REND. SEM. MAT. UNIVERS. POLITECN.TORINO

Vol. 47°, 2 (1989)

H. Cohen

ON W A V E P R O P A G A T I O N A N D E V O L U T I O N IN R O D S

Introduction

The problem of the propagation and evolution of shock and acceleration waves, within the framework of modern rod theory as proposed by COHEN [1] and GREEN k LAWS [2], has been extensively treated by COHEN and his co-workers [3-14]. These studies model a rod as a one-dimensional directed continuum. The same problem has been examined by PASTRONE and his as­sociates [15-17]. Our purpose in this paper is to introduce the results of these analyses to the reader. These works employ the method of moving singular manifolds on deformable continua to model the wave front of a propagating disturbance. For a rod a singular manifold is a point across which there are jump discontinuities in some field variables. The speed of this point on the rod defines the speed of the wave; the magnitude of the discontinuity defines the amplitude or strength of the wave. The studies referred to find the speed and type of the various waves that can propagate in a rod under a variety of constitutive assumptions. Further, these studies determine the evolution of the wave by providing the growth-decay law for the amplitudes of all wave types capable of propagation. Finally, they give the induced waves that are associated to a given primary wave and thereby specify the interaction that occurs between the various wave types. To achieve our purpose of introducing this material, we examine the wave propagation problem from a simpler per­spective than in these studies, namely, we consider waves in plane rods that are modelled by the Euler elastica and the Cosserat rod, the latter being a restricted version of a general directed rod.

In Section 1 we treat wave propagation on an inextensible elastica. In

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Section 2 we extend the problem to the extensible elastica. In both cases restriction is made to the plane elastica. With the same restriction in Section 3, we examine the problem of wave propagation in a Cosserat rod. The results clearly generalize those of the first two sections, while at the same time tliey give restricted but analogous versions of the more general analyses provided in the references.

1. The Inextensible Elastica

The modern theory of rods finds its foundation in the now classic con­cept of EULER-the elastica. EULER modelled slender bodies, capable of large changes in shape, by a mathematical curve which could carry load and resist deformation. To review the concept of the elastica, we will consider a plane curve C that lies in a fixed plane V of three-dimensional physical space S. S is an euclidean point space, the points of which will be denoted by italics x, y, ..., etc. V will denote the translation space of £; elements of V, vectors, will be denoted by boldface u, v, ..., etc. We reserver r to denote the posi­tion vector to a point x e S relative to fixed origin in S. Boldface upper case letters, A ,B, ... etc., will denote linear operators or tensors which act on the elements of V. ft will denote the real numbers. We introduce a rectangular cartesian coordinate system x,, z = 1,2,3, in £, with V coincident with the x\-X2 plane. Thus xa, a = 1,2, are rectangular cartesian coordina­tes in V. The orthdnormal basis of V associated to #,, will be denoted by e,. Furthermore, we shall employ the standard summation convention on repeated indices over their ranges, e.g., v = vaea = i^ei -\-v2e2, etc.

The curve C is to model the shape and motion of a slender and long material body that is acted upon by prescribed mechanical loads. To idealize the situation we assume that C is inextensible, yet capable bf bending. By bending, we mean that C can and generally will undergo changes in its curvature K. We make the motion of C mathematically precise by giving the parametric equation of a moving curve, viz.,

r = r (M) (1.1)

where s, 0 < s < oo,s G ft, denotes the arc length parameter of C and t, t G ft, denotes the time parameter. The assumption of inextensibility means that points on C are constrained to move in a way that keeps invariant the arc length between them.

We seek to study the dynamics of C, now to be regarded as a me­chanical entity. Thus we require a specification of the loads that act on C,

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the internal forces that can be sustained by C, and the equations of motion that govern C. For the sake of simplicity, we restrict consideration to applied loads that act at the ends sa,a = 1,2, of C. These are assumed to consist of forces n a and couples inff, which are vectors parallel and perpendicu­lar to V, respectively, Fig. 1. The loaded curve becomes a structural element,

Fig. 1 - Loads on the elastica

often called a beam, column, rod or elastica, if and when we assume a stress principle: at any section of C, defined by a point x(s) on it, one side of the section transmits to the boundary of the other a force n and couple m, the sfress, to represent the net interaction of the internal forces of the two sihdes, Fig. 2a. For the situation at hand these are assumed to have the same character as the end loads, i.e., n and in are parallel and orthogonal to V respectively. Fig. 2b shows the forces and couples that act on an arbitrary infinitesimal element of length As of C. This, in the terminology of engineering mechanics, constitutes a free body diagram for the element. Euler's laws of motion applied to the limiting situation in which As —• 0 result in the equations

n ' = i , m' + r ' x n = h , (1.2)

where ' denotes d/ds, denotes d/dt and 1 and h denote the linear and angular momentum per unit of length of C, respectively. Clearly, (1.2)!, and (1.2)2 express the balance of linear and angular momentum, respectively.

The preceding equations (1.2) speak to the universal mechanical laws, but not to the material behavior of the elastica. For this, we need constitutive

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relations, equations which relate the stress pair (n,m) to a measure of the deformation. For an initially straight, inextensible and planar elastica, the deformation is fully determined by the curvature K. Indeed, knowledge of K will provide a differential equation whose solution is a plane curve C with curvature K. We follow EULER and assume the linear elastic law

m~BK (1.3)

m-f Am

Fig. 2a - The stress principle Fig. 2b - Free body diagram

as the constitutive relation for the .elastica, where m = me3, B £ 11 is an elastic coefficient called the stiffness. For a homogeneous rod B is a con­stant; otherwise B = B(s). We note that there is no constitutive relation for ii, a consequence of the assumed inextensibility, and thus n is completely determined by (1.2)i and the motion (1.1).

