Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

e,

ressa

•

ng

Iy 980

Giorgio Ferrarese (Ed.)

Wave Propagation Lectures given at a Surruner School of the Centro Intemazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Balzano), Italy, June 8-17,1980

~ Springer

FONDAZIONE

CIME ROBERTO

CONTI

C.LM.E. Foundation c/o Dipattimento di Matematica ''D. Dini" Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11064-1 e-ISBN: 978-3-642-Il066-5 DOl: 10. 1007/978-3-642- I 1066-5 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1" Ed. C.LM.E., Ed. Liguori, Napoli & Birkhauser 1982 With kind permission of C.LM.E.

Printed on acid-free paper

Springer.com

CON TEN T S

c

0 U

r

8

e

8

A. JEFFREY

Lectures on nonlinear wave propagation

Psg.

Y. CHOQUET-BRUHAT

Oodes 8symptotiques .•..•••.•••••..•••

"

99

G. BOILLAT

Urti .•••••.•••••.••..•.....•.........

"

167

"

195

continui •••..•....•..•...............

"

215

Onde nei solidi con vincoli intern!

"

231

1

5 e min arB

D. GRAFFI

Sulla teoria dell'ottica non-linear.

G. GRIOLI

Sulla propagazione del calore nei Mezzi

T. MANACORDA T. RUGGERI B. STRAUGHAN

"Entropy principle" and main field for

a non linear covariant Byat••••••.••.

"

Singular surfaces in dipolar materials and possible consequences for continuUN mechanics .....•.........•••••.•••.•.•

"

275

CENTRO INTERNAZlONALE MATEMATlCO ESTlVO (C.l.M.E.)

LECTURES ON NONLINEAR WAVE PROPAGATION A. JEFFREY

CIME Session on wave Propagation Bressanone, June 1980

Department of Engineering Mathematics, The University Newcastle upon Tyne, NEl 7RU, England

9 CONTEIITS

Lecture 1.

Lecture 2.

Fundamental Ideas Concerning Wave Equations 1.

General Ideas

1-1

2.

The Linear Wave Equation

1-2

3.

The Cauchy Problem - Characteristic Surfaces

1-5

4.

Domain of Dependence - Energy Integral

1-9

5.

General Effect of Nonlinearity

1-13

References

1-15

Quasil1near Hyperbolic Systems, Characteristics and Riemann Invariants

2-1

1.

Characteristics

2-1

2.

WSvefronts Bounding a Constant State

2-6

3.

Lecture 3.

Lecture 4.

Riemann invariants

2-8

References

2-12

Simple Waves and the Exceptional Condition

3-1

1.

Simple Waves

3-1

2.

Generalised Simple Waves and Riemann Invariants

3-2

3.

Exceptional Condition and Genuine Nonlinearity

3-6

References

3-9

The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations

4-1

1.

General Considerations

4-1

2.

The Initial Value Problem

Time and Place of Breakdown of Solution

4-2 4-2

References

4-9

3.

Lecture 5.

1-1

The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying

5-1

Depth and Width

Lecture 6.

1.

Basic Equations

5-1

2.

The Bernoulli Equation for the Acceleration Wave Amplitude

5-2

3.

The Amplitude a(x) and its Implications

5-3

References

5-5

Shocks and Weak Solutions 1. Conservation Systems and Conditions Across a Shock 2. Weak Solutions and Non-Uniquenes&

6-1

6-1

6-4

10

Lecture 7.

3.

Conservation Equations with a Convex Extension

6-11

4.

Interaction of Weak Discontinuities

6-13

References

6-14

The 1l1emann Problem, Glimm.'s Scheme and Unboundedness of Solutions

7-1

1. 2. 3.

The Riemann Problem. for a Scalar Equation

7-1

Riemann Problem for a System

G1im1ll's Ilethod

4.

Non-Global Existence of Solutions

7-3 7-5 7-8 7-10

References Lecture 8.

Far Fields. Solitons and Inverse Scattering

8-1

1.

Far Fields

8-1

2.

Reductive Perturbation Method

3.

Travelling Waves and Solitons Inverse Scattering

8-3 8-6 8-9

References

8-13

4.

11

Lecture 1. 1.

Fundamental Ideas Concerning Wave Equations

General Ideas

The physical concept of a wave is a very general one.

It includes the

cases of a clearly identifiable disturbance, that may either be localised

or non-localised, and which propagates in space with increasing time, a timedependent disturbance throughout space that mayor may not be repetitive in nature and which frequently has no persistent geometrical feature

that can

be said to propagate, and even periodic behaviour in space that is independent of the time.

The most important single feature that characterises a wave

when time is involved, and which separates wave-like behaviour from the mere dependence of a solution on time, 1s that some attribute of it can be shown to propagate in space at a finite speed. In time dependent situattons, the partial differential equations most closely associated with wave propagation are of hyperbolic type, and they may be either linear or nonlinear.

However, when parabolic equations are

considered which have nonlinear terms, then they also can often be regarded as describing wsve propagation in the above-mentioned general sense.

Their

role in the study of nonlinear wave propagation is becoming increasingly important, and knowledge of the properties of their solutions, both qualitative and quantitative, is of considerable value when applications to physical problems are to be made.

These equations frequently arise as a result of the

determination of the asymptotic behaviour of a complicated system. Nonlinearity in waves manifests itself in a variety of ways, and in the case of waves governed by hyperbolic equations, perhaps the most striking is the evolution of discontinuous solutions from arbitrarily well behaved initial data.

In the case of parabolic equations the effect of nonlinearity

is tempered by the effects of dissipation and dispersion that might also be present.

Roughly speaking, when the dispersion effect is weak, long wave

behaviour is possible, whereas when it is strong a highly oscillatory behaviour occurs, though the envelope of the oscillations then exhibits some of the characteristics of long waves.

12 Waves governed by a linear wave equation arise in many familiar physical situations, like electromagnetic theory, vibrations in linear elastic solids, acoustics and in irrotational Inviscid liquids.

However,

these linear equations often arise as a consequence of an approximation involving small amplitude waves, so that when this assumption is violated the equations governing the motion become nonlinear.

Not only does this convert the problem to one involving nonlinear partial differential equations, but it also usually leads to the study of

a system of first order equations, rather than to a nonlinear form of the familiar second order wave equation.

This happens because the wave equation

usually arises as the result of the elimination of certain dependent variables from first order equations (like! or

~

in electromagnetic theory),

and this is often impossible when nonlinearity arises.

Our concern hereafter

will thus be mainly with quasilinear first order systems of equations - that 1s to say with systems that are linear in their first order derivatives, and for the most part we will confine attention to one space dimension and time. 2.

The Linear Wave Equation

Because of the importance of the linear wave equation (1)

let us begin by reviewing some of the basic ideas that are involved, though 1n the more general context of the variable coefficient equation

3

L

~,j·O

3

a

ij

U

i j x x

+

L

i-o

biu i x

+ cu

(2)

f

with aij' b i , c, f functions of the four dimensional vector

~

0 1 (x • x

x

2

Not all linear second-order equations of this form describe wave motion. and 00

account of this it is necessary to produce a method of classification which

readily allows the identification of wave type equations from amongst the other types that are possible (i.e. elliptic and parabolic). The form of classification to be adopted utilizes the coefficients of the highest-order

rl?-r!va~~ve~

and has an algebraic

~asis

but, as will be seen

3

x ).

13 in a subsequent section, this classification in fact effectively distinguishes between equations of wave type and those of other types.

Let us start by

attempting some simplification of the form of equation (2) by changing the independent variables through the linear transformation 3

I ""ijoJ i- where the GC

0,1,2,3

i

(3)

are constants.

ij

A transformation of this form gives an affine mapping of the

(xo, x1 , x 2 , x 3 )-space which Is one-one provided det I~j I

~

O.

Employing

the chain rule for differentiation we find that equation (2) may be re-wrltten 3

o•

I

(4)

i,j,k,l"O Hence the coefficients 8

1j

of the derivatives

U i

j' which are functions of

x x

position, transform to the new coefficients

of the derivatives

k l' which are also functions of position.

U

If, now,

f; f;

we confine attention to the set of coefficients a

specific point

~

z

!o

1n

o (x ,

appropriate to some ij 123 x • x • x )-space, we see that this is exactly

the transformation rule which would apply to the coefficients

8

ij

of the

quadra tic form 3

I

i,j-O

a ij nin j •

(5)

when the n i are transformed to Bit by the variable change 3

ni

I k-O

akiak'

Now it is a standard algebraic result that by a suitable transformation a quadratic form with constant coefficients may always be reduced to a sum of squares. though not all of the squared terms need be of the same sign. Furthermore, Sylvester's law of inertia asserts that however this reduction is accomplished, the number of positive terms m and the number of negative

14 terms n will always be the same.

To apply these results to the differential

equation (2) itself with the variable coefficients attention to a fixed poine

the

8

~

the specific values

1j

8

•

!o

ij

8

ij

, let us again confine

o 1 2 3 in (x , x , x , x )-space and attribute to

• alj(~)'

This then implies that some choice of the numbers G

ij

• d

ij

exists for

which 3

I

1,j-O

where m + n < 4.

The number pair (m.D) Is called the signature of the

quadratic form (5) and, being an algebraic invariant, 1s used to classify the quadratic form.

We shall use it to classify the

partial differential equation (2) at each point

~

-

va~lable

coefficient

!o.

1n the transformation The effect on equation (2) of us1ng these numbers Q 1j (3) is co yield at .! • m-1

I 1-0

!!o

a differential equation of the form

1Itkl-1 U

3

I

tit i

U

i-m

1 1 t t

+

I

i-o

biu 1 t

+

f

Equation (6) or, equivalently, (2) is called hyPerbolic

o ( -direction

(6)

0 a~ ~

z

!o

in the

when the signature is (1,3), elliptic when the signature is

(4,0) and parabolic when m + n < 4.

o

If an equation is hyperbolic in the ( -

direction at each point of a region O. then it is said to be hyperbolic in the CO-direction throughout O. Obviously, if an equation has constant coefficients, then one suitable transformation (3) will reduce it to the form of equation (6) throughout all space.

For example, aside from the trivial transformation to remove the

2 constant factor llc , the wave equation (1) is already seen to have the

signature (1,3).

Thus if a transformation is made at one point of space to

convert the factor lIe

2

to unity,then it does so for all points in the space.

The usual effect of variable coefficients and first-order terms in hyperbolic equations of the form (2) is to introduce distortion as the wave profile propagates.

This produces various complications, not the least of

which 1s the fact that the wave velocity becomes ambiguous and requires

15 careful definition.

Only when there is a clearly identifiable feature of

the wave which is preserved throughout propagation 1s it possible to define the propagation speed of this feature unambiguously.

Such is the case with

a wave froDt separating, say, a disturbed and an undisturbed region and across which a derivative of the solution 1s discontinuous.

3.

The Cauchy Problem - Characteristic Surfaces Fundamental, to the study of hyperbolic equations 1s the Cauchy problem,

and the associated notion of a characteristic surface.

In brief, when

working with four independent variables the Cauchy problem amounts to the determination of a unique solution to an initial value problem in which a hypersurface F 1s given, and on it the function u 1s specified together with the derivative of u along some vector directed out of F. directional derivative is

call~d

Such a

an exterior derivative of u with respect

to F, in order to distinguish it from a directional derivative in F which is known as an interior derivative.

In the Cauchy problem it must be

emphasized that the function u and its exterior derivative over the initial hypersurface F are independent, and can be specified arbitrarily. A hypersurface F for which the Cauchy problem is not meaningful because u and its exterior derivative cannot be specified independently is called a characteristic hypersurface.

Let us now see how characteristic hypersurfaces

may be determined. It is convenient to utilize curvi-linear coordinates (0, (1. (2. (3 and to let the hypersurface F on which the initial data is to be specified have the equation ( to (

o

o

~

O.

In terms of the new variables. a derivative with respect

is a directional derivative normal to F so that it is an exterior

1 2 3 derivative. whilst derivatives with respect to ( , ( • ( are interior

derivatives. We now utilize this by rewritine equation (2) in a form in Which the derivative u

~o~o

is separated froa the other second-order derivatives

16

3,

L

i,j,k,.e.""O 3

Uk

i,k-<J

,

Here

,

L

+

r

+

(7)

f •

Cll

signifies that the terms corresponding to k ~ 1 - 0 are omitted from

the summation.

,

Now if we specify u and u 0 independently on F, &s is required in the Cauchy problem, the substitution of their functional forms into equation

,,

(7) enables the determination of u 0 0' provided only that the coefficient of this derivative does not vanish.

Thereafter, the solution may be obtained

in the form of a Taylor series by determining the coefficients of the series by successive differentiation of the initial data and of equation (7) itself.

This is. of course, the idea underlying the classical Cauchy-Kowalewski theorem.

It is, however, very restrictive as an existence theorem since w

it demands that all functions involved are C .

In the event that the coefficient (8)

of

u 0 0

vanishes, neither this nor higher-order derivatives of u with

, ,

0

respect to

t can be found.

,,

Furthermore, the derivative u 0 0 may then be

specified arbitrarily on F, and even differently on opposite sides of F. This is not remarkable, because when the coefficient of

,

u and u 0 cannot be specified independently over

F.

u 0 0

0. It will be convenient to consider that (18) governs the motion of a membrane with density P, tension k per unit length, distributed springing under the membrane with spring constant q(x,y) per unit area and frictional coefficient

r. Then the potential energy within a fixed region R with boundary B of the (x.y)-plane comprises the energy stored in the springing qu

2

dxdy

and the energy stored in the membrane

~(t)

dxdy

with ~ the outward~wn unit normal to Band ds a length element of B.

The

first integral in ':(t) is the negative of the work done by the tension against the interior of R and the second integral the negative of the work done against the boundary. Green's theorem shows that

so that the total potential energy

21

(20)

The kinetic energy 1s (21)

so the total energy is

or (22)

It then follows after use of Greents theorem that

•

f atau B

au

-k-ds-

an

II

r

R

[:~r

(23)

dx dy ,

which 1s the outward flux of energy across the boundary and the loss due to friction.

7 t, f------,. 0 and recall that in general a

unique solution to (10) and (11) will only exist for a finite time.

As we

have seen in Section 1, a conservation equation possesses discontinuous solutioDs or shocks, corresponding to a non-unique solution along an arc.

Accordingly,

and with reference now only to a general function f and initial condition g, let us consider some strip 0


0 will be

called a weak solution of (10) 1f in this half plane it satisfies the condition

o •

(14)

66

for every twice continuously differentiable function w(x.t) that vanishes outside some finite region in the half plane t > O.

Such functions ware

called test functions and the closure of the region in which they are zero 19 then known as the support of the test functions.

