Finite amplitude elastic shear wave propagation

WAVE M O T I O N 11 (1989) 251-260 NORTH-HOLLAND

FINITE AMPLITUDE

251

ELASTIC SHEAR WAVE PROPAGATION

R.J. TAIT Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada

S.A. L O R I M E R Concordia College, Edmonton, Alberta, Canada

J.B. H A D D O W Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada Received 10 November 1987, Revised 1 July 1988

We consider the propagation of shear waves in an incompressible, isotropic, hyperelastic hollow cylinder, where the outer radius is m u c h larger than the inner radius, when shears are applied to the inner face. The cylindrical geometry introduces additional difficulties. We analyse the propagation of acceleration and shock waves. If the perturbations from the steady state deformed or undeformed configuration are small we also consider a linearised approach and shock formation in this case. A few numerical results are included.

1. Introduction When small spatially uniform perturbations of axial and torsional shear are applied at the inner surface of an inflated and stretched hollow isotropic hyperelastic incompressible cylinder the resulting radial wave propagation is governed by linear partial differential equations [1]. If finite displacements are applied the equations are, in general, nonlinear so that shock waves may occur. This is illustrated for a particular strain energy function in [2]. The form of the strain energy function has a central effect on the form of the equations. In the following we attempt to formulate the problem in a fairly general way. We impose sufficient conditions for the system of equations to be strictly hyperbolic and discuss the propagation of acceleration and shock waves into both unstressed and prestressed regions. Collins [3] considers homogeneous quasi linear hyperbolic systems which govern the propagation of transverse waves in an elastic half space. Since

only step boundary conditions are considered in [3] it is possible to find a purely analytic solution by considering adjoining constant states, Riemann invariants and similarity solutions. Cylindrical geometry eliminates many of these aspects so that a numerical treatment is then necessary, especially when dealing with smooth boundary conditions and internal shock development. There are of course similarities with [3] and we adopt many of the concepts and the notation. In the present note we consider only mechanical aspects of shock formation and restrict the discussion to isotropic materials. The analysis presented is valid until the first reflection occurs at the outer radius of the cylinder. We refer to [4] as a general reference on hyperbolic systems.

2. Governing equations We consider a very long isotropic hyperelastic incompressible hollow cylinder, inner radius Ri,

0165-2125/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

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R.J. Tait et al. / Finite amplitude shear waves

outer radius Ro with Ro ~ R~ > 0. Let (R, 0, Z ) , (r, 0, z) denote polar coordinates in the initial u n d e f o r m e d reference configuration and the current configuration, respectively. We consider the particular class o f motions described by

0 = O + c e ( R , t),

If F denotes the d e f o r m a t i o n gradient tensor and B = F F v the left C a u c h y - G r e e n tensor, where we e m p l o y physical c o m p o n e n t s of all tensors involved, the invariants of B are r2 R2 11 = a 2 + - - ~ + - - y - 7 +

r2oL'2 q-

W'2,

r

I3 = 1,

(2.2)

where a prime denotes a / ~ R . For a given strain energy function W ( I , , I2), the C a u c h y stress is given by T = - p I + 2 ve~/~B - 2 1,~72B-1 ,

(2.3)

where I denotes the unit tensor, p an arbitrary scalar which is not d e t e r m i n e d by the deformation, and

0¢¢

OIm'

Or

m = 1,2.

(2.9)

2T(rO) F- - pr~,

(2.10)

r

r(Ro)=Ro,

c~(Ro, t ) = o ~ ( R o , t ) = O

and ce(Ri, t) = % ( t ) ,

w ( R i , t) = wi(t)

or

=--V¢ tz

(2.4)

T(rO) = ~ R a ' ,

T ( r O ) ( R i , t) = r2(t).