As yet, we have left unspecified the definition of the momentum pair (i,h) that appear in (1.2). 1 view such a definition as a constitutive spe­cification, for an altered definition choice will certainly result in a different mechanical behavior. The standard definitions for these quantities, within the present context, result in the equations

l = pv, h = A'q, (1.4)

where the linear and angular velocities are defined respectively by

v = r, q = 0e3 . (1.5)

In (1.4) p denotes the linear mass density of the elastica, i.e., the mass per unit of length of C and K is the rotational inertia density of the "cross

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section" of the slender body that is modelled by the elastica. In (1.5) 0 denotes the angle between the unit tangent t of C and e i , Fig. 2a. In general both p and K are real valued positive functions of s\ in particular for a homogeneous rod, they will be constants.

If we write

t = r ; '= cos0ei+sin-0e2 , 0-6)

which is an equation that expresses the definitions of tlie quantities therein, then either by definition or by a calculation consistent with the FRENET formulation for curves we have

K = 0' . (1.7)

We suppose the elastica to be in a static or equilibrium state. By this we mean that

i = h = 0 . (1.8)

Then from (1.2)i,n = n0 = noe, where no, n0, e are a constant vector, constant scalar and costant unit vector, respectively. If we write

e = cos i]e\ + sin ije2 , (1.9)

then from (1.2)3, (1.3), (1.6) and (1.9) we obtain

DO"- n0 sin(0 - TJ) = 0 (1.10)

as the governing equation for the shape of a homogeneous elastica. The stan­dard transformation K' = Kdn/dO allows a first integration of (1.10) into the form

6' = (bcos(0 - ?;) + C2)1 /2 , b = -2n0/B , (1.11)

where C is a constant of integration. Standard transformations permit (1.11) to be integrated in terms of elliptic functions, an exercise which is now textbook material.

Elastica analysis has for the most part been restricted to the equili­brium theory, i.e, to the study of the shapes of the solutions of (1.11) and to the question of their stability. Here we are interested in the dynamics of the elastica, in particular the problem of wave propagation, but restricted to an analysis of the wave front. There is a well established body of knowledge associated with this aspect of the wave propagation problem, it being gene­rally developed within the framework of the theory of singular surfaces, cf TRUESDELL & TOUPIN [18]. For the elastica, the theory descends to that of

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a moving singular point on C. A singular point w = x(sw) is defined by the mathematical property that there occurs thereat a discontinuity or jump in some field variabile on C. If the point w moves on C then it is called a wave and it models the wave front, whilst the jump that it carries characterizes the disturbance being propagated into the elastica. For the elastica problem, we assume that K is discontinuous across w while 0 is continuous there. We introduce the notation

[«] = *;--/<+ (1.12)

for the discontinuity, where

« ± = \\m K(SW ±e) (1.13)

so that K+ and K~ are the limiting values of K just ahead and just behind the wave, respectively, Fig. 3. Let <j) be any field variable on C. We shall say/ that there exists an n-th order wave with respect to <j) if <j)(n) is discontinuous while <^n-1) is continuous at w. Here </>(") denotes an 7i-th order derivative, either with respect to s,t or some combination of the two. By virtue of (1.7) the elastica then bears a first order 0 wave.

Fig. 3 - The wave front

We assume the motion of w along C to be described by the equation

s w =sw(t) . (1.14)

Then the disturbance is propagated relative to the material points of the elastica with speed U defined by

U = sw ; (1.15)

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U is called the intrinsic sjxed of wave propagation. In order to obtain a straightforward and simplistic analysis of the calculus of the discontinuity we introduce an s-t diagram, Fig. 4, in which s and t are regarded as rectangular cartesian coordinates with es,et denoting the associated unit vectors. We define a world wave curve Cw in s-t space by the differential equation

w = [ /e ,+ e* , (1.16)

where w, the position vector in s — t space, corresponds to the space point w. 1 The solution of (1.16) that gives the wave curve Cw is evidently

w = sw(t)es +tet .

We observe that the limiting values n* of K at w are independent of the transversal to the world wave curve along which the point w is approached.

The directional derivative along Cw

of any continuous function on C, in par­ticular 0, is clearly given by

(1.17)

0° = 0 + U0' (1.18) s a

e<

and is itself contiuous across Cw despite the jumps on the right hand side of the equation. The ° operator has the kine­matic interpretation that it measures the Fig. 4 - The s-t diagram rate of change as seen by an observer that

moves with the wave2. The continuity of 0° is a consequence of the easily proved formula

[0°] = [ C (119)

and the assumption [0J = 0, which results in

[0] = -U[01 , (1,20)

A world curve Cw, dual to that defined by (1.16), may also be defined in a spacetime manifold W=£xTZ. Indeed, in this case the defining differential equation would be

w = r(«u ,(t) ,0+e> •

We do not develop this point of view here.

The operator is often referred to as a convective time derivative, in this case the convective velocity being U.

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a relation between the jumps in first derivatives of 0 which is commonly called the first order compatibility relation for 0.