000-

As a general

1 classical C solution to (10) subject to (11) has been found. we already 1

know that if a weak solution satisfying (14) is also piecewise C , then it

must be a classical solution wherever it Is ~. solution coincides with a piecewise

cl

1 Thus a piecewise C weak

classical solution, as would be

expected of any reasonable extension of the concept of a solution. Let us now show that there is a further common property shared between weak and piecewise

c1 classical solutions. This is that a piecewise Cl weak

solution satisfies the generalised Rankine-Hugoniot condition across a shock. Consider the region R bounded by the closed arc oR and traversed by the line L across which a shock occurs.

Denote the two sub-regions so defined

by R_ and R+ and their boundaries by 3R_ and 3R+ 1 and let the directed arcs along adjacent sides of L be oL_ and 3L+. as in the Figure.

t

L

Shock line L dividing R Then R - R_ UR+ and oR • 3R_uoR+.

The test functions w in (14) will be

assumed to have their support in R

80

on oR.

that the test functions w will vanish

Thus (14) may be written

II!:;

u

+ :: F(U») dxdt

o

(15)

67 Now multiply (13) by wand integrate over R_ to obtain

JJ

_ R

(w :~

+

W

:~)

dxdt

-

O.

which may also be written in the form

a(wu) + a(WF») [ at ax

dxdt -

II

dxdt

•

o.

(16)

R

Applying Green's theorem to the first terms in this result then transforms (16) to

I -wFdt 1aa vaL

o.

+ wudx _

(17)

However as the support of the functions w lie in R, w will be zero on itR

80

thOt (17) reduce. to

1 -wF(u_)dt

TaL

+ w u_dx

(18)

A similar result applies with respect to R+ where we find

taL-

F(U+) dt +

W

+

u+dx -

Ill:~

u + :: F] dxdt

-

o.

(19)

+

the integration along

at and 3L+ being oppositely directed, as indicated

in the Figure.

If (18) and (19) are now added. the sign of the line reversed with a corresponding replacement of aL

integral~ln

(18) is

by 3L+ and result (15) 1s

used we find (20)

where as the point (x,t) 1s now constrained to lie on 3L+ the term (dx/dt) represents the speed of propagation

~

of the shock along L.

As

w i8 arbitrary,

(20) can only be true if (21)

which 1s the one dimensional form of the generalised Rankine-Hugooiot condition.

This holds degenerately when u is continuous across L.

If, now, the support of w is allowed to be arbitrary, the same form of

68 argument proves that piecewise

cl

solutions of (13) satIsfying (21) across

a shock will also be a weak solution of (13).

We thus arrIve at 'the following

definition and theorem.

DefInition (Weak Solution) The function u will be called a weak solution of

o if for all twice continuously differentiable test functions w with support

In

t

> 0 the function u Is Buch that

ff"" { Jl at:

I)

aw PCu) ] dxdt + ax

o•

the integration being extended over the upper half plane

t

> O.

Theorem 3 (Properties of Weak Solutions) Let u be a weak solution of au + aF(u)

at

ax

_

o.

The following results are then true: (a)

If u Is piecewise C1 in additIon to being a weak solution it is also a piecewise C1 classical solution.

(b)

1 a piecewise C weak solution satisfies the generalised Rankine-Rugoniot condition

across a discontinuity moving with speed A; (c)

a necessary and sufficient condition for a piecewise

cl

classical

solution to be a weak solution is that across a discontinuity moving with speed l it satisfies the generalised Rankine-Hugoniot condition. The general objective when introducing a weak solution was to lift the requirements of strict continuity and differentiability that Deed to be imposed on classical solutions. solution is successful and,

In this respect the notion of a weak

furtbermore~

because of its method of definition

1 the class of weak solutions is even wider than the class of piecewise C

69 functions

80

that considerable

generality has been achieved.

However~

this generality has been obtained at the cost of the uniqueness of a weak solution.

MOre precisely. unlike a strict classical C1 solution. a weak

solution 1s not determined tmiquely by the initial data.

Th.1e is most easily

deDJostrated by means of a simple example.

Consider a Riemann problem for an equation of the form

[1 3}

au +...!. at a,,"3

•

u

0.

with

80

for x < 0

Ol'

•

ues.O)

{

for x > 0

that in (13) we have F(u) • u 313.

11leu,

the equation

a8

'!,8

ho.:)gea.eous, when it 1s differentiable a

non-constant solution u viII be a function of x/e, and it 1s easily verified that the function

for

0 u(". t)

•

(,,It)1

1 1

1s a

cl

"It for


0

1

< 0

are two artitrary constants.

More generally, it may be

extended to include a number of such discontinuities located along the ini~ial

line.

The characteristics of (1) are the curves (2)

along which the equation may be written in the form du

dt

o.

-

(3)

Rence for x < 0 the characteristics are parallel straight lines with slope

~

•

2 uo.

whereas for x > 0 they are parallel straight lines with

2

slope A • u • l If u~ < u~ these two families of characteristics diverge, as in Figure (a), when the wedge shaped region W is not traversed by any of these characteristics.

However, if u~ > u~ the two families of

characteristics intersect from the start, leading to non-uniqueness and shock formation of the type first indicated at the end of Lecture 1.

o.

76

t

lsI-it I i",e..

= ",,4,.:

(a)

I

(b)

Centred simple wave in W

Shock speed

we thus arrive at the result that the condition for a physically admissible ahock solution lor (1) i. (4)

Now (1) is invariant under tbe replacement of x and so that its solution depend. only on tbe ratio t • x/t. all pass through the origin, and along them. u • const..

the wedge shaped region W in (a), and

88

t

by ax and ~tt

The linea t • canst. They thus fill in

tbey are characteristics the wave

solution described by them iD W is called a centred simple wave.

In this

case the centre 1s at 0 which is the location of tbe discontinuity in the initial data.

Taking tbe particular case in (1)

leads

o•

U

to the differentiable solution for region W given in

Lecture 6, and illustrated there by

! 0

u(x, t)

0, u • 1 and setting u(x, t) • u(C) 1

for

(X/t) I

I

for

3

Figure:

x/t < 0

for

0 s xlt S 1

xlt > 1

Notice that the non-physical shock

(5)

(weak solution) liVeD in

Lec~ure

6,

_ly

u(x, t)

. {0 1

for xlt




1/3

(6)

77

lies in region Wand

is

80

~

produced by the intersection of characteristics.

It is for this reason that it is not physically realisable and

80

must

be rejected.

A physical shock occurs in the situation illustrated in Figure (b) however and emanates from the origin. u(x. 0)

•

{ :

for

z

for

x> 0

Using the initial data

< 0

as a typical example, we find from the Rankine-Hugoniot condition that ~

• 1/3.

Thus in this case the resolution of the initial discontinuity

merely involves its propagation along the shock line

3x.

t •

We conclude from this that for a centred simple wave (rarefaction fan) to occur, the characteristics must diverge from a point, leaving a wedge shaped region to be filled by the centred simple wave.

A shock will only

occur when the characteristics converge and intersect. ~.

Riemann Problem for a System Let us now consider the reducible hyperbolic system

Ut subj~ct

+

Va

0

to the initial data

U(x. 0)

where

A(U) U x

{::

for

x




0

and Un are constant n element vectors.

(8)

The Riemann problem DOW

becomes the resolution of the initial vector discontinuity at x • 0, though as with the scalar case it may be extended to include a number of such discontinuities along tbe initial line. We look for the solution of this problem in terms of generalized simple waves and shocks, which will be the analogue of the situation just discussed for a single equation.

The generalized Rankine-Hugoniot

condition is of the form [F] •

(9)

78

once (7) has been expressed in the conservation form

~+~(u).o. at _

(10)

This implies n possible types of shock with speeds i(l)~ ••• ~i(n) p~18ical

and we shall need a

we did in the simpler case.

admissibility criterion for them, just as The extension of our earlier result (4) that

provides the criterion we need i. due to Laz who requires that for some integer k with 1 :s; It S D

while

(11) A(k-1) (u )
'it)

separated by a

which baa the solution

for

x


),t

(17)

For ease of illustration, let U8 take all time Iteps equal and denote them by k.

'l'be first step of eliDa' 8 scheme gives

for (18)

for where now

[~

if

(19) if

The reault of n such stepa with the scheme is to give

[~

for

x < J h

for

x > J h

n

(20)

n

where we have •• t

•

number of

Q

j


0 for the solution v(x,t) when t.

(x > 0) •

1/2x

This is not due in any way to the intersection of characteristics within a family, for they are parallel straight 11nes. initial data becomes unbounded for large x

80

However, in this case the

that it might be considered

this 1s the cause of the unboundedness of the solution.

To show this 1s

not the caBe cODsider this next example.

_"Ple 2

2 Take uO(x) • a tanh x, vO(x) • 1, f • ltv and g • 1, when we find

ttl

u(x,t)

[tanh(x

+

t)

+

tanh(x - t»)

2 2+a[tanh(x+t)-tanh(x-t)]

vex, t)

Here u(x,t) remains finite for all x,t but v(x,t) becomes unbounded at an excape time t. given by t.

-1

tanh

2 2 4 asech x + a sech x + 4tanh2x 2 tanh 2 x

In this case, by making

a

suitably small, the deviation of the

initial data from constant values may be made as small as desired, but the finite escape time still persists.

84

References

[1] [2J [3] [4J [51 [6] [7J

Quasilinear Hyperbolic Systems and Waves. Research Note 1n Mathematics,S, Pitman Publishing, London, 1976. Lax, P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematic8, II, 1973. Gllmm. J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 13 (1965), 697-715. Godunov, S. An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521-523. Godunov, S. Bounds on the discrepancy of approximate solutions constructed for the equations of gas dynamics, J. Comput. Math. and Math. Phys. 1 (1961), 623-637. Charin, A. Random choice solution of hyperbolic systems, J. Compo Phys. 22 (1976), 517-533. Jeffrey, A. The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type~ Proc. Roy. Soc. Edinb. 77A (1977), 1-8. Jeffrey~

A.

85 Lecture 8. 1.

Far Fields. Solitons and Inverse Scattering

Far Fields

There are many different types of higher order equations and systems of equations that characterise nonlinear wave propagation in RX t, either with or without dispersion.

A simplification frequently takes place in

the representation of the solutions to inltlal value problems to such equations after a suitable lapse of time or. equivalently. suitably far from

the origio, particularly when the initla1 data Is localised and compact support.

80

has

These simplified forms of solution are often asymptotic

solutions, and are appropriately called far fields. The simplest examples of these are the types of far field behaviour exhibited by the ordinary linear wave equation and by a homogeneous quasllinear hyperbolic system with n dependent variables.

Due to its

linearity, the wave equation const. )

(c

(1)

may be written either in the form

+ c..l..] [au _ cau] [..l.. at ax at ax

o

(2)

_c..l..] [au + cau] [..l.. at ax at ax

0

(3)

or as

Then. if u(+) is the solution of

o.

(4)

it follows that u(t) is a degenerate solution of (2) and u(-) is a degenerate solution of (3).

The general 601ution of (4) is then

f(±) (x"+ ct) •

(5)

with f(±) arbitrary C1 functions. These travelling wave solutions are such that u (+) propagates to the right and u(-) to the left with speed c.

We thus have the situation that u(t)

86

are special simple types of solution to the wave equation (1), in the sense that they only satisfy a first order partial differential equation, whereas the wave equation itself Is of second order.

Such special solutions

become of considerable interest when the initial data

f~t)

(+)

with compact support, 60 that £0- (x) initial data lies in

Ixl

f

1

CO'

Is differentiable

Then, if the support of the

d, after an elapsed time dIe the interaction


/2 that 1s uniquely determined by the boundary conditions, but is invariant with respect to an arbitrary fixed

spatial translation.

In general, all solutions v(x,t) of Burgers' equation

are invariant with respect to a Galilean transformation.

This 801ut10n links

two different constant states at plus and minus infinity. The KdV solitary wave, as the solution (26) is called, is different and is a pulse shaped wave that, relative to the same constant value v. at plus and minus infinity, tends to zero together with all its derivatives as Itl~.

Its speed of propagation relative to

Veo

is proportional to the

amplitude a, and its width is inversely proportional to the square root of the amplitude.

In this travelling wave solution the speed is not determined

by the boundary conditions, but by the amplitude a > O.

Like Burgers' equation,

the KdV solitary wave 1s also invariant with respect to a Galilean transformation. zabusky and Kruskal found numerically that a KdV solitary wavj:!. behaves like a particle.

Specifically they

fo~nd

that when two different amplitude

waves of this type are such that the one with the greater amplitude starts to the left of the one with thelesset amplitude, then the larger one overtakes the smaller one and, after interacting with it, the waves have merely inter-

93 changed positions.

This is a nonlinear interaction yet the pulse shapes

are preserved exactly after the interaction, though the phases of the

pulses (the location of their peaks) is affected by this process.

On

account of this Zabusky and Kruskal invented the word "soliton" for a wave

that preserves its identity exactly in this sense after a nonlinear

interaction.

Thus KdV solitary waves are solitons.

The recent interest in solitons derives from the fact that the KdV equation is often found to arise as a far field equation and, furthermore, arbitrary initial data for the KdV equation evolves into a train of solitons together with, possibly, an oscillatory tail.

This means that solitons are,

in a sense, fundamental solutions of the KdV equation.

An

extensive

literature now exists on this topic, and we refer to the articles and to the references contained therein, in Jeffrey and Kakutani and in the various articles by Kruskal, Lax, Ablowitz, Newell and Segur 1n the AHS publication Nonlinear Wave Motion listed at the end of this lecture.

Many different

typea of nonlinear evolution equation have been found to possess soliton solutions and for more information on this topic we refer to the review paper by Scott,

Ch~_

and McLaughlin for both a good account of some of them

and also for the basic references, and also to the edited collection of papers by Bullough and Caudrey. 4.

Inverse Scattering The behaviour of soliton solutions to the KdV equation is suggestive

of linear behaviour and this motivated Gardner, Greene, Kruekal and Miura to try to find a linearising transformation of the type used by Hopf and Cole to transform Burgers' equation to the heat equation (see Scott, Chu, Mclaughlin).

No such transformation was found, but during their search

they discovered an important connection between the KdV equation and an eigenvalue problem for the scattering method used in

Schr~dinger q~ntum

equation in terms of the inverse

mechanics.

It 1s this result that has

94 come to be known as the inverse scattering method in the context of solitons. We can do no more bere than outline the ideas that are involved. The b.sic problem to be considered 1s how a general solution of the KdV equation

au

o

3

O.

Thus a soliton

On the other hand, in the case of

equation (33), it follows that

small luI, and we obtain.

p

e±ikx, l • k 2 •

-~

xx

~ l~

for sufficiently

This approximation is still

valid for an arbitrary value of u, provided k is thought to be sufficiently large.

