Acceleration waves in constrained elastic materials

Acceleration Waves in Constrained Elastic Materials

NIGEL SCOTT

Communicated by J. L. ERICI~S~

Abstract

Well known results on the propagation and growth of acceleration waves in Cauchy elastic materials are extended to materials which suffer one or two internal constraints. It is proved, under certain restrictions, that acceleration waves will not propagate in a material which has three or more internal constraints. The great simplifications deriving from an assumption of hyperelasticity are indicated. The present results could be extended to materials other than simple elastic.

1. Introduction

An acceleration wave is a surface of discontinuity, the unit normal to which we denote by n, which propagates through a given material in a manner such that the deformation gradient and the velocity u are both continuous across this surface but the acceleration /~ of a particle experiences a finite jump, denoted by [/~], as the wave traverses that particle.

Much work has been done on acceleration waves in unconstrained materials, for example by WRIGHT [1], TRUESDELL • NOLL [2], CHADWICK t~ CURRIE [3], FARLEY [4], VARLEY & CUMBERBATCH [5], VARLEY & DUNWOODY [6], CHEN [7], Strnum [8] and many others. In each ease a propagation condition has been obtained in the form

det(p o U 21 - Q ) = 0 , (1.1)

where Po is the material density in the reference configuration, U is the velocity of the wavefront relative to the material and Q is a 3 x 3 non-singular matrix termed the acoustic tensor. Except for a factor of P/Po, where p is the current material density, this (2 is the same as that defined in [2, w 71]. In general, the acoustic tensor is a function of deformation gradient, material particle, and the direction of propagation n of the wave at the point in question.

The possible wave speeds, for a given n, are given by the real, positive eigenvalues (if any) of (2 and the direction of [1i] is given by the corresponding right eigenvector R of Q:

[~i] = aR, (1.2)

58 N. SCOTT

where R is normalised so that R . R = 1 and a is the amplitide factor, having the physical dimensions of acceleration. Since U 2 is given by a cubic equation there is in general no simple explicitexpression for it and neither is there such an expression for R.

ERICKSEN [9] considers the case of homogeneous, incompressible, isotropic, hyperelastic materials and obtains a propagation condition again of the form (1.1). However, the constraint has the effect of introducing a spurious root P0 U2=0 : the matrix Q now has rank 2 rather than rank 3. Therefore, the determinant condition (1.1) reduces to a quadratic in Po U2 so that although explicit expressions for Po U2 and R now exist [9, (1.5)] they are a little complicated. These remarks are true also of a treatment of more general incompressible materials [2, w 78].

As far as the author is aware, the only other previous work on acceleration waves in internally constrained materials is that of CHEN & GURTIN [10], who consider materials which are inextensible but compressible. Again, an equation of the form (1.1) is obtained but it has a spurious root Po U 2 = 0 and the corresponding Q is of rank 2 (except in the case of waves propagating in any direction n which is perpendicular to the direction of inextensibility, in which case rank Q = 3). In the same paper these authors also consider the case where two families of curves, orthogonal in the reference configuration, are constrained to remain orthogonal after deformation. Similar conditions on Q are obtained. One purpose of this paper is to show that when any elastic material suffers one constraint the corresponding acoustic tensor Q* (that is, the Q of (1.1)) has rank 2, except possibly for certain exceptional n when rank Q* is 3, as in the unconstrained case.

The case of two or more simultaneous constraints has not been considered in the literature. It will be shown that, given certain conditions, no acceleration wave can propagate in a material having three or more constraints. If there are two constraints, we shall show that, as might be expected, there are two spurious roots P0 U 2 = 0 and that rank Q * = 1, where Q* is the acoustic tensor for the problem and takes the place of Q in (1.1). This has the result that simple explicit expressions for both Po U2 and R may be found, in contrast to the cases of no or of one constraint.

For unconstrained materials an expression for the growth of wave amplitude tr has always* been obtained in the form of a non-linear ordinary differential equation:

ha 2 p o - - + A a + B a 2 = 0 , (1.3)

fit.

where 6/6t, denotes differentiation with respect to time following a ray, or bicharacteristic. Rays are defined, in the usual way, in w 4. The functions A and B are defined on the wavefront and, in principle at least their variation along a ray is known. We shall see that an equation of the form (1.3) is valid also for materials with one or two internal constraints and exhibit the appropriate functions A and B.

* SuHum [8] differentiates along the wave normal n but also has present surface derivatives of tr. However, as WRIGHT [1, w 1 and following (4.10)] points out, these combine to form a derivative along the ray.

