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Acceleration waves and shock waves in transversely isotropic elastic non-conductors
ht. J. Engng Sci. Vol. 27, No. 11, pp. 1379-13%, 1989 Printed in Great Britain. All rights reserved
0020-7225/89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc
ACCELERATION WAVES AND SHOCK WAVES IN TRANSVERSELY ISOTROPIC ELASTIC NON-CONDUCTORS Y. B. FUt and N. H. SCOTT School of Mathematics,
University of East Anglia, Norwich NR4 7TJ, U.K.
Abstract-We consider the propagation of plane acceleration waves and weak nonlinear shock waves in an unstrained transversely isotropic elastic non-conductor which is incompressible but extensible along the preferred direction denoted by the unit vector e. It is shown that such a material may transmit two different shock waves in the direction of any unit vector n which is neither perpendicular nor parallel to the preferred direction e. One of these shock waves is polarised in the plane II spanned by e and n and carries an entropy jump of third order in the shock amplitude, whilst the other is polarised perpendicularly to II and carries merely a fourth order entropy jump. In the degenerate cases when the angle 0 between n and e is either n/2 or 0, the two different shock waves which can propagate both carry fourth order entropy jumps. In all of these cases, an asymptotic evolution law describing the decay of the shock amplitude is obtained from a reduced first order governing partial differential equation which is itself obtained by differentiating a certain Riemann invariant that is approximately constant under the isentropic assumption adopted here (in common with all studies of non-conductors). We consider also the limiting case in which the material is inextensible along the preferred direction whilst remaining incompressible. It is shown that when 9 = n/2, two shock waves can propagate as before, but when 19= 0, no shock wave can propagate, and when 0 + n/2, 0, only one shock wave can propagate. Shock waves which can propagate in such a constrained material are all of the second type. The shock wave travelling in the direction of inextensibility is allowed by the purely linear theory of such constrained materials but is forbidden by the weak nonlinear theory.
1. INTRODUCTION
Within the family of propagating singular surfaces of different orders, shock waves are unique in that their amplitudes do not obey a single exact evolution equation; their evolutionary behaviour is always coupled with that of the higher order discontinuities which accompany them. It is probably because of this that the global evolutionary behaviour of shock waves has been less studied in the literature than that of acceleration waves, most previous studies on shock waves having been confined to their general properties and their instantaneous growth and decay behaviour. Weak nonlinear shock waves in elastic non-conductors fall into two categories, the first consisting of those shock waves across which the entropy jump is of third order in the shock amplitude, and the second consisting of those across which it is of fourth order. In two previous papers [l, 21 we have studied these two categories separately, choosing a prototype for shock waves belonging to each of them, in order to show how an asymptotic evolutionary law for the shock amplitude may be derived. In [l], plane dilatational shock waves in elastic nonconductors were found to have an entropy jump of third order in the shock amplitude and so to belong to the first category, whilst in [2], plane transverse shock waves in unstrained incompressible elastic non-conductors were found to have a fourth order entropy jump and so to belong to the second category. Shock waves in the first category were found to be much more strongly affected by material nonlinearity. In the present paper, we consider an elastic non-conductor with more general constitutive law, namely, an unstrained incompressible transversely isotropic elastic material. In view of the increasing interest in engineering in this kind of material (which includes a continuum model for the fibre-reinforced composites), it is desirable to clarify which categories of shock waves this kind of material can transmit and how any such shock waves evolve. For a general review of the relevant literature we refer to [l, 21 but three further papers are mentioned here. Simple waves and shock waves in incompressible isotropic and transversely isotropic elastic materials have been studied previously by Collins [3]. Shock waves in t Present address: Department
of Mathematics,
University of Exeter, Exeter EX4 4QE, U.K. 1379
1380
Y.
