Waves in elastic rods

Journal of Sound and Vibration (1977) 51(2), 283-302

WAVES

IN ELASTIC

RODS

H. COHEN

Department of Civil Engineerhtg, Unit'ersity of l~lanitoba, tVinnipeg, 3Ianitoba, Canada AND

A. B. WHITMAN Deparmwnt of ~leehanieal Engineering, IVayne State University, Detroit, Mqchigan, U.S.A.

(Receiced 9 August 1976, and hz revised form 23 November 1976)

The problem of shock wave propagation within the framework of a linear director theory of elastic rods is examined for the case of uniform and non-uniform rods. For uniform rods the general result that all waves propagate with constant speed, without growth or decay, is obtained. Particular attention is paid in this study to the effect of rod curvature and twist on the wave speeds, wave modes and induced waves.

1. INTRODUCTION The theory of rods has received a resurgence of interest in recent years. This interest has no doubt been prompted by the so-called direct approach, which owes its origins to the brothers Cosserat [1]. Theories of rods, including those from the direct approach, are reviewed in the recent article by Antman [2]. The subject of wave propagation within the framework of direct theories of rods has received limited attention. We refer to the work o f Green, Laws and Naghdi [3], Shahinpoor [4] and Eason [5] who deal with infinitesimal harmonic plane waves. The work of Shahinpoor utilizes general linear hyperelastic constitutive relations within the framework of uniform rods, a notion introduced by Ericksen [6]. Green et aL and Eason restrict their considerations to untwisted straight and circular rods,'[" respectively, within the framework of a specific linear theory based on the study of Green and Laws [7]. The objective of this paper is to generalize the work of Green et al. and Eason in order to obtain insight into the problems of wave propagation in curved and twisted rods. We do not, however, deal with harmonic plane waves. Rather, we deal with the notion of a wave as a propagating discontinuity. Although we formulate the general non-linear shock and acceleration wave problems, our specific interest will be in the linearized results, pertaining thus to the propagation o f weak shock waves. O f particular interest to us in this study will be the effect of rod geometry on wave speeds and the associated wave modes. We begin in section 2 by giving the basic non-linear field equations, jump conditions and constitutive relations as proposed by Green and Laws [7] in the form recently put forth by Green, Naghdi and Wenner [8]. In addition, we add the appropriate jump compatibility "["Untwisted straight and circular rods are examples of uniform states. 283_

284

H. COHENAND A. B. WHITMAN

equations necessary for the analysis of wave propagation.l" Using an idea put forth by Ericksen [10],++ we rewrite the basic rod equations in a highly compressed and convenient form involving nine-dimensional generalized deformation, stress and force vectors. In terms of this setting we give the propagation conditions for non-linear shock and acceleration waves, as well as the decay-hl&tctionw equation for shock waves. In section 3 we carry out a linearization of the basic equations about a finitely deformed equilibrium state. This allows us to obtain, from the results of section 2, both the propagation condition and decay-induction equation for weak shock waves propagating into a deformed equilibrium state. The propagation condition yields an eigenvalue problem, set in a ninedimensional displacement gradient vector space. The squares of the speeds of propagation are the nine real eigenvalues of a symmetric acoustic tensor relative to a positive definite symmetric inertia tensor. The eigenvectors associated with the wave speeds define the wave modes mentioned earlier. For the case of propagation into a uniform rod we show that all speeds of propagation are constant and obtain an expression for the variation of the wave mode along the rod. These results are analagous to those obtained by Ericksen [10] for shells. In addition, we use the decay-induction equation to determine both that the strength of these waves are constant, as well as an expression for the induced waves. In section 4 we turn our attention to more specific results. Utilizing the form of the free energy function and inertia coefficients proposed by Green, Knops and Laws [11] and used in [3] and [5], we write the propagation condition and decay-induction equation for waves propagating into stress free rods o f arbitrary shape, with arbitrary initial twist. These basic equations are written in a form which clearly indicates the effect of rod geometry.II The propagation condition separates into two sets of equations; one is associated with deformation of the rod such as shear, extension and flexure, while the other pertains to cross-sectional deformations. Here we observe that the four wave speeds a n d associated modes pertaining to cross-sectional deformation effects are independent of rod geometry. This remark, however, does not apply to the induced waves associated with cross-sectional deformation waves, as these are in fact dependent on rod geometry. In section 5 we deal specifically with waves in straight untwisted rods. We classify the various waves that can propagate giving their speeds, wave mode vectors and induced wave vectors. We find that there can propagate two shear waves, S~, $2, one extension wave, E, twoflexure waves, F~, F2,82in addition to two cross-sectional extension waves, E+ and two cross-sectional shear waves, S+. The speeds of propagation are found to be in accord with those found by Green, Laws and Naghdi in [3]. Next, in section 6 we consider the case of an arbitrary untwisted plane curve. Here there is a coupling of extension and flexure effects due to the curvature of the rod and the speeds of these modes are curvature dependent. Specifically it is the E and F~ modes that are coupled to produce two new modes,I'I" which we term extension-bending waves, EFt+. The remaining seven modes that can propagate in the straight rod remain unchanged for the curved case. However, the curvature does serve to produce more coupling in the corresponding induced -~The general theory of jump compatibility equations is dealt with in considerable detail in [9]. :~This formulation has also been utilized by Shahinpoor in his paper [4], cited earlier. wThe induced wave associated with a shock wave is that acceleration wave coupled to it by the governing equations of motion. IIThe rod geometry is defined by a skew symmetric tensor A~ defined in section 4. The components Ajr, A23 and A~2 are called the components of curvature and twist, respectively. 82The shearing and flexural deformations associated with S~ and 1:~,respectively, lie in the same plane. A similar remark applies to S, and 1=2with their plane of deformation being orthogonal to that associated with St and Ft. In addition, these planes are planes of symmetry for the material behaviour of the rod. 1"I"The plane of curvature is chosen such that it is a plane of symmetry for the rod and FI denotes flexural deformations in this plane.