The primitive function in our theory is the motion (1.1). It cannot be discontinuous as this would imply a fracture of the elastica. However, first order waves in the motion r(s,t) are admissible. These are called shock waves. An equation completely analogous to (1.20) holds in this case. From (1.5)i, (1.6)i, this equation of compatibility may be written in the form

[v] = -V[t] . (1.21)

The situation in which v is continuous but v is not is said to define an accelemlion wave. In this instance a double application of (1.17) to r results in a second order compatibility equation for acceleration waves, viz.,

We remark that the elastica assumption of a first order wave in 0 is, by (1.21) and (1.6)2, equivalent to the assumption of an acceleration rather than a shock wave in the elastica motion. Nevertheless, as we shall see, this wave is characteristic of a shock rather than an acceleration wave.

The equations of linear and angular momentum balance (1.2) are invalid at a shock wave, for the derivatives appearing therein are meaningless at the wave. We therefore require appropriate expressions of these laws at the wave front. Consider the segment on C, shown by brackets in Fig. 3, defined by sw — As < s < sw + As. Since w is moving, the element consists of two parts, each with changing dimension. The net balance laws of linear and angular momentum for the element will then consist of two parts, with due account for the inflow or outflow of momenta to these parts due to the moving internal boundary. In the limit as As —* 0 the results become exact and yield the jump conditions

[»J = - t / [ H . [m] =-(/fA'wJ . . (1.23)

An analogous calculation for the balance of mass and inertia leads immediately t o 3

M = [A'] = 0 (1.24)

3 The conservation of mass balance law for an inextensible elastica may be written as

where po(s) is a prescribed function of a for all time. Clearly, it can not tolerate a density wave, otherwise stated, (1.24), must hold. The same is true for the inertia function K.

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so that (1.23) may be rewritten as

[11] = -Up[v]y [m] = -UKM • (1-25)

We are now in a position to examine the propagation of shock waves on the elastica. As we have already mentioned, we are restricted by the assumption

[0] = O, (1.26)

for to assume otherwise would result in a situation in which K and hence m are infinite at the wave and as a consequence fracture would occur at the outset 4. From (1.26), (1.6)2, (1.21) we obtain

W ' = M = 0 (1.27)

and from (1.25)i we conclude that

[»] = 0 , (1.28)

From (1.25)2, (1.3), (1.7), (1.5), (1.20) we obtain

(B-KU'2)c = 0, c={0'] , (1.29)

which is the propagation condition. Waves with a curvature jump, c ^ 0, will then propagate with speed

U = (D/K)l/2 . (1.30)

Clearly, these are not shock waves with respect to the motion; indeed they are acceleration waves. Yet they are "shock waves" with respect to 0 and as will be seen they are in general characteristic of shock waves rather than acceleration waves or "weak" shock waves 5. The natural nomenclature for these waves is that of a curvature wave.

This situation is forced by the assumed form of the constitutive relation (1.3). If it be altered to that of an elastic-plastic material then [0fl?*O would be possible for a finite bending moment m. We are unaware the treatment of such a problem in the literature.

Nevertheless, we do not treat that case here.

Shock waves are inherently a nonlinear phenomena. The simpler situation that arises in (1.29) is a consequence of the assumed linear constitutive relation (1.3). For a more complex law of elasticity the waves of small amplitude, weak shock waves, will also satisfy (1.29); but for fmte amplitude waves (1.29) will fail to hold. However, for finite amplitude acceleration or higher order

waves, (1.29) will remain valid even in the more complex case.

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We now investigate the growth-decoy behavior of a curvature wave. To do this we take the jump of the equation of motion (1.2)2 to obtain

> i ' J = A'[0] , (1.31)

wherein we have utilized (1.4)2, (1-5)2, (1.27)i and (1.28). To proceed we need a second order compatability relation for 0. It is obtained through a second application of (1.19) to 0 combined with a first application of (1.19) to 9'. The required result is

[0J = U2{0"} - U°c - 2Uc° . (1.32)

Unlike the situation considered in (1.10) in which B is constant, we assume here that B as well as K are functions of position. We substitute (1.3), (1.7) (1.30), (1.32) into (1.31) to obtain

2c° + {((J°/U) + B°/B}c = 0 , (1.33)

along with the conclusion that {0"} is arbitrary. Since (1.30) holds along the wave front we may differentiate it to find

B°/B = 2(U°/U) + K°/K . (1.34)

We substitute (1.34) into (1.33)...-The result is a differential equation that is called the amplitude equation, viz.,

2c° + {3(U°/U) + K°/K} c = 0 , (1.35)

an equation which can be solved by quadrature. The integration yields the growth-decay law •

M£f(fr; where the subscript 0 indicates evaluation at an arbitrary point on the elastica, possibly the starting point of the wave.

Equation (1.36) speaks to the variation in [K] as the wave progresses. The associated acceleration jump varies difTerently. To see this we rewrite (1.22) in component form with the use of the equation

r" = /ep, (1.37)

where p is the principal normal of c. We thus have that

K J = (/2c, a n = r - p , (1.38)

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which produces [an] ( UK \ 1 / 2

(1.39) . [flnlo \UoKoJ

We have discussed the fact that the linear constitute relation (1.3) has the effect of simplifying the essential nonlinear character of shock waves. To show the nature of the general situation we consider a simple special case of a nonlinear elastica, one governed by the elastic law

m = BK2 , (1.40)

where B as before is an elastic coefficient which is generally a function of position. We follow precisely the same analysis as in the linear case, with the exception that (1.3) is replaced by (1.40), in order to arrive at the propagation condition. The outcome of the analysis is the relation

BIK2}-U2K[K] = 0 . (1.41)