For a bound state, the eigenvalue becomes ). •

and. decreases exponentially at infinity. one bound state corresponds to one soliton. soliton solution

-K

2

p


,

e'est

1

dire admet I'equation

d'etat

p - p + cte

Dans ce cas les andes acoustiques sont exceptionnelles au sens defini precedeument.

Pour les fluides reels on a pI! p

~

0

done

a

5i

P~


~

0

0 on a raidissement des signaux.

On peut aussi trouver un deve10ppement asymptotique correspond ant aux andes

materielles; celles ci apparaissent comme multiples (en accord avec la theode generate de Bailiat, cf [4]). Le probleme de Cauchy oscillatoire est

144

resoluble~

au premier ordre d'approximation. pour les equations des fluides

parfaits.

II olen est plus de meme en presence de phenomenes dissipatifs, par exemple

pour un fluide charge

a

conductivite non nulle (c£ (3J) .

References.

[I] [2]

Lichnerowicz A., Hydrodynamique et magnetohydrodynamique. Benjamin 1967. Tauh A.H., High frequency gravitational waves and average lagrangian, General Relativity and Gravitation, Einstein Centenary volume, A. Held ed, Plenum.

[3J

Choquet-Bruhat Y., Ondes asymptotiques pour un syst~me d'equations aux

derivees pareielles non lineaires, J. Maths pures et appliquees, 48, 1969, p. 117-158. Coupling of high frequency gravitational and electromagnetic waves, Actes du Congres Marcel Grossmann, Trieste, Juillet 1975).

r4}

Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. Pura et Appl. ~,

[5}

1976, 31-44.

Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.R.P., vol.XXIX. n03, 1978.p.257.

[6]

Breuer R.A. and Ehlers J •• Propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. to appear in Proc. Roy. Soc. A.

145

VI DETERMINATION DE LA PHASE. BICARACTERISTIQUES. VARIETES LAGRANGIENNES.

I. D€finition

des variEtes lagrangiennes.

L'equation eikonale i. laquelle doit satisfaire 18 phase (1-1)

'f:

A(x. '!'x) • 0

est une equation aux derivees partielles du premier ordre non lineaire, A est un polynome de degre N, homage-ne, en

1 x."

NOllS

allons rappeler

CODIDent

on in-

tegre une tel Ie equation. en utilisant Ie langage de la geometrie symplectique. On

*

designe par T X l'espace fibre cotangent

a

Is variete x de dimension 1.

Un point de T*I est un couple (x,p) ou pest une I-forme sur llespace tangent

TxX a x en x, clest a dire un vecteur covariant. Une solution de l' equation I-I dans 8u-desSU8 de

Q

c. X

est une section du fibre T:l.X x lR,

n par

telle que

(d

'f) x

• < P. dx >

). - 0, .•. t-I

146

A(x,p)

et

o

sur une telle section on a

Soit n : T%X ~ X la projection canonique (x,p) l~ x de r*x sur X. On defiDit une J-forme sur TXx. appelee I-forme fondamentale ou forme de soudure par

u~T

6(x,p) (u) - p(fl' (u))

x,p

T"x

son expression en coordonnees locales est ;

6 La 2-forme

o -

d6

est fermee et de rang 2t, une telle 2-forme est dite symplectigue. La 2-forme cr munit

T*x

de sa structure symplectique canonique. Une sous-variete de T*x

qui annule a et qui est de la dimension maximum possible, c'est

a dire:

t •

est dite lagrangienne.

5i Vest une variete. de dimension t. immergee dans

T*x

par une application

f, on dit que (V. f) est une sous-variete lagrangieone [ilDl1ergeeJ de T*x si

sur V

au

£

2 Recherche dlune sous-vari€te lagrangienne (V.f) de TXx telle que A(x.p) ~ O.

Le probleme est celui de la recherche des varietes integrales (immergees

147 dans T~)de dimension t du systeme differentiel exterieur

o

(2-1)

o

(2-2)

A(x,p) '" 0

La fermeture de ce s~steme contient, outre les equations precedentes. l'equa-

T*x :

tion exterieure sur

o

dA •

(2-3)

Le systeme caracteristique de 2-1. 2-2 est le systeme associe de 2-1, 2-3. II est constitue par les champs de vecteurs v sur TXx tels que :

(2-4)

ia-kdA

avec

v

c'est

a dire

k€lR

en coordonnees canoniques (xA,p.>..) de TXX

ou

v-

A - O. • ..• 1-1 VA+!

(2-5)

- aA/a/'

Un champ de vecteurs VA possedant la propriete 2-4 est dit hamiltonien pour la structure symplectique a et l'hamiltonien A. On remarque que VA est tangent

a

Jt (sous-variete

A(x,p) • 0 de T*x)

Une trajectoire du champ de vecteurs hamiltonien VA est appelee une (courbe)

bicaracteristique de Itequation aux derivees partielles A(x. On

• O.

suppose que I'hamiltonien A nta pas de point critique sur cf};(c'est

dire que dA sur

~x}

cfC.

Theoreme

~

a

0 quand A • O}. Ie champ hamiItonien VA n'a alors pas de zero

et on demontre Ie theoreme fondamental suivant (cf par exemple

Ill):

Soit Y une sous-variete compacte de dimension t-I de T*x verifiant

o • 0 et A • O. transversale en chaque point au champ hamiltonien VA" Soit Ie flat du champ de vecteurs VA' alors (y x ~.f)

au

f : Y x:R ~T*x

par (y.t) .... ft(y)

f

t

148

est une sous-variete lagrangienne immergee de T~.

3. Determination de 18 phase.

Etant donnie une sous-variete. lagrangienne (V,f) immergee dans X, il existe dans chaque ouvert U C V simplement connexe une phase $ • de.terminee a une

constante additive pres, satisfaisant a l 1 equation d ~ •

(3-1 )

f'to

de

puisque l'on a. sur U. ft:

f"

e

_ o. v

Remarque : II existe toujours, globalement. sur Ie recouvrement universel V v

de V. projete sur V par

n.

v

une fanction v d ~

On de-duit de

IT

0

f

:

W•

COnDU

(f

$ telIe que 0

v IT)"

dans U C. V. une phase

f

e dans

nC

X si l' application

U + nest inversible.

L1 application f etant une immersion il existe toujours un sous-ouvert, encore note U. de U. tel que f soit un diffeomorphisme de U sur feU). L'applica-

tion

n0

fest alers inversible sur U si 1a projection IT : T*X ~ X restreinte

a £(U) est inversib1e : i1 en sera ainsi au voisinage de tout point OU f(U) n1est pas tangent

a

1a fibre de T*X, c 1 est

a dire

n 1 a pas un plan tangent

"vertical". Soit x Eo

n c.

X. tel que

n-I (x)

ne soit pas tangent :it f(U). Soit Y1 "··· Yk

les points de feU) tels que n(Yi) ex:

149 Le point x admettra un voisinage dans X, encore note Q tel que IT

-)

(0) soit

l l union disjointe d'ouverts de feU) : f(U )

U

(3-2)

i

i • I ••• _,k

A chaque U, pour une meme phase $ sur V. correspond une phase

f

i sur

n donnie

par :

If i

• ~

0

(IT i

0

f)

-)

•

Dans les applications la donnee physique est souvent la variete lagrangienne

V, provenant de la geometrie du probleme et de sa dynamique : les bicaracteristiques. c'est

a dire

les trajectoires du vecteur hamiltonien vA sont les

rayons lumineux (dans l'espace des phases) dans les problemes d'optique. les trajectoires des particules materielles dans d'autres problemes. Nous allons considerer Ie cas OU la p~ojection de V, supposee sous-variete de T*x pour simplifier, sur

n c:

T~X n'est pas bijective, mais

ble I: de V (son "contour apparent") tel que (3-3)

-I

n

au

(0). V

U

i • 1, ••• k

au

chaque restriction

J':i de n a

il existe un sous-ensem-

,1:

soit de 1a forme

U.

•

U i

u..... n

•

est un diffeomorphisme.

4. Solutions asymptotiques.

II est naturel de chercher une solution asymptotique du systeme differentiel

150

d'equation eikonale A(x.

~x)

a

- O. correspondant

une variete lagrangienne du

type 3-) sous la forme : k u(x)

(4-1 )

r

i ou les

~i

sont des phases. sur nc x. correspondant a la variete V. On sera

aide dans ce caleuI par la methode de la phase stationnaire qui montre (ef VII)

comment 4-1 liee a l'evaluation asymptotique d'une integrate. Des developpements du type 4-1, et les equations de transport correspondantes. sont utili-

sees pour determiner l'intensite lumineuse en presence de caustiques (enveloppe des rayons lumineux en projection sur l'espace-temps X). Remarque : Chaque phase

"Pi

n'est connue quia une constanCe additive pres.

puisque la variete lagrangienne V

De

determine

~

qu'a l'addition pres d'UDe

constante. depourvue de signification physique. La theorie des integrales asymptotiques et de ltincide de Maslov permet de determiner ces

constantes~

tification de

et

[51

relations entre

puis des conditions sur la variete V. dites conditions de quan-

Maslov~

pour qu'il corresponde

globale. avec une phase

[31

d~s

de VIII).

If

a v une

solution asymptotique

determinee globalement sur n • fi(V) (cf references

151

VII PHASE STATIONNAlRE. PARAMETRlSATION

D' UNE VARlETE LAGRANGlENNE.

1. Methode de 18 phase stationnaire (one variable).

On

se propose d'evaluer. pour w grand, une integrale de la forme l(w) m

au a et f sont des fonctions C , et a est )0)

On suppose que ~~

a

support compact.

ne s'annule pas sur Ie support de a; on deduit alers

de

a ( iwf)

aa

e

af

-iwaae

iwf

que l(w) -

1 iw

-

iw

l'integrale etant bornee par un nombre M independant de w on a

Mw

-1

152

Par iteration du

prod~de

on trouve, pour tout n €.

~

2°) On suppose que f s'annule en un point et un seul 0

if

a dire

que ---2 ' 0 pour a • a ' c'est o

aa

0

du support de a et

que f a un point critique, et un seu!,

non degenere sur Ie support de a. On montre que f peut alers s'ecrire dans un voisinage de ment de variable a --

t (0)

0

0

,

par un change-

tel que t (0 ) - 0, sous la forme 0

g( t) _ f(a(t» • g(o) +

f

t

2

e:

K

sign

done

I(w)

I

+

~

<Xl

elwE/2

t

2 b(t) dt

_00

_ iwg(o) ou b(t) - e .(a(t»

da

Cit

l'integrale existe, absolument convergente, puisque b a un support compact. On peut en particulier la calculer par passage

A

elwE/2

f

l(w)

t

a

la limite

2 b (t) d t

-A

On pose alars b(t) - b(o) " t e(t)

On sait que +A

lim A=oo

I

b (0) e iWE/2 t

2 dt-b(o)

-A

On va. estimer

I, (w)"

"A

lim A-+OD

I

e

iwe:/2 t

2 c(t) t dt

-A

Puisque b a un support compact on a c(A) - c(- A)

done

(31!.) 1/2

w

e

iITE/4

153

+A

( ) __I_

I,

lim

iwe:

W

A-

J-A

~

2

e iw£/2 t

c'(t) dt

2 l'integrale est bornee parce que c' est une fauction de classe C bornee ain2 iwE 2 / t c' (t) dt tend si que ses derivees 2 (on montre que e

J:

d'ordre~

vers zero quand A et B tendenr vers l'infini en faisant des integrations par

a nouveau

parties). Le raisonnement peut s'appliquer

pour l'estimarion de

I'integrate figurant dans 1 (00). 1

On a montre que :

l(w) On. b(o) -

On.

is.. at dt

+~ 30

do 30 dt

au point t - 0, a

=

a

o

on a done

2

do

E •

dt ) taO

done do

dt

I t-O

_

32~

1- 1/ 2

aa

n-o

o

D'oll finalement l'evaluation asymptotique

l(w)

ou 3°) Supposons que f a un ensemble fioi

OJ'

j Go J, de points critiques non

degeneres sor Ie support de a. En utilisant une partition de l'unite sur ce

154

support, isolant lea points critiques, on demontre que

ill£./4

•

E

I(w)

j

E

J

iwf(a.) J a(a.) J

•

J

ou

2. Methode de 1& phase stationnaire (plusieurs variables).

Considerons l'integrale I(w) •

JY

a(y) .iwf(y)

d~(y) m

oil a et f sont des fanctions a valeurs reelles. C

sur une variete riemanien-

ne C~J Y, d'element de volume d~(Y). de dimension n. to) Supposons que a est a support compact et que f o'a pas de point critique sur Ie support de a, on a alars, pour tout.N E. E

:

-N I(w)-O(w) En effet on a :

g

iwl

ij

•

d'oil

•Lwl

I

-

iw

iwl

•

en posane (si f n'a pas de point critique on a

g

dX v

i

af

ij H i

dX

j

i

0)

155 c'est

a

dire Vf v -

•

IVfl 2

i 0

et U ' 2

et de l'ensemble singulier :

r •

(0,0)

160

VIII DEVELOPPEHENT ASlMPTOTIQUE AD VOISINAGE D'1JNE CAUSTlQUE.

1. Caustiques du premier type.

La forme dlune parametrisation d'une variite lagrangienne est liee a la nature des singularites de l'inverse de la projection

n : T*x

+

X restreinte

a

V. Lea singularites des applications differentiables ont ite classifiees par Thorn. On montre que dans Ie cas Ie plus simple ( pliage, suquel correspond l'exemple

a 18

fin de VII) on peut parametriser V par une fanction de la

forme :

f(x,a) - a(x) + p(x)a ou a et p sont des fanctions regulieres du point x E X.

L'ensemble C est slars f

af - p(x) aa

- a

2

0

. a2 f L'ensemble singulier E de Vest l'image de l' ensemble --2 := - 2a - 0 de Cf" aa

II se projette sur X en p(x) - 0 et on a

U E

±

+ EP

1/2

V p)

x p(x) > 0

161

La,

figure represente la projection sur

X : l'ensemble singulier

~

se projette

sur l' ensemle C appete "caustique".

L'ensemble p(x) < 0 n'est la projection

p(x) < 0

d'aucun point de V, I'ensemble p(x) > 0 est recouvert deux lois par la projec-

p(x) > 0

tion de V : deux "rayons" psssent par

chacun de sea points.

2. Integrate asymptotique, fonction d'Airy.

Pour construire des solutions asymptotiques d'un systeme differentiel correspondant i une variete lagrangienne du type precedent on va etudier Is valeur asymptotique de 11 integrale

u(x) -

3

I

- e iwo(x) e iwf(x,a) a ( x,a ) da =

On sait que

(theor~me

I

iw(p(x)a ~ a ) 3

e

.(x,a) da

de preparation de Malgrange) il existe des Jonctions

regulieres 8 (X), al(x) et h(x,a) telles que: 0 2 + b(x,a) (p(x) - a )

. done puuque

Ie iwf

(x, a)

p(x) _ a 2 •

af an Jeiwf(X,Q)da +

a(x,a)da • •0 (x)

+

Le dernier terme peut s'ecrire

I

a)

(x)

Jeiwf(X,a) ada

e iwf(x,"'.';;'(A r t n

Ie (5.01) con s

grazi~

=~ .