The static solution can be readily obtained [5]. In this solution p is determined from the equilibrium form o f (2.8) to within an additive constant which d e p e n d s on a s u p e r i m p o s e d hydrostatic stress. C o n s e q u e n t l y a normal radial stress m a y be required at the inner surface which is assumed to be b o n d e d to a rigid cylindrical bar. For the d y n a m i c p r o b l e m p is determined from (2.8) after a ( t ) and w ( t ) have been determined. It is easily shown that T ( r r ) is continuous across any shocks that occur. We assume the cylinder is sufficiently long so that end effects can be neglected. The equations are n o n - d i m e n s i o n a l i s e d by setting

It follows from (2.2) and (2.3) that T(rz) = ~w',

- = pdJ,

T ( r z ) ( R i , t) = r l ( t ) ,

R 2 A2r 2 A2r2a,2+ r 2 w , 2 / R 2, /2 = A - 2 + r---T+ R---T-+

ITV,,

r

OT(rO)

(2.1)

z = A Z + w ( R , t).

A

T(rz)

where a s u p e r p o s e d dot denotes partial differentiation with respect to time and p denotes the constant density. The b o u n d a r y conditions are

R2 r 2 = - - + K, A

/~

aT(rz)

~ -fir+

T(ij) = T(ij) tx

0.)

r

Ri

Ri

(2.11)

(2.5)

where = 2a - ' I~, + 2 a I~2,

(2.6)

2 R I ~ 1 2rVv"2 qza r + AR

(2.7)

c3r

r

.2 - -prc~ ,

non-

0 Or (r~c° ') = rg;,

The equations of motion are OT{rr) q T { r r ) - T(O0)

w h e r e / ~ is the infinitesimal shear modulus. With the superpose ^ omitted, the dimensional forms o f (2.9) and (2.10) are,

(2.8)

9._ (rZqbRa,) = r3~.

ar

(2.12)

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R.J. Tait et al. / Finite amplitude shear waves

I f we introduce the vector uT= (P, Q, R, S),

(2.13)

where

P=a', Q=to', R=&,

S=o5,

(2.14)

We now simplify the system by excluding preinflation and extension, setting A = 1, K = 0 , so that r = R. These may be included once the simpler case is understood. Since W = W(I1, I2),equations (2.2) allow us to write A 2 = r2p2 +

we write (2.12) as the system

~+ A(u, R )-O-R+ Ou b(u, R ) =O,

Q2,

W= W(I1,12)= (2.15)

¢v'(A2) = W(A).

(2.21)

We consider properties of W below but note that • = ~=

where A has the partitioned form

A-1 ~dW -=

a - I W*(A),

(2.22)

where * -- d/dA. The components of C then reduce to

-clo/

Cll = ~ -F r2A -1 q~,p2,

and I is the 2 x 2 identity matrix, C a 2 x 2 m a t r i x with components Cll = ~ ,

C22 = ~ + A-Iq~*Q 2, C21 = r2C12 = r2A - l o * P Q ,

C12 = ~P~,Q,

~r ~P C21 = ~ Q .e,

,~r C22 = ~ - 7,

(2.23)

and (2.16)

b = - (0, 0, 3 r-lq~P + rA -I @.p3,

r-l@Q+ rA-l~*p2Q).

with s¢= q~ + pqb,p,

7/= ~ + Qqt, Q,

(2.17)

The eigenvalues of A are given by K,~ = W**(Z),

and

(2.24)

K~ = a -1 w * ( a ) = ~. (2.25)

b=_(0,0,[A(qb.

R + ~~)\ +-~52 R ~ } P,

and to ensure real eigenvalues we require W * * ( A ) > 0,

+_1 R

.R

(2.18)

r

The eigenvalues of A are given by

A-~W*(A)>O.

There are cases of interest where the order of the eigenvalues may change [6] but we confine attention here to the case where, if A # 0, K2> K2> 0,

4r

(o7 --

PQCrl~,op)} att, 1/2.

(2.19)

We note that for a Mooney Rivlin material W = ½{b(11 - 3) + (1 - b)(I2 - 3)}, 0 <~ b ~< 1

(2.20)

that the system uncouples and consists of linear wave equations with variable coefficients and is dealt with in [1].

