First Order Bifurcation Landscape in a 2D Geometry: The Example of Solid Friction

(1)

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First Order Bifurcation Landscape in a 2D Geometry:

The Example of Solid Friction

Arnaud Tanguy, Ph. Nozières

To cite this version:

Arnaud Tanguy, Ph. Nozières. First Order Bifurcation Landscape in a 2D Geometry: The Example of Solid Friction. Journal de Physique I, EDP Sciences, 1996, 6 (10), pp.1251-1270.

�10.1051/jp1:1996136�. �jpa-00247244�

(2)

First Order Bifurcation Landscape in

_a

2D Geo~netry:

The Exa~nple of Solid Friction

A.

Tanguy

_(~>*) and Ph. Nozières

(~)

(~) LPMMH, ESPCI, 10 _rue Vauquelin, 75005 Paris, France

(~) Institut Laue Langevin, BP. 156X, 38042 Grenoble Cedex 9, France

(Received

3 May1996, received in final form 17 June 1996, accepted 27 June 1996)

PACS.05.70Fh Phase transitions: general aspects

Abstract. The generalization to 2D of _a mortel for quasistatic friction reveals the impor-

tance of anisotropy ⁱⁿ the pinmng potential between the asperities _on slider and track. This

amsotropy, which breaks trie rotational symmetry of the potential, changes, qualitatively _as weII

as quantitatively, the dissipative behaviour of the asperity. This study _aims to show, with the

help of _an exarnple, the unfolding of bifurcation diagrams for first order transitions when trie

state space is changea from _one to two dimensions.

Résumé. La généralisation à 2D d'un modèle de frottement quasistatique met _en évidence

l'importance de l'anisotropie du potentiel d'accrochage entre _une aspérité du patin et _une _as-

périté de la piste. ^Cette amsotropie, _qui ^brise ^la symétrie de rotation du potentiel, change _qua-

litativement et quantitativement le comportement dissipatif de l'aspérité- Cette étude précise

à l'aide d'un exemple le déploiement du diagramme de bifurcation des transitions du premier ordre lorsque l'espace des états _passe de _une à deux dimensions.

Introduction

Reference

iii

describes _a model for solid friction based _on _a

hysteretic

elastic mechanism which involves the asperities at the surfaces of two solids in contact. This model is very

simple

in the

case of _a

single

contact. The contact is characterized

by

_a nominal coordinate _p, which _measures the relative

positions

of

asperities facing

each other, _on the track and shder

respectivelj,,

_m the

absence of _any elastic _response. When the slider _moves, _p is swept ^from -cc to +cc. Due to the pinning

potential acting

_on the contact, bath slider and track

undergo

elastic deformation and the relative coordinate _moves from _p to _r ₌ p + u. The total

potential

_energy of the

asperity

_may be written _as

~ ~~~~ ~ _~~ ~~~ ~~~

where

VIT)

is the

pinmng potential

and is _an elastic coefficient. TO _a first

approximation,

we

neglect

the interaction between contacts, _SO _we _cari

keep

_p _as _a central parameter. ^If is small

enough

"Soft"

system),

the

equilibrium position

_r has a range of

multistability, Ieading

to a

hysteretic

_regime

responsible

for the

dissipation

in the quasistatic Iimit.

(During

(*) Author for correspondence je-mail:

tanguyiÉmanet.pmmh.espar.fr)

(3)

1252 JOURNAL DE PHYSIQUE I N°10

v'(x)

(Bl)

(82)

Fig. I. Graphical construction of the equilibrium positions _m the lD _case. V'

(~)

is cut by _a straight

hne of slope

j-À

intersecting ^the x-axis at p. GK and IE _are the hysteretic jumps, FJ is the Maxwell

plateau.

the discontinuons

jump

from _Due stable

equilibrium

state to

another,

_a bunch of

phonons

_is radiated into the solid,

Ieading

to irreversible behavior.

The

analysis given

in

iii

is one-dimensional: it Ieads _to the classical bifurcation scheme of _a first order transition [2]. The aim of this paper is to

generalize

the discussion to the 2D _case.

Independently

of the

example

of

friction,

the issue is to

generalize

the "bifurcation

Iandscape"

from _a ID system to a 2D system. This is _a

question

of

general

interest, whose _answer is Rot _so obvions. We shall attempt to

give

_a

qualitative

_answer. For the sake of

simplification,

we first discuss the _case of _a

repulsive pinmng potential.

