Mathematical Problems in Elasticity

MATHEMATICAL PROBLEMS IN ELASTICITY

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Series on Advances in Mathematics for Applied Sciences - Vol. 38

MATHEMATICAL PROBLEMS IN ELASTICITY editor

Remigio Russo Istituto di Matematica Seconda Universita di Napoli italia

\\b World Scientific ••

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Library of Congress Cataloging-in-Publication Data Mathematical problems in elasticity / editor, Remigio Russo. vii, 192 p. ; 22.5 cm. -- (Series on advances in mathematics for applied sciences; vol. 38) ISBN 9810225768 I . Elasticity. I. Russo, Remigio. II . Series. QA931.M428 1996 95-48843 53I'.382'OI51 --dc20 CIP

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v

Preface

The relevance of the mathematical theory of Elasticity (in both the linear and nonlinear schemes) in the growth of theoretical knowledge and description of the physical world, as well as in the practical management of some important mechanical phenomena, is widely acknowledged and has been pointed out in most papers, so that it does not require to be emphasized here . Accordingly, the present volume collects five papers that may give a good sample of the problems and the results characterizing some recent trends and advances in this theory. Some of them are devoted to the improvement of a general abstract knowledge of the behaviour of elastic bodies, while the others mainly deal with more applicative topics . We shall now give a brief account of their main aims . The first paper, due to Ph. Boulanger amd M. Hayes, aims at giving a complete and systematic exposition of a body of selected results concerning the propagation of finite-amplitude plane waves in a deformed Mooney-Rivlin material. Such a system, when assumed to be homogeneous, isotropic and incompressible, may be envisaged as a "rubberlike material" The paper by C. O. Horgan provides a review of recent results concerning the decay at large spatial distance of solutions to (systems of) elliptic partial differential equations. This problem is of great relevance in connection with the well-known Saint-Venant principle in elasticity and also with entry flows for viscous incompressible fluids . In the third paper, by R. Russo and G . Starita, the well-posed ness (existence, uniqueness and continuous dependence of solutions upon the data) of the traction problem in incompressible linear elasticity for three-dimensional exterior domains is proved in the class of solutions with finite energy. In particular, the uniqueness classes there determined may be expected to be maximal, in view of analogous results obtained in the compressible case. The paper by T. Valent deals with an abstract formulation for boundary problems with symmetries, and with a study of a general perturbation problem with symmetries . This problem is of particular interest, in that it aims at a general treatment of any boundary-value problem in both linear and nonlinear elasticity with external forces explicitly depending on the unknown deformation .

vi

The fifth pap er , by L. T . Wheeler , owes its interest to a wide discussion of the applications of maximum principles for scalar-valued fun ctions to classical theory of elasticity. He also shows and discusses a numb er of counterex amples that may suggest a "boundary" to the possibility offinding a general maximum principle in the framework of linear elasticity.

R . Russo

vii

Table of Contents

Preface

. .... ....

Collected Results on Finite Amplitude Plane Waves in Deformed Mooney-Rivlin Materials . . . . . . Ph. Boulanger and M . Hayes

v

. . 1

Decay Estimates for Boundary-Value Problems in Linear and Nonlinear Continuum Mechanics C. O. Horgan

47

On the Traction Problem in Incompressible Linear Elasticity for Unbounded Domains . . . . . R. Russo and G. Starita

91

An Abstract Perturbation Problem with Symmetries Suggested by Live Boundary Problems in Elasticity T. Valent Maximum Principles in Classical Elasticity L . T . Wheeler

129

. . 157

2

1

INTRODUCTION

Here we draw together various results on finite-amplitude plane transverse waves propagating in a Mooney-Rivlin material which is maintained in a state of arb itrary finite static homogeneous deformation. The Mooney-Rivlin constitutive equation for homogeneous isotropic incompressible elastic materials is generally used to model the behaviour of ru bberlike materials (see, for instance, Beatty l for an account of different models used in nonlinear elasticity). Elastic waves in deformed Mooney-Rivlin mate rials are of particular interest because many explicit res ults may be obtained for finite amplitude waves alt hough the underlying theory is nonlinear. Because of the basic finite static homogeneous deformation of the material , the plane wave properties wi ll depend strongly on the propagation direction , and properties similar to those of crystal acoustics and crystal optics may be expected . For general incompressible homogeneous isotropic elastic materials maintained in a state of static homogeneous deformation , Green 2 showed that linearly polarized finite-amplitude plane waves may propagate along a principal axis of the underlying static strain . Later, Carroll' showed for a homogeneously deformed Mooney- Ri vlin material that such waves may propagate in any direction in a principal plane of the basic static strain. Then, Currie and Hayes 4 showed that two linearly polarized finiteamplitude transverse waves, polarized in directions orthogonal to each other and to the propagation direction , may propagate along any direction in a Mooney- Ri vlin material whi ch is subject to arbitrary finite static homogeneous. Later , Boulanger and Hayes 5 gave a simple characterization of the two polarization directions using t he ellipsoid associated with the tensor B- 1 , where B is the left Cauchy-G reen strain tensor of the basic static deformation. They also obtained simple expressions for the wave speeds and showed t hat the wave speeds are equal for propagation along special directions, called the "acoustic axes" There are determined only by the static deformation of the material. There are two such directions if this deformation is triaxial (t hree different prin cipal stretches) , and one if it is biaxial (two principal stretches equal). Using the acoustic axes, explicit expressions were obtained for the polarization directions and the wave speeds in terms of the propagat ion direction. In a furth er work , Boulanger and Hayes 6 investigated other propert ies of the finiteamp li tu de transverse waves, in the general case when the basic static deformation is triaxial. In particular, an appropriate energy- flux velocity vector has been introduced which turns out to be normal to t he slowness surface, and whi ch leads to t he concept of ray surface. A duality between the slowness surface and its properties and the ray surface and its properties has been exhibited. Also, results were formulated in terms of t he propagation direction (along the slowness vector), of the ray direction (along the energy flux velocity vecto r) , and of the polar izat ion direction (along the amp li Lud e vec tor). Here, we draw together t hese va ri ous res ulLs in a progressive and systematic pre-

3 sentation starting with the basic equations characterizing the Mooney-Rivlin model. We note in passing some further properties (see, for instance, section 10 on universal relations). The plan of this presentation is as follows . In section 2 , the basic equations for incompressible isotropic elastic materials of the Mooney-Rivlin type are recalled. The two constitutive constants C and D of the model are introduced and are assumed to satisfy the strong ellipticity conditions . Next (section 3), a finite-amplitude wave superimposed on a basic state of finite homogeneous deformation is considered, and the kinematics of the motion resulting from this superimposed wave is considered . In section 4, the equations of motion governing the propagation of such waves are obtained. It is shown that two finite-amplitude tranverse waves may propagate along any direction n, the waves being linearly polarized along the directions of the principal axes of the elliptical section of the ellipsoid associated with the tensor B- 1 by the plane n·x = 0 orthogonal to n . It is also noted that the fastest wave propagates in the direction of the greatest stretch and is polarized along the direction of the least stretch, whilst the slowest wave propagates in the direction of the least stretch and is polarized along the direction of the greatest stretch . Next (section 5) , an equation for the slowness surface is obtained. It is given a simple form by introducing the tensor E = (C/p)l + (D/p)B , where p is the mass density and C, D are the material parameters. Referring then the equation to the principal axes of the static deformation (principal axes of the strain tensor B), it may be written in the "Fresnel form" , a form analogous to that used by Fresnel in crystal optics 7 . Also using the tensor E, new expressions for the wave speeds are obtained. Generally, it will turn out that this tensor E is very useful for a simple formulation of most results presented here. In section 6, the energy-flux vector and energy density associated with a wave motion are considered. For time-periodic waves, the energy-flux velocity vector defined as the mean energy-flux vector devided by the mean energy density is introduced. Expressions are obtained for the energy-flux velocities of the two waves propagating in a given direction n. Section 7 introduces the "acoustic axes" , the special directions for which the two wave speeds are equal, which are along the normals to the planes of the central circular sections of the ellipsoid associated with the tensor B- 1 Assuming here that the basic static deformation is triaxial , there are two acoustic axes . The existence of acoustic axes is a feature reminiscent of the linear theory of crystal acoustics. However , some crystals may have in principle as many as 16 acoustic axes 8 For propagation along an acoustic axis, the polarization direction becomes arbitrary in the plane orthogonal to this axis and, by considering the energy-flux velocity of these waves, the phenomenon of internal conical refraction 7 is obtained. Then (section 8), the "ray surface" is introduced as the surface described by the extremity of the energy-flux velocity vector for all possible waves. It is seen that the energy-flux velocity is normal to the slowness surface while the slowness vector

4

is normal to the ray surface. The equation for the ray surface is given, both in the Fresnel form and in a form independent of the choice of coordinates. A duality between the slowness and ray surfaces is exhibited , reminiscent of t he duality known in crystal optics 9 and crystal acoustics 10 Also , it is seen that the maximum and minimum energy flux velocities occur for those waves with maximum and minimum wave speeds. An analysis (section 9) of the singular points and the singular tangent planes of the slowness and ray surfaces is presented. It is shown that both surfaces have four singular points (common to the two sheets of the surface), and four singular tangent planes (tangent together to the two sheets of the surface). The singular points of the slowness surface are determined by the directions of the two acoustic axes . Similarly, the singular points of the ray surface introduce two corresponding directions, called the "ray axes" Because of the duality between t he slowness and ray surfaces , the singular points of one surface correspond to the singular tangent planes of the other. Also , each singular tangent plane of the slowness or ray surface touches this surface in a circle, leading to t he concepts of cones of internal and external refraction . In section 10, some universal relations involving the wave speeds or energy flu x velocities and the components of strain or stress are presented. They are universal in the sense that they are independent of t he material parameters C and D . Some of these relations generalize results obtained by Ericksen ll and by Hayes 12 in the context of a linearized theory. However , here t he waves have finit e amplitude. Reciprocal pairs of orthogonal directions in the sense of Schouten 13 are also considered. Here, it is shown that such pairs of directions necessarily lie in a principal plane of t he basic static deformation. For a given propagation direction n , there are, in general , two possible polari zation directions for the amplitude vector. Similarly, it is shown in section 11 that for a given ray direction m (direction of the energy-flu x velo city vector ), there are, in general two possible waves and their polarization directions may be easily characterized. The inverse of the magnitude of the energy-flu x velocity vector, called the "r ay slowness" is obtained for these two waves . Next (section 12), results are expressed in terms of t he p ropagat ion direction n alone, and of the ray direction m alone. These are obtained using the acoustic axes and the ray axes, res pectively. The derivation is based upon the properties of the pair of common conjugate direct ions with respect to the ellipt ical sect ions of two concentri c ellipsoids by a cent ral plane 14 The ellipsoids considerd here are associated with the tensors EB - 1 and B - 1 (results in terms of the propagation direction) or with th e tensors E - 1 B and B (res ults in terms of the ray di rection) and all have t he same principal axes. F inall y, given a polarizat ion direction of a finit e-amplitude plane wave propagating in t he deformed Mooney- Rivlin material it is shown (section 13) that provided t hi s direction is not along a prin cipal ax is of the basic stat ic strain , then t he phase speed, the slow ness vecto r, the ray slowness, and the energy-flux velocity vector may all be

5

expressed in terms of this polarization direction . The properties of a wave propagating in a homogeneously deformed Mooney-Rivlin material are thus , in general, uniquely determined by its polarization direction. Section 14 concludes with a summary of the various results bringing out the duality between slowness and energy-flux velocity. The results for a given propagation direction, a given ray direction, and a given polarization direction are recalled in turn. Although several properties are reminiscent of crystal aco ustics, these results are unlike previous known results in classical lineari zed anisotropic elasticity where no simple explicit expressions are available for the various properties of the waves in terms of a given propagation direction or ray direction. Also, it is remarkable that here, in general, the various properties of a wave may be determined from its polarization direction.

2

THE MOONEY-RIVLIN CONSTITUTIVE EQUATION

Here, we consider incompressible isotropic elastic materials of the Mooney- Ri vli n type. These are characterized by a strain-energy density W per unit volume given by! :

2W

= C(I -

3)

+ D(II -

3),

(1)

where C and D are material constants and

2II = (tr B )2 - tr (B2)

1= tr B ,

(2)

Here B denotes the left Cauchy-Green strain tensor , whose components, in a rectangular Cartesian-coordinate system are (3)

where xi(i = 1,2,3 ) are the coordinates at time t of the point whose coordinates are XA(A = 1, 2,3) in the undeformed reference configuration. In t he special case when D = 0, the elastic material is said to be neo- Hookean. Because the m aterial is incompressible, possible deformations are restricted by the constraint (4) det (fJx ,/fJXA ) = 1, thus

(5)

I II = det B = 1. The constitutive equation for the symmetric Cauchy stress tensor Tis!

T = -pI

+ (C + DI)B -

DB2 ,

(6)

6 where p is an indeterminate pressure function of position x and time t. Alternatively, using the Hamilton-Cayley theorem (with IIl=l) ,

B 2 = B- 1

-

I I1

+ IB ,

(7)

this constitutive equation may be written in the form

T = -p. I

+ CB -

DB-I ,

(8)

where

(9)

p. = p - DIl ,

is also an indeterminate pressure. We also note that because det B = 1, we have

II=trB- I It will be here assumed that the strong ellipticity conditions hold, i.e. that C and D > 0, or C > 0 and D ?: O.

3

(10)

?: 0

FINITE AMPLITUDE WAVE SUPERIMPOSED ON A STATE OF FINITE STATIC HOMOGENEOUS DEFORMATION

Consider now a static finite homogeneous deformation of a Mooney-Rivlin material, defined by det( F ) = 1, (ll) where F,A is a constant deformation gradient satisfying the incompressibility constraint. The corresponding constant strain and stress are given by

(12)

T = -pI

+ (C + DI)B -

DB2

where p and p. = p - D I I are constant. Then , superimposed on the finite homogeneous static deformation (11), we consider the time-dependent deformation taking the particle at x in the static deformation to x, given by x=x+a!(ry , t) ry = n ·x, (13) where n and a are unit vectors , and let

p = p+q(ry,t),

(14)

7

be the associated pressure. The motion (13) and pressure (14) represent a finite amplitude wave propagating in the direction n , linearly polarized along the direction a . For this motion, the deformation gradient from the reference state, denoted by F , is given by

Ox, Oxi aXj ) FiA = aX = aXj aX = (Oij + ainJf,~ FjA , (15 ) A A denotes the partial derivative of f(TJ, t) with respect to "I. The motion (13)

where f,~ has to satisfy the incompressibility constraint

det F = 1 + f,~ n . a = l.

(16)

Hence a must be chosen orthogonal to n . n · a=O

,

(1 7)

n · n=a · a=l.

The possible waves are necessarily transverse. The left Cauchy-Green strain tensor corresponding to the motion (13) is given by

13 = FFT

(18) and , because

(1 + f,~ a

18)

n )(1 - f,~ a

18)

(19)

n ) = 1,

we have

(20) Also,

7 = tr 13 = I + 2 J,~ n

. Ba

+ F,~ n

I I = tr -B- I = I I - 2 f ,~ a . B- 1 n

-

The stress tensor

T

(21)

Bn ,

+ f 2,~ a . B -

1

a.

(22)

corresponding to (13), (14) is given by

(23) or , equivalently, by

T=

-p' l

+ CB -

DB-I ,

(24)

where p. = p - DII. From (9), (14) and (22) we note that p. may be written in the form (25) p. = p. + q.(TJ, t), where

(26)

8

4

PROPAGATION CONDITION. WAVE SPEEDS

The wave motion (13) (14) has to satisfy the equations of motion, which , in the absence of body forces, read (27) where p is the Let now b (ry,~, () be the to rectangular

constant mass density of the Mooney-Rivlin material. = n x a , so t hat n , a, b form an orthonormal triad. Let (Tj,~ ,

() and components of x and x (respectively) in this triad . Thus, with respect Cartesian axes along n , a , b , the motion (13) reads ~ =~+j(Tj , t),

ry=Tj,

( = (.

(28)

Because the components of the stress tensor 'I' are functions of Tj and t, or equivalently

17 and t , the equations of motion (27) read fJT"" Ory oT{" Ory oT," Ory

0

02j P fJt2 0 We now compute the components of have

T""

oT"" OTj oT{" , OTj

(29) (30 )

oT," OTj

(31)

'I' entering these equations. From (24) we

-Po + Cn· Bn - Dn· B -' n ,

= n · Tn =

(32)

and hen ce, using (18) (20) and (22), we obtain

T""=-p+D II+ Cn · Bn-Dn B-1n , or, using (9), (12) and (14) ,

T""

= -q

+n

(33)

Tn .

Also from (24) we have (34 ) an d hence, using (1 8) (20), we obtain

T{"

= ('a·

Bn -

Da·

B - 1n + f ," (C n Bn

+ Da · B -1 a ),

(35)

or, using (12) ,

(36)

9 Finally, we have

or

(37) _ Also, for the purpose of furth er reference, we note that the other components of T are given by

T~~ = -q

+ a· Ta + 2j,'I n T~( =

+ j 2,'1 (C n . Bn + Da · B-Ia), b · Ta + j,'I Cb . Bn, . Ta

(38) (39)

and

(40) Introducing (33) (35) and (37) into the equations of motion (29) to (31), and recalling that the stress tensor T corresponding to the basic static deformation is constant, we obtain

q,'1

pj,tt D(b · B-Ia)j,'I'I

0,

(41 )

(Cn' Bn + Da · B -Ia)j,'I'I'

(42)

0.

(43)

From (41), we conclude that the incremental pressure q has to be a function of time alone, and thus may be taken to be zero. Because B is constant, equation (42) is , for fixed n and a, the standard linear wave equation ("vibrating string equation") for the function j("I, t) . Assuming now that the material is not neo-Hookean , D f. 0, and looking for solutions j("I, t) nonlinear in "I(J,'I'I f. 0) , equation (43) yields the propagation condition

(44) For a given propagation direction n , it is a condition on the polarization direction a (recall b = n x a). The propagation condi tion (44) requires that a and b be along conj ugate directions with respect to the ellipsoid x . B-Ix = 1 associated with the tensor B-1 of the static homogeneous deformation. We call this the "B-I-ellipsoid". But a and b are orthogonal, and are both orthogonal to n . It follows that a and b must be along the principal axes of the elliptical section of the B-I -ellipsoid by the plane n . x = orthogonal to n . Hence, for every propagation direction n, two transverse linearly polarized finite-amplitude waves may propagate: the two possible directions of polarization are along the principal axes of the elliptical section of the B-I -ellipsoid by the plane n . x = 0, and thus along the unit vectors a and b satisfying

°

n · a = n · b = 0.

(45)

10

From the wave equation (42), it is clear that the speed of propagation of the wave propagating along n and polarized along a , denoted by v( a ), is given by (46) Similarly, the speed of propagation of the wave propagting along n and polarized along b , denoted by v(b), given by (47 ) R emark. Maximum and minimum wave speeds. Assuming that the basic static deformation is triaxial with principal stretches Al > A2 > A3 , satisfying Al A2A3 = 1, and denoting by i, j , k the unit vectors along the corresponding principal axes of B , we note that when n = i, then n Bn takes its largest value, Ai , whilst the largest value of a B- Ia is A3 2 with a = k. Thus, assuming C i= 0 and D i= 0, the fastest wave-speed Vrnax (say) is given by

(48) Also , the slowest wave-speed

Vrnin

(say) is given by (49)

Thus the fastest wave occurs for propagation in the direction of the greatest stretch and is polarized along the direction of least stretch , the slowest occurs for propagation in the direction of least stretch and is polarized along the direction of the greatest stretch. Ericksen 16 (p.336) noted t his fact for acceleration waves propagating in a neo-Hookean material. Also , from (48) and (49), we note that

(50)

5

SLOWNESS SURFACE

Here we obtain the equation of the slowness surface for the finite amplitude waves propagating in the homogeneously deformed Mooney-fuvlin material. We also write the equation for this surface in a form analogous to that used by Fresnel in optics. From (46) and (47), it is seen that the phase velocity v for the waves propagating along n must sat isfy

{pv 2

-

(Cn ' Bn

+ Da· B - 1 a)}{pv 2 -

(Gn · Bn

+ Db· B - 1b)}

=

O.

(51)

Because

(52)

11

this equation may be written (pV 2)2

_

pv 2 {2Cn.Bn+D(trB- 1 -n.B- 1n)} + C 2(n· Bn )2 + CD( n· Bn )(trB- 1 - n . B- 1n ) +D2(a· B-1a)( b . B-1b) = 0,

(53)

or, alternatively, (pV 2)2 -

PV 2 {2Cn . Bn + D(trB-l - n . B - 1n)}

+ (n · Bn) {C 2n. Bn + CD(tr B- 1

-

n · B-1n) + D2} = 0,

(54)

so that 2Cn . Bn + D(trB- 1

±D { (trB- 1

-

-

n . B-1n)

n . B- 1n )2 - 4n . Bn }

1/2

,

(55)

where we have used (45) and (5) to note that (a x b )· (B - 1a x B-1b) det(B-1)n· B (a x b ) = n · Bn.

(56)

For the slowness s = n /v,

(57)

we have the equation O(s)

== C 2(s· BS )2 + CD( s· Bs){ (trB-l )s · s - s· B -1s} + D2(S' s)(s· Bs ) -p

[2C(s . Bs ) + D{(trB-l)s· s - s· B-1s}] + / = O.

(58)

1, 2,3): the Here, O(s) = 0 defines a surface in the space of coordinates Si (i slowness surface. The slowness surface is made up of two sheets as appears from the factorization (51). Explicit equations for these two sheets were given in a previous papers Also the intersections of the two sheets with the principal planes of the basic static deformation were obtained. If the components of s , with respect to the principal axes of B , are denoted by Si (i = 1, 2, 3), then

where Ai> A2, A3 are the principal stretches. Then, it is easily checked, using AiAP~ = 1, that the equation (58) for the slowness surface may also be written

(A~s~

+ A~S~ + A~s~)(E2E3A~S~ + E3EIA~S~ + EIE2A~S~) +1 =

{(E 2 + E3)A~S~ + (E3 + El)A~S~ + (El + E 3)A;Sn

0, (60)

12

where E, are defined by

(i=I,2,3).

(61 )

Thi s notation suggests the introduction of the tensor E , defined by

pE = C l

+ DB ,

(62)

whose eigenvalues are E I , E 2 , E 3 . With this notation, the equation (60) for the slowness surface takes the form (det E )(s· Bs )(s· BE-I S) - s · [(trE)B - EB]s + 1 = 0 ,

(63)

and , using (57) , the equation for the phase speed v, may now be wri tten

v4

2

v n ·

-

[(t rE)B - EB]n + (det E )( n· Bn )( n· BE-In) = O. 2

(64)

2

We note that the solut ions v ( a) , v (b) of this equation may also be expressed using the tensor E. Indeed, using (56), we have v 2 (a) = (n · Bn )( b · EB-Ib )( b · B- Ibt l = (a· B - Ia )( b · EB-Ib),

(6.5)

v 2 (b) = (n Bn )( a EB -Ia)( a · B - Ia )- I = (b · B - Ib )( a · EB-Ia).

(66)

The fact that (65) and (66) are t he roots of (64) may be checked di rectly using (56) and the relations 5 (67)

(a · EB - 1a)( b . EB-Ib ) = (det E )n · BE -I n ,

trE = (a · EB -Ia)(a· B - 1at l +(b ·EB -Ib )( b· B-Ib )-I + (n · EBn )( n· Bn tl (68) Indeed , from (65) and (66), it follows that (n Bn ) {(b. EB -I b )( b · B- Ib )- I + (a · EB-1a )( a· B -Ia)-I} (trE )n . Bn - n · EBn ,

(69)

and v 2 (a) v 2 (b)

=

(n· Bn )2( b · EB -I b )( a · EB- 1a )( b · B - 1b )-I (a ' B - 1a)-1

=

(detE)( n · Bn )( n · BE - In) .

(70)

Now, it is easi ly checked that (60) may also be written in the "Fresnel form" r\

(

"F

)

'\i si

_

s =

E 1 s . Bs - 1

+E

'\~s~ 2

s· Bs - 1

+E

·\ 5s~ 3

-

0

s· Bs - 1 -

.

(71)

Using (57) , t hi s leads to t he "Fresnel form" of the equation for the phase speed v; ,\ 2 n 2

1 1 -::-_--'----"------,+ 2

E 1 n . Bn - v

,\2n2 2

2

E 2 n . Bn - v

2

+

,\2 n 2 3

3

E3 n . Bn - v 2

= 0 .

( 7~ )

13

6

ENERGY-FLUX AND ENERGY DENSITY

Here we consider the energy-flux vector R and the energy density £ associated with the two possible finite amplitude transverse plane waves that may propagate in any direction n. Here, as in 6 , the energy-flux is measured in the static state of homogeneous deformation. For time-periodic waves we consider the energy-flux velocity g defined as the mean energy-flux vector divided by the mean energy density. It will appear later that the vector g is normal to the slowness surface, and that a duality between the properties of the slowness vector s and the ray velocity g may be exhibited. Let T denote the time dependent stress field associated with the motion (13), and let P be the corresponding Piola-Kirchhoff stress

(73) Then , t he energy flux vector R corresponding to the motion (13), measured in the basic static state of deformation, is given by 17

(74) More precisely, Rk is the rate at which energy crosses an element of area normal to the xk-axis in the state of basic static deformation , measured per unit area in that configuration. The energy density corresponding to the wave motion is

£ = (1!2)pi".i" + W - W,

(75)

where Wand Ware the values of the strain energy density in the states of static deformation and of motion, respectively. The energy density and energy flux satisfy the energy conservation equation

(76) The superimposed dot denotes the time derivative at fixed x, or, equi valently, at fixed X (materi al time-derivative). We now compute the components in the triad n , a , b of the energy-flux vector R(a) of the wave propagating along n and polarized along a. Using (2S), we have -R~(a)

-R(a)· n =

-R€(a)

-R(a)· a =

-R«(a )

-R(a) . b =

i,t T<~, i,t (T« - i,~ T€~) , i,t T«.

(77) (7S) (79)

Hence, using (35) (3S) and (39), we have, recalling (S) and taking q = 0,

-R(a) . n -R(a) . a - R (a) . b

i ,t n· Ta + i,t i,~ (Cn ' Bn + Da· B-1a), i,t a· Ta + i,1 i,~ (Ca· Bn - Da· B-1n ), i,t b · Ta + i,t i,~ (Cb · Bn).

(SO) (Sl ) (S2)

14 Thus , recalling (46) ,

-R(a )

i ,l Ta + i ,l i,~ (CBn - Db x B -1a) i ,l Ta + i ,l i,~ {pv2(a)n + (a· Tn )a + C( b · Bn)b}.

(83)

For the energy density E(a), the sum of kinetic energy and stored energy densities of th e wave propagating along n , and polarized along a , we obtain , using (21) and (22) , (84) We now consider solu tions of (42) of the type i = F (1J - v( a)t) , where F denotes a ny periodi c function . Thus we restrict attention to time-periodic waves propagating with speed +v(a). Us ing a tilde to denote the mean over a period , we then note t hat

Hence for the mean energy density E(a), we obtain

(85) and for the energy- flu x velocity g(a) = R (a)/E( a) defined as the mean energy flux divid ed by th e mean energy density, we obtain

g (a )

p-1v-1(a)(CBn - Db x B-'a) v(a)n

+ p-1v-l (a){(a ' Tn )a + C( b· Bn )b}.

(86)

Analogo usly, for the energy flu x velocity g(b ) = R(b )/ E(b) of finit e amplitude periodic waves polarized along b , and propagating along n with speed +v(b ), we have

g(b)

+ Da x B -1b) + p- 1 V - 1 (b){ (b · Tn )b + e( a · Bn )a }.

p-1 V-1(b)( CBn v( b )n

(87)

From (86h and (87h , we immedi ately note that

g(a) · n =v(a )

,

g(b )· n= v( b ),

(88)

b · B - lg(b ) = O.

(89)

and, from (86h and (87) 1, that

a· Wlg (a ) = 0 ,

Property (88) is in ag reement with general res ults proved by Hayes l O in the context of lin ear co nservative sys tems. How ve r here the theory is non linear an d the waves have finit e a mplitude.

15

Also, using (56) we note that (86h and (87h may be written as

pv(a)g(a) = B-1a x (CB-1b + Db), pV(b)g(b) = (CB-1a + Da) x B-1b,

(90)

or, on using the definition (62) of the tensor E,

(91)

7

ACOUSTIC AXES

Here, we consider the possibility that the two wave speeds v(a) and v( b ) given by (46) and (47) are equal. Assuming D # 0, this will be the case if and only if

(92) which means that the principal axes of the elliptical section of the B - 1-ellipsoid by the plane n· x = 0 have the same magnitude, and so this elliptical section is a circle. Thus , the two speeds of propagation v(a) and v(b) are equal if and only if n is along the normal to a plane of central circular section of the B-1-ellipsoid. In the general case when the basic static deformation is triaxial , there are two such plane central circular sections with unit normal vectors denoted by n + and n - (say). Of course, if the basic static deformation is biaxial, the B -I-ellipsoid is a spheroid and there is only one such central circular section , whilst if B- 1 = 1 , every central section of the the B - 1-ellipsoid is circular. The directions of propagation for which the two wave speeds are equal are called the "acoustic axes" of the deformed Mooney-Rivlin material. From now on we will assume that the basic static deformatin is triaxial with the principal stretches ordered )1] > A2 > A3' Then , the acoustic axes referred to the principal axes i, j, k of the t ensor B , corresponding respectively to its eigenvalues Ai , A~, A~ , are along n + and n - given by ±

)1/2 • ± (A 3-2_A-2 ,-2_ A-2 )1/2 k I 2 1 22 . A3 - Al

-"2

n= ( 2 2 A3 - AI

(93)

Thus, n± do not depend upon the material parameters C and D . They are determined only by the basic static deformation. For propagation along an acoustic axis, for instance , n = n +, the vectors a and b may be two arbitrary orthogonal unit vectors in the plane orthogonal to n +. Thus we may take a = j and b = n + x j . In this case, n ± Bn± =A22AiA5 =A24,

a · B- 1a=b · B- I b=A2 2 ,

(94)

16

and the two wave speeds v(a) and v( b) reduce to V0 (say) given by (95) Thus, along an aco usti c ax is, linearly polarized waves of finite amplitude may propagate with speed V0 given by equat ion (95) , t he polarization direction being arbitrary in the plane orthogonal to the aco ustic axis . Recalling (48) and (49 ), we note in passing the relations

(96) which are independent of the values of the material parameters C and D . We now consider the energy-flu x velocity for waves propagating along an acoustic axis, for instance n = n+ For the waves polarized along a = j and b = n+ x j , equations (86) and (87) yield (97 ) and g(n+ x j ) = V0 n+ - E(pv0 tl(CA22

where

E

+ D )n+

x j == g(B )

(say),

(98)

is defined by (99)

Because the section of the B-I -ellipsoid by the plane n + . x = 0 is a circle, we may also, instead of the choice a = j , b = n+ x j , take a = ao (say) , b = b o (say), where

ao = cos Bj

+ sin Bn+

x j , bo =

-

sin Bj

+ cos Bn+

xj

= a8+rr/2'

(l00)

B being an arbitrary angle. With these a and b , equat ions (86) and (87) now yield g(ao)

V0 n +

g (b o)

V0

+ D) sin Bao + C A2 2 cos Bb o} , E(P V0)- 1 { ( C A22 + D ) cos Bb o + C A22 sin Bao}

- E( pv0 tl {( C A22

n+ -

(l 01 ) (102 )

But , using (100 ), and recalling (97), (98) , these become

g(ao) = (1/2) (g(A)

+ g (B)) + 1L(2B) , (103)

g (b e) = (1 /2) (g( A) + g(B) ) + 1L(2[B + 71" /2]) , where

1L(2B) = DE(2 pV0 )- I( - sin 2Bj

+ cos 2B n +

x j ).

( 104)

17

We note that g(ae) (or, similarly, g(b e)) gives the energy flux velocity of a wave propagating along the acoustic axis n+ , and which is linearly polarized in an arbitrary direction ae of the plane n+ . x = 0 (B is t he angle that this direction a e makes with j) . From (103) and (104) , it is now clear that when the polarisation direction is varied, that is when B is varied, the extremity of the vector g (ae) describes a circle in a plane orthogonal to n+ . Thus, the direction of the vector g( ae) describes an elliptical cone. By analogy with the situation encountered in crystal optics 7 and in crystal acoustics lO , this may be called the "cone of internal conical refraction"

8

ENERGY-FLUX VELOCITY AND RAY SURFACE

Here, we first show that the energy-flux velocity g is normal to the slowness surface O(s) = 0, where O(s) is given by (58) and that g . s = l. We then introduce the "ray surface", which is the locus described by the extremity of the vector g when the extremity of s describes the slowness surface. Consider for instance the wave propagating along n and polarized along a (the case of the wave polarized along b may be dealt with in a similar way). Its slowness is s(a) = n /v(a) , with v(a) given by (46), and its energy flux velocity is g (a) given by (86). The normal to the slowness surface at the point corresponding to s (a ) is along the gradient G (s) == (80/8s) computed at s = s (a ). From (58), we obtain (1/2)v 3 (a)G(s(a)) =

+ D(n · Bn) - pv 2(a)trB- 1 } n Bn) + CD [t rB- 1 - n · B-'nj- 2Cpv2(a)} Bn

D {C( n. Bn)trB-'

+ {D 2 + 2C 2(n

-D {C(n. Bn ) - pv2(a)} B-1n.

(105)

Now, using (46), (47) , (52) and (56), this may be transformed into (1/2)v 3 (a)G(s(a)) =

pC {v 2(b) - v2(a )} Bn

+ D2Bn + D2(a· B-1a)B-1n

_D2(a· B-1a) {(a. B-1a)

+ (n· B-'n )} n .

(106)

But, because n , a, b is an orthonormal triad (107) and, recalling (45),

Bn

B - 1a x B-1b = {(n' B-'a)n

+ (a · B-1a)a}

x {(n. B -1 b )n

+ (b· B - 1b)b}

(a. B-1a)( b · B-1b )n - (b · B-1b)( n · B-1a)a - (a · B-1a)( n· B - 1b) b .( 108)

18

Using these, (106) becomes

2p{v 2(b) - v2(a)} {CBn + D(a· B- l a)n - D(n · B- l a)a} 2p {v 2(b) - v2( a)} (CBn - Db X B- 1 a) . (109) Hence, comparing with (86), we have

(1l0) Thus the energy flux velocity g(a) is along G(s (a )) which is normal to the slowness surface at the point corresponding to s(a). Analogously, for the wave propagating along n and polarized along b , we have (I ll )

Thus the energy flux velocity g(b) is along the normal to the slowness surface at the point corresponding to s(b). Also, (88) may immediately be written in the form

g(a) . s(a) = 1

(1l2 )

g(b ) . s(b ) = 1.

Thus, the direction of the energy flux velocity g of a wave with given slowness s satisfying O( s) = 0 is that of the normal at the corresponding point of the slowness surface. Its magnitude is given by g . s = 1. When s is varied , the extremity of the vector g describes a surface w(g) = 0 (say), called the "ray surface" As explained for example in 9 (p.318), it fo llows from the above properties of g that there is a relationship of duality between the slowness surface and ray surface. Thus , the direction of the slowness s of a wave with given ray velocity g satisfying w(g) = 0 is that of the normal at the corresponding point of the ray surface. Its magnitude is given by g . s = 1. Hence,

00

00

as

as

g = -/(s·_)

oW

OW

og

og

s = - /( g · _ ).

( 113 )

Also, it follow s from this that the slowness surface is the envelope of the planes g·s = 1 (regarding s, as the independent variables) for all the possible g satisfying w(g) = 0 (ray surface). Conversely, the ray surface is the envelope of the planes s . g = 1 (regarding 9i as the independent variables) for all the possible s satisfying O(s) = 0 (slowness surface) . From the Fresnel form (7 1) of the equation of the slowness surface, it may be shown 5 that the equation for the ray surface is W ( )=

), -22 1 91

),-22 2 92

),-22 3 93

_

F g - Ei1g. B - l g _ 1 + Ei 1g . B - lg - 1 + E3 19. B - l g _ 1 - O.

