a properly invariant theory of small deformations superposed on large deformations of an elastic rod

Journal of Eiasticity 39: 97-131, 1995. 97 © 1995 Kluwer Academic Publishers. Printed in the Netherlands.

A Properly Invariant Theory of Small Deformations Superposed on Large Deformations of an Elastic Rod

OLIVER M. O'REILLY Department of Mechanical Engineering, University of California at Berkeley, Berkeley, California 94720-1740, U.S.A.

Received 22 August 1994; in revised form 6 February 1995

Abstract. In the context of the direct or Cosserat theory of rods developed by Green, Naghdi and several of their co-workers, this paper is concerned with the development of a theory of small deformations which are superposed on large deformations. ,The resulting theory is properly invariant under all superposed rigid body motions. Furthermore, it is also valid for elastic rods which are subject to kinematical constraints, and it specializes to a linear theory of an elastic rod which is invariant under superposed rigid body motions. The construction of these theories is based on the method developed by Casey & Naghdi [1] who established similar theories for unconstrained non- polar elastic bodies.

AMS subject classification (1991): 73K05, 73G05.

Table of Contents

1. In t roduc t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.1. Scope and outline of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 1.2. Notational conventions and terminology . . . . . . . . . . . . . . . . . . . . . . . . 100

2. S u m m a r y of the Theory of a Directed Elastic Curve . . . . . . . . . . . . . . . . . 100 2.1. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.2. Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.3. Kinematical constraints and response functions . . . . . . . . . . . . . . . . . . 104 2.4. Non-invariance of the linear theory of a directed elastic curve . . . . . 107

3. Development of Proper ly Invar ian t Infinitesimal Deformat ion Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4. A Proper ly Invar ian t Theory o f Small on Large . . . . . . . . . . . . . . . . . . . . . 114 4.1. Preliminary kinematical considerations . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2. Kinematics of the invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3. Balance laws, response functions and constraint equations of

the invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 4.4. Remarks pertaining to an invariant infinitesimal theory . . . . . . . . . . . 123

5. Consequences of an Alternat ive Selection of the "P ivo t" . . . . . . . . . . . . . 125 5.1. Effects of the choice of"p ivo t" on the motions 1X~ and 2X~ . . . . . 126 5.2. Effects of the choice of "pivot" on the difference motion X~ . . . . . . 127 5.3. Effects of the choice of "pivot" on the invariant infinitesimal

theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

98 o.M. O'REILLY

1. Introduction

This paper is concerned with the development of a theory of infinitesimal motions superposed on a given deformation of an elastic rod which is properly invariant under superposed rigid body motions. The resulting theory specializes to an invariant theory of infinitesimal motions and also incorporates kinematical constraints. The particular rod theory discussed here is known as a direct or Cosserat rod theory. It was developed by Green, Naghdi and their co-workers in a series of papers and encompasses various constrained theories of rods. For a review of the theory and additional references the reader is referred to the survey article by Naghdi [2]. Sup- plementary background material, on constrained theories of rods is contained in the paper by Naghdi & Rubin [3], and on invariance considerations under superposed rigid body motions in Naghdi [4, Sections 8 and 13] and Green & Naghdi [5].

The linear theory developed from the properly invariant non-linear theory of rods by Green, Naghdi & Wenner [6, 7] is similar to the classical (non-polar) theory of linear elasticity in that it is not properly invariant under all superposed rigid body motions. In the interests of brevity, we postpone a demonstration of this for an unconstrained elastic rod until Section 2.4 and for the constrained elastic rod until Section 4.4. For the classical theory this difficulty was overcome by Casey & Naghdi [1] who developed an invariant theory of infinitesimal deformations superposed on a given motion by introducing the concepts of a "pivot" and an auxiliary motion (cf. also Casey & Naghdi [8]). The concepts of their work are adapted here for a directed rod theory. However several significant differences exist between the deformation of an elastic body and a directed rod and these have to be incorporated into the development. To achieve this it is convenient to adapt a direct notation which was originally introduced by Naghdi [2, Sect. 13]. In addition to the variables defined in [2, Sect. 13], several others will be introduced in the course of this paper. Some of these variables will have obvious comparisons to similar quantities defined in classical (non-polar) continuum mechanics.

A theory of small deformations superposed on large deformations of an elastic rod was developed previously by Green, Knops & Laws [9]. Their theory (cf. [9, Sect. 3]), when reduced to the case of an infinitesimal theory, corresponds to the linear theory in [7, Sect. 8]. As mentioned earlier such a linear theory is not properly invariant. Furthermore, apart from a discussion of the Bernoulli-Euler rod theory (cf. [9, Sect. 6]) their work does not address the presence of general kinematical constraints. The main focus of this paper is the development of a properly invariant theory of small on large which can incorporate general kinematical constraints. To determine and accomodate the transformation of the constraint forces under superposed rigid body motions of the rod, it proves convenient to introduce an alternative approach to that outlined in Naghdi & Rubin's paper [3]. The difference between our approach and theirs may be attributed to our imposition of the requirement that the equations describing the kinematical constraints remain invariant under superposed rigid body motions.

THEORY OF SMALL DE, FORMATIONS 99

We mention that one of the primary reasons for considering the additional effects of kinematical constraints lies in the almost exclusive use of rod theories in the literature which may be considered as constrained theories of the direct theory under discussion here. In particular, we mention the rod theory of Bernoulli-Euler (or Kirchhoff-Love), and that of Antman [10], Reissner [11] and Simo [12].* By accomodating constrained theories, the invariant theory developed here applies to the aforementioned constrained theories, among others. As will become apparent in our development, the invariant infinitesimal theory is also applicable to studies on the dynamics of rods and strings where the motion can be considered as a small motion superposed on a large rigid body motion. Particular examples of these studies include flexible satellite components and travelling waves in rods and strings.

The corresponding development of a properly invariant linear theory for a directed surface is discussed in Naghdi & Vongsarnpigoon [14]. Although the present paper discusses a different theory, several of our kinematical results in Sections 2.1 and 3 are similar to those in [14, Sect. 3].

1.1. SCOPE AND OUTLINE OF THIS PAPER

In the forthcoming section, the balance laws in direct notation for the directed rod theory are outlined from the paper of Naghdi [2]. Several of his kinematical results are supplemented and the behavior of various quantities under superposed rigid body motions are discussed. In addition, an objectivity assumption on the kinematical constraints for the directed curve is introduced. The section closes with a demonstration of the lack of proper invariance of the standard linear theory of an unconstrained directed elastic rod. In Section 3 properly invariant infinitesimal deformation measures are developed by adapting the concepts of a "pivot" and an auxiliary motion developed by Casey & Naghdi [1] for a non-polar elastic body. Section 4 concerns itself with a properly invariant theory of small deformations superposed on large. The relevant balance laws, response functions and constraint equations are also discussed. This section concludes with a demonstration of the lack of proper invariance of the standard linear theory of a constrained directed elastic rod. In the last section of this paper the consequences of a different selection of the "pivot" on the results of Section 4 is addressed. This section and the paper concludes with a proof that the constraint equations and response functions of the invariant infinitesimal theory developed in Section 4 is independent, to the order of approximation considered, of the choice of "pivot".

* For details on the Bernoulli-Euler rod theory as a constrained Cosserat rod the reader is referred to Naghdi & Rubin [3, Sect. 4]. The justification of our remarks on the rod theory of Antman, Reissner and Simo may be obtained from Green & Laws [13] with the assistance of some material on constraint forces from [3].

100 o.M. O'REILLY

1.2. NOTATIONAL CONVENTIONS AND TERMINOLOGY

Relevant background on the tensor operations and terminology used in this paper may be found in standard texts on continuum mechanics. A concise summary which is pertinent to this paper is contained in [1, Sect. 2.1].

The summation convention over repeated lower-case latin indices will be employed. Lower case greek indices when displayed to the fight of a variable (e.g. d#, k r ) will also be subject to the same convention. Lower-case greek letters when displayed on the left of a variable (e.g. ~X, aM) will not be summed, even if repeated.

2. Summary of the Theory of a Directed Elastic Curve

2.1. KINEMATICS

We recall from Naghdi [2] the concept of a material curve £ together with 2 deformable vector fields or directors attached at each material point of the curve. The curve and its directors, which are embedded in E 3, is known as a Cosserat or directed curve ~ . To specify the kinematics of ~ , let the material point of £ be uniquely specified by the convected (Lagrangian) coordinate 0, and let l and t~(~) denote the present configuration of £ and R, respectively. It is assumed that the coordinate 0 is an element of a closed convex set E. The position vector r of an arbitrary material point and the directors d r (a = 1,2) associated with this material point in the present configuration are uniquely specified by the vector valued functions

r = r (0 , t ) , d r = da(0 , t ) , (2.1)

and it is assumed that the scalar triple product

[dld2a3J > 0, (2.2)

where

6~r a3 = ~-~. (2.3)

For notational convenience, the vector a3 will often be denoted by d3. A set of reciprocal vectors d i may also be defined:

d i . d k -- 6/k, (2.4)

where 6/k is the Kronecker delta. The velocity of the material point and the director velocities are denoted by

v = t , w r = / i t , (2.5)

THEORY OF SMALL DEFORMATIONS 101

respectively, where the superposed dot denotes the material time derivative. The reference configuration 0a;(7~) of the directed curve is specified by the

position vector R and the directors D~:

R = R(0) , D~ = D,~(0), (2.6)

and it is assumed the scalar triple product

where

[D1D2A3J > 0, (2.7)

OR A3 --- D3 - 00" (2.8)

In addition the reciprocal v e c t o r s D i may also be defined in a manner similar to the definition of the vectors d i.

