Loss of uniqueness and bifurcation vs instability : some remarks

1
Loss of uniqueness and bifurcation vs instability
some remarks

• René Chambon
• Denis Caillerie
• Cino Viggiani
• Laboratoire 3S GRENOBLE FRANCE

2
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability  definition 
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

3
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability definition
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

4
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

5
Introduction

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

6
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability definition
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

7
Lyapunov stability analysis

• Definition the motion of a mechanical system is
  stable if
• such that
• and
• implies

8
Lyapunov stability analysis

• Definition an equilibrium is stable if
• such that
• and
• implies

9
Lyapunov stability analysis

• 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

10
Lyapunov stability analysis

• The 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

11
Lyapunov stability analysis

• 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

12
Lyapunov stability analysis

• Lyapunov 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

13
Lyapunov stability analysis

• Comments 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)

14
Lyapunov stability analysis

• Comments on the Lyapunov methods
• 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

15
Lyapunov stability analysis

• Comments on the Lyapunov methods
• However this mechanical system is stable

16
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability  definition 
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

17
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

18
Hill criterion of directional stability (small
strain)

The positiveness of the second order work
everywhere implies the sufficient Hill condition
of stability
19
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

20
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability definition
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

21
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.

22
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 nonconservative
• 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

23
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability definition
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

24
Simple mechanical example 1
y
x
O
25
Simple mechanical example 1

• Kinetic energy
• Potential energy

26
Simple mechanical example 1

• Equation of movement
• Linearized equations in the vicinity of

27
Simple mechanical example 1

• Lyapunov stability first method
• Characteristic equation
• Stability threshold

28
Simple mechanical example 1

• Second order work criterion definite
  positiveness of the symmetric part of the
  stiffness matrix

29
Simple mechanical example 1

• 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.

30
Simple mechanical example 1

• Stability and bifurcation diagram

unstable
stable
31
Simple mechanical example 1

• 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

32
Simple mechanical example 2
y
x
O
33
Simple mechanical example 2

• Kinetic energy
• Potential energy
• Virtual power of force

34
Simple mechanical example 2

• 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

35
Simple mechanical example 2

• Lyapunov stability
• Characteristic equation
• Discriminant
• Stability threshold

36
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

37
Simple mechanical example 2

• 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

38
Simple mechanical example 2

• Comment here is a cycle which can supply
  mechanical energy to the system

DW0
DW0
DWgt0
39
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

40
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability  definition 
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

41
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

42
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.

43
Bifurcation studies

• Material bifurcation
• different classes of assumed modes
• Controllability (including invertibility)
• Shear band analysis.

44
Loss of uniqueness and bifurcation vs instability
some remarks

• Introduction
• Lyapunov stability analysis
• Hill approach
• Absi stability  definition 
• Simple mechanical examples and comments
• Bifurcation studies
• Concluding remarks

45
Concluding remarks

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

46
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

47
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.

48
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).

49
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.