WAVESIN ELASTIC

RODS

285

waves than in the straight rod case. The wave speeds found here are in agreement with the results of Eason [5] for circular rods. Of particular interest for the plane curve, which in general corresponds to a non-uniform state, is the decay equation. We integrate this equation in closed form and give a qualitative analysis of the possible effects of curvature change on wave propagation. Section 7 treats situations in which the rod has initial twist in its undeformed configuration. For the straight twisted rod we find that the twist will introduce two pairs of coupted modes, each of which couples shear and flexural deformations operative in planes at right angles to one another. These shear-flexure waves, Sa F2+, $2 FI+, propagate with speeds which depend on initial twist. The E mode and cross-sectional modes remain the same as for straight untwisted rods; however, as in the case of curvature, the twist has an effect on the form of the induced waves. The general situation of a spatial rod with initial relative twistt is such as to couple all five modes which can propagate in the straight untwisted rod. However, there arise two special cases in which some uncoupling occurs. The first of these is the case of restricted twist, where we take t.he twist of the cross-section to be equal to the torsion of the reference curve defining the rod. For this case two distinct coupled modes occur. These are the S~ F2+ modes found for straight twisted rods, in addition to three extension-shear-flexure waves (!:$2 F1)~.2.3. This latter mode can be regarded as one in which the $2 FI+ and E modes in a straight twisted rod are coupled together by the principal curvature of the reference curve. The other special case is the one we term zero twist, and it is analyzed in order to gain further insight into the effects of curvature and twist. Here, where the rod cross-section has essentially b=en untwisted in space, there is now no coupling between shear and flexure. The modes that can propagate are the two shear modes S~, $2, in addition to three new modes which we call the extension-doubleflexure waves, (EFx F2)~.2.3, These latter waves arise as a generalization of the plane curved case, the extension being coupled here to both flexure components by the two components of curvature. Finally, in section 8, we discuss briefly wave propagation within the framework of a constrained theory of rods. For this theory it is more convenient to use new strain measures. We define these, giving their relation to those used earlier, in order to obtain the restricted form of the free energy function utilized in the foregoing analysis. As is to be expected, the speeds and modes of propagation associated with rod deformations such as shear, extension and flexure are the same as found earlier. However, as the cross-section is now constrained to be rigid, we have eliminated three degrees of freedom, so that the four modes associated with cross-sectional deformations reduce to a single mode associated with a cross-sectional twist wave, T. 2. THE BASIC EQUATIONS We begin by giving the basic equations governing the dynamical behaviour of rods, within the framework of an iso-thermal hyperelastic theory. As mentioned earlier we shall adopt these equations from those given by Green, Naghdi and Wenner in [8]. According to these authors a rod consists of a curve ~ having two deformable directors d~ (~ = 1, 2) attached at each point. A motion or deformation of such a rod is specified by r = r(r

d~ = d~(~, t),

(2.1)

where ~ is a material or convected co-ordinate and t denotes time. We introduce the notation d3 = r'(~, t),

(2.2)

"t"By relative twist for a spatially curved rod we mean the twist of the cross-section over and above that due to the natural twisting of the rod, arising due to the torsion of its reference curve.

286

n. COHEN AND A. B. WHITMAN

where ' denotes a/a~ and the d~ (i-- l, 2, 3) are subject to the condition [dl, d2, da] > 0. We shall also use 9 to denote a/at. T h e basic equations governing the motion o f the rod are given by Inl = - ; . [ i ' l V, n' + 2 f = 2i:,

[p:l =--).)aB[da] V,

p~" _ n,, + 21 9 = ).y-,a/j~,

f a =yB~,

(2.3) (2.4)

where ).=1,112 r, *~33 P ' ~ 3 /4"1/2 3 Fo,

n = ). a~lad3,

h33 = d3.d3,

n: = ;. ar

p" = ;. aT,lad-,

(2.5) (2.6)

where ~, = ~,(~e'; ~

,0,

y r : r', d=,d',

3r

: R',D=, D~'.

(2.7)

Equations (2.3) and (2.4) express the balance laws for discontinuous and continuous motions, respectively. Equations (2.3)1, (2.4)1 and (2.3)2, (2.4)2 express the balance o f linear and director m o m e n t a , respectively. Equation (2.5)1 is a statement of mass conservation with p and Po denoting the linear mass densities o f t h e rod in the current and reference states, respectively. Equations (2.6) are the constitutive relations for the contact force, director force and contact director force, respectively; while (2.7) is the constitutive assumption o n the free energy function ip. The quantities Haa, R, D,, D" denote the reference values of h33, r, d~, d~, which we assume obtained by these quantities at t = 0. The free energy function is subject to the requirement of invariance under superposed rigid motions o f the rod,'[" which is fulfilled by taking Ill = ~(uT[, HiD A:a, ~),

(2.8)

where ql: ~u,

h'=i,

(2.9)

are the relative strain measures given by Yu = h u - His,

hu = dl-dj,

h'~ = ).=l - A~l,

(2. I0)

2=1 = d ' . d t ,

(2.11)

with H u , A,t denoting the reference values o f h u , 2~. We point out also that in equations (2.3) the j u m p o f a quantity ~b across a point of discontinuity is denoted by [$1- A moving point of discontinuity ~ constitutes a wave of order n with respect to ~, if q~ and its first n - I derivatives are continuous, while q5~"~ is discontinuous. The speed of p r o p a g a t i o n o f the wave is denoted by V. A shock wave corresponds to a wave o f order one with respect to the motion (2.1), while an acceleration wave is of order two with respect to the motion (2.1). The so-called first and second order compatibility equations for shock waves are Itl = - V l r ' l , li'l = V2lr"l - 2 V I ? l - 17lr'l,

IdA = - r i d ; l ,

Ilia = V210;,i- 2 Vld'=l- rid;I,

(2.12) (2.13)

where

(2.14) is the rate o f change of (75as seen by an observer moving with the wave. "i"It is for this reason that we do not require the moment of momentum equations given in [8], since they are equivalent to requiring invariance of V under rigid rotations when equations (2.4)2 hold.

287

WAVES IN ELASTIC RODS

We now introduce an abbreviated notation similar to that utilized by Ericksen [10] in dealing With wave propagation in shells. T o begin with we set n = 2 O~/&,

(2.15)

noting that since ~, is required to be invariant under rigid motions, Ip is independent of r and n is identically zero. We next define generalized deformation, stress and force vectors by =

(')

,

(')

.A" =

d~

p~

..t/= '

, n=

.~ =

(:)

,

(2.16)

where the vector quantities defined in (2.16) are column vectors o f ordinary vectors. We shall use E to denote the space of ordinary vectors and 6~ to denote the vector space consisting of column vectors of E. With this notation the balance equations (2.3) and (2.4) take the f o r m

1.4:1=

-;.x[#, I v,

(2.17)

,A," - ,.1[ + ;.~ = ).,x/',,~,

while the constitutive relations (2.6) and (2.7) become

.4:=;.a(,la~",

d,, =;.a~,/aa,,

~, = ~,(~,..q"; r

(2.18)

and we have suppressed the dependence of ~, on the reference values. In (2.17) we have utilized the following block matrix notation Yf=

(I

0a)

O~

K~a=y~q,

K~a

(2.19)

'

where I and 0= are the identity and zero operators on E, respectively.t In our new notation the compatibility equations become [~l =-V[~'I,

l#'l = VZl#~"]- 2Vl•'l-

IT[~'l.