A motion of the directed curve is denoted (for future convenience) by Xr~: 0~;(7~)

X.m = X.m(R(0), D~(0), t) = (r(0, t), d~(0, t)). (2.9)

For future reference we note that a motion X + differs from a motion X.m by a rigid motion if and only if

= (r+(0, t+ ), d+(0, t +)) (2.10)

= (Q(t)r(0, t) + q(t), Q(t)d~(0, t)),

where Q is a proper orthogonal tensor, q is a vector and t ÷ = t + a, where a is a real-valued constant (cf., e.g., [7, Sect. 7]).

The tensor F: Di -* di is defined as the linear transformation ([2, Sect. 13])

F = F(0 , t ) = dl ® D i. (2.11)

For future purposes, we recall that

d e t ( F ) - [dld2d3] [D1D2D3] > 0. (2.12)

From (2.11), the inverse of F may be determined:

F -1 = Di ® di. (2.13)

Using the polar decomposition theorem, the unique decomposition

F(0, t) = R(0, t)U(O, t), (2.14)

102 o.M. O'REILLY

exists, where R is a proper orthogonal tensor and U is a symmetric positive definite tensor.* It is also appropriate at this point to define the tensor H:

H = H ( 0 , t ) = F ( 0 , t ) - I = (di - Di ) ® D i, (2.15)

where I is the identity tensor. Supplementary to H, a symmetric tensor e and a skew symmetric tensor w may be defined,

e = e (0 , t) = I ( H + HT) , w = w ( 0 , t) = ½(H - HT) . (2.16)

The tensor E for the directed curve is defined by

E = I T ~(F F - I). (2.17)

Relative to the basis D i ® D k, the tensor E has components

E i k = l (d i " dk - Di" Dk). (2.18)

We remark that the components F-,ik (modulo a factor of 2) are 6 of the measures used to describe the deformation of a directed elastic curve (cf. [7, Eqn. (7.38)1]). The tensors Gs and 0Gs are defined by [2]

0ds 0Ds D3" G s - 00 ®D3' o G s - 00 ® (2.19)

Using (2.19) the tensors K~ and ks may be defined,

Ks = FTGs - oGs = ks + H ~ o G s

ks = loGs + HToGs, (2.20)

where loGs = Gs - oGs. The non-trivial components of the tensors Ks and ks relative to the basis D i @ D k are

0ds 0Ds K s i 3 = 0--0- " di - O--if-" D i ,

0(ds - Ds) 00Ds k s i 3 = O0 • Di + - - • ( d l - Di). (2.21)

It should be noted that the components K s i 3 (c~ = 1,2, i = 1,2, 3) describe the remaining 6 measures used to describe the deformation of a directed elastic curve (cf. [7, Eqn. (7.38)2]).

The transformation of F, R, H, Gs, U, E and I ~ under superposed rigid body motions of the directed curve may be determined from (2.10), (2.11), (2.14), (2.15), (2.17) and (2.20)1:

• To avoid possible confusion between the rotation tensor R and the position vector R, henceforth we shall always specify the position vector by R(0) while the rotation tensor will be denoted by R or R(0, t).

THEORY OF SMALL DEFORMATIONS

F + = Q F , R + = Q R , H + = Q H + Q - I ,

U + = U , E + = E , K + = K ~ .

G + = QGs,

103

(2.22)

Similarly, the related expressions for the kinematical variables e, w and ks may be determined with the assistance of (2.16), (2.20)2 and (2.22):

2e + = 2 e + ( Q - l ) e + e ( Q T - I ) + Q w - w Q T

- ( O - I ) ( Q T - I ) ,

2w + = 2w + (Q - I)w + w(Q T - I) + Qe - eQ T + Q - QT, (2.23)

k + = Qks - ((Q - I ) (Q T - I) + QH T - HTQT)0Gs.

We remark that the components of e and ks relative to the basis D i ® D k are used as the deformation measures for the linear theory of directed curves (cf. [7, Eqns. (8.2) and (8.3)]). The consequences of (2.23)1,3 shall be addressed in Section 2.4.

2.2. BALANCE LAWS

In this subsection we recall the local forms of the balance laws for a directed curve in direct notation from [2, Sect. 13]. Prior to recording them, it is convenient to recall the following 3 tensors [2],

1 1 k a M a = 1 m S N = v/a33 n ® d3, K = v/_d~ ® d~, ~/a33 ® d3, (2.24)

where

Or Or a33 = 0--0 " 0"-'0' (2.25)

n is the contact force, k s are the intrinsic director forces, and m s are the director forces. In the direct theory the vector fields n, k a and m s are assumed to be objective,* i.e. under superposed rigid body motions they transform as

n+(0 , t +) = Q(t)n(O,t), k~+(0, t +) = Q(t)k~(O,t),

m~+(0, t +) = Q(t)mS(0, t).

It follows from (2.10), (2.24) and (2.26) that

(2.26)

N+(0, t +) = Q(t)N(0, t )QT(t) , K~+(0, t +) = Q(t)K~(0, t)QT(t) ,

MS+( o, t+) = Q( t )Ma( o, t)QT( t ) (2.27)

* The reader is referred to Naghdi [4, Sect. 87] for further details on this requirement.

104 o.M. O'REILLY

Under suitable continuity assumptions, the balance laws for the directed curve are, mass conservation:

~/2 A = O, (2.28) jb + p div v = Pa33 =

balance of linear momentum,

dive N + pf = p(~ + gS~vs), (2.29)

(two) balances of director momentum,

dive M s + pl ~ - a3~/2k s = p(ySir + y~(v~), (2.30)

and the balance of moment of momentum,

[N + K + MS(GsF-1) T] = [N + K + MS(GsF-1)T] T, (2.31)

where f = f(0, t) is the assigned force, I s = lS(0,t) are the assigned director forces, yS = ys(O ) and yS~ = ys#(O ) are the inertial coefficients, p = p(O, t) is the mass density per unit length of the directed curve and

) 10a dive a ® d i - aV~O-O' (2.32)

for all vector valued functions a = a(0, t).

2.3. KINEMATICAL CONSTRAINTS AND RESPONSE FUNCTIONS

The expression for the mechanical power P of the directed curve in direct notation is (cf. [2, Sect. 13])

P = av/'d ~ tr((l~F-l)T(N + K) + (l~sF -1)TMa). (2.33)

With the assistance of (2.17), (2.20)1 and the moment of momentum balance law (2.31), the expression for P can be written in the invariant form:

p = a x ~ t r ( Y I F - T E F - 1 + MaF-TI~TF- I ) , (2.34)

where the symmetric tensor I I is defined by

2II = IN + K - MS(GsF-1 ) T] + [N + K - MS(GsF-1)T] T. (2.35)

The invariant expression (2.34) in component form is the standard expression for invariant mechanical power found in the literature on the direct theory of rods (cf. [7, Eqn. (6.31)], for instance).


Consider a directed curve whose free energy ~b has the functional forms (cf. [2, Sect. 13] and [7, Sect. 7])

( 0 d o 0D~ ) e = ~ dl, - N - ' Di, --N-' 0 = ~(F, Go, 060, D~, 0). (2.36)

As ~ is a constitutive response for ~ , it is assumed that ~b is objective:

~+ = ~(QF, QG~, 0G~, D~, 0) = ~}(F, G~, 0G~, V~, 0), (2.37)

where (2.37) was obtained with the assistance of (2.22)1,4. Using the polar decomposition (2.14), and selecting QT = R = FU -1, it follows that

~b = ~(U, U-1FTGc~, 0G,~, Di, 0) = ~)(E, K~, 0G~, Di, 0). (2.38)

We remark that in obtaining the final functional form of ~b (= ~), the invertibility of U and (2.17) were used. This final functional form is invariant under superposed rigid body motions and when written in the standard notation agrees with the result obtained by Green, Naghdi & Wenner [7, Sect. 7] who used an alternative approach.

Consider a single mechanical constraint ~ which restricts the deformation of 7~. The constraint equation qo is assumed to have the functional dependence ([3, Sect. 3])

q~ = ~ ~d/, Di, ,0 = ~(F, G~, 0G~, DI, 0) = 0. (2.39)

Because qo is a constitutive assumption on the response of ~ , it is assumed to be objective.* With the assistance of arguments paralleling those discussed in [7, Sect. 7]** or developments paralleling the derivation of (2.38)2 from (2.36), it follows that the objective functional form of ~p is

qo = 95(E, I ~ , oGa, Oi, 0) = 0. (2.40)

We now take the material time derivative of (2.40) and define ~b = 95:

q~ = t r ( r l ~ + A~K T) = 0, (2.41)

where the symmetric tensor r and the tensors A a are defined by

r = i~(E, K~, 0G~, Di, 0) = O~ 0E'

* Requirements of this type for the constraint equation for directed theories were discussed earlier by Green, Naghdi & Trapp [15].

** Although the development in Section 7 of [7] is concerned with the free-energy density ~,, it should be clear that their developments are also directly applicable to the constraint equation ~ = 0.

106 o.M. O'RE1LLY

A ~ =/k~(E, I~, 0G~, Di, 0) = 0~o 0K~"

It follows from (2.42) that under superposed rigid body motions

r + = F,(A'~) + = A ~.

(2.42)

(2.43)

A more general constraint equation of the form (2.41) can be postulated directly (i.e. ¢ is not necessarily the material time derivative of a function ~o). Henceforth, a function ¢ = 0 of the form (2.41) will be referred to exclusively as a constraint equation. It shall also be assumed that under superposed rigid body motions ¢+ = ¢ and that (2.43) holds.