After

reviewing

the

ID-case,

_we discuss the _case of _an isotropic

2D-potential

in which the

hysteresis disappears.

~Ve show that the anisotropy

plays

_a determmant rôle _m the restoration of

multistability.

We thus obtain

a coherent picture of the bifurcation

Iandscape.

The _case of _an attractive

potential

is

briefly

discussed in Section 4.

1. Presentation of the

quasi

lD-Model

The behaviour of _a

repulsive potential

_as _a function of _z is sketched in

Figure

2. We _con

take,

if _necessary, _a Gaussian

potential

V(z)

~2

= Ve~Z7

(2)

that _meets the condition of _no interaction at

infinity.

For _a

given

nominal position _p~ ₌ _p,

the elastic

displacement adjusts

itself in such _a way'as to minimize trie

potential

_energy U at

equilibrium according

to:

[

=

~ll~

₊

Ài~ Pit))

= 0 13)

(4)

v(x)

x

Fig. 2. Repulsive _pmmng potential.

x

'"

p

,:." iA)

(Bl)

Fig. 3. Behaviour of the equilibrium positions _x _as _a function of the control parameter _p

(catas-

trophe

mamfold).

The

equilibrium

is stable if

d2u d2v

w

⁼

w

⁺ ^> ^° ¹⁴⁾

Equation (3)

Ieads itself to _a

graphical

solution. _as illustrated in

Figure

1:

V'(z)

is cut

by

_a

straight

Iine of

slope (-À) intersecting

trie _x-axis at p. Two _cases may occur:

. If min V" ₊ _> 0, ^each ^nominal

position

_p

corresponds

to _a

single equilibrium position

x: this monostable system is called "hard". Such _a system does Rot

dissipate

_energy.

. If min V" ₊ _< 0, the system ^is ^"Soft ^it generates a zone of

multistability.

The behavior of _x _as _a function of _p is illustrated in

Figure

3.

Here,

G and I _are the

spinodal

Iimits

where trie metastable solution annihilates with the unstable solution. The

region

^GI is unstable. Points in the stable

regions

_occurring before G and after I _are referred to _as

(Bi)

and

(82) respectively,

while _a

point

in the unstable

region

GI is Iabelled

(A).

The total energy

Ue~(p)

is illustrated in

Figure

4, where the characteristic

points

of

Figure

3 _are shown. The double point Fi

corresponds

to the Maxwell

plateau.

^At this

(5)

point,

the _two metastable states

(Bi

and

(82)

hâve the _same energy.

Beyond

this

point (Bi)

is _no

longer

_a

global

_energy minimum. If the asperity ^coula

jump

over the barrier

IA) by

thermal

activation,

then the

jump

from _one branch to the other would _occur _at this location. In a first

approximation,

this

jump

is forbidden. The system _is

govemed by

the

"delay

mie" [3]. ^When p is

mcreasing

from _-oc, the slider travels up ^to G where the

asperity jumps

_on to the other branch at K.

During

^the retum

joumey,

the slider _goes to I before the

asperity jumps.

The behavior of the system is

hysteretic

and irreversible.

2.

Quasi-2D Isotropic Repulsive

Model

The

pmning potential (Fig. 2)

is

rotationnaly symmetric.

The real

position

_r and the nominal

position p(t)

_are _now _vectors. When

sweeping along

the

x-direction,

the comportent

py(t)

acts

as a constant

impact

parameter which _we control. Trie equations of motion,

expressed

in

polar

coordinates in T.space, are:

~~+À(r-p).ur=0

(Si

0+À(r-p).uô

=0

(6)

The

jumps

effected

by

the

asperity

con

only

be radial

jumps (r

_p _is

radial).

The effective energy is thus

rotationally symmetric

_m the parameter space described

by

(p~, py)

₌

(pcosÙ, psinÙ)

The

shape

of the effective _energy

(Fig. 4)

is

simply

obtained

by rotating

the 1D-curve of

Figure

4 around the vertical _axis.

The

stability

of

equihbrium

positions ^is now

given by

the sign of the

eigenvalues

of the Hessian matrix

d~U/du~,

which in this _case _are

~~~

V'(r)

~~~ ~ ~~ ^~~~

r

~ ~

Uefr

G

(Bl)

K

(82)

P

Fig. 4. Effective energy

(energy

of the equihbrium

positions)

_as

a function of _p. The upper branch IHG is unstable.

(6)

y

F J K

~

H

Îi'i'l)""..._ :.."""