.0,

-l-I}?h

I

in virtu di (7.09). Siccome ~ ~ nulla per l'urto nullo ne risulta '" (u

I

0

,~ ,~) !! 0

(8.01)

0

Pertanto, la sua derivate rispetto a u

o

e

anche nulla e tenendo conto dalla

(5.03) viene

"'1 '0

.-wV~ 0 0

che inserita nella (5.08) foroisee

= u'

h'

(8.02)

- u·

o

Adesso introduciamo un vettore a(u.~) cosl definito I • 1,2 ••.. ,m

(i)

(8.03)

La soluzione estate proprio perche vale la condizione di ecceziona1ita (7.11),

e

unica perche

e

ortogona1e ag11 autovettori d

puo mettersi sot to la forma

r .

11 saIto del campo princi~~

[151 (8.04)

Nel caso lineare g sarebbe nullo. 11 secondo termine rappresenta dunque 1a parte non lineare dell'urte caratteristico. I due vettori dipendona sol tanto I

delle state prima dell'urto e 90no cono9ciuti. Resta da determinare ..... (u ,u ). o I Deriviamo (5.04) rispetto a u =

';l

I

'"h'.h

187

cio~

• w• 1 I

10

.h/(l - g .h)

(8.05)

0

purch~

(8.06)

Per la derivata seconda

v +') h'H'~ hi I I'

•

(8.07)

9. Stabll! ti. dell' urto caratteristlco Sia (9.01)

01 = cost .•

una soluzione ovvia di (7.13). Scegliamo come parametri u sono del tutto arbltrari ) delle funzioni lineari di

Se supponiamo che

If.

1

(chef a priori,

Segue da (B.05. 07)

quat!he sia l'urta, (9.02)

l-gh>O o

come 10

e

quando l'urto

e

debale, ailars

la derivata prima cresce e siccome

e

nulla quando l'urto

Ne risulta che w tende all'infinito con h.h' ~ w

't .

e

nullo,

e

positiva.

Ma s1 vede facilmente che

188

e

e dunQue anche l'urto non rappresentata ad

e~empio

f e Questa

e

- '(

limitato appena s1 sposta 18 superficie d'urto

da un'equazione del tipo (1) i

'no.

)I·0 t

= x ",, -

i l caso dell'urto di contatto gil citato. In un urto stabile Is quan-

tita (B.D6) cambia di segno [161 . Vedremo come 51 traduce Questa condizione per Ie equazioni di Eulero.

10. Evoluzione dell' urto earatteristico di Eulero

DaIle equazioni di Eulero scritte nella forma

U

t

(vedi §3)

+

AIlu'

= Btu·

i

(10.1)

derivano Ie condizioni di Rankine-Hugoniot A'h' -I-h

(10.2)

= O.

" Da cui segue, tenendo conto del legame (§ 2)

e di (8.04). h = H'h' -

o

wV')./ 0

0

At

(10.3)

Definiamo gIl Butovettori 1'(A' - ~H')= O. I "

tale chef I' =- 1 I

.-

d'

I

I

(A' - ~H' )d' n I

H'd' I

(10,4)

0,

~II'

.

Allers da (3) viene semplicemente,

mentre g .h =

o

dove 0(.0 =o«u ' o

"1) e

0( w ,

(10.5)

0

una ftJn:z1one conosciuta del campo prima dell'urto

189

(10.6)

La (8.05) diventa u 1(1 -0( w)

'I

(10.7)

0

che s1 integra subito -' 2 (1 -",w) = - 0(

o

lui 2

0

+

a(u • ~). lui o

La costante di integrazione s1 determine sapendo che l'urto I

que anche g11 u • vedi 6.04) quando w

~

nullo

(5.05) e

~

nullo (e dun-

r17] (10.8)

Vediamo dB (5) che 1-ah-1-o(W o 0 L I instabi!i tlt corrisponde

guenzB l'urto

Non

e

e

8

- o. Iu I

e di conse-

~

continua. Derivando (4)

HI )V'd' - H'd' V'~ -~V'H'd' = O. I I I

Moltiplichiamo a slnistra per

vata del autovettore

W

cambia allore di segno .

difficile di vedere che Ie velocita radi31e (A' n

t

Ii

e a destra per ')i

di

che rappresenta Is deri-

rispetto a ~i

V't, ~id'

• -l ~

Eo Eo

- -

, (1.4) ~ = .. '" ./

1'"

dove E. '.Y' SOIlO rispettivamellte la cost ante dielettrica, la peraeabilitA magnetica, la conduttivitA del mezzo ael punta

X in

cui si cOllsiderano i vettori che cOlllpaiollo rispet-

tivamente in (1.3),(1.4),(1.5). Be poi il mezzo e anisotropo Ie £ ,

r-

Poiche E ,

' f vanno sostituite COil tensori -doppi.

r:J'"

dipendollo solo dal mezzo e nOll dal campo e-

lettromagnetico, le (1.1),(1.a),(1.3),(1.4) e (1.5) costituisCODO un sistema 1iaeare, perci6 l'elettromagnetiSJllo ordilla-._ rio si puO chiamare anche e1ettromagnetismo liaeare. Per6 in quei dielettrici (ai quali ci riferiremo sempre ill seguito) dove si manifestano i fenOllleni dell' ottica noa lineare J mentre restano val ide Ie (1.4) e (1.5). 1a (1.3) va sostituita

COD

UJla relazione non lineare ira 0 ~

E

che scriveremo:

(1.3') sicche i mezzi in cui vale (1.3') si possono chiamare dielettrici non 1ineari. Ia questa lezione cercher6 di.stabilire alcune proprietA

del~e

onde elettromagnetiche che si propa-

,,\\--..t.o

gano ne1 die1ettricorcosi da interpret are qualche fenomeno del1'ottica nOll lineare.

2.

Riferiamo i punti della spazio a un sistema di coordinate

cartesiane ortogonali (O,x,y.z) e supponiamo il dominio coincidente con una 1amiaa di spes sore

5

riempita da Ull

~

199

dielettrico non lineare omogeneo. Porremo ~

l'origin~

0 e l'asse

del sistema di assi in modo che le facce della lamina abbiano

equazione z=O, z=s. All'esterno della lamina supporremo il vuo-

to che, dal punto di vista elettromagnetico, si pu6 identifica_ re con I'aria. Indicheremo con £.la costante dielettrica del

vuoto, mentre ammetteremo la

r

che compare nella (1.4) ugua-

le a quella del vuoto (ipotesi non restrittiva dal punta di vista fisico)

ci~ ammetteremo~ identica

i. tutto 10 spa~io.

Nel semispazio z < 0 sia posta una sorgente che generi un' onda

elettromagnetica piana con all'asse

~.

dire~ione

di

propaga~ione

parallela

Supponiamo la lamina tagliata e disposta in modo

che il campo elettromagnetico dipenda solo da

e t; anzi, con

~

un'opportuna disposizione degli assi x e y si possa scrivere, per ogni punta dello spazio, : (2.1)

"E.

E. (l,lo)L"

laoltre supporrenioIi parallelo ad

E ci~; avremo:

(2.3) Allora la (1.3') diventa (sottintendendo le variabili z e t)· l'equazione scalare:

(2.4) e la fUnzitne D(E) verrA supposta di classe

c~

in qualunque

intervallo limitato dell'asse reale. Le equadoni di Maxwell nella lamina si riducono a:

200

La. (2.5) e (2.6) valgol1o aache all'esterno della lamil1a purchi!

r

si pOl1ga .0, e. ill luogo di ~ ~ • Ammetteremo inoltre, conPorme llesperienz~: (2.4')

cioi! D Pul1zione crescente di E e D(O)_O. Stabiliamo ora alcune condiziOl1i sui pian! che limitaao la lamina,pi~

precisamente sui piani z=+O, z=s-O ; si i! scritto +0 e

s-O per identi£icare Ie Pacce dei piani rivolte verso l'interno della lamina

o,pi~

brevemente,£acce interne.

Ora, nel semispazio z < 0 s1 avranno due onde, una che diremo

diretta, emessa dalla sorgente e che si propaga nel verso positivo dell'asse z, l'altra ri£lessa dal1a lamiaa e che si pro-

paga nel verso negativo dell'asse z. Detti E~(z,t), ~(z,t)

g~(z,t),

HdJz,t),

rispettivamente il campo dell'onda diretta e

il campo dell'onda rifles sa, si hal (2.7 )

Ora,

} c~e

e

noto, su un piano che separa due mezzi divers! so-

no continue Ie cOlllponenti taagenziali al piano del campo elettromagnetico (ovviamente z=-O, z=s+O sono Ie facce della lamina rivolte verso l'esterno

0

facee esterne). 5i ha cos!:

(~.:l)

£.olJ_O,!;)+fo",(-o,~). E (d,t)

(2.9)

Hd.(-o,~) .. I-1,,{_O,~). 1-1 (+o,~).

Ora, per note proprietA delle onde elettromagnetiche piane si ha:

(2.10) l4.l(-O,!;).~ Eol(-O,~) ./

Sostituendo (2.10) e (2.11) in (2.9) e sommando con (2.8) moltiplicata per \;~

.r

si eliminano E,.,. e HJ(... • Allora, riservando

201

il simbolo E{z,t), H{z,t) al campo entro la lamina ed ometteado, per semplicita di scrittura e perche ora non vi

e

luogo ad

equivoco, i segni + e - davanti allo 0, si hal

Hel semispazio z;> s si ha solo un'onda che diremo trasmessa e che si propaga nel verso positivo dell'asse z (non si possono avere riflessioni perche per z >s il mezzo e omogeneo) i cui campi indicheremo con

E~(z,t), H~(z,t).

Per la continuita del-

le componenti del campo elettromagnetico sul piano z=s (ora si possono evitare i simbcli +0 e _0) si hal

e poiche "~vale

Viii- If~ si ha subito:

(2.14) Le (2,12), (2.14) in cui

E~(O,t)

si suppone assegnato, cost i-

tuiscono condizioni alla frontiera per (2.§) e (2.6). Ad esse si possono eventualmente associare opportune condizioni iniziaIi, sicche i1 campo entro la lamina resta determinato.

Le (2;12) e (2.14) si devono al Frof.Cesari (11 (2) [3] [4] il quale ha dimostrato importanti teoremi di esistenza, di unieita, di dipendenza continua dai dati per Ie soluzioni delle

equazioni

(2.~)

e (2.6) corredate da (2.13) e (2.14), qual ora

sia noto Eo\,{O, t) per ogni t (positivo " negativo). Nel caso, importantissimo per le applicazioni, in cui co rispetto al tempo e con periodo T,

~nche

E~{O,t)

e periodi-

i1 campo entro la

lamina risulta periodico con 10 stesso periodo. Noto i1 campo

entro la lamina, mediante (2.8), (2.9) e (2.13)

e

colare i1 campo ri£lesso e trasmesso dal1a lamina.

facile cal-

.202

I teoremi di Cesari sono stati dimostrati per valori dello spessore s della lamina non trappo elevati. Torner6 in seguito su

[5]

questi risultati, per ora noterO che il Prof. Bassanini

ha dimostrato che i valori di s per cui sono validi i teoremi ora citati risultano superiori allo spessore delle lamine usate in pratica_

3.

Passiamo ora a

ricerca~

una soluzione di notevole interes-

se di (2.5) e (2.6) supponendo (come faremo sempre in seguito)

fhO. A questa scopo poniamo, ricordando (2.4'), (3.1) ~

Nel caso lineare (si ricordi (1.3»

= E.

(cost ante die-

lettrica) ed esiste una soluzione delle (2.5) e (2.6) per cui i1 campo elettrico ha l'espressione:

E(Z,t) dove G(u)

~

= G(u)

,

una fUnzione di classe C. della u per u variabile

in qualunque intervallo limitato dell'asse reale; G(u) se u=t vale il campo elettrico suI piano z=O e all'istante t,

sicch~

Ie proprietA della ru,zione di t,E(O,t), sono Ie stesse di G(t) o G(u).

Ora. nel caso lineare p(E)=

J8y' ;

viene perciO naturale con-

getturare valide Ie (3.2) anche nel caso generale sostituendo perO nell'espressione di u a V~, peE) come definita do. (3.1) e con seguo positivo. Si ha cosi:

(3.3)

E .. G(u)

(3.3')

u .. t - p(E)z.

203

~rimo m~mbro

di (3.3) si ha l'~quazion~ implicitam~nt~ E in £unzion~ di t ~ z :

Ora, portando G(u) al ch~ d~fiftisc~

E _ G(

(3.4)

t -

p(E)z ) • 0 z=o.

Qu~sta ~quazioft~ ~ ovviament~ risolubil~ p~r

risolubil~ p~r z~O ~ sufPic~ftt~, p~r ch~

implicite,

il

£Unzioni

)

condizione

c~rtam~nt~

discuter~

Cal n.5) di ~

t~or~ma d~ll~

sia

sin I

(3.5)

sto

AffiRCh~

intuitivo, che

z e (O,h), t

E:

soddisfatta

z=O. Ora,

p~r

amm~ttiamo, com~ d~l r~­

meglio la (3.5), ~sista

(-'J;,T) ( T,

~

Uft

h> 0

ftum~ro

~

T positivi avv~rt~nza

sia valida (3.5). Fino ad

ris~rvandoci

tal~

del

che

r~sto

p~r

ogni

arbitrari)

ift contrario,

amm~tt~r~­

i1

campo magne-

mo z e t Dei limiti ora indicati.

Cic

prem~sso, v~diamo

di

d~t~rminar~

tico H che, associato al campo

~lettrico ~spr~sso Maxv~ll

(2.5)

A questo scopo ricordiamo che i l Prof.

J~ffr~y

ha dimostrato,

n~lla

da

prima

soddisPa

d~ll~ Su~ l~ziofti, ch~

all'~quazion~

fi

(3.6)

soddisfar~ l~ ~quazioni

da (3.3),

di

sia

ta1~

valor~ d~l

~ rL~)

a

d~rivat~

%f "

E,

com~ ~spr~sso

~

(2.6).

da (3.3),

parziali:

0

che ora veri£icheremo direttameate.