To determine the response functions for N, K and M ~, the following standard assumptions are adapted (cf. [2, Sect. 12], [15], [16, Sect. 30]): (i) the responses I I and M ~ can be decomposed into the sum of a determinate or constitutive response IXde t and M~e t and an indeterminate or constraint response IIc and M~:

I I = Ildet + I I c , M a = M~e t + M~, (2.44)

(ii) the constraint response does not contribute to the mechanical power expression, (iii) that 1" and A a are independent of E and I ~ , and (iv) in the presence of multiple constraints the constraint responses due to each constraint are additive.

Let us now consider a directed curve subject to R kinematical constraint equations ¢L = 0 each of which is assumed to have the functional form (2.41). It follows from standard arguments using the 4 assumptions listed above that

av/-~ II = F ( A ~ - ~ +pLrL) F T,

( 0 ¢ ) a x / - d ~ M a = F ~ - - ~ + p L ALa F T,

(2.45)

where PL = PL( O, t) are indeterminate scalar fields which play the rSle of Lagrange multipliers. The resPonse functions for N and K are determined from (2.45) with the additional assistance of the moment of momentum balance law (2.31):

( 0~b ) FT av/-a-~ (N + K) = F A~--~ + pLr L

{)~ O~b + pLALC~ ) T F T. +Go C (2.46)

For completeness, we also note that the separate response functions for N and K can be determined from (2.46) with the assistance of (2.24)1,2:

L )


/ )3 + G~ A - ~ + pLA L~ FT(d 3 ® d3),

{A cg~b pLALa) T) + G ~ , ~ + FT(d ~ ®d~) . (2.47)

In the interests of brevity, we shall use the more compact form (2.46) in subsequent discussions.

If N, K and M ~ are specified by the response functions (2.45)2 and (2.46), then it is easily seen that the moment of momentum balance law (2.31) is trivially satisfied. The remaining balance laws (2.28), (2.29) and (2.30), and the R constraint equations eL = 0 (L = 1 , . . . , R), are then used to determine r(O, t), d~(O, t), p(O, t) and pL(O, t). This parallels the discussion for the unconstrained case in Naghdi & Rubin [3, Sect. 2].

For future reference, the assumed objectivity of N, K, M ~ implies from (2.22), (2.27), (2.43), (2.45) and (2.46) that PL are objective scalar functions:

P+ = Pt,. (2.48)

The objective nature of the multipliers will be of importance in the developments of Sections 4 and 5.

2.4. NON-INVARIANCE OF THE LINEAR THEORY OF A DIRECTED ELASTIC CURVE

As noted earlier, the variables e and k~ (cf. (2.16)1 and (2.20)2) are the deformation measures used in the linear theory of a directed elastic curve. In the interests of brevity, we shall restrict our attention here to the case of a directed elastic curve in the absence of kinematical constraints. A related discussion when kinematical constraints are present will be addressed in Section 4.4. Apart from some notational differences, the constitutive relations discussed in [7] for a linearly elastic directed curve in the absence of kinematical constraints may be written in the form (cf. [7, Eqns. (8.21)-(8.24)])

I I = .Al[e] + .A~[k~], M a = .A~[e] + .A~[k~], (2.49)

where .At, .,42 ~, .A~ and . A ~ are constant fourth order tensors*. The moment of momentum balance law and (2.49) are used to determine the constitutive relations for N and K. The invariance under superposed rigid body motions requirements

* The developments of [7] also includes material symmetry considerations. Such considerations have no bearing on the principal focus of the present discussion and are not discussed here.

108 o.M. O'REILLY

stipulate that for a motion of the directed curve which differs from a given motion by a superposed rigid motion (cf. (2.22)1,4, (2.27), (2.35) and (2.49)1)

I I + = Q I I Q T, M '~+ = QM~Q T. (2.50)

For a valid theory we require that the constitutive relations (2.49) are in agreement with the invariance requirements (2.50). In particular, the agreement should hold for all possible reference configurations, constitutive relations, motions and superposed rigid body motions.

Let us first consider H. From (2.49)1 and with the assistance of (2.23)1,3, the constitutive relations state that

H + = Al[e +] + A2~[k~]

= .Al[e + ½ ( ( Q - I ) e + e(Q T - I)

+ Qw - wQ T - (Q - I ) (Q T - I))] (2.51)

+ .A2#[Qk~ - ((Q - I ) (Q T - I)

+ Q(e - w) - ( e - w)QT)0G/3].

On the other hand, the invariance requirements under superposed rigid body motions discussed previously provides (cf. (2.22)1,4, (2.27), (2.35) and (2.49)1)

I I + = Q I I Q T = Q(.AI[e] + .A2~[k~])Q T. (2.52)

It is easily seen that when Q = I (i.e. the superposed rigid body motion is a pure translation), (2.51) and (2.52) are in agreement. Recalling the remarks following (2.50), we now choose a particular reference configuration and elastic rod:

.,41 = I, 0G;~ = 0. (2.53)

Furthermore, we choose the motion XR to be the identity motion:

0~(7~) = ~ ( ~ ) , e = w = k/~ = 0. (2.54)

With the assistance of (2.53) and (2.54), it is easily seen for this particular case that the constitutive relations (2.51) satisfy the invariance requirements (2.52) only for the restricted class of superposed rigid body motions where

Q + QT = 2I. (2.55)

Clearly, the constitutive relations for I I do not satisfy the invariance requirements in their entirety.

Similar results to those obtained for I I may be shown to hold for M ~, N and K. In summary, the linear constitutive relations (2.49) are not properly invariant


under superposed rigid body motions, and, as a consequence, the linear theory is not either.

3. Development of Properly lnvariant Infinitesimal Deformation Measures

We now begin to discuss the formation of a properly invariant infinitesimal theory of directed curves. We proceed as in the classical case considered by Casey & Naghdi [1] by selecting a point 0 E ~. The material point 0 = 0 is referred to as the "pivot". Consider a motion XT~: 0tc(7~) ~ tc(7~):

X.m(R(0), D~(0), t) = (r(0, t), d~(O, t)). (3.1)

Using the polar decomposition theorem we can calculate

1~ = R(/7, t) = F(0, t ) u - l ( 0 , t). (3.2)

An additional configuration to* (~ ) of the directed curve is now defined using the rigid motion x l : t¢(7~) ~ sc*(T~):

x~ ( r , d~, t) = (r*(0, t*), d*(0, t*)), t* = t + c*, (3.3)

where c* is a real valued constant and

r*(0, t*) = RT(r(0, t) - r(/7, t)) + R(0),

d~(0, t*) = l~Td~(0, t). (3.4)

For future convenience we define the motion X~ =- X~°Xg:ot¢(7~) ~ ~* (7~):

X.~(R(0), D~(0), t*) = (r*(0, t*), d;(0, t*)), (3.5)

where

r*(0, t*) = P,T(r(0, t) - r(/7, t)) + R(0),

d~(O, t*) = RTd~(O, t), t* = t + c*. (3.6)

We now establish certain results pertaining to the motion X~. Several of these results will parallel those obtained for the non-polar case by Casey & Naghdi [1] (cf. also [14, Sect. 2]).

THEOREM 3.1. Two motions IXre and 2Xze differ by a rigid motion if and only if 1 X . ~ ~ = * 2 ) ( . ' R .

110 o.M. O'REILLY

Proof We first establish the necessary part of the result. If 1XR and 2XTz differ by a rigid motion then, from (2.10),

2X.m(R(O),Da(O),t + a ) = (2r(0,t + a),2da(O,t + a))

= (Q(t) l r (0, t) + q(t) , Q(thaa(o, t)), (3.7)

where a is a real valued constant and Q is a proper orthogonal tensor. It may also be shown using (2.22)2 and (3.6) that

2R(0, t + a) = Q(t)lR(O,t),

2r(0, t + a) - 2r(0, t + a) = Q(t)(lr(O,t) - l r (0 , t ) ) . (3.8)

The motions 1X.~ and 2X.~ are (cf. (3.5))

ax~(R(O),D3(O),at* ) = (ar*(O, at + ac*),ad*~(O, at + ac*))

= (aRT(ar(0, a t ) - ar(/~, at))

+ R(O),j~rad~(O, at)),

(3.9)

where a R = aR(0, at), ac* are real valued constants, it = t and 2t = it + a. After substituting the relations, 2t = 1 t + a, (3.7) and (3.8)1,2 into the expression for 2X~ we obtain

2X.~(R(O), D3(8) , 2t* = it* + a + 2c* - i c*)

= lX.~(R(0) , Da(0) , it*). (3.10)

Modulo a constant additive time factor (a + 2 ¢* -- 1 C* ), the motion 2X.~ is identical to the motion IX.~. As in [1, Sect. 3.1], we do not distinguish between motions which differ by a constant additive time factor and consequently we may conclude that 1 X ~ = 2X.~-

To complete the proof, suppose that 1X.~ = 2X.~. By directly comparing the expressions for these two motions (cf. (3.9)) it follows that

2r(0, t + b) = 2R(0, t + b) lRr(0 , t ) l r(0, t)

- 2R(0, t + b)lRT(0, t ) l r(0, t) + 2r(0, t),

2da(O,t + b) = 2R(0, t + b)lRr(O,t)lda(O,t), (3.11)

where b is a real valued constant i t = t and 2t = it + b. The relations (3.11) are of the form (2.10) with

a = b, Q(t) = 2 R ( 0 , t + b)lRr(~i, t) , (3.12)

q(t) = 2r(0, t) - 2R(/~, t + b)lR:r(0, t ) l r (0 , t).


In conclusion 1X7~ and 2X7~ diff~ by a rigid motion and the proof of Theorem 3.1 is complete.

We now address how the deformation measures associated with a motion XT~ relate to those for the motion X~.