Fig. 5. Equilibrium positions _m the

ix,

y) plane for _an isotropic 2D pinning center. On the symmetry axis the thick Iines _are energy minima, the central dashed Iine GI is

an energy maximum,

FG and il _are saddle points. The circle is _a separatrix, corresponding to equilibrium positions when p~ = py = 0. When py # 0, the trajectory follows the dotted Iine, with _no discontinuity.

These

eigenvalues

characterize the

stability

of the system in radial and orthoradial directions

respectively.

With the

symmetric potential

of

Figure

2, the minima of

I"'(r)

and

V'(r)/r

coincide: _we retrieve the

Figure

3, trie radial

instability

_occurs _on trie _same

region (GI)

_as in trie ID _case, but trie

tangential

instability

occurs over a

Iarger domain,

which is Iimited

by

trie Maxwell

plateau (FJ),

1.e.

by

trie couic point of Ue~. In

Figure

4, trie _Upper surface of trie _cone

corresponds

to saddle

points

of

U, radially

stable, but uustable

tangentially.

Trie Upper surface of trie

potential

_energy Ue~

is

completely

unstable.

We _con _agoni follow the

equilibrium

positions r of the

asperity,

when _p~ is varied at _a constant py. ^Consider first the _case py = 0. A trivial solution _{is y} ₌ 0

(Eq. (6)

is

automatically satisfied).

We _recover the ID situation: the

equilibrium positions

remain

along

the z-axis which is _an

axis of symmetry ^for our system. The situation is illustrated in

Figure

5, where the

Iabelling

_is

trie _same _as in

Figure

3. The segments ^FG ^and IJ _are saddle

points

for trie

potential

U, GI _are maxima. If _p

= 0, another solution

corresponds

_to

IV' jr) /r)

+ = 0, ^1.e. to a circle of radius xF

corresponding

to _an

arbitrary

orientation of trie Maxwell

plateau.

If _we did Rot consider

the _transverse

instability, varying

_p~ from Ieft to

right

would

give

the _same behavior _as _m the

ID _case. The

jump

would then _occur at G and the

asperity

would _go _to K.

However,

_as _soon _as

py

#

0, ^the

asperity

^follows the dotted trajectory in

Figure

5: it avoids point F. The

asperity

goes around tire couic point and the

jump disappears.

The evolution of _r is continuons and it is _no

longer dissipative.

We conclude that the

tangentiai instabihty destroys

the

hysteresis.

This result is due to trie

assumption

^of

isotropy.

We _now

study

trie

anisotropic

_case, in which trie

disappearance

of trie

hysteresis

is _more subtle.

3.

Quasi-2D Anisotropic Repulsive

Model

3.1. INTRODUCTION. The introduction of

anisotropy

breaks the rotational symmetry ^of the effective potential ^and introduces _a _new control parameter into _our system.

By

_means of

(7)

a

change

of

scale,

and

assuming

decorrelation between trie elastic properties and trie _pmnmg,

we con introduce

anisotropy

into trie elastic stiffness of trie system instead of trie _range of trie

potential.

Trie

generalization

of trie

potential

_energy of trie equation

il)

to trie

anisotropic

_case

is thus

given by:

U =

Vir)

₊

jiT _P~itl)~

₊

)iv _P~it))~

₁₇₎

where Ày ^< À~. The amsotropy is characterized

by

the dimensionless parameter À~/À~ ^which is restricted between ₀ and _1. Note that in

comparison

with _a mortel which introduces the anisotropy m the range of the

pinning potential,

_a softer stiffness _m the x-direction

corresponds

to _a smaller _range of the

pinmng potential

in this _same direction.

3.2. EQUILIBRIUM PosiTioNs.

Equilibrium positions correspond

to an extremum of

U,

1-e- to

x

À~

+ ~'~~~~

=

À~p~(t) (8)

r

y

À~

+

~"~~~

#

Àyp~(t)

r

(where V'(r)

=

ôV/ôr

is the radial

derivative). Symmetry

is Iost except on the _x and y axis.

Consider for _instance _the _p~ axis, on which p~ = 0: there _two types ^of ^solutions _may coexist.

ii)

Either _y

= 0,

thereby preserving

the

(g

_-

-y) parity.

Then the first

equation (8)

relates

z to p~: we recover the lD situation,

multistability

_occurs if _mm V" ₊ _À~ _< 0.