B. FU and N. H. SCOlT
compressible isotropic and transversely isotropic elastic materials have been considered by Waterston [4] and simple waves in compressible transversely isotropic elastic materials have been considered by Howard [5]. The results presented here are new in two respects. Firstly, the above-mentioned studies are all confined to the particular case in which the direction of propagation is perpendicular to the preferred direction, whilst in our discussion the direction of propagation is arbitrary. Secondly, Collins’ [3] and Waterston’s [4] studies on shock waves are concerned with the stability problem, whilst our concern is the determination of the evolutionary behaviour of the shock amplitude. After setting up the basic equations in the first section, we devote the second section to the discussion of acceleration waves in incompressible transversely isotropic elastic materials. We have two purposes in including such a discussion here. Firstly, the propagation of acceleration waves in such materials does not seem to have been studied before. Green [6] has studied acceleration waves in a nearly inextensible tranversely isotropic compressible elastic material and has employed a perturbation procedure to investigate the limiting behaviour of the wave speeds when the direction of propagation n tends to a direction perpendicular to the preferred direction e. However, he did not study the evolutionary behaviour of these acceleration waves. (In the linear theory, of course, all such plane waves propagate with constant amplitude.) We show here that on introducing the constraint of incompressibility, whilst retaining a fully general nonlinear theory, we are still able to analyse the limiting behaviour of the wave speeds and amplitudes exactly. Our second reason for studying acceleration waves is the desire to compare the propagation of acceleration waves with that of shock waves in order to draw the following conclusion: along any direction of propagation n, if an acceleration wave travels with changing (constant) amplitude, then a shock wave with the same polarisation will carry a third (fourth) order entropy jump and its evolution will be strongly (weakly) affected by the material nonlinearity. In Section 3, we determine the types of shock waves an unstrained incompressible transversely isotropic elastic material may transmit by expanding the strain energy function in a Taylor series as far as quartic terms in the strain gradient and then using the reduced jump conditions (2.12)-(2.14). It is shown that this kind of material may transmit one of each category of shock wave along any direction n which is neither perpendicular nor parallel to the preferred direction e. The shock wave of the first category, carrying an entropy jump of third order in the shock amplitude, is polarised in the plane II spanned by e and n, whilst the shock wave of the other category is polarised perpendicularly to II and carries a fourth order entropy jump. In the degenerate cases when the angle 8 between e and n is either 0 or n/2, the shock waves which propagate are necessarily of the second category. In Section 4, we study the propagation of simple waves in order to derive for each shock wave a reduced first order governing equation by using the isentropic assumption to construct a Riemann invariant that is approximately constant. We then show how to obtain the evolution laws for these shock waves by transcribing our previous results [l, 21 making appropriate substitutions for the material constants. The final section deals with the propagation of shock waves in an unstrained incompressible transversely isotropic elastic non-conductor which is also inextensible along the preferred direction. Results are obtained for this material by taking the appropriate limits of the corresponding results in the previous two sections. It is shown that when 8 = ~r/2, two shock waves can propagate as before, but when 0 = 0, no shock wave can propagate, and when 8 # n/2, 0, only one shock wave can propagate. The shock waves which can propagate in such a constrained material all belong to the second category. It is of interest to note that the purely linear theory of such a constrained material allows a shock wave to propagate in the direction of inextensibility though this is forbidden by the weakly nonlinear theory.
2. BASIC
EQUATIONS
Let the preferred direction e of the material lie in the X,+X,-plane at an angle 0 with the X,-axis which is chosen as the direction of wave propagation. The most general form of a plane
Acceleration waves and shock waves
1381
transverse motion is then given by Xl
=x1,
x2=x2
+ U2(XI,
x3 =x3
t),
+ U3(XI,
o,
(2.1)
where xi, i = 1, 2, 3, is the position at time t of a particle which is at position X,, A = 1, 2, 3, in the undeformed reference configuration, both sets of coordinates being referred to a common Cartesian frame. The quantities Ui(X1, t), i = 2, 3, are the particle displacements. The corresponding deformation gradient tensor and the Green strain tensor are given by 100
F=
respectively,
m2 1 m3 0
0 1
E=;(FTF-I)=
9
(2.2)
where m2=mi+m:,
m3=u3,~,
m2=u2,~,
(2.3)
a comma signifying partial differentiation. For simplicity, we have written X for X1. It is clear from (2.2a) that the incompressibility condition det F = 1 is satisfied automatically by the plane transverse motion (2.1). The behaviour of an incompressible transversely isotropic elastic non-conductor is completely described by the specific internal energy function E, which depends on the deformation through the four deformation invariants and on the specific entropy q: E = a,
Z2,Jb
(2.4)
J2, rl),
where Zl = tr E, Using
Z2= i (I: - tr E’),
(2.2b) and noting that e has components
J1= e - Ee,
.I2= e - E2e.
