 (e3 ) 2 = ((us (e1 ) | e1 ) + (us (e2 ) | e2 )) 1 − [5.33] || ua (e3 ) ||4

Flutter KISS holds as long as

Δ21 (e3 ) < 1 or when || ua (e3 ) ||4

| Δ1 (e3 ) = det(e3 , ua (e3 ), us (e3 )) |⊥ is a critical point of Φ, namely solutions of [5.22] and [5.30] (or [5.23]) or equivalently of [5.24] and [5.25]. 5.3.4. Necessary and sufficient conditions: calculations in S2 To validate the previous results and to present the calculations with more usual tools of differential geometry, we use the parametrization of the problem

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Stability of Discrete Non-conservative Systems

by the sphere S2 of unit vectors of E meaning by the function h defined by [5.15] that is now recalled: 2



h(e3 ) = (tr us − (us (e3 ) | e3 )) − 4

Δ2 (e3 ) || ua (e3 ) ||2

2 [5.36]

The aim of this section is then to calculate the critical points of h on the sphere S2 . Put g(e3 ) = Tr us − (us (e3 ) | e3 ) and f (e3 ) = ||uΔa2(e(e33)||) 2 so that h(e3 ) = g 2 (e3 ) − 4f 2 (e3 ). Straightforward calculations give: h (e3 )() = 2g(e3 )g (e3 )() − 8f (e3 )f (e3 )()

[5.37]

g (e3 )() = −2(e3 | us ()) f (e3 )() =

Δ 2 (e3 )() || ua (e3 ) ||2 −2Δ2 (e3 )(ua (e3 ) | ua ()) || ua (e3 ) ||4

Δ 2 (e3 )() = det(, ua (e3 ), u2a (e3 )) + det(e3 , ua (), u2a (e3 )) + det(e3 , ua (e3 ), u2a ()) for all  ∈ Te3 S2 =< e3 >⊥ . Critical points e3 verify h (e3 )() = 0 on the basis of Te3 S2 meaning for example for  = ua (e3 ) and  = e3 ∧ ua (e3 ) (e3 ∈ / ker ua ). We do this in the following two subsections. 5.3.4.1.  = ua (e3 ) P ROPOSITION 5.5.– h (e3 )(ua (e3 )) = 0 implies relation [5.22].  We successively find: g (e3 )() = g (e3 )(ua (e3 )) = −2(e3 | us (ua (e3 ))) = −2(us (e3 ) | ua (e3 ))

Flutter Kinematic Structural Stability

145

Δ 2 (e3 )() = Δ 2 (e3 )(ua (e3 )) = det(ua (e3 ), ua (e3 ), u2a (e3 )) + det(e3 , u2a (e3 ), u2a (e3 )) + det(e3 , ua (e3 ), u3a (e3 )) = det(e3 , ua (e3 ), u3a (e3 )) = Δ3 (e3 ) = 0 because of Proposition 5.1. We deduce that: f (e3 )() = f (e3 )(ua (e3 )) =

Δ 2 (e3 )(ua (e3 )) || ua (e3 ) ||2 −2Δ2 (e3 )(ua (e3 ) | u2a (e3 )) || ua (e3 ) ||4

=

−2Δ2 (e3 )(ua (e3 ) | u2a (e3 )) =0 || ua (e3 ) ||4

because of the skew-symmetry of ua . From [5.37], we then deduce g (e3 ) = 0 (g(e3 ) = 0)) or (us (e3 ) | ua (e3 )) = 0. We again find [5.22] 5.3.4.2.  = e3 ∧ ua (e3 ) P ROPOSITION 5.6.– h (e3 )(e3 ∧ ua (e3 )) = 0 implies relation [5.23].  The calculations are a little more complicated. Calculations give: g (e3 )() = g (e3 )(e3 ∧ ua (e3 )) = −2(e3 | us (e3 ∧ ua (e3 ))) = −2(us (e3 ) | e3 ∧ ua (e3 )) = −2Δ1 (e3 ) Δ 2 (e3 )() = Δ 2 (e3 )(e3 ∧ ua (e3 )) = det(e3 ∧ ua (e3 ), ua (e3 ), u2a (e3 )) + det(e3 , ua (e3 ∧ ua (e3 )), u2a (e3 )) + det(e3 , ua (e3 ), u2a (e3 ∧ ua (e3 )))

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Stability of Discrete Non-conservative Systems

But from [5.16]: det(e3 ∧ ua (e3 ), ua (e3 ), u2a (e3 )) = det(e3 ∧ ua (e3 ), ua (e3 ), − || ua (e3 ) ||2 e3 +

Δ2 (e3 ) e3 ∧ ua (e3 )) || ua (e3 ) ||2

= − || ua (e3 ) ||2 det(e3 ∧ ua (e3 ), ua (e3 ), e3 ) =|| ua (e3 ) ||2 det(e3 , ua (e3 ), e3 ∧ ua (e3 )) =|| ua (e3 ) ||4

[5.38]

and one more time from [5.16] and the double cross-product formula: det(e3 , ua (e3 ∧ ua (e3 )), u2a (e3 )) = −(e3 ∧ u2a (e3 ) | ua (e3 ∧ ua (e3 ))) = −(e3 ∧ ( =(

Δ2 (e3 ) e3 ∧ ua (e3 )) | ua (e3 ∧ ua (e3 ))) || ua (e3 ) ||2

Δ2 (e3 ) ua (e3 ) | ua (e3 ∧ ua (e3 ))) || ua (e3 ) ||2

=−

Δ2 (e3 ) (u2 (e3 ) | e3 ∧ ua (e3 )) || ua (e3 ) ||2 a

=−

Δ22 (e3 ) || ua (e3 ) ||2

[5.39]

From [5.16], we get: Δ2 (e3 ) e3 ∧ ua (e3 ) =|| ua (e3 ) ||2 e3 + u2a (e3 ) || ua (e3 ) ||2 and thus successively: e3 ∧ ua (e3 ) =

|| ua (e3 ) ||4 || ua (e3 ) ||2 2 e3 + ua (e3 ) Δ2 (e3 ) Δ2 (e3 )

ua (e3 ∧ ua (e3 )) =

|| ua (e3 ) ||4 || ua (e3 ) ||2 3 ua (e3 ) + ua (e3 ) Δ2 (e3 ) Δ2 (e3 )

u2a (e3 ∧ ua (e3 )) =

|| ua (e3 ) ||2 4 || ua (e3 ) ||4 2 ua (e3 ) + ua (e3 ) Δ2 (e3 ) Δ2 (e3 )

Flutter Kinematic Structural Stability

147

and det(e3 , ua (e3 ), u2a (e3 ∧ ua (e3 ))) = || ua (e3 ) ||4 +

|| ua (e3 ) ||2 Δ4 (e3 ) [5.40] Δ2 (e3 )

and from Proposition 5.1 det(e3 , ua (e3 ), u2a (e3 ∧ ua (e3 ))) = || ua (e3 ) ||4 − β 2 || ua (e3 ) ||2

[5.41]

Finally, Δ 2 (e3 )(e3 ∧ ua (e3 )) = 2 || ua (e3 ) ||4 −

Δ22 (e3 ) − β 2 || ua (e3 ) ||2 || ua (e3 ) ||2

We deduce f (e3 )(e3 ∧ ua (e3 )) =

Δ 2 (e3 )(e3 ∧ ua (e3 )) || ua (e3 ) ||2 −2Δ2 (e3 )(ua (e3 ) | ua (e3 ∧ ua (e3 ))) || ua (e3 ) ||4

=

Δ 2 (e3 )(e3 ∧ ua (e3 )) || ua (e3 ) ||2 +2Δ2 (e3 )(u2a (e3 ) | e3 ∧ ua (e3 )) || ua (e3 ) ||4

=

Δ 2 (e3 )(e3 ∧ ua (e3 )) || ua (e3 ) ||2 +2Δ22 (e3 )) || ua (e3 ) ||4

= 2 || ua (e3 ) ||2 − =

Δ22 (e3 ) 2Δ22 (e3 )) 2 − β + || ua (e3 ) ||4 || ua (e3 ) ||4

Δ22 (e3 ) + 2|| ua (e3 ) ||2 − β 2 || ua (e3 ) ||4

= β 2 − || ua (e3 ) ||2 +2|| ua (e3 ) ||2 − β 2 =|| ua (e3 ) ||2 again from Proposition 5.1. h (e3 )(e3 ∧ ua (e3 )) = 0 then leads to: 0 = h (e3 )(e3 ∧ ua (e3 )) = 2g(e3 )g (e3 )(e3 ∧ ua (e3 )) −8f (e3 )f (e3 )(e3 ∧ ua (e3 )) = −4g(e3 )Δ1 (e3 ) − 8f (e3 ) || ua (e3 ) ||2

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Stability of Discrete Non-conservative Systems

and g(e3 )Δ1 (e3 ) + 2f (e3 ) || ua (e3 ) ||2 = g(e3 )Δ1 (e3 ) + 2Δ2 (e3 ) = 0

[5.42]

which is exactly [5.23]. 5.3.5. Summary of the results Because the examples will be handled through the parametrization by unit vectors e3 , we briefly summarize the results. The critical points e3,c are solutions of the following system: ⎧ ⎨

|| e3 ||2 −1 = 0 (us (e3 ) | ua (e3 )) = 0 ⎩ (Trus − (us (e3 ) | e3 ))Δ1 (e3 ) + 2Δ2 (e3 ) = 0

[5.43]

It is the intersection of three hypersurfaces, and it is then built by p ≥ 2 isolated points e3,c,k in E for k = 1, . . . , p (at least one minimum and one maximum thanks to the compactness of the domain and the continuity of the map). Among these points, there is the wanted absolute minimum e∗3 , and the flutter KISS condition can then be written as: | Δ1 (e∗3 ) = det(e∗3 , ua (e∗3 ), us (e∗3 )) | pf l,p may occur: the flutter KISS pf l,p ≤ pf l,C holds only for all C ∈ Gn1 but not for the unconstrained case (C = ∅). As defined in the previous chapter, the reserved word is “partial flutter KISS” and it is the reason for the notation pf l,p . As it has been recalled in Chapter 4, for a conservative system Σ with a divergence critical load pdiv , the KISS is ensured for p < pdiv and the KISS is universal, whereas for a non-conservative system with a divergence critical load pdiv , the KISS is ensured for p < pdiv,c = psow . But then, pdiv,c < pdiv necessarily holds because det Ks (p) ≤ det K(p) for any matrix K(p). As a result, the divergence stability of every constrained system ΣC also ensures the divergence stability of the constraint free system Σ: we may choose C = ∅ or r = 0 (in our reasonings, remember that the load is supposed monotonically increasing from p = 0 and often implicitly conservative stable for p = 0). On the contrary, the flutter stability of every constrained system ΣC no longer ensures the flutter stability of the free system Σ: for flutter KISS, to be sure that neither the free system Σ nor any constrained subsystem ΣC may be destabilized, we have to put p < min{pf l , pf l,p } without knowing a priori what the minimum value is. Lastly, the value pf l,p of flutter KISS also depends on the mass matrix as the value of the critical flutter load parameter pf l . The following examples will illustrate all these results. 5.4.2. Examples We now apply the above results to the usual three dof Ziegler column Σ as the one used in Lerbet et al. (2012) or Lerbet et al. (2015) for investigating the divergence KISS. Σ then consists of the three-degree-of-freedom Ziegler system Σ as in Figure 5.1 made up of three bars OA, AB, BC with OA = AB = AC =  linked by three elastic springs of the same stiffness k. The

150

Stability of Discrete Non-conservative Systems

non-conservative external action (the circulatory force) is the follower force P . In these examples, the numerical approximations of involved quantities are given to three decimal places.

