A note on infinitesimal shear waves in a finitely deformed elastic solid

Inl. J. Engng SCI Vol. 22, No. 7. pp. 823-827, Printed in the U.S.A.

1984 0

A NOTE ON INFINITESIMAL FINITELY DEFORMED

002@-7225184 1984 Pergamon

$3.00 + al Press Ltd.

SHEAR WAVES IN A ELASTIC SOLID

R. J. TAIT, J. B. HADDOW and T. B. MOODIE Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2Gl Abstract-In the present note we discuss a class of small dynamic deformations imposed on a large static deformation. A semi-inverse method is used to obtain the equations of motion for the small deformations in a simple manner. The results are used to extend known results for a tube of hyperelastic material subject to inflation and extension, when additional telescopic and rotational shears are applied.

INTRODUCTION

theory describing small dynamic deformations imposed on a large elastic deformation is well known [l-3], it can be complicated. If the particular class of deformation to be considered is known a priori it may be possible to simplify this by proceeding in a semi-inverse manner. That is, one can write down the full equations for deformations of the class to be considered, and by inspection, determine which perturbations are possible so that the invariants and equations determining the static elastic problem are not altered, to the same order of magnitude as the perturbation. We attempt to apply this procedure to a tube of hyperelastic incompressible material subjected to inflation and extension, together with deformations consistent with rotational and telescopic shears, in a number of different cases. In particular, in this way, we obtain an extension of results of Kurashige[4,5].

ALTHOUGH

THE

GOVERNING

EQUATIONS

We consider an isotropic incompressible hyperelastic solid with x, X denoting the spatial and material description, respectively, of a particle X. (I, 19,z) and (R, 8, z) are polar coordinates at x, X, respectively. We wish to examine a special class of motions, from an undeformed initial state, for which the displacements are given in the form r2 = R2/IZ + K,

8 = 0 + A(R, t),

z = 12 + B(R, t),

(2.1)

where 1, K are suitable constants. Evidently, apart from a simple extension and inflation, the displacements depend only on the radial coordinate R, and time t, through the functions A and B. It is a straight forward matter to compute, from eqn (2.1), the deformation gradient tensor F, the left Cauchy-Green tensor B = FF’, where the upper case T denotes the transpose, and the inverse B-l. Unless otherwise specified in the following we use physical components of the tensors involved. The invariants of B are given by Z, = I2 + r2/R2 + R2/A2r2 + r2A’2 + BJ2, Z2= Am2-I-R2/r2 -I- A2r21R2+ 122r2A’2+ r2Bf21R2,

(2.2)

and, in accordance with the incompressibility condition Z, = 1. Here we have written A’ = aAlaR. For future reference we introduce the notation 4, 4 for the values of Z,, Z2, given by (2.2) when A z B s 0, If W(Z,, Z2) denotes the strain energy function, we take the Cauchy stress in the form T = -pI + 2W,B - 2W2B-I, where we have written W, = i3W/al,, a = 1,2. p is a scalar undetermined 823

(2.3) function and I

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R. J. TAIT et al.

denotes the unit tensor. It now follows that the physical components

of T are given by

T( 11) = -p + 2W,(R2/J,‘r2) - 2W2(i12r2/R2+ A2r2At2 + r2B”/R2), T(22) = -p + 2 W&-‘/R’ + r2A”) - 2 W2R2/r2, T(33) = -p + 2 W,(n’ + B”) - 2 W,/L’,

(2.4)

T( 12) = @RA’, T( 13) = YB’, T(23) = 2 W,rA’B’, where @ = 2w,/n + anw,, Y = 2 W,RIlr

(2.5)

+ 2 W,r/LR.

With the stresses given by (2.4) the equations of motion may be written as dT( 1l)/ar + (T( 11) - T(22))/r

= - prk’,

aT( 12)/ar + 2T( 12)/r = pr& aT(13)/ar

+ T(13)lr

(2.6)

= pii,

where . E a/at, and p is the constant density, and we have taken p independent The last two equations in (2.6) may be written in the alternate form

of 8, z.

a(@RA’)/& + 2(@RA’)/r = pr& a(YB’)@r + (YB’)/r = pti.