(ii)

^Or _À~ +

(V'(r)/r)

= 0, which

yields

_r. Combined with the first equation

(8),

that condition

implies

Z " Px

~

~~

~

(9)

~ ~

Such solutions exist if _r _> _z, with _a limite value +y:

they

break the symmetry.

The bifurcation

corresponds

to _r

= z =

x*,

1-e- to À~ +

(V'(z)/x)

= 0. It is of the

usual

pitchfork

type,

corresponding

to a

tangentiai instability

of the

symmetric

solution when Ày ⁺

IV' ix) lx)

< 0. Note that the symmetry

breaking

solution

corresponds

to _r ₌ ~*: usmg

(9)

trie value of (y( follows.

In order to understand trie nature of that bifurcation _m _more

detail,

_we must follow trie evolution of trie varions roots _as trie central parameter _p~ is varied _away from the bifurcation

p]:

what _matters is the

sign

of dz

/dp~.

For the symmetric solution y =

0,

_z ₌ _xo, it follows

from

(8)

that

~~

Î

~

À~

ÎV"

~~~~

As _we shall _see _m the next section that ratio _is

positive

if the solution is

radially

stable: _we

Orly

consider that _case. _zo turns

tangentially

unstable if _zo _<

x*,

_p~ <

p].

In contrast the symmetry

breaking

solution _z+

obey (9)

and therefore

)~

= ~

~~

~

iii)

P~ ~ ~

Since _we assumed À~ ^< Àx that ratio is aise positive: ^it follows that trie bifurcation _on trie _z axis _is direct. The _two symmetry

breaking

solutions _appear when _p~ _<

p(,

_on the _same strie of the bifurcation _as the unstable branch _zo, _as shown _m

Figure

6a.

(8)

_p+

Py

Î°

~

é°

x

_.~

',

B_ .'B+

A~ ^~

"~_

". _.~

Py'

B A ~

. . . Px B~

Px~

(a) (b)

Fig. 6. The nature of transverse bifurcation _on symmetry axes m the radiaI1y stable _case. Dark

points correspond to the bifurcation. A

(Bl

in the _p axis _are mapped onto extrema _m the

ix,

vi plane.

In both _cases, A _are points before trie transverse bifurcation and B _are points after trie transverse

bifurcation for _a sweeping in trie decreasing p-direction. On the _~ _axis

(Fig.

fia), the bifurcation is direct (Ao, B+ _are minima, Bo is _a saddle

point).

On the _y axis

(Fig. 6b),

the bifurcation _is reversed

(Ao, B~ _are saddle points, Bo is _a minimum).

The situation is reversed if _we

study

the y axis, _p~

= 0, instead of the _x axis. Then _we _must

interchange

z and y m

(10,

ii

).

The bifurcation _occurs at _some y*,

p(,

^below ^which ^the x = 0 solution is

tangentially

unstable. The ratio

dy/dpy

remains positive ^for the solution _vo, while it is

negative

^for ^the two symmetry

breaking

solutions _y+,

corresponding

to

r+ = ~* _Y+ =

~~~~~

P~

i12)

It follows that trie bifurcation is inuerted.

Symmetry breaking

solutions _appear for values p~ ^>

p[

for which the symmetry ^axis ^is

transversally stable,

_as shown in

Figure

6b. While in trie _case 6a trie bifurcation _goes from _one minimum to a saddle

point

surrounded

by

two

minima,'in

_case 6b _one _goes from _a minimum surrounded

by

two saddle points

(p~

>

p()

to a

surgie

saddle point

(p~

<

p[).

Trie Iateral _extrema tum around trie

origin

_as

they

Ieave trie bifurcation

g*; they eventually

_merge at trie

symmetric

bifurcation

-y*. (Technically

trie transverse

stability

is reversed

by

the off

diagonal

elements of the

stability

matrix,

proportional

to

x).

Away

from the symmetry ^axis a

generic

bifurcation

corresponds

to coalescence of _a mode with _a saddle

point.

In trie direct _case the evolution is continuons

(in

close

analogy

with _a

ferromagnet

in _a limite externat

field).

In trie _reverse case, in contrast, trie

Orly

local minimum

disappears:

_a discoutinuous

jump

is unavoidable. That

jump persists

when p~ = 0. Note that

it carnet Iead _to trie above saddle

points: despite

trie fact that symmetry

breakiug

solutions

exist, trie jump is trot symmetry

breaking.