Acceleration Waves xn Constrained Elastic Materials 59

This paper is divided into sections as follows. Section 2 consists of definitions and preliminary results, whilst w 3 introduces the constraints and discusses their effect on the stress. Also, it is proved under certain conditions that acceleration waves will not propagate in a material suffering more than two constraints. Some examples of constraints are given and, in order to motivate the treatment of doubly constrained materials, a simple model of fibre-reinforced materials is discussed. In w 4 we define the ray velocity and stress its importance in acceleration wave theory, deriving some results on the geometry of the wave surface. The desirability of expressing the speed of propagation in a form homogeneous of degree one in n is emphasised. The propagation conditions corresponding to (1.1) and (1.2) in the case of one or two constraints are derived in w 5, whilst in w 6 the equations of growth for both these cases are shown to be of the form (1.3). It is demonstrated in w 7 how the results of w 6 may be written in terms of current variables rather than reference variables. The last two sections are devoted to the great simplifications induced by the assumption of hyperelasticity. In 88 we show that if there is one constraint in a hyperelastic material the two squared wave speeds are real and that their corresponding right eigenvectors are perpendicular. Also, the strong ellipticity condition is discussed, whilst in 89 the equations of growth are very much simplified for hyperelastic materials.

2. Notation, Definitions and Preliminary Results

Both the reference configuration and the current configuration are referred to the same rectangular Cartesian coordinate system, with respect to which a particle having material coordinates X A (A = 1, 2, 3) has the spatial (current) coordinates x i (i = 1, 2, 3). These are related at time t by a function of the form x = x (X, t) which we shall assume to be at least once continuously differentiable and invertible: X = X(x, t). We employ a variety of notations for the deformation gradient tensor p and its inverse p-1 :

~xl ~?XA = X a ,, (2.1) PiA-- c~X 4 - x i , A, Pai I = gx i ,

all the differentiations being at constant t. The convention whereby material suffixes are capital letters and spatial ones are lower case will be strictly observed. A superposed dot indicates time differentiation at constant X whilst ~?/~?t is at constant x. The velocity ~ will be denoted by u, so that d is the acceleration.

The right Cauchy-Green strain tensor C is defined by

c = p r p, CAn=piaPiB, (2.2)

where T denotes the transpose and the summation convention is employed. We define the Jacobian J of the deformation x (X, t) and note the following connexion between it and the material density p in the current configuration:

J = det p, P0 = P J- (2.3)

60 N. SCOTT

The Euler relation may be written in either of the forms

O~a (J p~it) = 0, ~3x-~. = 0. (2.4)

The Cauchy stress a and the first Piola-Kirchhoff stress n are interrelated by [2, w 43 A]

n = J a p -T, 7tia =Jaijp~), (2.5)

where, without ambiguity, - T denotes the transpose of the inverse. The equations of motion are, in the absence of body forces,

~, , l ,a = pot ) i , alj,~=-p~ti, (2.6)

but we shall work exclusively with the former. For unconstrained materials n is given by a constitutive law

n = nO(p, X), (2.7)

whilst in the important special case of hyperelasticity, where there is a strain energy* W(p, X) per unit volume of reference configuration, we have

OW n c = - - (2.8)

0p

The well known results corresponding to (2.7) and (2.8) for constrained materials are deferred until the next section.

The acceleration wave has already been defined and here we merely remark that the first derivatives of x (X, t), namely u and p, are continuous whilst some of the second derivatives, for example fi and p, suffer finite discontinuities at the wave surface.

Having quoted some necessary results on stress and strain we now consider the wave surface, which may be given in either of the forms �9 (X, t) = 0 or q~ (x, t) = O. Let n be the unit normal to this surface in the current configuration orientated so that the speed of propagation, V, of this surface along its normal is positive. If U is the speed of the wave relative to the material then

U = V - u . n. (2.9)

We define the column vector m by

m r = n r p, rnA=nlpla (2.10)

and note that, though not a unit vector, it is normal to the wave surface in the reference configuration. In fact, we may derive that

n 8q~/cg~b m cgq~ /~ (2.11) V - ~xx /&- and U 8X/ "

In discussing conditions at the wave front it is advantageous to introduce coordinates (x, qS) in place of (x, t) or (X, 4) in place of (X, t), provided that, in the

* To incorporate the principle of material frame indifference we must take W= W(C, X).

Acceleration Waves in Constrained Elastic Materials 61

first case, the wave front is not fixed in space (Odp/Ot4:0) and that, in the second case, it is not fixed in the material (~ 4: 0). This is done so that setting q~ = 0 or

= 0 in an equation immediately yields a relation valid at all points of the wave front as it varies with time. The derivative O/OXIo evaluated at ~ = 0 will be written O/aX and O/dxl4, evaluated at q~=0 will be written O/Ox. Consequently, for any differentiable function f of arguments (x, t) or (X, t), we have from (2.11) that

0 f Of n 0 f Ox Ox V Ot'

Of _ Of mr. OX OX U

(2.12)

We may define the wave front, or "hat" , derivative in an alternative but equivalent manner as follows. Let z (x) be the time of arrival at the place x of the wave front ~b(x, t )=0 and %(X) the time of arrival at the particle X of the wave front q~ (X, t)= 0. If x' is the current position of the particle X', these are related by x'=x(X', t) and z(x')=%(X'), where X' is a generic particle. For any f(x, t) and F(X, t) we may define f and P to be the values of f and F at the wave front: f(x) = f(x, z(x)) and F(X) =- F(X, % (X)). From the relations

Oz n &o m 0x =V- ' 0X - U ' (2.13)

we see that the hat derivatives may now be defined by

~f(x, t) Of(x) ~F(X, t) OP(X) - - = - - (2.14)