THEOREM 3.2. For any motion XT~ we can construct a motion X~, such that

(i) 30 = 0 such that x~(R(0 ) , Da(0), t*) = (R(0), U*(0, t*)D~(0)), (ii) qO = O such thatF*(O,t*) = U*(/~,t*),

(iii) E*(0, t*) = E(O, t) and K~(0, t*) = K~(0, t),

where t* = t + c* and c* is a real valued constant if and only i f X~ is o f the form (3.5)-(3.6).

Proof. We first establish the sufficiency part of this theorem. Condition (i) is established by first recalling the definition of the motion X~, (3.5)-(3.6), evaluating X~ at the "pivot" 0 = 0, and then using (2.11) and (2.14):

x~ (R(0 ) , D~(0), t*) = (R(0), U*(0, t*)D~(0)), (3.13)

i.e. the position vector of the "pivot" is invariant under the motion X~. To establish conditions (ii) & (iii), several kinematical variables associated with the motion X~ are first determined in terms of the corresponding quantities associated with the motion Xze:

F*(0,t*) = RT(O,t)F(O,t) = RT(o, t)R(O,t)U(O,t) , U*(0,t*) = U(0,t) ,

R*(0, t*) = RT(/~, t)R(0, t), G~(0, t*) = RT(0, t)Ga(0, t). (3.14)

Evaluating F* at the "pivot" establishes (ii). After recalling from (2.17) and (2.20) the definitions of E and I ~ , (iii) may be easily obtained using (3.14)1,4.

To establish the necessity part of the theorem we first suppose that condition (iii) holds. This implies that

F*(0, t*) = Q(0, t)F(0, t), U*(0, t*) = U(0, t), R*(0, t*) = Q(O, t)R(0, t),

R 'T(0, t*) 0d~(0,00 t*) _ RT(O ' t) 0 d ~ , t) , (3.15)

where Q is a proper orthogonal tensor and we have used (2.11), (2.14), (2.17), (2.20)1 and (3.6). With the additional assistance of (2.11) and (3.15)1,3,4, it follows that

d~'(0, t*) = Q(O,t)di(O,t), Od*(O,t*)o0 - Q(0, ~)~'0d~(0't)-~ , (3.16)

112 o.M. O'REILLY

which implies that Q is independent of 0. After setting i = 3 in (3.15)1 and integrating with respect to 0, then

r*(0, t*) = Q(t)r (0 , t) + q(t). (3.17)

To conclude the proof, we use conditions (i) and (ii), and evaluate (3.15)3 and (3.17) at the "pivot" to obtain

Q(t) = RT(0, t), q(t) = R(0) - RT(0, t)r(/~, t). (3.18)

It follows from (3.16)-(3.18) that the motion X~ is of the form (3.5)-(3.6). This completes the proof of Theorem 3.2.

The primary significance of the motion X~ may be seen by considering a superposed rigid motion of XT~. To elaborate, consider a motion of 7~ which differs from Xn by a superposed rigid motion:

x+(R(O),D~(O),t +) = (r+(O, t + ), d+(O, t + ))

= (Q(t)r(0, t) + q(t) , Q(t)da(0, t)), (3.19)

where t + = t + a. We wish to determine the motion X.~ associated with X +. From (2.22)2, (3.1)-(3.6) and (3.19) it may be shown that

R+(O, t +) = Q(t )R(0 , t),

r +" (0, t + ' ) = (Q(t)R(0, t ) ) T ( Q ( t ) r ( 8 , t ) - Q(t)r(0, t))

+ R(0) = r*(O,t*),

t * = t + c * , t +* = t + a + c +*. (3.20)

Similarly,

d+* (0, t +* ) = d*(0, t*). (3.21)

The relations (3.19)3.4 imply that t +* = t* + a + c +* - c*. As a + c +* - c* is a

+* = X* real valued constant, then from (3.19)2 and (3.20) we conclude that X~ 7¢. Using this result, the following relationships are immediately apparent

F+*(O,t * + b*) = F*(O,t*), U+*(O,t * + b*) = U*(O,t*),

H+*(8, t * + b*) = H*(O,t*),

E+*(O,t * + b*) = E*(O,t*), e+*(0, t * +b*) = e*(O,t*),

w + ' ( o , t * + r ) = w*(O,t*),


d +'(O,t* + b*) = d*(O,t*), k+*(0, t * + b*) = k~,(O,t*), (3.22)

where b* is a real valued constant. The proceeding theorem combines the results of the preceeding paragraph and Theorem 3.2.

THEOREM 3.3. Consider two motions IXTz and 2 X~ o f a directed curve. I f

1E*(0, it*) = 2E*(0,2t*), 1K*(0, it*) = 2K*(O, 2t*), (3.23)

* 2 * where 1 t - 2t = b* and b* is a real valued constant, then iXre = Xre and the motions 1X~ and 2Xr~ differ by a rigid motion.

Proof. The proof follows easily from Theorem 3.2 and the remarks following its proof.

As mentioned earlier, the variables ks and e are used as deformation measures in the linear theory of elastic rods. In contrast to the deformation measures I ~ and E of the non-linear theory, k,~ and e are altered by superposed rigid body motions (cf. (2.23)1,3). The following theorem provides a remedy for this lack of invariance which is achieved by using the motion X~ instead of XTz.

THEOREM 3.4. Consider two motions 1XTz and 2~(.7"~ o f a directed curve. I f

le*(O, it*) = 2e*(0, 2t*), lk*(0, it*) = 2k*(0, 2t*), (3.24)

where i t - 2t = b* and b* is a real valued constant, then 1X~ = 2X~'~ and the motions 1 ~(.T~ and 2XT~ differ by a rigid motion.

Proo f The definitions of It and e (cf. (2.15) and (2.16)1) and the fact that le* = 2e* imply that

B(0, I t ' ) = IF*(0, It*) - 2F*(0,2t*) = ((ld*(0, It*)

- 2d*(0,2t*)) • Dk)D k ® D ~ = -BY(0 , it*). (3.25)

Without loss of generality, we choose the directors Di to form a right-handed orthonormal basis in the reference configuration. Using lk~ = 2k~, and the definition of ks (cf. (2.20)3) we obtain

~0 (( ld* -- 2d*)" Di) = 0. (3.26)

Evaluating the partial derivative of B with respect to 0, and then using (3.26), it may be concluded that the skew symmetric tensor B = B(lt*). Recalling that at the "pivot" ~F*(0,~t*) = ~U*(/~,~t*), then, as ~U* is symmetric, B = 0. We conclude that 1F*(0 , l t * ) = 2 F * ( 6 , 2t*) and , after comparing the components of these tensors, that 2d,(O, 2t*) = ida'(0, it*). Integrating the identity 1d;(0, it*) =

114 o.M. O'REILLY

2d~(0, 2t*) with respect to 0 and evaluating the result at the "pivot", we obtain lr*(0, it*) = 2r*(0, 2t*). In conclusion, 1X~'~ = 2~I~'~. The final part of the theorem is established by invoking Theorem 3.1.

The balance laws, response functions and constraint equations need to be addressed in order to complete the discussion of the motion X~z. We shall address this material in the next section.

4. An lnvariant Theory of Small Deformations Superposed on Large

4.1. PRELIMINARY KINEMATICAL CONSIDERATIONS

We now consider two motions of the directed curve, 1XT~ and 2Xrz. Using (3.5)-- (3.6) we can define the two motions 1X~ and 2X~ and the difference motions

~ll~"~ = 2X'R.O(1XT"~.) -1 : 1/~(T~) --+ 2/~(~'~),

= 2x o(tx ) -1: (4.1)

where

X~(lr(O,t), ld~(O,t),t) = (2r(O,t),2d~(O,t)),

X~(lr*(O,t*), ld*(O,t*),t*) = (2r*(O,t*),2d*~(O,t*)), (4.2)

and it* = 2t* = t*, 1c* = 2 c* = c* = t* - t. For the future developments it is appropriate to define coordinates ( = ~(0, t) and (* = ~*(0, t*). The coordinates

and (* parameterize the configurations I~ (R) and 1~;*(~), respectively:

O( ~fl~Z33(O,t) ' 0(* = ~/lg;3(O,t.) ' (4.3) 0--0 = O0

where

Olr 01r 01r* Olr* (4.4) l a 3 3 = 00 0 0 ' l a ~ 3 = O0 00 "

The coordinates ~ and ~* can be conveniently identified as the arc length parameters of the material curve £ in the configurations I~ (R) and I~;*(R), respectively. As la33(0, t) = la~3(O, t*) > 0, ~ and ~* are monotonically increasing functions of 0. Consequently, for each instant t the inverse functions 0 = 0(~, t) and 0" = 0"(~*, t*) exist.

Consider a function f(#, t) then

f = f ( O , t ) = f ( O ( ~ , t ) , t ) = f ( ~ , t ) , (4.5)


and

Of O] O~ O], O] O~ O.[ (4.6) oo - - , / l a33 / - o i g i +

Similar relations pertain to f*(O, t*) and its derivatives. From (4.3), (4.4), (4.6)1, if f*(O, t*) = f(O, t) then

-* 0 ] ^ Of (i*(O t*~ , , t * ) = -~(((O,t),t). (4.7)

Using (4.3), (4.5) and (4.6) the vectors ~r, ~d~ and their derivatives can be defined as functions of ( and t. With the added assistance of (4.7), the vectors Br*, Bd~, and their derivatives can be defined as functions of (* and t*. In the interests of brevity, we only record a representative list of these results here

= i d a ( ( , t * ) = l D a ( ( , t ), 2 d a ( 0 , t * ) = 2 atg , ), ld*(0, it*) -* * * * * * d ' r e* t*'

ld;(0, t*) = ~/la;3(~* , t*)lD;((*, t*),

O~d~(O, t*) • 0 # d ~ ( ( , t*) -- ~ l t l ; 3 ( ~ * t* ) (4.8)

0 0 ' '

the other variables and material time derivatives being obtained in an obvious m a n n e r .