Figure 5.1. 3 dof Ziegler system

The elastic energy of the springs is k 2 ((θ + (θ1 − θ2 )2 + (θ2 − θ3 )2 ) 2 1 k = (2θ12 + 2θ12 + θ32 − 2θ1 θ2 − 2θ2 θ3 ) 2

U=

and the virtual power of P in any configuration θ = (θ1 , θ2 , θ3 ) is given by (P > 0 in compression): PP∗ = P (sin(θ3 − θ1 )θ1∗ + sin(θ3 − θ2 )θ2∗ )

Flutter Kinematic Structural Stability

151

Put p = Pk as dimensionless loading parameter and note that (0, 0, 0) is the unique equilibrium configuration. The stiffness matrix can then be written as: ⎤ ⎡ 2−p −1 p ⎥ ⎢ 2 − p −1 + p ⎥ K(p) = ⎢ ⎦ ⎣ −1 0 −1 1 Two cases of mass matrix will be investigated, but we do not systematically give the corresponding mass distribution. Only the second one is associated with a uniform mass distribution. In each case, we give the numerical approximation of the solution e3 (called here X with XT = (x1 x2 x3 )) of equations [5.46], and only for the first case, we give the expanded expressions of the quantities involved in equations [5.46]. We also validate the results by a direct numerical solution of the initial minimization problem. Because of the non-convexity of the criterium function, a specific algorithm is used. It relies on the conjunction of two classical numerical algorithms. First, the Nelder–Mead downhill simplex technique Forst and Hoffmann (2010) is used so as to find the minimum of a function f (X, p) with X on the unit sphere for fixed p. Then, this first optimization algorithm is piloted by a dichotomy procedure on p, which will converge to the value p∗ such that minX f (X, p∗ ) = 0. The sphere is parametrized by the usual spherical coordinates (x1 = cos(ψ1 ) · cos(ψ2 ), x2 = sin(ψ1 ) · cos(ψ2 ) and x3 = sin(ψ2 ) with −π ≤ ψ1 ≤ π and −π/2 ≤ ψ2 ≤ π/2) in order to lead to a minimization problem without constraint. The critical values pf l , pf l,s and pf l,c are then calculated in order to illustrate the above mechanical discussion. For the corresponding value of (C∗f l )T = (X∗f l )T = (a b c) to pf l,p , the destabilizing kinematic constraint can then be written as aθ1 + bθ2 + cθ3 = 0. The above analytic results give only the first-order equations for the critical points and then allow us to find a KISS critical value of the load parameter. In order to have the exact set of equations and inequalities for the minimums points, a second-order set of inequalities should be added to select among the critical points the minimum points. However, because of the non-convexity of the problem, we have to evaluate h on the critical points or, at best, on the set of the minimums to select the absolute one. The used simplex technique validates the analytic

152

Stability of Discrete Non-conservative Systems

approach by using a completely different way, which avoids any gradient method. 5.4.3. M = I3 ˜ In this first case, K(p) = K(p) and ⎡ ⎢ Ka (p) = ⎢ ⎣ ⎡

0

0

0

0

−1/2 p

−1/2 p

2−p

⎢ Ks (p) = ⎢ ⎣ −1 1/2 p

−1 2−p

1/2 p

⎤

⎥ 1/2 p ⎥ ⎦ 0 1/2 p

⎤

⎥ −1 + 1/2 p ⎥ ⎦

−1 + 1/2 p

1

The vector e3 parametrizing the problem is then the column vector X of the coordinates of e3 in the initial basis. ⎡ ⎤ x1 ⎢ ⎥ ⎥ X = ⎢ ⎣ x2 ⎦ x3 The above quantities can then be written as: Δ1 (X) = 1/2 x3 p2 x1 2 − x1 x2 x3 p − x1 px3 2 − 1/2 px1 3 − 1/2 px1 2 x2 −1/2 x1 2 x3 p + 1/2 x1 px2 2 + 1/2 px2 3 −1/2 x3 p2 x2 2 + 1/2 x2 2 x3 p + x2 px3 2 − 1/2 x2 x3 2 p2 −1/2 px3 3 + 1/2 x1 x3 2 p2 Δ2 (X) = −1/4 x1 p3 x3 2 − 1/8 p3 x1 3 − 1/8 p3 x1 2 x2 + 1/8 x1 p3 x2 2 +1/8 p3 x2 3 + 1/4 x2 p3 x3 2

Flutter Kinematic Structural Stability

153

and h, which must be minimized on the sphere, is given by: h (X) = 25 − 20 p − 5 p2 x1 2 − 5 p2 x2 2 − 4 x1 x2 x3 2 + 4 x1 x2 3 p −2 p2 x2 3 x3 − 6 px2 2 x3 2 + 8 x2 3 x3 p + 2 x1 x3 3 p + x1 2 x3 2 p2 +2 x2 x3 3 p + x2 2 x3 2 p2 + 4 p2 + p2 x1 4 − 4 x2 4 p + p2 x2 4 −8 x1 3 x2 + 12 x1 2 x2 2 + 2 p2 x1 2 x2 2 − 2 px1 2 x3 2 + 4 px1 3 x2 −2 p2 x1 3 x3 − 8 x1 2 x2 x3 − 8 x1 2 px2 2 + 4 x1 3 x3 p + 4 x2 x3 p2 +4 x1 x3 p2 − 8 px1 x2 − 2 x1 x3 p2 x2 2 + 18 px1 2 + 20 x2 x3 +20 x1 x2 − 8 x1 x2 3 + 4 x2 4 + 4 x1 4 − 2 p2 x1 2 x2 x3 +4 x1 2 x2 x3 p + 2 x1 x3 2 p2 x2 − 4 x1 x3 2 px2 + 4 px3 2 − 4 x1 4 p −20 x1 2 − 20 x2 2 − 10 x3 2 + 18 px2 2 − 8 x2 3 x3 + 4 x1 2 x3 2 −4 x2 x3 3 + 8 x2 2 x3 2 + 8 x1 x2 2 x3 − 18 x2 x3 p −10 x1 x3 p + 2 p2 x1 x2 + x3 4 The sow criterion is valid as long as p < 1 (psow = 1), and the flutter appears for the 3 dof (unconstrained) system Σ for pf l = 0.516. The necessary and sufficient condition given by [5.19] leads here to pf l,s = 0.699. Because pf l,s > pf l , no kinematic constraint may destabilize by flutter the flutter-stable unconstrained system Σ. The exact value of pf l,p is obtained by minimization of h or by solving directly [5.43] as well and leads to the same value pf l,p = 0.727 (ψ1 = −0.809 and ψ2 = 0.332) fortunately higher than pf l,s . In this case, the flutter KISS does not ensure the flutter stability of the free system Σ and the free system becomes unstable “before” finding a destabilizing kinematic constraint illustrating the fact that the flutter KISS is partial.

154

Stability of Discrete Non-conservative Systems

      

! 

  


   











(a)

(b)

Figure 5.2. (a) Minimum values of f (X, p) for X on the unit-sphere and p ∈ [0, 2] and (b) 2D map in spherical coordinates (ψ1 , ψ2 ) of f (X, p) on the unit-sphere for p = pf l,p . For a color version of this figure, see ww.iste.co.uk/lerbet/stability.zip

5.4.4. Uniform mass distribution No expansion of the quantities like h, Δ1 , Δ2 but only the result values are now given. For this mass distribution, the mass matrix can be written as: ⎛7 3 1⎞ 3

⎜3 M =⎜ ⎝2 1 2

In this case, ⎛

2 4 3 1 2

2 1 2 1 3

⎟ ⎟ ⎠

4.786 − 1.603 p

⎜ ˜ K(p) =⎜ ⎝ −7.160 + 2.087 p 4.013 − 0.630 p

−7.160 + 1.791 p 13.581 − 4.616 p −10.763 + 2.040 p

4.013 + 0.233 p

⎞

⎟ −10.763 + 4.210 p ⎟ ⎠ 11.863 − 2.551 p

The sow criterion is valid as long as p < 1 (p∗ sw = 1), and the flutter appears for the 3 dof (unconstrained) system Σ for pf l = 1.484. The necessary and sufficient condition given by [5.19] leads here to pf l,s = 0.950. Because pf l,s < pf l , there is a kinematic constraint that destabilizes by flutter the flutter-stable unconstrained system Σ. The exact value of pf l,p is obtained

Flutter Kinematic Structural Stability

155

by minimization of h (ψ1 = −0.974 and ψ2 = 0.666) or by solving directly [5.43] as well and leads to the same value pf l,p = 1.298 fortunately again higher than pf l,s . In this case, since pf l,p < pf l , the flutter KISS ensures the flutter stability of the unconstrained system Σ and the corresponding destabilizing constraint can be written as: 0.442θ1 − 0.650θ2 + 0.618θ3 = 0        

 

  



 






 

(a)





(b)

Figure 5.3. (a) Minimum values of f (X, p) for X on the unit-sphere and p ∈ [0, 2] and (b) 2D map in spherical coordinates (ψ1 , ψ2 ) of f (X, p) on the unit-sphere for p = pf l,p . For a color version of this figure, see ww.iste.co.uk/lerbet/stability.zip

5.5. References Bolotin, V.V. (1963). Non-conservative Problems of the Theory of Elastic Stability. Pergamon Press, Oxford. Challamel, N., Nicot, F., Lerbet, J. and Darve, F. (2010). Stability of non-conservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. Forst, W. and Hoffmann, D. (2010). Optimization – Theory and Practice, Springer, New York. Kirillov, O.N. and Verhulst, F. (2010). Paradoxes of dissipation-induced destabilization or who opened Withney’s umbrella? Z. Angew. Math. Mech., 90(6), 462–488.

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Stability of Discrete Non-conservative Systems

Lerbet, J., Absi E. and Rigolot, A. (2009). About the stability of nonconservative undamped elastic systems: Some new elements. International Journal of Structural Stability and Dynamics, 9(2), 357–367. Lerbet, J., Aldowaji, M., Challamel, N., Nicot, F., Prunier, F. and Darve, F. (2012). P-positive definite matrices and stability of nonconservative systems. Z. Angew. Math. Mech., 92(5), 409–422. Lerbet, J., Challamel, N., Nicot, F. and Darve, F. (2015). Variational formulation of divergence stability for constrained systems. Ap. Math. Model., 39(23–24), 7469– 7482. Lerbet, J., Hello, G., Challamel, N., Nicot, F. and Darve, F. (2016). 3 dimensional flutter kinematic structural stability. Non Linear Analysis: Real World Applications, 29, 19–37. Thompson, J.M.T. (1982). ‘Paradoxical’ mechanics under fluid flow. Nature, 5853(296), 135–137.

6 Geometric Degree of Non-conservativity

6.1. Introduction In the previous chapters, through the KISS analysis, we investigated the possibility to prevent the destabilizing effect of additional kinematic constraints. Both divergence and flutter have been tackled. The former leads to a conditional KISS, whereas the latter leads to a partial KISS. We only have to remember that such a paradox occurs only in non-conservative systems: for conservatives ones, the KISS is universal. In this new chapter, we question the dual issue. On the one hand, additional kinematic constraints may have a “negative” destabilizing effect, but on the other hand, they may transform a non-conservative system into a conservative one and present in this sense a more positive face. Using the same notations as the ones used for the KISS study, we suppose now that the constrained system ΣC is conservative for a certain C ∈ C. Then, any constrained system ΣC  with Im(C) ⊂ Im(C ) is again conservative. That means that the interesting quantity is the minimal number d of constraints that make ΣC conservative. We then put the following definition: D EFINITION 6.1.– The minimal number d of kinematic constraints, namely the cardinal of C that can make a corresponding system ΣC with C ∈ Gnd (R) conservative is called the geometric degree of non-conservativity (GDNC) of Σ.

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Stability of Discrete Non-conservative Systems

The aim of this chapter is to investigate the meaning of d, the way to calculate it and the set Cg of the suitable families of d constraints. In its formulation as well as for its solution, this issue appears as a dual issue of the KISS one. Moreover, because the signature of the non-conservativity is the non-symmetry of the stiffness matrix K(p), the mass matrix, and more generally the inertia terms, are without significance regarding this issue. The duality will be then clear regarding the divergence KISS issue that only involves the stiffness matrix K(p). In fact, for this dual issue, the extension to the nonlinear differentiable framework is possible, whereas the one for the KISS issue remains a perennial problem that is tackled in today’s research. It will involve differential forms on manifolds language, but is today definitively solved. Unfortunately, involving nonlinear differential equations, it cannot lead to an explicit analytic solution. This chapter is included in Lerbet et al. (2014) and Lerbet et al. (2016). It is organized as follows. First, we carry out a mathematical modeling of the GDNC issue and a solution for the linear framework is given involving usual mathematical tools, namely the same tools as the ones used for the divergence KISS issue: usual linear algebra on an Euclidean vector space E (namely Rn for us). Unfortunately, this approach fails to find the set Cg of all the suitable families of d constraints. In the following section, we then carry out another formulation thanks to the exterior calculus on E. This new approach then leads to a complete solution for Cg , involving the computation of the so-called Lagrangian planes of E. This allows us, in the third section, to tackle the nonlinear case whose solution is given thanks to the so-called Darboux’s theorem concerning the local structure of two-differential forms. It is better known as the theorem about the local form of symplectic manifolds. A short last section sums up the duality between divergence KISS and GDNC. 6.2. Modeling and calculation of the GDNC: examples 6.2.1. Calculation of the GDNC This section is partially included in Lerbet et al. (2014). We systematically use the tools that have been highlighted in Chapter 4 for dealing with the KISS issue. As in these previous chapters, the non-conservativity of Σ is characterized by the non-symmetry of the stiffness matrix K(p) so that ||Ka (p)|| is a primary obvious measure of the non-conservativity of Σ. Ka (p) = 12 (K(p) − KT (p)) is the skew symmetric part of K(p). This

Geometric Degree of Non-conservativity

159

measure is coarse because it depends on the chosen norm ||.|| and it does not take into account the relationship with the kinematic constraints. It then eliminates the possibility to have a geometric approach of the problem and to best control the non-conservativity of Σ. Note that numerous (real) numbers have been proposed to measure the non-normality of a matrix or an operator (see for example the chapter 28 of Trefethen and Embree (2005) but the proposed GDNC is a complete original measure and directly linked with the KISS issue. On the contrary, the measure proposed in [6.1] is a natural number. It will be proven that it is independent of the intensity of physical actions (measured here by p), independent of the chosen norm ||.|| and will lead to a set of solutions, namely a description of all the kinematic constraints that may remove the nonconservativity of Σ showing a possible way to control this non-conservativity. It is then a more appropriate measure of this non-conservativity of Σ. We will systematically use the tools that have been highlighted in Chapter 4 for dealing with the KISS issue. Recall then that, if u is the linear map describing a mathematical property of Σ and if C = {C1 , . . . , Cr } is the set of constraints, then the same mathematical property for the constrained system ΣC is described by the so-called compression u|F ⊥ of u on the orthogonal F ⊥ of F = Vect{c1 , . . . , cr }. For the rest of this chapter, we denote by a small bold letter, like x, a vector of E = Rn and by the capital bold letter, like X, the column vector of its coordinates in the canonical basis of Rn . Thus, C realizes the conversion of the non-conservative system Σ into the conservative one ΣC if the compression u|F ⊥ is symmetric or if the skew symmetric part u|F ,a of u|F ⊥ is nil. Because we are looking for the lowest r = card C satisfying this property and because dimF ⊥ = n − r, it is then equivalent to look for the subspaces F with the greatest dimension d such that its skew symmetric part u|F ,a vanishes. Any basis of F ⊥ then provides a solution for the problem of finding a family of appropriate constraints.

