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

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

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

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

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

Lyapunov stability analysis • Definition: the motion of a mechanical system is stable if such that and implies

Lyapunov stability analysis • Definition: an equilibrium is stable if such that and implies

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

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

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

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

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)

Coulomb cone weight 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 Lyapunov stability analysis Comments on the Lyapunov methods: However this mechanical system is stable

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

Hill criterion of directional stability (small strain) The positiveness of the second order work everywhere implies the sufficient Hill condition of stability

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

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.

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

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

OG1=mOA AG2=mAB la G1 a A B F i q G2 mass: m length:l Simple mechanical example 1 y x O

Simple mechanical example 1 • Kinetic energy • Potential energy

Simple mechanical example 1 • Equation of movement • Linearized equations in the vicinity of

Simple mechanical example 1 • Lyapunov stability: first method • Characteristic equation • Stability threshold

Simple mechanical example 1 • Second order work criterion: definite positiveness of the symmetric part of the stiffness matrix

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.

Simple mechanical example 1 • Stability and bifurcation diagram unstable stable

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

lu=BA F u OG1=mOA AG2=mAB B Cb G2 b -l(b-a) Ca G1 -la A a mass: m length:l Simple mechanical example 2 y x O

Simple mechanical example 2 • Kinetic energy • Potential energy • Virtual power of force

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

Simple mechanical example 2 • Lyapunov stability • Characteristic equation • Discriminant • Stability threshold

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

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

Simple mechanical example 2 • Comment: here is a cycle which can supply mechanical energy to the system DW=0 DW>0 DW=0

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

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

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.

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

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

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

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.

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).

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.

Loss of uniqueness and bifurcation vs instability : some remarks