From (4.2) and with the assistance of (2.11), (2.13), (2.15), (2.19) and (4.8), the kinematical variables pertaining to the difference motion X~ can be determined from the relevant quantities for the motions 1Xra and 2Xr~

F' = 2F(IF) -1 = 2d/@ ld I = 2d/• @ i DI, H ' = F ' - I ,

IG~ = 1G~IF -1 o~d~ 01D~ = 0"--'0- @ ld3 -- 0"----~ ® 1D3'

1Go~ = 2Go, mr_ 1 02d~ 02a~, -- 00 ® ld3 -- 0~ ® 1D3" (4.9)

Identically, the same variables may be defined for the difference motion X~ with the additional assistance of (3.5), (3.6) and (3.14):

F'* = 2F*(1F*) -1 = 2d~' ® ld *i = 2a~ ® 1D*i, H'* = F'* - I,

1 • G* F *-1 01d* 1d.3 01D* 1G~ = 1 c~l -- 00 ® -- 0~* ®lD*3'

116 o.M. O'REILLY

1 • i-~. F . -1 02d* @ ld.3 025* 1D.3" 2 G a -" 21dr°tl " ~ - 0---~ - 0~ -----~ ® (4.10)

From these kinematical variables the following relations and deformation measures can be defined,

E'* = I((F'*)TF'* -- I), 2E* - 1E* = IF*TEt*IF *,

1 * 1 * 1 * r* ~ I * T I ~ _ * 1 * 21Gc~ = 2 G a - 1 G a , Ks = ,. 2,-~a -- l G a ,

• F*TK I* F* 2K~ - 1 K * = 1 al • (4.11)

The deformation measures E ~ and K~ can be defined in a similar manner and the relationships between the kinematical variables and deformation measures for the motion X~ and those associated with X~ can be obtained using (3.14).

4 .2 . KINEMATICS OF THE INVARIANT THEORY

Preparatory to establishing the balance laws, it is appropriate to recall the deformation measures which arise in an infinitesimal theory (cf. (2.16)1, (2.20)2) and to define related measures which are suited to the present context:

= 1 * 2e'* = H'* + H '*T, k~ 21Gc~ + (Ht*)T~G *. (4.12)

These measures have the equivalent representations

e'* = ( l d * " (2d~ - l d ~ ) + l d ~ - ( 2 d * - l d * ) ) l d *i ® l d *k

= *- • - 1 D i ) ) I D ® I D *k, (1Di (2a~ - 1D~) + ID~ (2d,* * *;

0(2d* - ld*) , , O ld*~ id . / ® ld . 3 k~* = ld*" 0 ~ + (2di - l d i ) - O0 ]

(4.13) 0(2d* * -* 0 1 D ~ r~*i

= 1Vi •

where (4.13) was obtained with the assistance of (4.8). We first wish to establish the invariance of e t* and k~ under arbitrary and possibly independent superposed rigid body motions of 1Xrz and 2Xr~.

THEOREM 4.1. The kinematical variables e'* and k~ are unaltered by independent superposed rigid body motions of 1 ~(.7~ and 2 X.7"¢.

Proof. The proof follows trivially after recalling that each of the motions 1X~ and 2 ~ are unaltered by superposed rigid body motions of either 1~7~ or 2)[~7~


(cf. (3.19) and (3.20)). For it then becomes apparent from (4.13) that e'* and k~* are unaltered by superposed rigid body motions of either I~R or 2XR.

We now consider the case where the motion I X~ is known and the difference motion X~ is considered to be a superposed infinitesimal motion on 1X~. To precisely specify the smallness of the difference motion the following three order of magnitude parameters are defined:

e = sup {12d7 - 1D~l} , CH = sup {IH'*I}, ~*E E* i=1,2,3 ~*E E*

~G = sup {1211G;I}, (4 .14) ~*EE*

where the configuration 1~*(7~) of the material curve L is parameterized by ~* E E*. We recall that if a tensor function A(B) defined in a neighborhood of B = 0 where e = sup~.eE.{IBI}, is such that 3C > 0, where [A(B)I < Ce n then A = O(e '~) as e --~ 0.

We now consider the various kinematical variables and their asymptotic approx- imations. Although several scaling selections are readily apparent, we shall restrict ourselves to the case e H = gG = g. From (4.10)-(4.12), the following asymptotic expansions and estimates can be obtained by well-known methods (cf. [8, Sect. 3.1 and Sect. 3.21 for instance),

F ' * = I + W * , (F" ) - l = I - H ' * + O ( g 2 ) , e ' * = O ( f ) ,

k~ = O(g), E'* = e'* + O(g 2) K'* = k'* + O(g 2) (4.15)

as ~ ~ 0. For future reference it should be noted that the asymptotic expansions of F ~* and its inverse to O(E 2) are invariant under arbitrary and possibly independent superposed rigid body motions of 1Xr~ and 2X7~. These results follow from (4.15)1,2 and Theorems 3.2 and 4.1.

4.3. BALANCE LAWS, RESPONSE FUNCTIONS AND CONSTRAINT EQUATIONS OF

THE INVARIANT THEORY

We now establish the balance laws, response functions and constraint equations for the motions 1X~ and 2X~ and the infinitesimal difference motion ~* Xr~. The resulting balance laws for the difference motion will be expressed in terms of the kinematical variables associated with the motion 1XTe.* To proceed we assume that the directed curve is elastic and subject to R constraint equations ~b L = 0 of the form (2.41). Each of the constraint functions ~b L and the free energy ~b are assumed to satisfy the invariance requirements discussed in Section 2.3. Finally, the "pivot" used for both of the motions 1XTz and 2XTz will be the same.

Several additional results are required before the relevant laws and relations may be obtained. With the assistance of (4.15), the following 4th order tensors

118 o.M. O'REILLY

which occur in the Taylor series expansions of the functions arising in the response and constraint functions are obtained*

L • * ~ L * r ~*TM* ~ .1 2 FL* = r (2E ,2Ka) = 1FL* + l Y E [l r c 1 r J

~L* r ~w.T~.t. "1~'1 + I~'K~LI~ "Z l" J + O(~2),

2 A L a * A Lat E* K *x 1ALa* . . L a * r "m.T^t. m*l = ~2 ~2 a ) = + 1Jv~ E [1 r c 1 r J

t ~La*r ~ . T I . t . ~ . 1 ~ [ ~ 2 3 + 1JWtK~ [ l r ~-31 r J -{- " J k e ) ,

o¢ o¢, ,~. o¢ 02E* - 0E t2"~ , 2 K ; ) - 0IE* + 1£*[1F*Te'*IF*]

+ 1.~'3*[1F*Tk~l F*] + O(g2),

0 ¢ 0 ¢ (2E*, 0 ¢ + 1 ~ [1 r e 1 r ] O2K* - 0 - ~ " 2 K ~ ) - 01K;

+ 17"/a3*[1F*Tk~*lF* ] + O ( : ) (4.16)

as g --~ 0, where

02¢ 1~* - - 0 1 E . 2 , 1 ~g-t~* 01 E* 0 1 K ~ '

l '~Ot/3. __ 02¢

01K*01K~'

I ~ E L* orL L* orL , . L a * O A L a - 01E*' l ~ K o - - ~ , 1JWLE -- 0mE* '

OAL~ (4.17)

The non-trivial components of the fourth order tensors in (4.17)1,2,3 with respect to the basis Di ® Dk ® D1 ® Dn ale

1 ( 0 2 ¢ * 0 2 ¢ * 0 2 ¢ * l~*ikln =~ ~ , + +

\ 01 Elk O1Eln O1 E* t O1Ei* k O1Et* O1E~i

17~/3.i313 = 0 2 ¢ * 0"~'* ~ K* ' 1/x ai3Ul 313

• The dependence of the functions -yF L, -yA L°, ~b and their derivatives on the reference variables 0G,~, Di and O = 0" (~*, t*) is henceforth understood.

02¢* ) +

O1E*lO1E~i

THEORY OF SMALL DEFORMATIONS

1 ( 02¢ * 02~b" ~ l'~ct*i31n = -2 ~ E * + - - '

~ o 1 g a i 3 0 1 In O 1 g ; i 3 0 1 E * l }

and consequent ly the tensors 1 E* , 1,~ "ct* , 1"]'~ t~* possess the symmetries:

IE *ikln = 1 E-k i ln = 1E *iknl = 1~ *lnik,

17-/or, 6'*i313 = 17-//3 c~*/3i3,

1.~ a*i31n = 1.~r c~*i3nl = 1.~ a*lni3 = 1.~ t~*nli3"

119

(4.18)

(4.19)

Related symmetry considerations apply to the 4th order t e n s o r s 1~ L* , I~K~L*, laWtEA • La*

associated with the constraint equations ffL = 0 (L = I v . . . , R).* We define the fol lowing 3 tensors:

~N = 12~2N(~) - T , ~K = 12~2K(~) -T ,

21M '~ = 12/.,t2Ma(Ft) -T , (4.20)

where

1/2 2a33

12/3, -- 1/2 ~ lt~33

(4.21)

and

~ I N = 2 n ® ld3, x ~ - ~ l K = 2k '~ ® ida,

x/i-d~ 1M ~ = 2m ~ ® ld3. (4.22)

The purpose of these additional tensors is found by observing their intended roles in the balance laws for linear and director momenta (cf. (2.24), (2.29), (2.30) and (2.32)):

ldivc(1N ) _ 02n 1 02n 1 ~ - ~ 0 0 ' ~Kld~ = v/i_d~Ek~. (4.23)

2 * The balance laws and response relations for the motions 1X~ and X~ are obtained by first recalling the balance laws, response functions and constraint

* In total the 7 tensors IE*, 1 -~ 'a* , 1~'~ ~* contain 78 coefficients. We remark that if material symmetry conditions are imposed, as in [7, Sect. 8], for the free-energy ~ then the number of coefficients in the linear response functions reduces considerably. However, such considerations do not effect the primary focus of the work presented here and will not be addressed.