Let then F be any subspace of E . The condition can be written as u|F ,a (x) = 0 ∀x ∈ F

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Stability of Discrete Non-conservative Systems

or u|F (x) = u∗|F (x) ∀x ∈ F where u∗|F is the adjoint of u|F . That is given by (u|F (x)|y) = (u∗|F (x|y) = (x|u|F (y)) ∀x, y ∈ F or again thanks to the definition of the compression: (p(u(x))|y) = (x|p(u(y)) ∀x, y ∈ F which, by symmetry (as linear map) of the orthogonal projection, gives (u(x)|y) = (x|u(y)) ∀x, y ∈ F But u(p) = us (p) + ua (p) and the above last property obviously holds for the symmetric part. Thus, the condition can be written as: (ua (p)(x)|y) = (x|ua (p)(y)) ∀x, y ∈ F However, by skew symmetry of ua , then (ua (p)(x)|y) = −(x|ua (p)(y)) ∀x, y ∈ F . Therefore, the condition can be written as: (ua (p)(x)|y) = 0 ∀x, y ∈ F

[6.1]

We now omit the p-dependency and we report this issue at the end of this section. Let Fa = Im(ua ) and Ga = Ker(ua ). As every skew symmetric linear map, ua has an even rank r = dimFa = 2, and its kernel and its image are ⊥

orthogonal spaces. Thus, E = Rn = Fa ⊕ Ga . u2a is a symmetric linear mapping and so it is diagonalizable in an orthonormal basis. Moreover, Ga = Ker(ua ) = Ker(u2a ), the non-zero eigenvalues of u2a , are negative and the associated eigenspaces are two-dimensional and mutually orthogonal. Note these values −μ21 , . . . , −μ2 and E−μ2 the associated eigenspaces for i = 1, . . . , . Each of these spaces i are ua -stable. Because of the ua stability of each of the two-dimensional ⊥

⊥

⊥

⊥

planes of the decomposition Rn = Ga ⊕ Fa = Ga ⊕ E−μ21 ⊕ . . . ⊕ E−μ2 , 

Geometric Degree of Non-conservativity

161

we deduce (Cartan’s theorem) the existence of an orthonormal basis b of E = Rn such that the matrix of ua in b is: ⎛ ⎞ 0 ... ... 0 ⎜ .. . . .. ⎟ ⎜. . . ⎟ ⎜ ⎟ ⎜ ⎟ 0 . . . ⎜ ⎟ ⎜ ⎟ 0 −μ1 ⎜ ⎟ ⎜ ⎟ μ1 0 ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ .. ⎟ ⎝. 0 −μ ⎠ 0 μ 0 and we deduce the P ROPOSITION 6.1.– The GDNC d =  = 12 rank(ua ) and [6.1] hold when the family C = {C1 , . . . , Cd } of constraints is built by choosing ci in E−μ2 for i i = 1, . . . , .  The proof is left to the reader. Because of the lower semi-continuity of the rank, the loading interval I = [0, +∞[ may be decomposed as {0}∪]0, p∗1 ]∪]p∗1 , p∗2 ] . . . ∪]p∗r , +∞[ with p∗1 , . . . , p∗s non-zero singular values of loading. Each p∗k for k = 1, . . . , s is one of the finite (s + 1) discontinuity values of p → rank(ua ) = 2d(p). On each interval Ik =]p∗k , p∗k+1 ], the parameter p has no significance because the GDNC remains constant on ]p∗k , p∗k+1 ]. In the following examples, excepted for Bigoni’s system, 0 is the only singular value. 6.2.2. Examples In this section, we propose a collection of examples consisting of variations on the paradigmatic Ziegler column. The degree of freedom (parameter n) and the nature of the follower force (partially or completely follower force parameter γ) may change. In the most general case, the system Σ consists of n bars OA1 , A1 A2 , · · · , An−1 An with OA1 = A1 A2 = · · · = An−1 An = h linked with n elastic springs with the same stiffness k. As usual, the equilibrium position is θ = (θ1 , θ2 , · · · , θn ) = (0, 0, · · · , 0). P is the follower non-conservative load acting on An . As in the

162

Stability of Discrete Non-conservative Systems



other chapters, adopting a dimensionless format, we use p = Pkh as loading parameter. To show how the algebraic method is performing, it is confronted with the complete calculation only for the 3-dof Ziegler column. First, we investigate the pure Ziegler system and we note that the GDNC is 1 for any number n of rigid bars meaning for any degree of freedom: increasing the number of bars or the degree of freedom does not change its geometric degree of non-conservativity: from the geometric point of view, the Ziegler’s system is then weakly non-conservative. Second, a mechanical system, (which we called Bigoni’s system), is investigated. It is so called because it is a multiple-degree-of-freedom system using a device like the Bigoni’s one at each joint (Bigoni and Noselli 2011). This system appears as a generalization of the n-degree-of-freedom Ziegler column, where the load parameter is itself distributed along the system and may vary on each joint. It also may be considered as a discretized Leipholz column (Leipholz (1987)). It then leads to a multi-parameter dimensionless loading p = (p1 , . . . , pn ). In this case, the geometric degree of non-conservativity increases with the number of the bars and with the dof. Calculations are made only for n = 2 and n = 4. From a geometric point of view, the so-called Bigoni system is essentially stronger and more non-conservative than the Ziegler system. In each case, the stiffness matrix K(p) may be split into an elastic part and a geometric part denoted Kg (p). The elastic part involves only a symmetric contribution to K(p). It is then without significance and can be ignored in these developments. It is the reason why only Kg (p) is given. Moreover, for one-parameter loading, we may suppose Kg is one-degree homogeneous or linear in the variable p and we write Kg (p) = pKg . In this case, the GDNC may be investigated through the constant matrix Kg . In this case, we will write Ka (p) = pKa which is not valid for the full matrix K(p), and by construction, Ka = Kg,a . In order not to introduce a different notation, Ka = Kg,a = Ka (1) = Kg,a (1). 6.2.2.1. 2-dof Ziegler column with complete follower force The geometric part of the stiffness matrix is:  Kg =

1 −1 0 0



Geometric Degree of Non-conservativity

163

Its skew-symmetric part is:  Kg,a = Ka =

0 − 12 1 2 0



The square of Ka is: K2a

1 =− 4



10 01



−μ21 = − 14 , K2a is spheric and E−μ1 (u2a ) = R2 . C1 is then any vector of R2 . Obviously, any constraint converts the free system into a conservative one as a one-dof (elastic) system is always conservative because any continuous function has a primitive. 6.2.2.2. 2-dof Ziegler column with partial follower force The geometric stiffness matrix is:  Kg =

1 −γ 01−γ



Its skew-symmetric part is:  Kg,a = Ka =

0 − γ2 γ 2 0



The square of Ka is: K2a

γ2 =− 4



10 01



Conclusions are similar as above. 6.2.2.3. 3-dof Ziegler column with complete follower force The geometric stiffness matrix is: ⎛

1 ⎝ Kg = 0 0

0 1 0

⎞ −1 −1 ⎠ 0

164

Stability of Discrete Non-conservative Systems

Its skew-symmetric part is: ⎛

Kg,a

0 1⎝ 0 = Ka = 2 1

0 0 1

⎞ −1 −1 ⎠ 0

Obviously, rank (Ka ) = 2 and the GDNC is d = 1. The square of Ka is: ⎛

1 1⎝ 2 1 Ka = − 4 0

1 1 0

⎞ 0 0⎠ 2

⎛ ⎞ 0 Calculations give −μ21 = − 12 and E− 1 (K2a ) = Vec{α = ⎝ 0 ⎠ , Ka α = 2 1 ⎞ ⎛ −1 1⎝ −1 ⎠} leading to two generic constraints converting the system into a 2 0 ⎛ ⎞ 0 conservative one: C = {C1 } or C = {C2 } with C1 = ⎝ 0 ⎠, namely θ3 = 0 1 ⎛ ⎞ 1 and C2 = ⎝ 1 ⎠, namelyθ1 + θ2 = 0. In fact, any linear combination of these 0 two constraints lies in the corresponding plane and may be chosen as a possible constraint converting the system into a conservative one. We now propose to check in this case the results coming from this pure algebraic method, with respect to the direct approach of the problem. The virtual power of the follower force is given by: P ∗ (P ) = Q1 θ1∗ + Q2 θ2∗ + Q3 θ3∗ = P h(sin(θ3 − θ1 )θ1∗ + sin(θ3 − θ2 )θ2∗ )[6.2] The complete nonlinear condition in order to have a conservative system is that there is a function θ = (θ1 , θ2 , θ3 ) → U (θ1 , θ2 , θ3 ) = U (θ) (the potential ∂U function) so that Qk = − ∂θ , which is here obviously impossible without k additional constraint: the free system is non-conservative!

Geometric Degree of Non-conservativity

165

Suppose now the system subjected to a kinematic constraint φ(θ) = 0 which leads to the following condition on the virtual parameters: ∂φ ∗ ∂φ ∗ ∂φ ∗ θ1 + θ2 + θ =0 ∂θ1 ∂θ2 ∂θ3 3

[6.3]

Supposing the problem resolvable with respect to the variable θ3 meaning ∂φ that ∂θ

= 0, we deduce from the implicit functions theorem that (locally in a 3 neighborhood of θ = 0), θ3 = θ3 (θ1 , θ2 ), meaning that, up to order 1,



∂θ3

∂θ3

θ3 = θ3 (θ1 , θ2 ) ≈ θ1 + θ2 = c 1 θ1 + c 2 θ2 ∂θ1 θ=0 ∂θ2 θ=0

[6.4]

Thus, up to order 1, the expansion can be written as: Q1 = Q1 (θ1 , θ2 ) ≈ P h((c1 − 1)θ1 + c2 θ2 ) and Q2 = Q2 (θ1 , θ2 ) ≈ P h((c1 θ1 + (c2 − 1)θ2 ). The condition of conservativity of the loading can then be written as c1 = c2 = c . The kinematic relation is θ3 = c(θ1 + θ2 ), and the quadratic potential is: U (θ) ≈ −P h(

c−1 2 (θ1 + θ22 ) + cθ1 θ2 ) 2

[6.5]

For c = 0, we find the first generic kinematic constraint θ3 = 0, and for c = 0 it is, as expected, a linear combination of both generic constraints. Suppose now that the problem is not resolvable with respect to the variable ∂φ = 0. We then deduce that, linearly, the relation only θ3 , meaning that ∂θ 3 concerns θ1 and θ2 and reads linearly as: ∂φ ∂φ )θ=0 θ1 + ) θ 2 = a1 θ 1 + a 2 θ 2 ≈ 0 ∂θ1 ∂θ2 θ=0

[6.6]

and that a1 θ1∗ + a2 θ2∗ = 0. Resolving these relations, for example, with respect to the variable θ1 (θ1 = −bθ2 = − aa21 θ2 ) and reporting this relation in [6.2] shows that P ∗ (P ) = Q2 θ2∗ with Q2 = Q2 (θ2 , θ3 ) ≈ P h(−b(θ3 − bθ2 ) + (θ3 − θ2 )) ≈ P h((b2 − 1)θ2 + (1 − b)θ3 ) . The condition of integrability is then b = 1, the kinematic relation is θ1 + θ2 = 0 and the potential is nil up to order two. Then, we will come back precisely to the second generic kinematic constraint. To sum up, the direct calculations lead to both generic constraints obtained from our algebraic method.

166

Stability of Discrete Non-conservative Systems

6.2.2.4. 3-dof Ziegler column with partial follower force The geometric stiffness matrix is: ⎛ ⎞ 1 0 −γ ⎜ ⎟ −γ ⎟ Kg = ⎜ ⎝0 1 ⎠ 0 0 1−γ Its skew-symmetric part is: ⎛ 0 0 1 ⎜ 0 0 Kg,a = Ka = γ ⎜ 2 ⎝ 1 1 Obviously, rank (Ka ) = Ka is: ⎛ 1 1 ⎜ 1 1 1 K2a = − γ 2 ⎜ 4 ⎝ 0 0

−1

⎞

⎟ −1 ⎟ ⎠ 0

2 and the GDNC is again d = 1. The square of 0

⎞

⎟ 0⎟ ⎠ 2 2

Calculations give −μ21 = − γ2 and E γ 2 (K2a ) = Vec{α = − 2 ⎞ ⎛ ⎞ ⎛ 0 −1 ⎝ 0 ⎠ , Ka α = γ ⎝ −1 ⎠}, leading to the same two generic constraints as 2 1 0 above, which may convert the system into a conservative one: θ3 = 0 and θ1 + θ2 = 0. 6.2.2.5. n-dof Ziegler column with complete follower force (γ = 1) The stiffness matrix is: ⎛ 2−p −1 ⎜ −1 2 −p ⎜ ⎜ 0 −1 ⎜ K(p) = ⎜ . .. . ⎜ . . ⎜ ⎝ 0 0 0 0

0 −1 2−p .. .

