Loss of uniqueness and bifurcation vs instability : some remarks

Loss of uniqueness and bifurcation vs instability : some remarks

René Chambon

Denis Caillerie

Cino Viggiani

Laboratoire 3S GRENOBLE FRANCE

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• Introduction

• Lyapunov stability analysis

• Hill approach

• Absi stability « definition »

• Simple mechanical examples and comments

• Bifurcation studies

• Concluding remarks

3


• Introduction


• Hill approach

• Absi stability «definition»




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Introduction• Finally, the concentration of effort on stress strain relations so far has

been directed at representing the behaviour of stable materials –those exhibiting volume contraction on drained shear, or, at most small expansions. There has been a good deal of debate about unstable behaviour that develops in association with volume expansion. Loading of such a soil is accompanied by local inhomogeneities in the form of slip lines, shear bands, or « bifurcation » as they are now commonly called. Thus the single-element behaviour referred to in the foregoing breaks down as strains and displacements become localized in the shear zone. This behaviour has been examined by Vardoulakis (1978,1980) and worried about by other investigators. It occurs in real soil in nature very frequently, is the source of many soil engineering problems, and so far is not represented in a single soil model. At present, it is also difficult to see how a suitable model could be implemented in a finite element code, since each individual element must have the opportunity of developing shear bands as the loading progresses, and their position cannot be predicted in advance.

• R.F. Scott in his Terzaghi lecture 1985

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Introduction

• The concept of stability is one of the most unstable concept in the realm of Mechanics

A. Needleman

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• Introduction


• Hill approach





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Lyapunov stability analysis

• Definition: the motion of a mechanical system is stable if

such that

and

implies

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• Definition: an equilibrium is stable if

such that

and

implies

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The first method of Lyapunov: stability of a linear system:

• if every real part of the solutions of the characteristic equation is negative then the equilibrium position of the system is stable.

• conversely if the real part of at least one root of the characteristic equation is positive then the equilibrium is unstable

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Lyapunov stability analysisThe first method of Lyapunov: stability of a non linear system:

• if the real part of every solution of the characteristic equation associated with the linearized problem is negative, then the equilibrium is stable

• if the real part of one solution of the characteristic equation associated with the linearized problem is positive, then the equilibrium is unstable

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The procedure just mentioned certainly involves an important simplification, especially in the case where the coefficients of the differential equations are constant. But the legitimacy of such a simplification is not at all justified a priori, because for the problem considered there is then substituted another which might turn out to be totally independent. At least it is obvious that, if the resolution of the simplified problem can answer the original one, it is only under certain conditions and these last are not usually indicated

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Lyapunov stability analysisLyapunov second method:

• Let a system submitted to a set of forces,

• some of them are conservative and are then related to a potential energy

• the others are dissipative

• Then an equilibrium state corresponding to a minimum of the potential energy is stable

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Lyapunov stability analysisComments on the Lyapunov methods:

• The first method– Equations of motion have to be linearized– This is never the case in problem dealing with geomaterial except

if they are viscous, So this method can be used only for viscous materials, but neither for elasto plastic nor for hypoplastic nor for damage models

• The second method– Practically, it is only useful for fully conservative systems (i.e.

without dissipative forces)

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• Generally solid friction allows stability in the engineering meaning

but stability cannot be studied neither with the first method nor for the second method of Lyapunov

Coulomb coneweight

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However this mechanical system is stable

Coulomb coneweight

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• Introduction


• Hill approach





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Hill approach

• It is not clearly within the Lyapunov framework• It assumes that the more critical paths to compute

the excess of internal energy are monotonous linear loading paths. This defines according to Petryk and Bigoni the "directional stability".

• The studied materials obeys normality rule (or some equivalent property) which induces serious problems to apply this results to geomaterials

• External forces ares dead loads

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Hill criterion of directional stability (small strain)

The positiveness of the second order work everywhere implies the sufficient Hill condition of

stability

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Petryk contribution : material stability

• he defines clearly the studied system which allows him to specify the class of instability studied, namely the material instability

• he puts forward clearly the mathematical problem (equilibrium or deformation process) and the perturbation acting on the system

• he takes care of the deficiency of the linearized problem due to the incremental non linearity and tries to study the complete problem

• he provides a simple example which shows clearly that there is not a unique stability criterion

• many results obtained by Petryk can be proved only because the studied materials are associative

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• Introduction


• Hill approach





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Absi stability «definition»

• This work is representative of many confused works done all along the century about stability

• When submitted to a small perturbation (which can partly concerns the external forces and the positions of the system), the system goes to a new equilibrium position close to the previous one, the solution is unique (and the corresponding forces are finite), when the perturbation is removed, the system goes back to its initial position.