To proceed further we utilize the identity

[«2J = [K]2 + 2[K]K+ ' (1.42)

and for further simplification we use the assumption that the elastica be ini­tially straight, which has a consequence that

K+ = 0 , «'+ = 0 . (1.43)

We substitute (1.41), (1.42) into (1.40) to obtain

U2 = (B/K)c, c?0 . (1.44)

Clearly, the speed of propagation, unlike the situation in (1.30), now depends upon the magnitude of the discontinuity, and this is characteristic of the non­linear nature of the shock wave problem. As already mentioned in Footnote 5, an analysis of higher order waves based on the law (1.40) would result in a linear propagation condition. Indeed, if we assume

M = 0, M ^ 0 , (1.45)

then (1.31) with (1.3), (1.32) yields immediately

(B - KU2)a = 0, a = [K'J = [Q"\ . (1.46)

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The amplitude equation associated with the "strong" shock problem defined by (1.44) may be obtained a manner analagous to the derivation of (1.35) for the "weak" shock case. For added simplicity we assume that both A' and D are constant. We substitute (1.41), (1.32), (1.42), (1.43) into (1.31) to obtain

2Bc(T= K(U2<T-2UC° -U°c) . (1.47)

The convected time derivative of (1.44) results in

U° = \(B/Kyl2c-ll2c° . (1.48)

We use (1.44), (1.48) to rewrite (1.47) in the form

co + ?(2VA-)1 /2*c , '2 = 0 • ( L 4 9 ) 5

This is the amplitude equation; its solution is /

c = ( c ° / 2 " IL adi) ' (150)

where c0 is the amplitude at time reference zero. Dependent upon the sign of a, which measures whether the curvature of the elastica increases or decreases, the shock amplitude will either decrease or increase, respectively. The amplitude of higher order waves, such as defined by (1.46), can be shown to satisfy a differential equation which may be classified as of BERNOULLI type. The growth-decay behavior oP waves governed by this equation has been extensively sudied by CHEN [19].

Having digressed for a brief view of the nonlinear situation, we now return to the analysis of the curvature wave problem as based on the linear constitutive relation (1.3). While we have obtained the propagation condi­tion (1.29), the amplitude equation (1.35) and its solution-the growth-decay equation (1.36), the analysis is yet incomplete, for we have .made no use of the equation of motion (1.2)i. Equation (1.2)i does impose conditions on the wave propagation problem. These we now seek. We begin with the decomposition

n = rt + 7 p , (1.51)

where r and q are the tensile and shear components of the stress n, respectively. We substitute (1.51) into (1.2)i, take its jump, and use (1.4)i, (1.27), (1.28), (1.22), (1.37) to obtain

M t + r [t'J + [q'] p + q [p'J = pU2M p . (1.52)

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Note that we have also used the conservation of mass, which for an inexten-sible elastica takes the form

/> = 0 . (1.53)

For a plane curve the FRENET equations reduce to

t ' = «p, p ' = -Kt ; (1.54)

when these are substituted into (1.51), we find in terms of the notation (1.29)2 that

W]=qcy W\ = (p(BIK)-T)c. (1.55)

Clearly, the derivatives of the tensile and shear forces have jumps at the wave front.

By a continued differentiation of the balance laws (1.2) one can find a sequence of higher order jumps of all the field variables. In particular the growth-decay behavior of the curvature derivatives [K'J, [«"] etc. may be obtained. One can employ these to produce a Taylor's series for K based at the wave w which then may be used to find the wave motion behind the front. We refer the reader to COHEN k EPSTEIN [13] for further detail.

2. The Extensible Elastica

The mathematical treatment of an extensible elastica, one in which the arc length between the material points of C can change, is more complex than that of the inextensible case. Now, the motion of the elastica will generally alter the relative locations of its material points and thereby give rise to both extension and velocity distributions tangential to C. The concepts of extension and velocity distributions are generic to all deforming continua and must be accounted for in the analysis of the extensible elastica. In the mechanics of continua it is standard practice to single out one configuration of a body, not necessarily one assumed in its motion, and to use it as a reference relative to which all other configurations are compared. We shall adopt this procedure here; we let Cn be a fixed configuration of the elastica and select it as the reference configuration. The configuration C of the elastica at time t will be called the current configuration, Fig. 5. In the previous section we implicitly used the straight elastica as the reference configuration, but as there was no extension, its sole role lay in providing a reference with zero curvature, which thereby allowed K to measure the deformation of the current state.

As in the previous section we shall confine attention to the plane elastica. We note that much, but not all of the analysis, will carry over to the spatial

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elastica. The reference configuration Cn is given by a smooth curve

r/i = VR(SR) , (2.1)

Fig. 5 The reference and current configurations

where SR is the arc length parameter of Cn- The motion of the elastica is still defined by (1.1), provided that it is supplemented by the relation, invertible in sRr

s = s(sR,t), .' (2.2)

which allows one to follow the motion of the individual material particles in space. The numbers SR, which label the material particles, play the role of a converted coordinate system for the elastica, but we note that while SR is arc length in the reference configuration, unlike the situation in the inextensible case, it plays no such role in the current configuration. Indeed, the relation (2.2) between s and SR yields the extension of the elastica through the quantity stretch, which is defined by

\ = s' , (2.3)

where ' denotes (d/dsR)t. We shall also utilize the notations to denote (d/ds)t and ' to denote (d/dt)sR.,

I l l

If we combine (1.1) and (2.2) then the motion is given in terms of the convected coordinate by

r = r(sR,t) = r(s(sR,i)}i) . (2.4)