A

qu~sto

a z

~

(3.7)

scopo

poi ';IE

III

oss~rviamo

risp~tto

che, derivando (3.3) prima

a t, si hal

(-L~ c;.'(I. 0 si avrA un f'ronte d' onda, n0.lche si sposta col

temp~ di

ci~

Ull pia-

ascissa z. s z.(t) tale che per

z >z., E(z.t)=O. per z 0 e del resto qualsiasi. Poich~ E ~ uguale a zero per ogni t suI f'ronte d'onda. la sua velocitA sarA la velocitA del campo nullo,

ci~

il fronte d'onda si sposta con velocitA l/p(O).

5. Passiamo ora a discutere la (3.5). Anzitutto se G' (u) e 'df>P/; haJUlo (se diversi da zero) per ogni u e per ogni E 10 stesso seguo (per esempio G(u) e peE) sono

206

Punzioni crescenti, la prima rispetto a u, l'altra rispettG a E), la (3,S)

e

sempre soddisfatta e It

zOO

per ogni t, ~O

IB questa caso, se Ie condiziOBi iBiziali sonG Bulle per z • suI piaao z=O

e

assegnato per ogni t positive il campo elet-

trico, per UB teorema di unicitA del campo elettromagnetico, (3.3) e (3.10) (purche si asswu G(U)=O per u ... 0) rappreseJltaJlO 11 campo elettromagnetico caapatibile con Ie condiziOBi iJliziali e alIa frontiera e che si propaga nel verso positivo del-

l'asse

2..

Tornando al caso geJlerale, cerehiamo di dimostrare l'esisteJl_ za del Dumero h;> 0 di cui si

e

accennato al B.3.

A questo scopo, aggiungeremo un'ipotesi

pi~

che plausibile dal

punta di vista fisico. Ciee la funzione G(t) (0 che e 10 stesso G(u»

che rappresenta il campo E(O,t) sia limitata assieme

alla sua derivata G' (u) per t" (-- ,T); ia altre parole esistaao dUe lNIIleri positivi: H e H' tali che per ogni u .. (-..., ,'1') sia:

(61(....) I ~ M

(S,l)

Inoltre per Ie nostre

I g I'(E¥c>E \

Ci~

I ~ '(.. . ) I E; M'. ipotesi -fi.. o a c-k...

liJlitata da

U1l

I B \ .. H

sarA

numero II.

premesso, fissato un istante t, esisterA un numero positi-

vo h(t) tale che per z £ [O,h(t») , (3.S) e verificata e quiadi (3,3) risolubile. Allora per questi valori di z, t, che

I~

1< N,

I E(z,t)l::: IG(U)\ ~ H,

inoltre IG'(u)l~ H'.

Dimostriamo ora che esiste un numero h o tale che h(t);;. h o t E. (to che

0

e

sic-

'

,T). Infatti sostituendo h(t) in (3.S) e tenendo COnsoddisfatta se G' (u)

?lff) ~,= ~~

cCllle si era a1'i'e1'lllato. Assumeremo h~h. l'estremo ini'eriore delJli h(t) per te(-oc.T). T puc) essere anche ini'inito Perc) nel caso G(t)-O per soddisi'atte) e t non

~

purch~

t~

0

sia soddisi'atta (5.1).

(sicch~

(5.1) sono certamente

molto elevato, segue h-

sia p_ ~ 0 i l minimo valore di peE) per gono

le relazioai:

(5.4)

t .. T

t < P.

.-..

ho =

00.

I III ~ M,

lni'atti

aHora se val-

n"",..

N M'

i l valore di u che CClllpare- nella (5.2) ~:

-

u = t - p(ll)h(t) (; t - " h < P.

(5.5)

....

Ma allora il G'(U) della (5.2)

~

-

h - p h - O.

nullo e questa equazioRe noa

puc) essere soddisfatta per h(t) finito. Deve essere h= 00,0, che

~ 10

stesso, la soluzione (3.3)

~

valida,per valori di t

soddisfacenti (5.4), per ogni z, ed essa rappresenta il campo elettromagnetico in tutto il semispazio. Si noti che. come ved~~o

nel

num~ro segu~ntet

tempo in cui la (3.3)

~

N e malta piccolo; l'intervallo di

valida puc) essere sui'ficentemente gran-

de per le applicazioni pratiche. Hel caso in cui non siano soddisfatte Ie ipotesi ora esposte,

fissato t puc) esistere un valore z di z per cui la (3.5)

~

nul-

la, e se G(t - p(E)z) risulta diverso da zero. da (3.6) e (3.7) segue che

IH.'

te di E per z

:;l~J:E -to

z

1-

-1" DO

•

n.' BE 1~IM;lt' - ~~ --

,cioe Ie deriva-

tendono a di vent are infinite. 51 ha cioe,

conforme a una locuzione del Prof.Jeffrey, una catastroi'e. Si puc) cosi interpret are l'accennato risultato di Cesari per qui i suoi teoremi sono validi solo mina

~

sufficentemente piccolo.

~e

10 spes sore della la-

208

In seguito comvaque ammetteremo che (3.3) e (3.10) rappresentiao il campo elettromagnetico, almeno per valori di t e z suI£icentemente grandi per Ie question! pratiche.

6.

Nel casO s _... aotiamo che, mentre (2.12) rimae valida,

(2.14) non ha

pi~

signiEicato e si

pu~

sostituirla con la con-

dizione che il campo sia nullo all'inEinito, 0 meglio che il canapo rappresenti un'onda che si propaga nel verso positivo del-

l'asse z, condizione questa, come si

~

osservato al ••4, sod-

dis£atta dalle (3.3) e (3.10). Supponiamo ora I' onda Eot incidente suI piano z=O, col campo elettrico' (e quindi allChe i l campo magnetico) per t

a.. seali>t

Ie a

(

0

Ie (3.10) non

ha altre radici realijinfatti,in tal caso, la funzione A(V) ha derivate seconda seillpre positiva e i1 corrispondente diagramma

la concavita rivolta senpre verso Ie A positive. Se s1 riflette che V indica una velocita di yropasazione rispetto 81 mezzo continuo ed e, pertanto, espressa de V = VI_y.~, se con V' s1 denota Ie velocita di avanzamento nello spaziox della superficie d'onda e si 8uppone che 10 stato di riferimento coincida con quello attuale, s1 comprende corne possano esaere accet-

tabili anche valori

neg~tivi

di V. La scelta del segno di V pre-

euppone una discussione che ~inuncio a fare. Se Ie perturbazione a1 propaga in un mezzo inizialmente in quiete e a temperatura uniforme, nella regione imperturbata e per 000tinuit3 suI fronte d'onds risulta ~= cost., 1).1 = cost., ! = g =

°

e la (5.10), tenuto conto di (3.8), diviene zv~

0.12)

+

mzV 2 -

L

~

= 0,

la quaIe ammette due radici reali in V2 di cui Q3a sola positiva. Pert ant in un mezzo non perturbato e possibile solo una velocita di propa~azione.

°,

Os~ervezione.

Be i1 vincolo di ihcomprimibilita

e

assoluto,

cioe se nella (,;1) si suppone F = 0 e,pertanto, risulta ~ = cost., l'eq~ione (3.10) si identifiea con la (2.8), valida nel caso dei corpi rigidi.Pertanto, sotto questo sspetto il fluido non viscoso assolutamente incmpricibile si com?orta come un corpo

rigido. BIBLIOGRAFIA [lJ [2]

"~~.ell,J.C.Phil.Trans.aoy.30e.157A 49 (1867) Caotaneo,C.Atti del Se~inario natecatieo e fisico dell'universita di ';odena.~.(194E)

[3]

Vernotte,P.Compt.Rend.Acad.Sci.~'6(1958)

[41 [5] [6]

Cattaneo,C. Compt.Rend.Aead.3ci.247 (1958 Rettleton,R.F.Phys.Fluids 3 (1960) Chester,M Phys.Rev.13l (1963) Gurtin,r. and Pipkin,A.Arch.Rat.liech.and Anal.3l (1968)

[7J

229

La]

I;eixner Arch.Rat.~ech.andAnal.39 (1970) ~ Carressi,M e florro,A. Nuovo Cimento 9B (1972) ~~ Carrassi,M. Nuovo Cimento 4b B (1~78) r~ LindsaY,K.A. and Straughan,B. Arch.2at.Mech. and Anal. Ge (1978) Bressan,A. In corso di stampa nelle Eemprie dell'Accademia Nazionale dei Lincei Grioli,G. Nota r a , in corso di stampa nei Rend. dell'Accademia Nazionale dei Lincei Grioli,G. ibidem

CENTRO INTERNAZIONAlE MATEMATICO ESTIVO

(C.I.M.E.)

ONDE DEI SOLIDI CON VINeOlI INTERNI

TRISTANO MANACORDA

ONDE NEI SOLIDI COif VI1fCOLt IB1rERNI

Tristano Manacorda Universit~

di Pisa

I-Vincoli interni La nozione di vincolo di

e ben

incomprimibilit~ in

nota fin dai primordi;la stessa nozione

ta nella teoria dei solidi,ben piu

~luldo

un

e stata

ideale

introdot=

recentement~.Per quanto ~

mia conoscenza la si trova in Poincar~ [lll) nel ~ondamentele articolo di Hellinger

119]

a

e soprattutto

[i2J .Solo assai piu re=

centemente,sono state sViluppate considerazioni generali suI

v~

colo di incomprimibilitll. "uggerite inizialmente dallo studio del comportamento della gomma la quale

ea

dilatazione cubica nulla

in ogni sua tras~ormazione isoterma [5] colo cinematico studieto di·recente

~

[24] .Altro tipo diva:

quello della inestendibili

til. in una direzione introdotto da Rivlin [20] te studiato da Adkins

[2]

e poi ampiamen=

ad altri,anch'esso suggerito dal

comportamento della gamma rinforzata da una

~itta

trama di fili

di nylon.Vincoli piu complessi sono atati considerati da Wozniak [26] .Una teoria generale dei vinvoli cinematici non pub non far riferimento all'articolo di rruesdell e Noll

[25]

dello

Handbuch der Physik. E' quasi spontaneo, a questo punto,la introduzione di vincoli in= tarni non puramente cinematici me dipendenti anche dalla tempe= ratura assoluta (11 (1) Nella teoria termodinamica di MUller,sviluppata ampiamente da Alts 3] per solidi vincolati,l'esistenze della temperatu= ra assoluta ~ provata invece che ammeasa.Qui per semplicitll. ai accetterll. la temperatura assoluta come nozione primitiva.

r

Una prima est ens ions si ha quando ai smmetta cbe la dilatazio=

234

ne oubics s1s una funzione della temperatura aBsoluta la quale

ai riduce a zero nelle trasformazioni isoterme ( solidi incomp primibili secondo Signorini

[11], cfr. anche J.!anacorda [15].

Fiu in generale si pub ammettere come vincolo interno una

fra deformazione e temperatura,cfr.Amendola [4]

lazione finita e J.!anacorda

r~

(15] .Green,Haghdi,Trapp (91

hanno introdotto

vincoli espressi da forme differenziali non com)letamente

int~

grabill nelle.quali perb non compare la derivata temporale de! la temperatura. La teoria piu generale di vincoli termomeccanici
In1

.Easi hanno provato la

che il vincolo riguardi tutta la storia della de=

formazione e della variazione di temperatura Be a1 vuole

Boddi~

fatta la disuguaglianza di Clausius-Duhem da ogni proceeso pos= sibile.I lore risultati

sono stati ritrovati in modo piu ele=

mentare,ma con condizioni piu restrittive da Manacorda

[15] .

Reetrizioni suI gradiente di temperatura sono state consider:! te da Trapp [23] Benersl.

.Per 11 seguito,
di un solido vincolato nell'ambito della moderna mec=

canioa dei continuL

Di un continuo @.l

,fonnato da elementi

canto ad una configurazione di riferimento

X ,a1 considers ae=

Bo,la configurazio=

X indica la terna di coordinate di X in Bo rispetto ad un sistema cartesiano fiseo, mentre: ~ la terns

ne ietantanea

B.

corrispondente in II

B

B.

Data la corrispondenzs biunivoca

traB o


.!

( 1.1 )

=

b (~ ,..t)

mentre il gradiente di deformazione

, !=

X' 1 (~, 1)

- -X· 1'=

G~d

(2)ha Ie coml!

(2) le lettere maiuscole indicano che Ie derivate sono fatte r! spetto alle !. ponenti

cartesiane

235

t

1.2 )

r.

~

=~

PH· x 'H

Naturalmente ( 1.3 )

J pi l;l

J = det

e ai ammette

o,

J)

oX

0

durante tutto il moto del corpo.

Altri tensori di interesse cinematico nella meccanica

dei

continui Bono : i1 tens ore di Cauchy-Green l"~ £ = pT P

(

)

(!'f' a 11 trasposto

di

! ) e il tens ore di

( 1.5 ) E 1 e il tensore unitario. Lo stress di Cauohy solido

=1 ( C - 1 ).

'2 -

-)

rappresenta gli s~orzi interni nel

T

2:.!! , e ad esso corrispondono,11 tensore degli

j;~.

sforzi di

de~ormazione

Piola-Kirchhof~

T R

l

( 1.6 ) 3R = J 2: ( e il tensore lagrangiano degli sforzi

)-1 (

0

secondo tensore di

Piola-Kirchoff

Zo

( 1.7 ) Men~e

!R

non

a

[ 1 !R = J !-l

a simmetrico,l'equazione

! (

l

)-1

di bilancio

del mo=

mento della quantita di mota e l'assenza di coppie interne im= plica la simmetria di T e di !o' Le equazioni fondamentali di bilancio per un continuo sono: a)

l'eguazione di conaervazione della masea. Se ~

ta materiale di ll?>

in

per ogni sottodominio ( 1.8 ) dove

bo

e

11 , e

b

di

11.

la densita

di iB

dens!

in 110

Hla

S~~ dv"

l'insieme di

S. a

e la

)..S'. dV corrispondente a

b • Sotto

OV=

vie di regolarita ( 1.8 ) equivale a ( in forma rispettivamen= te lagrangiana ed euleriana ) ( 1. 8'

)

'i

J = ~o

236

b)

l'e uazione di bilancio della

~ r~

( 1.9 )

dt

:f, dv =

b

\.

Sotto opportune condizioni • ( 1.10 ) ~

~

in questa

.!

uantita di moto. Si Bcrive

~f.o...r ~

(i", cW

+

1,,;

di regolarita , ( 1.9 l equLvale a

=~! + div

!

-),V -;. ==X=;·; ..

la densita di forze di massa e

-

1a de=

v = X( X , t

rivata molecolere della velocita,

~

~

~

~

( 1.10' l + Div l R c) bilancio del momento della guantita di moto.In assenza

di

Alla

puo dare • ~! = ~.!.