(2.26)

(2.27)

SO that the system is strictly hyperbolic. A = 0 is a special case and will be treated later assuming that W * ( 0 ) = W * * * ( 0 ) = 0 , so that the eigenvalues coincide. We note some properties of W for later use. Clearly W is a symmetric function of A and • , using condition (2.27), is a monotone increasing function of A for A > 0, monotone decreasing for A < 0. If we set 2

2

U = r l -- K2, dU - W***(A)-A-1U, dA

(2.28) U(0)=0,

(2.29)

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R.J. T a i t et al. / Finite a m p l i t u d e s h e a r w a v e s

so that

U -~-A-1

I;

W:~*~:( A ) A da.

(2.30)

Since U > 0, A ~ 0 this implies a W***(A) > 0,

A#0.

(2.31)

We consider the m a x i m u m region in which (2.31) holds. Then A K * > 0 , A # 0 . If KZ>Kiz > 0, the analysis must be modified accordingly and the relationship between W * ( A ) and a is indicated in Fig. 1. We use the particular strain energy function W = I(A2-~- "~A4), 3,>0,

(2.32)

for illustrative purposes for the cases discussed above, and 3' < 0 in the alternative case. 3' < 0 introduces a restriction in the magnitude of 3' and a.

where [u] = u - u + and u_, u+ denote the values of u immediately behind and in front of C respectively. If K denotes the largest positive eigenvalue of A o = A ( u o , r), then C is the curve dr~dr = K and it is known that [u,]

=

err

(3.3)

where L, R denote the left and right eigenvalues of Ao corresponding to K and 0- is a scalar function. Then, if D / D r denotes differentiation along C, the transport equation for 0- is

Do--+ Dr

~'10- = ~r20-2,

(3.4)

where 1r~ = L . OA Riu ~+ K D r + - •

w**(a)>a-l~

W,

x (KL. R ) - ' ,

~

71"2 :

*(A)>0

A Fig. 1. Relationship between a and W*(A).

3. Acceleration waves

Consider the reduced form of equations (2.15) subject to equations (2.21) to (2.24) in the form

u,, + A(u, r)ur+ b(u, r ) = 0 ,

(3.1)

with a wave propagating into a steady state uo(r), where Uo is an equilibrium solution of (3.1). An acceleration front is a curve C in the (r, t) plane issuing from r = 1, t = 0 across which

[M] = 0,

[Mr ] ~ 0,

Ri

OU i

[18,t] ~a~0

(3.2)

(3.5)

L ~bli OA R , R / K 2 L

• R,

(3.6)

where ~i denotes the ith c o m p o n e n t of qt, the summation convention has been used and all quantities are evaluated at u0. If the wave propagates into an undeformed medium we have the special case A = 0 along C and the eigenvalues of Ao coincide. If W has a Taylor expansion about 0 so that W * ( 0 ) = W***(0) = 0, then K:= W**(0) and we still have a full set of eigenvectors. We take R above as a combination of the right eigenvectors corresponding to K > 0 a s R = 0-1(1, 0, -K, 0)T+ 0-2(0, 1, 0, --K) T,

(3.7)

the superscript denoting transpose. Since ~r2 = 0, we can substitute (3.7) in equations (3.4), (3.5). Operating in turn with L=(K, 0,-1,0)

and

(0, K, 0 - 1 ) ,

(3.8)

0"2 = 0 - 2 ( 1 ) r - 3 / 2 ,

(3.9)

results in 0-1 = 0-1(1) r-I/2,

ILJ. Tait et aL / Finite amplitude shear waves

along C: r = l + K t . It follows that [u t] remains b o u n d e d and shocks cannot occur on the front. Numerical experiments show that shocks do form and they must do so in the interior, see Fig. 2. In general A has two positive eigenvalues K1, K2 and we take K~ > K2 > 0. The eigenvectors are taken in the form R, =