The velocity distribution is then defined by

v = r = (dr/dt)9 -f ts , (2.5)

where the unit tangent t to C is

t r z ^ r ' A " 1 . (2.6)

The unit tangent to Cn is of course given by

tn = V'R • (2.7)

In Section 1 we presented a mundane treatment of the balance laws. A modern and more sophisticated version of these require that we give them in their global rather than their local form at the outset. We do this relative to the reference configuration C^, while leaving the dual formalism relative to the current configuration C for the reader to supply. The balance laws of linear and angular momentum are respectively

/_ n*+ / fn=( I \R) , JdCn JCn \JCn / / x

(2.8)

where Cn is an arbitrary portion of Cn, d£n is its boundary, i.e., the two end points sm, sR2 of the interval (sy?i, sR2) which defines Cji, and fR) cR

the referential body force and couple, respectively; the latter quantities had been omitted in the earlier discussion. Note that in keeping with modern usage we omit the differential elements from beneath the integral signs. The referential forms of the stress (n/?, m^) and momenta (\R, hR) are related in a simple way to these in the current configuration, viz.,

an = n, in/? = in, \R = 1A, hR = hA , (2.9)

so that all vectors are carried by parallel propagationTrom one configuration to the other with the latter two rules compensating for the effect of the stretch.

/ _ (mR + r x nR) + / (cR + r x fR) =

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Clearly, the transformations (2.9) are defined to keep the balance laws (2.8) invariant under change in configuration.

For completeness we append to (2.8) the global forms of the mass and inertia balance, viz., -

CW=o' (1**)=°' (2io)

along with the transformations to the current state

PR = pK KR = KX . (2.11)

These guarantee the conservation of mass and inertia in any motion. To extract the local consequence of (2.8)i we utilize the formula

/ nR = nR(sR2)-nR(sm) = I n'R (2.12) J dC-jz J C ft

as well as an analagous transformation for n\R. With the proviso of the continuity of all quantities under the integral sign, a standard argument which involves the shrinking of CR to zero results in

nk + f« = i/r. (213)

The analagous result for (2.8)2 is

m^ 4- At x nR + cR = hR , (214)

but to arrive at (2.14) we have utilized (2.13). Inherent in our treatment of wave propagation in the elastica shall be

the assumption that the singular point moves with a continuous speed through both the reference and current configurations and with a continuous current velocity in space. The moving point wR in CR may be located by its.arc length coordinate

sRw = sRw(t) . (2.15)

In an sR-t world space diagram the wave defines a curve whose slope UR is the speed of propagation in the reference configuration:

UR=sRw , (2.16)

where the superposed dot when applied to functions of t only, means the ordinary time derivative. Analagous to the situation in the previous section we define a convective time derivative of a function <j> = <f>(sR,t) by

<£o = 0 + c W . (2.17)

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The wave w in the current configuration C defines a continuous curve in s-t space whose abscissa is given by

sw=s(sRx0(t),t) } (2.18)

which is obtained by substitution of (2.15) into (2.2). The relative speed of displacement in C is defined by

sw = s + At/7? = C • . (2-19)

We note that s£, measures the wave speed relative to the time dependent arc length coordinate system in the current configuration. By assumption

[fl°] = [£/*] = () (2.20)

and hence from (2.19) we have the first order comparability condition

W = -UR[\] . (2.21)

The wave w on C has a path in space which is given by

Vw = r(s{sRW(t),t) = r(sRw(t),t),

whence its velocity \w has the forms

vw = rw = r(XUR + s) + (dr/dt), = s°t + (dr/dt)$ = Y'UR + v . (2.22)

We write the decomposition

v = t)(t + vpp, yw — tit 4- wPp . (2.23)

and call u the speed of displacement of the wave. From (2.22), (2.23), it is given by

u = s°w+(dr/dt)s.t = U + vt } (2.24)

where we have defined the intrinsic speed of wave propagation

U = XUK . (2.25)

If we rewrite (2.24)2 as

U = u - vt , (2.26)

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we see that U measures the tangential speed of the wave relative to the material particles of the elastica. By (2.24)i, (2.5)2, (2.23)i, equation (2.26) becomes

U = s°w-s, (2.27)

a result that is consistent with (2.19) and one which exhibits the intrinsic character of U.

The intrinsic speed of propagation is discontinuous. Indeed,

[U] = - W = [ti] - lvt] . (2.28)

For the inextensible elastica we have employed the requirement

[0] = O*=>[t] = [p] = O. (2.29)

From hereon, we invoke this assumption in the extensible case, so /that the subsequent results no longer hold in general, e.g., they would not apply in the problem of a general shock wave on a string or in the next section which employs an alternate model of a rod. Since [v^] = 0, we obtain from (2.22)4, (2.20)i that

b/dt),] = 0, M = KJ = 0, (2.30)

so that [U] = -Is] = -lvt] . (2.31)

To derive the consequence of the global balance laws (2.8) when the integrands have finite jumps at the wave front, we break the region of integra­tion into two parts CR = C^UC^ with a common boundary at the singular point WR. We employ the standard formula for the derivative of an integral with variable limits of integration to obtain

iti")=tty)=b[R-u«' (2.32)

/ ""l/H = L i*+ ««,'£ , (2.33) JSn, J JCR

We substitute (2.12) and the sum of (2.32), (2.33) into (2.8)i, assume that all integrands are bounded near the singularity and take the limit as both sR\ and SR2 approach SRW to obtain the first of