(114)

19 This equation for the ray surface has the Fresnel form, that is a form analogous to the form (71) of the equation for the slowness surface. We note that, formally (71) and (114) may be read off one from the other by the substitutions s ....., g, B +-+ B - 1 , )..~ +-+ )..;-2,

Ei

+-+

Ei- l .

On expanding (114), it is seen that it may be written, on discarding the factor g . B- l g , as

p- 2 w(g)

()..12g~

+ )..2"2g; + )..3"2g~) (E21E31)..12g~ + E3IElI)..2"2g; + ElIE21)..3"2gn {(E21 + E;·I) )..12g; + (E3 1 + Ell) )..2"2g; + (Ell + E21) )..3"2gn + 1 = 0, (115) which may also be read off from (60) by the substitutions just mentioned. Then, using the tensor E defined by (62) whose eigenvalues are EI, E2, E3, we note that the equation (115) for the ray surface also reads

Formally, the equation (63) for the slowness surface, and the equation (116) for the ray surface may be read off one from the other by the substitutions s +-+ g, B +-+ B- 1 , E +-+ E - I The form (116) of the equation for the ray surface is independent of the choice of the coordinate axes. Writing (117) g = mlu where m is a unit vector, and where u will be called the "ray slowness" (u- I represents the energy-flux velocity in the direction m) , equation (116) yields the quadratic equation

for the squared ray slowness u 2 Formally, the equation (64) for the phase velocity and the equation (118) for the ray slowness may be read off one from the other by the substitutions n +-+ m, v +-+ u, B +-t B- 1 , E +-+ E- I . Also, inserting (117) into (114) yields the "Fresnel form" of the equation for the ray slowness u: \ - 2 2 Al m l

Ellm. B-I m

-

u

2

+

).. -2

2 2

2 m E21m· B-l m

-

u2

+ _

).. -2 2 3 m3

E3 1m . B-I m

-

u2

=

o.

(11 9)

From the definition (62) of E , it follows, using the Cayley-Hamilton theorem, and recalling that II I = 1, II = i1' B-1, that

(120)

20 and also

(121 ) Hence, multiplying (116) by det E , and using (62), (120) and (121 ), we obtain, for the equation of the ray surface written in terms of the tensor B ,

p2( g. B - 1g)(C g . B - 1g + Dg. g) - p [2C 2g. B - 1g + C D {(trB )g · B -' g + g. g} +C3 + C 2DtrB + CD 2tr B -' + D3 = 0.

+ D2 {(trB)g · g -

g. Bg}]

(122) R emark. Maximum and minimum energy-flux velocities. Here we show that t he max imum and minimum energy-flux velo ci ties Igl = l / u occur for t he waves with maximum and minimum wave speeds 21 We first note that g g is an extremum under t he condition Il! (g ) = 0, with Il! (g ) given by (115) if g = ",o il! I og for some multiplier "'. But , recalling (113 ) we note t hat o il! I og is along the slow ness vector s . Hence, th e extrema for the energy-flux velocity occur for those waves for which g is along s. Using (90), it is easily seen that g(a) x s(a) = v-I( a)g (a) x n = when

°

(123) or, on taking the dot product wit h a and b , and assuming C

=f.

0, when

(124) But (124) means that B -' n is orthogonal to both a and b , thus along n . Hence , n must be a principal axis of the tensor B - 1 , or equi valently, of the tensor B . For propagation along a principal axis of the bas ic static deformation, it follows from (86) that the energy-flux velocity is along the propagation direction and thus g (a ) = v(a) n. Hence, maximum and minimum energy-flux velo cities occur for the waves with maximum and minimum wave sp eeds. Thus , recalling (-\S) and (49), we have, assuming C =f. and D =f. 0,

°

n

= I,

n = k,

a

= k;

a =

I.

( 125 )

(126)

The maximum energy- flu x velocity gmax occurs for propagation in the direction of the greatest st retch , with polarization along t he direction of least st retch. The minimum energy-Aux velocity gmin occurs for propagat ion in the direction of least stretch , with po lar ization along the direct ion of the greatest stretch.

21

9

SLOWNESS AND RAY SURFACES. SINGULAR POINTS AND TANGENT PLANES

Here properties of the slowness surface n( s ) = 0 and of the ray surface \]i(g) = 0 are studied. In order to determine the points on the slowness surface (60) in a given direction n , we write s = n lv, where v is the phase speed. This yields the equation

v4

v 2 {(E 2 + E3)A{n{

+ (E3 + E1)A~n~ + (E1 + E2)A;nn (A{n{ + A~n~ + A;n;)(E2E3A{n~ + E3E1A~n~ + E1E2A;n;)

-

+

= 0, (127)

2

which is quadratic in v It is equi valent to (72). For any given unit vector n , it has two real positive (non zero) roots given by (46) and (47). This shows that the slowness surface is closed and consists of two sheets. The fact that the roots of (127) are real may also be checked directly by computing its discriminant ~ n (say) . We have, recalling that E1 > E2 > E3 > 0,

~n

=

{A{(E2 - E3)n{

+ A~(E1 - E3)n~ + A;(E1 -

E2)n;f

-4A{ A;(EI - E 2)(E2 - E3)n~n; - E3)2~~~n,

A~(E1

with

~~

= { A1 (E2 - E3)1/2 n1

A2

E1 - E3

(128)

± A3 (E1 - E2)1/2 n3 } 2 + n~. A2

E1 - E3

Clearly, ~ n is positive. Analogously, to determine points on t he ray surface (115) in a given ray direction m , we now write g = m lu, where u is the ray slowness. This yields the equation u4

+

u 2 {(E;l + E31)A j 2m~ + (E3 1 + Ej 1 )A;2m~ + (El l + E;1)XJ2mn (Aj 2m{ + A;2m~ + A3 2m ;)(E;1E3 1A j 2m~ + E3 1El l A;2m~

+El1 E;l A32m;)

= 0,

(129)

2

which is quadratic in u It is equivalent to (119) . We note that (129) may be read off from (127) by t he substitutions n +-+ ill , V +-+ u, Ei <-> Ei- 1 Denoting the discriminant of t his quadratic equation in u 2 by ~ 1jI, we have

~1jI = {A j 2(E3 2 - E;2)m~ + A;2(E3 1 - El 1)m~ + A3 2(E;1 - El1 )m;f \ - 2 A3 \ -2(E- 1 _ E - 1)(E- 1 _ E-l)m 2m 2 - 4 A1 2 1 3 2 1 3 A;4(E3 1 - E11)2 ~~ ~ ;,

(130)

with ±

~ >f =

\-1( E - 1 _ E - 1)1 /2 {A1A2 E3 E2 \ -1

3

1

-

1

1

m 1

A-1 _1 E-1 m )1 /2 ± _3 _ ( E-1 2 A- 1 E 1 _ E 1 3 2

3

1

}2+ m2

2'

22

Because LlIjI is positive the roots of (129) for u 2 are real. They are also positive, because Ei > O. Thus , the ray surface, as the slowness surface, is closed and consists of two sheets. Singular Points Now we consider the singular points of the slowness surface and of the ray surface.

(a)

Slowness Surface

In order that the two sheets of the slowness surface have a common point , there must be some choice of n such that (127) has a double root for v 2 For t his to be so, the discriminant Ll\1 must be zero. From (128) , it clear that the only directions n such that Ll\1 = 0 are given by n = (}

{A3 (EI - E2)1/2 i ± Al (E2 - E3)1 /2 A2 EI - E3 A2 EI - E3

k} ,

(131 )

where (} is an appropriate scalar such that n· n = 1. The corresponding double root of (127) is v 2 = a2A~4E2 . (132) There are thus four points common to the two sheets, which are the intersections of the slowness surface with the lines through the origin in the two directions given by (131). These points are gi ven by

si =

A~2 E:;I(E1 - E 2 )/(E 1 - E3),

52

= 0, s~ =

A3 2E:;I(E2 - E 3)/( E I - E3). (133 )

Noting that El - E2 = D(Ai - A ~ ) and E2 - E3 = D(A ~ - A~), it is seen that the directions (131 ) are the directions n± of the acoustic axes (recall t he expressions (93 ) for n± and note that n · n = 1, and thus a 2 = 1). Indeed, t hese are the directions such that the t wo sq uared wave speeds (46) and (47) are equal, and the doub le root (132) is equal to v~ given by (95). The common points given by (133) are such that the gradient G (s ) == oD)os is zero and a re thus singular points of the slowness surface . It is also seen from (110), (111) , because v 2 (a) = v 2 (b) when n is along an acoustic axis.

(b) Ray Surface Consider now the ray surface. In order that the two sheets have a common point , m must be such that LlIjI = o. Th e onl y directions m such that ..J,~, = 0 are given by m -/3 -

{

A- I (E 2- I _ E.1- l)I /2 i± -.\ '-- 1 ( E-3 ' _ E-' )1 /2 k } 2 E-3 ' _ E-I ' \-1 E' _ E-1 l . 2 3

_3_ \ - 1 /\2

where /3 is a n a pprop ri ate scala r such that m root of (1 29) is

'

(134 )

m = 1. T he co rresponding double (135 )

23 There are thus four points common to the two sheets, which are t he intersections of the ray surface with the lines through the origin in the two directions given by (134) . These points are given by 9~

= A~E2(E21

- E11)/(Eil - Ell), 92

= 0,

9~

= A~E2(Eil

- E21)/(Eil - Ell). (136) The directions given by (134) are the directions of the "ray axes", these directions in which the two ray slownesses u are equal. From now on, they will be denoted by m± Recalling (61) , we obtain from (134) u;;;I(A~ -A;)1/2 m ± = Al {(A~ - A~)(C +D Aml/2i±A3{(A~ -A~)(C+DAml/2k, (137)

where

U0

denotes the double root (135) and is given by pU;;;2

= C(A~ +

A~ - A~) + DA~A;.

(138)

We note that unlike the acoustic axes, the ray axes depend on the constitutive coefficients C and D . Also, the common points (136) to the two sheets of the ray surface are such that the gradient 8iI1 / 8g is zero. They are thus singular points of the ray surface .

Principal Sections We consider, briefly, in turn, the sections of the slowness and ray surfaces in the principal planes of B . The section of the slowness surface by the x2x3- plane is 4 the pair of coaxial ellipses (139) Recalling (61) we note that the major axes of both ellipses are along the x3-axis. The ellipse (n l ) completely contains the ellipse (n2) ' The section of the ray surface by the x2x3-plane is the pair of coaxial ellipses

). ( ''T<'. 1 ) : E-1,-2 3 A2 922 + E-1,-2 2 A3 932 -_ l', (,T. '<'2·

, - 2 2 + ,- 2 2 A2 92 A3 93

-- E 1·

(140)

The major axes of both ellipses are along the x2-axis. The ellipse (iIl2) completely contains the ellipse (iIl 1 ). The section of the slowness surface by the Xlx2-plane is the pair of coaxial ellipses (141) The major axes of both ellipses are along the x2-axis. The ellipse (n3) completely contains the ellipse (n4)' The section of the ray surface by the Xlx2-plane is the pair of coaxial ellipses

-l,-2 2 E-1,-2 2 l' (,T.). ,-2 2 + ,-2 2 - E ( iII 3) : E 2 Al 91 + 1 A2 92 =, '<' 4 . Al 91 A2 92 - 3·

(142 )

24 The major axes of both ellipses are along the xI-axis. The elli pse (W 4 ) completely contains the ellipse (W 3)' Finally, t he sect ion of t he slow ness surface by the xl x3- plane is the pair of coaxial ellipses (143) The major axes of both ellipses are along the x3- axis. The major axis of (51 6 ) is greater than the major axis of (51 5 ), whereas the minor axis of (51 6 ) is smaller than t he minor axis of (51 5 ), The t wo ellipses have four common points. These are t he singular points given by (133). The section of t he ray surface by t he xl x3- plane is the pair of coaxial ellipses

The major axes of both ellipses are along t he x raxis. The major axis of (W6) is greater than t he major axis of (W6), whereas the minor axis of (W5) is smaller t han the minor axis of (W s) . The two ellipses have four common points. These are the singular points gi ven by (136) . The sections by the xl x3-plane are represented on Fig.1 (using a different scale for the slowness surface and the ray surface). Singular Tangent Planes Because of the relationship of duali ty between t he slowness and ray surfaces, it follows, as expl ained for example in g (pp.326-32 7) and in lo (pp .78-81), that to singular points of the slowness surface corres pond singul ar tangent planes of the ray surface, and , conversely, to singular points of t he ray surface correspond singular tangent planes of the slowness surface . Singular tangent planes of the ray or slowness surface are those plan es which are tangent togeth er to both sheets of t he surface . To a gi ven s satisfying 51 (s ) = 0, that is to a given point on t he slowness surface there corresponds a plane s . g = 1 (g being the variable) which is tangent to the ray surface w(g ) = 0. Consider now the four singular points (133 ) of t he slowness surface, namely s = ± v;; ln +, and s = ± V;; l n - To these points correspond the fo ur planes n+ . g = ± v0 , and n - g = ±v0 ' Here we check t hat these are the singular tangent planes to t he ray surface. Consider, for ex ample, the pla ne n+ . g = t'", correspondi ng to t he singular point given by (1 33 ) with 5 1 > and 53 > 0. Its equ ation also reads

°

Because t his plane is parallel to t he .r2-axis, and because t he ray surface is symmetri cal a bout the pl ane .r2 = 0, it is suffi cient to check th at in t he .r l x3-plane the line (1-15) is ta nge nt toget her to both elli pses (Ws) and (\[I .,) whose equ ations are given by (144). It is easily seen th at , in t he .t 1·r3-pla.ne, th e line (145) intercepts t he ellipse (\[I s) at a

25 single point A (say) whose coordinates g~A),g~A),g~A) are given by gjA)

=

)'I E3(EI _

E2)1/2{E2(EI _ E3W/2, g~A) = 0,

g~A) = A3EI(E2 - E3)1/2{E2(EI - E3W/2,

(146)

so that it is tangent to the ellipse (ws) at this point A (see Fig.l). The line (145) also intercepts the ellipse (W6) at a single point B (say) whose coordinates glB),g~B),g~B) are given by g;B)

=

g~B) =

Al {E 2(E I - E 2)}1/2(E I _ E2tl/2, g~B) = 0, A3{E2(E 2 - E3W/2(EI - E3)-1/2 ,

(147)

so that it is tangent to the ellipse (w 6 ) at this point B (see Fig.l).

Fig.1 Sections in the xlx3-plane of the Slowness and Ray surfaces Using (93) , (95) and (61) , it may be checked that (146) and (147) are respectively the components of the vectors g(A) and g(B) given by (97) and (98) . Thus, on Fig.l,

26

th e vector OA = g(A) represents the energy flux velocity of a wave propagating along the acoustic axis n+ , and which is polarized along the xTaxis . The vector OB = g (B) represents the energy-flux velocity of a wave propagating along the same direction 11+, but whi ch is polarized along the direction orthogonal to n+ in the x l x3-plane. Further, on multiplying (115) by 4E1EiE3, it may be seen that the equation of the ray surface may also be written in the form

{g. g - E 2(E 1 + E3)}2 - '\~(E1 - E3) 2 { (n+ . g )2 - v~ } { (n- . g )2 - v~} = 0, (148) where is the tensor defin ed by

It follows that the points where the singular tangent plane n + . g = ray surface are t he points where t his plane intersects the ellipsoid

V0

touches the

(150 )

From (149 ), it follows that n+ , which is norm al to a central circular section of the ellipsoid x . B-'x = 1, is also normal to a central circular section of the ellipsoid (150) (the two ellipsoids have the same planes of central ci rcular sections). Hence, the intersection of the plane n + . g = V0 with t he ellipsoid (1.50 ) is a circle . Thus, the singular t angent plane n+ . g = V0 touches t he ray surface in a circle. MoreO\·er. noting that

2'\~ { (g(A) x g . j )( g(B) X g . j ) + t'.~

+'\; (E1

+ E3) {( n + . g )2

- v~ }

,

it is seen that this circle is also the intersect ion of the plane n + g = cone (g (A) X g . j )( g(B) x g . j ) + t'~, g~ =

o.

gn (15 1) to.",

with the (1.5:.?)

T hi s cone gives all t he possible directions of the energy-flux \'elocity g corresponding to the singula r point s = vc;/ n + of the slowness surface . Its intersection with the .r1I3plane consists of t he lines 0 A and 0 B along g (A) and g (B), respect iwly. This cone, wit h equation (1.52), is precisely the cone of internal conical refraction introduced alread y in §4 when conside rin g propagat.ion along an acoust.ic axis. Indeed. using (1:3 1), (1:l:.?) , (146) and (1.5 1), it may be checked that., for eYE'n' angle e, the energyflu x velocity g (ao) given by ( 103) satisfies the equation (15"2) of the cone. No w, usin g th e dual ity between the slowness and ray surfaces, it is clear t hat, to t he four sin gular points (136) of the ray surfa ce, correspond t he four singular tangent pla nes m + s = ±u0 and 111 - S = ±u0 to t he slowness surface. In t he I [x3- pl a ne (see F ig.l ), t he pl ane 111 + . s = 1I ,,, touches the two sheets of t he slowness

27 surface (the ellipses (f!s) and (f!s)) at the points B' and A' (say) , whose coordinates (B') (B') (B') (A') (A') (A') 81 ,82 ,8 3 and 81 ,82 ,83 are given by

and

(154) Also, the singular tangent plane m+ ·8 = U0 touches the slowness surface in a circle, which is the intersection of this plane with the cone ( 8(B')

x

8 ·

j)(s(A') x

8·

j)

+ U~8;

=

o.

(155)

This cone intersects the XI x3-plane in the lines OB' and 0 A' along 8(B') and s(A'), respectively. By analogy with the situation encountered in optics, it may be called the "cone of external conical refraction"

10

UNIVERSAL RELATIONS

Here some universal relations involving the wave speeds or the energy-flux velocities and the components of strain or stress are obtained. These relations are universal in the sense that they are independent of the constitutive coefficients C and D, and are therefore valid for all Mooney-Rivlin materials. Some of these relations generalize relations obtained by Ericksen for principal wave speeds, in the context of the general theory of infinitesimal waves propagating in a deformed isotropic elastic material l l Waves propagating along a principal axis i,j, or k of the basic static deformation are called principal waves. When n is along d principal axis , a and b are along the two other principal axes, and we denote by Vkl, (I #- k) the squared wave speed of the finite amplitude wave propagating along the xk-axis and linearly polarized along the XI-axis (I,k = 1,2,3). Consider first an arbitrary propagation direction n( #- n± ). Subtracting pv 2 ( a) and pv 2 (b) given by (46) and (47) yields

(156) Hence, if n/(#- n± ) denotes any other propagation direction, and ai, b ' denote the corresponding polarization directions, we have

a / · B-Ia' - b ' · B-Ib '

(157)

28 This is a universal relation relating the squared wave speeds of the waves propagating in two arbitrary directions and the components of B- 1 along the polarization directions. In particular , n may be chosen along any prinipal axis of B , so that (157) also (158) which is one of the relations between the principal wave speeds obtained by Ericksen for infinitesimal waves ll . Also , multiplying pv 2 (a) and pv 2 (b) , given by (46) and (47), by b . B- l b and a· B- 1a , respectively, and subtracting, yields

Hence, using (56), and considering, as above, two arbitrary propagation directions, nand n /, we have the universal relation

[v 2 (a /)/ (a / B-la/)] - [v 2 (b /)/ (b' B-lb /)] b ' B-lb ' - a / · B- l a' 2 2 [v (a)/ (a B-la)] - [v (b)/ (b· B-lb)] = G/p.

(160)

b· B- l b - a· B-l a In particular, for propagation along the principal axes, (160) gives A~V?2 - A~V?3 AHA~

- A~)

A~V~3 - Aiv~1 A~(A~

- Ai)

Ai v51 - A~v52 AHA? - A~) ,

(161 )

which is also known to be valid for infinitesimal waves l l We note that (157) and (160) may be useful for the determination of the constitutive coefficients G and D from wave speed measurements. We now derive relations between the squared wave speeds v 2 (a) and v 2 (b ) and the shear components of the tensors T and B or B-1 Using n · Ba

= (a

x b) · B (b x n)

= -(n

B-la )(b· B- l b ),

(162)

Bb

= (a

x b ) · B (n x a)

= -( n · B-lb )(a· B-la),

(163)

n

and recalling the constitutive equation (8), we note that the shear components n Ta , n . Tb of the stress tensor T may be written n · Ta

= Gn

n · Tb

= Gn · Bb -

Ba - Dn · B -'a Dn · B- lb

= -p(b· EB-lb)( n · B-la),

(164)

= -p(a· EB-la)( n· B -lb).

(165)

Hence, it follow s from (65) and (66), using also (56), (162) and (163), that pv

2( a ) = -

n · Ta ( a · B - 1a ) =n Ta n·B - l a n · Ba(n· Bn )'

(166)

29 pv2(b)=- n · Tb (b.B- 1b)=n · B- 1b

n · Tb n · Bb(n · Bn )'

( 167)

Of course, (166) does not apply when n or a is along a principal axis of the basic static deformation, because then n . Ta = n Ba = n . B- 1 a = O. Similarly, (167) does not apply when n or b is along a principal axis of the basic static deformation. We now consider pairs of ort hogonal directions which are reciprocal in the sense of Schouten 13 : a pair of orthogonal unit vectors (n, a) is said to be "reciprocal" when a wave propagating along n and polarized along a , and a wave propagating along a and polarized along n are both possible. For waves propagating along a direction n , the two polarization directions a and b must satisfy (45). Now, the pair (n, a) is a reciprocal pair if and only if the two waves propagating along a are polarized along n and b , thus n·b=n·B- 1b=0

a· n

=a

·b

= O.

(168)

Hence, (n, a ) is a reciprocal pair if and only if b is orthogonal to both B- 1n and B-'a, which means that b is along B- 1n x B- 1a = Bb, and thus that b is an eigenvector of the tensor B . Hence, n and a are in a principal plane of B . Thus any pair (n, a) of orthogonal directions in a principal plane of the basic deformation is a reciprocal pair . We now consider a reciprocal pair and take here n and a in the Xl x3-plane. Thus b = j (b is along the x2-axis), and n B- 1n = ),~a . Ba.

(169)

Here, we denote by v~( a ) the squared wave speed of the wave propagating along n and polarized along a , and by v~( n) the squared wave speed of the wave propagating along a and pola rized along n. From (46) and (47), we now obtain , using (169),

(170)

where V8 is the speed of propagation along the acous tic axes, given by (95). Also, from t he constituti ve equation (8), we have, using (169), n · Tn - a · Ta = pv~),~(a. B- 1a - n B-1n) .

(171)

Hence, it follows that pv~( a )

a· B-I a

n · Tn - a · Ta a· B-l a - n . B-l n

pv~(n)

n . B-I n

(172)

30

For Mooney-Rivlin materials, this is a generalization of a result due to Ericksen ll for infinitesimal waves propagating along principal directions. Indeed, in the special case when n = k and a = i, (172) reduces to

PV 51

T33 -

A5

Tll

(173)

A5 - Ai

Note that (172) is valid for any pair of orthogonal unit vectors (n , a ) in a given principal plane. Also adding the two relations (170) yields the uni versal relation

More generally, we note that v~ (a) + v~( n) has the same value for any pair of orthogonal unit vectors (n, a) in any given principal plane. We now consider the energy- flu x velocities g (a) and g (b ) of the two waves propagating along arbitrary direction n , polarized along a and b, respectively. Taking the dot product of (S6) and (S7) with a and b , respectively, we have pv(a){a·g(a)}=n·Ta

,

pv(b){b·g(b)}=n·Tb.

(175 )

Using (166) and (167), this may also be written a·g(a)

n·B-1a n · Ba - v( a) a . B-1a = - v( a) n Bn '

( 176)

b g (b )

_v(b)n. B-1b = _ v( b ) n . Bb . b· B -1 b n · Bn

( 177)

We note that it also follows from (176) and ( 177) that a· g (a)

v( a)n Ba

b · g (b ) v( b )n· Bb

(17S )

Also, noting that (S6) and (S7) may be written as pv(a)g(a) pv( b)g (b)

+ D{(a · B-1a )n + D{(b· B-1b )n -

C Bn C Bn

(a· B -l n )a}, (b · B-1n )b },

( 179 ) (lS0)

a nd usin g (52) and (107), we obtain v(a)g(a)

+ v( b)g (b ) =

[n · g(a)] g(a)

+ [n · g(b)] g (b)

= fn,

(181)

where r is defin ed by pr = :.!( ' B

+ D{(trB - I )l

- S-I} = (trE) B - EB .

( 182)

31

°

°

Because C 2: and D > 0, or C > and D 2: 0, the tensor r is positive definite. As the sum {v(a)g(a) + v( b )g(b)} depends linearly on n , and as r is symmetric, uni versal relations relating t he energy-flux for waves propagating in different directions may be obtained exactly as in Hayes l2 Consider, for instance, two propagation directions n and n' , and denote by gn( a), gn( b) , gn,( a' ) and gn,( b' ) the corresponding energy-flux velocities. Using (181), we have

vn(a) [n'· gn(a)]

+ vn( b ) [n'· gn( b)]

=

vn,(a') [n· gn(a') ] + vn,( b') [n· gn,(b')]

=

n' rn,

(183)

which is valid for any pair of directions n , n l Taking now the dot product of (181) with n , we obtain

v 2 (a)

+ v 2 (b) =

(184)

n · rn .

Note that this may also be obtained direct ly from the phase speed equation (53) . Thus the sum of the squared wave speeds for a given propagation direction n is quadratic in n , and hence this sum takes constant values for n lying on coaxial cones 19 Also, if n , n' , nil is an orthonormal triad, we have, for the sum of the squared wave speeds corresponding to these three directions, {v~(a)

+ v~( b)} + {v~,(a') + v~,( b' )} + {v~,,(a") + V~,,( b")}

= trr.

(185)

Thu s this sum is the same for every orthonormal triad n , n' , nil We now consider a reciprocal pair (n , a) , for instance a pa.ir of directions in the Xlx3-plane, b being then along t he x2-axis. Here we denote by gn(a) and g.(n) the energy-flux velocities of the waves propagating along n and a, respectively, and polarized along a and n . Also, gn( b) and g.( b ) denote the energy-flux velocities of the waves propagating along n and a , respectively, and polarized along the principal direction b . Because now b· B-1n = b· B-1a = 0 , it follows from (179) and (180) that gn(a) and gn( b) and also g.( n ) and g.( b ) are all in the plane of n and a. From (175), using the symmetry of the stress tensor T , we obtain (186) Also, using (183) with n'

= a, a' = n, b' = b , yields the

universal relation (187)

11

POLARIZATION DIRECTIONS FOR A GIVEN DIRECTION OF THE ENERGY-FLUX

Let us recall that for a given propagation direction n , two plane waves may propagate. These are polarized along the directions a and b orthogonal to n and such that

32 a· b = a· B-1b = 0, or, equi valentl y (we ass ume D of 0), ( 188) Consider now a given direction m of the energy-flux velocity g = m /u . To t his direction m correspond two values of u 2 which may be obtained by solving the quadratic equat ion (11 8). Thus, two waves may propagate for a given m . Here, we determine the polarisation directions of these two waves. For t his, consider the expression (9 1)1 of the energy-flux velocity of t he wave propagating along n a nd polarized along a. From (188), we have (assuming a is not a long a prin cipal axis of B ) (189) for some scalar

0:.

(det E -l)v(a)g( a )

Inserting this into (91) yields

= o:B -1a

x (E -1a x a )

= o:{(a· B-1a)E- 1a -

(a· E-1B- 1a)a} , (190) which gives the direction of the energy-fl ux vector of the wave polarized along a in terms of a . Now, we seek the vectors a such t hat g (a ) has a given direction m . Because g (a ) is orthogonal to B- 1a , t hese vecto rs a must be such t hat (191) and are thus in the plane m· B -1 x = 0 conjugate to m with respect to the ellipsoid x · B - 1x = 1. Let p and q be two vectors in this plane, (192)

and such that (193 ) Thus, p and q are along the common conjugate direct ions of the elliptical sect.ions of the ellipsoi ds x · E - 1B-1x = 1 and x · B -1 x = 1 b~· the plane m· B- 1x = O. Then, any vector a lying in this pla ne may written as a linear combination of p and q : a = .x p

+ fl q .

(19-1 )

Insert ing th is into (190) yiel ds, on using (193), o:- I(det E - 1 )v( a )g(a) - {.xl p . E IB - 1p

+ J, 2q

+ {.x 2p . B - 1p + It

2

. E -1 B -1 q } (.\p

q . B - 1q} (A E -1 p

+ Jl q )

+ p E -1 q ).

( 195)

33 Now, the energy-flux velocity g (a) given by (195) is along m if and only if it is orthogonal to both B -1 p and B- 1q , that is )../12 {(p. B-1p )( q. E-1B-1q) - (q . B-1q)(p . B -1E-l p )} = 0, )..2/1 {(p . B-1p)(q. E-1B-1q) - (q . B-1q)( p . B-1E-1p)} =

Assuming (p . B-1E-1p )/( p . B-1p)

o.

i- (q . E-1B-1q)/(q. B-1q),

(196)

(197)

(196) yields).. = 0 or /1 = O. Thus the two possible polarisation directions a such that g(a) has a given direction mare p and q defined by (192) and (193). Thus they are along the common conjugate directions of the elliptical sections of the ellipsoids x · E-1B-1x = 1 and x· B -1x = 1 by the plane m · B-1x = 0 conjugate to m with respect to the ellipsoid x· B-1x = l. We note that the assumption (197) means that these two elliptical sections are not similar and similarly situated. It will be shown at the end of this section that they are similar and similarly situated when m is along a ray axis. Then, ).. and /1 are arbit rary and the polarisation direction may thus be any direction in the plane m · B-1x=0. We now show that the polarisation directions p and q of the two waves propagating with energy-flux in a given arbitrary direction m may be used to factor the ray slowness equation (lIS). From (192), we first note that (19S) where T/ = IB-1p

X

B -1 ql- l , because m is a unit vector. Thus, using (193h , (199)

It may then be directly checked that u 2 (p) and u 2 ( q ) given by u 2(p )

(m. B-1m)( q · E- 1B - 1q)(q . B-1qt 1 = T/2(p. B-1p )( q . E-1B-1q),

u 2(q ) =

(200) (m . B-1m )( p · E -1 B-1p )( p ' B-1p )-1 = T/2(q. B -1 q )( p . E -1 B-1p ),

are the roots of (llS), on using the relations 6

tr E -1

+

(p . E-1B -1 p )( p. B-1pt 1 + (q. E-1B -1 q )( q ' B-1qt 1 (m . E-1B- 1m )( m · B-1m t 1

(202)

34

Indeed, we check that

u 2 (p )

+

2

(q) (m · B - 1m) {(q . E -1 B -1q)( q . B - Iq r

U

l

+ (p . E -1B-Ip)(p ' B-1p)-I}

(tr E - I )m ' B - Im - m · E - IB - Im,

(203)

and

(m. B -l m )2(q . E-IB-1q)( p. E-IB-1p )( q. B-1q)-I(p . B- Ip )-l (det E-1 )(m· B - 1m)( m· B-IEm ).

(204)

Thus , (200) gives the ray slownesses of the two waves propagating with energyflu x in a given direction m. These waves are polarized along p and q. The fact that the first root (200h corresponds to th e wave polarized along p and the second (200 h to t he wave polar ized along q (and not the converse) will appear clearly from the results of §13. We may now complete t he interpretation of the ass ump t ion (197) . Indeed , from (200) it clearly appears this assumption means that the ray slowness equation (lIS ) has two different roots. Thus the direction m of the energy-flux is not along a ray ax is. When m is along a ray axis, t he equat ion (11 8) has a double root, (197) does not hold , and the ellipti cal sections of th e ellipsoids x· E - I B- Ix = 1 and x· B -1x = 1 by t he plane m· B -Ix = 0 are similar an d similarly situated. In th is case, all the waves polarized in the various directions of the plane m · B -I X = 0 have their energy-flux in the same direct ion m . However, all t hese waves have different propagation directions.

12

RESULTS IN TERMS OF THE PROPAGATION DIRECTION AND IN TERMS OF THE RAY DIRECTION

In Boulanger an d Hayes S , it has been shown that the polarisation directions a , b of t he finit e amplit ude waves corresponding to a given propagation direction n may be ex pressed in t erms of n , usi ng t he acous ti c axes n + and n - Consequent!\·, the squared wave speeds v 2 ( a) and v 2 (b ) may also be expressed in terms of n ..".ft er presenting anot her deri vation of these resu lts, we show here that similar results may be obtained for t he polar isation directions p , q of the finite amp litude wa\"es correspond ing to a given ray di rection m . These may be exp ressed in terms of m , now using the ray axes m + and m - Consequ entl y, the sq uared ra y slownesses ,,2(p) and u 2 (q ) may also be ex pressed in terms of m.

(a) R es ults in teI'I11S of a give n propagation dil'ect io n

35 The results obtained ins were derived using the fact that B- 1 may be written as

where n+ and n-, given by (93), are unit vectors in the directions of the acoustic axes of the deformed Mooney-Rivlin material. On the basis of this, the polarisation directions a, b characterised by n . a = n . b = 0 and a· b = a . B- 1 b = 0 were obtained in terms of n . The unit vectors a and b are along a* and b* , respectively, given by (see also Boulanger and Hayes 17 ) (206) with

oi = (n x n±) . (n x n±) = A~(n x n±)· B-

1

(n x n±) .

(207)

Note that o± = sin
= 4A12 A3 2 + (A3 2 -

A12)2( cos 2
+ cos

2


+ 2U3 4 -

A14) cos
(209) so that inserting this in (46) and (47) yields expressions for the squared phase speeds v 2 ( a), v 2 (b) as functions of the propagation direction (more precisely, in terms of the angles that the propagation direction n makes with the acoustic axes). Here, in order to be able to exhibit the duality between slowness and ray velocity, we derive other expressions for v 2 ( a) and v 2 (b) as functions of the propagation direction, using the tensor E defined by (62). For this we use the expressions (65) and (66) for v 2 (a) and v 2 (b). We first note that (205) is equivalent to (210) with ~

= A12(E1 - E 2) + A:;2(E2 - E3) = A~(E1 - E3).

(211)

Hence, for o± given by (207), we have

oi =

A;(n x n±) . B- 1 (n x n±) = E:;lA;(n x n±) . EB- 1 (n x n±) .

(212)

36

Then , using the results of17 , we obtain (b* EB-Ib*)( b *· B-1b*) -' = E2

+ (1/2)A22~(0+,L + 'lj;)/( n

Bn ).

(2 13 )

with 'lj; defined by

'lj; =

A~(n

x n+ )· B-1(n x n- ).

(2 14)

2

Thus, the expression (65) for v (a ) yields v2( a)

(n · Bn) (b* . EB-Ib*)( b * B-Ib*r l

E 2n· Bn

+ ( 1 /2)A22~('lj; + 0+0_).

(2 15)

Because with

we finally obtain 2v 2(a) = (EI

+ E3)n . Bn -

~ { (n+ Bn )( n- · Bn ) - A220+L}.

(2 18)

2

Similarly, t he express ion (66) for v (b ) yields

2V2( b) = (EI

~ {(n+. Bn )( n- Bn ) + A~O+ O_ }

+ E3)n · Bn -

(2 19)

Here o± and ~ are defined by (212) and (211) , respectively. Thus , for a given propagation direction n , the corresponding amplitudes a and b are along a* and b * given by (206) and the corresponding phase speeds v( a ) and v( b ) are given by (218) and (2 19). Then using (9.5), and noting that 2n+ . Bn = (Ai 2n - . Bn =

+ A~) cos r.p+ + (Ai -

(Ai -

A~ ) cos r.p+

A~ ) cos r.p - ,

+ (Ai + A~) cos r.p- ,

(220)

it may be checked that (218) (219) yield the res ults obtained ins for the squared wave speeds as fun ctions of the angles r.p + and r.p- that n makes with n+ an d n- . We note that (218) and (2 19) are analogous to results obtained by Boulanger &:- Hayes 20 for electromagneti c waves (see equ ation (10.13) of this paper) , and which generalize classical res ults obtained by Neumann for crystal optics. R emark Expressions (206) for the polar isation directions a* , b* are valid in the general case when the propagation direction n do es not lie in t he place of n + a nd n - When n lies in the plane of n + and n - (i.e. t.he plane of i and k ), (206 ) has to be replaced by l7

(22 1) but (2 18) and (2 19) remain valid .