(P, Q,

--KIP ,

255

where the suffixes 1, 2 refer to K~, K 2 respectively. If an acceleration wave is p r o p a g a t e d from r = 1, t = 0 by i m p o s i n g variable b o u n d a r y conditions at r = l , and Ao2=r2P2o+Q2o#0, then substituting (3.10) evaluated at Uo into (3.5), (3.6) we find, after some calculation, that equation (3.4) takes the f o r m

- - K , Q ) T,

R2 = (Q, - r 2 p , - K 2 Q , r2K2P) T, L1 = (r2K1 P, K,Q, - r 2 p , - Q ) , L2 =

(K2Q,

-

K*

i (trAo)2 ,

(3.11)

K

-K2P,- Q ,

P), where K ~ C: d r / d t = is

0.6

KI(A0)

KI(A0).

and [u.,]=crRl(Ao), along The solution o f equation (3.11)

o" = O'(1) Ao(1)r-'/2( K / K(1) ) -3/2

0'rll

,,

"

,,

,,

,

,

i

i

,

i

x ( { 1 - cr(1)Ao(1)K(1) 3/2

_

x

*

d r } A ) )( -r ' o

.

(3.12)

1

The condition that a shock occurs at r = rc on C is

,~ . . . . . Combined (b) " i ~,, --TorsionalShearOnly

O.4

i i

~

i i

,

0.2

i

I

~

i i ,

i i

I

I

1 = o-(1)A(1)K(1) 3/2 I i

rc

r-~/2K-7/2K * dr.

(3.13)

For e x a m p l e s u p p o s e we e m p l o y the simple strain energy function W = ½ ( A 2 + yA4), y > 0. Then the equilibrium solution is Qo(r)(1 + 2 y A 2 ( r ) ) = No~r, (3.14) r2po(r)( 1 + 2yA2o(r)) = 1Vo,

-0.2 -0.4 1.0

~

-

3.0

5.0

7.0 9.0 11.0 r Fig. 2. Propagation of T(rz), T(rO) into an undeformed medium as functions of r for t = 1.5 (1.5) 9, W=½(A2+ yA4), 7=0.1. - (a) and (b): T(rz)(1, O)= T(r, 0)(1, 0) = sin(qrt)H(t)H(1 - t). . . . . (a): T(rz)(1, O) = sinOrt)H(t)H(1 - t), T(rO)(1, O) = O. . . . . (b): T(rz)(1, 0) = 0, T(r0)(1, 0) = sinOrt)H(t)H(1 - t).

r

where No, No indicate suitable b o u n d a r y values at r = 1, which we take positive for convenience, and r = (1 +6yA~) '/2, K* = 67Ao/~,

Ao = r eg+

(3.15)

R.J. Tait et al. / Finite amplitude shear waves

256

C l e a r l y A o is b o u n d e d a n d a s y m p t o t i c a l l y Qo(r) N o / r , P o ~ I Q o / r 3. The integral in (3.13) is a m o n o t o n e i n c r e a s i n g f u n c t i o n o f r, b o u n d e d a b o v e a n d we d e n o t e the least u p p e r b o u n d by N1K(1)-3/2Ao1(1). A s h o c k will then form if ~r(1)N~ > 1,

(3.16)

a n d n o t otherwise. A n i l l u s t r a t i o n is given in Fig. 3. A s i m i l a r a r g u m e n t h o l d s if 3, < 0 with s u i t a b l e m o d i f i c a t i o n s . We note that the j u m p s in the c o m p o n e n t s o f Ou/Ot are c o u p l e d t h r o u g h e q u a t i o n (3.3). A s i m i l a r analysis c o u l d be c a r r i e d out a l o n g d r / d t = K2 t h r o u g h r = 1, t = 0 b u t this is p r o p a g a t ing into a d i s t u r b e d r e g i o n a n d the values o f P, Q are not k n o w n . This is so even in the case w h e r e 1°o -= 0, Qo ~ 0. The l o n g i t u d i n a l o r Q wave t h e n p r e c e e d s the P wave, a n d the b e h a v i o u r at the l e a d i n g front is the s a m e as that o f a p u r e longit u d i n a l Q wave. The c o u p l i n g with the Q wave a l o n g the slow c h a r a c t e r i s t i c prevents further analysis. 0.6,