M = -tfjiPid, [mul = -URlhR} . (2.34)

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The second of equations (2.34) is derived in an entirely similar fashion. The constitutive relations of the previous section require modification

for the extensible elastica. For the inextensible case no such relation was given for n, this stress being regarded as purely reactive. Here, we assume

iirt = E\t + qp , (2.35)

where E — E(SR) is an elasticity coefficient and q is a reactive shear force to be determined from the equations of motion. For the remaining constitutive variables mR, \R and \\R we employ (2.9), (2.11), (1.3), (1.4), (1.5) to find

mR = Bne3, \R = pR\, \\R = KR0e3 .. (2.36)

We emphasize that B,pR,KR are function of sR only. The procedure to acquire the propagation condition follows that em­

ployed in Section 1., but some changes are required. Whilst (1.27)i is still true, (1.27)2 is no longer valid. To see this we consider the first order com­patibility relation

[r] = -t/,*[r'] <=> [v] = -UR[\] t . (2.37)

From (2.37)2 and (2.23)i we conclude that

[vtl = -UR{\1 M = 0 . (2-38)

We substitute (2.23)i, (2.35), (2.3CJ, (2.38) into (2.34)i to find

(E-pRl/foe^O, M = 0, e = [AJ. (2.39)

An analagous process, which is entirely the same as that in the first section, derives from (2.34)2 the result

(B-KRU2R)c = Q . (2.40)

Equations (2.39)i and (2.40) are the propagation conditions for the ex­tensible elastica. They define two waves; stretch wave:

e^O, c = 0, Un = E/PR; (2.41)

curvature wave: e = 0, c-^0, U2

R = B/KR. (2.42)

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The decay of the curvature wave has already been analyzed in Section 1. For the stretch wave we employ (2.13) with [fa] = 0 to obtain

M = pair] • t, [T)K + fo'] = pRlr] p , (2.43)

since [q] — 0. We now require the second order compatibility relation for r, the dual of (1.32) in the reference configuration. It takes the form

(244) = U2

R{[\'\ t + K[A] p) - U°R[\] t - 2tfn([A]°t + [A] t°) .

We note further that t° • t = 0 , (2.45)

so that (2.43)i results in

(E' + PRU°f{)e + 2pIiU/leo = 0 (2.46)

upon the use of (2.39)i to eliminate the term in [A'J. Next we differentiate (2.41)3, and use E' = E°/UR to obtain

E' „U 0

VRPR UR PR

which, when substituted into (2.46) gives the reduced form of the amplitude equation

3 ^ + ^ + — = 0 . (2.48) UR PR e

Integration of the latter gives the growth-decay law for stretch waves, viz.,

An analysis that parallels the one of the previous section with {CR] — 0 in (2.14) yields the growth-decay law for curvature waves, viz.,

rO""(S)" Equation (2.43)2 has not been exploited at this stage. If we write

t ° p = a , (2.51)

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then the normal component of (2.44) used in (2.43)2 provides

h'] = -2URPRae\ (2.52)

for the stretch wave. Similarly, [q'] ̂ 0 for the curvature wave. Indeed, one can find the formula

fa'] = PRDXC/K . (2.53)

We remark that the amplitude equations for stretch and curvature waves are derived from (2.13) and (2.14), respectively. Thus for stretch waves, (2.14) is unexploited; likewise for curvature waves (2.13) is unused. The use of these equation will produce what we call the induced waves.

For the stretch wave, we obtain from (2.14), (2.35), (2.36) (2.39)2

(2.41)2, (1.32) that M = 0 , (2.54)

i.e., the induced curvature wave is zero. For the curvature wave, we obtain from (2.13) a similar result, namely.

[A'J = 0 . (2.55)

Thus, for the constitutive relations employed in this analysis the induced waves vanish l.

3 . T h e Cosse r a t R o d

We move now from the primitive model of a rod given by the elastica to a more sophisticated and modern model-the one proposed by the COSSERATS [20]. They recognized that a rod had more structure than could be accounted for by a geometrical curve. They realized that the transverse sections of the slender body, the cross sections of the rod, would play a key role in its mechanical behavior. Thus, they introduced the notion of a directed curve as a model for a rod. A directed curve is nothing more nor less than a curve with an orthonormal triad field assigned to it, the idea being that the curve would model the rod axis, while two of the directors define the plane and principal axes of the cross section, the third director being the normal to the cross section. ER1CKSEN & TRUESDELL [21] extended the COSSERAT'S idea to a deformable director triad, thereby removing the restriction that

These results implicitly assume that the region ahead of the waves is unstressed and undefor-med. For constitutive relations which couple the effects of stretch and bending these results would no longer pertain. We leave the details of the analysis for the reader to supply.

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the cross sections be rigid. Theories of rods based on these concepts have been elaborated by ANTMAN [22]. Here we follow the theory as developed by WHITMAN k DESILVA [23] for the rigid director case and we employ the terminology "Cosserat rod" to refer to the rod theory based on this model.