120 o.M. O'REILLY

equations for XTe (i.e. (2.28), (2.29), (2.30), (2.45)2, (2.46) and (2.41)).** Then with the assistance of (2.10), (2.22), (2.27), (2.43), (2.48) and (3.14) we obtain

7P*(0, t*)v/Ta~3(O , t*) = A(O),

---- _ 19/ ~ t * \ 7dive(7 N*) + 7P'7 f* 7P*(7 ~'* + Y 7 a),

Or* 7divc(TM ) + 7P'71 a* + 7K'7 da* = 7p*(yaT~ * + yaf~7@~),

._* ALa *~ F*T 7 ~ 3 7 Ma* =TF*(~O-~f~*+TPLT~ 07K~ ) 7 '

+7G*(A00-¢*\ 7K~ - - , --Lo~*~ T *TPLTa ) 7F *r,

7¢2 = tr(7 ALa* (7K*) r + 71"L'71~*) = 0, (4.24)

where L = 1 , . . . , R , andT, a,/~ = 1,2. The following transformations were used to obtain (4.24),

ffJ* : ¢ , 7 ( ~ ---- 7 ¢ L , 7/9* = 7P' 7 K* = 7Ka' 7 E* = 7 E'

7P*L = 7PL, 7 FL* = 7 FL, 7 ALa* = 7 ALa,

7 N* 71~TTNTI~, 7K* = 71~TTKTl~, 7M = 7RTTM(~71~. (4.25)

The transformations (4.25) are obtained by first noting that the motion 7X~ can be interpreted as a particular superposed rigid body motion of 7X7~, and then using the transformations (2.22)6,7, (2.27), (2.43) and (2.48), and the objectivity assumptions on ¢ and ¢L discussed in Section 2.3. A related procedure is used in [ 1 ] and [ 14].

The assigned forces 7f* and assigned director forces 71 a* are obtained from 7 f and 7 la as follows (cf. [4, Sect. 8,,/]). First consider the balance of linear momentum for 7XT~ and 7X~. With the assistance of (4.25)10 it may be deduced that

7diVc(TN*) = 71~TTdivc(TN), (4.26)

and consequently from (2.29), (4.24)2, and (4.25)3,

7f* - (7~'* + yC'7@* ) = 7RT(Tf-- (7~' + y'~Twa)). (4.27)

** We recall from the development of the response ftmctions (of. (2.45) and (2.46)), that the moment of momentum balance laws for the motions t Xn and 2Xn (of. (2.3.1)) are identically satisfied.


After evaluating .vt* and .yw* using (3.6), ~f* can be determined. An identical procedure using the two balance laws for director momenta is followed to obtain .yl a*. In summary the desired results are

.yf. = .yRTtf + .ya + y%yba, .yl ~* = -¢RT.ylC' + y%va + y~.yb#, (4.28)

where

.ya = (-/~/~ 2 • - , n n ) ( , r + ,c) + 2~n~(,v* + ~ ) - .y~.

,~b~, = (.y~/~ 2 * * - .~f~n).yd. + 2~fl~.~w~,

--T ~f t~ = 7R .~R, .yc = .vRT.tr(0, t) - a (0) . (4.29)

To complete the balance laws, response functions and constraint equations, they should be supplemented by boundary and initial conditions for the motions -yX~. These can be obtained from those for ~XTz using (3.6), (4.25) and (4.28). Finally, it should be noted that as a consequence of Theorem 3.3 and the results preceeding it, the balance laws, response functions and constraint equations for 1X~ and 2X~ remain unaltered under arbitrary and independent superposed rigid body motions of both 1XTz and 2~7~.

The balance laws, response functions and constraint equations for the difference motion are obtained from those for 1X~ and 2X~. The procedure used is adapted from Green, Knops & Laws [9, Sect. 3]. It is first assumed that the assigned forces and assigned director forces satisfy the expansion,

0~* 0¢* 2f* = lf* + gf'* + O(g2), 21 = 11 + e-l a'* + O(g2), (4.30)

and then that the Lagrange multipliers for the second motion may be expanded as

2P~ = l P ~ + g P ~ + O ( g 2 ) , (4.31)

where p~ is an objective scalar function (of. (2.48)). With the assistance of (4.16) and (4.31), the response functions (4.24)4,5 for

the second motion are expanded in a Taylor series about the first motion. Then the 1 * 1 * I c~* * tensors 2 N , 2 K , 2 M defined by (4.20), and the balance laws for the motion 1Xz¢

are used to establish the balance laws and response functions for the difference motion X~. The final form of the balance laws for linear and director momenta are

l d i v c ( N ' * ) + lp*f'* = l p * ( l l * 3 I- V/3z;) 3 I- O ( ~ ) ,

• ~ , . . ~ , * _ K ' * z d ~ * ldlvc(M ) + 1/9 1 = 1p*(y~h * + y~'~i;) + O(g), (4.32)

122

where

and

= - + o ( , 2 ) , , z ; = 2 w ; - i w ; +

O. M. O'REILLY

(4.33)

~N'* : IN* - 1 N* -[- O(E2) , g K t* = 21K * - 1 K* -1- O ( c 2 ) ,

gM ~'* = 1M~* - 1M a* + O(g 2) (4.34)

as g ~ 0. The response functions are

g 1 ~ 3 M'~'* = 1F*(V°~* + "po~*)IF*T + 1 ~ 3 Ht*IM °~" + O(e2),

1 Jr"~* ~,t'ot*T'~ g 1 ~ 3 (Nt* + K'*) = 1V~3 (Ht*(1N* + 1K*) + 21~oAJ.v.t )

+ 1F*( ) 'V* + ~ * ) 1 F * T

lJ,~. ~ . t , , c ~ * ~t~c~*)T1F.T + l t J a l r t`l," + + O(g 2) (4 .35 )

as g ~ 0, where

V a* = A( l~ 'C~*[ iF*Tet* l F*] + 17-~Lc*/3*[1F*Tk~*IF*]) ,

_*," A ~Lc~*r ~ , . T ^ t . -L-,.I ~ ALa*r F*Tnd* ~.1"~ -- t . , t L a * Da* = l_PLAldVt E [1 ~ t: 1 r J + IJWtK~ tl n~# 1 r j ) -4-PL12X E

• * = )~(le*[1F*Te'*lF *] + I~'~*[1F*Tk~IF*]),

• / ~ , . ,L*r ~ * T M * ~ . 1 L * T t . • - - t . r ~ L * ~ * = lPL( , IYE t l r v l r j + I~K~[1F* k~lF ]) "4- P L l X E • (4.36)

Finally the R linearized constraint equations for X~ are obtained using (4.16)1,2 and (2.41):

._/ a L a * / ~*Tnd* ~ , '~T t ,dLa*r ~ * T J * ~ .1 f2 "*T ti t lAX t`l r U t a l r ) -[" 1. /vt E [ l r e 11' J l l~ a

t ~La*r ~,*Tkt* F*I I ~ * T ~ - -~-l~'WtK~ tl r /31 Jl c~ ) - l -

,,.,L*r ~ . T ^ t . "m.3 ~ * t r ( l rL*(1F*Te '*W *) + I~'E tl~, ~ 1 r Jl L'

,,-,L* r ~*T~d* + l~aKatlr ~-~ tF*]ll~*) = 0 + O ( g 2) (4.37)


as ~ ~ 0, where L = 1 , . . . , R. In the interests of brevity, we have refrained from explicity writing the functional dependence of several of the quantities and their derivatives in the balance laws (4.32), response functions (4.35) and constraint equations (4.37) for X~ in terms of ~* and t*. This may be easily performed with the assistance of (4.5), (4.8) and (4.13).

It is appropriate at this point to recall the remarks following (4.15) and (4.29). The invariance under superposed rigid body motions of 1XTz and 2XTz of the response functions (4.35) is easily verified from these remarks. Similarly the constraint equations (4.37) satisfy the same invariance property and (from (3.20) and (3.21))

ti +* = ti*, ~v~* = @;. (4.38)

As lP* is invariant under superposed rigid body motions of 1XT~, the balance laws for director and linear momenta and (4.38) provide the relations,*

f'+* = f ' * , I s'+* = l s ' ' . (4.39)

In conclusion the theory of small deformations superposed on large deformations of a directed curve presented here is invariant under superposed rigid body motions of I Xn and 2XT~.

4.4. REMARKS PERTAINING TO AN INVARIANT INFINr r~IMAL THEORY

An infinitesimal theory for a directed curve which is unaltered by superposed rigid body motions can be obtained by specialization of the above results to the case where 1X~ is a rigid body motion and the motion 1X~ for this particular case is the identity motion. The relevant balance laws, response functions and constraint equations may be obtained from (4.32)-(4.37) by substituting

* * * llC~* la33 ---- A33, lPL---- 0 , 1v* = lWoe = 0, 1 f* ~--- = 0,

IF* = I, IE* = 1K* = O. (4.40)

In addition, the following substitutions are required

e* = e'*, k ; = k ~ , P~ = P ~ , 0P(•) - A v / ~ - lP*,

and the operator ~divc is replaced by Dive where, ([2, Sect. 13]),

a-1]20a Divc(A3-31/2a®D3) = 33 ~-~,

(4.41)

(4.42)

* These invariance results may also be obtained using the invariance of the balance laws (4.24)2,3 for the motions -~X~ under superposed rigid body motions of.vx~ and then using (4.30).