(1.10) si

forma lagrangiana. Si ottiene

coppie distribuite, si limit a ad imporre la simmetria

-

:!;

= T

TT

( 1.11 )

T

=

pT

!o

condizions per J ' R

e in conseguenza (cfr. ( 1.6 -R -

=

di

d) bilanoio della enarKie. In 8Bsenza di Borgent1 interne,st sorive per ogoi

(0 f -d J.)

( 1.12 )

ove estema a

e la b

b E:: B ,

dt

dv

= ( q. n d6" + ( tr( T grad X ldv ~L '" "V )1 "'bdensita di flusso ten:ico, n 1a normale

u b.

nei punti di

.

Questa equivale ,nelle consuete ipotesi di regolarita a ( 1.13 )

5 f.

= div q

S

+ tr (

! !! )

.£ = 1 (

grad v

+11,,.J·f)

+ ~( !grady ) = div

tenuto conto della simme-;ria

di

!

,con

Alle equazioni di bilancio va aggiunta

la

2

disuguaglian=

za dell'entropia.Qui ,per semplicita,in assenza di sorgenti,e a.sunt .. nella forma ( 1.14)

~ (~iJ..r - h:~~.I6"

:?-O

equivalente,nelle conauete ipotesi di regolarita,a ( 1.15 )

~6i

- ..d.:v~ + .~ .,...J.9 0 Q

237

e

In questa, d1v S

tna (

e

la temperatura assoluta.L'eliminaz1one d1 e

1.13·

) consente d1 scr1vere la

(1.15

) nella forma della d1suguagl1anza d1 Claus1us-Duhem.

(1.15

Le ( 1.13)

e

(1.15) s1 possono scr1vere in forma

grangiana,rispettivamente, . ( 1.13:) ~o £ = D1v ~R

e'1 -

+

*

T tr (~R

di scrivere

r

0

&-

"I ()

,conaente

e ( 1.15' ) nella forma

(1.15)

1 q. grad e ~ 0 T·"5"~ (:!'R!) - 1 ;J.R.Grad ~ 0 Le equaz10ni fondamental1 ( 1.10 e ( 1.13 ) vanno comple=

( 1.15' )

~(r' "I 8) -

•

!),

e;,

$. D1v ,gR + 9 R ' Grad L'introduz1one dell' energia libera =

( 1.15')

18=

tr (

"J ~o(r .. ,8 - tr

X .!! ) -

e

Y

tate da equazion1 cost1tut1ve. Un solido e detto termoelast1co

'f = If' ( !. e,!! )

se ( 1.16 )

-

ove g = grad

e

!

=

! ( !. e, ~ ),

~ = 9 ( !' e, § ) e sottintesa I'eventuale

ed

1=1( !' e, ~ ) ~.

d1pendenza da

La incond1z1onata valid1ta d1 ( 1.15 ) per oen1 processo am= m1se1b11e, implica che

!§

, e che s1 abbia

( 1.17 )

TR

'f' : 1

'.e

!

non

po;e~o

d1pendere

1 e ) - tr (.J R l ) = 0 = - vef • ! = J-~f!'T

~"(f +

=~o}!'"

i

da

c10e

ine1eme a

,s . .g;ro

( 1.18 )

( $R •

Q~ 0 )

3i richiamano infine le condiz1oni di d1scontinuita dovute alle equazion1 di b11ancio quando 11 solido sia attravereato da un'onda di diecontinuita del primo ordine; esse eono

[~u] = ( 1.19 )

[ ') ;!

[~"1 ( 3)

uJ +[!-e]=0 uJ + [:!! . ;3] = 0

0

[')'£

u] + ['!

.eJ =0

( 3 )

La validita di ( 1.19 )4 richiede che le eorgent1

entropia,eaterne e

di

intrinseche,siano limitate a1 tendere di

238

volume di b a zero. Green e Naghdi non ammettono tale condi zione t16] .S1 puc invece ammettere 11 esistenza di Wl flusso

=

intrinseco di entropia.ln tal caso esso dave essere sempre non negativo. h

e

i1 fluseo di entropia. E ' facile acrivare la versione la=

grangiana di ( 1.19 ) la quale fa intervenire la velocita

Bo

dell'immagine in Osservazione

1

UN

dell'onda. 8i assuma

-

h = q/9 ; se

~

G8

continua at=

traverso l'onda, la ( 1.19 )4 ' tenuto conto di ( 1.19 )3' di=

[S

viene

9, U.] -

[3£l.l] = 0

Ci08 [cfr. ( 1.19 )]

[£-'1 8]

=

['1"1

= 0

31 osserva che l'introduzione di un vincolo ha dei riflesai

anche meccanici , in quanto il tens ore degli sforzi non

e piu

completamente determinato da una equazione costitutiva

come

( 1.17 )1 ' ma contiene una parte che rappresenta gli

sfo~ziodi

reazione dovuti a1 vincolo.Per un vincolo termomeccanico,anche l'entropia e

S

non sono completamente determinati,mentre B1

puo assumere completamente nota la forma di Osservazione

2 -

In un solido

'Y

nelle ( 1.16 )

termoelastico non Boggetto

a vincali, Itessere 11 calare specifieD a configurazione stante

e

1 '

cv

positiVQ impliea corrispondenza biunivoca tra

co=

e

per cui s1 pub assumere come variabile termodinamica Ia

entropia al posto della temperatura. Tale corrispondenza viene a cadere per solidi vincolati,onde si potrebbero istituire due teorie paraIIele,assumendo in una ,come variabile termodinami= ca,percio soggetta al vincolo,la temperatura; nell'altra,invece l' entropia.

239

2 - Vincoli interni nei solidi

Esempi di vincoli a)

Vincoli ci..n:ematici

Vincolo di incomprimibilita ( 2.1 )

= det

J

F

~

= V det

C

= 1

~

Vincolo di inestendibilita 3i emmette che in

Bo

esista un campo vettoriale

~

linee vettoria11 conservano lunghezza inalterata in

(X) le cui

B. 3e

~

ha modulo unitar10,1l vincolo e tradotto in

Fe. _"..., F e

( 2.2 )

J'VIY

=1

c10e = 1

Vincolo superfic1ale : 31 emmette l'esistenza d1 una famig11a

1 di

Buperfici I. I dens a in

Eo' la cui dilatazione superfi=

B.

ciale e nulla nella trasformazione

--t BJ •

Poiche la dilatazione superficiale e ellS""

con

~

norn.ale nei punti

= J ()2'C-

di.2:.

l

:Q BoJla condizione di inell

in

-.oJ ) -L :r

stendibilita Buperficiale a1 acrive J ('V'C ~

per ogni punto d1 L ed ogn1

= 1

€.

b) Vincoli termomeccanic1 Sene 8stensioni di vinceli precedenti.

Vincolo di incomprimibilita ( 3ignorini ) J = f

( 2.4 ) ~

(e}",~)

f

(l:',7:j2S) = 1

e la temperatura ( supposta uniforme ) in

Bo '

Vincolo di inestendibilita (2.5)

F,3'

Vincolo anolonomo A

f(9;t',X); f(c,'l) = 1

(Green, Naghdi,Trapp ) tr ( /

2.6 )

ove

1:2

e

i

) +

1( !

,e) ..§

sono un tensore ad un vet tore funzioni di..!

=0

e

240

e

e di Si assumera ,d'OTa innanzi,l'esistenza di un vincolo nella f'onna

e ,X ) :

r ( -F,

( 2.7 )

(1) 11 principio di obiettivita

1,

de

solo per 11 trami te di

imp1i~a

.£ =! !

(1)

0

~

r

che t

possa dipendere

ma 91 conserve Is

forma ( 2.7 ) per comodita.

-----------------------------------Lungo un processo F: f ( t ), e: e(t) ,in condizioni • di sufficiente regolarita di r e identicamente r = 0, ~

~

~

qUindi

-

( 2.8

'I' •

tr ( r F ) + ~

ove

r

2.9

~

Osservazione

1

-

: -;)1"

r

fe

0

:

()e r r:l'(g ,f),xl; in una

~

51 assume

:

trasformazione linearizzata a part ire da Bo ~ LX) tr (k'l' E l") 0 ['I f ~ (") ill loT ove ~ e 1a 1inearizzazione di J, 2 E - : Grad~ +( Grad-II: ) ~~Ispostamento linearizzato, e e~) la linearizzazione della

+f, ell':

dO e

variazione di temperatura;

la determinazione in

~, quella di ~ 1'. In :r:srticolare, se

di ~ Te

B:o

#- : i:!. , 1a (~)

diviene

div con

a

costante.Cio accade

-u.'"1 : se r

a el!}

1.

dipende da

tramite dei suoi tre invarianti

per 11

IUE •

In un solido termoelastico soggetto ad in vinco10 del 2.7 ),10 stress

tipo

e l'entropia risultano non completamente

determinati da equazioni costitutive : precisamente si ha ( 2. 10 )

ove

p

T

e un

~R

:

-

p l' + -

5'..? 'f,;: i"

r

1"~

f- r 8

parametro 1agrangiano atto ad individuare 1a rea:

zione vincolare interne.

241

Qsservazione 2_

La potenza delle reazioni vincolari

da

e

data

-••

-tr(prl!')

e quindl

ed

~

nulla,in eorrlspondenza al vineolo,in ognl trasfoE

maziane isoterma .Piu in generale,essendo la

prod~ione

flea interna dl entropla data , da

5• ..y .. ~.1J

speci=

•

e

t)

la produzlone dl entropla dovuta alle reaziani interne - p

r

~

p

ed

e

ns assumendo

3-

- ..... 'F"

+

tr (r l!' »

qUindl zero in virtu del vine 010

Osserva.ione ta

(pe

( efr. ( 2.8 »

Una teoria perfettamente analoga si ottie=

1 come variabile

termodinamlea

e

da una equazione costitutiva. Naturalmente,

ta con

C.

e determina =

'r

va Bostitui

242

3 -

Onde di accelerazione

II quadro delle equazioni fondamentali in preeenza di un vin:

colo come(2.7),e in assenza di forze di massa e di sorgenti termiche e di entropia

~ei

= Div

( cfr. ( 1.10' ), ( 1.15' ), ( 2.10 »

.,

~o ~

~

,

:£R

= Div

of

~=J'!g

( 3.1 )

SR ='sR (2 ,§,G) G = Grad II 81 comincia ,qui, ad esaminare Ie condizioni per la propaga=

zione di secondo ordine ( onde di accelerazione ). A queeto scopo a1 pone

-a=[X] -

3.2)

Una notevole semplificazione e dovuta al fatto che,in un

co~

duttore definito,tale cioe che la parte simmetrica del tensore

~ = ~~R

e definita positiva, possono snche in presenza di

vinco11 (1)

(l),propagarsi solo onde omoterme ,per le quali,cioe

Per un solido non vincolato

[6 J

=

[ ~]

V.ad es.

o

=

'f

5i osservi ,infatti,che pur essendo ~

minata da un'equazione costitutiv8,

E

=

[B,71

completamente non 10

e

,mB a1 ha

'Y + 1 e = '1 - e~ + f" f e

tuttavia f. e da ritenersi continua attraverso l'onda (2) (2) La

(1.19)2

implica

[1,e]

= 0, cioe

Dalla equazione della energia si ha allora

[5'. 11k G]

= [SR

1

N

e perci/>

( 3.3 )

.!!

=

0

[p]

dete~

=

0

243

.l! e

ove

8ia

la normals all' immagine dell' onda in discontinuQ, perdo

G

~ = Grade

continua e

~;

=

[

~]

B. •

=o(!! '

e

dato che

a

; si ha aHora

,e )..

Sa ( ~+

Sa (~i""

+O(~, B )

qUindl

If'fx)=r.~a]·l!

= (Sa

(G-

+ ... ~,e)

-Sa(2- ,9) )~!=

0

Derlvando questa ldentlta rlspetto ado(

-- -

= 0

K+ N • N

che a impossibl1e per l'lpoteel che 11 conduttore eia definito.

31

puo

dunque assumere che Itonda sia omoterma.

Dopo dl cio,dalla ( 3.1)6 [ t;:

od anche

{ [

el ha

T'''p]

= 0

iJ= l.... ~ If i, ~

e

tr

l' T ( ~ lID

1i

UN

-- -

=l'N.a

-flN.a=O: - .--

( 3.4 )

le onde ( omoterme ) dl accelerazlone dl un eolldo termoelaet! co eono traevereall. 81 prende in eeeme,ora: la ( 3.1 )1 (

Tal

[dlV

=

-!... U II

L~a] = -[ :r] ove

g:

e 11

+

[Tal! )

Hi] _ei[e]

teneore quadruplo ()" ~ • 81 ha pol

([i]

= -

f (1"

-

ove

!:-

zlone dl

~ ~!! )

UN

=

-!... ~ ~ UN

indica l1n conveniente tensore del secondo ordine,fun = N,.

r p'T'r

=

[;JJ> p[d!'l'(i )J

e cloa,nelle condlzlonl presentl

( 3.5 )

ha

el

- [P!]!!. = -[pll'!

+ L

\r:' If

244

avendo poato

(

d P.2'( if 18l!J )

)N =

J~

Si ottiene infine la c-ondizione di c~patibilita dinamica

UN[pJ2'!+(Q-t~l)~ =

(3.6)

0

ove !!=~-p~

(3.7)

e un

tens ore doppio simmetrico.

Le implicazioni della equazione dell'energia si ottengono ra=

[2}

pidamente. Per essere

N

= 0 con

l.... [sR1 .Jl

[DiV SR1 = -

e

1lj.,

:j

=

a.9. 1. R . Essendo

L =

~

ottiene

-- aeN ) • JJi

Poiche

un tens ore

-

e una funzione doppio~tale che N L ( a 8 N )

-

e si ottiene,alla fine

~oe[iJ = A!J!.! -

( 3.8 )

lineare di L ( a

__

l....~!J

=f [pJ

con I

~

3.10

+

" - 5. d~! 't'

-

esiste

=.11. N.

a

"-"",,o.J

•~

UN

D'altra parte ,se B1 ricorda la ( 3.1)4

3.9 )

liP'; )

~

, s i ha

(~- p.f) .!!-.~

1 = - 1:! T'

Si ottiene quindi la condizione di compatibilita

3.11 )

"'!J!.J! -l....!!-.!!. ~= feEpJ

+fJ(e'- Pl) N.a ~

~

UN

~oiche

[p J

si

a1 limits ad esprimere

e di

p •

puo

ricavare dalla ( 3.6 ), la (3.11 )

A in funzione

di grandezze continue

Moltiplicando, infatti, la ( 3.6 ) per

T'

N

-~

si

245

0]

ottiene

G1

( 3.12 )

=

J.JL

~ =

50stituendo questa determinazione di

IT!I

nella ( 3.6 )

a1 perviene a ( 1 - \I ClII ) ( Q _~ U..

3.13 )

...........