0x 0x ' 0X 0X

The region ahead of the wave is designated the - r e g i o n and that behind the + region. The jump in a quantity g at a point x is defined in terms of its value g- just before the wave arrives at x and its value g+ just after the wave has left x by the equation [g] = g+ - g - . (2.15)

We use square brackets only to denote jumps. The following relation holds for any A and B:

[AB] = [A] [B] + A- [B] + B - [A]. (2.16)

The wave front hat derivatives owe their importance to the following properties:

The two hat derivatives are interconnected by

O _x[6 niu,] O O ( m~__) O (2.18, OXi'=PAj \ i j - - ~ - ] OX A , OX A : PiA + OX i"

3. Internal Constraints and their Effects

In this section we discuss the effects which scalar constraints on the components of deformation p have on the stress n and show that in general an acceleration

62 N. SCOTT

wave will not p ropagate in a material which has more than two internal constraints. We then in t roduce some specific examples of constraints and end with a possible applicat ion to fibre-reinforced materials.

A constraint has the form 2(p, X ) = 0 and the principle of material frame indifference implies that this is restricted to 2 ( C , X ) = 0 . Therefore we cannot have more than five independent constraints or the material would be rigid. Let n be the number of independent constraints where 0_< n _< 5. Denote these constraints by 2~(p, X ) = 0 , where ct ranges from 1 to n. Repeated Greek superscripts will be summed over, though no tensor character is indicated. In the case of an unconstrained material (n =0) we equate to zero all terms with the superscript and sums over ~ become vacuous.

We define tensors n ~ by c32~

rt " - (3.1) ~ p '

because if the constraints are postulated to be workless [2, w the stress may be written in the form

n=nC + p~n ~, (3.2)

where the functions p" = p~ (X, t) are arbitrary in that they are not directly dependent on the deformat ion p th rough any constitutive law. Of course, the p" must be selected so that the equat ions of mot ion (2.6) and the boundary condit ions are satisfied. The tensor rtC(p, X) is that part of the stress which is determined by the deformation. In the case of hyperelasticity we have no= ~W/~p and n is given [11, (6.16)] by it = ~?W*/Op, where W* = W + p ' 2 L Thus there is no need to postulate that the internal constraints in a hyperelastic material are w o r k l e s s - t h e y are automat ica l ly so [11, p. 82f].

Each constraint may be differentiated with respect to time to yield ni~PiA=0. N o w DiA:3~ll/~XA--U-lmA~l,, SO that rP/A] = - - g - l m A [ s Therefore, o n

associating with each constraint a vector v" defined by

v'=rc~m, (3.3) we have

v ~. [~i] = 0, (3.4)

for each e. The vector m may be chosen to lie in any direction by a suitable choice of wave normal n. For all those m such that the v ~ are linearly independent, it is clear f rom (3.4) that n < 3, that is, acceleration waves will not in general propagate in a material suffering more than two constraints. However, it is quite possible that two different constraints will produce v" which are linearly dependent for all posible m, but this case we exclude from the subsequent analysis.

If n = 1, we define a ~" by ~,~ = v~(v �9 v)-i (3.5)

and note that ~ | v ~ is homogeneous of degree zero in m (and therefore also in n) and that ! - ~ | v ~ is a projection operator .

If n = 2 there are two vectors v ~ which, in the case of linear independence, define a plane. We define the two vectors V to lie in this plane and satisfy

~ . v p = 6~t ~. (3.6)


The ~" are thus uniquely determined and the left-hand side of (3.6) is homogeneous of degree zero in m. Define also a vector p by

IU = y2 AII ~1 , (3.7)

The magni tude/~ of this vector and a unit vector/3 parallel to/~ are given by

#2 = v 1 " v 1 v 2 " v 2 _ (v 1. v2)2, /~ = ~u. (3.8) #

We may prove that /~ | 1 7 4 1 7 4 ~ (3.9)

and deduce that v ~. v a and ~ . ~# are mutual ly inverse 2 x 2 matrices satisfying

~ = (~ �9 ~#) v ~, v~=(v ~. v#)~ #, detv ~. Ylg ~--- ]22. (3.10)

Repeated Greek suffixes are summed but we note that (3.10)1,2 remain true if n = 1, where the summat ion is trivial. We also define a vector ~ and use (3.10) to prove the following result:

~ 2 A ~ I , ~ = # ' . (3.11) #

F r o m (3.4) we see that unless [ti] is to be identically zero it must be p ropor t iona l to/1. Therefore, in sharp contrast to the cases in which n = 0 or 1, in a doubly constra ined material, the only possible wave ampli tude is determined without reference to a condi t ion of the form (1.1).

If n - - 1, we require the vector v ~ not to vanish and if n = 2, the v ~ have to be linearly independent or ~ and p are not defined and the subsequent analysis fails.

Examples of Constraints

(i) Incompressibility. This is the most c o m m o n l y considered constraint and takes the form 2(p, X) = det p - 1 =0 . We have

n~ = p - l , v ~ = n ( 3 . 1 2 )

and clearly v" never vanishes.