124 o.M. O'REILLY

for all vector functions a = a(0, t). For added emphasis we note that the resulting infinitesimal theory is unaltered by superposed rigid body motions of 2Xrz.

In addition to providing an invariant linear theory, the invariant infinitesimal theory is also applicable to two topics which have received considerable attention in the recent literature. The first of these concerns the dynamics of rods and strings which are subject to a small displacement superposed on a rigid motion. The second topic concerns the linear behavior of travelling waves in rods and strings. For the first topic it is convenient to define an intermediate configuration by considering the motion 1XT~ as a general rigid motion of the rod or string. In studies on travelling waves a similar configuration may be defined by considering the motion 1XTz as a rigid translational motion. For both of these topics 1 ~ ' ~ is the identity motion and the invariant infinitesimal theory established here is applicable.

We now return briefly to the standard linear theory of a constrained elastic rod in order to demonstrate its lack of proper invariance under superposed rigid, body

• I

motions. From (4.35)-(4.37), (4.40) and (4.41), the response functmns for M s a r e :

Ev/-~33 M s' = ~(m s ® D3) : A(~'S[e] Jr 7-~Ls~[k~])

"4- pLALS(o) q- O(g 2) (4.43)

as g ---, 0. Let us now consider a motion of the rod which differs from a given motion by a superposed rigid body motion. Using the previously discussed conditions on m s (cf. (2.26h), one obtains

M s '+ = Q( t )M s ' . (4.44)

The response function for M s '+ may also be determined using (4.43), the invariance conditions on PL and A Ls (cf. (2.43) and (2.48)) and the transformation results for e and ks (cf. (3.23)):

1 e(Q T I) Qw wQ T M '~'+ : A ~ ' ~ [ e + ~ ( ( Q - l ) e + - + -

- (Q - I ) (Q T - I))] + AT~S#[Qk# - ( (Q - I ) (Q T - I)

+ Q(e - w) - (e - w)QT)oG#] q- pLALS(o). (4.45)

Using (4.44), (4.45) and arguments parallel to those used in Section 2.4, it may be shown that the constitutive relations (4.43) do not satisfy the invariance requirements. Similar results are also obtainable for the R linearized constraint equations and the response function for N' + K'. In conclusion the response relations and linearized constraint equations for the standard linear theory of a constrained elastic rod are not properly invariant under superposed rigid body motions.


5. C o n s e q u e n c e s o f an A l t e r n a t i v e Se lec t ion o f the "Pivot"

The results of Sections 3 and 4 are clearly influenced by the choice of "pivot". In this section we shall show, to the order of approximation adapted in Section 4, how the invariant theory of small deformations superposed on large is affected by a change of"pivot". This result when specialized to the case of the properly invariant linear theory shows that the theory, to the order of approximation considered, is independent of the choice of "pivot". Several of the results obtained here are identical in spirit to those obtained for classical linear elasticity by Casey & Naghdi [1, Sect. 5]. However our method of proof differs from theirs and there will be obvious differences due to the nature of the theory considered in this paper.

To proceed, we suppose that an alternative point 0' E E is selected as the "pivot". Then the motions ~X~o,: 0t¢(7~) ~ , ~ , ( 7 ~ ) ( a = 1,2) of the directed curve may be defined by

~X*.mo,(R( o), Do( o), t* ) = (~r},( O, t* ), ad*~,( O, t* ) ), (5.1)

where

and

~r~,(0, t*) = ~RT(0 ', t)(t~r(0 , t) - ~r(0', t)) + R(0') ,

~ d ~ , ( 0 , t * ) = ~RT"(O',t)t3d~(O,t), t* = t + c*, (5.2)

aR(0 ' , t) = aF(/~', t ) , u - l ( 0 ', t). (5.3)

For future convenience the subscripts 0' and 0 will be assigned on the right-hand side of all quantities associated with the choice of "pivot" O = 0' and O = 0, respectively. It should be noted that, without loss of generality,* we are considering

* * C* the cases where t = t~ = t#, + a~,, co = c~, = and a~, = 0. We may define the kinematical quantities associated with the motions ~X~y,

in the same manner as defined previously for the motions ~ X ~ . As our principal interest will lie in comparing these quantities for the different selections of the "pivot", we record here only the relationships between the various kinematical variables and deformation measures. For the kinematical variables, the following relationships are easily obtained

~RT F~, .~T..,..~ ,. .F~, = F~*, = 2~ r# 1~,, n#, = 2 ] ~ T H ~ * I ~ + 2 R T 1 ~ - - I,

1 * ^ T 1 * ^ 1 * ^ T 1 * ^ 1G/~, = 1R 1G/3~l R, 2G/3~, = 2R 2 G ~ l R,

* * * (5.4) . r a33 = ~ a 3 3 ~, = .7a33~,

* For addi t ional clarif ication on this point the remarks fol lowing (3.10) should be consul ted.

126

where

~R = ~RT(0, t)~R(tg', t). (5.5)

Similarily for the deformation measures we obtain the expected relations

t~E~, = oE~, #K*p = #K*o, (5.6)

O. M. O'REILLY

with the assistance of (2.17), (2.20)1, (4.10)3,4 and (5.4)1,4,5. For the deformation measures which appear in the balance laws for the infinitesimal motion we obtain with the help of (4.12) and (5.4),

2 ~,*TJ. ~,. ~T~t. lJe~t ¢~,1r~, ---- IF~T(1RO l l l I1~ + H~*Tlfi® T 1 RT

+ l l{(O + o T ) I ~ T - 21)IF~,

IF T(IRO1RrIG; _ • 1"¢~1 tta/~tlr~t = 1 a0

~TrjI*T 6r~Tx 6 T l f , * X ~* + 1R(O T + 1,- n~ 1,-,~ )1,, 1 ~ 0 ) 1 , ~ ,

where the proper orthogonal tensor O is defined as

O = O(0, 01, f* -- C*) = 2]{Tll{.

(5.7)

(5.8)

One of our principal future concerns will be the deformation measures (5.7) and it will be necessary, initially, to determine a measure of ® in terms of the smallness parameters ~HO and eHP"

5.1. EFFECTS OF THE CHOICE OF "PIVOT" ON THE MOTIONS 1 X,R* AND 2XR*

We now consider the balance laws, response functions and constraints equations for the motions 1X~ , and 2X~p. The change in "pivot" can be conveniently interpreted as superposed rigid body motions of 1~)~-~ and 2X~g by noting that

~r~,(0, t*) = #RT~r~(O, t*) + pRT(~r(0, t) -- t~r(0', t))

+ R(0') - e[ITR(0), (5.9)

and

^ T * "0 t* t~d*p(O,t*) = ~R t~da~( , ), (5.10)

is of the form (2.10). This observation greatly facilitates the transformation of the balance laws, response functions and constraints equations under a change of "pivot".


From (4.24)4,5,6, (5.6), (5.9), (5.10) and the objectivity of the scalar fields ~p~ (cf. (2.48)) i.e.

* * * ~ 0 t *~ /3P*L(O, t*) = /3PL~,(O, t ) ----/3PL~t , ), (5.11)

it follows that the fields associated with 1 X ~ and 2~;-~ may be obtained from their counterparts for the motions 1 X ~ , and 2 X ~ , using the relations (cf. (2.27))

~N~, = ~ R T a N ~ R , aK~, = ~RT~K~P, ,

/3* aRT M~*aR" (5.12) ~M~, =

For future use we record the relations

1 • ^ T 1 • ^ 1 • ^ T 1 . ^ 1 ~ / 3 " ^ T 1 /3* ^ ( 5 . 1 3 ) 2N~, = 2R 2N~IR, 2K~, = 2R 2K~l R, 2"~, = 2R 2M~ 1R,

which are obtained using (4.20), (5.4) 1,2 and (5.12). Using (5.9), (5.10), and (5.12), the assigned forces . ~ , assigned director forces .yI~* and inertial terms for the motions 1X7~ and 2X7¢~* may be obtained from these fields for the motions 1X7¢~, and 2 X ~ , by paralleling the derivation of (4.28). In conclusion, a change in the selection of the "pivot" for the motions 1X~ and 2X~ can be catered for in a similar manner to that used when considering superposed rigid body motions.

5.2. EFFECTS OF THE CHOICE OF "PIVOT" ON THE DIFFERENCE MOTION X ~

The effects on balance laws, response relations and constraints for the difference motion X~ due to a change in "pivot" are more delicate. To proceed we recall from (5.8) the definition of the tensor O. As ® is a proper orthogonal tensor it has the unique representation (cf. Truesdell & Toupin [17, Sect. 37] and Naghdi & Voonsarnpigoon [18, Sect. 2 and Sect. 4])

O = I + ~I, + ~ , (5.14)

where the symmetric tensor ¢I, and the skew symmetric tensor ~ are defined by

~I, = (1 - cos(v))( ' r ® -r - I), • = sin(v)e'r , (5.15)

where -r is a unit vector corresponding to the principal value 1 of O, e is the alternator and tJ is the angle of rotation which is determined from

tr(®) = 1 + 2cos(v) . (5.16)

The following result for IOl is readily obtained from (5.16) after noting that, IOl 2 = t r (OO T) = tr(l) = 3,

I01 = (5.17)

128

From (5.4)3 and with the assistance of (5.8) and (5.14) it follows that

'* ~ t = ® I = ~I, + ~I,. H~, - o I R T H ~ * I

O. M. O'REILLY

(5.18)

Taking the norm of both sides of (5.18), using standard inequalities (cf. [18, Sect. 1.2]) and (5.17):

IH~*,I + v ~ InTI >/I'I' + ,I,I = I'I' - OI. (5.19)

Recalling the definition of the smallness parameters gn~ and eli,, (cf. (4.14)2) and setting e l t = gnff = eli#,, we conclude that • + • and its transpose • - • are O(glt) as etl ---' 0. Consequently,*