37 (b) Results in terms of a given ray direction Now we derive similar results for the polarisation directions p , q and the squared ray slownesses u 2 (p), u 2 (q) corresponding to a given ray direction. We first note the identity (222) with (223) which may be checked directly using (61) and (137). This identity is the dual of (210), using now the ray axes m± instead of the acoustic axes n± (note that in (210) ~ is positive, whilst in (222) ( is negative). We note in passing that (222) shows that the plane orthogonal to a ray axis cuts the ellipsoids x· E -1 Bx = 1 and x· Bx = 1 in a pair of similar and similarly situated ellipses . Now recalling (192) and (193), we note that the directions B-1p, B-1q are characterized by m·B-lp = m ·B-1q = 0 and (B-1p )· E-1B (B-1q) = (B-1p )· B (B -1 q ) = 0: they are both orthogonal to m and are conjugate with respect to both ellipsoids x· E- 1Bx = 1 and x · Bx = 1. Thus, the unit vectors p and q are along p* and q *, respectively, given by (224) with 'Yi = E2A22(m x m± ) . E-1B(m x m± ) = >'2 2 (m x m±) . B (m x m±).

(225)

We note that (224) and (225) are the duals of (206) and (215), respectively: (206) and (212) give the polarisation directions a *, b* corresponding to a given propagation direction n whilst (224) and (225) give the polarisation directions p *, q * corresponding to a given ray direction. In order to derive expressions for the squared ray slownesses u 2 (p ) and u 2 (q) as functions of the ray direction, we now use the same procedure as for obtaining the expressions (218) and (219) for the squared phase speeds v 2 (a) and v 2 (b). Thus, using again the results ofl7 , we obtain

with () defined by () = >'2 2 (m x m+) . B(m x m- ).

(227)

2

Thus, the expression (200h for u (p) yields (m . B-lm)(q* E-IB -l q *)( q* B-lq*tl E;l(m· B-1m) - (1/2)>'~(h+'Y-

+ ()).

(228)

38 Because

with we finally obtain 2u 2 (p ) = (Ell

+ E3 l )m ' B -' m + ({ (m+ . Bm)( m-

. Bm) - Ah+'I'-}

(23 1)

Similarly,

2u 2 ( q ) = (El l

+ E3 1 )m

. B-'m + ( {(m+ Bm )( m · Bm ) + A~'I'+ '1'- }

(232)

Equations (231) and (232), wit h ( given by (222) are t he dual counterparts of equations (218) and (219). They give the squared ray slownesses as fun ctions of the ray direction m . Remark Expressions (224) for the polarisation directions p ', q ' are valid in the general case when the ray direction m does not lie in the plane of m+ and m- When m lies in the plane of m + and m - (i .e. t he plane of i and k ), (224) has to be replaced by 17 (233)

but (23 1) and (232) remain valid.

13

RESULTS IN TERMS OF THE POLARISATION DIRECTION

Here we show t hat the phase speed v, the slowness vector s , the ray slowness u, and the energy-fl ux velocity vector g of a finite amplitude wave propagating in the deformed Mooney-Rivlin materi al may a ll be expressed in terms of the polarisation direct ion of this wave (ass uming that this direction is not along a principal axis of the basic static deformation). Consider first the wave propagating along n and polari zed along a. Its phase speed v( a) is given by (65). Now using (189) , which is vali d for any a which is not along a principal axis of B , we note that

(~3-!)

b · EB - 1 b = a 2 (det E ) {(a , EB - 'a )( a ' E - 1 B - 1 a ) - (a· B - 'a )'} Also, IB - 'a x: EB - 1 a1 2

-

IB - 'a x EB - 'a I2 =

{a . (B -1a x: EB - 1 a) Q -2 ,

r (235)

39 on noting that pEB-la = C B- 1a + Da . Hence, with (234) and (235), (65) becomes 2 (det E )(a . B-1a) {(a. EB-la)(a . E - l B-1a) - (a· B- l a )2} v (a) = la x (B-la x EB la)12 .

(236)

This expression gives v 2 (a) in terms of a alone. Moreover, from (190), we obtain, using (235), v (ag a = ± (d et E ) ) ()

B-la x (E -la x a) . la x (B-la x EB-la)1

(237)

This expression, together with (236), gives g (a) in terms of a alone. The (±) sign is the sign of a in (189). If a is changed into -a, t hen b = n x a is changed into -b , and the sign of a in (189) is changed . Thus, it is always possible to choose the orientation of a such that (237) holds with the + sign. Consider now the wave propagating with energy-flux velocity along m and polarized along p. Its ray slowness u(p) is given by (200h- From (193), we have (assuming p is not along a principal axis of B ) (238)

where j3 is a scalar such that q . q = 1. Using this, we note that

Also, using (238) and (198), and recalling that m · m = 1, we have IB-'p x (p x E- l p)1 2 = j3-2I B- l p

X

B- l ql 2 = j3- 2 'Tf - 2

(240)

Hence, with (239) and (240), (200h becomes 2 (det E- l )(p . B-lp) {(p . EB-lp )( p . E- 1 B-lp) - (p . B-lp)2} U (p ) = IB-lp X (p x E-lp )12 .

(241)

This expression gives u 2(p ) in terms of p alone. We now show that the slowness vector s (p) of the wave polarized along p may be obtained in terms of p alone, by a formula analogous to (237) . First, from (189), we note that the slowness s (a) of the wave propagating along n = a x b and polarized along a is (242) Then, using (235) and (236), we obtain a x (B-la x EB - la) s (a) = ±

(det E )1/2( a . B- l a )1/2 { (a · EB- l a )( a . E-l B-la) - (a· B- l a )2}

1/2' (243)

40 In (243), a may be chosen to be any polarisation direction . Thus , applying this with a = p , and using (241), we finall y obtain u( p )s (p ) = ±(detE

_I P X (B - Ip x EB - Ip ) ) IB - lp x (p x E-lp )I'

(244)

Thi s expression, toget her wit h (241) , gives s (p ) in terms of p alone. We note that equations (236) and (241) may be formally read off one from the other by the subst it utions v( a ) +-> u( p ), B +-> B - 1 , E +-> E- I , a <-> B-Ip . The same duality applies to equations (237) and (244), wit h g (a ) +-> s(p ), but the sign has to be changed . Thus, choos ing an arbitrary direction d (say), and using (236) (237) wit h a = d , and (24 1) (244) wit h p = d , we obtain the phase speed v( d ), the ray slowness u( d ), the slowness vecto r s(d ), and the energy-flu x velocity vector g (d ) of the wave polarized along the direction d . Thus, the properties of a wave propagating in the deformed Mooney-Rivlin material are uniquely determined by its polarisation direction (assum ing that this direction is not along a principal axis of the basic static deformation). As a check, we note that for a = p = d , (237) and (244), together with (236) and (241), yield vecto rs g (d ) and s( d ) satisfying g (d )· s (d ) = l. Indeed, from (237) and (244), we have v( d )u( d )g (d ) . s(d )

(d B -ld){ (E - 1d x d ) . (EB-Id x B -1 d )} Id x (B - Id x EB Id )II B -l d x (d x E- ld )1 (d · B - 1d){ (d · EB -Id)( d · E-1B-ld ) - (d · B - l d )2} Id x (B- Id x EB -l d )II B- 1d x (d x E- ld )1 v( d )u( d ), (245)

on using (236) and (24 1). Also , from (237), we note that v2( d )lg (d W = (det E )2 IB - Id x (E-1d x ~ W = l,2( d )/ u 2(d ) Id x (B -I d x EB- d W '

(246)

on using (236) and (24 1). Similarly, from (244) , 2 u (d )ls(dW = (det

Et2IfB~I(:~~~d XX :~~~ir

2

2

= u (d )/v (d ).

(247)

This clearl y confirms that in (200), the first expression indeed corresponds to the wave polarized along p (and not to t he wave polarized along q ). Finally, we note that (244) may be written in a simpler form using both polarisation directions p and q corres ponding to a given m . Indeed , it fo llows from (244) a nd (238) th at s( p ) is along p x E - 1q. But s( p )· g(p ) = l. Thus s( p )· m = u( p ). Usin g th is, and recallin g (198) and (200) where 1J = IB -1 p " B -l q l-I, we obtain

41

Similarly, we have (249) Equations (248) and (249) are the counterparts of equations (9 1h and (91h in the duality v( a) f-> u(p), v( b) f-> u(q), B f-> B- l, E +-t E-I, a f-> B - lp , b f-> B-lq. (The reason no denominator appears in (9 1) is that la x bl = 1).

14

CONCLUSIONS

We conclude by summing up the main results, formulated in terms of the tensors B (left Cauchy-Green strain tensor of the basic static deformation), and E = (Cjp) l + (Dj p)B (related to B , but depending also on the material constants C and D of the Mooney-Rivlin material). We present results for a given propagation direction n and results for a given ray direction ffi . The duality between the two sets of results (n f-> m,B f-> B-l , E ...... E-l , a f-> B-lp, b ...... B - lq,v( a ) f-> u( p),v(b ) f-> u(q),g(a) f-> s (p) , g(b) ...... s (q )) is striking. Finally, given a polarisation direction d which is not along a principal axis of B , we have shown that there is just one wave (up to a reversal of the propagation direction) whose properties may be expressed in terms of d. These results are also recalled here.

(I) RESULTS FOR A GIVEN PROPAGATION DIRECTION n (n · n = 1) For a given propagation direction n , two transverse plane waves may propagate, with polarisation directions a and b , wave speeds v( a) and v(b), energy-flux velocities g (a) and g (b ). The polarisation directions a and b , which are easily characterized using the tensors B - 1 and EB- l , may also be given explicit expressions in terms of the acoustic axes n± (the basic static deformation is assumed to be triaxial with principal axes i, j , k and principal stretches Al > A2 > A3). Here the appropriate equations are recalled.

Polarisation directions (a , b : unit vectors) (Recall (188)) (250)

n · a=n · b=O

(or a · B-lb=a·b=O) Wave speeds (Recall (65) and (66))

v 2 (a) = (n Bn )( b · EB-lb)( b · B-lb t l

(or v 2 (a) = C n Bn

+ Da· B-la),

v 2 (b ) = (n· Bn )( a · EB-la)( a· B-lat l

(or v 2 (b) = C n · Bn

+ Db· B-lb). (251)

42 Energy -flux Velocities (Recall (91 )) v( a )g (a )

= B -1 a

x EB - Ib

, v( b )g (b )

= EB-Ia

x B-1b.

(252)

(note t hat la x b l = 1) Acoustic A x es (n ±: unit vectors) (Recall (2 10 ), (2 11 ) and (131)) EB - I = E 2 B- I

+ (U2)(n+ @ n - + n- @ n +) ,

(253)

wit h

(note that a = 1)

(254)

Explicit Expressions for the Polarisation Directions Using n ± (Recall (206), (207) and (22 1)). T he unit vectors a and b are along a " and b " given by (1) if n is not in the plane of n + and n- : r-I n x n - , WIt ' h vr2± = "2 \ 2( n x n ±) · B-I ( n x n ±) . ab "* } = V+ 0-1 n x n + ± v_

(255)

(2) if n is in the plane of n + and n -: (256) (II) RESULTS FOR A GIVEN RAY DIRECTION m (m · m = 1) For a given ray di rection m , two t ransverse waves may propagate, with polarisation directions p and q , ray slow nesses u( p ) and u( q ), slowness vectors s(p ) and s( q ). T he polarisation direct ions p and q , which are easily characterized using the tensors B - I and E - I B - 1 , may also be given expli cit expressions in terms of the ray axes m ± (the basic static deformation is also ass umed to be triaxial ). Here the approp riate equations are recalled. Polarisations Directions (p , q : uni t vecto rs) (Reca ll (192) and (193)) (257) R ay Slow n esses (Recall (200)) u 2(p ) = (m · B - 1m )( q· E -I B - Iq )( q . B - 1q )- I, (:::'58)

43 Slowness vectors (Recall (248) and (249)) u(p)s(p)

= p x E - l q / IB- l p x B- l ql ,

u(q)s(q) = E- l p x q / IB- l p x B- 1 ql . (259)

Ray Axes (m± ; unit vectors) (Recall (222), (223) and (134))

(260) with ( = Ai(E]1 - Ei l ) + A~(Eil - E;I),

m± =

f.l f>

l ) 1/ 2 l )1/2 } A-l ( E-2 I _EA-I ( E-3 I _E1 i ± _1_ 2 k l A-I E _ E-l \-1 E-l E-l { 2 3 1 "2 3 1 _3_

(note that

/3 #- 1)

(261) Explicit Expressions of the Polarisation Directions Using m± (Recall (224), (225) and (233)) The unit vectors p and q are along p ' and q ' given by (1) if m is not in the plane of m+ and m-; l

p' }=

BB-l q ,

-1 + - 1 -' 2 - 2 ± ± 1+ m x m ±I_ m x m , WIth I ± = A2 (m x m )· B (m x m ). (262)

(2) if m is in the plane of m+ and m- ; (263)

(III) RESULTS FOR A GIVEN POLARISATION DIRECTION d For a given direction d which is not along a principal axis of B , there is just one wave (and its corresponding wave propagating in the opposite direction) which is polarized along d . All the properties of this wave are characterized by its wave speed v( d ), ray slowness u(d) , slowness vector s(d ) and energy-flux velocity g (d ). Wave Speed (Recall (236))

V

2 (detE)(d· B- l d ) {(d. EB-1d )( d . E- I B- l d ) - (d· B- l d )2} (d ) = Id x (B-1d x EB-ld)12 .

(264)

Ray Slowness (Recall (241)) (det E- l )( d . B- l d ) {( d . EB- l d )( d . E-l B- l d ) - (d · B- l d )2} u (d ) = IB- l d x (d x E-ld)12 2

(265)

44

Slowness Vector (Recall (244)) -I

u( d )s( d ) = ±(detE

d x (B-1d x EB-ld ) ) IB-1d x (d x E-1d )I'

(266)

Energy-flux Velocity (Recall (237)) (267)

Acknowledgment Su pport from the F.N .R.S . (Belgium) for "Mission Scientifique de Bruxelles" (M. H. ) is gratefully acknowledged.

a l'Universite

Libre

References 1. Beatty, M.F., Appl. Mech . Rev. 40 , 1700-1734 (1987).

2. Green, A.E. , J. M ech. Phys. Solids 11 , 11 9-1 26 (1963). 3 . Carroll, M.M ., Acta M ech. 3 , 167- 181 (1967). 4. Currie, P. and Hayes, M. , J . Inst. Math s Applics 5 , 140-161 (1969). 5. Boulanger, Ph . and Hayes , M. , Q. Jl M ech. appl. Math . 45 , 575-593 (1992). 6 . Boulanger, Ph . and Hayes, M., Q. Jl M ech. appl. Math . 48 , 427-464 (1995). 7 . Born , M. and Wolf, E. , Principles of Optics, 6th edition (Pergamon , Oxford 1980) . 8 . Musgrave, M.J .P., Proc. R . Soc. Lond. A401 , 131-143 (1985). 9 . Landau, L.D. and Lifschitz, E.M. , Electrodynamics of Cont inuous Med ia (Pergamon, Oxford 1960). 10. Musgrave, M.J .P. , Crystal A coustics (Holden-Day, San Francisco , 1970). 11 Truesdell , C. , Arch. Ration. Meck . Anal. 8 , 263-296 (196 1). 12 Hayes, M., J . Acoust. Soc . Am. 56, 1-3 (1974).

45 13 Schouten, J.A., Tensor Analysis for Physicists, 2nd edition (Dover, New-York

1989). 14 Boulanger, Ph. and Hayes, M., Z. angew. Math . Phys. (to appear). 15 Ogden, R.W., J. Mech. Phys. Solids 18, 149-163 (1970) . 16 Ericksen, J.L., J. Ration. Mech. Anal. 2,329-337 (1953).

17 Hayes , M. and Rivlin, R.S., Z. angew. Math. Phys. 22 , 1173-1176 (1971). 18 Hayes, M., Proc . R. Soc. Land. A370 , 417-429 (1980).

19 Boulanger, Ph. and Hayes, M., Q. Jt Mech . appl. Math. 44 , 235-240 (1991). 20 Boulanger, Ph. and Hayes , M., Arch. Rational Mech . Anal. 116, 199-222 (1991). 21 Boulanger, Ph. and Hayes , M. (submitted for publication)

47

DECAY ESTIMATES FOR BOUNDARY-VALUE PROBLEMS IN LINEAR AND NONLINEAR CONTINUUM MECHANICS

C. O. Horgan School of Engineering and Applied Science University of Virginia Charlottesville, VA 22903, (USA)

48 1. INTRODUCTION

Considerable interest has developed in recent years in investigating the spatial decay of solutions of linear and nonlinear (primarily elliptic) partial differential equations. The impetus for much of this work arose from the need to properly formulate and analyze the celebrated "Saint-Venant's Principle" of elasticity theory. Extensive reviews of this research may be found in Fichera,20.21 Horgan and Knowles 48 and Horgan. 43 Earlier work is reviewed by Gurtin. 35 The problems of primary interest in 43 •48 are elliptic boundary value problems on long cylinders or strips, subject to homogeneous Neumann-type boundary conditions on the lateral surfaces (corresponding to traction-free boundaries in the elasticity context). As explained in detail 48 , for linear problems, superposition can be used to reduce the issues underlying SaintVenant's principle to consideration of the solution (and gradient of solution) of the relevant elasticity equations subject to arbitrary self-equilibrated tractions at one end of the cylinder or strip, with the remainder of the boundary being traction-free. One would expect that such solutions and their gradients would decay with distance from the loaded boundaries. Since explicit solution of many of the problems of linear elasticity (and more so, for nonlinear elasticity) is not feasible, qual itative arguments have been successfully developed to provide estimates for the rate of stress decay. Notable among the early results of this type are the energy-decay estimates of Toupin 11 9 for the three-dimensional linear elastic cylinder problem and of Knowles75 for linear plane elasticity. These results predi ct at least an exponential decay of stress and provide estimated decay rates which are lower

49 bounds on the actual rate of decay. For many second-order elliptic equations, these results are optimal in the sense that the estimated decay rates coincide with the actual decay rates. The foregoing results have been discussed extensively in previous review articles 20 •2I ,43,48, together with further results on the cylinder problem by Berdichevskii 7 and on the bihannonic problem (arising in isotropic plane elasticity) by Oleinik 104, \05, Oleinik and YosifianlO6-\08. More recent work on the cylinder problem may be found in 30,71,85,86,91,99,100,118 and on the bihannonic problem in 80-87 . Saint-Venant type principles for domains of general shape, including unbounded regions,

are also briefly discussed in 48 . For recent treatments of these classes of problems, see e.g. 74

For the connection between Saint-Venant's principle

and error estimates for plate and shell theory, see e.g. 32-34,88-90. In most of the studies alluded to above, the spatial decay of solutions is

examined under some a priori decay assumption at large distances from the boundary. Such an assumption occurs naturally when investigating SaintVenant's prinCiple (see e.g. Section 2 below). Recently, a number of investigations 28 ,29,S3,60,6l ,73,11I

have been concerned with establishing

asymptotic behavior in the absence of such an assumption, and the results are usually obtained in the fonn of growth or decay alternatives. These results are thus of the Phragmen-Lindelof type and are of particular interest for nonlinear problems. In this paper. we review a variety of recent results of both the classical Saint-Venant decay and Phragmen-Lindelof type. We begin (in Section 2) with a second-order linear elliptic partial differential equation. with variable coefficients. defined on a two-dimensional domain. Such equations arise in the theory of steady-state heat conduction for inhomogeneous anisotropic

50

materials as well as in the theory of anti-plane shear deformations for a linear inhomogeneous anisotropic elastic solid. Special cases include equations with constant coefficients and equations in divergence form not containing the mixed partial derivative. Further specialization leads to Laplace's equation, which was used48 •43 as a model equation to illustrate, in a particularly simple setting, several of the relevant issues arising in the analysis of Saint-Venant's principle. For the more general equation of concern here, we briefly describe the Saint-Venant principle and its connection with that of Phragmen-Lindel6f type. The domain is then specialized to be a semi-infinite strip and homogeneous boundary conditions (corresponding to traction-free conditions) are assumed on the long sides. Recent analysis of Horgan and Payne 61 is summarized leading to a theorem of Phragmen-Lindel6f type, providing estimates for the rate of growth or decay which are optimal for the case of constant coefficients. illustrative examples described in detail in Horgan and Payne 61 are reviewed. The results are applied to assess the influence of inhomogeneity and anisotropy on the decay of Saint-Venant end effects in

anti-plane shear. In Section 3, analogous results on the effects of anisotropy on stress decay for plane homogeneous elasticity are briefl y summarized. A detailed review of such results has been given recently67 In Section 4, a wide variety of nonlinear issues are discussed. First, a review of results on the formulation and analysis of Saint-Venant's principle in nonlinear elastostatics is provided. More general decay (and growth) results for second-order quasilinear and semi linear partial differential equations are also described. Spatial decay estimates for the stationary Navier-Stokes equations, with application to entry flow problems, are then revi ewed. Section 4 concludes with a summary of recent results of Horgan and Payne 60 on a different, but

51 related. issue. The asymptotic behavior of harmonic functions on a threedimensional semi-infinite cylinder was examined60 where the usual homogeneous Dirichlet boundary condition on the lateral surface is replaced by nonlinear boundary conditions. These boundary conditions are of the heat loss (or heat gain) type when the problem is interpreted in the context of steady-state heat conduction. It is shown60 that polynomial growth (or decay) or exponential growth (or decay) may occur. depending on the form of the nonlinearity. In Section 5. some recent results on spatial decay for timedependent problems (e.g. parabolic problems). are briefly described. The paper is a review and an extensive list of references to original work is given where details of the analyses may be found.

2. LINEAR SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

The traction boundary-value problem for anti-plane shear deformations of a linearly elastic inhomogeneous anisotropic solid has been shown49 . 45 to reduce to the problem of seeking classical solutions u (XI.

X2)

of the linear

second-order partial differential equation (1)

(where D is a bounded. simply-connected plane domain) subject to the boundary condition aaf3 u. a nJ3 where t = t (x I.

X2)

=t

on aD •

(2)

is a prescribed function on aD. Here we have used the

usual convention of summing over repeated Greek indices from 1 to 2 and the comma indicates partial differentiation with respect to the corresponding

52 Cartesian coordinate. We assume that the coefficients

~ (Xl, X2)

are

continuous in D and satisfy (3)

where, in the variable coefficient case, (3) is understood to hold at all

(Xl, X2)

in D. We observe that the conditions (3) ensure that (1) is elliptic. The

conditions (3) are a consequence of the usual assumption made in linear elasticity that the three-dimensional strain-energy density of the material is positive definite. When the elastic material is homogeneous, the aajl are constants. Second-order partial differential equations of the form (I) also arise in the theory of steady-state heat conduction for inhomogeneous anisotropic materials. In this context, u is the temperature field and

a(l~

components of the heat conductivity tensor. As described in

are the

49.45,

such

equations also arise in the theory of anti-plane shear deformations for a linearized inhomogeneous anisotropic elastic solid, in which case u is the axial displacement field and aajl are the components of the elastic stiffness tensor.

Anti-plane shear models with coefficients

a(l~

which vary

continuously with position have been used in fracture mechanics to model damage and local crack tip process zones. Elasticity models with variable elastic coefficients are also widely used in geophysics. One of the main motivations in

61

was the desire to assess the influence of material

inhomogeneity on the decay of Saint-Venant end effects. Of course the class of elasticity problems governed by a single second-order partial differential equation is limited. Nevertheless, the analysis provides some guidelines for investigating the effects of inhomogeneity in the more complicated problems

53 governed by systems of second-order partial differential equations or by fourth-order equations. In the case of constant coefficients, the role of the second-order problem in elucidating the issues underlying Saint-Venant's principle in a simple setting has been discussed extensively in

43,48 .

The

boundary-value problem (I), (2) for the displacement u is one of obliquederivative type for a second-order linear partial differential equation. On employing the divergence theorem , it is readily shown that the prescribed traction t must be such that

J

aD

t

ds =0

(4)

for a solution of (1), (2) to exist Of course, (4) also follows from the restriction on the data (2) necessary for overall force eqUilibrium . With the above hypotheses, standard arguments show that the solution to the above boundary-value problem is unique to within a constant (corresponding to a rigid-body translation). The only non-zero in-plane stresses arising due to an anti-plane shear are Ta = aaj3 u , ~, (u= 1,2) .

(5)

Suppose now that D is the interior of a rectangular domain D = {(xl, X2) I 0 < Xl < I , 0 < X2 < h) , and consider the special class of

problems for which the long sides are traction-free and the ends Xl

=0,

Xl

=I are subjected to prescribed tractions so that (2) now read aa2 u ,a = 0 on x2 = 0, h; 0::; Xl ::; I ,

where gl and g2 are sufficiently smooth, and, by virtue of (4), must satisfy

(6)

54 h

h

fo g\ (X2) dX2 = f0 g2(X2) dX2 .

(8)

A "Saint-Venant" principle for (I). (6). (7) concerns a comparison of u (or. more physically. the stresses T a) with the solution

u to the problem obtained

by replacing g\. g2 with the statically equivalent pair g\. g2 for which b

h

fo g\ (X2) dX2 = f0 g\ (X2) dX2 .

(9)

By superposition. this comparison may be done by considering (1). (6). (7) under the assumption that g2 == 0 and

(10)

Henceforth write

gfor g\.

Thus one is concerned with estimating the solution

(and gradients of solutions) of (1). (6). (7) for g2 = 0 and where the prescribed boundary data

g== g\

satisfies

the

self-equilibration condition (10).

Intuitively. one would expect rapid decay of solution from the end X\ = O. It has been shown61 that. for a wide class of coefficients aajl(x). this decay is in fact exponential, thus quantifying the anticipated boundary layer character of the solutions. For the linear problem of concern here. the rate of decay of end effects from either end X\ = 0 or XI = I is the same and so. without loss of generality. 0

can be taken henceforth to be the semi-infinite strip

0= (XI. X2) I XI > O. 0 < X2 < h) . In this case. the stipulation that g2 = 0 in

the second boundary condition in (7) is now to be interpreted as an a priori decay assumption as XI -t

00 .

Thus. in the context of Saint-Venant's

principle. the adoption of such an assumption is a natural one. As pointed out in the Introduction. there has been a developing interest recently in establishing more general results of Phragmen-Lindelof type where such an a

55 priori assumption is not made. In this case, the results obtained are generally in the form of growth or decay alternatives.

Such a result has been

established recently61 for the equation (1). In what follows, we review these results. As shown in61 , since (1) is in

divergence form, a conjugate function may be introduced and so it is sufficient to consider (1) subject to the Dirichlet boundary conditions (11)

(12) where the prescribed end data is assumed to have a square-integrable first A

derivative on [O,h] and to satisfy f(O)

A

= 0, f(h) = O. No a priori assumption is

made about the asymptotic behavior of u as

XI ~

00,

so that a Phragmen-

Lindelof type alternative is established 61 . The usual energy method

43,48

for deriving a Saint-Venant type decay

estimate for finite energy solutions of (I), (I I), (12) involves the energy E(z) stored beyond the position

XI

~

E(z) =

= z; it is defined by

h

f f aap u,a u,p dA,

z~

XI

(13)

>0.

z 0

This function can be shown to satisfy a first-order differential inequality whose solution furnishes an explicit decay estimate for E(z), showing in particular for constant aap that E(z) must decay exponentially (see e.g.

10 1,

pp.

33-35). However, in contrast to the situation for Laplace's equation, for example, the estimated decay rate, a lower bound for the actual decay rate, is not optimal. The actual decay rate may be readily obtained using separation of variables. It is given in (34) below. To motivate the method used

61

to

treat problem (1), (11), (12), the constant coefficient case was first considered

56 and a technique was developed which would yield the optimal PhragmenLindelof growth and decay rates for an appropriate energy. This method was then generalized to the variable coefficient case 61 We note that when the

aa~

are constant. the change of variables (14)

transforms (1) into the Laplace equation in the

(C,. ll) variables. and any

bounded rectangular region in D is transformed into a parallelogram in the

(C,.

ll) plane. Then. using the standard energy arguments for harmonic functions for the energy in the strip to the right of C, = constant. one is led to a Phragmen-Lindelof result in which the growth and decay rates are optimal . We note that in the case of constant coefficients

a~.

the curve C, = const. is

just a line (15) on which a12 f, I + a22 f.2 = O. For the variable coefficient case. the family of curves (15) is also used but now f is to satisfy (16)

It is the family defined by (15). (16) that plays an imponant role in the investigation described in

61 . 61.

Geometric propenies of the family of curves (15). (16) are From the theory of first-order partial differential equations it

is seen that along any such curve

(17)

The aap

(XI. X2 )

are all continuous. a22 is positive and it is assumed that the

quotient 31 2/a22 is bounded and Lipschitz continuous in

XI

for

( Xl . X2) E

D.

From the theory of ordinary differential equations it is known that through

57

each point of D there exists one and only one solution f (x I. X2) = constant. Thus curves f (XI. X2) = constant do not intersect one another. Since dX2/dxI does not vanish in D this means that if the curves are labeled by a parameter q as (18)

then for some range of values of q each curve of the family will enter and leave R through opposite long sides of the strip. e.g. enter through x2 = 0 and exit through x2

=h.

It is assumed also that f.1 E

(19)

C (D) .

The notation (20)

and (2 1)

is also introduced.

The following result is established in 61 :

Theorem 1. Let U(XI. X2) be any solution of (1). (11). (12) with C(D)

coefficients satisfying (3) and the quotient a12/a22 bounded and Lipschitz continuous in xI. Then,for the family of functions f (XI. X2) satisfying (16), (19) and a region Dq defined by (20), either

lim q ..... ~

[

f

aap u.!l u.P dA] exp ( - 21t

DIDq

for some positive constant K or

J [B" (t) AM (tW

qo

I

dt )

~K

(22)

58

f an~ U,n u,~ dA:S; [f ~ U,n u,~ dA] exp ( - 21t J [B\t) AM(t)r l dt ) .

~

~

~

(23)

Here qo is the smallest value of q for which the curve [(XI. X2) = q enters and exits D through X2 = 0 and X2 = h. and B' and AM are given by h

B·(q) =

f (I aii dX2.

(24)

a

(25)

where all quantities are evaluated on Lq. and A has been defined in (3) . Note that the coefficient a 11 (x I. X2) is not involved in the definition of the family of curves f = constant. Also in the differential equation (1) only the XI-derivative of all occurs. Thus the theorem would remain valid under less restrictive assumptions on all. The main concern 61 is with the case in which the solution decays as XI (or q) tends to infinity. i.e, with inequality (23). For the special case in which (26)

where max

O<X2 < h

I g(x2) I = G .

(27)

from (23) for XI > 2G. one obtains the explicit result

/

an~ u,n u,~ dA:S;

[J a~

u,n

".{ -h

u,~ dA 1

'T

(28)

[B'(t) AM(t)r'

dtf

In passing from (23) to (28) a number of inequalities have been used which have the effect of increasing the multiplicative factor in the decay estimate;

59 however they do not alter the estimated rate of decay as a function of XI. Of course, in specific examples, sharper results follow on using (23) as it stands. Another special case of interest is when al2 = 0 in (1) . From (17), it is seen that the curves f XI

= constant.

= constant

are now just the vertical straight lines

Thus g == 0 in (26) and G == 0 in (27) and (28). In this case, the

decaying exponential on the right in (28) has the form

(29)

where

(30)

and (31)

Equations of this special form occur in numerous applications of linear continuum mechanics. A Saint-Venant principle for such equations was investigated in 76,121 See 48 for a discussion of application of these results to the axisymmetric torsion of elastic shells of revolution. Returning to the general situation where al2

* 0, we note that in the

special case of constant coefficients the inequality (23) becomes

where XI and X2 lie on L q • There are two cases to consider:

60 1) Case 1, al2

> O.

'10 = O. and X2

In this case

of (32). Also in this case

may be replaced by h on the right hand side

Oq::> OXl + alzh/a22 '

Thus replacing the variable

- 21t 822X l

41ta12

J aajl u.

Cl

u, ~ dA ~

D~

[

J aajl

U,Cl

0

u, ~ dA ] e

A

e

bA

(33)

The exponential decay estimate (33) shows that solutions of (1). (11). ( 12) decay in energy norm at least as fast as e- 2ki l • with estimated decay rate k given by (34) The decay rate (34) is optimal, as may be readily verified by solving ( 1) by separation of variables. 2) Case 2, al2 < O. I a121 h

In this case q = - - - . and

o

by zero. Since

Oq ::> OXl

au

X2

may be replaced on the right of (32)

one finds in this case that. for

Xl ~

I a121 h/au.

again an explicit inequality with the optimal decay rate (34). If al2 = O. the estimates (33) and (35) are identical and yield the estimated (and exact) decay rate

(36)

valid for an orthotropic homogeneous elastic solid in anti -plane shear. The

61

isotropic case can be found on then letting a22 = a1l (= 11) so that (1) reduces to Laplace's equation and (36) yields

k=~

(37)

h

which is the well-known exact decay rate for harmonic jUnctions on the semi-infinite strip D. Suppose now that the homogeneous material is highly anisotropic so that (38)

This may be viewed as modeling the case of a fiber-reinforced material with fibers parallel to the x I

-

axis. Introducing the dimensionless quantity e by a22

e = - . (e < < 1) ,

(39)

a1l

and expanding the right hand side of (34) in a Taylor expansion (for small e), it is shown in Horgan and Miller49 that (40) thus predicting a very slow decay rate in this limit. The characteristic decay length (the distance over which the stress (at fixed height X2) decays to 1 % of its value at

XI =

0) is then given by (41)

which is large compared with the strip width. The asymptotic estimates (40), (41), which are obtained directly in Horgan and Miller49 from the explicit result (34), have counterparts in the plane strain case. Their derivations in this case (see e.g. Choi and Horgan l6 , Horgan

39,40)

are much more

62 complicated than the relatively simple arguments outlined above. Returning to the inhomogeneous material, with governing boundaryvalue problem given by (I), (11), (12) we illustrate some applications of the result (23), described in detail in 61. To begin with, suppose that the a~ (suitably nondimensionalized) are given by

(42)

Then solutions of (1), (11), (12) decay in energy norm at least as fast as

(43)

as Xl

~

00 .

We observe that the power of Xl in the decaying exponential is 1f2

instead of I, as in the homogeneous case. A second example is a modification of (42) where all has quadratic growth in xl so that

(44)

Then it is shown 61 that solutions now decay in energy norm at least po[ynomially as

(45)

where kl is given by (43). In the next example, al2 and azz are taken to be constant and all = all (X I) has polynomial growth. Thus let

(46)

whe re 0. is a positive constant. It is shown by Horgan and Payne

61

that if 0. >

2 there may not be any sol ution of (1), (11), (12) which decays to zero in

63 energy norm as

XI ~

00,

for a < 2 all decaying solutions decay at least

exponentially in energy norm at least as fast as

exp (-21t

while if a

~] [

(I-a/l)

/ (1 - a/2) }

(47)

= 2 all decaying solutions decay at least polynomially in energy

norm at least as fast as -211

(48)

This result for a XI -

=

2 is consistent with that for (44) where the

dependence of all was also quadratic. Finally, we describe an example

for which the decay in energy norm is Jaster than exp(-y x I) for any positive

y. Thus let

Then, it is shown by Horgan and Payne 61 that the energy decay as

XI

~

00

is

at least as fast as

(50)

It is clear from the foregoing examples that the precise form of material

inhomogeneity has a significant influence on the decay character of SaintVenant end effects in anti-plane shear. The results also suggest the interesting possibility of 'tailoring' the material to produce a desired decay character. To make the result (28) fully expliCit, one requires a bound for the total energy

J aa~ u. a u.~ dA in terms of the data f(x2) D

at

XI

= O. Such a result is

64 obtained 48

61

on using the Dirichlet principle. Similar results are described in

(and the references cited therein) for the variety of partial differential

equations (and systems of equations) arising in elasticity theory.