(ii) Inextensibility. Let the unit vector A (X) define the direction of inextensibility in the reference configuration, which we term a fibre. It is k n o w n [11, w that the current fibre direction a is given by a = pA and that the constraint takes the form a �9 a = 1. We have

7t~=a | v~'=ya, ( 3 . 1 3 )

where ~ = a - n = A - m . The vector v ~ vanishes only if the fibres lie in the wave surface and the exceptional nature of this case was also noted by CHEN & GURTIN [I03.

It is possible to apply these two constraints s imultaneously and the resulting material may be regarded as an idealised model of fibre-reinforced materials in which inextensible fibres are embedded in an incompressible matrix. A general

64 N. SCOTT

account of much of the work done in this field has been given by SPENCER [11] and a certain measure of agreement with experiment has been claimed [12]. We note that in this example the r are in general linearly independent, this not being the case only for waves such that V = 0 or ~ = 1, that is, for waves such that the fibres lie in the wave surface or are normal to it.

In a later paper I hope to apply the results of this paper to acceleration waves in idealised fibre-reinforced materials and also to consider other examples of constrained materials.

4. Wave Geometry and Rays

Here we discuss those properties of the wave geometry and rays necessary both to show the importance of the ray velocity in acceleration wave theory and to derive the equation of growth in w We begin by defining ~/~3x by

( 1 - n | ~?x ==- ~?x = - ( 1 - n | c3--x-" (4.1)

These definitions are equivalent, as may be seen from (2.12)1. The derivative ~/~x (or I~) is performed at q~(x, t )=0 and at constant t and is evidently a surface derivative. We state the result

~r/i = ~ni (4.2) Oxj 0x~ '

which is easily proved by writing n = Vc/)/I V491 and applying (4.1)2.

We denote by s this symmetric surface tensor ~n/Ox and remark that it contains information on the surface geometry. In fact, if e is a unit vector in the wave surface at fixed time t, the curvature* x e of the curve in the wave surface to which e is tangent is given by

tr e = S i j e i e ~ . (4.3)

The eigenvalues of s are the principal curvatures, together with an irrelevant zero eigenvalue, and the mean and Gaussian curvatures are given respectively by

f2= �89 K=�89174 (4.4)

Of course, det s = 0.

Let a point on the wave front have velocity v. Then in order that that point stay on the front, v must be such that

n . v = V, (4.5) where

V= V(n, x, t) (4.6)

is assumed to be a known function.

We define h/fit to be the time rate of change moving with the wavefront along the normal n. This is the f-derivative of THOMAS 1-13, p. 40f.]. Define also fi/ft v to be the time rate of change moving with the wavefront in the direction of v.

* We adopt the convention that curves concave to n have negative curvature.


Implicit in this definition is the requirement that n . v = V We then have

6 6 6t = V n . O-x' 6to =v" 8x (4.7) .

We now derive some results on the rate of change of n. Since the left-hand side

of (2.13) depends on x only we have =--~-xi , which on contraction

with n~ and by use of (4.7)1 yields

6n ~ V 6t =0' (4.8)

a result equivalent to that of THOMAS [13, p. 45, (4.11)]. F rom

6t o = 6 t -+v . O~ (4.9)

and (4.8) we may derive a result valid for all v such that n . v = V:

6n ~VI [ OV~ ~n + - - = v j - - - , (4.10)

6to Ox , [ One) 8xj

where [, denotes differentiation holding n fixed.

We may eliminate the surface geometry from (4.10) and considerably simplify its form if it is possible to take v=OV/On. But from (4.5) and EULER'S result on homogeneous functions, this is possible if and only if V is homogeneous of degree one in n. Now this has always proved to be the case in acceleration wave theory but even had it not done so we would simply define a V*, equal to V but homogeneous of degree one in n, by

(" ) V*(n, x, t)=(n, n) ~ V ~ - , x, t (4.11)

and use it in place of V. We define the ray velocity v* in the usual way (see for example [4, 5, 14]) by OV*

v* = - - (4.12) On

and a ray is the path x = x(t) of a point with velocity e*, that is dx/dt = v*. If v is chosen to be v*, equation (4.10) becomes

6n t ~V* 6 t , Ox ,, = 0 , (4.13)

where 6/6t , has been written for 6/6to, . This is the result of VARLEY & CUMBER- BATCH [5, (2.11)] and gives the variation in n along a ray. It is a result easier to use than (4.8) because the derivative at constant n is easier to work out than that in (4.8) and the surface geometry is not directly involved.

Over and above the fact that its use simplifies (4.10), the ray velocity owes its importance in acceleration wave theory to the fact that the equation for the growth in wave amplitude is best expressed in terms of a derivative along a ray (1.3). 5 Arch Rat Mech Anal, Vol. 58

66 N. Scorr

Since we shall work in terms of material coordinates we note that the image X = X(t) of the ray in the reference configuration is given by [4, (4.16)]

dX 0 U def. , ~ - = ~ - m = V , (4.14)

where U must be written in a form homogeneous of degree 1 in m. We shall need the result

v* . . . . V * . - - (4.15) Ox fit , OX "

As an aside, we differentiate (4.11)to obtain

OV* ~V - - = v * = V n + ( 1 - n | On On (4.16)

and remark that, if in some particular example it is inconvenient to rewrite V in a form homogeneous of degree one in n, we may nevertheless determine the ray velocity, using (4.16)2 rather than (4.16)1.