E

I

I

I

I

I

I

0.4

O 0.2

0.0 0.2

I

I

I

I

I

I

I

I

I

I

p

I

I 2.0

I

I 4.0

I

1

System (3.1) m a y be written in c o n s e r v a t i o n f o r m u,+G.r+d=O,

(4.1)

with G(u, r) = - ( R, S, P 4 , Qqb) r, d(u, r ) = - ( O , O , 3 P 4 / r , Q 4 / r ) .

(4.2)

If a s h o c k forms a l o n g F : d r / d t = V, t h e n V[u] = [ G ] ,

(4.3)

w h e r e as a b o v e [u] d e n o t e s the j u m p in u across F, u_ d e n o t i n g values b e h i n d a n d u+, values a h e a d o f F. T h e n P4

a n d c o m p a t i b i l i t y requires (P+Q_ - P _ Q + ) [ 4 ] = 0.

(4.5)

The last c o n d i t i o n has the s a m e form as the p l a n e case [3] a n d we a d o p t the n o t a t i o n used there. If the first b r a c k e t is zero, we have a t r a n s v e r s e s h o c k a n d if we write r P = A cos 0, Q = A sin 0, then 0 d o e s not j u m p or does so by ±Tr. I f [ 4 ] = 0, we have a c i r c u l a r d i s c o n t i n u i t y across which A d o e s not j u m p . This d i s c o n t i n u i t y c o i n c i d e s with a characteristic. I f a s h o c k p r o p a g a t e s into an u n d e f o r m e d m e d i u m it d o e s so a l o n g the slow c h a r a c t e r i s t i c b e h i n d it, t h a t is

I

(b)

0.0

-0.2

4. Shocks

I

6.0

8.0 r Fig. 3. Propagation of P, Q into a deformed medium as functions of r for t=l.5 (1.5) 7.5, W=½(zl2+yzl4), 3,=0.1. T(rz)(r, 0) = 0.5, T(rO)(r, O) = O. T(rz)(1, t) = r(r0)(l, t) = i sin(~rt)H(t)H(l - t).

V 2 = K~(A_) = 4 _ .

(4.6)

C o n d i t i o n (2.27) ensures that K~(za ) > V 2 a n d the r e m a i n i n g stability c o n d i t i o n that V2>K~(0) follows f r o m the m o n o t o n i c i t y o f 4 , the values b e h i n d the s h o c k b e i n g given b y e q u a t i o n (4.3). Next, if the s h o c k p r o p a g a t e s into a region w h e r e o n e o f P o , Qo is zero, say Uo = (0, Qo(r), O, 0) T, t h e n the o n l y p o s s i b i l i t y given b y c o n d i t i o n (4.5) for a stable s h o c k to p r o p a g a t e is P_=0,

Q_~O,

V2=[Q4]/[Q],

(4.7)

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R.J. Tait et al. / Finite amplitude shear waves

and the stability condition is K~(Q_) > V2 > K2(Q-),

V2> K~(Qo).