0 0 −1 .. .

0 0

0 0

··· ··· ··· ··· ··· ···

0 0 0 .. . 2−p −1

p p p .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −1 + p ⎠ 1

Geometric Degree of Non-conservativity

167

so that ⎛

0 0 0 .. .

⎜ ⎜ 1 ⎜ ⎜ Ka = ⎜ 2 ⎜ ⎜ ⎝ 0 −1

0 0 0 .. .

0 0 0 .. .

0 0 0 .. .

0 −1

0 −1

0 −1

··· ··· ··· ··· ··· ···

0 0 0 .. . 0 −1

⎞ 1 1⎟ ⎟ 1⎟ ⎟ .. ⎟ = Kg,a .⎟ ⎟ 1⎠ 0

Obviously, rank (Ka ) = 2 and the GDNC is again d = 1. The square of Ka is: ⎛

1 ⎜1 ⎜ 1⎜ ⎜1 2 Ka = − ⎜ . 4 ⎜ .. ⎜ ⎝1 0

1 1 1 .. .

1 1 1 .. .

1 1 1 .. .

1 0

1 0

1 0

··· ··· ··· ··· ··· ···

1 1 1 .. .

0 0 0 .. .

1 0

0 n−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and E− n−1 (K2a ) = Vec{α = Calculations give −μ21 = − n−1 4 4 ⎛ ⎞ ⎛ ⎞ −1 0 ⎜ −1 ⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −1 ⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ , K2a α = 12 ⎜ .. ⎟}, leading to two generic constraints converting the ⎜ . ⎟ ⎜.⎟ ⎜ ⎟ ⎜ ⎟ ⎝ −1 ⎠ ⎝0⎠ 0 1 system into a conservative one: θ1 + . . . + θn−1 = 0 meaning that the motion of An−1 is constrained to remain on the axis OY (see Figure 6.1) and θn = 0 (see Figure 6.2). 6.2.2.6. n-dof Bigoni system Bigoni and Noselli (2011) or n-dof discretized Leipholz column Leipholz (1987) The system Σ consists of n bars OA1 , A1 A2 , · · · , An−1 An with OA1 = A1 A2 = · · · = An−1 An = h linked with n elastic springs with the same stiffness k. Adopting the same device at the end of each bar of Σ leads to a family of follower forces P1 , · · · , Pn (see Figure 6.3). The pure follower forces P1 , P2 , · · · , Pn are applied at the ends of OA1 , A1 A2 , · · · , An−1 An (resp.).

168

Stability of Discrete Non-conservative Systems



Adopting a dimensionless format, we use pi = Pki h for (i = 1, · · · , n) as loading parameters. The stiffness matrix K(p) = K(p1 , p2 , · · · , pn ) can be written as: ⎛  ⎞ n pi −1 + p2

2−

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ K(p) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

i=2

−1

2−

n 

p3

pi −1 + p3

i=3

0

−1

2−

n 

0

−1

.. .

.. .

.. .

0 0 0

0 0 0

0 0 0

p

p5

···

pn−1

pn

p4

p5

···

pn−1

pn

p5

···

pn−1

pn

pi −1 + p5 · · ·

pn−1

pn

.. .

.. .

.. .

.. .

0 0 0

0 0 0

pi −1 + p4

i=4

0

p4

2−

n  i=5

.. .

· · · −1 + pn−1 pn · · · 2 − pn −1 + pn ··· −1 1

Y

An

n k A n
1 n
1

An
2

k

An
3

A4 k

4 k

3

A2 k

A3

2 k 1

A1 X

k

O Figure 6.1. n-dof Ziegler column with complete follower force: θ1 + . . . + θn−1 = 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Geometric Degree of Non-conservativity

p

Y

An n  0 k

An
1

n
1

An
2

k

4

A3

k

3

A2

k

2

A1

k 1

X

k

O Figure 6.2. n-dof Ziegler column with complete follower force: θn = 0

Y

pn

An

pn
1  n

k

An
1

n
1

An
2

k

4 k p2  3

A2

p3

A3

k

2 k

1

p1

A1 X

k

O Figure 6.3. n-dof Bigoni system

169

170

Stability of Discrete Non-conservative Systems

Its skew-symmetric part is: ⎛

0 −p2 −p3 −p4 .. .

p2 0 −p3 −p4 .. .

p3 p3 0 −p4 .. .

p4 p4 p4 0 .. .

p5 p5 p5 p5 .. .

··· ··· ··· ··· .. .

pn−1 pn−1 pn−1 pn−1 .. .

⎜ ⎜ ⎜ 1⎜ ⎜ Ka (p) = ⎜ 2⎜ ⎜ ⎜ ⎝ −pn−1 −pn−1 −pn−1 −pn−1 −pn−1 · · · 0 −pn −pn −pn −pn −pn · · · −pn

⎞ pn pn ⎟ ⎟ pn ⎟ ⎟ pn ⎟ ⎟ .. ⎟ . ⎟ ⎟ pn ⎠ 0

Contrary to the case of a one-parameter loading, we cannot a priori restrict the rank analysis to one of a constant matrix.  rank(Ka (p)) =

n n−1

if n even if n odd

thus:  =

n/2 (n − 1)/2

if if

n even n odd

For n = 2, calculations give:  1 1 0 Ka 2 (p) = − p22 0 1 4 1 −μ21 = − p22 4 K2a (p) is spherical and E−μ21 (K2a (p)) = R2 . α is then any vector in R2 . The geometric degree of non-conservativity is equal to 1, and the constraint is a linear combination of the two generic constraints θ1 = 0 and θ2 = 0: this is any linear constraint! (see Figure 6.4). For n = 4, calculations give: 1 1 1 3 −μ21 = − p24 − p23 − p22 + a 8 4 8 8 3 1 1 1 −μ22 = − p24 − p23 − p22 − a 8 4 8 8

Geometric Degree of Non-conservativity

171

Figure 6.4. 2-dof Bigoni system



9 p44 + 12 p23 p24 + 2 p24 p22 + 4 p43 + 4 p23 p22 + p42 and

  E−μ21 (K2a (p)) = Vec α1 = 2 − p23 p24 a + 2 p3 p2 p24 a − p24 p22

where a =

a + p3 p32 a + p33 p2 a − p24 p42 + 3 p3 p32 p24 + 5 p33 p2 p24 + 2 p3 p2 p44 + p23 p22 p24 + p44 p22 + 7 p44 p23 + 6 p64 + 2 p24 p43 − 2 p44 a + 2 p53 p2    + 3 p33 p32 + p3 p52 , − 2 p33 p2 + 5 p3 p2 p24 + p3 p32   + 3 p24 p22 + p3 p2 a + 3 p44 + 2 p23 p24 − p24 a − p24 + p22 + a ,   − 3 p44 − 2 p23 p24 + 2 p24 p22 + p24 a + 2 p23 p22 + p42 + p22 a    − p4 2 + p2 2 + a , [0] [6.7] 

 E−μ22 (K2a (p)) = Vec α2 = 2 p4 p33 a + p32 a + p23 p2 a + p3 p22 a + 2 p24 p2 a + 2 p24 p3 a + p3 p42 + 2 p44 p2 + 3 p24 p32 + 3 p23 p32 − 7 p33 p24 − 6 p3 p44 − 2 p53 − p24 p22 p3 + 5 p24 p23 p2 − p22 p33    + 2 p43 p2 + p52 , − p4 2 p23 p2 + 5 p24 p2 + p32 − 3 p3 p22 + p2 a      − 3 p24 p3 − 2 p33 + p3 a − p24 + p22 + a , 0 , − 3 p44 − 2 p23 p24   + 2 p24 p22 + p24 a + 2 p23 p22 + p42 + p22 a − p24 + p22 + a [6.8]

172

Stability of Discrete Non-conservative Systems

for pi =

c , we have ih

 √ c2  −95 + 7729 1152 h2  √ c2  −95 − 7729 −μ22 = 1152 h2 √ 7729 c2 1 a= 144 h2

−μ21 =

so √ ⎫ ⎞ 35 7729 + 137 ⎪ ⎪ √ ⎪ ⎜ ⎪ ⎟ ⎬ 6 6 27 + 7729 ⎜ ⎟ √   c 2 1 ⎜ ⎟ E−μ21 (Ka ) = Vec α1 = 6 ⎜ − 24 281 − α , K 7729 a 1 √  ⎟ ⎪ ⎪ h ⎝ 1 ⎪ ⎪ ⎪ ⎪ 57 + 7729 ⎠ ⎪ ⎪ 8 ⎩ ⎭ 0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

√ ⎫ ⎞ 35 7729 + 37 ⎪ ⎪ ⎪ ⎜ 3 27 + √7729 ⎟ ⎪ ⎬ 6 ⎜ ⎟ √   c 2 1 ⎜ ⎟ E−μ22 (Ka ) = Vec α2 = 6 ⎜ − 12 1 − 7729 ⎟ , Ka α2 ⎪ ⎪ h ⎝ ⎪ ⎪ ⎠ ⎪ ⎪ 0√ ⎪ ⎪   ⎩ ⎭ 1 57 + 7729 8 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

In this example, the geometric GDNC is equal to 2: two additional kinematic constraints φ1 (θ1 , . . . , θ4 ) = 0, φ2 (θ1 , . . . , θ4 ) = 0 are then necessary to convert the system into a conservative one, each constraint φi being chosen in E−μ2 (K2a ) for i = 1, 2. Calculations give i

√  √ 35 7729 + 137 1  √ 281 − 7729 θ2 φ1 (θ1 , . . . , θ4 ) = θ1 − 6 27 + 7729 24   √ 1 + 57 + 7729 θ3 8 √  √ 35 7729 + 137 1  √ 1 − 7729 θ2 φ2 (θ1 , . . . , θ4 ) = θ1 − 6 27 + 7729 12   √ 1 + 57 + 7729 θ4 8

Geometric Degree of Non-conservativity

173

or equivalently C = {C1 , C2 } with ⎛ ⎜ ⎜ ⎜ C1 = ⎜ ⎜ ⎜ ⎝

√ 35 7729+137 √ 6 27+ 7729 √ − 281−24 7729 √ 57+ 7729 8

0

⎞

⎛

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎜ ⎜ ⎜ C2 = ⎜ ⎜ ⎜ ⎝

√ 35 7729+137 √ 6 27+ 7729 √ − 1− 127729

0

√ 57+ 7729 8

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

6.3. Calculation of Cg In the previous section, we have computed the GDNC d = 12 rank(Ka ) = 12 rank(ua ) and we have given a way to get an appropriate family of d constraints C = {C1 , . . . , Cd }, making the constrained system ΣC conservative. A natural issue is to question the building of the set Cg of all the families of d constraints C converting Σ into a conservative system ΣC . Such sets of kinematic constraints transforming the physics of the problem is then significant to have information about Cg . The present section is devoted to this problem. It will lead to a description of Cg as a manifold whose dimension is a first measure of the “size” of Cg . We also remark that, as presented in the previous section, the computation of a solution C = {C1 , . . . , Cd } necessitates the calculations of the eigenvalues of u2a . More accurately, we now question the effectiveness of the building of the constraints as proposed above. It used the spectral theorem for K2a . What does the effectiveness for the spectral theorem mean? The usual proof is done by induction on the dimension of the space. For initializing the induction reasoning, the D’Alembert Gauss theorem is used for finding an eigenvalue of the characteristic polynomial of K2a , and this theorem is not effective in the sense where only a numerical method may lead to (an approximation of) the eigenvalues. Therefore, with these tools, the solution just in the linear framework is not effective. Remark however that the constraints are also the critical points of the Rayleigh quotient R associated with K2a , and that only the eigenspaces are interesting and not the eigenvalues −λ2i , i = 1, . . . , s. The use of Rayleigh quotient is then especially relevant and the constraints may be evaluated by successive minimizations of

174

Stability of Discrete Non-conservative Systems T

2

aX R(X) = − XXK T X . By minimax theorem, the constraints are also the solutions of