As in the previous two sections, for simplicity of analysis, we confine attention to a plane rod. Thus, the directed curve V consists of a plane curve C that lies in V, and an orthonormal triad field T on C, one vector of which is always orthogonal to V. The reference and current configurations are defined respectively by

*R = *n(sR,t), dRa = dRa(sR,t) , (31)

and r = r(sn,t), da = da(sn,t) , (3.2)

where da}a = 1, 2, 3, are the directors. In (3.1), (3.2) the quantity SR is a convected coordinate that measures arc length only in the reference confi­guration DR. As before we shall use s to denote arc length in the current configuration V and define the stretch A by (2.3). The directors being orthonormal of course satisfy

dRa • dRf, = Sab, da db = Sab , (3.3)

where Sab is the Kronecker symbol. Furthermore, the restriction to the plane rod case requires

dR3 = d3 = e3 , (3.4)

so that (d2 ,d3) define the cross section; d! is orthogonal to it. We let 7 be the angle between the director and Frenet triads, Fig. 6. Then

di = cos7t — sin 7p, d2 = sin yt -f cos 7p . (3.5)

IR = dm

Fig. 6 - The various bases and angles

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We assume the cross sections to be orthogonal to the axis in the reference configuration, viz.,

dm = t'n, dfl2 = PR • (3.6)

Clearly, j measures the trasverse shear deformation. We define the flexure f of the rod by

/ = 1 ^ = 0 - 7 , (3-7)

fR = KR = 0'R, (3.8)

where ip defines the orientation of the directors relative to the fixed frame e a , Fig. 6. The flexure is a measure of bending. But it is based on the rate of change along the axis of the cross sections, rather than the curvature of the axis. From (3.7) we see that it involves both the curvature of the axis and the angle of shear. For the analysis of wave propagation we shall employ "linearized" constitutive relations that are based upon the assumption of small strain variables. We introduce the extension e by the definition

e = A - 1 , (3.9)

and assume that \e\ <C 1, |T| <€. 1. Next we define a displacement pair (u,b a) such that x

r = TR + u, da = dRa + b a . (3.10)

The restrictions at hand require that

U3 = 0, b a = (pe3 x dRa, <j) = t/}- OR , (3.11)

where \<j>\ <C 1 is the angle of rotation of the cross sections from their reference positions, Fig. 6.

If we write the decomposition

u = uttR -f UPPR , (3.12)

then

e = (r ; • r ' ) 1 / 2 - 1 « tu • u ; = u't - KRUP , (3.13)

7 WS1117 = t • d2 « PR - u ' - <t> = (u'p + KRut) - (j> , (3.14)

where (3.5), (3.6), (3.9), (3.10), (3.11), (3.12) have all been used.

There should be no confusion between the u used for the wave velocity in Section 2 and that used here for the displacement vector.

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We propose the following constitutive relations for the case at hand:

ii/? = Eetn + Gypn, ITIR = Bfe3 . , (3.15)

These can clearly be seen to be an extension of the elastica constitutive equa­tions to include the effect of small shear as well as a restriction of the same equations to small extension. Generalizations of (3.15) that couple the strains (e,y,f) in linear relations can be made in an obvious way, but considerably complicate their application. We note that such analyses have indeed been carried out in the references of the introductory section. The constitutive relations (2.36)2,3 for the momenta are taken to be

lft = PRV, lift = A'ft^e3 ; (3.16)

only the second of the above is changed from its previous form, being propor-tional now to the director angular velocity. The precise interrelation between (3.16) and (2.36)3 can be found from (3.7), (3.11)3, viz.,

<j> = 6-j, (3.17)

a relation which depends not upon the size of the deformation. The field equations to be employed in the wave propagation analysis are

precisely (2.34) and (2.13), (2.14). This means that the stress principle that was invoked in the case of the elastica is assumed to apply in this case as well-the implication being that the pair ,(11,111) that act on points of the axis are assumed dynamically equivalent to the interactions of the cross sections of the rod. As we are restricting consideration to small deformations the analysis to be carried out belongs to the realm of weak shock waves., We begin discarding (1.26). This requirement is no longer valid-the model has been changed. We replace it by the assumption

[Vj = ty] = 0<=>[da] = 0 , (3.18)

a condition which guarantees that the rod not be split. From the first order relation

W = -Unlv'l (3.19)

and (2.23)!, (3.10)i, (3.13), (3.14), (3.18)2 we find component relations

[vt] = -URlu'.tR} = -URle] , (3.20)

[vP] = -URW • Pftl = -UR[J] . (3.21)

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We note from (3.18)i and (3.7)2 that a jump in shear is accompanied by a corresponding jump in the tangent vector to the rod axis prescribed by

N = [7] = K 1 + **[«(]. (3.22)

For this reason wave that carries a jump in shear will be called a shear-kink wave. This fact holds as well in the strong shock case wherein it contributes to the overall difficulty of the nonlinear problem. In the case of an extension or stretch wave one has

[e] = [A] = [ti'J - KRIUP] . (3.23)

The propagation condition is obtained as before, this time from the substitution of (3.29), (3.20), (3.21), (3.15), (3.16) into (2.34). The result is

(E - pRU2R) [e] = 0, (G - PRUR)[I] = 0, (B - KRUR) [/] = 0 . (3.24)

To obtain (3.24) we have employed the compatibility relation

M = M = -URW\ = -URW\ = -UR[f] , (3.25)

as follows from (3.7) and (3.11)3. For notational convenience we write

e = [el g = bl c=U), (3.26)

which apart from the last is consistent with previous notation and entirely consistent when 7 = 0.

The propagation condition yields the following classification of waves; extension wave 2:

e?0, g = c = 0, UI=E/PR; (3.27)

shear-kink wave:

0 5*0, e = c = 0, UR = G/PR ; (3.28)

flexure wave:

c^O, e = g = 0tUR = B/KR . (3.29)

The amplitude and growth-decay equations are obtained as before. We omit the details, e.g., for the shear wave the result is

JL = (ilm)3l2(em)u* . ( 3 . 3 o) ffo \Un / \Pn J

2 We observe that for an extension wave there exists an associated density wave, its form being

precisely [p]=-»p.|{[e].