(4.8) To satisfy (4.8) we must have Q_ > Qo > 0 or Q_ < Qo < 0. To see this suppose first of all that Qo > 0. The inequality K~(Q_)> KE(Qo) then follows since W**(Q_)> w * * ( Q o ) . Equation (2.31) then implies either Q_ > Qo > 0 or by symmetry of W, Q changes sign and Q_ < - Q o < 0. The condition V2> K22(Q_) can be written as Do< ~_ i f 0 < Qo< Q_ and as ~bo> @_ if Q _ < - Q o < 0 . Since @*= A-I(K~--K~)>0 if d > 0 , the first inequality is valid. For the same reason since qb(-A) = ~ ( A ) , the second cannot hold. It remains to show that K~(Q_) > V2> K2(Qo) if 0 < Qo < Q_, but this follows immediately on writing out the definitions and using a mean value theorem and condition (2.31). The Second case Qo < 0 follows by a similar argument. Equation (4.3) requires P_ = R_ = 0 and since the shock F is non-characteristic and one solution of system (4.1) behind the shock, taking the values u_ on the shock, is u = ( 0 , Q, 0, S) T with Q, W determined from the reduced system (4.1). This solution, assumed unique, will exist in some region behind the shock up to some suitable K2 characteristic or until a second shock forms. As in the linearised case considered in Section 5, a purely longitudinal Q wave precedes the torsional wave and there is no coupling. The shock speed V given by (4.7) is the same as that if a pure longitudinal Q disturbance is propagated. Consider the situation when a discontinuity is propagated at r = 1, t = 0. Initially the material is in a deformed state described in the above paragraphs and boundary conditions are imposed on r = 1 such that P(1, O) --/5,

V2 = [Qqb] [Q] '

P-Qo - PoQ- = 0,

(4.10)

so that 0 does not jump but A does. The stability condition is

K~,(A_)> V2> K~(A_),

V2> ~,(Ao), (4.11)

which is satisfied if and only if (4.12)

A_ > Ao> O,

and either (i) 0 < Qo< Q_ or (ii) Q_ < Qo
5. Perturbations of the steady state If system (3.1) is perturbed from the steady state

Q(1, O) = (~,

,~ = (/52+ (~2),/2 # Qo(1).

also possible to have a shock and a circular discontinuity as in the case of a rectangular half space [7]. We suppose Qo(r)> 0 and a consider a shock propagating into the steady state region, issuing from r = 1, t = 0. Then t~_ = Q_(1 ÷, 0 +)/> z~, where again t~_ is the value of Q immediately behind the shock as r--> 1, t ~ 0. If not, then the monotonicity of qb implies K~(Zi)>K22((~_) and by (2.31), K2(z~) > K~((~_). This would imply a second shock through r = 1, t = 0 at least as fast as V and the jump conditions would be violated. Then either t~_ = z~ and it is possible for a circular discontinuity to propagate, [ ~ ] =0, or Q _ > z~ and no second discontinuity propagates. In general if Po # 0, Qo # 0, the jumps across a transverse shock are coupled and equations (4.5), (4.6) give

(4.9)

In the linearised version of this problem (see Section 5) two discontinuities propagate along the two positive characteristics through r = 1, t =0. It is

Aou! ° ) + b o = O ,

r > l,

where Ao = A( u (°), r), bo=b(u (°), r), (Po, Qo, 0, 0) T, by setting u = u(°)(r)+ uO)(r, t),

(5.1) u(°)=

(5.2)

R.J. Tait et al. / Finite a m p l i t u d e s h e a r w a v e s

258

where u °) is a small perturbation induced by the b o u n d a r y conditions, we obtain the linear system U(D ,t

• (1)+ + Aou , Bou(~) =O, (5.3)

r>l,

t>O,

with (Bo)o

\ k = , ouj ar

Ouj],=,(o, (5.4)

i,j = 1, 2, 3, 4,

P(t)(1, t ) = F ( t ) , (5.5)

Q(i)(1, t ) = G ( t ) ,

g(O)= ¢~R(o, ,

h(°)= r2R(o2)

(5.7)

where R(o~), i = 1, 2, are given by (3.16) evaluated at Po, Qo, and

subject to the conditions u(l)(r, O) = O,

present p u r p o s e s if we consider j u m p discontinuites in u (1> we take /2o(¢) = H ( ¢ ) , the Heaviside unit step function, whereas if u (~> is continuous but the first derivatives j u m p /2~)(~o) = H ( ¢ ) . The r e m a i n d e r terms for U (1), U (2) are regular functions of x, t, st, s2. Linearity allows U (1), U (2) to be treated separately when (5.6) is substituted in equation (5.3) so that equating coefficients of/~_~, / 2 o , . . . to zero gives

D~'/+ ¢ri~'i = 0, Dr

i = 1, 2,

(5.8)

t>0.