5. Speeds of Propagation

We shall treat the problem by regarding (3.2) as an inhomogeneous and explicitly time dependent constitutive law and follow, as closely as possible, VARLEY'S analysis of the unconstrained problem I-4, w Even if the constitutive law nc and the constraint tensors n" are homogeneous and time independent, the stress n will not be so because, in general, p" =p ' (X , t).

Define a fourth order tensor c by

c = d +p~c% (5.1)

where ~CA . . . . OrCi~A (5.2)

CiAj B -- ~-n--- ' CiAjB - - " ~FjB OpjB

From (3.1), c ~ has the symmetry

Ci~AjB : C3BiA (5 .3)

and for hyperelasticity c ~, and hence c, have the same symmetry.

In what follows we may recover the unconstrained case by putting n~=c~= v ~ = p~ = 0 and replacing c by c c, so that unless otherwise stated the following results are valid for n = 0, 1, 2.

For ease of reference we repeat some of the basic equations which are needed:

7~iA,A = [ 0 t l i , (5.4)

aul (5.5) Pia = ~Xa '

~,A - " "" ~ (5.6) - - C i A j B P j B W P 7~iA.

This last follows from (3.2) and (5.1). We may use (2.12)2 to rewrite the first two of


these as �9 ma ~ i A (5.7)

P ~ -~iA= aXA

and ~ui mA . PiA= aXA U ui. (5.8)

Since p is continuous we have that a ~ and n ~ are continuous. Now the integral form of the equations of motion demands that [n] m = 0, which from (3.2) yields I - f ] v ~= 0. We are taking v ~ to be linearly independent, so E l ] - - 0 implying that n itself is continuous. Therefore, on taking jumps of (5.7) and (5.8),

�9 mA P~ +---U-- [#iA] = 0 , (5.9)

rn A ~,A] = - - ~ - [t~,] (5.10)

and with (5.10) we see that (5.6) becomes

1 [~iA] -~" - - y CiAjB ma [/~j] + [P'] 7~'A �9 (5.11)

On eliminating [9] between (5.9) and (5.11) we have

{Po U21 - Q } [~] = - U[P ~] v~, (5.12)

where the acoustic tensor Q is defined by

Qij = CiAjB mA mB. (5.13)

It is possible to define tensors QC and Q~, analogous to Q, based on c c and c ~, respectively: then

Q=Q~+f 'Q~' . (5.14)

For unconstrained materials Q=Qr and (5.12) yields the usual form (1.1), whilst for constrained materials (n = 1 or 2) we contract (5.12) with ~a and use (3.5), or (3.6) as the case may be, to derive

U ~ ] = ~ - O [~]. (5.15)

This result is true for both n = 1 and n--- 2 and vacuous for unconstrained materials, n=0 . We use (5.15) in (5.12) to obtain

{Po Ug 1 - Q*} [ri] = 0, (5.16)

where Q* is defined by

O* = (1 - ~ | Q. (5.17)

Clearly, the problem has been reduced to the eigenvalue problem discussed in w 1 and both (1.1) and (1.2) hold if the Q of (1.1) is taken to be the Q* defined by (5.17).

Using (1.2) and (5.17) we write (5.16) as

Po U2 R - QR +~'(v ~'. QR) = O, (5.18)

5*

68 N. SCOTT

from which it is clear that unless Po U 2 = 0 ,

~'. R = 0 = v ". R, (5.19)

as demanded by (3.4). Since det Q*=0 , if n = l or 2 we see that p o U 2 = 0 is a single or double root, respectively, of (5.16), but it is spurious because it can be shown that its right eigenvector R cannot satisfy (5.19).

If n = 1, Q* has rank 2 and so there is the possibility of two real wave speeds each with a right eigenvector satisfying (5.19). If n=2 , Q* has rank 1 and there is only one possible wave speed in this case. This and the corresponding right eigenvector will now be derived explicitly. We note from (3.9) that

Q* =/~ | (5.20) from which it is clear that

g = ~ , (5.21) a result predicted in w 3, and

po U2 =/~. Q/] =/~. QC~+p,~. Q,~. (5.22)

From (5.16) and (5.19) we see that for any eigenvalue po Uz we have

Po U2 =R. QR, (5.23)

provided R is normalised to be of unit length, but (5.22) has more content than this because/~ is known explicitly whilst R in general is not.