It would be desirable to obtain pointwise estimates for u valid throughout D. As is well-known (see e.g. the discussion in

48)

this usually

requires considerable technical effort using higher order energies or arguments based on maximum principles.

Techniques based on the

maximum principle can be used in the case where

~

are constants leading to

pointwise estimates with optimal decay rates. However, when the coefficients are variable, such methods are not readily applicable except in special circumstances. The arguments based on higher order energies will, in general, require additional smoothness hypotheses on the aafl. For the case when the aa.1l are independent of XI, such arguments have been

61

applied to

yield the desired pointwise estimates. For the special class of problems, with variable coefficients, for which a12 = 0, pointwise estimates for u and its gradient were obtained using energy arguments

principle

121.42

See also Section II of

48

76

and using the maximum

for a general discussion. For the

special case of Laplace's equation, improved pointwise gradient decay estimates which are explicit in terms of the boundary data (12) have been obtained recently by Horgan et al.

62

Other decay results for second-order linear elliptic partial differential equations in plane domains. not necessarily rectangular. are described in the review article

48.

In particular, the problem of axisymmetric torsion of

isotropic elastic shells of revolution is discussed in detail.

65 Results comparable to the foregoing for three-dimensional domains are relatively scarce. Analogs of the decay results described above for harmonic functions on cylindrical domains of arbitrary cross-section are discussed in

48

Pointwise gradient decay estimates for harmonic functions on such domains are also treated in 48 and in the recent paper 62 .

3. PLANE ELASTICITY: LINEAR FOURTH-ORDER ELLIPTIC PDES.

The fourth-order analog of (1) arises in the theory of plane

deformations of linearly elastic inhomogeneous anisotropic solids.

An

analogous result to Theorem I for such equations on a semi-infinite strip has

not been established to the author's knowledge. Considerable work has been carried out on the homogeneous problem with constant coefficients in connection with Saint-Venant's principle i.e. in establishing exponential

decay results under appropriate decay assumptions as XI -7 00 • In this case. the governing boundary value problem for the Airy stress function <1> (XI. X2) reads 1lz2 <1>.1111 - 21326<1>.1112 + (21312 + (366) <1>.1122 - 21316<1>.1222 + 1311 $.2222 = 0 on D.

(51)

subject to $=0• $=F(X2). <1> . ~

(52)

$.2 = 0 on X2 = O. h. <1>.I=G(X2)

onxl=O.

-70 (uniformly in X2) as Xl -7

00 •

(53) (54)

where F (X2). G (X2) are prescribed sufficiently smooth functions satisfying appropriate continuity conditions at the corner of the strip. The constants 13pq

66 are the elastic constants. The associated strain-energy density is assumed positive definite so that the partial differential equation (51) is elliptic. For the special case when the material is isotropic. one has

PI6

= 0 . 1326 = 0

and

the remaining elastic constants are such that (51) reduces to the bihamwnic equation 4>.llll

(55)

+ 24>.1122 + 4>.2222 = O.

Results comparable to (33). (35). with optimal decay rate analogous to (34). have not been obtained for the problem (51)-(54). Indeed an explicit formula for the exact decay rate in terms of the elastic constants is not available. The separable solution of (51)-(54) is elaborate

16.

involving

generalization of the Fadle-Papkovich eigenfunctions for (55). (52)-(54). The exact decay rate is given by the real part of a complex eigenvalue which. in tum. is the root of a transcendental equation. These roots depend on the elastic constants in a complicated way (see the recent paper

102

for a

discussion). Energy decay techniques have been applied to (51-54) in

36. 102 .

resulting in decay estimates of the form (33). (35) but the estimated decay

rate k is generally conservative. underestimating the exact decay rate. Indeed. this is even the case for the biharmonic problem (55). (51)-(54). first investigated by Knowles 75 . See

20-22. 43.44.48. 7~87 . 106. 107

and the references

cited therein for the extensive literature on this problem . Asymptotic results analogous to (40). (41) for highly anisotropic orthotropic materials have been obtained which do provide an accurate estimate for the exact decay rate. As is the case with (40). (41). the results for the plane problem predict a very slow decay rate with large characteristic

decay length . For a detailed review of these results and their practical

67 implications for the mechanics of composite materials, see 39,40,43, 102 and the recent review article of Horgan and Simmonds 67 . In a recent paper

lll,

the a priori assumption (54) has been removed in

conjunction with the biharmonic problem and Phragmen-Lindelof type alternatives have been established for general plane domains. The biharmonic problem also arises in the theory of stationary slow viscous flows (Stokes flow)--see the discussion in Section 4. Energy decay results for the biharmonic (and polyharmonic) equations in three-dimensions were obtained by Lin 92,93

4. NONLINEAR PROBLEMS 4.1 Nonlinear Elasticity

The formulation and analysis of Saint-Venant's principle in finite elastostatics is an elaborate issue. Early work on this issue is that of Roseman 114

and Breuer and Roseman9 • Versions of Saint-Venant's principle within

the framework of finite anti-plane shear were established by Horgan and Knowles

47

and Horgan and Payne

and in Horgan and Knowles

48,

55 .

As was discussed in these references,

Horgan 43, the issues of concern in connection

with Saint-Venant's principle in the nonlinear theory of elasticity are considerably more involved than those arising in the linearized theory. One difficulty is that the appropriate Saint-Venant solutions need to be carefully characterized. Secondly, in the absence of superposition, consideration of

self-equilibrated end loads is no longer sufficient. Furthermore, instabilities may have to be taken into account. Also the decay rate for end effects, even if exponential, might depend on the overall loading as well as on geometry and

68 material characteristics. The mathematical issues considered in Horgan and Knowles

47

are concerned with the spatial decay of solutions of a Neumann-

type boundary-value problem for a second-order quasilinear elliptic partial

differential equation on a two-dimensional semi-infinite strip. The physical problem is that of finite anti-plane shear of an isotropic nonlinearly elastic cylinder with a semi-infinite strip as undeformed cross-section. The long sides of the strip are traction free. while the short side carries a prescribed shear traction directed parallel to the generators. This given traction is not necessarily uniformly distributed and the associated average shear stress p (and hence the resultant shear force) does not necessarily vanish. At infinity in the strip. the displacement is that of a simple shear with shear stress p. It is shown in Horgan and Knowles

47

that. along the long sides of the strip. the

difference between the nonvanishing component of shear stress and its average value p is bounded by an exponentially decaying function of the distance from the end. A lower bound is given for the rate of decay for materials for which the governing partial differential equation is uniformly elliptic and which either "harden" or ··soften" in simple shear. This lower bound depends on the strip width. on the average stress p. on the strain-energy density of the material and on a measure of the departure from uniformity of the given end traction. The latter quantity enters into the estimates obtained in Horgan and Knowles

47

as a result of the main analytical technique

employed there. namely. a comparison (or maximum) prinCiple for secondorder quasilinear elliptic partial differential equations. In Horgan and Payne 55. an analog of the problem considered by Horgan

and Knowles

47

was investigated using an energy approach. A finite

rectangular region of width h is considered. with traction-free lateral sides.

69 prescribed shear traction at the near end with resultant shear force ph, and subjected to a uniformly distributed shear stress p at the far end. Under rather general constitutive assumptions (which do not, for example, necessarily require that the governing second-order quasilinear partial differential equation be elliptic), it is shown in Horgan and Payne

55

that an energy-like

quadratic functional, defined on the difference between the deformation field and a state of simple shear corresponding to p, decays exponentially with distance from the near end. The estimated decay rate (which is a lower bound for the actual rate of exponential decay) is characterized in terms of the load level p, the domain geometry and material properties. The special situation when the load is self-equilibrated (corresponding to p=O) is also considered and the results compared with those of Horgan and Knowles

47 .

Furthermore,

a specific class of incompressible materials, namely those of the power-law type, is examined for which the estimated decay rate can be written down explicitly, as was also possible in Horgan and Knowles 47 . The energy method employed by Horgan and Payne

55

has certain

advantages and disadvantages when compared to the technique based on maximum prinCiples used by Horgan and Knowles

47 .

In addition to the

wider applicability of energy methods, it was not necessary in

55

to convert

the basic Neumann problem to a Dirichlet problem as was the case in

47 .

Furthermore, decay results are obtained for a finite domain. The estimated decay rates obtained in

55

good as those derived in

are in some cases better and in some cases not as 47.

Also pointwise estimates are more readily

obtained using the maximum principle approach. Results analogous to those obtained in

55

have been established by Horgan and Payne

58

for a restricted

theory of nonlinear plane elasticity, where the governing equation is a

70 fourth-order nonlinear elliptic partial differential equation. We now briefly review the rather extensive body of results obtained on the spatial decay of solutions (and gradients of solutions) of second-order quasilinear elliptic partial differential equations on semi-infinite strips (and more general domains) in the special case of the foregoing when p

= 0, i.e.

when the end loading is self-equilibrated. While this special case leads to considerable analytic simplification and thus to more explicit results than the case when p

*" 0, the latter must be considered when a complete analysis of a

nonlinear elastostatic Saint-Venant's principle is required. Of course, the results when p = 0 are of interest, not only for the guidelines they provide for the case of nonzero p but also from the perspective of the theory of partial differential equations. The second-order quasilinear equation governing finite anti-plane shear deformations u (x 1, written as

X2)

of an isotropic nonlinearly elastic material can be

45

[p (k2) u.al. a = 0 on D ,

(56)

where (57)

and W is the strain energy density per unit undeformed volume. It is assumed that (56) is elliptic so that p + 2 p' k2 ~ 0 ,

(58)

for all solutions u of (56) and at all points of D, where p' ;: dp/dk 2 The Cauchy stresses in the plane are given by

Ta =2 P (k2) u,a'

(59)

71

On linearizing fonnally by assuming k Z«

1, one recovers Laplace 's

equation. For power-law materials for which (60) the explicit fonn of (56) is given by

[

I UZI + -b Uz] n-l I + b (2 - -) 2 U II + 4b - u I U 2 U 12 n ' n " n' "

2

(61)

2]

I b + 1+-UI+b(2--)U2 U22=O onD. [ n ' n" When n = I, it is readily verified that (61) reduces to Laplace 's equation. When n =

1/2 ,

(61) becomes

(1 + 2bu~z) u,ll - 4bu, I U,Z u,IZ + (1 + 2bu~l) u, 22 = 0,

(62)

which, on rescaling u (or x a ), is the minimal surface equation (1 + u~z) u,ll - 2 U,I U,Z U,IZ + (1 + U~I) U,22 = 0,

(63)

which also governs the flow of a Kannan-Tsien gas. It is readily verified that (61) is elliptic if n 2:

1/ 2•

The material (60) hardens or softens in shear

according as n > 1 or n < 1 respectively. The energy nonn used by Horgan and Payne 55 may be written as

f

k2 == u,~u,~ ,

(64)

where D z denotes the subdomain D z = {(XI, X2) I 0 $ z < XI

, 0 < X2 < h)

E(z) =

P (k2) u,au,a dA ,

Dz

of the semi-infinite strip D. The functional (64), defined on solutions of (56), is the nonlinear analog of (13), defined on solutions of (1). Under various hypotheses on p, it was shown in Horgan and Payne 55,51 that solutions of

(56), subjected to nonzero boundary conditions on the end XI = 0, zero

72

Neumann-type boundary conditions on the long sides and a decay assumption as Xl

~

00,

decay exponentially with xl in energy norm at least as fast as

solutions to Laplace's equation. For the power-law materials (60), for which the explicit form of (56) is (61), the estimated decay rate obtained by Horgan and Payne

coincides with that given by (37) for Laplace's equation. A

55,51

similar result was predicted by the maximum principle approach of Horgan and Knowles

47,

Horgan and Abeyaratne

46

when n ~ I, although when Y2 < n

< I, the maximum principle arguments gave an estimated decay rate slower

than that for Laplace's equation. In particular, when n =

In

in (60) so that

(56) reduces to the minimal surface equation (63), the estimated decay rate obtained by Horgan and Payne

55,51

coincides with that for Laplace's

equation. A similar (pointwise) result was obtained by Knowles

78,

using the

maximum principle, for solutions of the minimal surface equation on a semiinfinite strip subject to homogeneous Dirichlet boundary conditions on the long sides and a decay assumption as Xl

~

00

(for earlier work on related

problems, see Roseman 115, Horgan and Wheeler 68) . This result was generalized by Horgan

41

to apply to equation (56) when p' ::; 0 on D. It was

shown by Horgan and Siegel

65

that the exact decay rate for the minimal

surface equation subject to homogeneous Dirichlet boundary conditions on the long sides is precisely that for Laplace 's equation. Moreover, a decay condition as Xl

~

00

need not be assumed a priori. Explicit gradient decay

estimates for the minimal surface problem were obtained recently by Horgan et a1 62 . Several generalizations of the foregoing have been established. Horgan and Payne

53

extended the results of Horgan and Payne

51

to Dirichlet

problems for (56) on bounded (or unbounded) regions of arbitrary shape. For

73 unbounded wedge-shaped domains. the decay is shown to be of power-law type rather than exponential. Quintanilla Horgan and Payne

51

Breuer and Roseman

113

generalized the results of

to a wider class of quasilinear equations. See also

15.

Inhomogeneous equations of the form [p (k2) u,al,a + 2g = 0 •

(65)

where g > 0 is a constant. were considered by Horgan and Payne

54 .

Here 2g

may be viewed as a constant body force in the anti-plane shear problem. Horgan and Payne

54

established the exponential decay of solutions to (65).

on thin rectangular domains subjected to homogeneous Dirichlet boundary conditions on the long sides. to solutions of the corresponding onedimensional problems for ordinary differential equations on the cross-section of the rectangle. Such results are of interest in assessing the approximate

nature of one-dimensional theories compared to exact two-dimensional theories. and have played an important role. for example. in establishing plate and shell theories in solid mechanics. Generalizations to the equations governing capillary surfaces and extensible films were obtained by Horgan and Payne

56

An alternative approach to these problems. using the maximum

principle. is described in Payne and Webb

112 .

alternative (i.e. with no a priori assumption made as

A Phragmen-LindelOf Xl ~

00) for (65) was

established recently by Knops and Payne 73 . The effects of constitutive law perturbations on solutions to Dirichlet or Neumann problems for (56) on a semi-infinite strip were examined by Horgan and Payne

59 .

Solutions to (56) were compared to solutions v of a differential

equation arising from a small perturbation of the constitutive function p. subject to the same boundary conditions as u. Such a comparison is of

74 interest when the problem for v is much simpler than that for u, for example, v may be the solution to a linear problem. Illustrative examples are given in Horgan and Payne 59, two of which involve comparison of u with hannonic functions, while the third example compares solutions for two softening power-law materials, i.e. (60) with n < I. The results are of interest given the practical difficulty in constructing constitutive models that provide an exact

description of material behavior. Comparable results for nonlinear second-order problems in threedimensions are not as readily obtained. It was shown by Horgan

41

that

solutions of the analog of equation (56) in three independent variables over a semi-infinite cylinder R with simply-connected cross-section S, vanishing on the lateral surface of R and at infinity, decay exponentially in the axial direction at least as fast as do hannonic functions subject to the same boundary conditions, if p'

:0;

0 on R and provided that S is convex. Thus, in

particular, this result holds for the three-dimensional minimal surface equation on such domains.

Other decay results for three-dimensional

second-order quasi linear and semilinear elliptic partial differential equations on cylindrical domains are discussed by Horgan and Payne 52 , Breuer and Roseman 11, Flavin et aI, 28,29, Roseman and Zimering 11 6, and Shenker and Roseman 117 . Roseman

Non-cylindrical domains are considered by Breuer and

13.

Returning to nonlinear elasticity. Breuer and Roseman

10

considered

end loads statically equivalent to uniform tension or compression in a theory of plane strain with sufficiently small strains and strain gradients, while Knops and Payne

72

gave a treatment of a Saint-Venant principle for the

three-dim ensional nonlinearly elastic cylinder. Further results on asymptotic

75 behavior in a nonlinearly elastic beam subject to self-equilibrated loads were obtained by Galdi et al 31 and Levine and Quintanilla 91. A generalization of the results of Oleinik and Yosifian obtained by Orazov

109 .

108

to nonlinear elasticity has been

Results on small deformations superimposed on

large for plane deformations of an incompressible semi-infinite strip were obtained by Abeyaratne et all. Since this problem was amenable to a linearized analysis, exact results for the decay rate in terms of load, geometric, and constitutive parameters were obtained. For certain ranges of load and constitutive parameters (e.g., for large loads), extended Saint-Venant edge zones similar in character to those occurring in highly anisotropic and composite materials were observed. Further results on this problem were obtained by Durban and Stronge

17 .

Saint-Venant's principle within the

theory of finite strain plasticity was discussed by Durban and Stronge

18. 19

for

the problem of axial loading of a circular cylinder and for plane strain. Vafeades and Horgan von

Karmin

120

obtained energy decay estimates for solutions of the

equations on a semi-infinite strip. A comparison with the

linearized problem for the biharmonic equation was also made. Extended edge zones, due to nonlinearity, are predicted by this analysis. Exponential decay estimates for solutions in a nonlinear theory of elastic edge-loaded circular tubes (cylindrical shells) were obtained by Horgan et al

63

An

approach to a Saint-Venant type prinCiple in nonlinear elasticity from the viewpoint of center manifold theory is described by Mielke

100

The axial

variable in a cylindrical domain is viewed as time in an evolutionary setting. Results from center manifold theory are then used to establish spatial decay of end effects. The analogy between dynamical stability arguments and the energy arguments used in connection with Saint-Venant's principle was

76 pointed out in

48

(p. 204). In fact, this analogy was remarked on by Saint-

Venant in his original work. The relevant observations of Saint-Venant are quoted verbatim in the footnote on p. 204 of 48 .

4.2. Viscous flows in pipes and channels.

A principle of Saint-Venant type involving nonlinear effects also arises in connection with incompressible viscous flows in pipes and channels. Of concern is the classical entry problem of laminar flow theory involving the development of velocity profiles in the inlet region. Horgan and Wheeler

69

investigated this issue within the general framework of the Navier-Stokes equations, governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross-section. The end effect studied involves a comparison between two distinct solutions of the Navier-Stokes equations, namely a base flow with arbitrary entrance profile and the corresponding fully developed solution (e.g., Hagen-Poiseuille flow for the case of a circular pipe). The flow development is analyzed by consideration of the spatial evolution of the difference between these flows . The velocity field associated with this difference satisfies a condition of zero net inflow, corresponding to the "self-equilibration" condition arising in elasticity. Using energy decay arguments, Horgan and Wheeler

69

establish an exponential

decay estimate for the energy dissipation function associated with this difference velocity field for a small enough Reynolds number. The estimated decay rate is characterized in terms of the Reynolds number, the prescribed entry profile of the base flow, and cross-sectional properties of the pipe. These results yield upper bounds for the "entrance length" through which the flow develops. A generalization of these results is carried out by Ames and

77

Payne

3.

The two-dimensional version of the foregoing problem, concerning flow development to the (parabolic) Poiseuille flow in a semi-infinite parallel-plate channel, is investigated by Horgan

38 .

In this case a more explicit treatment

is possible; moreover, the analogy with considerations of Saint-Venant's principle in elasticity becomes more apparent. Introduction of a stream function 'I' (XI, X2) for the difference velocity field leads to a boundary-value problem

for

a single fourth-order nonlinear elliptic

equation.

In

nondimensional form, this equation reads:

(66)

where d is the two-dimensional Laplacian and R is the Reynolds number (based on channel half-width). On the long sides of the channel, one requires

'I' and '1',2 to vanish, 'I' and '1'.1 are prescribed at the entrance Xl

= 0, -1 :5 x2 :5 I,

and suitable conditions are imposed at infinity. (In the

formal limiting case of vanishing Reynolds number, one obtains a boundary value problem for the biharmonic equation governing the development of Stokes flows. This problem is formally equivalent to that discussed in Section 3 in connection with Saint-Venant's principle in linear isotropic plane strain. Thus results for plane elasticity may be immediately applied to yield estimates for the two-dimensional Stokes flow problem) decay arguments, Horgan

38

44 .

Using energy

establishes an exponential decay estimate for the

energy dissipation associated with solutions 'I' of (66), provided the Reynolds number R is small enough. The estimated rate of decay is given in terms of the Reynolds number and the prescribed entry profile of the base flow. The

78 axisymmetric analog of this problem (for entry flows in a circular pipe) has been considered recently by Ache 2 . Issues related to those just described have been considered by Yosifian 122, 123

and by Amick

5,6

in the course of their general studies of solutions of

the stationary Navier-Stokes equations in domains with boundaries extending to infinity. Yosifian 123 (see also Oleinik 105) established an energy decay

estimate for weak solutions of Dirichlet problems for the Stokes system in three-dimensional

(not

necessarily cylindrical)

domains, again

with

boundaries which extend to infinity. He used this result as an a priori estimate to establish a uniqueness theorem for velocity fields with possibly unbounded energies. Similar results for the Navier-Stokes equations, as well as a result concerning the behavior of solutions near irregular boundary points, were announced by Yosifian

122 .

The work of Amick 5 was concerned

with steady flows in two-and three-dimensional domains which are cylindrical outside some bounded set. Methods of functional analysis were used to establish existence, for small enough Reynolds number, of weak solutions of the Navier-Stokes equations with finite energies which tend to appropriate Poiseuille flows at infinity. Estimates for the rate of approach, shown to be exponential, was provided by Amick 6 4.3. Steady state linear heat conduction with nonlinear boundary conditions.

A nonlinear problem of somewhat different type has been investigated recently by Horgan and Payne

60

This paper concerns the asymptotic

behavior of harmonic Junctions defined on a three-dimensional semi-infinite cylinder when homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder The motivation for a study of this issue

79 comes from the theory of heat conduction where nonlinear heat loss or heat gain is assumed to occur on the lateral surface. The classical PhragrnenLindelof theorem states that harmonic functions which vanish on the cylindrical surface must either grow exponentially or decay exponentially with distance from the finite end of the cylinder: It is shown in Horgan and Payne

that the results are qualitatively different when the homogeneous

60

Dirichlet boundary condition is replaced by the nonlinear boundary condition. The steady-state temperature field

U(XI' X2 , X3)

satisfies Laplace's

equation (67)

U,ii=O .

A

nonlinear boundary-value

R = (Xl,

X2, X3) I (Xl, X2) E

s ,

problem X3

Simply-connected region in the

on

the

semi-infinite

cylinder

> OJ is considered where S is a bounded,

(Xl, X2)

plane with Lipschitz boundary as,

and the origin is taken at a point inside S in the plane

X3

=O.

The specific

boundary conditions imposed on u are au an + f (u) = 0

on as x (0, 00) ,

(68)

and either (69)

or (70)

where

~ an

denotes the normal derivative directed outward on as and f (u) is

assumed to satisfy u f(u) ~ y lul 2p

(71)

for some positive constants yand p with p > V2. In Horgan and Payne 60, no a

80 priori assumption is made about the asymptotic behavior of u as X3 --+ 00, and a Phragmen-Lindeliif type alternative is established. It is assumed throughout that a classical solution of (67), (68) with (69) or (70) exists. The notation (72) with So = S, is also used. The functional (z) considered by Horgan and Payne

60

is

(z) = -

f

u

U

s.'

d 2 3 dA = - ~ dA dz S. 2 .

f

(73)

The notation used indicates that the integral is taken over D in the plane X3 = z. For instance, the data (69) or (70) is given on Do, the interior of D at x3 = O. For p in the range

Ih < p
~

00

(74)

either (z) decays to zero or - (z) becomes

unbounded. The rate of decay (or growth) depends on p, and explicit bounds for the decay (or growth) rates are obtained. The decay (or growth) rate of (z) for p in the range V2 < P < I is quite different from that in the range p > I . The case p = I is treated separately.

The simplest example of an f(u) which satisfies (71) is f(u)=y luI 2 (P- l ) U,

(75)

a form of f(u) which is often assumed in heat conduction problems. For black

body radiation conditions in hea t conduction, the value of p ari sing in (1 .11) is P = 5{2, while for some natural convection problems p = 9/8. The main re sults established by Horgan and Payne

60

are as follow s.

For p E (1/2 , I), it is shown that the L2 integral of u ove r the cross-section Sz

81

must either grow polynomially or decay exponentially in z as z ~

00 .

Bounds

for the rate of growth or decay are obtained explicitly in terms of p. When p

= I, it is shown that the corresponding ~ integral must either grow or decay exponentially in z as z

~

00.

Here it is shown that the

Analogous results are established for p > 1.

~

integral of u over Sz must either grow

exponentially or decay polynomially in z as z

~

00.

Again bounds for the

growth or decay rate are obtained explicitly in terms of p.

S. TIME-DEPENDENT PROBLEMS A Saint-Venant principle for transient heat conduction in an isotropic material (governed by a parabolic differential equation) was formulated and analyzed in

48.

The feasibility of a Saint-Venant principle in this context was

first suggested by Boley. An analysis due to Knowles77 is described in

48

concerned with an initial-boundary-value problem for the heat equation in a three-dimensional cylinder subject to nonzero boundary conditions only on the ends. An energy estimate was established in Knowles77 which showed that end effects for the transient case decay spatially at least as rapidly as do their counterparts in the steady-state case (governed by Laplace 's equation). This result was also obtained by Horgan and Wheeler

70

using arguments

based on the maximum principle. A stronger result was established by Horgan et al

64

which shows that the spatial decay rate in the transient

problem is actually Jaster than that for the steady-state problem . This confirms observations made by Boley in related contexts. Explicit pointwise results of this type for the temperature and its gradient have been obtained recently by Payne and Philippin

110

using the maximum principle. Further

references for spatial decay results for linear (and some nonlinear) parabolic

82 problems may be found in arguments

70

48,43 ,

More recently, the maximum principle

have been generalized to a wide class of nonlinear parabolic

equations by Breuer and Roseman 14

Phragmen-Lindelof alternatives for

quasilinear second-order parabolic equations have been established recently by Lin and Payne95 ,96 The time dependent analogs for some of the viscous entry flow problems described in Section 4.2 have been examined recently. Spatial decay for non-stationary three-dimensional Stokes flows (slow viscous flow) in a cylindrical pipe of arbitrary cross-section is considered by Ames et al

4

while the two-dimensional analog is investigated by Lin 94 In the latter study, a stream function formulation is used as in the stationary problem

38

so that

the problem considered involves a linear fourth-order initial boundary-value problem. The influence of perturbations in the diffusivity coefficient on spatial decay for the linear transient heat equation was considered by Lin and Payne 98 . The issue is the parabolic analog of that examined by Horgan and Payne 59 for finite anti-plane shear. The effect of domain perturbations (i.e. perturbations on the cross-sectional domain) on spatial decay was also investigated by Lin and Payne 98 , the issue here being the analog for transient heat conduction of related questions examined by Horgan and Payne 57 for the elliptic system of linear isotropic elasticity. Spatial decay for some ill-posed parabolic problems (e.g. the backward heat equation and backward Stokes flow equations) is discussed by Lin and Payne 97 , The asymptotic spatial behavior of solutions for elastodynamics (governed by hyperbolic differential equations) is quite different. Boley used

83 a simple one-dimensional beam model to show that Saint-Venant's principle is applicable provided the end loads are applied sufficiently slowly. More elaborate models, more closely modeling three-dimensional elastodynamics, were described by Boley

8

where it was pointed out that the issue is pan of

the more general question of assessing the validity of using quasi-static solutions in dynamic problems. Thus restrictions to low frequency range might be expected. Further references to this earlier work are given in

48,43

More recently, spatial decay estimates of the type described in Horgan and Knowles

48

have been obtained by Flavin and Knops

23

for damped acoustic

and displacement initial-boundary value problems in elastodynamics in the low frequency range which substantiate the early work of Boley. It should be

noted that all of the dynamic investigations mentioned in the foregoing were concerned with linear isotropic materials. Extension of these results to nonlinear elastodynamics and linear anisotropic elastodynamics are described in Flavin et a1 27 • 6. CONCLUDING REMARKS The emphasis of this review has been on properties of solutions of boundary-value problems on cylindrical three-dimensional domains or on rectangular regions in two dimensions. A variety of results on asymptotic behavior of linear elasticity solutions on more general domains have been obtained in recent years

30,74,84-86,108,

although comparable results in the

nonlinear theory have not yet been developed. While many of the questions discussed in previous reviews of Saint-Venant's principle in its classic interpretation have been answered, a complete analysis of the broader physical and mathematical issues arising in connection with the asymptotic

84 behavior of solutions in elasticity remains to be developed. Acknowledgments

This research was supported by the U.S. National Science Foundation under Grant No. MSS-91 -02155. by the U.S. Army Research Office under Grant DAAH04-94-G0189. and by the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-F49620-92-J-0122. It is a pleasure to acknowledge numerous discussions over the years with J. K. Knowles and L. E. Payne. BIBLIOGRAPHY 1.

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Oleinik, O. A., "Applications of the energy estimates analogous to Saint- Venan t' s principle to problems of elasticity and hydrodynamics. " In Lecture notes in physics, vol. 90, Springer-Verlag, Berlin, 422-432 (1979).

89 106.

Oleinik. O. A. and Yosifian, G. A., "On Saint-Yenant's principle in plane elasticity theory," Sov. Math. Dold. 19,364-368 (1978).

107.

Oleinik. O. A. and Yosifian, G. A., 'The Saint-Yenant principle in the two-dimensional theory of elasticity and boundary problems for a biharmonic equation in unbounded domains," Siberian Math. J. 19, 813-822 (1978).

108.

Oleinik. O. A. and Yosifian, G. A., "On the asymptotic behavior at infinity of solutions in linear elasticity," Arch. Ratl. Mech. Anal. 78, 29-53 (1982).

109.

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110.

Payne, L. E. and Philippin, G. A., "Pointwise bounds and spatial decay estimates in heat conduction problems," Math. Models & Meth. in Appl. Sci. (in press).

111.

Payne, L. E. and Schaefer, P. W., "Some Phragmen-Lindelof type results for the biharmonic equation," ?AMP, 45, 414-432 (1994).

112.

Payne, L. E. and Webb, J., "Spatial decay estimates for second order partial differential equations," Nonlinear Analysis, 18, 143-156 (1992).

113.

Quintanilla, R., "Some theorems of Phragmen-LindelOf type for nonlinear partial differential equations," Publicacions Matematiques, 37, 443-463 (1993).

114.

Roseman, J. J., "The principle of Saint-Yenant in linear and nonlinear plane elasticity," Arch. Ration. Mech . Anal. 26, 142-162 (1967).

115

Roseman, 1. 1., "The rate of decay of a minimal surface defined over a semi-infinite strip," J. Math. Anal. Appl. 46,545-554 (1974).

116.

Roseman, J. J. and Zimering, S., "On the spatial decay of the energy for some quasilinear boundary value problems in semi-infinite cylinders," J. Math. Anal. Appl. , 139, 194-204 (1989).

117.

Shenker, Y. and Roseman, 1. 1., "On the asymptotic uniformization of the temperature in a laterally insulated rod for the steady-state nonlinear model," Canadian Applied Matheml11icsQuarterly, 2, 115-149 (1994).

118.

Shenker, Y. and Roseman, J. J., "On the Saint-Yenant principle in the case of infinite energy," J. Elasticity, 30, 55-68 (1993).

119.

Toupin, R. A., "Saint-Yenant's principle," Arch. Ration. Mech. Anal. 18,83-96 (1965).

120.

Yafeades, P. and Horgan, C. 0., "Exponential decay estimates for solutions of the von

Karnuin equations on a semi-infinite strip," Arch. Ration. Mech . Anal. 104,1-25 (1988). 121.

Wheeler, L. T. and Horgan, C. 0., "A two-dimensional Saint-Yenant principle for second-order linear elliptic equations," Quart. Appl. Math ., 34, 257-270 (1976).

122.

Yosifian, G. A., "Saint-Yenant's principle for the flow of viscous incompressible fluid," Russian Math . Surveys, 24,166-167 (1979).

123.

Yosifian, G. A., "An analog of Saint-Yenant's principle and the uniqueness of the solutions of the first boundary value problem for Stokes' system in domains with noncompact boundaries," Soviet Math . Dold. 19, 1048-1052 (1978).

91

ON THE TRACTION PROBLEM IN INCOMPRESSIBLE LINEAR ELASTICITY FOR UNBOUNDED DOMAINS

R. Russo and G. Starita

Istituto di Matematica Seconda Universita di Napoli Via F . Renella 98, 81100 Caserta (Italy)

92 •

Introduction. The present paper deals with some mathematical problems associated with

the equilibrium of a linearly elastic incompressible body identified either with an exterior of a three-dimensional bounded set or with a half--space. The main purpose is to extend to these domains some basic theorems usually stated for bounded regions: existence and uniqueness of regular solutions to the boundary- value problems, work and energy theorem, Betti 's reciprocity relation and tbe principle of

minimum potential energy in function classes defined by weak condition at infinity on the displacement field. Special attention is devoted to the traction problem which, as is well- known, is of some interest in the applications. The paper is organized as follows. In Section 2 we state the traction boundaryvalue problem and make our main hypotheses; in Section 3 we collect the preliminary results we shall use to prove our results; in Section 4 we extend the basic theorems of incompressible linear elastostatics to ext erior domains ; in Section 5 we prove a well-posed ness theorem for the traction problem in the class of all vector fields having finite energy; in Sections 6 we deal with the the cases of a body identified with a half- space; finally, in Section 7 we apply the results previously established to prove a uniqueness theorem in the dynami cal theory of incompressible linear elastodynamics. It is worth observing that, except for the uniqueness of solutions to the mixed

and the traction problems, no hypothesis is required on the behavior at infinity of the pressure field.

Notation - Letters in ordinary form denote scalars; lower case bold face letters different from 0 and x denote vectors (in JR3), while 0 and x stand respectively for the origin of the reference fr ame and a generic point of JR3; r = x - 0 , r = r( x ) =

Ir l

a nd e r = r- 1r ; upper case bold face letters indi cate second-order t ensors (linear transformations from JR3 into JR3); Lin denotes the set of all second-order tensors L , Sym = {AI A = AT}, LinD = {A E Lin I tr L = O} and S~'ffio = {A E

Sym I tr L a nd

= O} ; V'u is the second- order tensor with components (\'U )i] = Bu. IB.r] Vu, Vu indicate its symmetric and skew parts respectively ; SR(X o), S~(xo)

st and respectively for the ball a nd the half- ball centered at Xo with radius R,

SR = SR(O) , S~ = SR(O) and TR = S2R \ SR , Tii. = SiR \ S~ Let 'B be a domain of JR3 We set. 'BR = 'B n SR; V('B) stands for the set of all Coo vector

93 fields u with compact support on

~

such that divv = 0,

Lq(~)

(q 2: 1) is the

space of all measurable fields 'I/J on ~ such that I'll I'l/Jlqdv < +00, LToc( ~) is the space of all measurable fields 'I/J on ~ such that 'I/J E L2(K) for any compact K of ~ , Hl(~) :J{(~)

= {u

= {'I/J

E L2(0): \l'I/J E L2(~)} , H6(~)

E LToc(~): \lu E L2(~) and divu

L2(~) and div u

= O}

= O} ,

= {'I/J E Hl(~): 'l/Ja'll = O}, Q;(~) = {u E LToc( ~) : Vu E

and 91 denotes the set of all infinitesimal rigid displacement,

i.e ., the fields w = a + b x r , with a and b constant vectors . If 0 is a bounded domain of R 3 , we set 'l/Jn = (meas 0) -1 In 'I/J da. Finally, to avoid use of several symbols to denote positive constants whose numerical values are unessential to our purpose, we suppose that all the physical fields we consider have been put in a nondimensional form and we let c,

Ci

(i

= 1,2 ... ) indicate generic positive constants;

their numerical values may change from line to line. We stress that, when c,

Ci

are used in inequalities involving SR(O), S~(o), TR(O) and Tit (0), it has to be understood that c,

•

Ci

are independent of R.