(cos 8, sin 8,0), we have
J1 = m2 sin 13cos 8 + k m2 cOsze,
(2.6)
J2=~m~sin2~+~m2cos2~+~m2m2sin~cosB+$m4cos2~. Therefore,
(2.7)
we can write E = 4Z, JI,
where ZdgfZ1= -2Z2. The first Piola-Kirchhoff given by Q = p &ldFti -pFi!, which gives
J2,
s),
stress in an incompressible
d&
n”=p=ip
(2.8)
elastic material is
(i = 2,3),
(2.9)
density and p is an arbitrary pressure. The reduced equations of motion are
where p is the constant
jdil,X
=
(i = 2,3),
Pi.t,
where v, = u,,~ are the velocity components,
(2.10)
whilst the reduced energy equation is ?j = 0.
(2.11)
1382
Y. B. FU and N. H. SCOTT
Finally, the following reduced jump conditions must be satisfied across a shock wave: [%I = P[s] = $ (JrJi:+
-P”Advi19
(2.12)
(i = 2,3),
nil)[W],
(2.13)
-“N[mi],
(2.14)
[VI2 09
(2.15)
[vi] =
the last being a consequence of the second law of thermodynamics. Here U, is the shock velocity in the reference configuration. In equation (2.13), summation over the repeated subscripts 2 and 3 is implied.
3. ACCELERATION
WAVES
In this section we study the propagation of acceleration waves in an unstrained incompressible transversely isotropic elastic non-conductor. Following the standard procedure we first derive the propagation condition and then determine the evolution laws. The basic kinematical compatibility relation for any function G(X, t) appropriate to one-dimensional acceleration waves is
-&[Cl =[G,xl +&N [GA
(3.1)
where now UN denotes the referential propagation speed of the acceleration denotes the space derivative following the wave. Replacing G by v, gives [P?li,t]= [Vi,x] = - & [tit] = - &
aij
wave and 3/3X
(3.2)
where a,, defined by Uj = [tii],
(3.3)
is the acceleration wave amplitude vector. To obtain the propagation condition, we first take the jump of the equations of motion (2.10) and use (3.1) and (3.3): [nil,,] = -puNai* (3.4) Recalling (2.9), we have
[Jhrl = q~,Jl?
(3.5)
where d2&
dnil =----&=
cij
-
I and a superscript “+” signifies evaluation
’
d??li
am,
(3.6)
’
in the natural strain-free
state. On substituting (3.5)
into (3.4) and using (3.2), we arrive at (p@/S,, - CC)% = 0, which is the desired propagation
condition.
It is straightforward
(3.7) to show that
(3.8)
Acceleration waves and shock waves
1383
Equation (3.7) then implies that two acceleration waves can propagate propagation speeds and amplitude vectors given by PGx = G,
in any direction,
with
az+O,
a3 = 0,
(3.9)
a2 = 0,
a3#0.