(2.20)

F r o m equations (2.17)1, (2.18),, (2.20)1, and (2.5)1 we find

[a~,/a..q"l

=

av[~"! vL

(2.21)

while from equations (2.17)2, (2.18)2, (2.20)2 and (2.5), we have

I(a(da~")'l - la~,/a~'l

=

X l # ' " ! v" - 2 X ' l # ' l v - . : * l # " i

P,

(2.22)

since in the analysis we assume ~ is at least a C2 function of its arguments, while ~ " and . X are at least Co. Equations (2.21) and (2.22) are the p r o p a g a t i o n condition and decay-induction equation for finite shock waves, respectively. F o r acceleration waves [~"l = 0, (2.23) so that ~k is a C2 function of continuous variables. F r o m equations (2.22) and (2.23) we obtain the propagation condition for finite acceleration waves, {(a 2 tp/O.~' a.~") - V 2 ~ } [ ~ " i = O, (2.24) an eigenvalue problem whose characteristic equation det {(O2 ~k/0~' 00 ~') - V 2 ,X"} = 0

(2.25)

determines the speeds o f propagation. ? .At is a linear operator on 8 which we call the inertia tensor. It is a 3 x 3 matrix of linear operators on E. If we define an inner product on d' by ~to~2 = rl.r2 + d,~.d, v then with respect to it ~ is both positive definite and symmetric: i.e., ~ o .~0'~> 0 and o'r =Y{"r where ~ o .,qr = ~2ooX:r~.

288

H. COttEN AND A. B. W H I T M A N

3. LINEARIZATION AND UNIFORMITY In this section we begin by linearizing the basic shock equations (2.21) and (2.22). We consider an equilibrium state denoted by ~ and assume a small deformation q/superimposed upon it, i.e. ~=#+~Z,

(3.1)

(~

(3.2)

where q/=

b~ "

In equation (3.2) u and b~ denote the displacement of cO and of d~, respectively. We now expand the first order partial derivatives of ~, in a Taylor's series about ~ , retaining only the linear terms in ql and ~ Thus

ar

d~'

+ .,-r

ar

= a~ r qg' + J'oh,,

(3.3)

where d , ~ and ~- are bilinear operators defined by

{r

d =

ta

'a

;

(3.4)

In equations (3.4) the comma denotes partial differentiation and we employ the parentheses { } to denote evaluation ofthe partial derivatives in the equilibrium state ~ . Here, our notation is borrowed from a similar one utilized by Naghdi and Trapp [12], in studying shells. We term d the acoustic tensor and note that it is symmetric. On substituting equation (3.3) in equation (2.21) and (2.22), using the continuity of q/, ~ ' and Ip, we find ( d - V2 arC') l~z'l = 0, (~r - v2.,~") I ~ " ! + (~r + ae',, + F'.,~c) I~t'l = - 2 x ' i ~

v,

(3.5)

where we have set a~('a = ~ - acgr = -.-~ar.

(3.6)

Equations (3.5)1 and (3.5)2 are the propagation condition and decay-induction equation governing the propagation of weak shock waves into an arbitrarily deformed hyperelastic rod. We see that the squares of the speeds of propagation are the real eigenvalues of the symmetric acoustic tensor, relative to the positive definite and symmetric inertia tensor. In addition the modes of propagation are defined by the eigenvectors belonging to the aforementioned eigenvalues. We note that if .~t is positive semi-definite, then the squares o f the speeds of propagation are all non-negative leading to nine real wave speeds and associated modes of propagation. A uniform rod in the sense of Ericksen [6] consists of, (a) a uniform state and (b) a uniform material. The first of these properties is a geometric one and is characterized by -7" = R~r

R R r : R r R = I,

Ro = I ,

(3.7)

289

WAVES IN ELASTIC RODS

where ~" denotes the state variables defined in equations (2.7), R is an orthogonal transformation depending on ~ and the subscript 0 denotes evaluation at an arbitrarily chosen reference point ~ = ~o on the rod. Uniform states consist of straight, circular and circular helical rods, in which the cross-sections can have constant twist and shear. The second property associated with uniformity is a physical one and pertains to material properties. It is characterized by ~ ( ~ ; 4) = ~ ( ~ o ; ~o),

X" = X'o.

(3.8)

When equations (3.7) and (3.8) both hold the rod is said to be uniform. For the purpose of analyzing equations (3.5) we can utilize the setting in 6" and characterize uniform rods by = ~o,

#,, = ~ ,

~ P ( ~ , ~ ' ; ~) = ~(~o, ~ ;

~r ~o),

= ~r~

= •,

~o = J,

.X" = ..X"o,

(3.9)

where ~ is an orthogonal transformation on o~ defined by = Ror

(3.10)

In equations (3.9) and (3.10) J is the identity operator on 8, defined as a 3 x 3 matrix of operators on E, with I as the diagonal elements and 0 as the off-diagonal elements. Utilizing the relationst = Ado ~r,

Jd" = #~oY('o~ r ,

(3. i I)

we have from the characteristic equation associated with equation (3.5)t that det (a~ro - V2.X'o) = 0,

(3.12)

so that V~ = V~o,

(ct = 1. . . . 9),

(3.13)

and the speeds of propagation are constant for uniform rods. Furthermore, from equations (3.5)1 and (3.1 I) one finds for the eigenvector associated with V~ the expression l ~ ' L = a ~ [aa' Lo,

ao = 1,

(3.14)

where a is an arbitrary scalar. These results are completely analagous to those obtained by Ericksen [10] for shells. We now turn our attention to the decay-induction equation (3.5)2. Utilizing equations (3.11), (3.9)3, (3.13) and (3.14) in equation (3.5)2, we find for uniform states that (or

V~oV)l~"l,+(St'+a~a+2V~of"#'+2v,

oe'e/a)i~'l,=O,

where [ ~ ' L is the induced wave corresponding to the wave mode vector are defined by Ego = 5ar = qlrd _ ~ q t , , q f , = _~r = ~, ~r.

(3.15)

i~'I~ and ,.9' and qt" (3.16)

On taking the scalar product of equation (3.15) with [ ~ ' ] , and employing the relations [~'],o(d-

V~o~c) I q t " l , = i ~ " l , o ( ~ -

[ql'l,o.Ct'A [ ~ ' l , = O,

v,~ x ' ) l ~ ' l , = 0 ,

[ ~ ' L o ( ~ + 2V~o~c~r) [ ~ ' I , = O,

(3.17)

we find ~=0,

:> a = l ,

(3.18)

"["Equation (3.11)1 follows from equation (3.9) and the chain rule for differention, while equation (3.102 follows from equation (2.19), (3.10) and (3.9)3.