If the solution of equations (5.3), (5.4) subject to (5.5) is known, the linear a p p r o x i m a t i o n to the actual characteristics can be corrected using W h i t h a m ' s rule [8] to estimate the occurrence of a shock. The variable coefficient matrices Ao, Bo m a k e this difficult but we can use techniques on estimating the growth o f singularities [9] and procede as outlined below. The case o f p r o p a g a t i o n into an u n d e f o r m e d m e d i u m , Po = Qo = 0, is omitted. This has been dealt with above; the linear system uncouples and the reduced separate systems can be treated in a straightforward way. C o n s i d e r then the general case, Ao: = r2p2o + Qo 0, and following [9] write

~(i)

a' (0) l ' ( *-'0 i)i

" R(o i)

'

give the t r a n s p o r t equations, L(oi) given by (3.16) and D / D r denoting differentiation along ~o~ = Sl, ~2 = s2 for i = 1, 2 respectively. T h e solutions of equations (5.8), after some c o m p u t a t i o n , are

,.,-l(S0 (,,~°>(1))-~/2,ao(1) r,

=

(K ~O)(r))3/2Ao(r)r '/2 ' (5.9) r2(sg(~°>(1))~/2ao(1)

rz = (K~O)(r))3/2Ao(r)r3/2" In particular, if the initial discontinuity occurs at r = 1, t --- O, then (5.6), (5.7), (5.9) give F ( 0 ) = r,(0)Po(1) + ~'2(0)Qo(1), (5.10)

U(l)(r, t) = U (1)-~- U (2)

G(0) = rt(0)Qo(1) - r2(0)Po(1),

= (no(q~)g (°) + n , ( ~ , ) g ( ' ) + • • .)

+ (t'/o(~2)h (°) + n,(~o2)h (') +.

_

7Ti- 4(0)

•

.),

(5.6)

with g ( k ) = g(k)(x ' t), h ( k ) : h(k)(x, t), k = 0, 1 , . . . . The characteristic curves q~l = Sl = constant, ~2 = s 2 = c o n s t a n t c o r r e s p o n d to the positive eigenvalues K~°), K(°) of Ao and we parametrise t h e m by their values on the r = 1 axis so that (pl(1, S) ----S ---~2(1, s). In the general t r e a t m e n t /2k(~p) are generalised functions with /2~,(~)=/2g-l(q~) and for

if the discontinuity is a j u m p in u (~), while if the j u m p is in O#(1)/Ot, F(O) and G(0) are replaced by F'(O), G'(O) respectively. C o n s i d e r the case when we have an initial discontinuity in Ou°)/Ot at r = 1, t = 0 , but u (~) is continuous. T h e characteristic curve of the nonlinear system corresponding to /<1 is d r / d t = K~. If the p e r t u r b a t i o n is small we have ~1 = K ~o>+ (K * ) ( ° ) ( a - a o ) + " • •

= K~°)+ (Kl*)(°)(r2Pop")+ Q o Q " ) ) / A o +. • .. (5.11)

R.J. Tait et al. / Finite amplitude shear waves Ahead of the characteristic s2 = 0, equations (5.6), (5.7) and (5.9) allow us to write, along ~ = s~,

and Bo(42)

Q(,)] = r, Oo(S,)

+ A,

(5.12)

259

=

-

(r(K~°))2)'/r,

with

(K~°))2= W**(Qo), where the regular terms have been collected into A with A and all first derivatives zero on Sl = 0. I f we substitute (5.12) into (5.11) and integrate d r / d t = K], we have t = Sl +

(K~0)(q))-l{1 -- (K ~°)(q))-](K*)(0)(q) 1

x (,,Oo(S0+(q2Po, 0o). A/1g')} dq, (5.13) as the improved approximation to K~characteristic. The condition that the characteristics meet on the front and that a shock form is obtained from (5.13) by formally differentiating partially with respect to s~ ~ 0. The last term in the bracket is then zero and the condition is 1 = r,(0)(K~°)(1))3/2/10(1)

x

(K~°)(q))-7/2(K*)(°)(q)q -1/2 dq.