(2.7)

Consider, first of all, the case where A, B are small perturbations, of order E, on a static extension and inflation. If we consider the equations correct to order c, it is clear from (2.2) that Z,, Z2take the values 4, &, since A’, B’ appear only as quadratic terms. Similarly the expansion of W,, W, about the static case reduce to p,, w2 where fi = a(&, 4). As a result (2.4) shows that the normal stresses T(ii) take their static values, provided p has been chosen to satisfy the static problem. The first of (2.6) to this order is then automatically satisfied. As an example consider the case of a cylinder, inner radius Ri, outer radius &, so that 0 < Ri < R < R,,, which is inflated and extended, and appropriate values for 1, K, p are given. Small torsional and telescopic shear stresses are then applied to the inner surface R = Ri at t = 0. If K = 0, the case of simple extension, (2.7) reduces to simple wave equations, and the axial and telescopic shear waves travel outward with constant Lagrangian velocities [(2V, + 2n9v2)lp]“2,

[(2Dv, + 2F2)/p3,3’211’2,

(2.8)

respectively. If K # 0, we have the static case of extension and inflation and the eqns (2.7) reduce to the cases considered by Kurashige[4,5]. To obtain the equation in [4], for example, set

and note that 2m,/n + 2nmz = (r(l1)

- ~(22))/(R21~2r2 - lr2/R2)

Note on infinitesimal shear waves in

the first of (2.7). For this case the Eulerian wave speed is Ci = (2m,/n + 2A.~z)R2/plr2,

and the Lagrangian wave speed is C; = 12r2/R2C;, this latter speed C,, reducing for a neo-Hookean material to the shear wave speed (p/p)“’ where ,U is the shear modulus for infinitesimal deformation from an undistorted state. Next, we consider the effect of A(R) being finite and B(R, t) of order 6. We consider the same long cylinder as above 0 < Ri < R < R,,, which is extended, inflated, and whose inner surface Ri is rotated by a finite amount A(Ri). A small telescopic shear B(R, t) is now imposed by deforming the inner surface by an amount B(R, t). We refer to the xarious quantities involved, when evaluated at B = 0, by superimposing a double bar, thus Z<1)= Z,le=o. We look first at the static problem, B E 0. In general the deformation will not be a controllable one, and it will then be necessary to specify the form of the strain energy function W(Z,, I,). Since A is a finite static displacement the first of (2.6) is satisfied by writing T(11) = -

R (T(11) s4

- T(22))R

dR/Ar2.

(2.9)

Taken in this form, (2.9) implies that the inner face is free from normal stress, and a constant tension or compression must be applied to the outer face. Alternatively the constant of integration can be chosen so that the outer face is free of normal stress. With 2, K fixed (2.9) may be written in the alternate form

T(l1)

= +

L2r4- R4 r2A’2 +

12r2R2

R@(R) dRlr2,

(2.10)

where @ is given by (2.5) evaluated at ?,, T2. It is clear that even if K = 0 in this case T( 11) # 0. In addition, from the nature of the specified deformation (2.1) it is clear that the ends of the long tube must be maintained plane. The second of eqns (2.6) requires T( 12) = k/r2 for suitable constant k. Thus the deformation @(R)RA’(R) For a Mooney-Rivlin

(2.11)

is possible if A(R) can be chosen to satisfy = k/r2.

(2.12)

material w = p/2(b(Z, - 3) + (I - b)(Z, - 3)},

(2.13)

we have 0 = p{b/l + A(1 -

b)},

where b is a constant, 0 I b s 1. Equation (2.12) then gives A =

A,, + (k/K@) ln(R/I”‘r), A, - lk/2R2@,

K # 0 K = 0,

(2.14)

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R. J. TAIT et

al.

so that if A, K are fixed, A,, k can be chosen to give A a prescribed value at R,, and zero value at 4. If a small telescopic shear is now applied to the face at Ri, the resulting wave motion is governed by the third of (2.6) or equivalently the second of equations (2.7). Since, for the Mooney-Rivlin material !I’ is independent of A we recover the same wave equation as in the case when A is zero. Since B is small of order t, the static solution is not affected to order c by the terms involving B. Thus, as far as wave propagation of the perturbing telescopic shear is concerned, it is not affected by the additional rotational shear, for a Mooney-Rivlin material. In order to consider a properly non-linear material we look at a second order strain energy function and take this in the form suggested by Ishihara et a1461 (2.15) with p as before and b, y are positive constants with 0 I b I 1. In this case the eqn (2.12) reduces to the cubic equation q3+Hq+G=0,