290

H. COHEN AND A. B. WHITMAN

by virtue of equation (3.14)2. The result follows for propagating waves where V~ # 0, a # 0, and since .X" is positive definite. Thus t.he magnitude or so-called strength of waves in a uniform rod are constant. In order to find an expression for the induced wave [q/"]~ we write

[~"h = a I ~ ' L + [~'"IL

(3.19)

where 1r lies in the subspace d~ c 6' complementary to [~"l~. Note that if V2 is a root of multiplicity r then [o//' L spans a space of dimension r, and dim6'~ = 9 - r. If we set ~

= d - r'~ X',

(3.20)

then .L~'~is an invertible linear transformation on 6"N and ..L~';"~ exists. Using equations (3.5), , (3.18) and (3.19) in equation (3.15) we find

l~"i'~ =

- ~ ; - ~ (5," + a'g a +

2V~.~//:) [q," L,

(3.21)

while a is indeterminate. If we now make use of equations (3.5h, (3.14)x, (3.16)2, (3.18) and the fact that ~ and "W" commute, i.e., ..,~'t" = ~/t",~,

(3.22)

we can write equation (3.20) in the form I~"l~ = ( ~ ' - W ; * ( s r

+ 2 V~2,Yg'~')} I~"Lo,

(3.23)

giving the induced wave in terms of the initial or reference value of the wave vector. 4. WAVES AND GEOMETRY We now focus our attention on the specific linear theory of rods proposed by Green, Knops and Laws in reference [11]. Following these authors we begin by assuming that the free energy function ~ is a constant coefficient quadratic form in the strain variables qb.i We next assume (a) that ~ and . ~ are form invariant under the symmetry transformations

--*+4,

d,--* +d~,

(4.1)

for all combinations of + and - , (b) that the rod is stress free in its reference configuration, and (c) that Dl are an orthonormal triad. The last assumption coupled with equation (2.2) implies that D~ are perpendicular to ~ and that ~ is arc length in the reference configuration. These assumptions result in the following form for the free energy function: 2 2 ~ = A~Ya* 7~~ ),a* + B~'~"]a3"]#3 + CaYTay?33 + k3"~323 + BzB~~ K=yKaa + D~# K~t3K,~3' (4.2)

where the coefficients in equation (4.2) are defined by the following 2 x 2 matrices: A x~ll = D ( k , , 89 B z)'ll

=

A ~22 = D( 89

D(k,o, 89

B ~'2` = Dv( 89

B ~'22= D( 89 k,s),

B :'y = 89

k2), kn),

ks),

A ~''~ = A "~x =

88

k,),

B ~''2 = o a k , , , 89 C ~y = D(ks, k9),

D ,y = D(kt6, kts). (4.3)

In equations (4.3) D denotes a diagonal matrix, D v a diagonal matrix in which the columns have been permuted and the k's are all material constants. In addition, ~ invariant under 1"We note that there is no explicit dependence on reference geometry and that the linear theory resulting from this assumption, coupled with a particular choice of strain variablesq/, is highly specialized. We remark also that the resulting theory is uniform in the sense described in section 3.

291

WAVES IN ELASTICRODS the t r a n s f o r m a t i o n s (4.1) requires the first two o f yl2 =y21 = 0;

tX1 = y l l ,

(4.4)

tX2 =y22,

while the l a t t e r two e q u a t i o n s are i n t r o d u c e d for n o t a t i o n a l convenience. I f we set D~ = A u D~, A u = -As.

(4.5)

then the A u characterize the g e o m e t r y o f the rod.~" In fact Ant = r: sin f ,

A23 = t~cosf,

(4.6)

A~2 = z + f ' ,

where x a n d r are the principle curvature a n d torsion o f ~ , respectively, w h i l e f d e n o t e s the angle between D~ a n d the b i n o r m a l to ~ . T h e quantities A~2 a n d f ' are called the twist a n d relative twis~ o f the rod, respectively. W e n o w utilize e q u a t i o n s (4.2)-(4.5) in e q u a t i o n (3.5), to o b t a i n the linearized wave equations associated with the linear t h e o r y o b t a i n a b l e f r o m (4.2). T h e result is a set o f e q u a t i o n s governing linear wave p r o p a g a t i o n into a stress free rod. W e write these e q u a t i o n s in the following m a t r i x f o r m + : L W = 0,

LW t + MW = KW,

(4.7)

where W a n d W * are defined by

w = (w,, w2Y,

w , = (wl, w~')L

W, = (lull, Iu~l, lu;l, lb[3l Ibm31),

W2 = ([b~l, Ib~l, lb[21 [b~l),

W[ = (l',;I, lu;I, lu;l, lbx~l, lb231),

W~ = ([bl~l, 1b221, lbxd, [b2tl).

(4.8)

The coefficient matrices a p p e a r i n g in equations (4.7) will be expressed in a f o r m which clearly indicates their d e p e n d e n c e on the material coefficients, as well as on the initial g e o m e t r y o f the rod. W e begin b y m a k i n g the following definitions: V~ = k d p o ,

P~ = k1412poOq2,

V~ = kslpo,

P~ = kt~12po~,2,

p2 = ~, v ~,~2 + ~ vg ~1, fl~ = A22 + A23,

V~ = 4 k g p o ,

~1~ = ~ ,

f12 = A22 + A~,.

VZ4 = k,6/po ~,,

V~ = 2kdpo,

~ = (~)-1 =

V~ = k , d p o ~2;

V92 = 2k9/po;

~V~i~,l~, (4.9)

"1"The Au include the Al~ defined by (2.11)2 as a subset. We note also that For uniform states A u = ( A L I ) o = matrix R'R T, with respect to D~ as basis9 :~These equations follow from equations (3.5) on taking their components with respect to Dl as basis on each of the component subspaces E of oa. We point out, that while W are the components of [ ~ ' l with respect to this basis, W ~are not the components of [~,/"l. The choice of definition here is a matter of convenience and at any rate once W and W J are known, any other quantities of interest may be calculated. We shall use the terminology of induced wave for W ~.