(3.10)
and P&
= G3,
It is clear that the two acceleration waves defined by (3.9) and (3.10) are polarised along the X,-axis and the X,-axis, respectively. We now proceed to establish the relations between the material constants (a&/%)+ etc. used here and the more familiar linear elastic constants. From (2.5) to (2.9) we see that 1 a& -JrZ1 =-m2-tdl P
$
1
(sin 8 cos 19+ m, cos2 e)
1 aa de -11-31=dlm3+dJm3cosze+1 P
- m2 + z (m’ + 2mz)sin e cos 8 + m’m, cos’ 8 ,
(3.11)
- m3 cos2 8 + m2m3 sin 8 cos 8 + m’m, co? e .
(3.12)
Since for an unstrained material there should be zero stress whenever both m2 and m3 vanish, we must take
a&+
o
(aJ,>=* Linearisation
(3.13)
of (3.11) and (3.12) then yields (3.14) (3.15)
For incompressible transversely isotropic elastic materials, given, for example, by Spencer [7] become
the linear stress-strain
relations
x21 = pLm2 f Bm2 sin2 e cos28,
(3.16)
3631= yTm3 + (iuL - k-b3 cos2e,
(3.17)
where pL and j.+ represent shear moduli (measured along and transverse to the preferred direction, respectively) and /3 is related to the Young’s modulus EL in the preferred direction by (3.18) B=&.-~PC~L+PT Comparing (3.14) with (3.16) and (3.15) with (3.17) gives =
2(PL- PT)P
B-
(3.19)
On introducing (3.19) into (3.8), we find that the squared wave speeds (3.9) and (3.10) become pi7& = pL + p sin2
e ~0s~e,
PO& = PL cos’ 8 + pT sin2 8.
(3.20) (3.21)
The material being nearly inextensible corresponds to EL being large, and hence, by (3.18), to /.I being large. Equations (3.20) and (3.21) show that in the limit of inextensibility, the acceleration wave which is polarised in the X,-direction and which has velocity ON3can still propagate, whilst the other acceleration wave with velocity ON2may propagate only if 8 = 0 or ~12 since otherwise ON*tends to infinity. However, when 8 = 0, the constraint relation J1 = 0 reduces to m2 - 0, which means that no disturbance actually propagates along the direction of
Y. B. FLJ and N. H. SCOTT
1384
inextensibility, although ON2is finite. Therefore, the acceleration wave which is polarised along the X2-axis can propagate only if 8 = ~r/2 and we have the exceptional case of two waves capable of propagating in any direction orthogonal to the direction of inextensibility. In any other direction only one wave can propagate except that no wave propagates in the preferred direction. This seemingly singular nature of the inextensible material can be resolved by considering a material that is almost inextensible. When p is large but not infinite, it can be deduced from (3.20) that oNz takes a large value of order (~/~~i’2 if both ~12 - 8 and 8 are of order one; it takes a value comparable with the other wave speed UN3if either 3r/2 - 8 or 8 becomes of the motion must order (pL//I)1’2. However, although ON2is finite when 8 is of order (p,/p)ln, necessarily be of vanishingly small amplitude. This is made clear by considering (2.6) which can be rewritten as (mz + tan
e)2 + rn: = tar?
8 + 2J1/cos2
8.
(3.22)
This relation confines m2 and m3 to lie on a circle in the m2,m3-plane of radius (tan2 8 + 2Jr/cos2 @)rn which is of order (~~/~)1’2 when 0 is of order (~~/~)~‘2. (J1 is of order &/3 since J,/3 remains of order one in the limit /3---, m). This circle contracts to a point when 0 = 0 and when the material is inextensible. Green [6] has considered acceleration wave propagation in compressible inextensibie and almost inextensible elastic materials and has used a perturbation procedure to study the behaviour of the wave speeds in the limit of inextensibility. In the inextensible limit our wave speed UN3is the same as his wave speed U, given by [6, equation (2.16)]. But since in our analysis the material is also incompressible, his wave speed U2 has no counterpart in this paper. We now turn to the derivation of the evolution laws. First, differentiating (2.10) with respect to time and using (3.6) and (2.11), we have (3.23) where (3.24) Repeated use of (3.1) gives [mj,x]
=
&
(3.25)
ajt
N
[mj,*t]
=
-$$ +& N
[Vj,c*]*
(3.26)
N
On taking the jump of (3.23) and using (3.2), (3.25) and (3.26), we arrive at (3.27) CASE
i: ~0% = p&
= Cz2, a2 # 0, a3 = 0.