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Absi stability «definition»

• the occurrence of instability has invariably been taken as synonymous with the existence of infinitesimally near positions of equilibrium; this may be quite unjustified when the system is non--linear or non—conservative

• It is not however, the present intention to review a confuse literature nor to attempt any correlations with experiments but to make a fresh start and establish a broad basic theory free at least from the objections mentioned

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• Introduction


• Hill approach

• Absi stability definition




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AB

G1

G2

OG1=OA

AG2=AB

Simple mechanical example 1

O

mass: mlength: l

x

y

F i

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• Kinetic energy

• Potential energy

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• Equation of movement

• Linearized equations in the vicinity of

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• Lyapunov stability: first method – Characteristic equation

– Stability threshold

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Simple mechanical example 1• Second order work criterion: definite

positiveness of the symmetric part of the stiffness matrix

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• Lyapunov stability: second method – equilibrium conditions:– stability threshold:

– for which gives (fortunately) the same threshold as the other method

– for the other solutions of the equilibrium conditions (which are available as soon as )

which is always met.

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• Stability and bifurcation diagram

stable

unstable

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• Comments– this is a simple model of elastic buckling– such a situation is typical of elastic media– around the stable equilibrium positions the movement is a

vibration with exchange between kinetic and potential energy

– instability means here that the kinetics energy is growing, due to the transformation of potential energy into kinetics one; this is possible because the system is not in a position corresponding to the minimum of potential energy

– viscous damping does not change essentially the results

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mass: mlength: l


O

A

B

x

y

C

C

G1

G2

OG1=OA

AG2=AB

F u

lu=BA

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Simple mechanical example 2• Kinetic energy

• Potential energy

• Virtual power of force

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• Equation of movement

• Linearized equations in the vicinity of

• Notice that the stiffness matrix is not symmetric, this is due to the fact that is not conservative

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• Lyapunov stability – Characteristic equation

– Discriminant

– Stability threshold

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Simple mechanical example 2• Second order work criterion: definite positiveness

of the symmetric part of the stiffness matrix

• Comparison:– if we can have

– if we can have

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• Comments– the instability encountered in this example is called flutter

instability (it is very important in aircraft mechanics)– it consists of a quasi periodic movement with a growing

amplitude– instability means here that the kinetics energy is growing,

this is possible because some external force is not conservative and can supply energy to the system

– as seen before there is a link between this property of external forces and the non symmetry of the stiffness matrix

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Simple mechanical example 2• Comment: here is a cycle which can supply

mechanical energy to the system

W=0 W=0W>0

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Are these studies useful for cohesive frictional materials?

• The second method of Lyapunov is almost useless (friction sliding)

• Unfortunately the first method is also almost useless– equations are not linearizable, there is only “directional

linearization”– there is no proof that such a linearization gives us some indication

for the complete (non linearized) problem– In these linearized problem the non symmetry of the stiffness

matrix is due to the non associativeness of the constitutive equation and not to a non conservative external force, so where is the energy coming from in the so called flutter instability sdudies?

• Linearization can only be consider as an heuristic method

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• Introduction


• Hill approach





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Bifurcation studies• General bifurcation studies

– It has been proved that the positiveness of the second order work for any point of the computed structure, and for any strain rate ensures the uniqueness of the solution of the rate problem.

– This is proved for associative materials, and for non associative materials

– We proved that the same result holds for a rate boundary value problem involving hypoplastic models

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Bifurcation studies

• Material bifurcation

The system is an homogeneous piece of material (geomaterial) and non uniqueness is searched. We propose to call such problems material bifurcation problems.

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Bifurcation studies

• Material bifurcation :

different classes of assumed modes– Controllability (including invertibility)– Shear band analysis.

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• Introduction


• Hill approach





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Concluding remarks

• A result at the material level does not imply the same result at a global level

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Concluding remarks

• All these studies can be useful but we have to know what is studied

• Here is a check list asked to people speaking about stability or bifurcation

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Concluding remarks

• What are you studying?– stability– uniqueness.

• In any case which is your system?– a complete system– an element and so your study is a material study.

• In any case, do you use?– a justified linearization– a partial linearization (unjustified)– the complete non linear model.

• In any case,– precise the interactions with the outside.

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Concluding remarks• In any case if you end up with a condition or a criterion is

it?– a sufficient condition– a necessary condition– both.

• In any case– do you assume a specific mode– do you restrict your study to a class of modes– do you perform a complete study.

• If you are studying stability– give the perturbation (input) and a measure of its magnitude– give the measure of the criterion (output).

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Concluding remarks

• Let us use the same word for the same concept and only one word for one concept, this will avoid useless, endless and boring discussions.

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Loss of uniqueness and bifurcation vs instability : some remarks