292

H. COHENAND A. B. WHITMAN

In terms of the definitions (4.9) the matrices L, M and K are given by

L=(L' 0) 0

L2 '

[ Mn M=(M2t

Mt2~ M22}'

K =-2VD(1, 1, l,r

0t2,~Xt, a2, ~t, 0~2),

(4.101

where V~A~2-- V

0

L: =

0

--~ V~At~A~

0

V] + ' q V,:~A~z2 _ V ~

--.~t V[ At: A ~

"q V~,At:~

--~V~AtzA~t

--.~ V]A~A2:~

V~+at V,A~t ~ ~ + ot:~Vs~A2~ 2 - V2

-otz V~At2 ~

0

-.-~tV~A~t

a~V~A:~

0

~ VIA12

--~t V2,A3t

~I(VI - V ~)

0

-a~ V~ A ~

0

~2 VIA2~

0

, q ( V ~ - Vq

L= '

L22

[a,(lT~-

V 2)

~,~ 1'52

~

~1~ e~

~(f'~-

V')

,

(4.1~ I

'

a,2 P~ ).

(4.121 I

~(~'~ - v ~)

The component M I I of M is given by M n = Mfi + MSt + M~t,

M~=~'D(l,l,l,a,,ot2),

(4.13]

where M~~ MS, M~t are diagonal, symmetric and skew-symmetric, respectively, with thl first given by equation (4.13)2 and the latter two by

M~t =

-~2 V~(A2~2)"

0

-ctz V~(At2 A23)'

0

.-.a2 Vs A~

0

r,I V[(A~)'

-cq V[(At2A~t)"

eq V[A't2

0

--ct2 VJ(At2 Az,)'

.-at V2,(Atz A,,)'

---a, V2,A;t

ot~ V~ Ai~

0

cq V~A~z

- : q VIA'~

0

0

0

or2 VIA'23

0

0

"-'~2 V~ A't2

~t V~(A],)" + ~2 V~(AJ~)"

(4.14]

293

WAVESIN ELASTIC RODS o

_a.(v,~+vi+p,)

A , , ( v i + v i + ~ ~) v ~ - , . , v i # ~

V22 - ~2 Vs1 fl~

Mi*, =

-A,,(vl + vi + v')

.h~(v] + v] + ~'~)

- ( V [ - cq V[fla)

--~t V[A3~Az3

oh V~Az2 A3~

~ V]At~A~

o

"-='t V ~ A ~ A ~

(4A5)

0

The remaining components of M are given by 0

"a2 V]AI2A23

0

a2 V~Az~A3t

--vq V[ As2 A~t

0

eft V~ At2 A23

0

(4.16)

Ml2 = - M r

-atV~A~

0

0

~tz V~A23

=/~D(~,,:,) M2z

~ M2221

cq(V2,+ff3-=x

2

2

2

2

~ t ( P l -- ~t ~5)A3t

"~2(V5 +

~ 2

P42 -- "'2t ~5)2 A~t

M2,,2 ) VDCaI,~2)

'

{ -~,(e ~,- ~i lTg) ~2( 17,~- ~ r M22,2 "-:-M2r2, = A2, k"-a,C~?g _ a~ e~)

/

a2(e~ - ~2t Fg)l"

(4.17)

We observe from equations (4.7)1 and (4.10)1 that the propagation condition separates into two distinct equations Lt Wl = 0,

L2 W2 = 0.

(4.18)

From equations (4.8)3., we see that equation (4.18)1 governs the propagation of waves involving shear, extension and flexural deformations, while equation (4.18)2 governs wave propagation associated with cross-sectional deformation. From the form of equations (4.1 I) and (4.12) we see that the former set (4.18)x will involve five geometry dependent speeds and modes, while the latter set (4.18)2 will give rise to four speeds and modes entirely independent of rod geometry.

294

H. COHEN AND A. B. WHITMAN 5. W A V E S I N U N T W I S T E D

STRAIGHT

RODS

We begin our analysis of waves, as governed by equations (4.7), by considering the straight untwisted rod. This corresponds to taking

A~j = 0.

(5.1)

Our purpose in considering this case at the outset is to provide a setting for the consideration of waves in curved and twisted rods. On substituting equation (5.1), where necessary, into equations (4.10)-(4.17) we find from equations (4.7) and (4.8) the wave speeds, wave modes and induced waves. Since the rod is uniform, there is no growth or decay and the wave strength remains constant. Based on the deformation associated with each of the waves, we arrive at the following classification of wave types: (i) shear wave,t S~: V2, = V~, Wa = ( 1 , 0 , 0 , 0 , 0 ) ltdlo,

W2 = O,

W~ = (a,O, O, 1, O)lu~lo V~/~(V~ - V?),

WJ = O;

(5.2)

Wg = 0;

(5.3

(ii) shear wave, 52: V~ -- V~, W~ = (0, 1,0, 0, 0) lu~]o,

W~ = 0,

Wxt = (0, a, 0, 0, 1) ltdh V]/..(V] - V~), (iii) extension wave, F: V~ = Va2, W l = (0, 0, 1,0, 0) [u] 1o,

wJ = (o,o,=,o,o),

W 2 = 0,

w ] = (r ., o, o) lu] lo,

(5.41

where -= { ~ v ~~( ~ '", - v~) - ~,, ug ~'g}/=,=d(171 - v l ) ( ~ i - vg) - ~Tg},

= { ~ v ~ ( f ~ f - v ~ ) - ~ , ~ v~ r

~((e~_ v~)(V~- v~)- ~};

(5.5!

(iv) flexure wave, Fa: VF, = V~, W~ = (0, O, O, 1,0) [b~3lo,

W,=O,

W[ = (l,O,O, =,0) lb~3lo V~/(V~ - V~),

W~ = O;

(5.61

WJ = O;

(5.71

(v) flexure wave, 1:2: V~2 = V5z, w~ = ( o , o , o , o , l) lb~;lo,

W~=0,

W[ = (0, l,O,O,a) [b~31o Z~l(Z~ - Vg), (vi) cross-sectional extension waves,~ E+: V~• = 89 WL+ = 0 , W[+

+ p~ + {(17~ _ I722)+ 41762},,21, W2+ = (1, ~2(Ve• 1 2 ---2 2 V1)/P6,0,O)lbul+_o,

= (O,O,d+,O,O) [bii]_+o, '

~'

W2+_I _ (l,~2(vv: t 2 _ PO/P6,o,o),~, 2 2

(5.8

"j"In addition to the jump in the linearized shear strain [Y,3] = [u~], the quantity lug] produces a propagatin: discontinuity in the slope of the deformed reference curve. Hence the terminology, shear-kink wave, migh be more appropriate than, shear wave, which we adopt here for convenience. .+ Although we use the terminology cross-sectional extension and shear waves, to be more precise we shoul term these cross-sectional extension gradient and shear gradient waves, respectively.