Taking i = 2 in (3.27) and noting (3.8), we obtain (3.28) which has a solution a2(0)
(3.29)
a2(x) = 1 + (~~,/2p@&~(0)X where ~(0) is the initial amplitude. Using the definition (3.24), the expression
for C-&
can be calculated
with the aid of
Accelerationwavesand shock waves (2.5)-(2.8)
1385
and is given by
&=psinI!ZcostI
3 1 &‘+I
c&j+ +
+3
(
$
+
2>
3 cos2 8
3
d2& +
+
+ sin’ e cos2 e 2 . ( aJ: > ( 1)I
(3.30)
It then follows from (3.29) and (3.30) that the acceleration wave which is polarised along the X2-axis propagates with changing amplitude unless 0 = 0 or 1r/2. CASEii: ~0% = PI!?:, = Cz3, a2 = 0, a3 f 0. For this acceleration
wave, we take i = 3 in (3.27) giving the evolution equation ,.
-+
C da,+xai=(). dX
(3.31)
2pl&
It can be shown with the aid of (2.5)-(2.8) that c”333- 0 and therefore the acceleration which is polarised along the X3-axis always propagates with constant amplitude.
4. THE SHOCK
VELOCITY
AND THE ENTROPY
wave
JUMP
This section is devoted to discovering the precise nature of the shock waves which an unstrained incompressible transversely isotropic elastic material may transmit. In view of (2.8), the dependence of the internal energy function E on m2 and m3 is through I, J, and J2. Assuming that the region ahead of the shock is unstrained, we can expand E about this strain-free state in a Taylor series as far as fourth powers of the small quantities: PE = Carl + (YZJ:+
~y3J2 + p8o(Tj
+
~0) +
@lZJl+ /32JlJ2 + 835: + ~11~
y2ZJ: + y3ZJ2+ y4J’: + ysJfJ2 + yhJg + 6,( 7 - qo)Jl+ * * *,
(4.1)
where 4, /I,, yj and f3r are material constants and e0 and rlo are respectively the equilibrium temperature and entropy. In (4.1) we do not include the linear term in J1 since it gives a term in the expression for nzl which does not vanish with m2 and m3 [see (3.13)]. Only the linear terms in the entropy excess r,r- q. have been included in (4.1) in anticipation of the result that the entropy jump is at least of third order in the small quantities for a non-conductor. On substituting (2.5) (2.6) and (2.7) into (4.1), we obtain
+ i &mf: + peo(q - rlo) + &r4), where 1 c1 = (Y~+ 2 cu3+ 2a2 sin’ 8 c0s2 8, 1 E2 = al + 2 a3 CO? 8, E3 = 3 sin
8 cos e( 2 ( a3 +
j3, + i p2) + p3 sin’ 8 ~08~8 + @2~0s~e},
g4 = sin 6 cos e[ i ( a3 + @r) + ( a2 + a @2)~~s2e), E5=y,+~y3+$y6+(~3+~l+$32)~0s2e+~2c0s4e + (2P2 + 2y2 + ys)sin2 8 cos’ 8 +
6f?3
sin’ 8 cos4 8 + 4y, sin4 8 COST8,
(4.2)
1386
Y.
B. Fu and N. H. SCQn
+ Cu,COS40 + 4 & + y2 Sm 0 C0s2 8 + k (($3 + y&in* 6 cos4 8, (” P2
(
57= Yl + cu, + /3*+ ; y3>cos* 8 + ; 2&, + ,62f ; y6>cos4 8.