ˆ min E

max R(X)

dim F =k X∈F \{0}

for k = 1, . . . , n avoiding by this way the use of the D’Alembert Gauss theorem. However, this minimization process gives no analytic explicit result. Removing the calculation of the eigenvalues of u2a will be a real advantage and it is easy to do it by following this building of C. First, we choose a not nil element c1 of Im(ua ). Let F1 = Vect(c1 , ua (c1 )). Then, we choose c2 , not nil in the orthogonal of F1 in Fa = Im(ua ) and c3 not nil in the orthogonal of F1 ⊕ F2 in Fa = Im(ua ) and so on. That produces an appropriate family C = {c1 , . . . , cd } with 2d = rank(ua ) = dim Fa . However, such an approach cannot be used for both describing the set C of all the suitable families and extending this approach to the nonlinear framework. It necessitates another approach of the problem and the use of other mathematical tools we now deal with. This section is mainly in Lerbet et al. (2016). 6.3.1. The GDNC and the exterior calculus To date, a linear constraint has been viewed as a vector c of Rn or C of Mn1 (R) and a family of constraints C has been equivalently viewed as a set {c1 , . . . , cr } of vectors of Rn , a set {C1 , . . . , Cr } of vectors of Mn1 (R), a matrix (C1 . . . Cr ) ∈ Gnr (R) or a vector subspace Vect{c1 , . . . , cr } of Rn . In fact, towards the generalization to a nonlinear framework, it is fruitful to view a kinematic constraint Cj as a linear form c∗i on E = Rn . The canonical scalar product on Rn or E, – namely the Euclidean structure, – allows us to canonically associate a unique vector noted cj with the given linear form c∗j by (in E) c∗j (x) = (x|cj ). This canonical scalar product, which makes canonically isomorphic E and its dual space E ∗ , is here more a trap than a help. From now on, we do not use it. Now, before addressing in a following step the nonlinear case, we still focus on the linear case with the help of the exterior p-forms: we look the matrix Ka

Geometric Degree of Non-conservativity

175

(skew-symmetric part of the stiffness matrix) no longer as the matrix of a linear map of Rn , but as the matrix of an exterior 2-form on Rn . We indifferently note E = Rn and E ∗ its dual space, the vector space of the linear forms on E. Thus, let φ be the exterior 2-form defined on E = Rn by: φ(x, y) = XT Ka Y = (ua (x)|y)

[6.9]

with the usual correspondence between a vector x = (x1 , . . . , xn ) of Rn and ⎛ ⎞ x1 ⎜ .. ⎟ the column vector X = ⎝ . ⎠ of Mn1 (R). Thanks to a basic theorem of xn linear algebra (see (Godbillon 1969), for example), there is a basis B = (e1 , . . . , en ) of Rn and a number r = 2s ≤ n so that φ(e2i−1 , e2i ) = −φ(e2i , e2i−1 ) = 1 for i ≤ s and φ(ei , ej ) = 0 for the other values of i and j. In the dual basis (e∗1 , . . . , e∗n ) of (e1 , . . . , en ), the form φ is then given by: φ = e∗1 ∧ e∗2 + . . . + e∗2s−1 ∧ e∗2s

[6.10]

where ∧ denotes the usual exterior product between exterior forms. According to the previous section and the condition [6.1], we have to find a subspace F of Rn so that φ(x, y) = 0 ∀x, y ∈ F and the constraints, viewed now as linear forms c∗1 , . . . , c∗s ∈ E ∗ , then belong to F ⊥ , the orthogonality being then understood in the sense of the duality: F ⊥ = {c∗ ∈ E ∗ | c∗ (x) = 0 ∀x ∈ F }. Choosing each constraint c∗i in the subspace < e∗2i−1 , e∗2i > spanned by e∗2i−1 and e∗2i in E ∗ leads to the wanted result. Indeed, suppose to simplify that c∗i = e∗2i−1 for all i = 1, . . . , s and that G is the vector subspace spanned by (e∗2i−1 )1≤i≤s . Let x, y be in F = G⊥ where the bidual E ∗∗ is canonically identified with the space E itself, this identification being here possible, because of the finite dimension of E. Thus, by use of [6.10], e∗2i−1 ∧ e∗2i (x, y) = e∗2i−1 (x)e∗2i (y) − e∗2i (x)e∗2i−1 (y) = 0 − 0 = 0, and thus φ(x, y) = 0. If c∗i is any vector in < e∗2i−1 , e∗2i >, a similar proof as hereafter for differential forms may be used and is not reproduced.

176

Stability of Discrete Non-conservative Systems

The effectiveness of the building of the constraints is now brought back to the one of the basis B = (e1 , . . . , en ). The proof is done by induction on the dimension n of E (see, for example, Godbillon 1969, pp. 30–31). This proof is effective, and the following paragraph will highlight how it is performing on an example. Before dealing with the example, the interesting issue is to characterize all the solutions, namely to find Cg . 6.3.2. Set of solutions For describing the set of solutions, usual concepts of symplectic geometry are used. For this purpose, we first brought back the issue in the usual framework of symplectic geometry. The exterior 2-form φ does not necessarily define a symplectic structure on Rn because it has not necessarily a maximal rank, namely φ may be degenerate. For instance, that necessarily ˜ is a occurs when n is odd. Let then F be the kernel of φ. Then, (Rn /F, φ) ˜ 2s-dimensional symplectic vector space, where φ is canonically defined by ˜ x, y ¯ ) = φ(x, y) with x (resp. y) any vector of the class x ¯ (resp. y ¯ ). φ(¯ n Remark that thanks to the canonical scalar product (. | .) on R , we could choose the orthogonal F ⊥ of F for the scalar product as a “canonical” supplementary space of F in Rn and φF ⊥ the restriction of φ to F ⊥ . Then, (F ⊥ , φF ⊥ ) could be also a 2s-dimensional symplectic vector spaces and there should be three possible meanings for orthogonality in F ⊥ : duality, scalar product and φ-orthogonality. However, the scalar product has no meaning on ˜ and only φ-orthogonality and orthogonality for duality keep useful (Rn /F, φ) n ˜ Moreover, towards generalizing the reasoning from vector on (R /F, φ). spaces to manifolds, the natural euclidean structure of Rn does no longer exist in the tangent and co-tangent spaces. It is then more judicious to avoid the use of this structure. However, the orthogonal for the duality of any subspace G will still be denoted by G⊥ . If W is any subspace of Rn /F , the φ-orthogonal or symplectic orthogonal of W is the vector subspace of Rn /F noted W ⊥φ defined by: ˜ x, y ¯ ) = 0 ∀¯ ¯ ∈ W ⊥φ ⇐⇒ φ(¯ y∈W x Because φ˜ is non-degenerate, the map φ : Rn /F → (Rn /F )∗ defined by ˜ x, y ¯ ) ∀¯ ¯ ∈ Rn /F is a canonical isomorphism and then = φ(¯ x, y φ (W ⊥φ ) = W ⊥ for any W subspace of Rn /F . φ (¯ x)(¯ y)

Geometric Degree of Non-conservativity

177

A subspace L of Rn /F is called Lagrangian if L = L⊥φ . It is also often called a Lagrangian plane, even though dim L = s. A straightforward calculation shows that the dual basis of any basis of any Lagrangian subspace is a solution of the initial problem. More accurately, if L1 is a Lagrangian plane, then there is a Lagrangian supplementary space L2 of L1 in Rn /F . If (e2i−1 )1≤i≤s (resp. (e2i )1≤i≤s ) is any basis of L1 (resp. L2 ), then the dual basis (e∗2i−1 )1≤i≤s of (e2i−1 )1≤i≤s in L∗1 is a family of constraints solution of the mechanical issue. In fact, this process realizes any solution. Thus, by this process, the set of all the solutions of the problem is in a bijective relationship with the set of the bases of all Lagrangian planes of Rn /F , but the geometrical meaning of the solutions lies in the set of Lagrangian spaces ˜ Λ(φ) has been deeply investigated, especially in noted Λ(φ) of (Rn /F, φ). relationship with the theory of Maslov index (see Souriau 1976, for example, for a highlighting presentation of the construction of this index). Λ(φ) is an s(s+1) 2 -dimensional submanifold of the Grassmannian manifold of all s-planes ˜ of Rn /F and called the Lagrangian Grassmannian of (Rn /F, φ). Let Λ(s) be the Lagrangian Grassmannian manifold of the usual R-symplectic vector space (Cs , ω) with its canonical symplectic structure. This manifold may be explicitly described by and identified with the set Us (s) of unitary symmetric complex matrices of Ms (C). A Lagrangian plane L ∈ Λ(s) is identified with the matrix UL ∈ Us (s) by the following way: ¯ where X ¯ is the conjugate column vector X of x ∈ L ←→ X = UL X, s ˜ then x ∈ C . If u is a symplectomorphism from (Cs , ω) onto (Rn /F, φ), Λ(φ) = u(Λ(s)), which achieves the complete and explicit description of the set of solutions, namely Λ(φ). Note also that there is an explicit representation of matrices of Us (s). If U belongs to Us (s), then U = X + iY with X, Y two real symmetric matrices of size s. Since U is unitary, X2 + Y2 = Is and XY = YX. There is a common basis that makes X and Y diagonal, and an R-orthogonal matrix O so that X = OT diag(xi )O and Y = OT diag(yi )O. We deduce that x2i + yi2 = 1 for all i = 1, . . . , s, and the problem can be parametrized by xi = ri cos αi and yi = ri sin αi . We deduce that U = OT RVO with R = diag(ri ) and V = diag(eiαj ). 6.3.3. An example We now illustrate these results with the four-degree-of-freedom Bigoni system, which is shown in Figure 6.3 with n = 4. The usual case of a unique

178

Stability of Discrete Non-conservative Systems

follower force at the extremity (namely the usual Ziegler system) is not very interesting since the geometric degree of non-conservativity is then reduced to 1 (see the previous section) with an obvious solution θ4 = 0. In this case, the direction of the external force remains constant and this force becomes conservative! According to the previous notations, it means that dim F = 2, s = 1 and that the Lagrangian spaces are one-dimensional subspace of the ˜ Finding the one-dimensional two-dimensional symplectic space (R4 /F, φ). Lagrangian subspaces of this symplectic space is equivalent to finding the set of linear kinematic constraints so that when the system undergoes one of these constraints, it becomes conservative. Consider now the general case. The force system is then set up by p = (p1 , . . . , p4 ) (see Figure 6.3 with n = 4). The skew symmetric matrix Ka (p) can be written as ⎛ ⎞ 0 p2 p3 p4 1 ⎜ −p2 0 p 3 p4 ⎟ ⎟ Ka (p) = ⎜ ⎝ 2 −p3 −p3 0 p4 ⎠ −p4 −p4 −p4 0 meaning that if (1 , . . . , 4 ) is the canonical basis of R4 , then 1 φ = (p2 ∗1 ∧ ∗2 + p3 ∗1 ∧ ∗3 + p3 ∗2 ∧ ∗3 + p4 ∗1 ∧ ∗4 2 + p4 ∗2 ∧ ∗4 + p4 ∗3 ∧ ∗4 ) Here, det(Ka (p)) = p22 p24 , showing that φ is not degenerate when p2 p4 = 0, which is now supposed. Thus, with the previous notations, F = {0} and R4 /F = R4 and (R4 , φ) becomes a four-dimensional symplectic space. We want to find a basis (e1 , . . . , e4 ) so that φ = e∗1 ∧ e∗2 + e∗3 ∧ e∗4

[6.11]

Let us choose e∗1 = ∗1 , e∗2 =

p2 ∗ ∗ p3 p4 2 , e3 = ∗1 + ∗2 + ∗3 , e∗4 = ∗3 + ∗4 2 2 2

[6.12]

Geometric Degree of Non-conservativity

179

Then, [6.11] holds and [6.12] means that the constraints x1 = 0, x1 +x2 +x3 = 0 convert the system into a conservative one. We focus now on the set of all solutions, namely here on the set of Lagrangian planes. Let ⎛

0 ⎜ −1 J=⎜ ⎝ 0 0

1 0 0 0

0 0 0 −1

⎞ 0 0⎟ ⎟ 1⎠ 0

be the matrix of the R-symplectic four-dimensional vector space (C2 , ω) in its canonical basis v1 = (1, 0), v2 = (i, 0), v3 = (0, 1), v4 = (0, i). Let A be the matrix of a symplectomorphism u from (C2 , ω) onto (R4 , φ) in their respective canonical bases. Then, the relation ω(x, y) = φ(u(x), u(y)) for all x, y ∈ C2 leads to the usual relation J = AT Ka A. However, if Q denotes the changeof-basis matrix to pass from (∗i )1≤i≤n to (e∗i )1≤i≤n , then P = (QT )−1 is the corresponding change-of-basis matrix to pass from (i )1≤i≤n to (ei )1≤i≤n , and the formula [6.11] is then J = PT Ka P. [6.12] means that ⎞ 1 0 1 0 ⎜ 0 p2 1 0 ⎟ 2 ⎟ Q=⎜ ⎝ 0 0 1 p3 ⎠ 2 0 0 0 p24 ⎛

It follows that A = (QT )−1 and calculations give: ⎛

1 0 1 ⎜ 0 ⎜ p2 A = 2⎜ 1 ⎝ −1 − p2 p3 p4

p3 p2 p 4

⎞ 1 0 0 0⎟ ⎟ ⎟ 1 0⎠ p3 1 − p4 p4

[6.13]

2×3 It now remains to parametrize the three (= 2 )-dimensional 2 Grassmannian Λ(2) of Lagrangian planes of C , which is done as above in the general case through a parametrization of Us (2). Let

 U=

u1 u2 u3 u4

∈ Us (2)