295

WAVES IN ELASTIC RODS

where

~+_=

{ ~ Vg(V~-" - v ~ ) -

2z ~ 172~, 6 . 172~ 6~ V~

V~_+);

(5.9)

(vii) cross-sectional shear waves, S+:

V~_+= 89

+ 172 _+{(I732_/Tz) + 417~}~/2],

Wl•

W:_+ = ( 0 , 0 , l,c~2(Vs• t , _ 17~)117~)[b~2l+_o,

w;• = 0,

wL+ = (0,0, 1, ~2(Vs_+ , ~ - e~)/e~),~.

(5.10)

In the above set of equations a denotes an indeterminate second order wave amplitude, while the subscript 0 on a jump indicates its initial value. The coupling of wave types in the induced wave can be summarized simply as follows. Shear waves induce flexure waves and vice versa, while extension waves induce cross-sectional extension waves and vice versa. We point out for cases (vi) and (vii), where the wave modes have more than one non-zero component, that the decay-induction equations for these components comprise a system, rather than a single differential equation as in cases (i)-(v). Finally, we observe that the speeds ofpropagation found in equations (5.2)-(5.10) are in agreement with the corresponding phase velocities found in reference [3], provided we evaluate them at zero wave length. 6. WAVES IN UNTWISTED PLANAR-CURVED RODS Here, we focus attention on the effects of curvature on wave propagation in rods, by taking ~ to be an arbitrary plane curve. In addition to the rod being untwisted, we assume that one of the planes of symmetry of the rod lies in the plane of the curve ~. We remind the reader that the planes of symmetry of the rod are, in view of the symmetry requirements (4.1), defined by the three planes associated with the triad Dl. For this situation we have (6.1) On substituting equations (6.1) into the appropriate equations in the set (4.10)-(4.17), and utilizing equations (4.7) and (4.8), we arrive at a classification of the waves in such a rod. The primary effect of the curvature is to introduce two new waves, which couple extension and bending in the plane of the rod. The speeds of these waves are curvature dependent. All other waves remain the same as in the straight rod case. The secondary effect is to introduce more coupling of different wave types in the induced waves, than in the straight rod case. The specific nature of these effects is given in the following classification of wave types:t A31 = K,

A23 = AI2 = 0.

(i) Sl:Wf = (a, 0, Kd, c, 0)lu;lo,

(6.2)

(ii) $2: (same as equation (5.3)), (iii) F2:WJ = (0,0, rd, K~) [b~lo,

(6.3)

(iv) E+ :Wl_+ = (0, 0, d_+,Kc_+,0) [bi~ 1+_o,

(6.4)

(v) S+ :Wt~+_= (0, 0, 0, 0, h'd_+)[bh I,_o,

(6.5)

(vi) extension-flexure waves, EFI+: V2_+=-~[V 2 + Z42(I + cq h.2)+ {[V~ + V~(I + cq K2)]z- 4Vaz Vz}x/2], Wx+ = (0,0, I,KV~/E+,0) [u;l+, W~_+= (-Kd+_,0,.+, .--/+_,0),

W,_ = (0,0,al r:V,2/E-, 1,0) [b'131-, W2t_+= 0c~+_,~•

[u;]_+,

W2• = 0, (6.6)

t We give only the results that differ from the straight rod case, referring the reader back to equations (5.2)(5.10) for wave speeds and mode shapes not listed here. We also group the results so that we give first the waves whose speeds and mode shapes are unchanged.

296

H. COHEN AND A. B. WHITbIAN

where E~ = V~ - V~,

~_ = V] + ~, tr 2 V] - V~z, 2

E•177177177177177

r

(6.7) =

- 2 V•

-~:V~,+_ + E• d~. - ~' V2,[uAl• + if'• [bhl._ = -2V• [b]'31•

(6.8)

In equations (6.2) to (6.6) ,z denotes an indeterminate wave amplitude, while #, c, ~ and denote the components of the induced waves. These latter quantities, which are different in each of the above cases, are given by rational polynomial expressions in the V2's, ~ , ~, and h-. The forms o f these expressions are similar to that of expression (5.5), but involve h-, and reduce to the straight rod results when rc = 0. Equations (6.8) are the decay-induction equations which determine (a) the strength of the IZF~wave components [uA]_+and [b~3]+ and (b) the direction in 6" defined by the partially indeterminate components ~+, d+_ of the induced wave vectors. On carrying out an analysis of equation (6.8) we find that e + = =,

+ KV~a/E•

d• = ~•177

75 = [(x2) ' {89V~(E• - E• + 2 V~) + E z E• - 2 V•2 E• E•,

[uAl•177 i

2 2 = (Vzo/V•

3/4,

V~]/KE•177 + ~•

{1 + (~,•

t

+ L~•177 r

Ibm3IU [bx315o = (~clxo) (E•177

t

(lu3 l• [u31•

(6.9)

1/2 ,

(6.10)

For brevity we have combined the results for both waves, mentioning that special care and use o f equations (6.10) is necessary, in order to evaluate the wave corresponding to V_ at ~r = 0. We note that in equations (6.9)Lz the quantity ,~ is indeterminate. There are two cases to consider corresponding to uniform and non-uniform rods respectively. For circular or uniform rods, r ' = 0, the wave speeds (6.6), are constant and we see from equations (6.10) that the wave strengths are constant. From equations (6.9) we find that the induced waves are indeterminate in magnitude, but parallel to the primary waves. We mention that all wave speeds given in cases (i)-(vi) above, correspond to the phase velocities at infinite frequency

44

v~

0

2

V E F I --

Curvoture K

Figure 1. Variation of wave speed with curvature.

297

WAVES IN ELASTIC RODS

,z.•l

/

~

'~ -

_~

.. I

/

/

/

[b,;]-'w~

0

"~

/

/

/

//

,/I

///

*

iI

[U31_--'~

/'1

lI ///

~

urvQtur:K

Figure 2. Variation of wave strength with curvature. given in reference [5]. For non-circular or non-uniform rods, K' # O, both the wave speeds and wave strengths vary in a manner given by equations (6.6)t and (6. I 0), respectively. In addition, from equations (6.9) we see that the induced wave is no longer parallel to the primary wave, but in each case lies at an indeterminate point on a straight line parallel to the primary wave vector. In order to illustrate the effects of curvature on wave propagation we consider an illustrative example. Consider a rod which is initially straight, gradually becoming curved, with ever increasing and eventually unbounded curvature. A simple example of such a rod is defined by the cornu spiral, whose curvature is given by ~: = A~, where A is a constant. We pose the question as to what happens to either an extension or a flexure wave initiated at the straight end. A qualitative analysis of equations (6.6)t and (6.10) yields the results illustrated in Figures 1 and 2. We see in Figure 1 that the speed VEF,+increases monotonically with u, while the speed FEE1-decreases to zero with increasing x. Note that initiated extension and flexure waves travel with the speed VEFt§ and //EFt-, respectively. Figure 2 illustrates the variation of wave strength for both the initiated component as well as for the wave component coupled to it by the curvature. For the initiated extension wave (solid curves), the flexural component has a relative maximum, and both wave components die out at sufficiently large curvature or wave speed. Conversely, an initiated flexural wave (dashed curves) has both components increasing indefinitely with increasing curvature or as the wave slows down. This behaviour leads one to ponder the possibility of fracture for such a situation.