(2.16)

where q= RA’, H = R2f5/2~y~3r2,

(2.17)

G = - kR2/2pyR3r4, and & is again evaluated at 4, &,.There is an explicit solution to (2.16), positive or negative as k is positive or negative, and for large positive R q=RA’

N 1k/QoR2,

(2.18)

where Q0 = ~(bA_’ + (1 - b)lZ + 2y1,(21+ A-* - 3)). We note also that the quantities @, Y evaluated at _zrk, fk, k = 1, 2, are positive for the strain energy function given by (2.15). Consider now the effect of a small perturbation B(R, t) on the configuration. We have, from the second of (2.7) Z” + R(2 - ~r’iR~~~‘l~r2 - R%/1’r4 = pR~l~r~.

(2.19)

where we have written z = T(13) = PB’.

(2.20)

If initially the tube is quiescent and a small telscopic shear is applied at the inner face we have T(R,t)=O,

t r0

(2.21)

z(Rf, t) = QWX with H(t) the unit step function. It is now clear from eqn (2.18) that in all cases the Lagrangian wave speed will depend on A through p. (2.19) does not appear to have accessible solutions. Some information can however be obtained from a wavefront expansion. We express r in the form

827

Note on infinitesimal shear waves

with (dF,(z)/dz) = Fn_l, for n 2 1, F. denoting the wave form. On substituting in eqns (2.19) and (2.21) we find F,(t - S(R)) = H(t - S(R))(t - S(R))‘@!, S(R) =

R (pR/Ir~)1’2 dR, s Ri

(2.23) (2.24)

with the Lagrangian wave speed being given by c2 = h!?/pR,

(2.25)

and the wavefront by t = S(R). The leading coefficient is given by z,(R) = zo(Rr’~/Rir3Zi)“‘,

(2.26)

where the index i denotes the value at RP Further coefficients can then be obtained from the recurrence relation

G+

dR) = i (~/P)1/2(R~/r3)1~4Q,+ dR),

(2.27)

where Q,+,(R) =

R (r5~/R3)‘14{~~ + R(2 - Ir2/R2)~Jh2 s4

- R2q,/12r2) dr.

(2.28)

The expressions given above described outgoing waves. If the outer boundary & is sufficiently large we can use the estimate (2.18) to see that for large R, the speed given by (2.25) reduces to that given previously in the case when A = 0.

CONCLUSIONS

In the preceeding paragraphs we have shown that for a certain class of small dynamic deformations imposed on a large static elastic deformation the equations of motion may be obtained in a simple manner. It is also clear from the examination of the invariants given by eqn (2.2) when the procedure will not work. Thus if B = 0 and a further small deformation A1(R, t) is imposed on a finite deformation A,(R), we would have A(R, t) = A,(R) + cA,(R, t) which introduces terms of order 6 in Zr, Z2,rather than terms in c*, making the problem more difficult. There are of course other situations when this technique will work if the form of the additional deformation is known a priori. If simplifying information of this type is not available one would then have to revert to the general theory of small deformations on large.

REFERENCES [I] A. E. GREEN and W. ZERNA, Theoretical Elasficity, pp. 113-144. Clarendon Press, Oxford (1968). [2] M. A. BIOT, Mechanics of IncrementalDeformations. Wiley, New York (1965). [3] C. TRUESDELL and W. NOLL, The Non-Linear Field Theories of Mechanics. Encyclopedia of Physics (Edited by S. Flugge), Vol. 11l/3, pp. 260-267 (1965). [4] M. KURASHIGE, Znt. J. Engng Sci. 12, 585-596 (1974). [5] M. KURASHIGE, J. Appl. Mech. 96, 83-88 (1974), [6] A. ISHIHARA, N. HASHITSUMA and M. TATIBANA, J. Chem. Phys. 19, 1508-1520 (1951). (Received 3 September

1983)