2 - Basic equations. Consider an incompressible linearly elastic body identified either with the exte-

rior

~

of the set

~o

= Uj~j, where ~j (j = 1, ... , m) are m open connected bounded

sets of R3 , or the half-space R! = {x I X3 2: O} . If ~ is an exterior domain, we denote by d'll o the diameter of The equilibrium of

~

~o .

is governed by the system div C(\lu ) + b = \lp

in

~

(2.1)

divu = 0 (see, e.g., Gurtin4) , where b is the body force Eeld, u and p are the (unknown)

displacement and pressure Eelds and C is the elasticity tensor, i.e., a map ~ x Lino Sym, linear on Lino and such that C(W)

= 0,

-+

for any skew W

Unless otherwise stated, we assume that the boundary a~ of ~ is of class C 2 , C is continuously differentiable on

13 and b is differentiable on 13.

Moreover, we assume that (2.2)

94 Set

c(A) = A . C(A),

V A E Lino.

We assume that C satisfies the inequalities (2.3)

VA E Symo ,

for some positive constants J.Lm and J.LM independent of A. C is symmetric iff

A . C(B) = B . C(A), Let

s be

VA , B E Lino .

a continuous field on 8'B. The traction problem of incompressible

linear elastostatics consists in finding a solution (u, p) to system (2 .1 ) which satisfies the boundary condition (*)

s=s

(2.4)

at 8'B,

where s

= -pn + C(V'u)n

and n denotes the unit outward normal to 8'B.

•

3 - Some lemmas. We collect in this section the preliminary results we shall need in the sequel.

Lemma 3.1. Let D be either the ball SR or the half-ball S~ or the shell TR and let f E L 2(D) . If fD = 0, then the equation divv =

f

in D

admits a solution v E HJ (D) and there exists a positive constant

iD (V'V)2 dv ::;

Cl

1

f 2 dv.

(3. 1) Cl

such that (3.2)

Lemma 3.1 is proved in Bab uska & Aziz 1 (see also Galdi 2 ).

(*) By a solution to system (2.1) we mean a couple (u, p) which satisfies (2.1) - (2.3) pointwise.

95

Lemma 3.2.

(Poincare and Korn inequalities). Let u E Hl(D) , where D is

either the shell TR or the half-shell Tii. Then, there exist two positive constants C2 and C3 such that

h h

(u - UD)2 dv ::; C2R2h (\7U)2 dv

(3.3)

[\7u - (VU)D]2 dv ::; c3h ('~u)2 dv.

(3.4)

Moreover ifu E HJ(SR) , then (3.5)

(3.6)

Lemma 3.3. U

Let '13 be an exterior domain and let u E :J-C('13). Then lim uas, = R~oo

m and, VR > d'B o ,

(3.7)

Lemma 3.4.

(Korn 's inequality). Let U E lE('13). Then limR~+oo(Vul'BR = Woo,

with Woo skew tensor, and (3.8)

Moreover, ifu = o(r), then Woo =

o.

Inequalities (3.1), ... ,(3.7) are well-known; Lemma 3.4 is proved by Kondratiev & Oleinik5 (see also Russo 7 ). If rp E lE('13), we denote by rpm and Woo(rp) the constant vector and skew

tensor appearing in (3.7)-(3.8), written with u = rp . Moreover , we set (3.9)

96 If u E Q: (13) , by the arithmetic-geometric mean inequality, (2 .2), (3. 7) and

(3.8) we have that

r

l~R

lu. bl dv::;

r

l~R

I(u - wCXl ) . bl dv +

r IWCXl . bl dv

l~

: ; l~R r r- 2(u - WCXl)2 dv + cl~r(r b )2 dv ::; c

(l

(Vu )2 dv

+

l

(3. 10)

(r b)2 dV) .

Therefore, letting R -; +00 in (3 .1 0) , we have that that u· b E L1 (13 ). Le mma 3.5.

(Caccioppoli 's inequality). Let (u ,p) be a solu tion to system (2.1)

in S2R(X o )' Th en

Inequality (3. 11) is proved by repeating t he steps in t he proof of Theorem 1.1 in Giaquinta & Modica 3 and making use of (3.5)-(2.2). Lemma 3.6.

Let 13 be an exterior domain and let (u. p) be a solution to s.,·stem

(2.1). Th en V R »

r

l~R

d~ o'

Vw E 91 ,

(VU )2 dv ::;

+

c(R- 2 r

(u - w )2 dv

lTR

+

r (r b)2 dt ,

l~

r I(u-w) , slda+ bl)

(3. 12)

la~

where I =

~ -hr

r

la~

U·

nda.

(3.13)

Proof - We fo llow Giaquin ta &: Modica 3 (Remark 1.6). Let 9 be the cut-off

function (3. 1-1)

97 Let i.p be the field defined by in TR in ~R in R3 \ S2R, where, is given by (3.13) and h E HJ(TR ) is a solution to equation (3 .1) in TR corresponding to f = div[g2(u - ,r- 2 e r )]. Multiplying (2.1h scalarly by i.p and integrating by parts , we have

l l c: (V'u) dv

=, l r- l(1 - 3er @e C(V'u) dv + r V'h . C(V'u) dv - 2 rg(u -,r- 2e JTR JP, + r dv + r JP, JaP, s da 3

r )·

r )·

i.p' b

C(V'u)V'gdv

(3.15)

i.p'

By the arithmetic- geometric mean inequality, we have, V~ > 0, 2r- 3 1r(1 - 3e r @e r ) . C(V'u)1 :::; ~(VU)2

:::; ~(VU)2 2lg(u

-,r-

+ C 1 bf.lMr- 3 (1 -

+ clrlr-

2e )· C(V'u )V'g l :::; ~(gVu)2 r

6

,

+ C1(f.lMlu

+ (f.lMVU)2 , 21i.p ' bl :::; ~r-2i.p2 + C 1 (rb)2

21V'h· C(V'u)1 :::; (V'h)2

Then , (3. 15) implies

By (3.2) - (3.5)

3e r @ erW

-,r-

2e llV'gl)2, r

98 Moreover , by (3.5H3.6)

h

h (f, h (f, h

r-2 rp 2 :::: C

(Vrp )2 dv :::: c

:::: c

(gV U)2 dv

:::: c

(f, h[V( g2u)f dv + hR('Vh)2 dv + "'? hr- 6 dV) + R- 2

(g VU) 2 dv + R -

2

hR hR

2 u dv

+

hR

2 u dv

+

hi)

('V h )2 dv

Therefore, from (3.16), by properly choosing 9

= 1 on 'B R , it

+ III)

f, a nd taking into account that

follows that

Hence, taking into account that TR can be covered by a finite number of balls and ,

Vw E 91 , u - w is a solution to system (2. 1), by (3. 11) we have (3.12). Remark 3.1 - If s

= 0 , choosing (3.17)

from (3. 12), by virtue of (3.3)- (3.4) it follows t hat

Lemma 3.7.

Let (u ,p) be a solution to system (2.1 ) with b

= O.

If s

=0

at 8'B ,

then , Vp, R > 0: R > p » d13 o '

(3. 19) where Cl'

=

{ (l - f,d[(l + Jcl)(C2C3J.1.AfJ.1.;;/)~ + 2f,j - l (1 - f, 1)[(1

+ Jcl)y'C2C3J.1.MJ.1.;;/ + 2f,j - l

for ".nllIlJt'tric C

(3.20) otherwise

99 for som e p ositive constants ~ and 6 . Proof - Let 9 be t he function (3.13) and let h be a solution to equation (3. 1),

with f = div [g(u - w - ,r- 2e r )], and " w given by (3. 13), (3 .17) respect ively. Multiply (2. 1) scalarly by

g(u - w -,r- 2 e r ) - h , in TR g(u-w -,r- 2 e r ), in23 R { 0, in R3 \ Then, an integration by parts yields cp=

1

gE(\lU ) dv

'B

=,1

r- 3g( 1 - 3e r Q9 e r )· C(\lu ) dv

'B

+ R- 1

+

S2R .

JTR

\lh· C(\lu ) dv (3 .21)

r (u - w -,r- e r ) . C(\lu )e r dv iTR 2

If C is symmetric, by Schwarz's inequality, t he arithmetic-geom etric mean inequalityand (3. 2)- (3.3)-(3 .4) we have R-

1

liR

1

(u - w )· C(\lu )erdv l :::: (R -

2

i RE((U -

w ) Q9 er) dv

i RE(\lU ) dV) , 1

:::: (J.L MR -

2

i R(u -

:::: (C2 C3J.L MJ.L ;;,1)! R- 1

I,iTRr

r- 2e r · C(\lu )e r

liR

\lh· C(\lU )dVI ::::

r

w )2 dv

i RE(\lU ) dV ) ,

E(\lu )dv,

iTR

dvl : : ~ iTR r e:(\lu ) + c

1

r

(r J.L M)2

iTR

r- 6 dv,

(iRE(\lh ) dv iR E(\lU)dV) ! 1

:::: ( CIJ.LM R - 2

2 (u - w - , r- er )2 dv

iR

:::: ( 2C IJ.L MR- 2

i RE(\lU )dV) ,

[r (u-w) 2dv+,2 r r iTR iTR

:::: ( 2CIJ.L M[c2 C3J.L ;;,1

4

r E(\lU ) dv +,2 iTR r riTR

dvj

r

1

e:(\lU)dV ) '

iTR

1

6

dvj

r E(\lU) dV ) ' iTR 1

:::: ([2CIC2C3 J.LMJ.L;;,1

i Re:(\lu dv + q2 j iR E(\lU ) dV) ,

:::: [2CIC2 C3J.L MJ.L ;;,lj!

iTR

r e:(\lu) dv +~ iTR r E( \lu) dv+C 1clr l,

100

I,l

r- 3 g(1 - 3e T

@

eT ) . C(Vu) dvl

:s:

6l

gc(Vu) dv

+ ~lIIrI·

Then, from (3.21) we have

f(R) =

rgc(Vu) dv :s:

(X-I

i'B

Since

J'(R) = R- 2

r c(Vu)dv + clrl.

(3.22)

iTR

r c(Vu)rx) dv ,

iTR

from (3.22) it follows that

f(R)

:s:

(X- I RJ'(R)

+ clrl ·

(3.23)

Then, integrating (3.23) over (p , R) , we have

l

g(p)c(Vu) dv

:s:

C:1ir l g(R)c(Vu)dv + clrl·

Hence (3.19) follows , V R > 2p, taking into account the properties of g. If p 2' R/ 2, (3.19) follows from the obvious inequality

r (Vu)2dv:S:(UR)Q(R/p)Q .'Bl

i'Bp

(Vu)2dv:S: 2Q(U R)Q

R

r

(Vu)2 dv

i'BR

The proof of (3. 19) for nonsymmetric C is analogous to the above one so it is omitted.

Lemma 3.8.

I

Let C be constant and let (u ,p) be a solution to the system div C(Vu) = Vp

in R3

divu = X

'

(3.24)

where X is a constant. Then , V R , p > 0 : R > p,

r (Vu)2 dv :s: (:1i) iSr (Vu) 2dv. R

isp Proof

C4

3

(3.25)

Under the hypotheses of the lemma, the following inequality holds

(see, e.g ., Giaquinta & Modica 3 )

r (VU)2 dv :s:

isp

C4

(~f

r

(VU)2 dv.

iSR

101

Hence, since, Vw E ryt , u - w is a solution to system (3.24) , we have

r (Vu)2 dv ::;

C4

) Sp

Hence, choosing V'w

= ('\7U)SR

(~) 3

r (V'u - V'w)2 dv .

} SR

and making use of (3.4) , (3.23) follows.

I

Let C be constant and let (u,p) be a solution to the system

Lemma 3.9.

divC(V'u)

= V'p

divu = X u

=0

in R3 in R3 at {x I x3

= O}

where X is a constant. Then , V R, p > 0 : R > p,

h+

(Vu)2 dv ::; c

(~) 3

h+

(Vu)2 dv.

R

p

The proof of Lemma 3.9 is analogous to that of Lemma 3.8. Lemma 3.10.

Let u E HI(SR)' lfu

=0

in {x Ir(x) = R}, then

h~ u dv ::; c (h~ (Vu)2dV) 4

2

(3.26)

Let ¢ be a nonnegative and nondecreasing function on [R I , +00) (RI :::- 0) and assume that

Lemma 3.11.

¢(p)::; [CS(~)3 + (]¢(R), If

VR:::-p>R I .

« ( -3cs(3 )(3/(3-{3) (1 - (3) - , 3

for some (3

E

(0,3), then

¢(p) ::; c

(~){3 ¢(R),

V R:::- p

> RI .

Lemma 3.10 is well- known. Lemma 3:11 is proved in Meier 6

102

•

4 - The basic theorems of incompressible linear elastostatics in exterior

three-dimensional domains.

In this section we extend the basic theorems of incompressible linear elastostatics: Work and energy theorem , Betti 's reciprocity relation , uniqueness of solutions to the boundary- value problem s and th e principle of minimum potential energy,

to exterior domains in function classes defined by weak conditions at large spatial distance on the displacement field u. Accordingly, throughout the section we assume that 13 is an exterior domain. It is worth stressing that , except for the uniqueness of solutions to the traction and mixed problems, no hypothesis is made on the behavior at infinity of the pressure field p. Let (u , p) be a solution to system (2 .1 ). We set

!

= -

r b dv - J&'B r s da,

m = -

J'B

Theorem 4.1.

r r x b dv - Jr&13 r x s da .

(4.1 )

J'B

(Work and energy theorem). Let (u,p) be a solution to system

(2.1) such that u E Q:(13). Th en , limR_+oop'B R = Pp. E R, p - Pp. E L 2( 13 ) and

r c(\7u)dv= J'Br u·bdv + Jr

J'B

u·sda+um· ! +woo· m+pp.

&13

r

J &13

u·nda , (4.2 )

where Woo is the axial vector corresponding to the skew tensor W oo (u ) appearing in the inequality (3 .8) .

Proal - Let TJ be a regular cut- off fun ction on R3, vanishing on Sfi. (R > d'Bo) and equal to 1 outside S2f1.' Then the couple (v = TJU, r:v = TJp ) is a solution to the system div C (\7v) + 1jJ = \7r:v (4.3)

divv = () where

1/Ji = - CijhkChu/tojTJ - OJ (CijhkU hOkTJ) + TJb;

+ PO,TJ

() = u · \7TJ .

Let t.p E HJ (SR) (R» R) be a solution to equation (3. 1) with an integration by parts yields

1 = r:v-r:vSR'

( 4.4)

Then

103

Since by (3.2) and (3.5)

lisR \1'1"

C(\1v) dVI

~ PM (isR (\1'1')2 dv isR (\1V)2 dv ) ~ ~ PM.,foi (isR (tv -

R

tvS )2 dv

13

(\1V)2 dV) !

,

lisR cp . 1jJ dvi ~ (isR r-2cp2 dv isR (r1jJ)2 dV) ! ~ 2 (isR (\1cp)2dv isR (r1jJ) ~ 2C1 (isR (tv -

R

tvS )2 dv

2

!

dV)

13

(r1jJ) 2 dV) ! ,

from (4 .5) it follows that

(4.6) Integrating the inequality

over Sm and taking into account (4.6), we have

(tvs m

-

~ 21Sml-1

tvsJ2

(1

Srn

(tv - tvs m ? dv +

~ 21 Sml- 1 (ism (tv -

m

tvs )2 dv

r (tv - tvsJ2 dV)

JS

m

+ isn (tv -

tvsJ2 dV)

~ cm- 3 .

is a Cauchy sequence in JR so that limR_+ oo tvs R = Pp. E JR. Moreover, making use of Fatou 's lemma in (4.6) we realize that tv-Pp. E L 2(JR3) Hence it follows that {tvs n

}

so that, taking into account that tv = P outside S2R we see that P - Pp. E L2('B) . Hence it follows that limR_+oo P'B R = PI'-" In order to prove (4.2) multiply (2.1) scalarly by g(u - w oo), where 9 is the function (3.13) and

l

Woo

is given by (3.9). Then, an integration by parts yields

- Woo

l (l

gr x b dv

+ R- 1

(hR

(u - w oo) . C(\1U )€r dv

gc:(\1u) dv

=

gu· bdv

(l l

+ ia'B U · sda -

Um

+ ia'B r

+ Pp.

x

S

da )

+

hR

gbda

+ ia'B Sda)

u . n da

(4.7)

(p - Pp.)(u - woo) . €r dV)

104

Since by Schwarz's inequality and (3 .7)-(3.8)

R-

1

IlR

(U - w oo ) . C(V'u)e r dvl :::; J-lM ( R:::; J-lM

:::; c

R- 1 letting R

IlR --->

2

(lR

lR

(U -

r- 2(u -

W

OO

lR lR

)2 dv

W,xY dv

(l (VU)2 lR (VU)2 dV) dv

(p - PI")(U - w oo) . e r dvl :::; c

1

(VU)2 dV)

1

(VU)2 dV)

"2

1

"2

(l (VU)2 lR (p - PI')2dV) dv

"2

1

"2 ,

+00 in (4 .7) , taking into account that Vu , P-PI' E L 2('B ), u·b E Ll('B )

and making use of Lebesgue 's dominated convergence theorem , we get (4.2). Note that

um f

+ W oo

. Tn

+ PI"

I

r U· n da,

J81J

represents the work made by th e force and moment at inilnity. Remark 4. 1 - If we require that C is twice continuously differentiable in 'B , V'C

is bounded , b is continuously differentiable in 'B and r-1V'b E L2('B ), we can show that the second derivatives of u are square summable over 'B so that the pressure field p belongs to the space Hl('B). Indeed , under these hypotheses, we have that

(Ehu,fhp) (k = 1,2,3) is a solution to the system div C(V'8k u)

Let r.p =

{

+ div(8k C)(V'u) + 8k b = V'8 k P

in'B.

(4.8)

g2(8kU - (3r- 2 e r ) - h , on TR g2(8 k u - (3r- 2 e r ), on 'BR 0,

on IR \ S2R ,

where 9 is the function (3.14) , h E HJ (TR) is the solution to the equation divh = di v[g2 (fhu - (3r - 2e r )] on TR a nd (3 = (47T)- 1 f81J 8 ku · nda. Multiplying both sides of (4.7) by r.p. repeating the steps in the proof of (3. 12), we get

(4.9)

105

Since U E <1:(13), letting R by (2.1) 'Vp E L2(13).

-+ +00

in (4.8), we have that 'VVu E L2(13) . Therefore,

(Betti's reciprocity relation). Let C be symmetric and let (u,p) , (u,p) be two solutions to system (2.1) corresponding to the body force fields b and b respectively. If u and u E <1:(13) , then

Theorem 4.2.

r u· b dv

l~

+

r

la~

= r u. b dv

h

+ Urn . f + Woo . m + PJi.

u· s da

+

r

~

U· s da

r

la~

u· n da

+ Um . f + Woo . in + PJi.

(4.10) r

~

U· n da,

where PJi. is defined as in the Th eorem 3.1. Proof - Let 9 be the function (3.14). Then , an integration by parts yields

~9u.bdV+ la~ u·sda-um (~9u.bdV+ la~ u.Sda) (~gr x b dv + la~ r

- Woo

= ~ gu· bdv + la~ U· - Woo .

x

S

da)

s da - Um

(~gr x b dv + la~ r

+ PJi.la~ u

. n da

(~gbdV + la~ s da)

x s da)

+ PJi.la~ u

. n da

+ R- 1 r (u - w oo) . [C('Vu) - (p - pJi.)l]e r dv

lTR

- R- 1 r (u - w oo) . [C('Vu) - (p - pJi.)l]e r dv.

lTR

Hence (4.10) follows by letting R

-+ 00,

making use of the Schwarz 's inequality,

(3. 7)-(3 .8) and taking into account that u , u E <1:(13) , P - PJi., P - PJi. E L2(13 ) and u.b , b ' UELl(13).

I

Consider the functional on <1:(13) defined by


2

r c('Vu)dv- r(v-vm) · bdv- r (v-vm)·sda

l~

l~

+woo(v)·

la~

(~r x bdv+ la~r xS da)

106

(Principle of minimum potential energy). Let C be symmetric.

Theorem 4.3.

a) Let (u, p) be a solution to the traction problem of incompressible linear

elasticity such that u E <1:('13 ) and p E L2('13). Then, p(u) ::; p(v) ,

Vv E <1: ('13) ,

( 4.11)

where the eq uality holds only if u - w E !R. b) Let u E <1:('13 ) satisfy (4 .11 ). Then , there exists a function p E Hl ('13 ) such that the couple (u, p) is a solution to the traction problem of incompressible linear elastostatics. Proof - a) Let

Pg(V)=~ 2

r

J'B

gc(\7v)dv -

(l

+ w oo(V)'

r

J'B

g(v -vm)· bdv -

gr x bdv

+ fa'B gr

r

J813

g(v-vm ) ·sda

x sda) ,

where 9 is the fun ction (3 .14). Note t hat by Lebesgue's dominated convergence

theorem (4.12)

Let (u , p) E <1: ('13) x L2('13 ) be a solution to the traction problem of incompressible linear elastostatics and set z = v - u . Then, an integration by parts gives Pg(V) - Pg(u) =

~

l

gc(\7z)d v

-l

g(d iv C(\7u ) - \7p + b )· (z - w oo( z )) dt,

r (z - w oo (z)).(-pn+ C(\7u )n-s ) da +R- r (z- woo( z)).(-pI + C(\'u)) erdv. J +

J8 'B

J

TR

Hence, taking into account t ha t u satisfi es (2 .1) we have

r gc( \7z ) dv + Jr (z-woo(z))·(s-s)da + R - r (z -- w " ,(z)) . (C(\7u) - pI )e r dv. J

Pg (V) - Pg (u) = ~

2

IB J

TR

813

(·1.13)

107

Since

8

= 8 on 8'B, U E


E

L2('B) so that by Schwarz's inequality, (3.7) ---> + 00, the desired result

and (3.8) the last integral in (4.13) tends to zero as R follows letting R ---> + 00 in (4.13). b) Let z E V . Since v integration by parts gives

=

U + z E <1:('B) , taking into account (4 .11) , an

~l C:(VZ)dv- lz.[diVC(VU)+b]dV?O,

VzE V.

(4.14)

Since az, Va E 1R, is permissible in (4 .14) , we get

~a2l c:(Vz) dv -

a l

z · [div C(Vu)

+ b] dv ?

0,

Vz E V.

Hence it follows that

l

z . [div C(Vu)

+ b] dv = 0,

Vz E V.

Then, by appealing to a well- known result by G. de Rham (d., e.g., Temam lO ) and to Theorem 4.1, there exists a function p E L2 ('B) such that (u, p) is a solution to system (2.1) . Therefore, letting R

--->

+00 in (4.13) and taking into account (4 .11) ,

we have

~

rc:(Vz)dv+ Ja~r (z-woo(z))·(s -s )da?O,

2 J~

VZE<1:('B)

(4.15)

Since az , Va E 1R, is permissible in (4.15) , we get

r (z - w oo(z)) . (8 - s)da

Ja~

= 0,

V z E <1:('B)

(4.16)

Hence

zm '

r

Ja~

(s-s)da+woo(z) ·

r

Ja~

r x (s-s)da=O,

Choosing first z E JC('B) (so that woo(z)

VZE
= 0), then z = 0(1)

(4 .17)

in (4.17), from (4.16)

we deduce that Sa~ = sa~ and (r x 8)a~ = (r x s)a~ . Then, from (4.16) it follows that

r

Ja~

z.(s-s)da=O,

VZE<1:('B).

(4.18)

108

By well- known results (see, e. g., Solonnikov & Scadilov 8 ), (4 .1 8) implies that the tangential component of s - 8 vanishes on 8'B. Then, from (4 .18) we get

r [(s-s)·n]z·nda=O,

\iZE <E('B ).

Ja'B

(4. 19)

Let v be a continuous fun ction on 8'B. It is well-known that the Newmann

problem /::"cr = 0 ('Vcr)·n=v

in 'B at8'B

admits a unique solution cr E J-C('B) (modulo a constant) . Choosing implies that

r [(8 -

s) n]vda ,

Ja'B

Z

= 'Vcr , (4. 19 )

\iv E C(8'B ).

Hence the desired result follows.

I

In the following theorem we give some sufficient conditions on the elasticity tensor C and on the behavior at infinity of the solutions u to system (2. 1) assuring that u E <E('B).

Let (u,p) be a solution to system (2.1) and assume that one o[ the [allowing conditions is satisfied. a) C is constant and

Theorem 4.4.

u -

W

= o(r) ,

(4.20)

[or some wE 91. b) There exists a constant elasticity tensor Coo such that

(4.21)

[or some constants 6 E (0 , 1) , and ( 4.22)

[or some w E 91 , where c; E (0,3) is such that 166

2

<

Co~cJ ~

(

1-

D'

(4.23)

109

and

C4

is the constant appearing in (3.25). Then

U

E Q!('B).

Proof - Let C be constant. Let (v = 'TIU, w = 'TIp) be a solution to system (4.3) and let (VI, wIl, (V2 , W2) be the solutions to the systems

divvI = 1'J TR

in SR ,

Vj=V

at8SR,

divC(\7v2)

+ Iff = \7W2

(4.24)

in SR,

(4.25)

V2 = 0 Observe that v = VI + V2 and VI satisfies (3 .25). An integration by parts yields

r c(\7V2) dv JrSR W2 div V2 dv + JSrRV2· =

J SR

Iff dv .

(4 .26)

Hence by (3.2)

lisR W2 div V2 dvl

~ isR c(\7v2) dv + lisR V2· Iff dvl ~ c (13 (1'J -1'J TR )2 dv + lisR V2 . Iff dV/) .

(4.27)

Therefore, from (4 .26)-(4.27), taking into account (3.5)-(3.6) , we have Pm isR (9V2)2 dv

~ C (13 (1'J -1'JTpY dv + lisR V2 . Iff dV/) ~c(

~C

r (1'J-1'J

k3

(13

Hence, by properly choosing

TR

)2dv+E

[(1'J -1'J TR )2

r r-2v~dv+cl Jvr (rlfffdv)

J~

+ (rlff)2] dv + EisR (9V2)2 dV)

E, it follows that (4.28)

110

where we set

K =

C

(l3

[(iJ - iJTfY

+ (r~Pl dV)

Making use of the inequality (VV)2 ::; 2(Vvd 2 + 2(VV2)2 and (VVI)2 ::;

2(VV)2

+ 2(VV2)2,

(3.25)-(4.28) imply that

f (Vv)2dv::; 2 f (Vvddv+2 f (Vvddv

is

is

p

is

p

p

::; 2C4 (

!!... ) 3 f (Vvdd v + 2K R

::; c (

(4.29)

iSR

(~) 3 isR (VV )2 dv + K)

Now , by (3. 12), t he properties of T] and (4.20) , we have

f (Vv)2 dv ::; 2 f (Vu)2 dv + c f

iSR

IBR

iS2f1.

(Vv )2 dv

::;C (R -2 f (u-w )2 dv+ f (rb)2 dv+ f

iTR

i'B

i 8'B

lu -w lls lda

+ f _(VV)2 dv + hi + K) = o(R 3 ) .

iS2R

Therefor e, letting R --;

+00 in (4.29) , we conclude that u E \!: ('B ).

Let C satisfy (4.2 1). Consider the system div Coo (''VVI ) + div [C - Coo]('V'v ) + l/f = 0 divv = iJ where v =

T]U,

W=

T]P

and l/f, iJ are given by (4.4).

Let (VI , WI) a nd (V2 ' W2) be the solutions to t he systems

divVI = iJTfI. ' VI =

V,

in SR , at

asR ,

111

Integrating by parts, we h ave

r \7v2'

iSR

=

COO (V'V 2) dv

+

r

W2 div V2 dv

iSR

r V'V2' [C - Coo](V'v) dv + iSrRV2' IJi'dv. iSR

( 4.30)

Hence by (3.2) it follows that

Ir

iSR W2 div V2 dvl ::; CI/.LI irm3(19 -19 Y dv T2i

+ where /.LI

= ICoo l·

IiS r V'V 2' [C R

Coo]( V'v)dvl +

IiSR r V2' IJi'dv l,

(4.3 1)

Making use of (4.31) , the arithmetic- geometric mean inequality

and (3.5) - (3 .6)-(4.21) in (4.30) , we have /.La

r (VV2)2 dv ::; 2/.La8 iSRr IVV211Vvi dv + 21 iSRr V2 ' IJi'dvl +CI/.LI r (19 - 19 Y dv ::; /.La r (VV2)2 dv i m3 4 iSR

iSR

T2f

+4/.La 82 ::; /.La 2

r (VV)2 dv + ~~ iSrRr-2v~ dv + M

iSR

r (VV2)2 dv + 4/.La 82 iSrR(VV)2 dv + M,

iSR

where

Hence ( 4.32) Since v

= VI

+ V2 , by reasoning as we made in deriving (4.29) and making use of

(3.25)-(4.32), we get

r (Vv)2 da ::; 2 isrp(V VI) 2 da + 2 ispr (VV2)2 da ::; 2C4 ( £) r (VVI)2 da + 168 2 r (Vv)2 dv + 4M R iSR iSR ::; 4c4(1 + 88 2) ( £) r (Vv)2 da + 168 2 r (Vv)2 dv + MI R iSR iSR

isp

3

3

::; [36c4

(£) R

3

+ 168 2 ]

r (VV)2 dv + M

iSR

I,

(4.33)

112

where MJ = 4M(1

+ 2C4) .

Vv E L2(JR3) To prove this , we proceed p er absurdum. If 2 L (JR3) , in virtue of (4.23) there exists a positive RJ such that

We affirm that

\7v

tf.

so that (4.33) implies that

Then , in virtue of Lemma 3. 11 and (3.12), we get

(4.34)

Letting R

-+

+00 in (4.34) and taking into account (4.22) we have that the LHS

of (4.34) is nonpositive.

Vv tf.

Therefore , since this contradicts the extra assumption

L2(JR3), we conclude that

Vv

E L2(JR 3). Hence the desired result follows at

once.

Remark 4.2 - Observe that, if C - C oo = 0(1),

(4.35)

then (4.21) is satisfied by a ny positive number b, for some RJ (\). Hence it follows tha t in this circumstance we can choose as .; any arbitrar y positive number less than 3.

113

Theorem 4.5.

(Uniqueness of solutions to the traction problem). Let (UI,PI),

(U2' P2) be two solutions to the traction problem of incompressible lin ear elastostat-

ics. Set U

= UI

- U2, and assume that one of the following conditions is satisfied.

a) (4.36)

where

Q

is given by (3.20).

b) C is constant and (4.37)

c) (4.21) holds and

r

JTR

(VU)2 dv = o(R'),

(4.38)

for some <;" satisfying (4.23).

d) (4.35) holds and (4.39)

for some positive constants c. Then U E <1:('13). Moreover, if (PI - P2)!" = 0, then UI - U2 E 91 and PI = P2. Proof

Of course, since system (2.1)-(2.3) is linear, (u,p) is a solution to

system (2.1 )-(2.2) with zero body force b and zero surface traction s. U

If U satisfies (4.36), letting first R E <1:('13). Then, from (4.2) we have

rc(\7u) dv =

J~ Hence, if (PI - P2)!"

= 0,

---t

+00 in. (3.19), then p ---t +00, we see that

(PI - P2)!"

r

Ja~

by (2.2) it follows that UI - U2 E 91 and PI - P2 E R .

Finally, by (2.4) we deduce that PI = P2· If C is constant, using (3.18) (of course, with b

Hence, letting R

---t

U· n da.

= s = 0)

in (4 .29) we have that

+00 and taking into account (4.37) we get the desired result.

Finally the proof of the theorem in the cases c) and d) is analogous to the previous one (with obvious modifications) so it is omitted.

•

114

Remark 4.3 - From (3.12)-(3.18) it follows that (4.37) holds if either (4.20) or

Vu =

0(1).

(4.40)

is satisfied. Moreover, (4.38) holds provided either (4.41) or (4 .42) holds. Finally, (4.36) [resp. (4.38)] is implied by one of the following conditions

(u - w)2 = o(rQ-l)

(VU)2 = 0(r

Q -

[resp. (4.22)]'

3) [resp. (VU )2

= 0(r,-3)].

In particular, we have the following Liouville's theorem.

Let 13 = ~3, let (u,p) a solution to system div C(Vu)

=

Vp (4.43)

divu = O.

and assume that one of the following hypotheses is m et: i) C is constant and u satisfies either (4.20) or (4.40). ii) (4.20) holds and u satisfies either (4.41) or (4.42).

Th en u E 91 and p E

~.

Remark 4.4 - Let C be constant and 13 = ~3. If (u ,p) satisfies system (2. 1), then so does the couple (Dku , DkP) , where Dk = 8k / 8x~l.r;O.r;3, (k = kl Therefore, from (3.25)- (3.12), taking into account. that 13

=

~3

+ k2 + k3 ).

=> ') = 0, we get

115

Hence it follows that U equal to k - l.

= o(rk)

implies that U is a polinomial of order at most

Let {al~,a2~} be a partition of a~ and let iL, al~

oS

be two continuous fields on

and a2~ respectively. The mixed boundary-value problem of incompressible

lin ear elastostatics consists in finding a solution (u, p) to system (2.1) which satisfies the boundary conditions (.) U = iL at al~' (4.44) 8 = oS at a2~' If a2~

= 0,

then the above problem is known as displacement problem.

Let

'I(r,JL) = {(u,p): Theorem 4.6.

8813

= rand (r x 8)813 = JL}.

(Uniqueness of solutions to the mixed problem) . Let

(UI'

pd,

(U2,P2) E 'I(r,JL) be two solutions to the mixed problem of incompressible linear elastostatics. Set U = UI - U2, and assume that one of conditions a), b), c) or d) is satisfied. Then

9u E L2(~).

Moreover, 11 (PI - P2)J.L = 0, then

UI -

U2 E 91 and

PI = P2·

The proof of Theorem 4.6 is analogous to that of Theorem 4.5 so it is omitted. Remark 4.5 - Note that if Um

= 0 and

Woo

= O.

UI

and U2 assume the same value at infinity, then

Therefore, in the present circumstance, we see that in

Theorem 4.6 the hypothesis that (UI ' PI) ' (UI' PI) E 'I( r , JL) is unessential. In the light of this observation (see also Remark 4.3) we can state the following (Uniqueness of solutions to the displacement problem). Let Theorem 4.7. (UI,PI), (U2,P2) be two solutions to the displacement problem of incompressible

linear elastostatics. Set U =

UI -

U2 and assume that one of the following condition

is satisfied:

j) C is constant and U = 0(1). jj) (4.35) holds and:Jc: > 0: U = O(r-E). (.) By a solution to system (2.1) we mean a couple (u , p) which satisfies (2.1)(4.42) pointwise.

116

Th en

•

UI

=

U2

and PI = P 2 modulo a uniform pressure.

5 - A well-posed ness theorem for the traction problem. In this section we aim at showing that the traction problem of linear incom-

pressible elastosta tics in an exterior domain admits a unique solution (u , p) E

(E CB) x L2('B ) which depends cont inuously upon t he data (b, s) wit h respect to suitable norms. To this end we say tha t a displacem ent field u E (E ('B ) is a weak solution to syst em (2. 1)-(2.3) iff

D(cp , u ) =

l

\lcp . C(\lu ) dv

= ![cp-Woo(cp )]· bdv+

J'B

!

J 8'B

(5. 1)

[cp-woo( cp )]·s da,

for every cp E (E('B ), where w oo( cp ) is given by (3.10). The following t heorem holds. Theorem 5.1.

System (2. 1)-(2.3) admits a unique solution (u , p) E (E ('B ) X L2('B ).