7. WAVES IN TWISTED SPATIAL RODS In this section our interest lies in generalizing the situations already considered, so as to include the effects of initial twist and spatial curvature. Obviously, a spatially curved rod has associated with it a natural twist induced by the torsion of the reference curve ~. However, before considering this and other situations, we shall focus our attention on the straight rod with initial twist. Our aim in this respect is to acquire a feeling for the effect of twist in the simplest possible setting. Moreover, for the sake of simplicity we shall restrict our consideration to a rod with uniform twist. This situation isdefined by taking Ast = A 2 3 = 0 ,

At2 = f ' ,

f'=O.

(7.1)

298

H. COHEN A N D A. B. W H I T M A N

On substituting equations (7. l) into the appropriate equations in the set (4.10)-(4.17) and utilizing equations (4.7) and (4.8) we arrive at a classification ofwaves in straight twisted rods. As in the case of curvature the twist produces a coupling between the modes which can propagate in a straight untwisted rod. Specifically, we find that the two shear modes, $1 and $2, are coupled to the two flexure modes, F2 and FI, respectively, to produce two pairs of coupled modes. The speeds of propagation of these modes are twist dependent. Also, similar to the case of non-vanishing curvature, we find that the twist produces more coupling in the induced waves than that arising in the case of the untwisted straight rod. The detailed nature of these results are given by the following classification of wave types :-~ (i) I:: (same as equation (5.8)), (ii) E_+:W2t_.= (a,a~?2(V~+_- V-2, ) / V-26 , f , d+,f , c+), (iii)

S__.:W2t.= , s t r ' d

'•

r'

1 2 c+_,a,ac%(Vs+_

_

(7.2)

~.2)/~,2),

(7.3)

(iv) shear-flexure waves, 51 Fz+: Z 2- = ~[Z 2 + V~(l + ct2f '2) + -- {[Z 2 + Z~(l + .2f'2)12 - 4V~ V2~1/21 5J J' Wl+ = ( l , 0 , 0 , 0 ,

v ~ f ' / ( v g - v~)) lull.o,

W2+ = 0,

W~'+ = ( . , f ' t+,o,,~+,~v~J2 .,[(Vs2 _ V2+))lull+o,

W~+ = 0,

w , _ =(~2 v g f ' / ( v ~ + ~ 2 f '2 z] - z_2), 0 , o , 0 , l) lbhl-o, WL=(~

W2_ = 0,

V s~f ' ](V~~ +o~2f '~ V 2 - V~),d_,O,f ' ~-,,~)Ib231-0, '

W2'_ = 0,

(7.4)

(v) shear-flexure waves, 52 I:1+: Vz = 89[ V2 + V2(1 + c q f '2) + {[ V22+ V~(I + ~f,2)]2 _ 4 V2 V2}atz], w , + = (o, l, o, - z ~ f ' / ( z l - v~), o) 1,61+o,

w 2 + = 0,

W~t+ = ( f ' g + , a , O , - a V 2 f ' ] ( V 1 - V2),c+) lull+o,

w L = 0,

w , _ = (o,-~1 z L f ' / ( v ~ + ~ l f '2 v~ - v~), 0, l, o) [b;3 I-o,

W2_ = 0,

W l _ = ( g _ , - a ~ l V ~ f ' / ( V ~ + ~ q f '2 V 2 - V2-),O,a,f'c_)[b~3]-o,

W~_=0.

(7.5)

In the above set of equations (7.2)-(7.5) the quantity ~ denotes an indeterminate wave amplitude. The quantities g and c, which are different in each of the above cases, denote rational polynomial expressions in the V2's, ~1, r a n d f ' , all of which reduce to the straight rod results f o r f ' = 0. The explicit form of these induced wave amplitudes is similar to that for non-vanishing curvature and irrelevant for our understanding of the situation. We note also that due to the twist, a cross-sectional I: wave now induces a cross-sectional S wave and vice versa. The situation for non-uniform twist may be handled in precisely the same way as the case of non-uniform curvature. The wave speeds and modes remain unchanged in this more general situation. However, the decay and induced wave equations will now be more complex leading to results analagous to t.hose in the variable curvature case. We now turn our attention to spatial rods. In general such a rod will have arbitrai'y initial twist. For this situation equations (4.18)~ do not simplify and yield five speeds of propagation, whose associated waves couple all five modes, that can propagate in an untwisted straight rod. In order to make some progress we seek situations in which some degree of uncoupling of "i" See a ~ a i n the f o o t n o t e t o the first p a r a g r a p h o f s e c t i o n 6.

WAVESIN ELASTICRODS

299

these modes takes place.'~ The first of these corresponds to the situation mentioned at the beginning of this section. For this case, in addition to requiring that the twist of the rod correspond to the torsion of q~, we also require that the osculating plane be a plane of symmetry for the rod. This situation, which we call the case of restricted twist, is defined by taking A31 = h',

A23 = 0,

A I2 = r.

(7.6)

For this situation we find that the characteristic equation associated with equation (4.18)1 factors into the following quadratic and cubic equations in V2:

V ' - V2{Vf + Vg(1 + ~2 r2))+ Z~ V~ = 0, V 6 - V4{V~+ V~+ V~(l + c q K 2 + c q r 2 ) } + V2(V~ V]+ U~ V~(I +cqtr 2) + V~ V2(I + ~x r2)} - V2 V3: V2 = O.

(7.7)

The first of equations (7.7) corresponds to the 51 F2_+waves which can propagate in the straight twisted rod, producing in this case a coupling between shear and flexure which are parallel and perpendicular to the osculating plane, respectively. The three modes defined by the second of equations (7.7) are new. We see that these modes couple extension with shear and flexure, which are perpendicular and parallel to the osculating plane, respectively. We term these modes the extension-shear-flexure waves, (E$, FI)~, (I2 = 1, 2, 3), and remark that their existence is consistent with our earlier acquired knowledge on the effects of curvature and twist. We can regard the principal curvature as coupling the E and S2 F~+_waves, which can propagate in the straight twisted rod. Finally, in order to gain added insight into the effects of curvature and twist we consider a situation in which a spatially curved rod has essentially been untwisted. This case, which we term zero twist, is defined by taking A23 = •sinf, A12 = 0, (7.8) A31 = K cosf, so that the twist a n g l e f i s given by f = - I r ds.