5i rieordi ora che pendicolari a

a

( 3.15 )

-

Q a l

~

II problema

=

0

posto

'"

( 1 _::!Ilf~ .8.1 = la condizione ( 3.13 ) a1 scrive ( 3.14 )

a:

deve essere Bcelto tra 1 vettori per=

~

\I

1 )

"\J)ON---"",

~

e quindi

) ,g ,

= ~ U'l a "

N

-

-

a • \l =

~

0

ridotto a bidimensionale,ed esistono

pereio due autovettori almeno diatinti e,ae i eorrispondenti autovalori sono poaitivi, easi individuano Ie direzioni di di' ,= aeontinuita e Ie rispettive velocita di propagazions. In particolare,sia per

--

e

A

u

A

A =

v'l.

u

un Rut ovett ore comune, ( se esiste )

11 ~

!

u

=

:;2.

U

si ottiene

~l ~ =l]oU; ~= (v.. _pV2.)u

(3.16)

la quale mostra che,anche Be

ehe

U 'N

-

Segue dunque

2fl[~

.!! 1.

N

T

l)

e infine (e N 1 .N

=

i grad ~ +

( grad

,... -

l::. )

-

L~ J= ~~ ~411 ~ 1 ;>. • N = 0

2" .... 0

,...,

+

!!;illI~,}

248

( 4.8 )

frtl =

0

C"L(" 1 =

0

(4.4 )

e,da~la

( 4.9 ) Aneora dalla

4.6 )1 ,molt1pl1cando scalarmente per

iI"

a1 ottiene

e percH. ( 4.9 ) che aasegna la 'i,(')

=

la ( 4.6 )2

d1 propagaz1one dell'onda.Infine,po1che +/f1l.'i) [t e.)] = 0

veloc1t~

(f)~)

"'It: •

e

~~ •

e

e aodd1afatta da

'

[~'.!! 1

sodd1afatta anche La ( 4 .6 )3 (2)

=

0, che rende

aubordinatamente

1

["It GlJ.1 • N = 0'11 gra= temperatura normale deve es~ere continuo .•

Se vale 18 lagge d1 l'OURIER , 4iente

di

alla

( 4.9 ):11 fluaao normale d1 calore

e continuo.

249

[e 1 f

b)

0

(onde term1che ) ,

a

~ 0

In questa caso,11 vincolo essendo rappresentato da (4.5 ),s1

ha

-

( 4.10 )

".'

~

( 4.11 )

= a@

N

~

da'ora,molt1pl1cando scalarmente per

La(4.6)1

u;

= [- 'It +

a @

A

[~1.1I

Se

~.

non

e

=,.u.~.

parallelo ad

~

UN la determinaz10ne

la ( 4.12 )

e

~

-(~.'.!!)!i.)

=~ •• ! = a e .

t

.. " ~UN~.·~

mentre,molt1pl1cando per (4.12)

+ 2)LA Gl

= t(~.

!!.].

[!.

eJ

~

e -

N,

~

t

,s1 ott1ene ancora per

(4.9). Se invece l'onda

U:

una 1dent1til. e

Per onde non longitudinal1

(e

e

e longitudinale

indeterminato.

neanche trasversal1, v. (4.10)

s1 ott1ene dalla ( 4.11 ) per la ( 4.9 )

[}GJ

( 4.13

a

=

e (;\

+)J. -

["l"11 =

ofr.n.l Oss .1 ) Segue.po~endos1

4 .15 )

-a(~+).l»

( .!L. ( 2'{

r.'t-

D' altro canto,la ( 4.1 )4

)

~

e ( 4.14 )

~

+!!...

implica

che

1f

sia continua

e quindi ahe sia continua anche assumere

(€("] =

)@.

~.'t;

"!'('J .

=0

~

L'P '!] l

e quind1, da ( 4.6 )2 ( 4.16 ) che coincide con

(q~11.

N,

=

m. 'l:

['1("].

(4.6)3' Le cond1zion1 di comp~tibilita

termomeccaniche Bono qUindi atte a determinare

UN ed insie=

250

-

me le diseontinuita di

e

e

di '7.. (Ii in funzione di ® .

3i ha snehe ( efr. ( 1.19 )1

c~ con

uJ

=

U =

0

u

n

- v

n

u;..c veloeita dl avsnzamento ( locale dl propagazlone )

dell' onda. Linearlzzsndo

quindl

[~l"

UN' =

-t, (#lJ

=

~o[l\J

= -

go af> UN

e quindi

[ ~ ('ll =

4.17 ) l'onda

e

compresaiva

se

-\,oa6:

>

a @

O. espansiva nel

caso

opplilsto (3) (3) 31 osserva ehe ( 4.17 ) vale snehe se l' onda e longltudl= ( ® = 0 ) impliea lOa nale, mentre,per onde meccaniche cont inulta dl ~(". Per onde longitudinall,

UN

non pub eSsere determinato dal=

le sole eondlzlonl di diseontinuita,mentre per le dlseontinul= ta di )C , di

"'L 1') [}Cl

( 4.18 )

sl ha

=

a@(?I+21J-1l'

r

-

a

-oU") I~ N

251

mentre

e

anche

BDlLIOGRAFIA A

Vincoli

1 - J.E.

Adkins

: A three-dimensional problem for

highly

elastic materials subject to constraints,Quart.J.Mech.Appl. Math,11,88-97 ( 1958 ) 2 - J.E. Adkins - R.S.Rivlin : Large elastic deformation of isotropic materials : X • Reinforcement by inextensible cor: ds, Phil.Trans.Roy.Soc.London A, 248 201-223 ( 1955 ) 3

T.Alts -

Termodynamics of thermoelastic bodies With

kinematic constraints : fiber reinforced materials.

Arch. Rat.Mech An. 61 (1976) 253-289. 4 - G.B. Amendola

A genaral theory of thermoelastic solids

restrained by an internal thermomechanical constraints,

Atti Sem. Mat. Fis. Univ. Mod. 27 (1978) 1-14. 5 -

J. Bell

: The experimental foundations of solid mecha=

nics.Encycl.of Phys.Vol

VI a/I ,Springer Verlag ,Berlin

1973. 6 - G.Capriz - P.Podio- Guidug1i

Formal structure and

classification of theories of oriented materials,Ann.mat.

pura appl.(4) 115 (1977)17-39. 7 -

H.H. Erbe

W~rmeleiter

mit thermomechanischer in;

neres Zwangsbedingung,ZAM,Sonderheft

76-79 (1975)

252

8 - J.L. EriksGn- R.S.Rivlin

: Large elastic deformations of

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elastic deformations. and

non -linear continuum mechanics,. Oxford,Clarendon l'ress 1960 ~O-

A.E. Green- P.M.Naghdi-J.A.Trapp : Thermodynamics of a con= bnuum with internal constraints,lnt.J.Enw1g.

.Sei.8 (1970)

891-908. 11- M.F.Gurtin-P.Podio-Guidugli : The thermodynamics of constrai ned materials, Arch.Rat.Mech An.51 (1973) 192-208. 12 -E. Hellinger , Die allgemeinen Ansatze der Mechsnik der Kon= 4 tinua,Enz.math.Wiss. 4 , 602-694 (1914). 13 -S.Levoni:Sulla integrazione delle equazioni della termoela= sticita di solidi incomprimibili,Ann.Sc.Bbrm.Sup. Pisa (3), 22,(1968),515-525.. 14 -T.Manacorda:Una osservazione sui vincoli interni in un soli= do,Riv.Mat.Univ.Parma (3),3,(1974),169-174. 15 -T.Manacorda:Introduzione alla termomeccanica dei continui, QUaderni UMI,n.14,Pitagora Ed. ,Bologna 1979. 16 -S.L.Passman:Constraints in Cosserat surface theory,lst.Lomb. Sc.Lett. Rend. A,106,(1972),781-815. 17 -A.C.Pipkin:Constraints in linearly elastic materials,J.of El. 6,(1976),179-193. 18 - H.Poincare:Le90ns sur la theorie mathematique de la lumie= re,Paris, 1889. 19 -H.Poincare:Le90ns sur la theorie de l'elastcite,Paris,1892. 20 -S.Rivlin:Plane strain of a net formed by inextensible cords, J •Rat.Mech.Anal. 4, (1955) ,951-974. 21 -A.Signorini:Trasformazioni termoelastiche finite,Mem.III, Ann.Mat.pure app1. (4),39,(1955),147-201.

253

22 - A.J.M.Spencer:Finite deformations of an almost incompres= sible elastic solid,Second-order Effect in Elasjicity, Haifa 1962,Oxford,Pergamon Press,1964. 23 - J. A. Trapp: Reinforced materials with thermo-mechanical

COIl=

straints,Int.J.Engng.Sci. 9,(1971),757-778. 24 - L.R.G.Treloar:The physics of rubber elasticitY,Clarendon Press,Ox~ord,1949.

25 - C.Truesdell - W.Noll:The non-linear field theories of Me= chanics,Encycl.of Phys. Vol.III/3,Springer Verlag,Berlin, 1965.

Bl

) Omie.

1 - A.Agostinelli:Sulla propagazione di onde termoelastiche in un mezzo amogeneo e isotropo,Lincei Rend. (8),50 (1971) 163-171,304-312. 2 -G.B.Amendcla : On the propagation of first order waves in incompressible thermoelastic solidS, B.U.M.I. (4) 1 267-284. 3 -

G.B. Amendola rials. Atti

Acceleration waves in incompressible mate.

Sem.Mat.Fis.Modena

24

(1975) 381-395.

4 - C.E.B.evers : Evolutionary dilational shock waves generalized

(1~73)

in

a

theory of thermoelasticity,Acta Mech. 20(1974)

67 -'19.

5 -

B.R.Bland: On shock waveS in hyperelastic media,IUTAM, Second order effects in Elasticity,Plasticity and Fluid Dynamics,Pergamon Press,1964.

6 - P.Chadwick - P. Powdrill:3ingulllr surfaces ilL linear thermo=

254

elasticity,Int.J.Engng Sci. 1 (1965) 561-596. 7 -

P.Chadwick-P~.Currie

, The propagation and growth of acce=

leration waves in heat-conducting elastic materials,Arch.

Rat.Mech.Ah 8 - P.J.Chen

49

(1972) 137-158.

Growth and decay of waves in solids,Enc.of Phy=

sics, VI a/ 3 Springer Verlag,Berlin, 1973. 9 - P.J .Chen - M.E.Burtin

: On wave propagation in inextensi=

ble elastic bodies, Int.J.Solids Struct.1Q

(1974) 275-281.

10 - P.Chen-J.W.Nunziato :On wave propagation in perfectly heat conducting inexteneible elastic bodies,J.of El.5(1975) 155-160. 11 - B.D.Coleman, M.E.Gurtin,I.Herrera,C.Truesdell : Wave prop! gation in dissipative materials,A reprinj of five memoirs, Springer Verlag, New York 1965. 12 - WEiD.Collins : One-dimensional non linear wave propagation in incompressible elastic materials,Q.J.Mech.appl.Math.19 (1966) 259-328. 13 - J •Dunwoody

: One dimensional shock waVeS in heat conduct!

ng materials with memory : 1- Thermodynamics-Arch.Rat.Mech. An.~7

(1972) 117-148; 2-Shock analysis, ibidem, 192-204;

3- Evolutionary behaviour, ibidem 50;278-289. 14 - J.Dunwoody:On weak shock waves in thermoelastic solids,Q. J.Mech.appl.Math.30 (1977) 203-208. 15 - J.L.Eriksen:On the propagation of waves in isotropic incom= pressible perfectly elastic materials,J.Rat.Mech.An. 2 , (1953) ,329-338. 16 - A.E.Green-P.M.Naghdi,A derivation of jump condition for entropy in thermomechanics,J.of Elast. 8 (1978) 179-182. 17 - E.Inan:Decay of weak shock waves in hyperelastic solids, Acta Mech. 23 (1975) 103-112.

255

18 - T.Manacorda:On the propagation of discontinuity wave~ in ther-moelastic incompressible solids,Arch.Mech.Stos.(2),

24 (1971) 277-285. 19 - T.Manacorda: Zagadnienia Elastodynamiki,Ossolineum,Varsa=

via,1978; Cfr.anche A - 15. 20 -

R.W.Ogden:~rowth

and decay of acceleration waves in incom=

pressible elastic solids,Q.J.Mech.appl.Math. 27 (1974) 451-464. 21 - N.Scott:Acceleration waves in constrained elastic materials

Arch.Rat.Mech.An. 58 (1975) 57-75. 22 -

N~Scott:Acceleration

waves in incompressible elastic so=

1ids,Q.J.Mech.appl.Math. 29 (1976) 295-310. 23 - N.Scott-M.HaYes:Small vibrations of a fiber-reinforced composite,Q.J.Mech.app1.Math. 29 (1976) 467-486. 24 - C.Trimarco:Onde di accelerazione in materiali termoelast±:

ci con vincolo di inestendibi1ita,in pubbl. su Atti Ace. Sci.Modena. 25 -

C.Truesdell:~enera1

and exact theory of waves in finite

elastic strain,Arch.Rat.Mech.An. 8 (1961),263-296.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(C.I.M.E.)

"ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON

LINEAR COVARIANT SYSTEM

TOMMASO RUGGERI

INTERNATIONAL MATHEMATICAL SUMMER CENTER (C.I.M.E.) 10 1980 C. LM.E. Session: "Wave propagation".

Bressanone 8-17 giugno 1980. "ENTROPY PRINCIPLE" AND MAIN FIELD

FOR A NON LINEAR COVARIANT SYSTEM by

TOMMASO RUGGERI Istituto di Matematica Applicata -

Universita di Bologna

Via Vallescura 2 - 40136 Bologna (Italy).

1.Introduction My lecture is complementary to the lectures given by G.Boillat in the first

part of this course. In

part~cular

I am shall deal with some problems concer-

niog quasi-linear hyperbolic system compatible with a supplementary conservation

law; relativistic theories will be considered with special emphasis. I start with a brief bibliographical introduction to the subject I shall be

concerned with. In 1970-71 1. MUller, in Bome works [1] on "rational" thermomechanics of continuous media, proposed the "entropy principle" as a criterion for selecting the constitutive equations. This author considers the equation governing the evolution of a thermomechanic system: a) balance of momentum, b) balance of mass and c) balance of energy eqlJations. Adding the constitutive equations to the previous system one gets a system of 5 equations in 5 unknowns. Each solution of this system is called a IIthermodynamic process". Then MUller postu- lates the existence of an additive function cr (entropy) such that: +

(PSv

i

+

~

i

) = cr

>

o

¥ thermodynamic process.