It

necessarily

ends _on _{trie y} axis, at _a

point

_vi such that

V'lgf)

=

À~lPi gf) l13)

The

graphical

solution is the _saine _as in ID

(Fig. 4).

This feature is crucial in

understanding

hysteresis.

(9)

125s JOLiRNAL DE PHYSIQUE I N°10

3.3. STABILITY ANALYSIS. The _nature of bifurcations is controlled

by

local

stability

of the

equilibrium solutions,

1.e.

by

the

eigenvalues

of the Hessian matrix

~ ~

Àx ~~VÎÎ _Î~~~

~x~ ~Y ^~ ~YY~~

Bifurcations _occur when _one of trie

eigenvalues vanishes,

1-e- when trie determmant A of that matrix _is _zero. The

eigenvector

of H with _zero

eigenvalue

^dermes trie soft mode which

changes stability. Noting

that

ii

=

vii

il

₊

1'1

,

ii

=

[vii 1' ₁₁

we

easily

calculate trie determinant A

vl ~rl

A = V" ₊

V"(Ày

^cos~ + À~

sm~

_Ù) ₊ _{((À~ cos~} ₊

Ày

sin~

_Ù) ₊

ÀXÀ~

(14)

r r

(when

is the

polar angle

m the

II, y) plane).

Trie roots of

(14)

separate

regions

of different stabilities.

Along

each of trie symmetry _axes, equation

(14)

^factorizes with

decoupled

radial and tan-

gentiaI

instabilities. Trie

corresponding

equations are:

(15)

x axis y axis

radial V" ₊ Àz = 0 V" ₊ _À~

= 0

tangential

^~' + À~ ^" °

1'

+ À~ " °

The solution of

(15)

is shown

graphically

_m

Figure

7. We note that:

ii)

Bifurcations

only

_occur for soft systems m which at Ieast trie smallest stiffness _{À~ is} _such that

ç'+Ày<0

m which

VI'

is trie _common minimum of V" and V'

I.r

at _r ₌ 0.

iii)

Since _we assumed Ày ^< Àz, Az

(Fig. 7)

is

always

closer _to trie

origin

than Bz: the

tangential instability

_occurs ^first _on ^the _x-axis. In contrast, ^the ^relative

positions

of

A~

and

B~ depends

on

jthe

anisotropy.

This situation is summarized _on the

phase diagram

of

Figure

8, which

displays

four

regimes:

. In region

I,

there

il

no

instabiiity

ai ail:

Orly

_one

equilibrium

position _r exists whatever _p.

.

Region

II

only

has _one

instabiiity,

which evolves

continuously

from _a radial

mode,

_on trie y-axis, to a

tangential

mode, _on the x-axis. In between trie soft mode has _no

particular

symmetry.

. In _region

III,

_a second

instabiiity

_appears closer to the

origin

_r

= 0. The

corresponding

soft mode evolves

frqm tangeniial

_on trie

y-axis

to radial in trie x-axis.

(10)

~

~l'

r

~V'/r -Ày

À~ A~

Vo"

Fig. 7. Graphical solution of

iii).

B~ and By _are radial instabilities respectively _on the

x and _y

axes, A~ and Ay are tangential instabilities.

~

~~

z' ."#~ _l'.~~.

.$', 'i~"

'~ ^~' .4

~~

~> .j

'~~ _j~., ~f'~

~'"'

'." 2«

'. _/ ~' ~Î

."'

~

Î'~

~

n

.l~ iv iii

~ ^Àx

v~"1

Fig. 8. Phase diagram displaying the various regimes as a function of À~ and Ày.

. In

region IV,

the roots

along

the

y-axis

bave crossed. Since the roots of

(14)

contrat

cross for intermediate

angles

6

(in

the absence of symmetry

they repel),

that _means _an

mterchange

of eigenvectors as one moves from _z _to _y. The outer bifurcation _curve in the

ix-y) plane

is _a

strictly tangential instability

_on trie _two _axis and

primarily

_so in between.

The muer _curve is

primarily

^radial.

The varions situations _are

displayed

ⁱⁿ

Figure 9,

which shows trie locus of bifurcations in the

ix, y) plane

for trie three _cases

II, III,

IV.

(11)

x x x

A~ B~ A~ B~ A~

Case II Case III Case IV

Fig. 9. Locus of bifurcations in the (x, y) plane. The Iabelbng of points is the _same _as _m Figure 6.

The _arrows denote trie polanzation of the soft mode at trie bifurcation.