(5.14)

1

This is exactly the same condition obtained in the fully nonlinear case and is obtained here on the assumption that the perturbation is small. The initial j u m p rl(0 ) is given by equation (5.10) with F'(0), G'(0) again replacing F(O), G(O). The argument is inconclusive along the characteristic corresponding to s2 = 0 since in the correction to r2 the regular coupling terms are nonzero. Note that equation (5.7) implies the jumps in aP(])/at, aQ(~)/at are coupled across Sl = 0 and no shock occurs on the front if F ' ( 0 ) = G ' ( 0 ) = 0. Once a shock has formed these arguments no longer hold. I f P0 = 0, Qo ~ 0 the two linearised waves are uncoupled, and A0 has components given as before but here having the values

c,,=(r~,°)) 2, c==(K~°)) ~, c,~=c~,=0 and the only nonzero components of Bo are B0(31) = -(ra(K(2°))2)'/r3

(K(2°))2 = Qo' W*(Qo).

The Q wave preceeds the P wave, and the arguments on the leading front are as in the previous paragraph. A shock cannot occur on the characteristic corresponding to s 2 = 0 ; since /I 2= r2p(l)2+ (Qo+ Q(1))2, then aA2/ar, 0/12/0t are continuous functions behind the leading front. To this degree of approximation, since P(~), QO) are solutions of the linear system, they remain bounded and the differential equations d r / d t = K2(/1) cannot have more than one solution through a given point and so the characteristics cannot intersect. If there are initial discontinuities in u ~), the linearization provides a first approximation but the characteristic speed is slower than the actual shock speed. The j u m p conditions obtained from equation (5.9) across characteristics, namely across d r / d t = K~°), d r / d t = K~°) through r = 1, t = 0, are respectively

P°)Qo - Q(l)po = 0, (5.15)

r2p(])Qo+ Q(1)Qo = O, and are linearizations of the shock conditions (4.5).

6. Conclusion In the preceding sections we have analysed the behaviour of cylindrical torsional and longitudinal shear waves propagating into a deformed and undeformed medium and have obtained conditions that shocks will or will not form on the front. In general these estimates are upper bounds in the sense that if shocks do not form in the interior earlier they must do so by the time given for shock formation on the front. An analysis similar to that in [7] would be useful in understanding the general behaviour of such waves but so far this has not been obtained and numerical procedures appear to be the simplest way to proceed.

260

R.J. Tait et al. / Finite amplitude shear waves

References [1] R.J. Tait, J.B. Haddow and T.B. Moodie, "A note on infinitesimal shear waves in a finitely deformed elastic solid", Int. J. Eng. Sc. 22(7), 823-827 (1984). [2] J.B. Haddow, S.A. Lorimer and R.J. Tait, "Non-linear combined axial and torsional shear wave propagation in a incompressible hyperelastic solid", Int. J. Non-linear Mechanics 17(4), 297-306 (1987). [3] W.D. Collins, "One-dimensional non-linear wave propagation in incompressible elastic materials", Q.J. Mech. and AppL Math. XIX, 259-328 (1966). [4] A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, London (1976).

[5] A.E. Green and W. Zerna, Theoretical Elasticity, 2nd ed., Oxford (1968). [6] J. Wegner, J.B. Haddow and R.J. Tait, "Finite amplitude wave propagation in a stretched elastic string", in: M. McCarthy and M. Hayes, eds., Proceedings I U T A M Symposium on Elastic Wave Propagation (March 1988), in press. [7] W.D. Collins, "The propagation and interaction of one dimensional non-linear waves," Q.J. Mech and Appl. Math. XX(4), 429-452 (1967). [8] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). [9] R. Courant, and D. Hilbert, Methods of Mathematical Physics, Vol. 2, Wiley, New York (1962).