Loss of uniqueness and bifurcation vs
instability : some remarks
René Chambon
Denis Caillerie
Cino Viggiani
Laboratoire 3S GRENOBLE FRANCE
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
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
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. 4
• R.F. Scott in his Terzaghi lecture 1985
Introduction
• The concept of stability is one of the most
unstable concept in the realm of Mechanics
A. Needleman
5
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
6
Lyapunov stability analysis
• Definition: the motion of a mechanical
system is stable if
such that
and
implies
7
Lyapunov stability analysis
• Definition: an equilibrium is stable if
such that
and
implies
8
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
9
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
10
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
11
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
12
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)
13
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
Coulomb cone
weight
14
Lyapunov stability analysis
Comments on the Lyapunov methods:
Coulomb cone
weight
However this mechanical system is stable
15
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
16
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
17
Hill criterion of directional
stability (small strain)
The positiveness of the second order work everywhere
implies the sufficient Hill condition of stability
18
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
19
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
20
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.
21
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
22
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
23
Simple mechanical example 1
y
OG1=mOA
AG2=mAB
la
G1
q
O
mass: m
length: l
a
A
G2
B
Fi
x
24
Simple mechanical example 1
• Kinetic energy
• Potential energy
25
Simple mechanical example 1
• Equation of movement
• Linearized equations in the vicinity of
26
Simple mechanical example 1
• Lyapunov stability: first method
– Characteristic equation
– Stability threshold
27
Simple mechanical example 1
• Second order work criterion: definite
positiveness of the symmetric part of the
stiffness matrix
28
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.
29
Simple mechanical example 1
• Stability and bifurcation diagram
unstable
stable
30
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
31
Simple mechanical example 2
y
Fu
OG1=mOA
AG2=mAB
B
Cb
-l(b-a)
-la
Ca
G1
a
O
mass: m
length: l
lu=BA
G2 b
A
x
32
Simple mechanical example 2
• Kinetic energy
• Potential energy
• Virtual power of force
33
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
34
Simple mechanical example 2
• Lyapunov stability
– Characteristic equation
– Discriminant
– Stability threshold
35
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
36
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
37
Simple mechanical example 2
• Comment: here is a cycle which can supply
mechanical energy to the system
DW>0
DW=0
DW=0
38
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
39
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
40
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
41
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.
42
Bifurcation studies
• Material bifurcation :
different classes of assumed modes
– Controllability (including invertibility)
– Shear band analysis.
43
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
44
Concluding remarks
• A result at the material level does not imply
the same result at a global level
45
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
46
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.
47
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).
48
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.
49