6. Growth of Wave Amplitude

We now find an equation of the form (1.3) for the acceleration wave amplitude a appearing in (1.2). So far as possible we treat the cases n=O, 1, 2, together. To simplify the analysis we assume that the region ahead of the wave is at rest. That is, on the wavefront

u -+ = z i - = p - =~b "- = 0 , U = V. (6.1)

We have not assumed the region ahead of the wave to be undistorted; this would necessitate the requirement p = 1 at the wavefront. Certain results simplify considerably with the assumptions (6.1). Equations (2.18) become

Ox-=P~ 1 c?Xa, -OX a =P,a Ox i (6.2)

and it can be shown that the Euler relations (2.4) are valid in the form

( j p ~ l ) = 0, (6.3)

Define a sixth order tensor e by

e = e c + p" e ", (6.4)

where 0 O c __ c ~t __ c e iAjBkC--~CiAja , eia~akC--~pkcCialB. (6.5)

~ F k C

Acceleration Waves m Constrained Elastic Materials 69

We have eta ~-- eiA jB kC PkC + P ~ C~A jB" (6.6)

The equations (5.4)-(5.6) may be differentiated to obtain

~tA,A : DO iii , (6.7)

Off, (6.8) i~ta = OXA '

~iA = CtA jB Pin + eiA jB kC P ~BPkc + 2 p~ Ci~ j8 Pin + if' lh~" (6.9)

We rewrite (6.7) and (6.8) in terms of hat derivatives and take jumps to obtain

Po [fit] + - - ~ [iiiA] ----O-~A [~ia] (6.10)

and rn~

[/ita] + --U- [/i i] = ~ [fit]' (6.11)

From the assumption of quiescence (6.1) and (2.16) we have

0"2 0" -a ~j~Pkc]=-~msmcRjRk, ~[~js]= - ~ - m s R j [ ~ ]. (6.12)

On taking jumps of (6.9) we may eliminate [/~] and [~] from (6.10) and (6.11) in favour of [//] and [~']. From the resulting equation we may eliminate [~], ~ ' ] , and [p] in favour of [fi], which is written as O"R, and deduce that

{P0 U21 - Q} [//] + U ~ ' ] v" = - K, (6.13)

where K is defined by

K~ - ~X z U {CiLia m s Rj + CiA jL ma R j - ~ f,~. QR}

2 ( ~ / 1 \ ~ [ 1 ~_~ 1 ~ g j ] +0"U l~-X-S [ ~-ctAiam.RO-- -~X~A t-~-rqAv .QR) +-~CiAi, mA-~X~B; (6.14)

0"2 +--U-- {eiA jBkC rnAmBmCRjRk-- 2 Q~j Rj~" . QR}.

We contract (6.13) with ~P and use (3.6) in order to eliminate [~'] from (6.13):

{Po Uz 1 - Q * } [~] = - ( 1 - ~ ' | ~') K+po U2f ,'. [~] r (6.15)

Let L be the left eigenvector of Q* corresponding to the right eigenvector R and normalised so that L. R = 1. On contracting (6.15) with L we see that [~] is elim- inated from the left-hand side and this is the usual method of obtaining the equation of growth (1.3) for unconstrained materials. However, for constrained materials [~] remains on the right-hand side, though in the next paragraph we show that ~ - [/~] may be written in terms of [li].

70 N. SCOTT

Just as the constraint 2" = 0 may be used in the form 2"= 0 to obtain (3.4) it may be used in the form )~ = 0 to obtain

v" . [i i] = U Ci~a W ~iA [~jn-] + U n,~ ~ [ill] %f'g'. (6.16)

From (6.12), we see that

�9 g a , , , ~ R , a 2 g = ~ (UR i lria ) + a U lriA ~ a + U - R. Q~R. (6.17)

Defining ~ by ~ = V . [~] we remark that ~,~ is determined in terms of g~ and v ~ and state the result that

~,~v" =g~V. (6.18)

This result is vacuous if n = 0, trivial if n = 1, but has some content if n = 2.

Before applying (6.18) to (6.15) we define a vector L p by

LP= L. ( 1 - v ~ | (6.19)

and note that L p. R = L. R. From the fact that L is the left eigenvector of Q* we may deduce that

Po U2 L" i" = LV " Q v . (6.20)

On contracting (6.15) with L and using (6.18)-(6.20) we see that the following equation for a holds:

LV . K - LV . Q V g~=O. (6.21)

By inspection this is of the form A

ZL ~ x L + A a + B a 2 = O ,

where U Z L = L ] {CiLjB m~ + C,A jL ma -- 7rill V~, Qkj -- Irj~L Qik V~,} R j,

1 �9 ~, tC~Rk) and U QoVj ' ka OXA, ~

1 B = ~ y {e,AjBacmAmBmcL~RjRk -- 2(L". Q~R)(f' ~" Q R ) - ( R . Q~R)(f' ~. QTLV)}. (6.25)

Notice that only L p and not L is involved in (6.21)-(6.25). These equations are valid for any of the cases n = 0, 1, 2.

For an unconstrained material the usual results may be obtained by equating identically to zero all terms bearing the superscript a, so that LV=L, (6.21) becomes simply L. K = 0 and (6.23)-(6.25) are also much simplified.