Moreover (5.2)

Proof - Consider the quotient space

e

= (E ('B )/ 9t Of course

e

space with norm Ilcp 'll = U'B( Vcp )2 dv } ~ Setting, V cp o E e ,

D(cp' , u ') = D(cp ,u ), ! [cp - w oo(cp)]· b dv+ ! [cp-woo(cp)]·sda, J'B J 8'B equation (5. 1) is equi valent to I: (cp ') =

D(cp' , u *) =

~(cp*) ,

By So bolev's imbedding th eorem we have

V cpo E

Q; '

is a Hilbert

117

Then, by Schwarz 's inequality and (3.7)-(3.8)-(5.3), we get 1L;(cp*)I::;

(l +

r- 2[cp - Woo (cp)]2 dv

l

(rb)2 da)

(h'B r- 2[cp - Woo (cp)]2 h'B (rs)2 (l (rb)2 + h'B (rs)2 ~ dv

::; cllcp*1 1

dv

~ da)

~

da)

Hence it follows that the linear functional L; is continuous on E*. Then, since f2 is coercive on <1:* , the Lax-Milgram lemma implies that (5.1) admits a unique solution

u (to within an infinitesimal rigid displacement) in <1:('13) which satisfies (5.2) too. Now, from (5 .1) we have

f2(cp,u) = 0,

\:Icp

E

V

so that in virtue of a result by G. De Rham (cf., e.g., Temam lO ) and Theorem 4.1, there exists a function p E L2('13) such that (2.1) is satisfied at least in the distributional sense on 'B. However, since C E GleE) well- known regularization results (Giaquinta & Modica3 assure that the solution (u , p) we found is a classical one on 'B. Finally, by repeating the reasoning we used in the second part of Theorem 4.3, we realize that the boundary condition (2.3) is pointwise satisfied at 8'13.

I

Remark 5.1 - Let t be a continuous vector field on 8'13 such that t . n = O. The contact problem of incompressible linear elastostatics consiste in finding a solution (u, p) to system (2.1) which satisfies the boundary conditions u · n =0, 8 -

Let <1:0('13)

= {u

E <1:('13): u ·n

(8'

n)n

= 0 at

= t,

at 8'13.

(5.4)

8'13} . By a weak solution to system (2 .1)- (5.4)

we mean a field u E <1:0('13) such that

f2(cp , u)

=

f[cp-w oo(cp)]· bdv+ f [cp-woo(cp)] · tda ,

)13

)813

\:ICPE<1:o('13).

Repeating the steps in the proof of Theorem (5.1), we can show that system (2.1) admits a unique solution (u,p) E <1:0('13) x L2('13). Also, by appealing to

118

Theorem 4.4 , we can prove that this solution is unique in the class defined by

(4.20) [resp. (4.22)] provided C is constant [resp. C satisfies (4.2 1)]. Remark 5.2 - In this paper we required that the data satisfy strong regularity assumptions. This has been made only for the sake of simplicity. Indeed, it is not difficult to realize that our techniques can be reproduced to work with weak solutions to system (2. 1) (see, e.g., Kondratiev & Oleinik 5 , Russo 7 )

,

i.e., for dis-

placement fi eld u E HI~c('B ), with divu = 0 such t hat

r \7
h

'V

stance we have to assume that C is bounded, b satisfies (2.2), S E L ~(a'B) and every 'B J is a Lipschitz domain .

•

6 - The traction problem in the half-space.

In this section we assume that 'B is the half-space IR.t an consider the traction problem divC (\7u)+b=\7p inB divu = 0

in B

(6. 1)

at aB, where the fields C and b satisfy assumptions (2.2), (2.3) respectively and the hypotheses

s obeys (6.2)

For the sake of simplicity, we assume t hat the elasticity tensor C is constant. However , the more general case C = C(x) can be t reated by the methods of Section 4. Theorem 6 .1 (Caccioppoli 's inequality) . Th en , 'V R > 0, 'V W E ryt ,

Let (u ,p) be a solution to System (6. 1).

119

Proof - Let cp be the field defined by

cp=

g2(U-W )- h , in T~ (u-w ), in S~

{ 0,

in]R3 \ SiR '

where 9 is the function (3 .14) and h is the solution to equation (3.1) in Tii. corresponding t o

f = div[g2(u - w )]. Then, an integration by parts gives

rld \7u)dv = JT~r \7h · C(\7u)dv - J~rg(u - w ) . C(\7u )\7gdv + r cp . b dv + r cp '.9 da. J~ Ja~

J~

(6.4)

Since, by Schwarz-Holder inequality and Lemma 3.10 1

Ifa~g2(U-w) . .9dal ~ (fa~ [g(U-w)]4dar 1

~ c (l [\7(g(u -

W) )]2

~c (l [V (g( u -

w )) f

dV)" (fa~ S~da)

~ c (~ l (gV u )2 dv + C and

I1"

3

(fa'B (9s) 1d ar 3

4

dV)! (fa'B s~daf 1

l

(u - W)2(\7g)2dv

+

(fa~ S~da) %)

rr

h'dal ~ (1,T;/da (1,;1 da) 1 ~, ([/Jh)'dV (1,.'lda) I ~ c (h~ (u - W)2(\7g)2dv + (fa~ s~da) %) ,

repeating step by step the proof of Lemma 3.6 from (3 .15) on (putting f = 0) and properly choosing~, from (6.4) we obtain (6 .3).

I

120

Remark 6.1 - Choosing

from (6.3 ), in virtue of Lemma 3.2, it follows t ha t

Let (u ,p) be a solution to System

Theore m 6.2 (Work a nd energy theorem).

(6. 1). If (6.6)

then

Vu , p E L2('B ) and

r

c( \7u ) dv =

JB where

f

and

Tn

U2

U·

bdv

+

r

JaB

U·

s da

+ U rn f + W (X)

(6. 7)

. Tn ,

are defin ed in (4.1).

Proof - Let ER Ul and

r

JB

= {x

E

- +

SR: r(x)

= R}

and let

r R = {x

E

- +

SR:

X3

= OJ.

Let

be the solutions to the system div C(\7ud = \7Pl divUl = 0 Ul =u -Pl e3

+ C(\7u )e3

= 0

divu2 = 0

in S~

atoE R at Or R

in S~

= 0

at oER

+ C(\7u )e3 = s

at OrR

U2

-Ple3

in S~

By Lemm a 3.9 we have tha t

(6.8)

121

On the other hand, an integration by parts gives

Hence, taking into account t hat

IhAU2 . bdv l ::; (h AT- 2U2? dv hA(Tb)2dV) ~ ::; C

(h A(~U2)2dv hA

(Tb)2 dV)

and

/IRU2 S d a / ::; (IRU~da) t (IRs~ da s,

(lt

r(L

1U )'dV

~

f

,1 da )',

it follows t hat

Putting together (6.8) and (6 .9) and taking into acco unt that

(6. 10)

Hence, choosing w it follows that

= (VU )y+RT + [u + (V u )T+RTj T+Rand making use of Lemma 3.2 , (6. 11 )

Letting R ----;

+00 in (6. 11 ) and taking into account (6.6), we have t hat Vu

E

L2('13 ).

122

Next, let us prove that the pressure p is square summable over:E . To t his end, set H~R(S~) = {u E Hl (S~ ) : u = O} ,

let Hr:(S~) be its dual a nd denote by

t he divergen ce operator. The range of the adjoint

L:

L2 ( S~ ) ---. Hr: ( S~ ) of L is

closed and its kernel is {O} , so that t he following inequality holds (6.12 )

where

11 ' ll wr R'(s+) stands R

for the norm in Hrl ( S~ ). These properties are proved R

in Giaquinta & Modica 3 Since the stand ard Poincare inequality

holds in H~R(S~) , the scalar product (V'v , V'cp ) =

r

Js~

V'v · \ cpd t"

defines a norm wh ich is equivalent to that of Hl ( S~ ) . Therefore. b~· Riesz's th ea-

rem ,

where <, > denotes the duality between H~R (S~) and H i: ( S ~ ) . defin es a can onical

isomorphism between H~R (S~ ) and Hr:(S~) such that (6. I·!) Now, a n integration by parts giw>s (6.15 )

123

Choosing X

= L(p)

in (6.13), from (6.15) , taking also in account that

r

ls+ pdivcpdv =< L(p),cp >, R

it follows that

r

ls~

\lcp. \lvdv

=

r

ls~

\lcp. C(\lu)dv -

r

lrR

cp. sda ,

V cp E

H~R (S~).

Hence, setting cp = v and making use of the inequalities

obtained by Schwarz's inequality, we have

Hence, taking into account (6.12) , (6.14) it follows that

(6.16)

Therefore, letting R

-7

00

we deduce that p E L2('13).

Finally, (6 .6) is proved by repeating step by step the proof of (4.2) from (4.9) on. I Remark 6.2 - By repeating the arguments in section 4 we can prove that Theorems

4.2, 4.3 hold for the half-space too in the functional class defined by hypothesis (6.6). Also, existence of solution is proved by means of the arguments we used in Section 5. The only difference consists in the proof that the last member of (5.1)

124

defines a continuous functional in

e

Indeed, this property follows now from the

estimates

1~(cp)1 :::; +

(1

r- 2(cp - W oo (cp) r - CPoo)2dv

1

(r b)2 d v ) t

1

3

4

4

(1 r- (cp - W oo(cp)r - cpoo)4da) (la'B 84 da) 2

:::; cllcp'lI

((1

(r b)2

dv)t + (la'B 8~da)~)

Remark 6.3 - From (6.10)-(6 .11) it easily follows that the traction problem of incompressible linear elastostatics in the half-space has a unique solution in the classes {u:u=u*+o(r)}

{u:Vu=S'+o(l)} ,

where u' and two assigned vector and second-order tensor fields , respectively. Also, this result is sharp, as it is inferred by the the following counter-example. Let 13 be homogeneous and isotropic and assume that b

=

0,

s = O.

Then,

System (2 .1)-(6.1) becomes j..l~U

= \lp

divu = 0 -pe3

+ 2j..lfh u + j..l€3

x curlu

=0

in 13 in 13

(6.17)

at 813.

It is easy to check that System (6.17) admits the solution

as well as the trivial one and u = 0 (r),

Vu =

0 (1).

R emark 6.4 - Reproducing a technique outlined by Giaquinta and Modica3 we can prove that, if u is a weak solution to system (2.1) corresponding to zero traction at the boundary, then

125

Hence, taking into account Korn and Caccioppoli inequalities, it follows that

(6.18)

Then, if u

= 0 (r 2 ), letting R -> 00 in (6.18), we have that Vu =

A so that u

=

AT,

with A and a constant vector and constant second-order tensor, respectively.

•

7 - Uniqueness theorems in dynamical theory. We aim at showing now as the results of the previous sections can be used to

get uniqueness for the boundary-initial value problem associated with the system of linear incompressible elastodynamics in exterior domains. We assume, for the sake of implicity, that the density field of B is equal to 1. As is well-known, in the absence of body forces the motions of :E are governed by the system (Gurtin4) ii = div C(Vu) - Vp

divu =

in:E x [0, +00)

a

(7.1)

where the superimposed dot means partial derivative with respect to time. Let

u and

S be two continuous fields on 8 1 :E x [0, +00) and 8 2 :E x [0, +00)

respectively, and let uo, Uo be two smooth fields on:E. The mixed boundaryinitial value problem of linear incompressible elastodynamics consists in finding a solution (u,p) to System (7.1) which satisfies the boundary conditions u=u

at8 1 :Ex[0,+00)

=s

at 8 2 :E x [0 , +00)

s

(7.2)

and the initial conditions u

If 8 1 :E =

0

[resp. 8 2 :E

placement] problem.

=

= Uo,

0]

U

= Uo

in:E x {a}.

(7.3)

the above problem is known as traction [resp. dis-

126

A

COUNTER-EXAMPLE TO UNIQUENESS -

As has been pointed out in Starita9

we cannot expect uniqueness of solutions without making suitable hypotheses at infinity either on the displacement field or on the pressure field. Indeed , assume that the body is homogeneous and isotropic. Then System (7.1) becomes

it = 6.u .- \lp.

(7.4)

If :E = IR.3 , (7.4) admits the solution

(7.5) as well as the null one, both satisfying the initial conditions u = 0 , it = 0 in IR. 3

X

{O}. Hence it follows that , as far as the Cauchy problem is concerned, uniqueness of solutions might be find in the following class it = {(u,p): u = o(l)} .

In Starita9 is proved, among other things , that , if C is symmetric, then uniqueness of solutions to System (7 .1) , (7.2), (7.3) in exterior domains holds in the set

{(u,p) : 30> O, p = O(r l -f)}. Of course, if the pressure in System (7.1) has to be understood as a constraint, it is preferable to get uniqueness in functional classes defined only by a suitable behavior at infinity on u. Then, the uniqueness problem is solved if we are able to show that the sole solution to System (7.1) corresponding to zero data is the trivial one in U. Let X be a (scalar, vector or second order) tensor field in :E x [0, +(0 ). "\ is said to have a Laplace transform \::', if there exists a positive real nwnber aD such that , 'rj a ::::: aD, the integral

converges uniformly in :E.

127

Let £ denote the whole set of the couple (u,p) such that u , V'u, V'V'u a nd V'p have Laplace transform. Of course, if (u, p) is a solution to System (7.1) corresponding to vanishing data, then the couple (u' , p') is a solution to the stationary system a 2 u' = divC(V'u*) - V'p' in :B divu' = 0, with the boundary conditions

=0

at 81 :B

s· = 0

at 82 :B.

u*

The following theorem holds true.

Let C be constant and positive deEnite. Then , the displacem ent problem of linear elastodynamics in exterior domains has at most one solution (modulo a constant pressure) in the class II n £.

Theorem 7.1.

Proof - By repeating step by step the technique outlined in the previous sections we easily obtain that u' == O. Then the desired results follows from the obvious fact that u*(x , a) = 0 ,

'if a 2: ao

{===>

u(x , t)

== o.

I

It is worth stressing that in Theorem 7.1 is not required that the elasticity tensor C is symmetric.

Acknowledgement - Work performed under the auspices of C.N .F.M. of italian C.N.R.

•

References 1. Babuska, I. and Aziz, A.K., "Survay lectures on the mathematical foundation of the finite element method", in The Mathematical Foundation of the Finite

Elements Method with Applications to Partial Differential Equations, A .K. Aziz ed., Academic Press, New York (1972) .

128

2. Galdi , G.P., "An introduction to the mathematical theory of the Navier-Stokes equations", vol. I, Springer Tracts in Natural Philosophy 38, Springer-Verlag (1994). 3. Giaquinta, M. and Modica, G., " Non linear systems of the type of the stationary Navier- Stokes System", J. reine angew. Math. 330, 173- 214 (1982). 4. Gurtin , M. E. , "The linear theory of elasticity", in Handbuch der Physik, Truesdell, C. ed., vol. Vla/2, Springer (1972). 5. Kondratiev, V.A. and Oleinik, O.A., "Asymptotic properties of solutions of the elasticity systems", in Applications of Multiple Scaling in Mechanics (ed. Ciarlet , P.G. and Sanchez- Palencia, E. ed. , Masson , Paris (1986). 6. Meier , M. , "Liouville theorems for nondiagonal elliptic systems in arbitrary dimensions", Math. Z. 176 , 123- 133 (198 1). 7. Russo, R. , "Linear elasticity in unbounded domains" , forthcoming. 8. Solonnikov, V.A. & Scadilov, V.E. , "On a boundary value problem for a stationary system of Navier- Stokes equations", Proc. Steklov In st. Math . 125 , 186- 199 (1973). 9. Starita, G. , "Some theorems in linear incompressible elastodynamics for exterior domains" , SAACM 2 , 445- 462 (1992) ,. 10. Temam, R. , "Navier- Stokes equations", North-Holland, Amsterdam (1976).

130

•

1 - Introduction. Many problems arising in various branches of mathemat ical physics and applied

ma thematics lead to an equation of the form

(1.1)

A(x,c:) = 0,

where x belongs to a Banach space X , c: is a real parameter , a nd A is an op erator t aking its values in a Banach space Y . Suppose that A(x,·) is differentiable at O. Then , putt ing

(1. 2)

Ao(x) = A(x, 0), and

B (x, O)

= Bo (x),

B (x,c:) = c:- 1 (A(x,c:) - A(x,O)),

(1.3)

if c: =/:. 0,

we can write

A(x,c:) = Ao(x)

+ c:B (x,c:).

( 1.4)

Moreover , we suppose t hat t he (unperturbed ) operator Ao: X

--+

Y has two affine

symmetries of t he type described and discussed in Valent 171 In Section 2 these symmetri es will be defi ned by using two affin e representations of a Lie group G , one on X and the ot her on Y , related by a linear mapping

T:

X

--+

Y which , "'hen

t he operator Ao describes a boundary value problem , may be a trace operator. T he abstract symmetries considered for Ao have been suggested to us

b~'

what. happens

in concrete sit ustions: in particular in the mat hematical theory of fin ite elasticity, where t he frame-i nd ifference of t he material response and the balance of the angular moment um im pose affine sy mmetries wh ich are a particular case of those proposed hare (as we sha ll see in Section 9). In Valent lSI , where t he case when A is affine in

E

has been treated. we have

seen how t he presence of symmetries for .-I,) crt'ates serious diffi culties in obt.aining local existencp t heorems for t he eq uation A (.1', c:) = O.

\\'e shall

how that t he

a pproach dev ised there t.o a local analysis of solutions of t. he equat illn A (.r, c:)

=0

in t he presence of affine symmet ries for Ail. also works in the more general context ('()lls idered here'.

131

A first result (in Section 6) concerns a local theorem on existence, uniqueness and analitic dependende on c in the case when no element of the tangent space TeG to the Lie group G at its identity element e is "critical" for the operator Bo at a pair (~o, go) E X x G. In Section 7 we associate to the (perturbation) operator B : X x R at any

~o

--t

Y,

EX, go in G and c E R, certain linear subspaces of TeG that serve to

discriminate the situation of essential singularity from those in which the singularity is apparent. Subsequently, a local theorem on existence, uniqueness and analitic dependende on c is proved in absence of essential singularities (see Theorem 8.1). In Valent & Bertin l9 ] and Bertin & Valent ll ], a strategy for studying local existence, also when essential singularities are present, is developed in the context of finite elastostatics with general live loads. It may be interesting to see whether some crucial ideas contained in these papers continue to apply in the abstract perturbation scheme considered here. As is well known, the literature about perturbation problem in the presence of symmetries and the role of symmetry in bifurcation theories is very exstensive. However, we beliewe that our formulation of symmetries and definition of potential for an operator and our approach to the local existence problem for the equation

A(x, c)

= 0 may constitute a small contribution in this area,

and be a starting point

for further investigations. Among the papers that are closest to our contents and point of view we mention a recent article of Chillingworth I2 ], the books of Golubitsky, Stewart and Schaeffer I3 J,I4], and of Marsden and Hughes]5}, as well as the works of Signorini, Stoppelli, Capriz and Podio Guidugli, Bharatha and Levison, Spector, Chillingworth and Marsden and Wan, Marsden and Wan, Ball and Schaeffer, Podio Guidugli and Vergara Caffarelli, Podio Guidugli and Vergara Caffarelli and Virga, Le Dret, and the books of Grioli, Fichera, Wang and Truesdell, Truesdell, Gurtin, Ciarlet quoted in Valent I6 }.

•

2 - Potentials for a mapping F: X

-+

Y.

In this Section X, Y will be real linear spaces , T

:

X

-+

Y a linear mapping and

{3y an inner product on Y. Let ((H,{3H),
132

(i.e., H is a Hilbert space with the inner product /3H, a nd 'P H is a linear isometry of (Y, /3y) onto a dense subspace of H) , and let j H be the canonical isomorphism of H onto its dual H' . We now propose a definition of potential relative to

/3y for a mapping F : X Definition 2.1.

---+

T

and

Y

A C1-function fH : H

---+

lR is a potential for F: X

-+

Y with

respect to T and /3y if (2. 1)

where

f~

is the differential of f H·

It is no difficult to prove (see Valent l71 ) that if (( H ,/3H),'PH)) and (( K , /3K) ,

'PK) ) are two Hilbert completions of (Y ,/3y) and'P is the canonical linear isometry of (K , 13K) onto (H , /3H) such that 'P 0 'PK = 'P H, then the fun ction fH 0'P : K -; lR is a potential for F with resp ect to T and /3y whenever f H : H

-+

lR is so.

Now, we suppose that T is one-ta-one and that T(X) is dense in (Y , /3y) , and consider the inner product /3x on X defined by putting

X. To any completion (( H , /3H) , 'P H)) of (Y ,/3y), we associate the Hilbert completion ((H, /3H) , 'PH 0 T) ) of (X , /3x). Then , it is easy to see that the following three properties are equivalent:

for all

X l , X2 E

i) the fun ction f H : H ---+ lR is a potential for F with respect to T and /3y, ii) the fun ction f y : Y ---+ lR related to fH by f y = fH 0 'P H satisfies \:Ix E X ,

ijj) the fun ction f x : X

---+

lR related to f H by

I.,

(2.2 )

= f H 0 ~ H OT satisfies

\:Ix E X .

(2.3)

Thus (when T is one-to-one and T(.\ ) is dense in P ",cld) any C 1-function fy Y -+ lR satisfy ing (2.2) and any C 1-function fx : X -; lR satisfying (2 .3 ) can be called a potential for F with respect to T a nd /3y.

133

We observe that it may occurs (see Section 9) that Y ~ H and 'PH : Y is the identity function. In this case (2.3) takes the simpler form

f'x(x)

•

= jH(F(x)) 0 T.

--->

H

(2.4)

3 - Some notations and technical preliminaries. Henceforth X and Y will be real Banach spaces with norms II . Ilx and II . Ily.

Let A(X) to denote the (Banach) space of all continuous, affine mappings from X into itself endowed with the norm 0: f-> 1100(0)llx + sup{lIo:(x)llx : Ilxllx S I}, let L(X) to denote the subspace of A(X) whose element are (continuous) linear

mappings, and let A(Y) and L(Y) to denote the analogous spaces for Y. Moreover, let G be a Lie group, let e be its identity element, and let TgG be the tangent space to G at the element 9 of G. Consider an affine representation

Pg of G on X and an affine representation 9 f-> pg of Y (namely, 9 f-> Pg is a homomorphism of the group G into the group of invertible element of A(X), and 9 f-> pg is a homomorphism of the group G into the group of invertible element of A(Y). The linear part of Pg and the linear part of pg will be denoted bi 19 and 19, respectively; so pg(x) = 19(x) + Pg(O) Vx E X, and Pg(Y) = 19(y) + Pg(O) Vy E Y The mappings 9 f-> Pg and 9 f-> pg are supposed to be analytic . The differentials at e of the mappings 9 I-> Pg, 9 f-> pg, 9 >-t 19 , 9 I-> 19 are denoted by v >-t Rv, v>-t Rv, v I-> Lv , v f-> Lv , respectively: they send TeG into A(X), A(Y), L(X) , L(Y), respectively. It is easy to see that

9

I->

(3.1) for all v E TeG, x E X, Y E Y In Valent 18] we proved that, for any 9 E G, (3.2) Let us set R = {Rv: v E TeG},

and, for any x E X and y E Y ,

R(x)

=

{Rv(x): v

E

TeG},

R(y)

= {Rv(y):

v E TeG}.

134

Note tha t (3.1) yields

(3.3)

VT E R .

•

4 - Symmetries tor the operator A o. The first symmetry we consider for the operator Ao : X

->

Y introduced in

Section 1 consists in t he following property: an affine representation 9 ,...... Pg of a Lie group G on X , an affine representa tion 9 ,...... pg of G on Y and a continuous , linear mapping

T :

X

->

Y exist such that Vg E G,

(4.1)

and Vg E G.

Ao 0 Pg = [g 0 Ao

(4 .2)

Moreover , we suppose that there is an inner product (Yl , Y2) ,...... Yl . Y2 on Y such that

Vv E TeG and Vx E X. We observe t hat , since (4.1) easily implies (4.3) is equivalent to the condition

Ao(x), T(Rv (x)) = 0

Rv

0 T

=

TO R,.,

(4.3)

Vv E TeG , condition

Vv E TeG and Vx E X

(-1.3)'

which can be put in t he form

Ao(x) E N(x) O

Vx E X,

( -1.3)"

where N(x) = R (T(X)) , i. e.

N(x)

= {Rv(T(X)):

v E TeG }

= {T(Rv(x)):

v E TeG} ,

and N(x)O denotes t he ort hogonal to N(x) in t he pre-Hilbert space (1', .), Note t hat, in view of (3. 1), fr om (4.3)' it fo llows that

Ao(x) . T(Rv(O)) = 0

(,U )

We sha ll see t hat symmetry (4.3) plays a crucial role when ~:,I1lmet r~' ( -I .~) is present. However , under sui table hypot heses on

..1.\)

and on the representations

g""'" Pg and 9 ,...... pg of G related by (-1 .1), condition (-1.3) is a consequence of the

symmetry (4.2), provided (4 .-1 ) holds. In fact t he fo llowing t heorem has been proved in Va lent I7).

135

Let the mapping Ao : X ---> Y have a potential of class C 2 with and the inner product on Y and satisfy (4.4), and assume that the

Theorem 4.1.

respect to

7

affine representations g

>->

Pg and g

>->

pg of G have the following two properties \:I g E G and \:IYI , Y2 E Y,

the derived algebra of {Lv: v E TeG} is pointwise dense in the subspace {Lv: v E TeG} of L(x).

(4.5) (4.6)

Obviously, in (4.6) TeG has to be regarded as a Lie algebra, with the Lie algebra structure carried by the Lie algebra of the Lie group G. We emphasize the fact that (clearly) condition (4.6) is satisfied if TeG coincides with its derived algebra, and that this occurs whenever the Lie group G is semisimple (efr. ,for instance, Varadarajan[IO), p. 313).

*

*

*

Now, we make some remarks about the case when the operator Ao "describes" a boundary problem (see an important example in Section 9). In this case Y is a product of Banach spaces, let Y = YI X I,

.. . ,

X .. . X

Ym , and there are Banach spaces

Xm containing X as a dense subset; moreover, for each j = 1, ...

,m, there

onto Yj. We set

is a continuous, linear mapping

7j

from

for all x EX; the meaning of

7

may be that of a trace mapping. When one

Xj

deals with a boundary value problem for differential operators the further four facts happen: 1) X, XI, . . . ,Xm , Y, YI , ... , Ym are space of ]Rn-valued functions, for some n. On each of then, the norm (defining the product topology) descends from a norm prescribed on the linear space of real valued functions that are the n component functions; 2) X is invariant under composition with all affine mappings from ]Rn into itself; 3) if an element x of X takes its values in a one-dimensional, linear subspace of ]Rn , then

71

(x), ... , 7 m (x) take their values in that subspace;

136

4) the affine representation 9 sentation 9

f->

I:1 g

f->

Pg of C on X descends from an affine repre-

of C on lRn in the following way:

pg(x)

= I:1 g 0 X.

We remark that, when this occurs , for any 9 E C , the mapping Pg : X

f->

X

remains continuous-and hence a homeomorphism-too for the topology on X defined by each of the seminorms x

f->

IITj(x)lIy" j = 1, ... , m, and also for the

topology induced on X by that of each X j, j

= 1, ... , m;

thus for every j

= 1, .. . , m

and for every 9 E C the mapping Pg can be extended to an affine homeomorphism

Pb from

X j

onto itself and there is

Cj (g)

E lR such that

Vx E X.

Then it is easy to see that, putting for every

(Xj)j=l , .. ,m

E Xl X ... X Xm

/g is a one-to-one, continuous, linear mapping (hence a linear homeomorphism) from X Ym onto itself such that /g 0 T = TO I g. Evidently, /g is the linear part of the affine mapping pg from YI x ... X Y m onto itself defined by setting YI x ...

for all

(Xj)j=l , .. ,m'

Therefore, we can conclude that if the conditions (1 ), (2), (3) are fulElled, then

for every representation 9 affine representation 9

•

f->

f->

Pg

Pg of C on X of the type described in 4) there is an ofC on Y which satisfies (4.1 ) .

5 - Further assumptions on the operator Ao . A basic Lemma. We suppose that the (unperturbed) operator Ao has the symmetries stated in

the previous Section with

T

one-to-one, and supp ose that there is (0 E .\ such that

Ao((o) = 0,

(5 .1 )

Ker A~((o) ~ R((o ),

(5.2)

dim Ker A~((o) :::: codimlm A~((o),

(5.3)

137

where A~(~o) denotes the differential of Ao at ~o. Note that, since from ((4.2),(5.1)) it easily follows that Ker A~(~o) 2 R(~o),

((4.2),(5.1),(5.2)) implies (5.4) Note also that ((4.3),(5.1)) implies A~(~o)(x) E N(~o)O '
<; N(~o)O; then, as Y

= dimN(~o), Ker A~(~o)

=

= N(~o) E9 N(~o)O, we have codimImA~(~o)

2: codimN(~o)O

whence codimImA~(~o) 2: dimKer A~(~o), because (by (5.4)) dim dim R(~o) = dimN(~o). In view of (5.3) this implies dimKer A~(~o) = codimImA~(~o).

Therefore, codimN(~o)O = codimImA~(~o) N(~o)O) implies 1m A~(~o) = N(~o)O Thus

< +00 , which (being

ImA~(~o) <;

(5.5) Using (3.3) it is possible to deduce from (5 .5) that, for any gEe,

where the subspaces N(P9(~O)) and 1m A~(P9(~O)) of Y are orthogonal for the inner product on Y (see Remark 2.1 in Valent I8l ).

*

*

*

Let ih, . . . , vr E Tee be fixed such that (R V1 " define a mapping I : X x Y -+ Rr by putting

for all (x, y) E X x Y Since Rv

0 T

=T

O

'"

Rv r ) is a base of R , and

Rv '
I(X , y) = (Y'T(Rv,((x)))j=l " , The mapping I will be an important role. We observe that it depends on the choice of the base (Rv 1 ,

Rvr ) of R, but the points where I vanishes are independent of the choice of the base of R. We shall be interested to such points ..• ,

138

and not to the particular function "( vanishing at them. Evidently, "( is linear in y and affine in x, and we have

,,((x, y) = 0

¢=}

Y E N(x)O

(5.6)

Later , we shall need that (5. 7) We remark t hat, by using (3.3), it is easy to prove that (5.7) holds if

The following simple lemma will be crucial in our approach to the local analysis of t he solutions of the problem (1.1 ) Lemma 5 .1.

For each

xE X

there is a neighborhood Uj; of x in X such that

N( x) nN(x)O = {O}

"Ix E Uj;.

Proof - N(x) is a (closed) linear subspace of Y of dimension r. For any x E X ---t ~r be the (continuous, linear) mapping defined by setting

let a(x) : N( x)

a(x)(y) = ,,((x, y) for all y E N( x). T he mapping a(x) is one-to-one, because if a(x)(y) = 0 with y E N(x) , then (by (5.5)) Y E N(x) n N(i)O , i.e. y = O. Therefore each element

of a suitable neighborhood of a(x) , in the topological linear space q,V(.rl . ~r) of all linear mappings from t he (r-dimensional) linear space N (i) of }' into

~r

endowed with the bounded convergence topology, is a one-to-one mapping . On the other hand , one can easily see that x

t-+

a(x) is a continuous mapping from ,\ into

L(N(x) , ~r) Then a neighborhood [Ii' of .1' in.\ exists such that a( :r) is one-to-one for every x E Uj;; thus if y E N( x ) n N(.r)" with 1: E Uj; , i.e . if (x, y) E Uj; x N(.'t) and a(x)(y) = 0, then y = O.

139

• 6 - A first theorem on existence, uniqueness and analytic dependence on c for the equation A( x, c) =

o.

To the operator B : X --+ Y (efr. Section 1) we associate, at any t; E X, the mapping M{ : G --+ R defined by putting

for all 9 E G. Moreover , for any go E G, we denote by go the differential at e of the translation 9 ....... ggo of G; thus go is a continuous , linear mapping from TeG into the tangent space TgOG to G at go.

Definition.

Let t;o E X, go E G, and 0 =I- v E TeG. We say that v is critical at

(t;o,go) for the operator Bo if the diHerential M~o(go) of M{o at go vanishes at the element go(v) of TgoG.

The following lemma, together with Lemma 5.1, are the principal tool towards the proof of Theorem 6.2.

Lemma 6.1.

L et (t;o,go) E X x G be such that M{o(go) = O. Suppose that (5.7)

holds and that B is of class

en, n ~ 1. If no element =I- 0 ofTeG is critical for Bo at

(t;o, go) then neighborhoods Uo of t;o in X and Wo of go in G and a number co ~ 0 exist such that for each (t; , c) E Uo x ~ with Icl < co there is a unique element

§(t;,c) in Wo such that

(6.1) Uo and Wo can be chosen such that the mapping (t;, c) ....... §(t; , c) is of class mapping is analytic at (t;0,0) provided B is analitic at (SgO (t;o), 0).

en; this

Proof - Suppose that no element =I- 0 of T eG is critical for B at (t;o , go) , and

for any (v , t; , c) E Te G x X x ~ set (6.2)

r

is a

en mapping from TeG x X

x ~ into R; it is analytic at (0, t;o, 0) provided B

is analytic at (Sgo(t;o),O) . Since B( x,O) r(v,t;,O)

= Bo(x),

in view of (5.7) we have

= M{((expv)go).

140

Note that r(O , (0 , 0) = M {o (90) = O. Since the differential a t 0 of the mapping v ...... r( x, (o , O) is the mapping

(6.3) fr om TeG into R , it is one-to-one because no element oj:. 0 of TeG is critical for Eo at ((0 , 90) . As dim R :::; dim TeG , this implies tha t dim R = dim TeG and that (6.3) is

an isomorph.ism from TeG onto R . Then, by virtue of the implicit function theorem applied to the equa tion r( v , ( ,0: )

= 0, open neighborhoods Uo of ( 0 in X , Vo of 0 in

TeG and a number 0:0 > 0 exist such t ha t for each ( E Uo and 0: E R wit h 10:1

< 0:0

there is one and only one element iJ(Co:) of Vo such that r(iJ(Co:) ,( ,o: ) = 0; Uo and Vo can be chosen such that the mapping (( , 0:) ...... iJ (( , 0: ) is of class and analytic

en,

at ((0 , 0) if E is analytic at (590 (( 0) , 0). Since the equality r( iJ(( , 0:) , ( , 0:)

= 0 means (6.4)

the proof is complete if we put, for ( E Uo and 10: 1 < 0:0,

g(( ,o:) = (exp iJ(( , 0: )) 90 ,

(6.5)

and we take as Wo a neighborhood of 90 in G such tha t (exp V)90 E Wo wit h v E TeG implies v E Vo·

I

Now, consider the equation (1.1) and let the operators E Eo : X

-+

Y , and Ao : X

-+

Y related t o t he operator A

X x R X x R

-+

-+

Y,

Y by

(1.2 ) and (1.3); thus (1.4) holds. The following theorem concerns (local) existence,

uniqueness, and a nalytic dependence on 0: of a solution Xe of (1.1 ) near 5 9 0 (( 0) when no element oj:. 0 of TeG is cri t ical for E o at (( 0, 90) ' Theorem 6.2.

A ssume th at A o and E are of class

en,

11

2: 1, that (4.1 ), (4. 2).

(4.3) , (5 .7) hold , and that th ere is ((0 , 90) in.\ x G such that (5.1), (5.2), (5. 3) are satisfied and M{0(90) = 0, and fixe a topological supplem entary E~o of R (( o) in X . If no elem ent oj:. 0 of TeG is cri tical for E o at ((0 , 90) then neigllborh oods U of ( 0

in X and W of 90 in G and a num ber E > 0 ex ist such th at for each 0: E R such that 10: 1 < E th ere are a unique ( , in U n (~o put ting

+ E~o )

and a un ique 9, ill W such th at

141

we have

A(x.,c:) = o. The mapping c: ....... x< is of class en in a suitable neighborhood of 0: it is analytic at 0 provided Ao is analytic at ';0 and B is analytic at (S 90 (';0), 0).

Proof - Suppose that no element f. 0 of TeG is critical for Bo at (';0,90), and let c:o, Uo, Wo and 9 be as in the statement of Lemma 6.l. By Lemma 5.1 there is a neighborhood U~o of ';0 in X such that 'd'; E U~o'

(6.6)

Let p~o denote the projection of Y onto its (closed) subspace N(';o)O (= 1m A~(';o)), and set

for 1c: 1 < t and T) E E~o such that ';0

+ T)

belongs to the interior [;0 of Uo.

A is a mapping of class en from the open subset (([;0 - ';0) n E~o) x R of the Banach space implies A(O,O)

E~o

x R into the closed subspace

= 0,

ImA~(';o)

of Y. Note that (5. 1)

while from ((5.4),(5.5)) it follows that the differential at 0 of

the mapping T) ....... A(T), 0), i.e.