® = l + O ( g H ) as * a ~ 0 . (5.20)

From the relationship for the trace of O, (5.16), it follows that

= 0 + 0 (~H) , ¢ = O(~ 2 ) , ~ = O(~a) ,

O = I + q + 0(~2) , (5.21)

i.e., ® is of the form corresponding to an infinitesimal rotation (cf. [16, Sect. 55]). In what follows we now set (cf. (4.14))

= ~ = g~, = gr i = ~ a ~ = gn~, = gG~ = gG~'" (5.22)

Furthermore, we presuppose that the balance laws, response relations and constraints for the difference motion X7¢~, ~* are given by (4.32), (4.35) and (4.37) with the inclusion of the subscript ~1. Our goal is to derive the corresponding laws and

l* t* from those for the motion X7¢~, relations for the motion XTe~ To proceed we use the estimates (5.21)3,4 to obtain the relations

Hi* = IRTH~*I ~ + ~ + O(~2),

~ * T l* ~* ~*T^t* ~,* lr~, e~,lr~, = l r ~ ~ l r ~ -[- O(g2),

~ w . T , . t . * ~7*Tkt* F* - - O ( 8 2 ) (5 .23) l r ~ , ~ - ~ , I F ~ , = l r ~ c~l ~ "l-

as g --+ 0. The calculations of (5.23) were performed with the assistance of (4.11)3, (4.12), (4.14)3, (5.4)3 and (5.7). Recalling the invariance considerations for the constraint equations ~b L = 0 discussed in Section 2, (5.6) and (5.23)1,2, it may be concluded that the R linearized constraints (cf. (4.37)) for the two motions ~* XT~,

i. are identical to the order of approximation considered here. and XTe~

1 * • This result is independent of the norm of either 211G'~'b or 2~G~,.


Let us now address the response functions. Using (4.34), (5.4)1,2,6, (5.12) and (5.13), it may be shown that

o¢ l *

= (5.24)

4 I* Returning to the response function for M0, (cf. (4.35)1)), the invariance results for the motions ~X~o and 4X~, discussed earlier, and the assumed objectivity of p)2 (cf. (4.31)):

p~ = t* l* PLO' = PLO, (5.25)

then clearly from (4.36)L2, (4.17) and (2.43),

4 * V~* = VO, + O(g2), ~ * = "P~,* + O(g 2) (5.26)

as g --+ 0. It now follows from (4.35)m and (5.4) that

g l ~ 3 M f . ^T . 4" .,-.4", ~ .T I ~ 3 H , 1Mo = mR (1F~(V~ + r ~ )lrff + . t, 4*)mR ^

+ ff'mRT1M~ * IR + O(g 2) (5.27)

as E ~ 0. After equating (5.27) to (5.24), then with the assistance of (4.34)3 we find that

-- . 43/I* e I ~ 3 M 0 * 4" ¢,4"~ ~*T _/-'-S--, ~,t, ,,,4" = 1F~(V~ + r ~ )mr~ +~/ma~3n~llVX~ + O ( g 2)

as g ~ 0. Alternatively, we record for future use

OtO* ^ T ~ 4 t * ^

Mg, = mR (M~)mR + O(~ 2) as ~ ---+ 0,

(5.28)

(5.29)

4 1 . 4 I * where the double tilde in (5.29) is used to emphasize that M~ and M~, are given by response functions of the form (5.28). With the added assistance of (5.4)4,5 and identical arguments it may also be shown that

_- ?* "- I * ^ T z I * .,~ I * ^

N~, + K~, = 1R (Nj + K# )IR + O(g 2) as g ~ 0. (5.30)

Returning to the balance laws for linear and director momenta (of. (4.32)):

• I * * • * 4 - * md, + lPb * = + v

130 o.M. O'REILLY

• ot t* * od* m-l* ~¢ 0 I* ~* t'. (~11" ldlV~0,(M~, ) + lP0,1#, + l~ , lU ---- lp~,l,Y ~, + Yaf~i~0, ). (5.31)

From the earlier discussion of the balance laws for the motion 1X~ following (5.13), recall t ha t lP~ = lP~,, and 1 a330* = 1a33 y,* , these results in conjunction with (5.9), (5.10), (5.29) and (5.30) imply

" 1 . 7` l *

ldiVco(N 0 ) = 1~, ldivc0,(N0, ) + O(e),

O f f * 7` O~ o *

ldivco(Mo ) = 1R ldivd,(Mo, ) + O(g),

7` - ^ 7. I * - I *

K ~ l d aO" --1 R I ~ , l d ~O + O(e) (5.32)

as g ~ 0. From the balance laws (5.31) and (5.32) the relations

6 * - ug'* - v '*z 0 = , R ( 6 * , - - +

I / O L * O/ " * Ot " * " - = IR(I~, - y u 0, - y~f~z~,) + O(g) (5.33)

as ~ ~ 0, may be obtained. The inertial terms in (5.33) for the "pivot" selection 0 = 0 can be described in terms of the corresponding quantities for the "pivot" selection 0 = 0~ using (5.9) and (5.10). Then the assigned forces 6" and assigned

O f f * director forces I 0 can be determined from (5.33).

5.3. EFFECTS OF THE CHOICE OF "PIVOT" ON THE INVARIANT INFINITESIMAL THEORY

The results for the difference motion specialize to the case of an invariant infinitesimal theory by setting

1 ~ 1F~, = 1 F ~ - - I, 1E~ = 1E~, = K* = 1 ~ = lK*~, ---- 0, (5.34)

in (5.23)-(5.33). From (5.8) and (5.21), it follows that

2R = I - gt + O ( g 2 ) . (5.35)

The R linearized constraints have the form (cf. (2.43), (4.37) and (5.23))

La -t* A L a ( o ) I ~ o ) tr(FL(0)6~ *, + A (0)k~0,) = tr(rL(0)6~ * +

+ O ( g 2 ) = 0 + O ( g 2) as g ~ 0 . (5.36)

Similarily the response relations are unaltered by the change in the "pivot" selection to the order of approximation considered here (cf. (5.30)):

= Ot l * = O~ l * = l * = l * 7` l * 7` I *

Mg, = M 0 +O({2) , N 0 , + K 0 , = N 0 + K 0 + O ( ~ 2) as ~ 0 . (5.37)


For the balance laws for linear and director momenta, the change in the inertial terms can be determined using (5.9), (5.10) and (5.35). Then in a manner paralleling the discussion following (5.33), the assigned force and assigned director forces can be calculated.

In summary, to the order of approximation considered here a change in the "pivot" selection does not effect the constraint equations or the response relations. The inertial terms, assigned force and director forces will however be effected in a manner which can easily be determined.

References

1. J. Casey and P. M. Naghdi, An invariant infinitesimal theory of motions superposed on a given motion. Arch. Rational Mech. Anal. 76 (1981) 355-391.

2. P. M. Naghdi, Finite deformation of elastic rods and shells. Proc. IUTAMSymp. Finite Elasticity (Bethlehem PA 1980), edited by D. E. Carlson and R. T. Shield, Martinus Nijhoff Publishers, The Hague, The Netherlands (1982), pp. 47-103.

3. P. M. Naghdi and M. B. Rubin, Constrained theories of rods. J. Elasticity 14 (1984) 343-363. 4. P. M. Naghdi, The Theory of Plates and Shells. Fliigge's Handbuch der Physik, Via/2, (edited

by C. Truesdell) Springer Vedag, Berlin (1972). 5. A. E. Green and P. M. Naghdi, A note on invariance under superposed rigid body motions. J.

Elasticity 9 (1979) 1-8. 6. A. E. Green, P. M. Naghdi and M. L. Wenner, On the theory of rods, I. Derivations from the

three-dimensional equations, Proc. R. Soc. London A 337 (1974) 451-483. 7. A. E. Green, P. M. Naghdi and M. L. Wenner, On the theory of rods, II. Developments by the

direct approach. Proc. R. Soc. London A 337 (1974) 485-507. 8. J. Casey and P. M. Naghdi, Physically nonlinear and related approximate theories of elasticity,

and their invariance properties. Arch. Rational Mech. Anal. 88 (1985) 59-82. 9. A. E. Green, R. J. Knops and N. Laws, Large deformations, superposed small deformations and

stability of elastic rods. Int. J. Solids Structures 4 (1968) 555-575. 10. S. S. Antman, Kirchhoff's problem for nonlineady elastic rods. Quart. Appl. Math. 32 (1974)

221-240. 11. E. Reissner, On finite deformations of space-curved beams. J. Appl. Math. Phys. 32 (1981)

734-744. 12. J. C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I.

Comp. Meth. Appl. Mech. Engng. 49 (1985) 55-70. 13. A. E. Green and N. Laws, Remarks on the theory of rods. J. Elasticity 3 (1973) 179-184. 14. P. M. Naghdi and L. Vongsampigoon, A theory of shells with small strain accompanied by

moderate rotation. Arch. Rational Mech. Anal 83 (1983) 245-283. 15. A. E. Green, P. M. Naghdi and J. A. Trapp, Thermodynamics of a continuum with internal

constraints. Int. J. Engng. Sci. 8 (1970) 891-908. 16. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics. Fliigge's Handbuch der

Physik, III/3, Springer-Verlag, Berlin (1965). 17. C. Truesdell and R. A. Toupin, The Classical Field Theories, Fli~gge's Handbuch der Physik,

III/1, Springer-Vedag, Berlin (1960). 18. P. M. Naghdi and L. Vongsampigoon, Small strain accompanied by moderate rotation. Arch.

Rational Mech. Anal. 80 (1982) 263-294.

a properly invariant theory of small deformations superposed on large deformations of an elastic rod

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