180

Stability of Discrete Non-conservative Systems

Because U is symmetric, u2 = u3 and because U is unitary, the following three independent relations hold: | u1 |2 + | u2 |2 = 1, | u4 |2 + | u2 |2 = 1, u1 c(u2 ) + u2 c(u4 ) = 0. Then, | u1 |=| u4 | and the problem is parametrized by u1 = cos αeiα1 ,u4 = cos αeiα4 , u2 = u3 = sin αeiα2 . The third relation u1 c(u2 ) + u2 c(u4 ) = 0 then becomes cos α sin α(ei(α1 −α2 ) + ei(α2 −α4 ) ) = 0. 4 Generically that leads to α2 = α1 +α + (2k + 1) π2 : the parametrization is 2 given by α, α1 , α4 and the corresponding matrix U (α, α1 , α4 ) can be written as:   α1 +α4 cos αeiα1 sin α iei 2 U (α, α1 , α4 ) = α1 +α4 sin α iei 2 cos αeiα4 A vector v = (x, y) ∈ Λ(2) if and only if  α1 +α4   cos αeiα1 x c(x) (−1)k sin α iei 2 = α1 +α4 y c(y) (−1)k sin α iei 2 cos αeiα4

[6.14]

which gives 

x = cos αeiα1 c(x) + (−1)k sin α iei y=

(−1)k

sin α ie

Writing v = (x, y) =

i

α1 +α4 2

4 

c(x) +

α1 +α4 2

c(y)

[6.15]

cos αeiα4 c(y)

xi vi in the canonical basis of the R-symplectic

i=1

four-dimensional vector space (C2 , ω), c(v) = (c(x), c(y)) =

4 

(−1)i+1

i=1

xi vi and [6.15] then becomes: X = (x1 x2 x3 x4 )T ∈ ker B(α, α1 , α4 ) with B=

⎛

cos α cos α1 − 1

cos α sin α1

⎜ cos α sin α1 − cos α cos α1 − 1 ⎜ ⎜ 4 (−1)k sin α cos α1 +α4 ⎝ (−1)k+1 sin α sin α1 +α 2 2 (−1)k sin α cos

α1 +α4 2

(−1)k sin α sin

α1 +α4 2

α1 +α4 2 α1 +α4 2

(−1)k+1 sin α sin k

(−1) sin α cos

(−1)k sin α cos k

(−1) sin α sin

α1 +α4 2 α1 +α4 2

cos α cos α4 − 1

cos α sin α4

cos α sin α4

− cos α cos α4 − 1

⎞ ⎟ ⎟ ⎟ ⎠

[6.16]

Geometric Degree of Non-conservativity

181

These equations define a plane P = P (α, α1 , α4 ) of R4 , and L(α, α1 , α4 ) = A(P (α, α1 , α4 )) (with A given by [6.13]) is then the Lagrangian plane of (R4 , Ka ) defined by α, α1 , α4 , which achieves the parametrization of the set of all solutions of the problem. To sum up, this “symplectic” solution is simple in comparison to the one proposed in the previous section that involved tedious calculations for the calculation of only one solution. This method allows us moreover to get the set of all solutions as well. Thus, it shows that this way is strongly more efficient than the way using the spectral theorem for K2a , by avoiding the calculations of the eigenvalues and the eigenvectors of K2a . Having a more efficient method of building the set of appropriate constraints by converting the non-conservative system into a conservative system is certainly a good (at least a better!) way for the extension to the nonlinear case, which is addressed in the following section. 6.4. Extension to the nonlinear framework We address now a very significant issue that will be crucial regarding the direction of future investigations and research works. It deals with the extension to the nonlinear framework. Actually the set of significant results that have been obtained for the last couple of years about KISS and GDNC are, on the one hand, analytic and general but, on the other hand, limited by the linear framework of their implementation. To improve their impact, one of the most important tasks is to generalize the framework for their applications. If the ultimate goal deals with elastoplasticity, the natural first step will consist of extending these results to the discrete nonlinear differentiable framework, namely for us, to the nonlinear discrete hypoelasticity (Truesdell 1955, Truesdell 1963). The other parallel extension will involve continuous media and thus infinite dimension spaces. To investigate nonlinear discrete mechanical systems, one of the most useful languages is the language of analytic mechanics and more specifically the Lagrangian mechanics language. It is in fact underlying in the linear approach as quickly mentioned at the beginning of Chapter 4 devoted to divergence KISS. Indeed, linear investigations take place in the tangent space Tq∗ M to the configuration manifold M at the equilibrium configuration q ∗ . Because this vector space Tq∗ M is the only one involved during the linear

182

Stability of Discrete Non-conservative Systems

investigations, there is no real difficulty to identify Tq∗ M with Rn . However, since we intend to extend our results to the nonlinear case, we must admit to make differential calculus on the manifold configuration M. This also means that the objects involved in the linear framework must be eventually replaced by other ones adapted to the manifold language. The whole extension to the nonlinear framework remains still today a challenge, especially concerning the KISS issue. We may mention, for example, the recent paper Lerbet et al. (2018) that lays the foundations for dealing with the nonlinear KISS. However, surprisingly, the GDNC issue which is purely geometric may be “quite naturally” extended to the nonlinear framework. A short last section explains why the extension to the nonlinear framework is so direct for the GDNC issue and so tedious for the KISS issue. Before starting, we emphasize that we will deal here with purely local investigations. Global nonlinear issues fall in the differential topology field and are beyond the scope of this chapter. For the basic notions of differential geometry that are used, see, for example, Lang (1999) or Godbillon (1969). 6.4.1. Nonlinear issue and notations We then consider now a mechanical discrete system Σ so that its configuration manifold M is a C ∞ real n-dimensional manifold and (q = (q1 , . . . , qn ), U ) denotes generically a local coordinate system. That means that U is an open set of M and there is a function φ : U ⊂ M → Rn (C ∞ ) with for all m ∈ M, φ(m) = q. We suppose that the system is subjected to a positional force system Π so that Π is described by a differential 1-form QΠ on M whose local expression in (q, U ) is given by: QΠ =

n 

QΠ,k (q)dqk

[6.17]

k=1

According to Godbillon (1969), the exact vocable for describing Π is a semi-basic 1-form. That means that Π is described by a 1-form (or a Pfaff form) ωΠ on the total space T M of the tangent bundle τ M such that ω is on the image of the canonical vertical operator. The local expression of a such semi-basic form is then ωΠ =

n  k=1

ωΠ,k (q, q)dq ˙ k

[6.18]

Geometric Degree of Non-conservativity

183

Thanks to the positional property of the forces Π, ωΠ,k (q, q) ˙ only depends on q, meaning only on the projection T M → M of a point of T M and may be viewed as a function on the basis M of the tangent bundle. Then, [6.18] takes the form [6.17]. If there is no ambiguity, we omit the force system Π and write Q instead QΠ . Any p-dimensional (C ∞ real) submanifold N of M may locally in (q, U ) be described by a family (f1 , . . . , fn−p ) of n − p independent (C ∞ real) functions defined on φ(U ) so that for all q ∈ φ(U ), −1 −1 m = φ (q) ∈ N ∩ φ (U ) ⇔ f1 (q) = . . . = fn−p (q) = 0. The mechanical system whose N is the configuration space is called a subsystem ΣC of Σ and functions (f1 , . . . , fn−p ) are called the local (nonlinear) expressions of the constraints C. We indifferently note ΣC or ΣN . The positional force system Π acting on Σ is said to be conservative if there is a function h on M so that in any coordinate system (q, U ): QΠ,k (q) =

∂h(φ−1 (q)) ∂qk

That is equivalent to Q = dh on M. It also means that Q is locally exact on M. Let us emphasize that we will note by the letter d the differential of a map and by d the exterior derivative of any k-form on M. When h is a function on M, namely a 0-form, its differential dh coincides with its exterior derivative dh. It implies that Q is a closed differential 1-form and then dQ = 0 on M. Locally, by use of Poincaré’s theorem and after having chosen an appropriate coordinate system, we may suppose the reciprocal property valid on each U of ˆ covering M. As mentioned above, the global issue involving the the atlas EA topology of M is out of the scope of this work. In Godbillon (1969), the vocable “conservative” means only that the semi-basic Pfaff form Q is closed. The term “Lagrangian system” is reserved for the case where this form is exact in the framework of positional force system. In this section, only the local extension to the nonlinear case is investigated. The issue is then the following one: is there a submanifold N of M so that the physical action Π on ΣN is conservative? If so, find all the possible N with the highest possible dimension p so that the number of constraints is the smallest possible (because every subsystem built by adding kinematical constraints to a conservative system is again a conservative system!).

184

Stability of Discrete Non-conservative Systems

6.4.2. Link with the linear framework We focus now on the link with the linear framework involved in the previous section. Suppose that there is an equilibrium configuration m∗ ∈ M of Σ so that the linear approximation may be used at m∗ . Let (q, U ) be a local coordinate system of M at m∗ so that, investigating the evolution of the system in the vicinity of m∗ , we may suppose that m ∈ M remains in U . We are then brought back, thanks to the coordinate system (q, U ), to calculations in the V = φ(U ) open subset of Rn . The fundamental point will be to ensure that the involved tools will not depend on the choice of the coordinate system (q, U ) at m∗ . We will discuss this point at the end of this chapter. Let x = q − q ∗ = (x1 , . . . , xn ) ∈ V . We then suppose that QΠ may be ˜ Π , leading to: approximated by its Taylor expansion to the first-order Q ˜ Π (x) = Q

n n   ∂QΠ,k (q ∗ ) ∗ (QΠ,k (q ) + x )dxk ∂q k=1

[6.19]

=1

which means that, if u = (u1 , . . . , un ) ∈ Rn ≈ Tq∗ M ˜ Π (x)(u) = Q

n 

(QΠ,k (q ∗ ) +

k=1

n  ∂QΠ,k (q ∗ ) =1

∂q

x )uk

˜ Π (x) at 0 becomes: and the exterior derivative of x → Q ˜ Π (0) = dQ

n   ∂QΠ,k (q ∗ ) ∂QΠ, (q ∗ ) ( − )dx ∧ dxk ∂q ∂qk k=1 j

Calculations then give (for the sake of simplicity  = 1): dQΠ (θ) =



pj cos(θj − θi )dθi ∧ dθj

i 0

[8.48]

with the boundary conditions ⎧ ⎪ ⎪ ⎨

w(0) = 0 w (0) = 0 P w() + EIw () = λw (), λ > 0 ⎪ ⎪ ⎩ P 2 (3) (3) 2 w () + EIw () = λw (), λ > 0

[8.49]

In Appendix 8.7.2, the calculations are done and allow us to exhibit a sequence ((λn (P ))n≥0 , (wn (P ))n≥0 ). Because of the structure of A˜s given by [8.44], we deduce from Proposition 8.8 that: P ROPOSITION 8.9.– For P < P2∗ , the solution of the EP is a sequence of ((λn (P ))n≥0 , (wn (P ))n≥0 ) and the family (wn (P ))n≥0 is an orthogonal Hilbert basis. C OROLLARY 8.1.– P1∗ = P2∗ and the solution expanded above is the optimal one in the whole.  It directly results from Propositions 8.3 and 8.9.

Continuous Divergence KISS

249

8.6. Stability issues As was clearly claimed at the beginning of this book, only linear divergence-type stability is investigated, which means that mass effects are not taken into account. Nevertheless, investigating links with linear dynamic stability more accurately is an interesting challenge. For both finite (Ziegler column) or infinite (Beck column) unconstrained mechanical systems subjected to a complete follower force, no linear divergence-type instability may occur and only linear flutter-type dynamic instability occurs. For example, for a 2 dof Ziegler system, the flutter-type instability occurs for the value p∗f ≈ 2, 54, whereas for the Beck column, it occurs for p∗f ≈ 20.05 with dimensionless load parameter p. The instability for p > p∗f ≈ 20.05 may be found in a 1952 paper by Beck (Beck 1952), whereas the stability for p < p∗f ≈ 20.05 has been proved by Carr and Malhardeen in (Carr and Malhardeen 1979). Note however that the proof in (Carr and Malhardeen 1979) does not use a direct Lyapunov method. Here, obviously, we do not pretend a priori that the constrained system is stable for P < P ∗ but that it is divergence-stable for P < P ∗ and that it is divergence-unstable for P = P ∗ . Nevertheless, thanks to a special property of c∗ , the constrained Beck column will turn out to be stable (flutter-type and divergence-type as well). We stress again that we investigate only linear stability. Let us start by the 2 dof Ziegler system. Its (linear) dynamic equation is ¨ + K(P )X = 0 MX

[8.50]

M is the mass matrix. It depends on the mass distribution and is symmetric positive definite. K(P ) is given by [8.3]. When the system is subjected to the kinematic constraint c described by the column vector C, then the dynamic equation of the constrained system can be written as: 

¨ + K(P )X + λC = 0 MX CT X = 0

[8.51]

The optimal constraint C∗ is given by C∗ = K(P ∗ )X∗ with X∗ = 0 ∈ ker Ks (P˜ ∗ ) = {0} because Ks (P˜ ∗ ) is degenerated

250

Stability of Discrete Non-conservative Systems

(see section 8.2.3). Multiplying the left-hand side by Ks (P ∗ )K−1 (P ∗ ) removes the Lagrange multiplier and [8.51] then becomes: ¨ + Ks (P˜ ∗ )X = 0 M∗ X