A~(';o),

is a bijection of E~o onto 1m A~(';o). Then the

implicit function theorem applied to the equation A(T), c:)

= 0 says

that there are a

neighborood U1 of 0 in X contained in Uo - ';0 and a number t, with 0

< t < C:o,

such that for each c: E R, with 1c:1 < t, a unique T)< exists in U1 n E~o such that A(T)< ,c: ) = 0, and that the mapping c: ....... T)< is of class en; this mapping is analytic at 0 if Ao is analytic at ';0 and B is analytic at S 90 (';0) . We set by (4.3)", 1(';<, A o(';<)) = 0, in view of Lemma 6.1 we have

which means

On the other hand, the condition A(T)<,c:) = 0 yields

.;<

= ';0

+ T)<.

Since,

142

Then , if t is small enough to have~, E U~o whenever

lei < co , from

(6.6) it follows

tha t

for all

c E IR

wit h

Icl < t . Therefore,

putting

and recalling (4.2) , we easily get Ao(x,) + c B (x"c)

= 0,

c E IR with Icl < t. Thus the proof is concluded if we (~o + U n U~ o' a nd we recall that 9 is of class en and that

namely A(x"c) = O,for all t ake W = Wo and U =

j )

it is analytic at (~o , 0) if B is a nalytic at ( Sgo(~o ), 0).

•

I

7 - Some subspaces of R induced by the operator B . If

f :

X -. E is a mapping with X a topological space and E a finite di-

mensional real linear space equipped with an inner product (., .) (i.e. an Euclidean space) , and Xo E X , let us denote by Kf (xo) a ny linear subspace of E having t he following property: "K f(xo) is a maximal element of th e set of all linear s ubspaces K of E such th at if y E K and y is orthogonal to f (x) for each x belonging to som e neighborhood of xo in X , then y = 0." It is easily seen t hat, if (a j , ... ,a m ) is a

base of E anf

J

is a maximal element of the family of subsets J of {I , . .. , m} such

that the set of fun ctions j E J,

is linearly independent on every neighborhood of Xo, t hen LJ EJ IRaj is a K f(xo); conversly, any 1\f (xo) is of this type for a suitable choice of t.he base (aj, ... , am) of E. Now , id us

t ilk, '

as E t he space R defined in Section 3 endowed with the hiller

product (-, .) carrieci by t. he inner product of IRm when t he base of R is that one

143

used for defining ,; moreover let us take as defined by putting

f(x ,E)

f the mapping from X

x JR into R

= ,(X,B(X,E))

for all (X,E) E X x JR. In this case, the spaces Kf(Sgo(~O) ,EO), with ~o E X , 90 E G and EO E JR, will be denoted by KB(~0,90,EO) . Thus KB(~0,90,EO) is any maximal element of the set of linear subspaces K of R having the following property: "if

Rv

E

Rand

(Rv,,(x,B(x,E))) = 0 for every (X,E) belonging to some neighborhood of (Sgo(~O),EO) then Rv = 0". In the following lemma, which generalizes Lemma 6.1 , we allow Bo to admit critical points in TeG.

Let (~o, 90) E X x G be such that M~o (90) = 0, and let T be a

Lemma 7.1.

linear subspace ofTeG such that the set {Rv : VET} contains some KB(~O, 90, 0). Suppose that (5. 7) holds and that B is of class en, n 2: 1. If no element # 0 of T is critical for Bo at (~o, 90), then neighborhoods Vo of ~o in X and Wo of 90 in G and a number EO> 0 exist such that for each ((,E) E Vo x JR with lEI < EO there is fj(~,E)

in Wo such that (6.1) is fulfilled. Vo and Wo can be chosen such that the mapping (~,E) >-+ g(~,E) is of class en; this mapping is analytic at a unique element (~o,

0) provided B is analytic at (SgO (~o), 0).

Proof - Suppose that no element KT Let

7rT

# 0 of T

is critical for Bo at

(~o, 90),

and set

= {Rv : VET}.

be the projection of R onto KT with respect to the inner product (.,.)

considered on R , and define the mapping rT . T x X x JR':"'" KT by putting

Evidently, (Sgo(~o),O).

rT

is a

en

mapping; it is analytic at (0, ~o, 0) if B is analytic at

Since rT(O,~O,O) = 7rT(M~0(90)), we have rT(O,~O,O) = O.

that , in view of (5.7) we have

Note

144

Note also that, since KT contains some KB(1;0 ,90,0), from the definition of

KB(l;o , 90 , 0) it follows that there is a linear mapping LT : KT

---t

n

such that

i(X, B( X,E)) = (LT

0

7rT

0

i)(X, B( X,E))

for all x b elonging to a sui table neighbor hood N of 5 90 (1;0) in X and all with lEI suffiently small. Thus, if

VI

E

E R

is a neighborhood of 0 in TeG and UI is a

neighborhood of 1;0 in X such that

then

for all (v, 1;) E

VI

X UI and all E E R with

lei sufficient ly small.

From (7.1) it follows

that

LT

0

OvfT(O, 1;0, 0) = M~o (90)

0

go

where OvfT(O, 1;0 , 0) denotes the differential at 0 of the mapping v

>-->

f T(V,l;o , O).

Hence, if OvfT(O ,l;o , O) vanishes at Vo E T , then M~0(90)(go(v)) = 0, and this implies Vo = 0 because no element =I=- 0 of T is critical for Bo at (1;0, 90 ). Thus Ov fT(O ,l;o, O) is a one-to-one (linear) mapping from T into K T ; therefore, since dimKT ::; dimT , we have dimKT = dimT and the mapping o" f T(O, 1;0 , 0) is a bijection from T onto K T · Then the implicit fun ction theorem applied to the equation f T (v , 1;, E) = 0, says that open neighborhoods \ . of 0 in TeG and U of 1;0 in X and a number Eo > 0 exist such that for each I; E U and each E E R with lEI < Eo there is one and only one element v(l;, E) of T n F such that

r T(v(I; ,E), I;,E) = O. Moreover , U, V a nd

EO

can be chosen such t hat the ma pping (E;,E)

(7. 2) f---'

i' ( ~,E )

is of

class en; it is analyt ic at (1;0,0) provided B is analyt ic at (S'h,(E;o), 0). Combining (7.1) with (7.2) we obtain

145

which, by (5.7) is equivalent to (6.4) . Since (6.4) becomes (6.1) when §(E"E) is defined as in (6.5), to conclude the proof it sufficies to set Uo = un U1 and take as Wo a neighborhood of go such that v E TeG, (expv)go E Wo ===> v E V

n VI. I

•

8 - A stronger theorem on existence, uniqueness and analytic dependence on 10 for the equation A(X,E) = O. Theorem 6.2 requires that the operator Eo does not admit critical points in

TeG at some pair (E,o, go) E X x G. In the following theorem we weaken this

hypothesis, and we ask operator Eo to admit at (E,o , go) no critical points belonging to a subspace T of Te G such that {Rv : VET} contains some K B (E,o, go , 0) . Suppose that Ao and E are of class en, n ~ 1, that (4.1), (4.2), (4.3) , (5.7) hold, that there is (E,o,go) E X x G such that (5.1), (5.2), (5.3)

T heor em 8 .1.

are satisfied and M€o(go) = O. Let T be a subspace ofTeG such that {Rv : v E T} contains some KB(E,o,go , O), and suppose that no element for Eo at (E,o,go).

-I-

0 ofT is critical

Then, fixed a topological supplementary E€o of'R.(E,o) in X ,

neighborhoods U of E,o in X and W of go in G and a number [ > 0 exist such that 10 E R with 1101 < [ thete are unique E,€ in Un (E,o + E€o)and a unique g€ in n {(expv)go : vET} such that, putting

for each W

we have

The mapping

10

I-t

x€ is of class

en

in a suitable neighborhood of 0; it is analytic

at 0 provided Ao is analytic at E,o and E is analytic at (S90 (E,o) , 0). Proof - One can proceed as in the proof of Theorem 6.2, and use Lemma 7.1

instead of Lemma 6.1 which has been utilized in proving Theorem 6.2.

I

146

An easy consequence of Theorem 8.1 is t he fo llowing Corollary 8 .2 .

L et the hy potheses of Theorem 8.1 be satisfi ed, and set Xo

Sgo (~o ) and

D~O,go (T)

=

U S(exp v)go(~o + E~o)' vE T

If th e m apping (~ , g )

>->

Pg(O from (~o

+ E~o) x {(exp v) go :

VE T} onto D~o,gO (T )

is a local hom eomorphism a t (~o , go ), th en a neighborh ood Uxo of Xo in X exists such th at for each E '" 0 with lEI sufficiently small there is on e and only one x, in Uxo

•

n D~o,go (T) satisfying A( x" E) = O.

9 - Examples from elasticity. As we have observed in Sect ion 1 and declared in the ti tle of t his article, t he

abstrac t setting in which we have placed our analysis (in part icula r t he formulation of affine symmetries proposed in Sect ion 4) was suggested to us mainly by t he geometric approach t o the "traction problem" in fini te elasticity in t he presence of loads which depend on the unknown deformation ( "live" loads). The present section is devoted to show how such boundary problems arising in elasticity are included , as particular cases, in our abstract approach . To t his purpose we start by making a list of some symbols we shall use. D will be a bounded , open su bset of

]Rn,

aD its boundary, D its closure a nd v t he outward , unit nor mal to aD. \\'e

denot e by ·MI n the set of n x n real matrices, by MI;; t he set of elements Z of MIn such t hat det Z > 0, by Sym n the set of symmetri c elements of MI n. by Skew n t he set of skewsymmetric elements of MIn , and by (());; t he set of f'lement Z of MI;; such tha t ZT = Z- l , where ZT is the transpose of t he matrix Z . ,\ a nd Y1 will be t he Ba nac h space of sufficient ly smooth functions a Banacg space of ]Rn -valued fu nct ions defi ned on Y1 x

r 2·

I

:

aD.

D

-+ ]Rn ,

\\'hile } '2 will be

and }' will b e t he product

Later we shall propose two concrf't.e examples of spaces .\ . Y1 ,

Let (t, Z)

>->

1''2.

s(t , Z) be a given function frol1l D " MIn such t ha t

s(t , R Z ) = Rs(t . Z ),

s( t , Z)ZT E SY Il1",

V(t,

z, R)

E

D x MI n

X (());;,

V(t, Z, ) ED" MIn.

(9.1) (9.2)

147

For any x E X consider the function S(x) : n

--->

MIn defined by putting

S(x)(t) = s(t , ox(t)),

'it EOn,

where ox(t) denote the gradient at t of the function x: n

(9.3) ---> ]Rn ,

and set

Ao(x) = (- div S( x), S(X)18nLJ).

(9.4)

We can say that Ao is the (n-dimensional version of the) finite elastostatics operator. In the physical context (n = 3)

n may be thought as a reference config-

uration of an elastic body, the function x as a deformation of the body (from the reference configuration), and the function s as the elastic response function for the first Piola-Kirchhoff stress, in the sense that s(t, ox(t)) is the first Piola-Kirchhoff stress at the point tEn when the body is deformed by x. Moreover , symmetries (9.1) and (9.2), written with MI~ instead of MIn , follow from the principle of material indifference and from the balance of angular momentum. We observe that , as our results will be local, no loss in generality occurs if we deal with symmetries (9.1) and (9.2). Upon the function s : n x MIn

--->

MIn we consider the further two

hypotheses , concerning the behaviour of s at the deformation in, where in denotes the identity function from

n into ]Rn: s(t, I) = 0,

'it E

n,

(9.5)

n

l:=

OZhkS(t,I)ZhkZij > 0,

'it E nand 'iZ E Symn \ {O} ,

(9.6)

i,j, h,k= l where I is the unit element of the ring MIn (hence I = Oin) · The mechanical meaning of (9.5) (in the case n = 3) is that the reference configuration of the elastic body in unstressed. Let us consider the follows two choice of the spaces X , Y1 , Y2 within Sobolev and Schauder spaces:

148

where kEN , 1 < p E JR, ), E]O , 1[, and (p(k

+ 1) > n.

The definitions and the analysis of some properties of this spaces can be found in Va lent 16] , Chapter II. It is importa nt to remark that these choices of the Banach spaces X , Yj , Y 2 work for operator A o because it is possible to prove that , if X , Yj

,

Y2 are the spaces (9. 7), or (9 .8), then Ao is a smooth op erator from X into Yj x Y2 , provided

n and

the function s are smooth enough. Indeed , the following theorems

holds (see Va lent 16] , Chapter III).

Let n be of class Ck+2 and let s E C k + 2(D x MIn, MIn) [resp ectively of class Ck+2,). and s E C k + 3 (D x MIn , MIn)}. If X , Yj , Y 2 are deEn ed by (9.7)

Theorem 9.1.

n

[respectively (9.8)}, th en Ao is a C j mapping from X into Yj x Y 2 . Furth ermore, if

s E CoorD x MIn , MIn) and the fun ctions (t , Z) r--. or; s( t, Z ), lal :::; k + 1, are analytic in Z at I uniformly with respect to t , then Ao : X ---> Yj X Y 2 is analytic at in. As regard the symmetries of Ao we take as G the group of all isometries of

BbbR n (i.e. functions 9 : JRn

--->

JRn of the type

g(t ) = c + Rt with c E JRn and R E

o;t), and we define

Pg and {>g \;;/g E G and

T :

X

--->

Y by

setting pg(x) =g ox,

(9.9)

and

T(X) = (X,XI8n)

(9. 10)

for all x E X and all (Yj, Y2) E Yj x Y2.

Let Ao X -> Yj X Y 2 , be defined by ((9.3), (9. -4 )), with X , Yj , Y 2 as in (9.7) or (9.8) , let G be the gro up of isom etries of JRn, and let Pg, {>g and Remark 9.2.

defined by (9.9) and (9. 10). Then: (4. 1) holds (obviously) , the property (9.1 ) of the fun ction s implies symmetry (-4 .2) , while the property (9.2) of s implies (4.3).

T

Proof - It is easy to see that (9 .1 ) implies (4.2). Let us show that (9.2) implies (4.3). Here TeG is the set of (affine) fun ctions v : JRn ---> JRn of the type

v(t) = c + Wt

149

with cERn and WE Skew n ; moreover

Rv(x)=vox,

Vv E TeG and Vx E X.

Hence, condition (4.3), (which is equivalent to (4.3)) assumes the form

- Jnr ((div S(x)(t))· (c + Wx(t)) dt + Jan r (S(x)(t)v(t))· (c + Wx(t)) da(t) = 0 for all cERn, WE Skew n and x E X, where· denotes the inner product of Rn By the divergence and (9.3), this condition reduces to

W in8x(t)s(t, 8x(t)f dt = 0

VW E Skew n and Vx E X ,

namely to

in8X(t) S(t,8x(t)f dt E Symn

Vx E X ,

which evidently is satisfied if (9.2) holds.

I

The following remark concerns hyperelasticity, namely the case when t here is a CI-function w : n x MIn

s(t, Z) Remark 9.3.

-+

Rn such that

= 8 z w(t, Z) ,

V(t,Z) En x MIn .

If (9.11) holds, then the mapping fx : X

fx(x) =

in

-+

(9 .11)

R defined by

w(t,8x(t))dt

is a potential for the operator Ao defined by ((9.3),(9.4)) with respect to the linear

mapping

T :

X

-+

Y defined by (9.10) and the inner product

!3 on Y

= YI

X

Y2

defined by

!3((YI,Y2),('fh,Y2))

=

r Y2(t) ' Y2(t)da(t), JnrYI(t)'YI(t)dt+ Jan

where · is the inner product ofRn (See Section 2).

Proof - We suppose that (9.11) holds. For any x, x E X we have

f~(x)(x)

=

in

8zw(t,8x(t))8x(t)dt.

(9.12)

150

Then , recalling (9.3) and (9 .9) and using the divergence theorem , we obtain without difficulty that

j' (x)(x) = - ( ((div S(x))(t) . x(t) dt

Jo.

+ (

J80.

(S(x)(t)v(t))· x(t) dO'(t).

(9. 13)

We observe that , if we take as completion of YI x Y2 the pair (H ,

equipped with the inner product (9.12), and

jH(YI, Y2) , (:ih , ih) = j3((Y I' Y2) , (iiI , ih)) for all (YI ,Y2), (:ih ,'!h) E H.

From (9.14 ) it follows that the mapping

potential for Ao with respect to

T

fx

and j3 (see (2.4) in Section 2).

is a I

By Theorem 4.1, a consequence of Remark 9.3 is that if (9.11 ) bolds with w of class C 2 , then symmetry (4.2) implies (4.3). We observe that t his fact is also a consequence of the following simple remark. Remark 9.4 .

If (9.11 ) holds with w of class C 2 , then symmetry (9.1 ) implies

(9.2). P roof - A consequence of (9.1) is that, for any fi..xed tED and Z E MI" ,

s(t , (ex p W) Z)

=

(exp W) s(t, Z)

which easily implies

f}zs( t, Z)ZW

= W s(t , Z),

Vn- ESkew",

namely Vl-! ' ESkew".

(9. 15)

On the other ha nd , from (9. 11 ) it follows that 8 z s(t, Z) E Sym". Therefore, since Z WZ T E Sy mn , if (9.11) holds with of class C 2 then (9.5) implies

WS (t,Z)ZT which is equivalent to (9.2).

=0

VII ' E Skewn , I

151

Till now we have been discussing the symmetries (4.2) and (4 .3) for the operator

Ao defined by ((9.3),(9.4)). The following theorem, proved in Valent 6 ] (Chapter III) , concerns the properties ((5.1),(5.2),(5.3)) of Ao. Theorem 9.5.

Let

n and

the function s be as in the statement of Theorem 9.1.

If s has the symmetries (9.1), (9.2) and the properties (9.5), (9.6) then

(9.16)

and hence (9.17)

where

LO

is the identity function from

n into IRn,

Of course, here it is understood that n(LO) and N(LO) have to be related to the group G of isometries of ]Rn and to the representation 9

l-->

Pg and 9

pg of

l-->

G defined by (9.9). It is also understood that (9,16) (9.17) holds for both choices (9.7) and (9,8) of the spaces X , Y1 , Y2 . We remark that , for a suitable choice of the base of n , we have tEO,

(9 ,1 8)

for any x E X and Y = (Yl, Y2) E Y , where x /\ Yl and x /\ Y2 are pointwise defined , i.e, (x /\ yd(t) = x(t) /\ Yl(t) \:It E n and (x /\ Y2)(t) = x(t) /\ Y2(t) \:It E an , with /\ the external product on]Rn Therefore , for such a choice of the base of n , here the mapping M'n : G

M'n(g)(t)

---7

n is defined by putting

= (j~ BO,I(g(T)) dT +

fao BO,2(g(T)) da(T)) + (l g(T) /\ BO,I(g(T)) + fao g(T) /\ BO,2(g(T)) da(T)) dT

for all isometry 9 of ]Rn and all t

E ]Rn ,

where

BO , 1 :

X --; YI and B O,2

are the mapping for which Bo(x) = (B O,I(X),Bo,2(X)) \:Ix E X .

*

*

*

t

X --; Y2

152

Let BI : X x JR

-+

Yl and B2 : X x JR

-+

Y2 denote the mapping for which

V(x,c) E X x R We will say that B = (B 1, B 2) is a load operator. An interesting example of load operator independent of c (namely with B = Bo) is the following:

BI(X)(t) = bl(t ,x( t),ox(t)) ,

(9. 19)

{ B2(X)(t) = b2(t , x(t), (cofox(t))v(t)), where cofOx(t) denotes the matrix of cofactors of matrix ox(t), and

are given functions. Note that (cofox(t))v(t) is an element of JRn parallel to the normal to the boundary of x(O) at x(t). Note also that example (9.1 9) includes the simple, but important case when B describe the load acting on a heavy elastic

body submerged in a quiet, homogeneous, heavy liquid. In this case we have Bdx)(t) = J.11(t) e, tEO { B2(X)(t) = - J.12((X(t) . e)cofox(t))v(t), where e is a fixed element of JRn with

lei = 1, J.11

is a real valued positive function

defined on 0 , and J.12 is a positive constant. For a detailed analysis of load operators of the type (9. 19) we refer the reader to Va lent 161 , Chapters V and VI. Here, we confine ourself to make some remarks on the simplest case of load operator: the case (of a dead load) when B is a

constant operator. If B is constant , we denote by (b l , b2 ) the constant value taked by B = (BI, B 2); so, now , bl is an JRn-valued fun ction defined on 0 a nd B2 is an JRn-valued functi on defined on 00. Then

M'n(g)(t) = ( l bl(T) dT +

lao b

2(T) dO"(T)) +

( l 9(T)l\bl (T) dT + for every isometry 9 of JRn and every t E JRn

lao9(T) l\ b (TldO"(T)) 2

t

V,'ith the statement of Theorem 8.1

in mind, we suppose that there is go E IR" such that M'n (go) = 0, i.e.

(9.20)

153

Any pair (b1, b2 ) satisfying (9.20) will be said equilibrated with respect to go· From (9.20)1 it immediately follows that any KB(~n,go, 0) is a lin ear subspace of all (linear) fun ctions v : IRn -+ IRn of the type v(t)

= Wt,

t

E IRn

(9.21)

with W E Skew n (see Section 7); in order to semplify our notation, we will write )(gO) instead of KB(~n ,go, O). On the other hand, it is evident that any constant function t >-> c from IRn into IRn is critical for B at go (= Sgo(~n)}. The

W(b 1 ,b 2

elements v of TeG of the type (9.21) that are critical for B at go can be characterized in the following way. Remark 9.6. G defined by

Let 0

#- W

E Skew n , and consider for any A E IR the element g>. of

g>.(t)

= (expAW)go(t)

t E IR n

Then the element v of TeG defined by (9.21) is critical for B at go if and only if (b 1 , b2 ) remains equilibrated with respect to g>. VA E IR. For the proof of this remark we refer to Valent[6), Chapter V. The meaning of the condition expressed by Remark 9.6 is very clear when n = 3. Indeed for any W E Skew3 \ {O} the set {t E IR3 : W t = O} is an one-dimensional linear subspace of IR3 (the axis of W) , which coincides with the axis of the rotation t >-> (exp W)t of IR 3, i.e. with the set {t E IR3 : (exp W)t = t}. Hence Remark 9.6, in the case n = 3, states that, the element v ofTeG defined by (9.21) is critical for B at 90 if and only if (b 1 , b2 ) is equilibrated with respect to any isometry ofIR3 obtained from go by an arbitrary rotation about the axis of W By using Remark 9.6 it is possible to prove (see Valent (6), Chapter V, Lemma 6.2) the following

Remark 9.7.

Let

W I, ... , Wn

be the eigenvalues of the matrix

154

(which is symmetric because of (9.22)) . Then, for no element v#-O ofTeG of the type (9.21) is critical for B at go it is necessary and sufficient that (9.22) ~j=l, ., , 1t

iij

In Valent 16] (Chapter V, §6) , for various choices of a linear subspace W of the

space of all elements v E TeG of the type (9.21) with W E Skew n , there is an algebraic characterization of the fact that no element

#- 0 of W is critical for B at

go· A further remark is necessary to make explicit Theorem 8.1 for a constant load operator B. Remark 9.8.

A topological supplementary ofR(oo) in X is the space

{x

E

X

l

x(t) dt = 0,

l

8x(t) dt

E sym n

}

A proof of this remark can be found in Valent 16], Chapter III , Section 1.

We are now in a position to state a consequence of Theorem 8.1 in the particular case when Ao is the finite elastostatics operator and B is a constant (load) operator. Theorem 9.9.

Let Ao : X

->

Y1

X

Y 2 be defined by ((9.3),(9.4)) with X , Y 1 ,

Y 2 as in (9.7) or (9.8), and let S1 and s be as in the statement of Theorem 9.1 , with s satisfying (9.1) , (9.2) , (9.5) and (9.6). Moreover, let (b1 , b2 ) E Y 1 x Y 2 be equilibrated with respect to some isometry go oflI~n , and let T be a linear subspace n >-> Wt from IR into itself with WE Skew n . Suppose

of the space of all functions t

that T contains some W ( b 1 ,b2 )(gO) and that no elem ent #- 0 ofT is critical for (b 1 , b2 ) at go . Th en , a number co > 0 and some neighborhoods U of 00 in X and W of go in G exist such th at, for each c E IR with

l~« t)dt=O

lEI < EO,

and

th ere BJ'e a unique ~£ E U with

l 8~«t)

dt = 0,

(9.23 )

155

and a unique isometry ge of IR n belonging to W n {( exp V)90 : VET} such that, setting Xe=geo~e ,

we have div S(xe) + cb 1 { - S(xe)v + €b 2

The mapping €

f-+

= 0, = 0,

in r! on or!.

(9.24)

Xe is of class C 1 ; it is analytic at 0, provided s E Coo (n x MIn , MIn)

and the functions (t, Z) with respect to t .

f-+

a;:s(t, Z),

lal S k + 1,

are analytic in Z at I uniformly

In view of Remark 9.7, from Theorem 9.8 it follows

Let Ao : X ---> Y1 x Y 2 be defined by ((9.3),(9.4)) with X, Y1 , Y 2 as in (9.7) or (9.8), and let r! and s : r! x MIn ----> MIn be as in the statement of Theorem 9.1 , with s satisfying (9.1), (9.2), (9.5) and (9.6). Moreover, let (b 1 , b2 ) E Y1 x Y 2 and 90 be an isometry ofJR.n such that (b 1 , b2 ) is equilibrated with respect to 90 and (9.22) holds. Then, for each € E JR., with I€I sufficiently small, there are unique ~e near l.n satisfying (9.23) and a unique isometry ge of IR n near go in {( exp v)go : v E TeG} such that, setting Xe = ge 0 ~e, the equalities (9.24) are satisfied. The mapping x >---> Xe is analytic at 0, provided the function s is Coo in x MIn and the functions (t , Z) f-+ a;: set, Z), lad S k + 1, are analytic in Z at I uniformly with respect to t . Corollary 9.10.

n

•

References [1] Bertin G . and Valent T ., "Local existence for live traction problems in finite elasticity", Rendiconti di Matematica (VII) 9. 625- 644, 1989. [2] Chillingworth D.R.J., "The Signorini perturbation scheme in an abstract setting" , Proc. of Royal Society of Edinburgh 9 A, 373-395, 1991. [3] Golubitsky M. and Schaeffer D.G. , "Singularities and groups bifurcation theory" Vol. I, Springer-Verlag, New York 1985. [4] Golubitsky M. , Stewart I. and Schaeffer D.G. , "Singularities and groups bifurcation theory" Vol. II , Springer-Verlag, New York 1988.

156

[5] Marsden J.E. and Hughes T.J.R. , "Mathematical foundations of elasticity" , Prentice-Hall, New Jersey 1983. [6] Valent T. , "Boundary value problems in finite elasticity", Springer-Verlag, New York 1988. [7] Valent T., "An abstract setting for boundary problems with affine symmetries" , to appear. [8] Valent T. , "A perturbation problem in the presence of affine symmetries", to appear. [9] Valent T. and Bertin G. , "On local existence, uniqueness and analytic dependence on a parameter for the traction boundary problem in finite elastostatics" ,

Memorie Mat. Ace. Lincei IX , 1 31- 58 , 1990. [10] Varadarajan V.S. , "Lie groups, Lie algebras and their representations", Springer-Verlag, New York, 1984.

157

MAXIMUM PRINCIPLES IN CLASSICAL ELASTICITY

L. T . Wheeler

Department of Mechanical Engineering University of Houston Houston , Texas 77004-4792, (USA)

158

1. INTRODUCTION. One of the most basic qualities of the solutions of Laplace's equation, dU=O,

is the absence of relative extrema at interior points. Closely related are sub- and superharmonic functions, which when sufficiently smooth obey dU ~ 0 (subharmonic)

du

~

0 (superharmonic)

and which fail , respectively. to have relative maxima and minima at interior points. The consequences are far reaching. For the quantity represented by u, the principle yields direct physically significant consequences about the maximwn and minimum values. For problems of the Dirichlet type, in which u itself is prescribed on the boundary, these extrema are of course part of the data. Not only do they yield direct physical implications, but maximum principles also lead to highly explicit comparison results for solutions and their gradients. Further, they lead to results about the uniqueness of solutions and their continuous dependence upon the data. Maximum principles generalize in a straightforward manner to second order elliptic operators. Systems of partial differential equations, on the other hand, are an altogether different matter. Efforts to derive a suitable generalization of the remarkably powerful and simple results of scalar second-order operators to higherorder systems have thus far been unrewarding. The success of the standard maximwn principle is reason enough to search for extensions to the system of partial differential equations governing the fields of displacement, stress, and strain in classical elasticity theory. As we shall see in the sequel, there are many ad hoc applications of the conventional maximum principle in elasticity. However, the effort to find a principle of significant generality raises many questions and leads to many dead ends. We have on the one hand the knowledge that the dilatation, mean stress and rotation satisfy Laplace's equation and are thus subject to the maximum principle. Of course, these quantities are of limited physical significance. On the other hand, a maximum principle for quantities such as the displacement magnitude, the principal stresses and strains as well as the strain energy density would presumably have important physical implications. Unfortunately, not only are such results not available. but there are counterexamples which indicate that such a principle is not likely to enjoy the simple direct form that it possesses for Laplace's equation. In spite of this daunting conclusion. there are many special classes problems in elasticity theory where physically important harmonic, subharmonic, or superharmonic functions occur naturally and are thus governed by the maximum principle. The present article has a twofold purpose. One objective is to give an account of maximum principles for scalar valued functions and an extensive discussion of their applications in classical elasticity theory. A second purpose is to discuss counterexamples

159 which have a bearing on the problem of finding a general maximum principle in elasticity theory.

2.

MAXIMUM PRINCIPLES.

In this section, we discuss certain maximum principles for the Laplace operator which have proven to be useful in elasticity theory. Throughout this section, 'lll, denotes a bounded open region in n-dimensional euclidean space ffi" unless otherwise indicated. -

2

-

Theorem 1. Let u E C€('lll,) nC€ ('lll,), where 'lll, denotes the closure of 'lll, and assume 8U = 0 on 'lll, . If u assumes its maximum (or minimum) value at an interior point of 'lll, then u is constant on

iiii .

This is perhaps the most basic form of expressing the maximum principle for harmonic functions. The conventional proofl of this result uses the mean value theorem.

In certain instances, and especially in the construction of comparison functions, it is advantageous to deal with subharmonic functions. A continuous function u is said to be subharmonic if for every harmonic function v such that v ~ u on

a0't we have v ~ u on

'lll,.

It is not difficult to show 2 that a function u E C€2('lll,) is subharmonic on 'lll, if and only if 8U ~Oon 'lll,.

(I)

Although the relationship between subharmonic functions and harmonic functions is important, subharmonic functions satisfy a maximum principle which for sufficiently smooth functions admits a straightforward direct proof. Theorem 2. Let 'lll, denote a bounded open region in n-dimensional space, let -

2

u E c€('lll,) nC€ ('lll,) and assume that 8U ~ 0 on 'lll, . Let x E 'lll, . Then, u(x) S; max u .

'*" This result is a straightforward consequence of Theorem l. On the other hand, a direct proof can be based upon the obvious fact that if 8U >Oon 'lll" then u < max u on 'lll,. Applied to

'*" u = v +EX· x, this idea yields the conclusion in the limit as

E

~ 0 that if 8v ~ 0 on

'lll" then v S; max v .

'*" In the sequel, we discuss applications involv~n& exterior regions, in other words, regions which contain a complete neighborhood of mfmltY· These results depend upon dimension. For the three dimensional case, we have

160

Theorem 3. Let WI, be an open exterior region in assume du a.

=0 on WI,.

Let u(x)

=c + 0(1) as Ix I~

00 •

lJe, let u E ~(m) (J~2(WI,), and Then either

Ic I< max Iu I' ern

or b.

u

=c on WI, .

This result is easily proven with the aid of a Kelvin inversion, which takes the the region WI, into a bounded region

ril . . Under such an inversion, the function u is transformed to a

harmonic function having the value c at an interior point of WI, . The desired conclusion thus follows from Theorem 1. For two dimensions, the corresponding result differs only in the assumption as to the rate of approach of the solution to its value at infinity. We have

Theorem 4. Let WI, be an open exterior region in two dimensional euclidean space ffi2 , let u E

~(m) (J~2(WI,) , and assume dU =0 on WI,. Let Iu(x) I~ c as Ix I~

00.

Then either

or b. u = c on WI, .

Approximations based upon the theory of subharmonic functions often involve the construction of a comparison function , which in one way or another resembles the unknown function . This construction can ordinarily be made easier by relaxing the smoothness requirements on the comparison function. The definition of a subharmonic function mentioned earlier does not require smoothness beyond continuity, and indeed even continuity is not required in the general theory of subharmonic functions 2 , though the present work does not involve functions which fail to be continuous. This definition in and of itself establishes a comparison between a harmonic function and a class of subfunctions, but does not directly provide a characterization reminiscent of the condition (1). There are a couple of ways out of this dilemma. One approach is to observe that subharmonic functions are sub-mean-valued 2 , which for n = 2 requires that corresponding to each x E WI, and each p > 0 such that np(x) c WI, , where np(x) denotes the open disk of radius p centered at x, there follows u(x)

~

def 1 M{u ,x, p) = -2-

ltp

1 dOp(x)

uds .

161

This leads to the introduction of a generalized Laplace operator, ,& , defined by '&u(x) = 4 lim inf ~[M {u,x,p} - u(x)] . p.....o

p

It is easy to confirm that if u is in C€2 on a neighborhood of x, then '&u(x) = ~u(x) . The key result here is Theorem 5. Let u be continuous on!JJI.. Then u is subharrnonic on !JJI. if and only if '&u;:: 0 on!JJI.. For the proof, see Section 3.7 of Rado' s monograph 2. To illustrate the use of this idea in connection with gradient bounds, we include here results which in the sequel will be applied to stress bounds for Saint-Venant torsion. Lemma 1. Assume: av (a) v E C€(!JJI.) , v = 0 on a!JJI., an exists on a!JJI. , and ~v;:: M on !JJI. , where M is a positive constant; (b) u,W E C€(0k) nC€2(!JJI.) , u = w on a!JJI. and - M':5 ~u:5 M", ~w = 0 on!JJI., where M' and M" are positive constants. Then,

aw

au M" av au M' av - - - < - < - + - - on a!JJI. an Man-an-an Man .

-

For the proof, see the paper3 by Fu and Wheeler. Subject to some rather technical restrictions on !JJI., Cimmin04 (see also the paper by Fu and

WheeJe~) constructs a function v E C€(0k) . which depends only upon dis tance from a!JJI. and has the properties av 2 v =Oand- = I ona!JJI.,~v;::-(2) on!JJI.. p -Kp an

(2)

Here, K denotes the minimum curvature of a!JJI. , subject to the convention that the curvature of the boundary is positive at points where the center of curvature lies within !JJI. and negative if the center is external to !JJI.. The remaining quantity. p , denotes a number

162

greater than or eq ual to the radius of the largest disk contained in 'lit, and if 1C>O, less than 11K. The maximum princi ple holds for elliptic partial differential equations5, and indeed we use such a principle in Section 3.2 in connection with the problem of axisymmetric torsion.

3. APPLICATIONS OF THE MAXIMUM PRINCIPLES IN ELASTICITY THEORY. There are a number of well kn own special cases in classical elasticity in which maxi mum principle has rather straightforwa rd applications. Thus, for example , in the absence of body force the dilatati on and rotation satisfy Laplace's equation. As a result, the dilatation has to have its maximum and minimum values on the boundary. Moreover, the square of the magnitude of the rotation field is readily seen to be subharmonic, and thus must assume its maximum value at a point of the boundary . Of course, these simple observations are not particularly significan t from a physical point of view. At the expense of increased complexity, applications of the classical maximum principles can be broadened to encompass a larger range of problems. The purpose of the present section is to discuss a number of such problems - of vary ing degrees of physical significance - which have received attention in the literature.

3.1. Saint- Venant Torsion. Consider a shaft of simply-connec ted cross sec tion typified by a plane domain qIJ bounded by a curve C€. The torsion problem is conveniently formulated in terms of the Prandtl stress function , which is governed by L\ = - 2 on qIJ, =

°

on C€ .