(1)

Furthermore he supposes that both the entropy density S and the flux .1 are ~

constitutive functions (p and v are respectively the mass density and the velocity). Hence, from (1) further constrains arise for the constitutive rela-

260 tions,

besides

the

usual

ones

which

can

be

imposed

according

to

the

principle of material objectivity. In particular

which

he

the

author shows

with

identifies

the

the existence of' a universal function,

absolute

temperature;

hence,

he

deduces

the first principle of thermodynamics. In a different conte>d: in 1971 K.O.Friedrichs and P.D.Lax the

:former

in

1974

[3] Friedrichs,

[3J

examined

a

similar

in a covariant formalism,

problem.

In

[2] and later particular

in

considers a conservative quasi(0)

linear hyperbolic system of r first order equations of the type ,

a

3

a

(2)

(U) = .2(U)

In (2), N eqs. may be identified with the field equations, while the remaining r - N are supplementary conservation laws. Then

comp~

tibility conditions are required in order that the system has a solution. In particular, when r=N+l (one supplementary law), as the system is quasi-linear, compatibility is ensured by the existence of an r-vector

z(U), such that: a l"a!!

Introducing the operator V

• U,

= l".2

= a/au

• , U•

a

we have = 0

(3)

(condi tion I)

Moreover Friedrichs supposes another condition holds: it exist at least a time-like covector

{~

a

l, independent of the field, such that the quadratic

form (condition II)

is positive definite.

Here 6U is an arbitrary variation of the field and

a

=6U,VV!!

6U.

Using condition I and II Friedrichs shows that the system of the field equations is a hyperbolic symmetric system.

(*) To avoid misunderstanding the vectorErn

r

are underlined.

261 Later several authors [4J,

[5].

[6J provided further contributions on this

subject, especially concerning shock waves in non-covariant formalism.

Now we shall obtain the above mentioned results in an explicitly covariant formalism, dealing with

the physically relevant case of one supplementary

law. The covariant formulation allows to apply the results to explicitly covariant

theories and, moreover, to emphasize some conceptual aspects that, in our opi-

nion, have not yet been pointed out. 2. Main field and Covariant convex density.

Let V~ be a C·, 4-dimensional manifold and x a point of V~. x a being local

coordinates of x. The manifold is supposed to endowed with a pseudo-Riemannian a

metric. In the local coordinates x ,g

aB

represents the components of the

metric tensor of signature (+ - - -). On V4 we consider a quasi-linear conservative system of N first order partial differential equations for the unknown N-vector U(x

Q )

ERN (5)

a

the components of F

and U are contravariant tensors and

~Q

is intended as a

covariant derivative operator. We suppose that the system (5) is hyperbolic. i.e.: ~ a time-like covector (tal, such that the following two statements hold: det(Aat ) ~ 0

i)

•

a

(A.a •• Fa: • • a/aU)

ii) V covector{t a } of space-like the eigenvalue problem (t-."t)Aad.O a a'

(7)

has only real proper solutions p(k) and a set of linearly independent eigenvectors d

(k)

(k=1.2 ••••• N).

The covectors{C

- pta} • where p is solution of (7) are called "characteristic", a while the {tel} fulfilling 1). 11) are said "subcharacteristic ll •

262

When a differentiability conditions holds, let us suppose that (5) is compatible with a supplementary conservation law

a. h

G

(8)

h ·(U) = g(U).

being a contravariant vector and g a covariant scalar.

In this case we may write the conditions I and II of Friedrichs in a more convenient form. We have:

Since by (3) 1 is defined up to a scalar factor, we may write

l

-

then Friedrichs conditions lock like U'·VFO= Vho,

(9)

U'·C

(10)

g.

(11)

We remark that (9)

I

multiplied by 15 U, can be written

equivalently~

(12)

"6 U.

The identity (12) show the first important result:

u'is invariant mth respect to field transformatiors:in fact d!'! fa and

{t and then

U' depends only

it does not depend on the choice of the field variables.

By applying the operator li

to (12) and replacing into (11) we get

Q "" OUI • oyQ

~ >0

•

(13)

Hence (13) too is independent of the choice of the field; we may then choose the field in the convenient form U"" yQf;

•

(14)

263 We put also (15)

and contracting (12) with

to we

U' ·6 U

have

= 6h

+4

U' = Vh

(16)

For the particular choice (14) of the field variables, U' is espressed by the

gradient of a covariant scalar function h only. In the case of continuum mechanics, the expression (16) is equivalent to

the first principle of thermodynamics. We point out that the components of U' play the same role of the Lagrange

mul tipliers introduced by I-Shih Liu [7]

in the context of entropy principle of MUller. Condition (13) is now equivalent to (17 )

i.e. to the convexity of the covariant scalar density h=ha~ with respect to a the field U ::: F

a

to'

If for a system (5) there exists a vector U' and at least a caveator (£';a}such that

(12)

hold~ we say

and (17)

that the system is a convex covariant density

system. Conditions (16) and (17) ensure also that the mapping U' +-+U is globally invertible, becauseVU' =VVh and this gradient matrix is symmetric and positive definite; then for a theorem about globally univalence ([8J) ur +-+ U is gloN bally univalent in every convex open domain D ~ R Therefore it is possible to choose the vector U' itself as field variables and prove that in this case system (5) has the form L (U·)

where the operator

~

(t)

(18)

flU' )

is given by A' ().

-

a

a

(19)

(t) For the proof of the statements proposed in this lecture one may see [9]

.

and (20)

System (18) is symmetric hyperbolic: in fact a system A,a a U' :z: f is sima a aT metric hyperbolic if ~' is positive definite, and in our ~' a case we have ='Q' V'h', but from (20) h,Q t = hi = •A' I;a a = U'· U - h is the Legendre conjugate function of h and then it is a convex function of U' •

We remark also that the differential operator in (19) depends only on oneiQ

four-vector h

and this justifies our definition of "four vector generating func-

tion II for the symmetric system. We have seen that any convex covariant density system is endowed with a vector U' that may be expressed as a function of the field variable and is invariant with respect to transformations of field. In fact it is determined completely law

(8).

the

system

is

only

by

Moreover

well

assumes

posed.

the we a

conservative

pointed

out

system

that.

symmetric form,

Such

remarkable

so

(5)

and

the

supplementary

when

V'

is

that

the

local Cauchy

properties

suggest

chosen

us

to

as

field, problem

call

VI

the

"main field" of the system. We remark possess a the

that

not only on

the

mathematical

special role with respect

physical

point of view,

point of view U'

to other quantities,

and h I a

but also from

they are privileged, since they are related

to the Itobservables" of the physical system, as we shall see later. System a

(5),

sui table

easy

;-.0

compatible

choice of

prove

that

with

the

the

field

system

(8),

is

riducible

variables (18)

to

the

and vicevers8;

form

(18)

in fact

it is

provides always a supplementary law

(8):

let h

Q

= U'

• V'h lQ

_

h l o, we have a

U'·!' = g.

Finally we have shown that

for

a

h

Q

aa u'

U"f{U'}

265

A necessary and sufficient conditions for the system (5) to be compatible IJ'ith a supplementary conservation Z<w peat to

a

(8J

with

choice of the fieLd

convex function with res-

sY8tems)~

(convex covariant density

U = F (a

h(U)

is that there exists a

U' (invariant respect field transformations and indepen-

dent of the congruence defined by the time-like caveator {, )), so that the a

system (5) assumes the syrrrnetric form (18) with h'=h,QE; This

proposition

is

a

first

contribution

of

convex function of U

a questlon

the

proposed

I

by

I.MUller (liS challenge to mathematicians" [1]). At least we point out that if in (8) we impose the condi ticn g

>

O.

then, by (10) all solutions of (18) satisfy U' • ~ {U·}~O.

3. Shock Waves Theory for Convex Covariant Density Systems.

i) Entropy growth across a shock wave. Let

a

[1

connected

open

into two open subset Gh of r :

we shall

set

G2 _ Let".

indentify r

\1+

of

a

r

and

a hypersurface

) = 0, ~ E em

cutting II:

(m> 2). be the equation

with a shock hypersurface for the field U.

It is known that the Rankine-Hugoniot conditions must hold

[Fa]~

=O,onf a where brackets denote the jump across rand q. a = 0a q. • Formally

the

Rankine-Hugoniot

equations

are

(21)

obtained

from

the

field

eqs. (5) through the correspondence rule

aa However

this

rule

does

+

~

[

(22)

]

a not work

when

applied

to

the

supplementary

equation (8); in fact (23)

does

not.

in general,

n is

non negative.

vanish.

This

Furthermore

result was proven

1. t

is

possible

to show

that

in

a non covariant formalism

by P.D.Lax [4J introducing an artifical viscosity in the field equations; a different proof was given in [51. It is know that the positive signature of

n for the non relativistic perfect fluid implies the growth of thermo-

dynamic

entropy across

n

is

> 0

often

the

called

in

shock. the

That

is

the

literature

reason why the condition

"entropy growth condition"

•

266

aoo

is

assumed as

a criterion to pich ur> the physical shocKs among the

solutions of the Rankine-Hugoniot equations. In this

section we

suggest the

proof of the fact that Let It

I

main steps of an explicitly covariant

r.

is non negative on

Tl

1 and a a

be a 6ubcharacteristic covector such that

" covariant scalar

defined 8S:

r. a +a

a = -

(24)

th8'l there exists a space-like covector {t )

such that

"

(25)

Let

cp(x

Cl

)

= 0 be the equation of a characteristic hypersurface which

locally has the same Itdirection of propagation" Le.

det(A" -

(jl )

det{A"(~

•

a

-

a

t )}

-"

a

to of the shock surface. (26)

0,

(27) where

Ik)

are

(k-=l,2 •...• N)

~

the

solution of

(26);

these

eigenvalues

are real by the hyperboliclty condition. Now

we

consider

a

solution

U U.

r

U* (in

field

being the

the

U(U',9),

perturbed

following

computed

~

for

*

"

and

will

U ::

of

the

denote

U*).

Here

the

U

Rankine-Hugoniot ~

U'

we

values take

(21) (28)

unperturbed the

equations

fields of

any

respectively

on

function of the

only k-shocks

according to

the following

Definition of k-shock

We

shall

say

that

a

shock

is

a

k-shock

if

there exists a number k (=1.2 •...• N) such that lim C"'Il (k)

•

U

:: U·

(29)

267

Roughly speaking a k-shock

approaches

is a

shock that

to a characteristic velocity

weak Sh DC k 5 Wh en a

We suppose

is near to

to know

the

(k) )

~*

solution

vanishes

(of course,

(28)

for

a

shock

speed

these shocks become

k-shock

and

replace

it

a

a

differentiating

the

.

into (23): then we get n as function of U* and ¢I

By

when

a

"(U*'.a) • h (U(U*'.e».a - h (U*).a

(30)

to

ifa

(9).

a

J -

(30)

respect

and

taking

into account

after some calculations we obtain [h

Vh· [F

a

)

Thus (31)

Since h is a convex function of U, defined in a convex domain D, we have: w(U,U-) = -h(U) ... h(U·) + Vh·CU - Ufo) > 0,

lJ. U

f. U* oS D

So the r.h.s. in (31) is equal to -w, restricted to r. Hence 3 n Ja¢la

(a

Furthermore.

in

the




(33)

0

As an/CIa is a scalar quantity, inequality (33) is independent of the

frame; So n is a strictly increasing function of

in any frame.

0

Since our shock is supposed to be a k-shock we have lim

n

= 0

0"" (k)

•

hence we get

For a convex covariant density system and a k-shock one Iuw when

0

;:.

(k)

~*

(on rJ.

268

1i) Jl

If

as generating f'unction of' the shock.

is a know function of u· and

1)

+" a

it is easy to prove that the

to llo»iYl{/ re lations ho lde on r

v· '1

A~ • a

J

[U'

..

(34)

whsre

EQ. a

(34)

that if ve kno... only the scalar function

D'le8nS

function oC U·

Jump

of

U'

and

and

• a ' w1 th

(which 1s

non characteristic) we may find the

•Q

therefore of U;

Tl

"behaves

like a

"potential"

for

the

shock. Of course,

in practice,

ll(U·, +0)

is computed when the shock is known

8S a solution of the Rankine-Hugoniot equations. However it is interesting the

fact

that.

were

it

possible

to

determine

n

through

experimental

tests, we should be able to have all information of the shock. iii) Relativistic bound of the shock speed.

The Rankine-Hugoniot equations Fa (U)+

(35)

a

provide N equations for the perturbed field U if U· and fa are knaYn. Eqs. (35) are equations of the kind

which always possess

feu. +) = f(U·.+ ) a a the trivial solution U

have also non trivial solutions U .solution)

which

in turn are

~

(36) :=

U"

for any

~a

They may

U* (branching solutions of the trivial

physically

acceptable only

aB

if g

tate

~

o.

so that the speed of' the shocks does not exceed that of light, according to relativity theory. We put no'"

the followin&; question: function

f

is

does it exist a set of values of

+0 such

that

the

globally

invertible

U for a

fixed

to? If the answer is affermative,

with

respect

to

then only the trivial

269 solution U

= U·

is allowed.

The problem has been examined by G.

[6]

SoHIat and T. Ruggeri

I

who proved

that non vanishing schoks take place only if their speed is greater

than

the

smallest

characteristic

speed

and

smaller

than

the

greatest

one. It is possible

to provide an explicitly covariant formulation of the

[6J

proof given in

and show that:

Fop hyperbolic convez covariant density systems the speed of the non vanishing shoek fuLfa the eondition: (37)

where m

mini \l

in!

UfD

As a consequence the

(k) }

I

M

sup

Max {\l

UfD

k

8h~ck

(k)

k

manifoLds are time-like or light-like if so are

the characteristic manifotds. In fact

if

(37)

holds,

and the characteristic manifolds are time-like

or light-like we have: .B (k) (k)} ( gcp~ Ma x


,Q

when

conseq~ence

of i) since it is possible to show that

n:: _r·ua~ • a where

p*

is a

Q

""pu /8

[s J

r'(a_~')ua• 0.

If now, the potential energy is positive definite in the following sense

(' ., OJ,. we deduce from (4.7) that (4.9)

E(O)

~

•

J

u. jk U

n However, for n c JP.

w%,2

Ul)

1- a

,

'k

t..J

az.

C COCO), see Gilbar-g and i'rudinger

[16], corollary 7.11, and so there is a constan~ ~ depending on

n.

A such that

(4.10)

[E(O)]!

~

" sup

n

1~(~.t)l.

Thus, by making F(O) arbitrarily small

luI

is likewise a~bi~ra~lly

small and so the solution is stable in the CO norm~

Acknowl edgments

I should like to express my sincere thanks to Professor D. Graffi for his kind invitation to study in Italy in June - July 1990 and to both Professor Graff! and Professor G. Ferrarese for the opportunity to hold a seminar in Bressanone.

finally, I

wish to express my appreciation to Professors D. Graffi and M.

Fabrizio and to Dr. F. Franchi for many stimulating discussions.

292

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3.'

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