Knowing

trie positions of bifurcations in trie

(z,y) plane,

_we _can infer trie

configurational

coordinates _p from

(8), thereby constructing

trie bifurcation

diagram

_m trie

(pz, p~) plane.

Trie latter

diagram

is trie _one _we need, since _p is trie

physical

coordinate which is controlled from outside. Detailed results

depend

_ou trie

shape

of

VIT),

but

qualitative

features _can be

easily

found

using

a

Taylor

expansion of V' and V":

~'~~~~

=

l§"(1 ar~) ₍₁₆₎

r

V"(r)

=

lfi"(1-3ar~) (Anyhow (16)

is valid _near _r

=

0). Expansions

_near trie symmetry _axes _are

straightforward,

but somewhat tedious: _we sketch trieur in the

appeudix

and here _we

give only

the results sho~vn

m

Figure

10. These results call for _a number of _comments:

ii)

The threshoids

Az,

_A~ for

tangential

instabilities _on the symmetry axes are

imaged

into cusps in the

(pz,

_py

plane (see

the

appendix

for _precise

results).

This is trie

expected

behaviour for

pitchfork

bifurcations. In contrast radial saddle point

bifurcations, Bz, B~,

are

imaged

into

regular analytic points

in the

(p~, py) plane.

iii) Continuity

arguments

provide

a

simple description of muitistability.

Consider for instance

case II and _assume that py ^is decreased from +cc, _say at p~ = o. Trie stable solution

persist

up ^to

B~

m the

II, y) plane,

1-e- _up to _a

negative

value of _p~. Hari _we started from _p~

= -oc,

we,would

proceed

to the opposite value of p~ before

meeting instability.

It follows that the shaded _area of

Figure

10 is

multistable,

with two stable sheets and

necessarily

_one unstable sheet in order to connect them. The total energy

U(p)

bas two

spinodal

limits that _merge _at the cusps Az, and in between _a double fine

(here

the _pz _axis

by symmetry)

that

corresponds

to trie Maxwell

plateau.

Trie _same situation repeats _a second time in _case III: another fold

develops,

limited

by

the cusps

Ay.

^In that case, this

secondary

fold

develops

_on the unstable

sheet of the primary bifurcation: _it is

physically

irrelevant.

(iii)

In _case

IVa,

the

bifurcation

_curues _cross _m trie

(pz,

_p~

plane,

which _may look surprising.

Such _an intersection would be

impossible

in the

(~, y) plane,

where there is _no

multistability (according

to

(8),

_pz and _p~ _are

uniquely

defined given x and

y).

But in the

(pz, py) plane

the two threshold _curves of

Figure

10

correspond

to

dijferent

sheets of the

equilibrium

states:

(12)

y e~

'

' ' ÎÎ

1

' ,

' '

, '

',, _-'

y

' '

1 '

l

,' ^' ~'

" '

'

gj , ^'

,

_,

' ' , X ', _i

p

, ^' '

,

' 1

, , '

1

' ~ '

" ., ,_

~~' '

'

,-_

'

i

, '

i

, ,

<

,

i

' '

, ⁱ ^'

'

,

, ¹

' ',

',

' '

) ~

'

~ ' Î

X

'

, '

, ~ '

~ ' '

~ ' ,

'

,' '

, ' _,

, ~

j ,' ,

i X ^'~ ^', < X

' t ' '

j

' ' ~ i

,

' _'

' '

' _' _' ,

' ~ '

",

"..

" ~'

.,"' ~

Fig. 10. Locus of the bifurcations in the (x~vi and (p~, py) planes for trie _vanous _cases of Figure 8.

For each _case the Iabelling of symmetry

powis

is the _same _m the two figures and also the _saine _as

in Figures 7 and 9. As _a guide for trie _eye, _one quadrant of the

(~,

vi plot is shown _m fuII Iines, and the _image of that branch in the (p~, py) plot is also shown _m fuII Iines: _in 1.hat _way _one _can follow

continuously the evolution of the _vanous sheets _m the (p~, py) plane.

(13)

____,...,__ _,...,____

:""' "".,o

.""

"".,*

;" f". _." r.

'.

x x

y

(a)

ib)

Fig. ii. The distribution of discontinuous jumps. Figure lia corresponds to the _cases II _or III of

Figure 10. Figure 11b corresponds to the _case IV of Figure 10. The fuII _curve _is the spinodal Iimit of

Figure 9 and the dotted _curve the corresponding final position.

there is no

repulsion

constramt and the bifurcation _curves _ignore each other:

they

_may weII

cross. ~Ve also note that

Ay

^and

By

^do Rot

exchange along

^the _py

axis,

while

they

do _m the

(x, y) plane.