Considerable simplifications are possible also for a doubly constrained material, n = 2. Here we may easily derive explicit expressions for L and LP:

L = QTfl LP=~=R. (6.26) PoU2 '

(6.22)

(6.23)

~x~ ~i (6.24)

Acceleration Waves in Constrained Elastic Matermls 71

We use (6.26)2 in (6.23)-(6.25) to obtain

U ZL = (C,L ~B mn + CiA jL ma) ft, ft j-- ft~ rC~L ~ " ((2 + Q r)/), (6.27)

^ ~ CiAjBmBPj. + l c i A j B m A

(6.28) 1 ~ftk U [*" Qv~n~A cOXa "

and

B = ~ {eiAjSkC m A marn c ftftjft k -- (ft. (~ft) ~ . (2 Q + Qr)/)}. (6.29)

We complete the proof that the equation for tr is of the form (1.3) even for constrained materials by showing that

dU Z = 2 P ~ 0m (6.30)

and using (4.14) and (4.15) in (6.22) to deduce (1.3). Employing a method used for unconstrained materials (see for example I-4, 5]) we take 8/dm L of (5.16) and contract with L to eliminate dR/PmL:

L~Rk ~ {Po U2 (~ik - - (~ij --V~ V~) Qjk} = 0 . (6.31)

We note that (2* is homogeneous of degree two in m as required in w 4. In the unconstrained case the result (6.30) is immediate since L. R = 1 and

~Qjk ~m L = CjL kS ma + CjA kL ma- (6.32)

It now becomes necessary to treat separately the cases of singly and doubly constrained materials.

(a ) One constraint: n = 1. From (3.3) and (3.5) we have, writing v for v ~,

av~ = av~ - Zi~L --2~i VinYL (6.33) amt - n~z, amL v- v v .-----v-"

which may be used, together with (6.32) in (6.31), to obtain (6.30), where Z is given by (6.23) and Po U2 is an eigenvalue of Q*.

(b) Two constraints: n =2. Rather than work with the appropriate form of (6.31) we note from (5.22) that

U2/./2 t~m---~- {Po -- Qij#ipj} =0 , (6.34)

where p rather than/i is used for ease in differentiation. From (3.7) and (3.9) we have

~ ^ ^ v,v~)(v~ ~pL-- v~ ~pL)" 8mL--ei'q(#'PP+ ~-~ i 2 2 , (6.35)

72 N. SCOTT

On writing the terms in/~ ^ v ~ in terms of ~ we see that

so that, finally,

Also

(6.36)

0m r = # , ~ ~ z - ~ #p n~L. (6.37)

O m L --- 2 # v i n iL . (6.38)

From (6.34), (6.37) and (6.38) we are able to deduce (6.30), where now Z is given by (6.27) and U by (5.22).

This concludes the proof that the growth of wave amplitude a of acceleration waves in constrained materials satisfies (1.3), where for a singly constrained material A and B are given by (6.24) and (6.25), respectively, and for a doubly constrained material by (6.28) and (6.29), respectively.

7. The Spatial Form of the Growth Equation

With the assumptions (6.1) it is easy to write the equations of the last section in terms of the spatial variables x. Define a quantity d in terms of c by

d i m j n : j - x ci a .i8 Pma PnB (7.1)

(7.2) and note that

Define also a vector z by

Qij = Jdlm.in nm nn "

z = J - 1 p Z . (7.3)

We now rewrite the equations of growth in terms of the current variables x, n, d and z, rather than the corresponding material variables X, m, c and Z. To this end we use (6.2) and (6.3) to show that for any differentiable function f

(7.4)

and define a reaction stress tr ~ in terms of rt ~ by (2.5)

a~ = j - 1 n~ pr. (7.5)

On dividing (6.22) by J and using (6.2) and (7.3), it is possible to rewrite (6.22) as

~ 0 " 2 z . ~ x + a a + b a =0 , (7.6)

where a = j - 1 A , b = J - 1 B . (7.7)

We see that the quantities z and a may be written in terms of current variables by using the equations of this section in (6.23) and (6.24):

- - p ct --~t ~t - -~ U z 12 d n + d n a v l - - i{ iljn n imjl m- - il k Q k j - - a j l Q i k V k } R j (7.8)


and { ~ ~R~

1 -~ �9 ~Rk ) (7.9)

Also, b may be written in terms of n rather than m by using (2.10) and (7.2) in (6.25). These results are valid for n = 0, 1, 2 but as in the last section we may obtain more explicit results for n = 2 by replacing both L p and R by ~ in (7.8) and (7.9).

It is easy to show from (6.30) and (4.15) that

z = j _ 1 d u an (7.10) and hence to rewrite (7.6) as

fia 2p ~ . + aa + ba2=O, (7.11)

where now the current density p is employed in place of P0. For future reference we state the result

OZ---A-a- = J ~zi (7.12) ~XA ~x~"

8. Orthogonality, and Reality of Wave Speeds

In view of (5.19) we may rewrite (5.16) as

{Po U21 - Q**} [~i] =0 , (8.1)

where for hyperelastic materials

Q** de-~f'Q--va{~vaQ--Qva ~ ~a (8.2)

is symmetric.* It is immaterial that equations (8.1) and (5.16) do not yield the same right eigenvector corresponding to their respective spurious eigenvalues. In the case of a singly constrained hyperelastic material the fact that O**(n) is symmetric implies, from the formulation (8.1), that the other two eigenvalues are real with orthogonal eigenvectors. This result of course is the counterpart of the result well known for unconstrained hyperelastic materials that Q (n) is symmetric and consequently has three real squared wave speeds with amplitudes R mutually orthogonal. It may be that one (or both) of these eigenvalues is negative, in which case a wave cannot propagate.