(7.9)

On substituting equations (7.8) into equation (4.18)1 we find again that a certain degree of decoupling occurs. In fact, we find that the characteristic equation corresponding to equation (4.18)1 yields the two shear speeds V1 and /I2, with their associated modes S1 and $2, respectively, in addition to three new coupled waves whose speeds ofpropagation are given by v 6 - v~{v~ + v~(l + ~ A h ) + vg(1 + ~ Ah)} + v~{v~(v~ + vg)

+ V2V2(I +~qA~t +ct2A23)} - V 2 V 24 V s2 =

O.

( 7 . I O)

We call the waves associated wi.th the roots ofequation (7. i O) extension-double flexure waves, (EF1 Fu)n, (f2 = 1,2, 3), and note that here extension is coupled by the components of curvature to both components of flexure. We note that another interesting special case accrues if we consider an untwisted planar rod, having a plane of symmetry making a fixed angle f, with the normal to c6'. For this situation we find that precisely the same waves can propagate, as in the spatial rod with zero twist. 8. WAVES IN CONSTRAINED RODS In Eason's study [5] of harmonic wave propagation in circular rods he considers several "'simplified" cases. These cases correspond in fact, to the so-called constrained theory of rods. 1"It is somewhat surprising that the case of a twisted planar rod has coupling equally as complex as in the general spatial case.

300

H. COHEN AND A. B. WHITMAN

Such theories of rods are more akin to the classical rod theories in that they disallow crosssectional strains. The resulting field equations corresponding to the constrained rod take on a form which is more easily correlated with the classical field equations. The constrained theory is developed by requiring the directors to be rigid and to constitute an orthornormal triad. Thus d~.dl=611,

d 3 = d I x d 2 : ~ r'.

(8.1)

It is important to notice that the definition of d 3 is no longer the same as in the previous secti6ns. Ife t is a fixed orthonormal basis in E, then we can characterizethe deformation of the rod directors by writing d, = Qe,,

QQT = Qr Q = I ,

(8.2)

where Q = Q(~,t) is a proper orthogonal linear transformation. From equations (8.2) we have d; = ; . d , = ( x d,, ,~ = - , U = Q, QT (8.3) where ( is the axial vector corresponding to the skew symmetric tensor ).. Based on the above kinematical restrictions on the deformation of the directors, the mechanical theory of such a rod is found to be governed by the set of equations Iml = - ) . 1 o ' I V,

(8.4)

m' + r ' x n + 21= ;./r,

(8.5)

[ni = - ; . l i ' i V,

n' + 2 f = 2~, where m = d ~ x p ~,

l = d ~ xl ",

a =y~ad~ x da,

(8.6)

and n = ). a~/Or',

m = ). a$/a~,

r = Co(z,, K,), zt = Yl -

Yt,

Yt = r'. d ,

Ks = ~'l - Z .

~i = ~. di.

(8.7)

(8.8) (8.9) (8.10)

Equations (8.4)2 and (8.5)2 are the balance of moment of momentum equations in the discontinuous and continuous situations, respectively. The free energy function in equation (8.8) satisfies the requirement ofinvariance under superposed rigid motions, as do the relative strains (8.9) and strains (8.10), respectively. The quantity a represents the angular momentum density of the rod, and Yt and Z~ are the reference values of Yl and (~, respectively. Constrained theories of rods of this type have been considered by Whitman and DeSilva [13, 14] and by Green and Laws [15]. They have been utilized to solve boundary value problems by Whitman and DeSilva [16] and by Antman [17]. It is not our intention to enter into a discussion of the relative merits and interrelationships between the various formulations of the constrained theories, although a few remarks are in order. The development of such theories may stem from either the two director [15], or three director [13] models. Moreover, the theory may be developed by considering constraints within a broad kinematical context, or by starting at the outset with a more restricted kinematical setting. Any of the approaches which might accrue from the possibilities mentioned will introduce slight differences. In particular, using constraints will introduce a number of indeterminacies, which may be found from the director momenta field equations, not usually written explicitly in the set of field equations. Our choice of the strain variables used in reference [14] is a consequence of their easy kinematic interpretation.

301

WAVES IN ELASTIC RODS

The foregoing equations can be utilized in the same way as the field equations in section 2 to formulate the non-linear and linear wave propagation problems. We do not enter into the details of these calculations, but rather mention only relevant procedures and the final outcome. In order to construct a constrained linearized theory of wave propagation based on the form of the free energy function ~P adopted in section 4, we introduce the aforementioned constraints by setting ~B ----0,

~ct,B)= 0,

(8.11)

]r = k12 + kla - k14.

(8.12)

in equation (4.1), in addition to defining

In order to utilize the constitutive equations and strain variables given by equations (8.7)-(8.10), we need the transformation equations between these strain variables and those defined by equations (2.10) and (2.1 l). These transformation equations are given by y , = h,~,

Ya ----"(h33 - Y~t.l,',) 1/2,

).~ = e~(a,

2 ~ = e~f~yt(j,

(8.13)

where e, B and e~jk are the two- and three-dimensional permutation symbols, respectively. Finally, we introduce the infinitesimal rotation vector ~b, defining the director displacements b~ according to bf = ~k x Dl, (8.14) and leading to a six-dimensional generalized displacement vector ~ c in the form

Upon carrying out the analysis based on these considerations, we find that the wave vector again separates into two components given by We, = (lufl, lull, lu;l, [4';1, lq~il),

We, = ([4';!).

(8.16)

The five speeds and modes associated with W q are precisely the same as those governing W~ in section 4. On the other hand the constraints introduced here have served to remove the modes W2 associated with cross-sectional strains. However, as the cross-section can still rotate rigidly, we now find a new mode of propagation associated with a cross-sectional rotation gradient W~2. We term this mode a torsional or twist wave, T, its speed of propagation being given by V~ = fc/po(~l + ct2). (8.17) ACKNOWLEDGMENT The authors wish to acknowledge the support of the U.S. National Science Foundation, the National Research Council of Canada and the University of Manitoba Research Board.

REFERENCES 1. E. COSSERATand F. COSSERA'r1909 Th~orie des Corps D~formable. Paris: Hermann. 2. S. S. ANTMAN1972 in Handbuch tier Physik, Bd. VI a/2. The theory of rods. Berlin: SpringerVerlag. 3. A. E. GREEN,N. Laws and P. M. NA~IqDI 1967 Archi~'es of Rational Mechanics and Analysis 25, 285-298. A linear theory of straight elastic rods.

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