We show in the

appendix

that

they approach

each other _in second order _as trie

boundary

of

regions

^III and IV is

approached: they

coiucide _at trie

boundary

but

they

do not intersect.

(iv)

The _case IVb

corresponds

to very small anisotropies, _Ày ^and Àz

being

_very close. In the isotropic ^iimit À~ = Ày, ^trie ^four cusp central _curve of

Figure

10 shrinks to a

point

_at trie

origin:

_we _recover the results of the

preceding

section: the

origin

_p

= o is the map of the whole outer circle

(A~, Ay)

which is _an attractor in trie

ix, g) plane.

The multistable

region

shrinks to _zero.

3.4. HYSTERETIC BEHAVIOUR. In practice,

Orly

stable solutions _are

physically

relevant, 1-e- trie part ^{of trie}

(x, y) plane

outside trie externat bifurcation _curve in

Figure

10.

Everything

that _occurs inside that first bifurcation is irrelevant: there is _no need to map these

unphysical,

uustable sheets. One should trot

forget

however that trie control parameter is

(pz, py),

not

(~, g)!

One should _map the

trajectories

back into trie

p-plane,

which is

easily

clone

using

arguments ^of

continuity.

In that _case _one _can locate trie muitistabie

regions: hysteretic jumps

occur when _one ieaues these multistable regions, due to trie

disappearance

ofa local minimum.

Each _case _must be considered

separately.

The

simplest

situation is that of _case Ilin

Figure

10.

Coming

from py = +oc, one reaches the

spinodal

Iimit in the Iower half of trie

(p~, py) plane.

^It

corresponds

to _a _curve _m trie _Upper half of the

ix, y) plane. Multistability

_occurs in the shaded _area. Whatever the direction of

approach,

the

hysteretic jump

_occurs _upon

Ieaving

that _area. The coordinate

ix, y) jumps

to another stable

position pertaining

to the _same value of the control parameter

(pz, py).

^This is illustrated in

Figure

lia which sketches the _one to _one

correspondence

between the

spinodal

(14)

initial

position

r* and the final position ri after the jump. _ri lies in the stable part of the

ix, y) plane,

_as it should. Note that in that

plane

the

vicinity

of Az is

regular.

In that

regime

the

physics

is

essentially

_one

dimensional, parametrized by

the

impact

_parameter _py.

Case III is identical to _case II: the

secondary

fold

BzA~

lies _on the unstable sheet and is irrelevant. On trie other hand the situation is somewhat dilferent in _case

IV,

_as the bifurcation is

primarily tangential.

The

spinodal

Iimit

corresponds

to the four _cusps _curve

AzAy

^of

Figure

10, whether that _<:urve intersects _or not the other bifurcation

(trie

^latter ^sits _on another unstable sheet which is

physically

irrelevant: _case IVa and IVb _are

identical).

Once

again

discontinuons

jumps

_occur

only

when _one leaues the inside of that

région.

Thé

vicinity

of thé _cusps Az

(in

thé "hard"

direction)

is similar to that of _case II. In the "soft" direction thé cusp Ay ^shows

up in the _curve of thé

corresponding

final positions _rf, as

displayed

in

Figure

11b where

A[

designates

the final position

corresponding

to a

jump

at

Ay.

^As could be surmised, the

jumps

are

priniarily

in trie soft direction _y, with _no spontaiieous symmetry

breaking (see

Sect.

3.21.

As for _as friction is

concerned,

thé

quantity

of

physical

interest is the

dissipated

_energy,

1.e. thé

discontinuity

in total energy U due to thé

jump:

AU =

U(rf) U(r*) (17)

An

analytic expansion

is

possible only

_near the ends A~ elsewhere AU must be calculated

numerically

for each

pinning potential V(r).

Here _we

only

make _a few

qualitative

comments on the

dependence

of AU _on the elastic stiifness _À~ and Ày, which ^is

markedly

dilferent in ID and 2D. In _one dimension the energy

dissipation

_goes to oc as goes ^to 0. This somewhat

paradoxical

result is obvions _m

(3

for small

pinning

con

produce

_a

huge

elastic

displacement

u =

lx p)

~t

1/À.