[8.52]

with M∗ = Ks (P ∗ )K−1 (P ∗ )M. It is then easy to check that t → U(t) = tX∗ is an unbounded solution of [8.52]. It is also interesting to check that U is an unbounded solution of [8.51] with the Lagrange multiplier λ(t) = −t at each time t, which proves the linear dynamic instability by divergence of the corresponding equilibrium of the constrained system. Indeed, [8.51] can also be written in the form  ¨ + X + λX∗ = 0 K−1 (P ∗ )MX [8.53] ∗ T ∗ T (X ) K(P ) X = (X∗ )T K(P ∗ )X = 0 and at each time t:  −1 ∗ ¨ + U + λ(t)X∗ = (t + λ(t))X∗ = 0 K (P )MU ∗ T (X ) K(P ∗ )T U = t(X∗ )T K(P ∗ )(X∗ ) = 0

[8.54]

¨ = 0. with λ(t) = −t ∀t ≥ 0 because U However, it is worth noting that the direct Lyapunov method is ineffective here because of the non-symmetry of M∗ or of K−1 (P ∗ )M. Another way to keep a symmetric mass matrix for the constrained system might consist of using the compressions of the mass and stiffness matrices as in Lerbet et al. (2015), for example. In this way, the size of the dynamic system is reduced and the compressed mass matrix remains symmetric positive definite. Unfortunately, the compressed stiffness matrix generally remains non-symmetric, which again prevents us from using the direct Lyapunov method. However, we may extract, by a similar way, an unbounded solution of the constrained linear dynamic equation of the Beck column: ρ

∂2w ∂4w ∂2w + EI 4 + P 2 + λc(w) = 0 ∀x ∈]0, [ ∀t ≥ 0 2 ∂t ∂x ∂x

[8.55]

with the same above boundary conditions [8.2], we may rewrite it as: w(0, t) =

∂2w ∂3w ∂w (0, t) = (, t) = (, t) = 0 ∀t ≥ 0 ∂x ∂x2 ∂x3

[8.56]

Continuous Divergence KISS

251

and with the linear constraint c(w) = 0, where c is a linear functional. Suppose now that P = P1∗ = P2∗ = P ∗ and that c = c∗ is built as above: π2 2 c∗ (w) = w (). Let (t, x) → tu(x) = t(1 − cos(π x ) + 2 2 x ) be a displacement field suggested by the function w∗ ∈ ker As (P ∗ ) given by [8.39]. The reader may readily check that u satisfies the boundary conditions [8.56] (whereas tw∗ (x) does not satisfy [8.56]!). However, ρ

2 ∂4u ∂2u ∗∂ u + EI + P + λc∗ (u) ∂t2 ∂x4 ∂x2   x  x  π2  2 π4 π2 ∗π + P 2 cos π + = t −EI 4 cos π + λ(t)      

=t

π2 π2 + λ(t)  

= (t + λ(t))

π2 

=0 for the value λ(t) = −t of the Lagrange multiplier and because of the critical given by [8.38]. That proves the linear value of the loading P ∗ = P2∗ = π 2 EI 2 dynamic divergence instability of the constrained system by c∗ for P ∗ = P2∗ = π 2 EI . However, the constraint c∗ surprisingly leads to a 2 conservative constrained system equivalent to the system [8.1]+[8.4] with Pc = P ∗ constrained by c∗ . The stability of such a system is easier to study because only divergence-type instabilities may occur. It follows that the constrained system is stable (both divergence and flutter-) if P < P ∗ = π 2 EI 2 obviously in the linear framework. Besides, an energy functional E may be built for this constrained system:  ˆ    ∂w 2 1 1 E(w) = ρ dx + φs (P )(w, w) ∀w ∈ F = ker c∗ [8.57] 2 ∂t 2 0 To conclude, we may remark that, contrary to the finite-dimensional case and because of the boundary conditions, we cannot choose exactly u(t, x) = tw∗ (x) with w ∈ ker As (P ∗ ), but we have to introduce a corrective π2 2 term t 2 2 x to build a non-bounded solution of dynamic equations. We also < 20.5 EI . However, no general stability may observe that P ∗ = π 2 EI 2 2

252

Stability of Discrete Non-conservative Systems

analysis of any constrained Beck column has been done, which should lead to a general flutter KISS investigation and which is out of the scope of this paper. A general flutter KISS investigation was conducted for 3 dof Ziegler system in Lerbet et al. (2016a). Finally, the fact that c∗ leads to a conservative system is pure chance and is related to the so-called geometric degree of non-conservativity (GDNC), which was deeply investigated in Lerbet et al. (2014) for linear systems and also in Lerbet et al. (2016b) for nonlinear systems, but only in the discrete mechanic framework. 8.7. Appendices 8.7.1. Equivalence constraints

between

different

forms

of

kinematic

In this appendix, we prove the equivalence of the forms [8.15] and [8.14] of any kinematic constraint c ∈ V  . First, because a ∈ L2 ([0, ]) if a ∈ V, it is obvious that [8.15] is a particular case of [8.14] with a1 = a , a´2 = 0, a3 = 0. Reciprocally, by linearity,´ it is sufficient to show, ´     apart that 0 a1 (x)v (x)dx, 0 a2 (x)v (x)dx and 0 a3 (x)v(x)dx can be ´   written as 0 a (x)v  (x)dx with a an element of V. We do it by two steps. ´ First, we show that the last two types of integrals 0 a2 (x)v  (x)dx and ´ a3 (x)v(x)dx may be written with the same form as the first, namely ´ ´0  (x)dx, and we do it explicitly only for the first,  a (x)v  (x)dx. a (x)v 1 0 0 2 We omit justifying the validity of the derivatives and integrals. The fact that the is bounded is fundamental. In the second step, we show that any ´  interval  0 a1 (x)v (x)dx has the form of [8.15], which achieves the proof. Step 1 ´  be any integral of this form. Put Let 0 a´2 (x)v (x)dx  a1 (x) = − x a2 (t)dt. Then, a1 (x) = −a2 (x) and a1 () = 0. After integrating by parts: ˆ 0



a2 (x)v  (x)dx = [−a1 (x)v  (x)]0 − ˆ =

0



a1 (x)v  (x)

ˆ

 0

(−a1 (x))v  (x)dx

Continuous Divergence KISS

since a1 () = 0 et v  (0) = 0. For an integral like ´ ´ put a1 (x) = x ( x a3 (u)du)dt.

´ 0

253

a3 (x)v(x)dx, we have to

Step 2 ´ Now let 0 a1 (x)v  (x)dx be any integral of the considered form with a1 ∈ ´x ´t L2 ([0, ]). Put a(x) = 0´( 0 a1 (u)du)dt. Then, a is two times differentiable x and a(0) = 0, a (x) = 0 a1 (u)du, a (0) = 0 and a (x) = a1 (x) (almost ´ everywhere on [0, ]). We deduce that 0 a1 (x)v  (x)dx has the appropriate ´ form 0 a (x)v  (x)dx with a(0) = a (0) = 0, namely that a ∈ V. 8.7.2. Eigenvalues problem for A˜s (P ) In this appendix, we solve the EP given by [8.48] and [8.49]. [8.48] can also be written as: (EI − λ)w(4) (x) + P w (x) = 0

∀x ∈]0, [, λ > 0

[8.58]

If λ = EI and since P = 0, then w (x) = 0, and because w ∈ V, the natural boundary condition w(0) = w (0) = 0 leads to w = 0. Thus, λ = 0. Two different cases may occur: λ < EI and λ > EI. 8.7.2.1. E(A˜s (P ))∩ ]0, EI[ P , namely λ = EI − ωP2 and, because λ > 0, then Put here ω 2 = EI−λ P ω 2 > EI . Then, the solution w has the following form:

w(x) = ax + b + c cos(ωx) + d sin(ωx)

∀x ∈]0, [

[8.59]

The two first boundary conditions of [8.49] lead to b = −c and a = −dω, which gives w(x) = −dωx − c + c cos(ωx) + d sin(ωx)

∀x ∈]0, [

[8.60]

We successively deduce w (x) = −dω − cω sin(ωx) + dω cos(ωx)

∀x ∈]0, [

[8.61]

254

Stability of Discrete Non-conservative Systems

and w (x) = −cω 2 cos(ωx) − dω 2 sin(ωx)

∀x ∈]0, [

[8.62]

and lastly w (x) = cω 3 sin(ωx) − dω 3 cos(ωx)

∀x ∈]0, [

[8.63]

Thus, the last two boundary conditions of [8.49] are given by: P P w() + 2 w () = 0, 2 ω P  P w () + 2 w(3) () = 0 2 ω or 1 (−dω − c + c cos(ω) + d sin(ω)) 2 1 + 2 (−cω 2 cos(ω) − dω 2 sin(ω)) = 0, ω 1 (−dω − cω sin(ω) + dω cos(ω)) 2 1 + 2 (cω 3 sin(ω) − dω 3 cos(ω)) = 0 ω or, again, 1 (−dω − c + c cos(ω) + d sin(ω)) − c cos(ω) − d sin(ω) = 0, 2 1 (−d − c sin(ω) + d cos(ω)) + c sin(ω) − d cos(ω)) = 0 2 Thus, (c, d) must be a non-trivial solution of 

c(cos(ω) + 1) + d(ω + sin(ω)) = 0 c sin(ω) − d(cos(ω) + 1) = 0

[8.64]

Continuous Divergence KISS

255

Put σ = ω. In order to have a non-trivial solution, the determinant of [8.64] must vanish, which can be written as: σ sin σ + 2 + 2 cos σ = 0

[8.65]

Remark that [8.64] is not equivalent to [8.35], whereas [8.65] is the same equation as [8.37]. Contrary to the computation of P2∗ , we now have to investigate all the solutions σn for n ≥ 1. The subset of positive solutions of [8.65] is built by an increasing sequence (σn )n≥1 whose ((2n − 1)π)n≥1 is a sub-sequence of solutions. The approximations of the four first solutions are: σ1 = π, σ2 ≈ 5, 596, σ3 = 3π, σ4 ≈ 12, 242. We then have σ2n−1 = (2n − 1)π and (2n − 1)π < σ2n < (2n + 1)π ∀n ≥ 1 The corresponding eigenvalues are λn = EI − corresponding eigenvectors are

P 2 for n ≥ 1, and the σn2

 x x 1 1 + − cos σ2n  2 2   x 1 ∀x ∈]0, [ ∀n ≥ 1 + sin σ2n σ2n 

w2n (x) = −

[8.66]

and x w2n−1 (x) = cos((2n − 1)π ) − 1 

∀x ∈]0, [ ∀n ≥ 1

[8.67]

because, from [8.65] for ωn  = (2n − 1)π, we deduce that d = 0 and c is any. Remark that, thanks to P < P2∗ , λn > 0 for all n ≥ 1. 8.7.2.2. E(A˜s (P ))∩ ]EI, +∞[ Here put ω 2 = following form:

P −EI+λ ,

namely λ = EI + ωP2 . Then, the solution w has the

w(x) = ax + b + c cosh(ωx) + d sinh(ωx)

∀x ∈]0, [

[8.68]

The two first boundary conditions of [8.49] lead to b = −c and a = −dω, which gives w(x) = −dωx − c + c cosh(ωx) + d sinh(ωx)

∀x ∈]0, [

[8.69]

256

Stability of Discrete Non-conservative Systems

We deduce successively w (x) = −dω + cω sinh(ωx) + dω cosh(ωx)

∀x ∈]0, [

[8.70]

and w (x) = cω 2 cosh(ωx) + dω 2 sinh(ωx)

∀x ∈]0, [

[8.71]

∀x ∈]0, [

[8.72]

and, lastly, w (x) = cω 3 cosh(ωx) + dω 3 sinh(ωx)

Thus, the last two boundary conditions of [8.49] can be written as: P P w() − 2 w () = 0, 2 ω P  P (3) w () − 2 w () = 0, 2 ω or 1 (−dω − c + c cosh(ω) + d sinh(ω)) 2 1 − 2 (cω 2 cosh(ω) + dω 2 sinh(ω)) = 0, ω 1 (−dω + cω sinh(ω) + dω cosh(ω)) 2 1 − 2 (cω 3 sinh(ω) + dω 3 cosh(ω)) = 0 ω or, again, 1 (−dω − c + c cosh(ω) + d sinh(ω)) − c cosh(ω) − d sinh(ω) = 0, 2 1 (−d + c sinh(ω) + d cosh(ω)) − c sinh(ω) − d cosh(ω)) = 0 2

Continuous Divergence KISS

257

Thus, (c, d) must be a non-trivial solution of 

c(cosh(ω) + 1) + d(ω + sinh(ω)) = 0 c sinh(ω) + d(cosh(ω) + 1) = 0

[8.73]

Put σ = ω. In order to have a non-trivial solution, the determinant of [8.73] must vanish ,which can be written as: −σ sinh σ + 2 + 2 cosh σ = 0

[8.74]

which has only one solution σ0 ≈ 2, 399 in R∗+ . There is only one P 2 corresponding eigenvalue λ0 = EI + 2 , and the corresponding σ0 eigenvector is x x 1 x 1 1 sinh(σ0 ) w0 (x) = − + − cosh(σ0 ) +  2 2  σ0  8.7.3. Figures

c P P θ2

k

θ1

k

Figure 8.1. 2 dof Ziegler’s system

∀x ∈]0, [ [8.75]

258

Stability of Discrete Non-conservative Systems

P c P

w(x)



E, I

x

Figure 8.2. Beck’s column

8.8. References Absi, E. and Lerbet, J. (2004). Instability of elastic bodies. Mechanic Research Communication, 31, 39–44. Allaire, G. (2005). Analyse numérique et optimisation: une introduction à la modélisation mathématique. Les Éditions de l’École polytechnique, Paris. Beck, M. (1952). Die Knicklast des einseitig eingespannten tangential gedrückten Stabes. Z. Angew. Math. Phys., 3, 225–228. Bolotin, V.V. (1963). Non-conservative Problems of the Theory of Elastic Stability. Pergamon Press, London. Carr, J., Malhardeen, Z.M. (1979). Beck’s problem. SIAM J. Appl. Math., 37(2), 261– 262. Challamel, N., Nicot, F., Lerbet, J., and Darve, F. (2009). On the stability of nonconservative elastic systems under mixed perturbations. EJECE, 13(3), 347–367.