(3)

All cartesian components of the stress vanish except T31 and Tn' which are expressed in terms of by (4)

where m denotes the applied torque and k denotes the geometric torsional rigidity. which can be cast in the fo rm s

(5)

From the first of (3), it follows th at 1V'1 2 is subharm onic on r;]" and as a result, the seco nd of (3) yields, for sufficientl y smoo th and C€ . a max lV'1 = max I-a I . '2JJ iEf, n

(6)

163

Let

and let v be the function alluded to in (2). Then, Lemma 1 of the preceding section yields

n::

d

I~I ~ p(2 - Kp) .

(7)

Thus, and by (4) and (5), we have (8)

The torsional rigidity k obeys the inequality6 k ~

4

1ti ' which is isoperimetric in the sense

that equality holds if the cross section is circular. Hence 21ml

1: ~ - 3(2

1tp

- Kp) .

(9)

Thus, we have a simple upper bound on the shear stress which depends upon the geometry of the cross section as represented by K and p . It is easily verified that this bound is isoperimetric in the sense that the upper bound coincides with the exact result when 2IJ is a disk. For further discussion and related results, see the article3 by Fu and Wheeler. The bound (9) also has application to thin sections. In articles by Weinberger and Serrin? and Payne and Wheeler8 , a lower bound on 1: is given which serves to characterize the circular shaft as the one of least maximum stress among all competing cross sections of the same area. This result is based upon the following result of Miranda9.

Theorem 6. Let u E «&\2IJ) and let u denote a biharmonic function on 2IJ, (10)

Define

-~u.

(11)

w ~ sup w on 2IJ .

(12)

w=IVu1

2

Then

'€

The proof is remarkably elementary. Direct computation yields

164

Consequently, w is subharmonic on 9l and (12) thus follows. To apply Miranda's result, we take u = <1>, so that 1Y'1 2 + 2<1> ~ suplY'1 2 9lJ

Integrate both sides of this inequality over 9l and substitute from (4) and (5) to arrive at

"t

22m2 > --leA

(13)

It is readily confmned that equality holds in (13) if 9l is bounded by a circle. The torsional rigidity k obeys the classical inequality

( 14)

in which equality holds for 0 a disk. This inequality expresses the result, conjectured by Saint-Venant and proved by Polya lO , that of all cylinders of given cross sectional area, the circular cylinder yields the greatest torsional rigidity. Combining (13) and (14), we arrive at 2m1t1l 2

(1 5)

"t~--3-12-

A

It is easily verified that equality holds for 9l a disk. The inequali ty (1 3) appears in Payne ll , in a different form.

Miranda's result has been employed extensively by Pay ne and his collaborators in connection with the torsion problem. In Payne l l , it is shown that fo r 9l convex, I

max IY'I -< -K

'

which is isoperimetric for a slender domain. Further results are found in Pay ne and Philippin 12, which includes a discussion of the location of the boundary point at which the max imum val ue of the shear stress occurs. Related applications of the maximum principle are also discussed in the book 13 by Sperb. 3.2. Axisymmetric Torsion.

Here we are l'oncerned with the stresses arising from the torsion of an elastic solid lyin g in a region of revolution 'lJ/. having pl ane ends Let (r,8,z) stand for cylindrical

165

coordinates such that the z-axis coincides with the axis of symmetry and let (x l ,x2,x3) denote Cartesian coordinates such that x l = z, x 2 =r cos e, x3= r sin e

o: ; r <

00,

0 ::;; e < 27t, -

00

}

(16)

< z < 00.

Body force is assumed to be absent and the lateral surface is assumed to be free of traction. The boundary conditions at the ends n (a) (a = 1,2) have the form

a

Tn = Tzz = ,Tz9 = t

(a)

(r) on n

(a)

,

(17)

where t(a) are given functions. Let M denote the open meridional half section lying in the half-plane

e= 0

and La stand for the segments formed by the ends n (a). See Figure 1.

~

____~__________________~__~Xl

(a,a)

(b,O)

Figure 1.

We employ the Michell torsion function, which is denoted by'll, assumed to have the smoothness 1-

2

(18 )

'V E <+6 (M) (1<+6 (M) , and to obey

3

~ == '1"11 + '1"22 - -'1"2 = a on M' x2

a : ; X2 ::;; r I' 'V(b,x 2) = m (2)(X2) for a : ; X2 ::;; r 2' 'V(a,x 2) = m (l)(X2) for

'I'(xl,Q) =

M

a for a::;; XI::;; b, 'V = 27t on r,

(19)

166 where (20)

We assum e th at tea) ~ 0 , so that mea) are non-decreasing functi ons. The cylindrical components of stress are given by

(21 )

Of particular interest is the shear stress magnitude (22)

It is a simple matter to show from (21 ) and (22) that

1<1\jl x2 an

1:=2"I-lonr

(23)

The operator ~ is elliptic on M . Despite the singular behavior of th e coefficients on x 2 = 0 , the maximum principle may be applied to the boundary value problem to conclude :

~Oon r .

(24)

Thus, (23) may be written

1<1\jl

1 : = - - onr 2

X2

an

(25)

A direct calculation based upon the first of ( 19) yields (26)

Thus, 1: is subharm onic and as a result it follows th at t assumes its maximum value at a point of the boundary of the open domain M. Fo r problems where r contains a notch, it is sensible to anticipate that 1: assumes its max imum value at a point of the notch, although a rigorous demonstrati on of this conjecture is lacking.

167

Horgan 14 gives the following comparison theorem which is applicable to the problem of fmiling upper and lower bounds on the maximum value of 't on i:

Theorem 7. Let and let .:£<1>

'JI

be a solution of the problem (19). Suppose that

1-

2

<1> E C(& (M) n C(& (M)

SOon M,

°,

The proof l4 is simply a matter of applying the maximum principle to the function

(j)

= 1jI-

Consider the domain depicted in Figure 2.

i (-I,R)r--_ _......----t

(- 1,0)

(I,R)

(1,0)

xl

Figure 2.

Let the shearing tractions on the ends have the values appropriate to the Saint-Venant distribution in a straight shaft of radius R, so that

(27)

and define a stress concentration factor through (28)

168

Applied to the function

(29)

the foregoing Theorem yields the lower bound 2

3

Sa l a a a K > 2 - S - +S(-) --(-) + -(-) R R 2R 2R

4

(30)

Notice in particular that the lower bound approaches 2 as aIR tends to zero. which is of course the accepted limiting value. It is shown in Horgan 14 that for a significant range of values of aIR near zero. (30) compares favorably with values based upon photoelastic and analogue methods. Also given in Horgan 14 is a somewhat better but considerably more complicated choice for <\>. Finally. it is menti oned in Horgan 14 that although the foregoing theorem is capable of delivering an upper bound. none has yet been fou nd. The maximum principle is applied to the torsion of ax isy mm etri c shells in an article 15 by Horgan and Wheeler to es timate stress and in an other article 16 to obtain estimates which support a principle of the Saint-Venant type. In an article 1? by Wheeler. the problem of minimizing th e maxim um stress by varying a segment of r is studied. Such a segment might corres pond to a notch or fillet. The main result enables one to compare the maximum stress for two profiles and to conclude that if a profile can be found along which 't is constant. then it will yield the leas t maximum value of't over the profile. Consider a shaft which ex tends to infmity in both direc tions. as depicted in Figure 3.

Figure 3.

169 Here, the boundary value problem for the torsion function assumes the form 3 :£'¥ == ,¥, II + '¥'22 - -'¥'2 = 0 on .M., x2

( ,¥xl ,x2) ~m

(I)

(X 2) as xI ~-ooforO~x2~rl'

'¥(XI,x 2) ~ m

(2)

(X 2) as x I ~ 00 for 0 ~ X2 ~ r2'

,¥(X I,O) =0 for (I)

where m and m the followin g

(2)

00

< xI <

00,

(3 1)

M '¥ = 21t on r,

.

are gIVen by (20). A comparison principle for this problem is given in

-

A

1-

2

Theorem 8. Suppose that '¥, '¥ E <{6 (.M.) n<{6 (.M.) and satisfy :£

'if =0, :£ 0/ =0 on .M., 'if ~ lim 0/,

lim

XI ~±-

x]---7±OO

'if ~ 0/ on r. Further, let x

o E r and assume that '¥(x - 0) = '¥(x 0 ) . Then,

d\iI d\jI an an

- ~ - atx

0

This result is an elementary consequence of the maximum principle for elliptic operators. Next, we note that the equation :£ '¥ = 0 is invariant unde r co ntractions and translati ons which keep the axis fixed. Thus, let (3 2)

x' =kx+ce l , where k > 0 and c are constants, and set '¥'(x')

='¥(x) =,¥(~x' - ~et) on.M.' ={x' lx' =kx + ce l , x E

.M. }

(33)

Then :£ '¥' =0 on .M,'

(34)

It is well known that straight lines are taken into straight lines parallel to their antecedents under a mapping of the form (32). Let .M. and.M,* be two domains whose cylindrical parts agree for sufficiently large lXI I,

170 and let \jf* obey

3

.;e \jf* == \jf*, II + \jf* '22 - - \jf* '2 = 0 on M. *, X2

\jf*(X I'X2) ~ m(l\X 2) as XI ~ \jf*(X I,X 2) ~ m(2\x 2) as xI ~

\jf*(X 1,0) = 0 for -

00

00

for 0 ~ X2 ~ r l ,

M

00

(35)

for 0 ~ X2 ~ r 2,

< XI < 00, \jf* = 2n on r

*.

Let k E (0,1] and c be chosen such that M.' c M.* and such that r, r* have a point common. Our objective is to demonstrate that

xO

in

(36) We assume outer boundaries have a continuous normal except for a finite number of points and that at such points the stresses vanish. This assumption evidently rules out the kind of stress singularity which would be ex pected a sharp re-entrant groove. We assume that the point

xO

is a point of continuity of rand r*

We further assume that

m (a)

are

nondecreasing functions. The following properties of \jf' and \jf* are easily inferred:

lim \jf'(x I,x2) = mO\x' 2/

XI~-oo

\jf'(XI,O) = 0 for -

k),

lim \jf'(x l ,X 2) = m(2)(x' 2/

XI~

M

00

M

k)'} (37)

< xI < 00, \jf' = 2n' \jf* ~ 2n on r

Defme

'V

= \jf' -

\jf* on M.'

(38 )

Then, .;e iif =0 on M.', iif~on r, lim iif~o, 'V(xO) =0.

(39)

I XI I ~ oo

The refore the maximum principle yields a\jf' an

a\jf* an

O~-~--atx

0

(40)

and as a consequence there fo ll ows (4 1) Next, it is easy to velify that

171

(42)

The desired result (36) now follows from (41). We are finally in a position to apply the foregoing results to the optimization problem. The two domains .At, .At * depicted in Figure 4

A

A

Figure 4.

have boundaries which coincide except for segments A, A * corresponding torsion functions , and assume "t

="to (constant) on A ..

Let \jf, \jf*

be the

(43)

The purpose of the present analysis is to demonstrate that the profile which corresponds to a constant value of the stress is optimal. In other words , we wish to demonstrate that (44) There are two cases to consider, A c.At * and A
Then, by Theore m 8,

o: ; ~ : ; ~* at the endpoints of A. Therefore, and in view of (25), "to::;; "t

* at the endpoints of A,

which yields the desired conclusion (44).

172 Suppose now that A
We assume there exists k

E

(0,1), c, and a choice of

the origin 0 such that .M' c.M * and A *, A' have a point xO in common. Apply (36). Then

and we see that the optimality condition (44) follows at once.

3.3. Antiplane shear deformation. Here, the displacement is uni-directional and independent of distance in the same direction, so that in suitably chosen cartesian coordinates (45)

where q]j denotes a plane domain. Further, the stress components reduce to (46)

where (47 )

11 being the shear modulus. As a result, the equilibrium equations yield L'iu = 0 on q]j.

(48)

We introduce the stress magnitude 1: defmed by 2

1:

2 1/2

= (T13 + T 23 )

= IVul .

(49)

Consider the traction boundary value problem . In the present circumstances, the tractions reduce to a single component t in the x3-direction. In view of (47) and (48). there follows au an =ton ~.

(50)

where aq]j denotes the boundary of q]j . Hence. if the tractions are prescribed, the stress function u is the solution of the 2-dirnensional Neumann problem au _ i';u = 0 on q]j. -a = t on (YjI . n

(51)

For present purposes. it is more convenient to work with v. the complex conjugate of u. than to work with u itself because v represe nts the solution of a Dirichlet problem. Thus.

173

we introduce a function v related to u through the Cauchy-Riemann relations

au

av au

(52)

For !lJJ simply-connected and bounded, we may take v to be the solution of the Dirichlet problem ~v

= 0 on!lJJ, v = f(s) on

a!lJJ .

(53)

where f(s) =

!ost(cr)dcr

(54)

and s denotes arc length. As for the stress magnitude, it follows from (49) and (52) that't has the simple form 't = IVvl on!lJJ.

(55)

The maximum principle lies at the heart of an interesting optimization problem for minimizing maximum shear stress. To formulate this problem, let a!lJJ be partitioned into two parts, one fixed part and the other a complementary arc C{6 which is variable. This arc is traction free. The problem is to determine the shape of C{6 to minimize the stress concentration. More precisely, we consider the problem of finding within a class g' of competing curves, the one which minimizes the quantity 1:M {«6}

=sup't .

(56)

'(5

Thus, we seek an arc «6* 't* {C{6 ) = sup'!:

E

g' for which 't* defined by

(57)

'(5*

satisfies 't*

= inf 1: .

(58)

'(5E~

It is natural to associate the idea of minimizing the most intense stress along «6 with the idea that if the stress is spread as evenly as possible over C{6 , then the most favorable distribution has been found. Carried to its ultimate conclusion, this scheme suggests that we seek «6 so as to render't constant on C{6. This reasoning leads directly to a class of freeboundary problems which has been thoroughly investigated in the hydrodynamics of jets and cavities. In the present discussion, our main task is to illustrate how the association between the two problems can be made mathematically precise. A key result in the reduction of the foregoing optimization problem to classical free streamline problems in hydrodynamics is the "Min-Max" principle of Garabedian and Spencer l8 The drift of this principle is that the free streamline connecting two points yields, among all connecting streamlines, the one bearing the least maximum speed. We

174 include here for the sake of completeness a version of this principle in the form of a theorem given in Wheeler 19 , the proof of which is fashioned after the one found in Gilbarg 20 As a prerequisite, we introduce a specific version of the class ::f of admissible arcs. The class introduced here describes the problem of determining the optimum bottom for a groove in an elastic halfspace. Accordingly, we take as admissible any C€ for which '2il is the intersection of the halfspace x2 > 0 with the complement of a closed region which is starshaped with respect to a point x2 = -a (a~) of the x2 -axis (see Figure 5).

flv = 0 on '2il

C€(v =0) Fixed part of a'2il

v=O

(O,-a)

Figure 5.

The conditions imposed on the function v ensure that the surface traction vanishes at all points of d'2il and the loading at infmity is given by (59)

Theorem 9. Let ::f be as defined in the previous paragraph, assume 2

flv = 0 on '2il, v = 0 on d'2il, v = x2 + 0(1) as xl + xi --t 00 1

•

(60)

and assume there exists a curve C€O in ::f such that (6 1)

where 'to is a constant. Then (62)

175

Proof. Choose a curve «6

E

:J', and let

2/)

denote the domain detennined by this choice.

Suppose first that 2/)0 c 2/). Since v is non-negative on 2/) and v0 = 0 on (¥.!bo' the function v

=v -

Vo

(63)

satisfies (64) Therefore and since -

/).v

=0 on 2/)0' and v- =X2 + 0(1) as Xl2 + X22 ~

00 •

it follows that (65) The curves «6 and «60 have common endpoints, and at these points they are tangent. Therefore, and by (65),

av
-

at the endpoints. At points of «60' (66)

and at points of «6,

av an = - t . Therefore, and by (66), we see that t 2: to ' and consequently

This disposes of the case 2/)0 c 2/). Assume now that 2/) 0 is not contained in 2/), and map 2/) into a new domain 2/), through a contraction x

-

- =x= (x j ,x2) ~ (kx1,kx2) = (Xj,x2)

176

such that 2110 c 21l and the curves «6 , «6 have a point ~ = that at

~ , the normals to «6 and

c€

(X I ,x2)

in common. It follows

coincide.

Defme ~

~

I~

~

~

(67)

vex) = kv(k - x) for all x E 21l . Then

(68 )

and 't(x) = 1V'~(kx)1 for all x E 21l .

(69) (70)

Set Then. -

-

-

ll.v = 0 on 2il o• v ~ 0 on d21l(» and v --70 as

2 xI

2

+ x2 --7

00

(71 )

Consequently, the maximum principle implies (72)

Therefore. and since (73)

there follows

av

-

(74)

an :<;; 0 at x.

so that (70) yields

(7 5) Since

a~ _ 1_ avo an (x) = - 't(k x) and an(x) = - 'to .

(76)

(75 ) gives 't o :<;; 't(k

- 1-

x) .

Accordin gly. and becau se k

-Ix

is a point of

asl. the desired co nc lusion now foll ows.

Becau se of the forego ing theo rem. we are able to obtain res ults for this case from well -es tabli shed results 2 1. 22. 23. 24 on the Riabouchinski cavity problem of free streamline hydrod ynami cs. In a parcr 2~ by Wh eeler. Tezduyar. and Eldiwan y. profil es of the

177 foregoing type are detennined along with the corresponding stress concentration factors and compared with the stress concentration factors corresponding to certain standard profiles. It is shown that the ideal profile results in a reduction of as much as 30%. The case in which the flats are obliquely inclined to the halfspace yields readily to the hodograph method. Results for this case are found in Eldiwany and Wheeler2 6 . The problem of estimating stress concentration factors for prescribed geometries does not appear to have been addressed by means of maximum principles for the linear theory, perhaps because of the somewhat inconsequential nature of antiplane shear deformation. However, for finite antiplane defonnation, where explicit results are wanting, stress concentration factors have been estimated 27 • 28 with the aid of maximum principles and comparison methods. In an article 29 by Horgan and Silling, computational work is reported which show these results to be quite sharp. Here is an example in which analytical and computational methods yield highly accurate results even though an exact fonn of the solution is not available.

3.4. Elastic Inhomogeneities. It is well known that an inhomogeneity may serve as stress concentrator which induces self failure or may induce failure in the matrix or the interface. For a matrix and an inhomogeneity of given elastic moduli, an interesting question is thus raised as to whether it is possible to configure the inhomogeneity to minimize the stress concentration. It is shown in an article30 by Wheeler that under reasonable conditions on the loading, such a minimizer is furnished by a suitably proportioned ellipsoid. A byproduct of this research is an isoperimetric lower bound on the stress concentration which is easily deduced with the aid of the maximum principle. Let 2ll denote an exterior region bounded internally by a single surface o211 . Within

2ll lies a homogeneous and isotropic elastic material having modulus of rigidity Il and Poisson's ratio v. Let (x l ,x2,x3) denote rectangular cartesian coordinates and let T denote the stress tensor. The material in 2ll is in equilibrium, and as a result the governing equations may be taken as

(77)

where

e =Tick . The first of (77) represents the local balance of force and the second are the

Beltrami-Michell equations. We denote by 2ll the domain complimentary to the closure of

2ll. The stress in §i is denoted by

T and is governed by (78)

The interface between the matrix and the inhomogeneity coincides with the surface

o211. We assume perfect bonding, so that

= Tn on 02ll, u = ~ on 02ll, Tn

(79) (80)

178 where 0 denotes the unit outward normal to arzll, and u stands fo r the displacement of the material lying within rzll. Let E denote the strain,

I T E = i(gradu + (gradu) ).

(8 1)

The loading is represented by the stress applied at infinity, (82)

where

T~

is prescribed . We henceforth assume that the underlying cartesian frame is

principal for

T~,

so that (83)

The matrix strains are related to the stresses through

v

1 21l

E=-(T--81)

l+v

(84)

'

and the analogous expression is assum ed to hold for the inhomogeneity. Let T(k) denote the principal stresses, ordered so that T(l) ~ T(2) ~ T(3)

Our objective is an isoperimetric lower bound for the quantity (3)

~ (3)

TM = max( supT ,supT

}.

The interface conditions (79) yield

Te = Te

(85)

Ej ,

(86)

Ej =

where the subscripts e and i denote . respectively. the external and internal parts 31 For S a second order symmetric tensor. these components are defined by S j = (1 -

0

®

Se = S - Sj =

0) 0

S (1 -

0

®

0) •

® So + So ®

0 - (0 .

So) n ®

0 .

The conditi on (85) is merely a restatement of (79). whereas (86) follows from (80) by taking delivatives in the tangent plane to Cflh.

By (85).

179 def

def

= n · Tn =n . Tn = N. 8 = trT = trT + N • e=trT =trT + N.

N

j

j

(87) (88)

From (86). one draws (89)

Accordingly. (84) yields

I 211

2v I +v

-(trT - --8) I

I ~ =--,-(trT 211

I

2v ~ . I +v

~ 8)

(90)

Substitute from (87) and (88) to conclude

~ (11-I1)N

~ I-v

I-v -

=11- 8 -11----". 8 on a9Jl . 1+v l+v

(91)

Note first that ~(3) ~(3) 8~3T ~3supT ~3TM .

(92)

§b

We now assume that ~

(93)

11>11· Therefore. and by (91). (92).

(94)

Since

Theorem 3 (Section 2) yields (95)

sup 8 ~8~. i1!l! Therefore and by (94), (J-v)( J+v)~

TM~ (l+v)[(l+v)~+2(l-2v)111

8~ .

180

or, in terms of the bulk moduli, TM~

(l-V)K

~

e~

(96)

(I+V)lC + 2(l-2v)1C

This inequality is isoperimetric. In other words, under certain conditions on the material properties and the applied stresses, there is a choice of geometry for which equality holds 3o .

4. COUNTER-EXAMPLES FOR A MAXIMUM PRINCIPLE GOVERNING THE PRINCIPAL STRESSES

The obstacles presented by the system of partial differential equations governing the stresses in elastic solids to achieving a suitable version of the maximum principle are legendary. By means of counter-examples, it is shown in a paper32 by P61ya that the maximum principle fails for a variety of quantities of physical interest. In particular, it is concluded by means of an example involving conditions of plane strain that the stress may possess interior values at least as great as those at the boundary. The example leading to this conclusion does not rule out the possibility that the maximum value is always present at the boundary as well. The door was thus left open for a weak form of maximum principle, one which asserts that the maximum stress always occurs at the boundary, but might also be present at interior points. A three dimensional example in which the greatest principal stress has a strict maximum at an interior point was presented later by Langhaar and Stippes33 for a particular value of Poisson's ratio. This example, which appears to have been constructed by trial and error, involves a linear combination of axisymmetric solutions generated with the aid of spherical harmonics. The presence of a maximum is confmned by expanding the principal stress in a neighborhood of the point where it occurs. This process presupposes that it is possible to represent the principal stress explicitly in a neighborfwod of the critical point. Turteltaub and Wheeler 34 have derived expressions for the first and second gradients of the principal stresses which yield a systematic method constructing further three dimensional examples where the principal stress has a relative maximum at an interior point. These expressions are summarized below and an example is discussed.

4.1. Polya's counter-example. In a paper32 by P6lya, counterexamples to the maximum principle are given for se veral quantities of physical interest. One of them in particular deals with the largest principal stress. A brief discussi on is included here. For states of plane strain in a simplyconnected domain in two dimensions. the in -plane stresses are represented in terms of an Airy function, (97)

The re maining stress components are given by

181 (98)

P61ya [4) introduces the function

(99)

where a and b are arbitrary constants. From (97), (98), upon taking a = 3 and b = -1, there follows

The largest principal stress is found to be

(101)

whereas the remaining two principal stresses are

By (101),

T

(3)

2 2.. = 2 - (x 1 + X2) + o~ x

I2)as Ix l-t 0 ,

(102)

and thus T (3) exhibits the desired relative maximum. The foregoing example serves the purpose of a counter-example from a two dimensional point of view, but it fails to yield a relative maximum in three-dimensions because the maximum value occurs at all points of the x1-axis. There is no explanation in P61ya's paper32 of the rationale behind the construction of the example.

4.2. The counter-example of Langhaar and Stippes. Consider axially symmetric deformations, and let (r,e,
=cos e, q = sin e , and let Pn(p)

stress components

stand for the Legendre polynomial of degree 2. The

182 TIT = 3p2 - 1 - 6(2 + v)r2r4, 2

2

Tee = 3q - 1 + [3(21 - 2V)P2 + 12P4 - 8(2 - v)¥ 3}r , 2

(104)

T
2

is advanced in the paper33 by Langhaar and Stippes, which represent the nonvanishing components of an elastostatic stress field in the absence of body forces. By (103), for v = 1 /4 , the principal stresses may be expanded in powers of r to yield 1 TO ) = T(2) = _ 1 _ 24 (1 - 7p2)r2 + o(r\} T

(3)

= 2-

21 4(3 - p 2)r2 + oCr2) as r ~ O.

(105)

Accordingly, and since (3 - p2) > 0 , it follows that the greatest principal stress, T3 , has a strict relative maximum at r = O. It is not difficult to show that the maximum shear stress and the octahedral shear stress also possess a strict relative maximum at r = O. Examples are given in the article 33 by Langhaar and Stippes in which two of the principal stresses have relative extrema, in one case a maximum for one stress and a minimum for the other and in the other, two maxima. Of course, the fact that . ru Ies out th e POSSI' b'l' . . al stresses T O) + T(2) + T (3).IS harmOnIC I Ity th at all th ree pnnclp possess a relative extrema of the same kind. The authors of this article 33 disc uss the possiblility of constructing an example where two of the principal stresses possess an extremum of one kind and the remaining principal stress has an extremum of the other kind. This issue does not appear to have been resolved.

4.3. Expressions for the gradient of the principal stress. The following result is proven in the paper34 by Turteltaub and Wheeler:

Theorem. Let T be a second-o rder sy mmetric tensor field which is defined on a domain

01. Assume that T and its principal vectors e (p) are differentiable on 01. Then the principal values T(P) are differentiable on 01. and the derivatives are given by (106) If T is twice conti nu ous differe nti able on 01 , then so are the principal values and the second derivatives are given by

183

(107)

.

. The main significance of (106) and (107) is that while the principal values and

pn~clpal vectors as. w~ll as certain derivatives of T appear in the right hand members,

denvatIves of the pnnclpal values and vectors themselves are absent. It is clear that similar results for higher order derivatives may be obtained recursively. The following cartesian components are arrived at in Turteltaub and Wheeler34 with the aid of the Papkovich-Neuber stress functions expressed in terms of spherical harmonics T II = T22 = - a + 3b[(1+2v)x32 - v(x 2I+x 22)], T33 = 2a - 3b[(1-v)(xi+x;) + 2vx~, TI2 =0, Tl3 = 6bvx l x 3,

(108)

T23 = 6bvx 2x3· For this field, one finds T(O) = 2ae 3 ® e 3 - a(e l ® e l + e 2 ® e 2) , and hence

(109)

(110) Accordingly, and by (106) there follows gradT(3)(O) = 0 ,

(Ill)

while (107), (108), and (110) yield (112) Clearly, if b>O, the conditions for T(3)(O) to possess a relative maximum at x = 0 are fulfilled .

5.

CONCLUDING REMARKS.

What are the possibilities for finding maximum principles of significant generality for elasticity theory? The difficulties posed by the counterexamples have no doubt considerably dampened enthusiasm for working on general maximum principles. Perhaps so negative a view is unwarranted. The knowledge that th~ largest principal stress can have an interior maximum does not by Itself preclude the possIbIlIty that It IS a rare event, and

184

indeed, the counterexamples are difficult to construct and interpret. In this connection, one might wish to examine the question as to by how much the stress at an interior point may exceed the stress at boundary points. A first step in this direction might be to examine the traction problem for an elastic sphere and attempt to exploit the mean value theorem 35 Despite the counterexamples of the preceding section, there are one or two cases in which a degree of generality is achieved. One is the result of Miranda 9 mentioned in Section 3.1, which has implications for biharmonic functions in two dimensions. Another result, not previously mentioned in this article, is a theore m due to Fichera 36 which subjects the displacement magnitude to an inequality of the fonn sup lui::; H sup lui , '!lJ

CBJ

where H is a constant which depends only upon the domain £/) and the elastic properties of the solid. Perhaps this inequality, which in particular yields results for uniqueness and continuous dependence upon the data for the displacement boundary-value problem , is the most that can be said. Or perhaps it serves as a model for other boundary conditions and other quantities of physical interest. In this co nnection, it is worth me ntioning that not much is known about the factor H. References I. Kellogg, O. D., Foundations of Potential Theory, Dover, New York (1953).

2. Rado, T. , Subharmonic Functions, in Ergebnisse der Mathematik , Springer, Berlin (1937). 3. Fu, L. S. and L. T. Wheeler, "Stress Bounds for Bars in Torsion," 1. Elast., 3(1), (1973) . 4. Cimm ino. G., "Formale di Maggiorazione net Problema di Dirichlet per Ie Funzio ni Armonicbe," Rend.. Sem . MaJ, Un . Padova, 3, 46·66, ( 1932). 5 Protler, M. H. and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey (1967). Reprinted by Springer·Veclag, New York, (1984). 6. Payne L. E., Isoperimelric Inequalities and Their Applications, SIAM Review 9(3) 453-488, (1967). 7. Weinberger, H.F. and J. B.Serrin, "Optimal Shapes for Brittle Beams under Torsion:' in Complex Analysis and Ils Applications, Anniversary Volume in Honor of I. Y. Vekua's 70th Birthday, Steklov Mathematics Institute. Academy of SCIences, USSR, Nauka, Moscow, 88·91, (1978). 8. Payne, L. E. and L. T. Wheeler (1984), "On the Cross Section of Minimum Stress Co ncentration in the Saint. Venant Theory o f Torsion," 1. ElO$t., 14, 15· 18. (1984) . 9. Miranda , C., "Formuo le di Maggiorazio ne e Teorema di Esistenza per Ie Funzioni Biarmo mche di Due Variabili," Giarn. Mat. Battaglini, 78,97-118 , ( 1948) 10. P61ya. G .. "Torsional rigidity. Principal Frequency , Electrostatic Capacity a nd Symmelnzation:' Quart . Appl. Marh .. 6, 267·277, ( 1948)

II. Payne, L E.. "Bounds for the Maximum Slress in the Saini Venant Torsion Problem, " Indian J. oj Mec hs. and Math. , Special iss ue devoted B. Sen, 5 1·59, ( 1968). 12 Payne. L E. and G. A. Philippin. "Some remarks o n tbe prol.liern s of elastic torsion and torsional creep," Some Aspects of Mechanics of Continu a, PART I, Depasunenl of Mathematics. Jadavp ur University, Calcutl a (1977) .

13 . Sperb . R .. Max imum Princip les and Their Applications, Mathematics in Scie nce and Engineering, 157. Academic Press, New York, (1981) . 14. I-Iorga n, C . 0 .. Max imum Principles and Bounds o n Stress Concencration Factors

In

the Torsion o f Grooved

185 Shafts of Revolutio n," J . Elast., 12, 281 -291 , (1982). 15. Horgan, C. 0" and L. T Wheeler, "Maximum Principles and Pointwise Error Estimates for Torsion of Shells of Revolution," J . Elast" 7, 387-410, (1977). 16. Horgan , C. 0 ., and L. T. Wbeeler, "Saint Venants's Princi ple and the Torsion of Thin Sbells of Revolution," J. App!. Mechs., 98, 663-667, (1976). 17. Wheeler, L. T. , "On the Rol e of Constant Stress Surfaces in tbe Pro blem of Minimizing Elastic Stress Concentration," Int. J. Solids Struct., 2, 779-789, (1976). 18. Garabedian, P. and D. C. Spencer, "Extremal Metbods in Cavitational Flow," J . Rat. Mech. Anal., I. 359409, ( 1952). 19 . Wheele r, L. T. . "Applications of the Maximum Principle to the Minimization of Stre ss Conce ntrati on in Elastic Solids Subj ect to Antiplanc Sbear," in Partial Differential Equations and Dynamic al Systems, W. E. Fitzgibbon III ed, Pitman Research Notes in Mathematics, 101, 353-366, (1 984) .

20. Gilbarg, D., "Jets and Cavities," Encyclopedia of Physics, lx/III, S. Flugge (ed), Springer, New York, (1960). 21. Riabouchinski, D., "On Steady Fluid Motions with Free Surfaces," Proc. London Math. Soc., 19, series 2, 206-215, (1920). 22 . Riaboucbinski, D., "On Some Cases of Two-dimensional Fluid Motions," Proc. London Math. Soc., 25 , series 2, 185 -194, (1925). 23 . Birkboff, G . and E. H. Zarantonello, Jets, Wakes, and Cavities, Academic Press, New York, (1957). 24. Gurevicb, M. I., Theory of Jets in Ideal Fluids, Academic Press, New York, (1965). 25. Wbee1er, L. T , Tezduyar, T E. and B. H. Eldiwany, "Profiles of Minimum Slress Concentratio n for Antip1ane Deformation of an Elastic Solid," J. Elast., IS, 271-282, (1985). 26. Eldiwany, B. H. and L. T Wbee1er, "Groove Bottom Contours of Minimum Stress Concentration for Antiplane Sbear Deformation," J. Appl. Mechs., 52, 379-384, (1985). 27. Abeyaratne, R., and C. O . Horgan, "Bounds on Stress Concentration Factors in Finite Anti-plane Sbear," J . Elast .. 13, 49-61. (1983) . 28 . Jafari, J. H., Horgan, C. 0 ., and R. Abeyaratne, "Finite Anti-plane Shear of an Infinite Slab witb a Tractionfree Elliptical Cavity : Bounds on the Stress Concentration Factor," Int. J . Nonlinear Me chs., 19, 431-443, (1984) . 29. Horgan, C. 0., and S. A. Silling , "Stress Concentration Factors in Finite Anti-plane Shear: Numerical Calculations and Analytical Es timates," J . Elast., 18, 83-91 , (1987). 30 . Wbeeler, L. T, "Inbomogeneities of Minimum Stress Concentration," in Anisotropy and Inbomogeneity in Elasticity and Plasticity, The American Snciety of Mecbanical Engineers, AMD -Vol. 158, (1993). 31. Hill, R. Continuum Mecbanics and Related Problems of Analysis, N. I. Mu skb elisbvili 80tb Anniversary Volume (edited by L. I. Sedov), p. 597 . Academy of Sciences, SSSR . 32. G. P6lya, "Liegt die Stelle der grassten Beansprucbte an der Oberfliicbe?," ZAMM, 10, , 353-360, (1930). 33 . Langhaar H. L. , and M. C. Stippes, "Location of Extreme Slresses," J. Elast., 6,

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Series on Advances in Mathematics for Applied Sciences Editorial Board N. Bellomo Editor·in·Charge Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy

C. M. Dafermos Lefschetz Center for Dynamical Systems Brown University Providence , RI 02912 USA

J. G. Heywood Department of Mathematics University of British Columbia 6224 Agricultural Road Vancouver, BC V6T lY4 Canada S. Lenhart Mathematics Department University of Tennessee Knoxville , TN 37996-1300 USA P. l. Lions University Paris XI· Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France S. Kawashima Department of Applied Sciences Faculty Eng.ng Kyushu University 36 Fukuoka 812 Japan V. Kuznetsov Institute of Chemical Physics Russian Academy of Sciences Kosygin Str. 4, Bid. 8 Moscow 117 977 Russia

G. P. Galdi Editor·in·Charge Institute of Engineering University of Ferrara Via Scandiana 21 44100 Ferrara Italy

B. Perthame Laboratoire d'Analyse Numerique University Paris VI tour 55-65 , 5ieme etage 4, place Jussieu 75252 Paris Cede x 5 France K. R. Rajagopal Mech. Eng .ng Department University of Pittsburgh Pittsburgh, PA 15261 USA

R. Russo Dipartimento di Matematica Universita degli Studi Napoli II 81100 Caserta Italy V. A. Solonnikov Institute of Academy of Sciences SI. Petersburg Branch of V. A. Steklov Mathematical Fontanka 27 SI. Petersburg Russia F. G. Tcheremissine Computing Centre of the Russian Academy of Sciences Vasilova 40 Moscow 11 7333 Russia

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Impulsive Differential Equations with a Small Parameter

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