Necessary and sufficient conditions on Q* (n) for the existence of real wave speeds are not known but in order to provide at least a sufficient condition we now apply the strong ellipticity (SE) condition [2, (71.15)] to constrained materials. Again in view of (5.19), we may replace the acoustic tensor Q**(n) of (8.1) by Qt (n) where Q, ~f Q** +(v ~, . Qva) ~ | (8.3)

* I am indebted to Professor J. L. ERlCKSEN for suggesting this approach.

74 N. SCOTT

has the property that its null space is the space spanned by the v ~. The SE condition, therefore, is that the symmetric part of Qt(n) shall be positive definite on the orthogonal complement of this null space. But this reduces to

S . Q ( n ) S > O , VS such that S .v~=0.

If the SE condition holds two results are immediate:

(1) There is a real wave speed corresponding to each direction of propagation which is a real right eigenvector, other than r of Q* (n).

(2) For a hyperelastic, singly constrained material we may further state that the eigenvectors of Qt (n), excluding v ", are both real with positive eigenvalues, so that two waves may certainly propagate.

The sufficiency of the SE condition is clear also from (5.23). Equations (8.1)-(8.3) are valid also for n=2, but the SE condition yields no

more than is obvious from (5.22).

9. The Hyperelastic Case: Equation of Growth

We shall show that the assumption of hyperelasticity greatly simplifies the expressions for Z, A and B given in w 6. First we prove that for these materials

LP=R. (9.1)

This result is well known in the unconstrained case, since it merely states that the left and right eigenvectors of a symmetric matrix coincide, whilst from (6.26)2 it is always true in the case in which n---2. It is sufficient, therefore, to consider only the case in which n = 1.

Choose a coordinate system where v ~ is parallel to (1, 0, 0), so that Po U21 - Q * becomes 0

Po U2 0 /

-Q21 Po U2-Q22 -Q23 �9 (9.2)

- 031 - O3~ po t 1 ' 2 _ 0 3 3 /

We have R=(O, R2,R3) and since Qaz=Q23 we have L 2 = f l R 2 , L 3 = f l R 3 , for some ft. The normalisation L. R = 1 requires fl = 1 and, on return to a general coordinate system (9.1), follows.

On use of both (9.1) and the symmetry of Q and c, (6.23)-(6.25) simplify down to

U Z L = 2 cia~L m a R i Rj - 2 R i 1rill V~ " QR, (9.3)

1 c~ZL= ~ ( ~ X L ~mL) A = - - f OXL Po , (9.4)

and 1

B=-U~ {eiamkcmamAmcRiRiRk-- 3(R. Q~'R)(T, ~'. QR)}, (9.5)

which are valid for n = 0, 1, 2. The current form of these is given from (7.8) and (7.12) by

U zl= 2di,,,jln,,,RiRi- 2 Ri~t~ ~ QR (9.6)


and

a = 2 d x l - - d x t (9.7)

In the case n = 2 we may replace R by ~ in all these results. Though (9.1) holds whether or not a doubly constrained material is hyperelastic, the equations of this section may not be derived for a Cauchy elastic material from (6.27)-(6.29) because the symmetry of Q and c is essential for this derivation.

R e f e r e n c e s

1. WRIGHT, T.W., Arch. Rational Mech. Anal., 50, 237-277 (1973). 2. TRUESDELL, C., & W. NOLL, Encyclopedia of Physics III/3. Ed. S. Fliigge. Berlin Heidelberg New

York: Springer 1965. 3. CHADWICK, P., & P. K. CURRIE, Arch. Rational Mech. Anal., 49, 137-158 (1972). 4. VARLEY, E., Arch. Rational Mech. Anal., 19, 215-225 (1965). 5. VARLEY, E., • E. CUMBERBATCH, J. Inst. Maths. Applics., 1, 101-112 (1965). 6. VARLEY, E., 8~ J. DUNWOODY, J. Mech. Phys. Solids, 13, 17-28 (1965). 7. CHEN, P.J., Arch. Rational Mech. Anal., 31, 228-254 (1968). 8. SUHUBI, E. S., Int. J. Engng. Sci., 8, 699-710 (1970). 9. ERICKSEN, J.L., J. Rational Mech. Anal., 2, 329-337 (1953).

10. CHEN, P.J., & M.E. GURTIN, Int. J. Solids Structures, 10, 275-281 (1974). 11. SPENCER, A. J. M., "Deformations of Fibre-reinforced Materials", Oxford, 1972. 12. SPENCER, A.J.M., J. Mech. Phys. Solids, 22, 147-159 (1974). 13. THOMAS, T.Y., "Plastic Flow and Fracture in Solids". New York-London: Academic Press 1961. 14. COURANT, R.,&D. HILBERT, "Methods of Mathematical Physics", Vol. II, Holden-Day, 1965.

Department of Mathematics The University

Dundee, Scotland

(Received February 18, 1975)

Acceleration waves in constrained elastic materials

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