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Challamel, N., Nicot, F., Lerbet, J. and Darve, F. (2010). Stability of non-conservative elastic structures under additional kinematics constraints. Engineering Structures, 32, 3086–3092. De Langre, E. and Doaré, O. (2015). Edge flutter of long beam under follower loads. JoMMS, 10(3). Doaré, O. (2010). Dissipation effect on local and global stability of fluid-conveying pipes. J. Sound Vib., 329(1), 72–83. Doaré, O. (2014). Dissipation effect on local and global fluid-elastic instabilities. In Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations. Kirillov, O.N. and Pelinovsky, D.E. (eds), 67–84, John Wiley & Sons, Hoboken. Doaré, O. and de Langre, E. (2002). Local and global instability of fluid-conveying pipes on elastic foundations. J. Fluid. Struct., 16(1), 1–14. Elishakoff, I. (2005). Controversy associated with the so-called “follower forces”: Critical overview. Applied Mechanics Review, 58(1–6), 117–142. Herrmann, G. (1971). Dynamics and stability of mechanical systems with follower forces. Monography, Report NASA CR-1782, 234. Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. J. Mech. Phys. Solids, 7, 209–225. Jurisits, R. and Steindl, A. (2011). Mode interactions and resonances of an elastic fluid-conveying tube. PAMM, 11(1), 323–324. Kirillov, O.N. (2013). Nonconservative Stability Problems of Modern Physics. De Gruyter, Berlin. Kirillov, O.N., Challamel, N., Darve, F., Lerbet, J., and Nicot, F. (2014). Singular divergence instability thresholds of kinematically constrained circulatory systems. Physics Letters A, 378(3), 147–152. Kirillov, O.N. and Verhulst, F. (2010). Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella? Z. Angew. Math. Mech., 90(6), 462–488. Koiter, W.T. (1996). Unrealistic follower forces. Journal of Sound and Vibration, 194, 636–638. Knops, R.J. and Payne, L.E. (1968). Stability in linear elasticity. International Journal of Solids and Structures, 4(12), 1233–1242.

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Lebedev, L.P., Vorovitch, I.I., and Gladwell, G.M.L. (2003). Functional analysis. Application in Mechanics and Inverse Problems. Kluwer Academic Publishers, Berlin. Lerbet, J., Aldowaji, M., Challamel, N., Nicot, F., Prunier, F., and Darve, F. (2012). Ppositive definite matrices and stability of nonconservative systems. Z. Angew. Math. Mech., 92(5), 409–422. Lerbet, J., Kirillov, O., Al-Dowaji, M., Nicot, F., Challamel, N., and Darve, F. (2013). Additional constraints may soften a non-conservative structural system: Buckling and vibration analysis. International Journal of Solids and Structures, 50(2), 636– 370. Lerbet, J., Challamel, N., Kirillov, O., Nicot, F., and Darve, F. (2014). Geometric degree of non conservativity. Math. and Mech. of Complex Systems, 2(2), 123–139. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2015). Variational formulation of divergence stability for constrained systems. Appl. Math. Modell., 39(23–24), 7469–7482. Lerbet, J., Hello, G., Challamel, N., Nicot, F., and Darve, F. (2016a). 3-dimensional flutter kinematic structural stability. Nonlinear Analysis: Real World Applications, 29, 19–37. Lerbet, J., Challamel, N., Nicot, F., and Darve, F. (2016b). Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable non Linear case. Appl. Math. Modell., 40(11–12), 5930–5941. Lerbet, J., Challamel, N., Nicot, F., Darve, F. (2016c). Kinematical structural stability. Discrete and Continuous Dynamical Systems – Series S (DCDS-S) of American Institute of Mathematical Sciences (AIMS), 9–2 June. Mennicken, R. and Möller, M. (2003). Non-self-adjoint Boundary Eigenvalue Problems. North-Holland Mathematics Studies, 192, Elsevier. Movchan, A.A. (1959). The direct method of Lyapunov in stability problems of elastic systems. PMM Journal of Applied Mathematics and Mechanics, 23(3), 686–700. Movchan, A.A. (1960). Stability of processes with respect to two metrics. PMM Journal of Applied Mathematics and Mechanics. 24(6), 1506–1524. Moiseyev, N.N. and Rumyantsev, V.V. (1968). Dynamic Stability of Bodies Containing Fluid. Springer, Berlin. Nemat-Nasser S. and Herrmann G. (1966). Adjoint systems in nonconservative problems of elastic stability. AIAA J., 4(12), 2221–2222.

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Index

1-form, 182, 183, 187, 189, 191, 195 2-form, 175, 176, 184, 185, 193, 195 B, C, D Beck column, 48, 223–226, 230–232, 239–241, 245, 246, 249, 250, 252 Beck’s problem, 258 bifurcations, 24, 25, 28, 57, 99, 200, 259, 261, 262 Bigoni system, 162, 167, 169, 171, 177, 188 Bresse–Timoshenko beam, 201, 206 buckling, 1, 9, 11, 13, 18, 22–24, 27, 28, 33, 39–41, 45, 47, 48, 199–202, 205–210, 215–218, 225, 260 Cauchy stress tensor, 59, 66 compressions, 83, 105, 109, 120, 121, 126, 127, 150, 159, 160, 194, 236, 237, 250 conditional KISS, 108, 157 conservative systems, 1, 2, 4–6, 8, 18, 28, 38, 45, 49, 104, 108, 116, 122, 123, 156, 260 constitutive relation 59, 67, 81 continualization, 211–214, 216

continuous linear form, 232, 236 system, 1–3, 28–30, 38, 84, 85, 87, 105, 109, 115, 121, 210, 211, 218 Cosserat, 199–201, 209, 218 chain, 1, 9, 18–20, 23, 25, 26, 32, 89–92, 99, 112, 199–203, 206, 209, 215, 218 rod, 18, 28, 200 critical load, 40, 93, 98, 103, 107, 149, 225, 243 Darboux’s theorem, 158, 186, 187, 191, 194 destabilizing constraints, 114, 155, 228, 229, 244, 246 diagonalizable, 81–83, 127, 160 discrete beam interactions, 19 systems, 1–3, 37, 38, 70, 80, 84, 86, 201, 208, 224, 225, 244, 247 disturbance, 77–79, 92, 94, 95, 99 velocity, 2, 3, 66, 78, 79, 82, 91, 92, 94, 95, 96, 99

264

Stability of Discrete Non-conservative Systems

divergence instability, 7, 8, 39, 81, 104, 107, 108, 125, 225, 245, 251, 259 KISS, 104, 108, 109, 111, 114, 118, 119, 121, 125, 149, 158, 181, 194, 223, 237 stability, 38, 48, 104, 107, 110, 114–116, 118, 121–123, 149, 156, 196, 224, 226, 227, 260 duality, 158, 175, 176, 194 dynamic instability, 7, 8, 125, 249, 250 E, F, G eigenvalue, 7, 8, 11, 13, 22, 74, 79, 81, 83, 86, 87, 92, 94, 99, 127, 135, 138, 160, 173, 174, 181, 187, 230, 238, 246, 248, 253, 255, 257, 260 eigenvector, 74, 86, 135, 138, 181, 238, 246, 247, 255, 257 orthonormal, 83, 117, 128, 131–136, 139, 160, 161 elastic structures, 69, 122, 155, 196, 259 energy function, 202–204, 212–215, 251 enriched continuum, 200, 201, 215 Euler–Bernoulli beam model, 215 Eulerian formulation, 67, 68 exterior calculus, 158, 174 derivative, 183, 184, 187, 195 external forces, 42, 69, 72, 73, 77, 78, 81 failure, 57, 68, 230 flutter instability, 8, 48, 104, 107, 109, 125–127, 240 KISS, 119, 126–128, 137–139, 142, 143, 148, 149, 153, 155, 194, 252 load, 13, 15, 17, 35–37, 42, 49, 149

follower load, 9, 15, 28, 29, 38, 41, 42, 44, 48, 88, 190, 259 generalized stress, 68 geometric degree of nonconservativity, 119, 157, 162, 170, 178, 188, 190, 252, 260 GDNC, 157–159, 161, 162, 164, 166, 167, 172, 173, 181, 182, 194, 252 gradient elasticity, 201, 209, 211, 215, 216, 218 granular, 199–203, 206–211, 215–218 structural systems, 199, 216 Grassmannian, 126, 128, 129, 131, 132, 177, 179 H, I, K, L Hencky bar-chain, 9, 19, 29, 200 Hilbert basis, 238, 246, 248 eigenvectors, 85, 86, 135, 138, 181, 238, 246, 255 space, 121, 226, 233, 235–237, 262 Hill’s stability, 104, 111, 121, 122 hyperelastic, 9, 71 hypoelastic, 122, 181, 197 infinite dimension space, 119, 181, 223 instability, 2, 7, 8, 13, 37, 40, 41, 43–45, 49, 70, 84, 86, 94, 99, 100, 103, 104, 106–109, 125–127, 199, 201, 218, 225, 240, 245, 249–251, 258, 259 interaction, 49, 80, 202, 206, 208–211, 218, 223, 259, 261 isotropic cone, 85, 86, 111, 113, 137 kinetic energy, 5, 57, 60, 63, 64, 67–69, 74, 77–79, 84–86, 91, 92, 94, 99, 100

Index

KISS, 49, 104, 105, 107–109, 111, 113–115, 118, 119, 121, 122, 125–128, 132, 137–139, 142, 143, 148, 149, 151, 153, 155, 157–159, 181, 182, 194, 195, 223, 235–237, 239, 240, 252 Lagrange multipliers, 105, 116 Lagrangian planes, 158, 177, 179, 181 lattice shear system, 200, 201 limit states, 68 local stability, 87 loss of uniqueness equilibrium solution, 81, 92, 94 static problem., 92, 94 Lyapunov stability, 2, 11, 38, 49, 78 M, N, O, P, R material point scale, 67 non-associate, 122 non-conservative systems, 2, 8, 19, 28, 31, 37, 38, 46, 49, 76, 79, 87, 104, 116, 125, 149, 157, 139, 240 objective time derivative, 59, 66 partial KISS, 108, 157 Piola–Kirchhoff stress tensor, 60, 62, 66 rate-independent, 3, 59, 81, 104 S, T, U second-order work criterion, 38, 40, 45, 47–49, 103, 111, 114, 115, 121, 134, 230, 232, 237, 239 semi-Lagrangian formulation, 67 shear interactions, 19, 199, 200, 210, 217, 218 skew-symmetric part, 163, 164, 166, 170, 175, 185, 194 sphere, 115, 126, 128, 129, 132, 144, 151, 153 Sphere manifold, 126

265

stability, 1–5, 7–9, 25, 29, 30, 36–40, 42, 44, 47–49, 57, 69, 74, 78, 81–84, 87, 90, 93, 103–108, 110, 111, 114–116, 118, 121–123, 125–127, 149, 153, 155, 156, 160, 195–197, 201, 209, 210, 218, 223, 224, 226, 227, 249, 251, 258–262 static instability, 7, 8, 125 Stiefel manifold, 126, 128, 132 stiffness matrix, 6, 8, 10–12, 37, 42, 70, 73, 79, 81, 87, 89, 99, 106, 109, 126, 151, 158, 162, 163, 166, 168, 175, 185, 225, 227, 247, 250 strong formulation, 227 symmetric part, 8, 38, 70, 76, 79, 87, 91, 99, 111, 121, 134, 158–160, 184, 194, 195, 228, 234, 239, 240 symplectic structure, 176, 177 tangent bundle, 182, 183, 188 universal KISS, 108 V, W, Z variational formulation, 105, 109, 115, 119, 123, 156, 196, 226–228, 230–232, 235, 239, 240, 260 kinematic constraints, 37, 38, 40, 48, 49, 69, 82–84, 86, 100, 104, 106–108, 110, 116, 118, 157, 159, 172, 173, 178, 193, 194, 232, 233, 237, 243, 252 weak formulation, 226–228, 231, 232, 235, 239 Winkler foundation, 203, 205–208, 215, 218 Ziegler column, 41, 70, 88, 90, 91, 94–98, 100, 109, 149, 161–163, 166, 168, 225, 249 Ziegler system, 38, 109–111, 125, 149, 150, 162, 178, 188, 190, 223, 224, 226–228, 244, 246, 249, 252