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The induced waves may also be obtained as before. The induced wave associated to the extension wave has components

[VI = 2«ne/(l - G/E), [/'] = 0 ; ^ (3.31)

the induced wave associated to the shear wave has components

[e1 = -2**0/(1 ~ G/E)% [ / I = g/{(KR/pR) - (B/G)} . (3.32)

We note that to obtain these results we have used

t-R - URKRPR* P°R = -URKR^R • (333)

Finally, the wave induced by the flexure wave has components

M = 0, h ' ] = c/{(GKR/PRB) - 1} . (3.34)

These results can be extended to spatial Cosserat rods where there exist six wave types which include the effect of rod twist. More generally, for directed rods which allow cross sectioned deformation, the results extend to the situation of nine wave types. In all cases, for weak shock waves, the formulas for growth-decay have the format given in equation (3.30). Similar results have been obtained for acceleration waves; the growth-decay laws in that situation for the linear problem all being of the form

where a denotes the amplitude of an arbitrary second order spatial gradient jump.

Acknowledgement

The autor wishes to acknowledge the support in part of The University of Manitoba and the Natural Sciences and Engineering Research Council of Canada. The author also expresses his thanks to the Seminario Matematico di Torino for inviting him to prepare this paper.

R E F E R E N C E S

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[2] GREEN, A.E. k N. LAWS, "A General Theory of Rods", Proc. Roy. Soc. A293, (1966), pp. 145-155.

[3] COHEN, H. k A.B. WHITMAN, "Waves in Elastic Rods", Journal of Sound and Vibration, 51, 1, (1977), pp. 283-302.

[4] COHEN, H. "Shock Wave Decay in Elastic Rods", Iranian J. Science k Technology, 7, (1978), pp. 83-91.

[5] COHEN, II. "Decay of Acceleration Waves in Elastic Rods", Archive for Rational Mechanics and Analysis, 67,2, (1978), pp. 151-163.

[6] COHEN, H. "Wave Decay in Nonhomogeneoiis Elastic Rods", Archive of Mechanics, 30, 2, (1978), pp. 175-188.

[7] WHITMAN, A.B. k II. COHEN, "Constitutive Equations for Curved and Twisted, Initially Stressed Elastic Rods", Acta Mechanica, 30, (1978), pp. 237-257.

[8] BACHMAN, R.C, k H. COHEN, "Wave Propagation in Elastic Rods with Multiple Wave Speeds", Math. Proc. Camb. Phil. Soc , 86, (1979), pp. 179-191.

[9] COHEN, II. k M. EPSTEIN, "Acceleration Waves in Constrained Elastic Rods", Archive for Rational Mechanics and Analysis, 72, (1979), pp. 141-154.

[10] COHEN, H. k M.EPSTEIN, "Acceleration Waves on a Class of Rods", Recent Developments in the Theory and Application of Generalized and Oriented Media, Calgary (1979) pp. 191-194.

[II] WHITMAN, A.B. k II. COHEN, "Waves and Instability in Simple Ortho-tropic Elastic Rods", Acta Mechanica, 34, (1979), pp. 257-262.

[12] COHEN, II. k A.G. TALLIN, "Waves in Thermo-viscoelastic Rods", Acta Mechanica, 42, (1982), pp. 85-97.

[1.3] COHEN, II. k M. EPSTEIN, "A Note on Nonlinear Wave Propagation", Mechanics Research Communications, 10, (1983), pp. 37-44.

[14] COHEN, II., "Dynamics of Elastic Strings", Developments in Engineering Mechanics, Elsevier, (1987), pp. 25-49.

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[15] PASTRONE, F., Onde di discontinuita nei continui elastici sottili, Qua-derni Inst. Mat. Appl., Univ. di Pisa, (1983/3).

[16] PASTRONE, F. k M.L. TONON, "Wave Propagation in Elastic Rods", Journal de Mecanique Theorique et Appliquee, 5, 4, (1986), pp. 615-627.

[17] MOLINA, C. k F. PASTRONE, "Wave Propagation in Thin Elastic Bo­dies", I.U.T.A.M. Galway, (1988), pp. 1-6.

[18] TRUESDELL, C. k R.A. TOUPIN, The Classical Field Theories, Hand­buch der Physik, Vol IIJ/1, Springer-Verlag, (1960).

[19] CHEN, P.J. Growth and Decay of Waves in Solids, Handbuch der Physik, Vol. VIa/3, Springer-Verlag, (1973), pp. 303-402.

[20] COSSERAT, E. k COSSERAT, Theorie des Corps Deformable, Paris, Her­mann, (1909).

[21] ERICKSEN, J.L. k C. TRUESDELL, "Exact Theory of Stress and Strain in Rods an Shells", Archive for Rational Mechanics and Analysis, 1, (1958), pp. 295-323.

[22] ANTMAN, S.S., The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Berlin, Heidelberg, New York, Springer, (1972), pp. 641-703.

[23] WHITMAN, A.B. k C.N. DESILVA, " Stability in a Linear Theory of Elastic Rods", Acta Mechanica, 15, (1972), pp. 295-308.

Harley COHEN Department of Applied Mathematics Faculty of Science - The University of Manitoba Winnipeg, Manitoba, Canada, R3T 2N2

Pervenuto in redazione il 7.5.1990