The

corresponding

elastic energy is _~t u~ _x

ci

1/À:

it is

basically

released upon the jump. ^The reason for that is that there is _no way ^to avoid the

pinning

center in ID

trapping

cannot be

escaped.

In contrast, _a second dimension _opens _a _new

possibility:

_turning

around the pinning center in order to avoid metastable states that

ultimately

build _up

large

elastic

energies:

_as _a result AU _gros

through

_a maximum _as _a function of À,

retuming

to 0 when gets smaller

(as

sketched in

Fig. 12).

Such _an évolution _con be studied

quantitatively:

here it is

enough

to look

quickly

at the Iimit _- 0. A

glance

at

Figure

7 shows that such _a Iimit

always corresponds

to case IV: trie first bifurcation is

always tangential,

whether _on trie

z or on trie _y _axes. Consider for instance trie _z axis: at bifurcation _z and _p~ _are related

by (9).

They depend

_on trie amsotropy

Ày/À~,

but net on trie absolute value of _À~ and Ày. ^Since z

lies

necessarily

within trie _range of trie

potential V(r),

it follows that p~ aise remains

limite,

with _a

logarithmic

correction that _we

ignore,

when _- 0: trie released energy upon

jumping

is of order instead

of1/À!

In

principle

_very small values of and extreme anisotropy could

conspire

to

produce

_cases II and III. Then trie

primary

bifurcations _are

radial, corresponding

to (À~

+V")

= 0.

Equation (9)

no

longer

froids: while _z is

small;

_p~ _con become _very

large, implying

_a

large

elastic

displacement

u we recover trie ID situation in which AU grows when become smaller. But such _a _case

is artificial: for _any reasonable

anisotropy

small stilfness _means _case IV!

4. Trie Attractive Case

The

pinning potential

V is _now

negative: V'IT) /r

is

always positive

_as shown in

Figure

^13.

Tangential

instabilities _can _no

longer

_occur _on trie symmetry axes, while radial instabilities

occur in

pairs symmetric

with respect ^to trie

origin.

Trie relevant

quantity

is

again

trie minimum

value Vo'" Three _cases _may _occur:

ii _(VI'

< Ày ^< Àz: trie system ^is

fully

stable and there _is _no bifurcation _at aII hence _no

hysteresis.

(15)

( (

Î Î

o $

Il£

~

o

~

/ i Î

0 ^'

Stiflness

Fig. 12. Dependence of the dissipated _energy /hU _on stiffness

Àya~/(V('(

for p~ = 0. The Iower

curve corresponds to the anisotropic _case À~/Ày =

1/2.

The _upper _curve corresponds to the ID _case.

Zones III and IV _are related to the corresponding regions of instability.

V'/r

r

~ Il

~lJ Ù~

Fig. 13. Geometrical construction of bifurcations _on the symmetry axes m the attractive _case.

iii)

_Ày < Àz < (VI'( ^radial instabilities _occur _on both trie _z and _y _axes. In

between,

trie bifurcation _curves

interpolate

_as shown _m

Figure

¹⁴

(trie

exact

shape

is obtained

solving

Eq. (14)). Regions

and 3 _are

locally stable,

while _region 2

corresponds

to a saddle

point

of trie total _energy

Ue~.

When carried eut in trie

(p~, py) plane (using (8)),

trie two _curves of trie

ix, y) plane interchange,

_as shown in

Figure

14b. This is most

easily

seen m trie

(16)

y i Ii iii

Si

2

3

X Px

(a) (b)

Fig. 14. Bifurcation diagram _m the attractive _case when À~ < Ày ^<

Vi'

(. The hatched _area _is that multistability. I, II, III represent three different sweepmg paths at constant p~. Spinodal jumps occur at Si and 52 for path I only.

V~

B

B'

Fig. là. Geometrical construction of _p~ for _a radial instability along the _x _axis _in the attractive

case. The IabeI1ing of points _is the _same _as

m Figure 14.

graphical

construction of

Figure

15, which

yields

_pz

given

_z at the bifurcation point. ^Such an

interchange

^is

crucial,

_as it _meaus that the hatched _area in

Figure

14b is muitistable. As usual the argument ^relies on

continuity

_as the representative

point

is

brought

from

infinity

in trie

(z, y) plane.

Assume for instance that _we _corne

along

the

y-axis

downward: _we meet

instability

at

point

^A ^and _we recover another stable branch at B. In between

py(y)

^is reentrant: _an unstable branch _goes backward from A to B. The _same situation occurs in trie Iower half of the