(4.3)
We have omitted the final term of (4.1) from (4.2) even though it is of fourth order when 77- pinis of th ir d order. This is because when q - q. is of third order we need only expand the strain energy as far as third order terms and so our omission does not affect the final results. Similar remarks apply to other approximate expressions in this section. Using (4.2), we have from (2.9) that x*1 = 5,m* +
53mI + E4m: + 5,m;
JE31= $$?I3 + Zj4rn2m3
+ $,mmZ
+ o(m3),
(4.4)
+ &jf?&Yl3 + &?2~ + o(t?Z”).
(4.5)
On inserting (4.2), (4.4) and (4.5) into (2.13), we arrive at 2&[77] = ; g3[m213 + 64[m21[m312 + ; &b214
+ h[m2]2[m312 +;
‘&b31”
+ o([m14>,
(4.(j)
and on substituting (4.4) and (4.5) into (2.12) with the use of (2.14), we obtain &‘[m,]
= 51[m2] + E3[m21* + 54[m312 + 55b21’ &b31
= [m31(g2 + 2$4b2]
+ 56[m21[m312 + o([m13)>
+ 56[m212 + t%[m31”) + o([m]“).
(4.7) (4. f-9
CASE i: 8 = z/2. In this case E3( 5) = g4( 5) = 0, and (4.7) and (4.8) reduce to
(4.9)
Pmm31=
rm31( E*(5)
+ &i(;)[m212
+ s,(;)rm31*}
+ ~([~I”).
(4.10)
These two equations are compatible only if either [m2] or [m3] vanishes. These two possibilities are now studied separately. CASE i(a): [mJ = 0, [m3] f 0. Equation (4.9) is satisfied trivially, whilst (4.10) gives (4.11) and from (4.6) we have [rl
The second law of thermodynamics this shock may propagate only if
= +&
e,( $314
+ WIm316)*
requires the entropy jump to be non-negative
e,(f)=Yi 2 0.
(4.12) (2.15) and so
(4.13)
Acceleration
1387
waves and shock waves
CASEi(b): [m2] # 0, [m3] = 0. In this case, expressions analogous to (4.11)-(4.13) Pug=
51(t)
[rll = &
are
+ e@m212+
e,(;)[m214
(4.14)
OW214),
(4.15)
+ o([m21”),
(4.16) CASEii: 13= 0. In this case &(O) = g4(0) = 0 as before. In addition, we have cl(O) = f2(0), &(O) = 56(O) = E,(O). Equations (4.7) and (4.8) therefore reduce to PGh21
=
b21(51@)
+ 5&9M2)
+ o(b13h
(4.17)
PW4
=
b31(51K9
+ MO)M2)
+ o(b13)7
(4.18)
from which we have PG
= 51(O)
+ EmM2
(4.19)
+ o([ml”),
while from (4.6) we obtain [VI=&
0
55(oM~1”
(4.20)
+ o(bl”>.
Equations (4.19) and (4.20) are both independent of the ratio [m2]/[m3]. This implies that the polarisation vector can be along any direction in the X2,X3-plane because of the symmetry. The second law of thermodynamics (2.15) requires 55(O)=n2+~3+B,+~~,+ul+~~3+a~6~0.
(4.21)
More generally, it can be shown that if the material is prestrained, two shock waves can propagate when 8 = 0. One of them is plane polarised and has the same character as the one discussed above. The other is circularly polarised and does not exist in an initially unstrained material. CASEiii: 8 # n/2,0. In this case, the only two possibilities are [m,] = 0 and [m,] # 0, with [m2] being always non-zero [as is clear from (4.7)]. To simplify the notation, we shall suppress the explicit dependence of the g’s on 8. CASEiii(a): [m2] # 0 [m3] # 0. Equations (4.7) and (4.8) yield Pv,
= 51 + 53[m2] + 54%
P%
+ ‘&[m212 + f6b312
= E2 + 254[m21 + t6b21”
+
+ o([m12),
57b31’ + o([m12)*
(4.22) (4.23)
These two expressions for pU$ are compatible only if (4.24) that is,
b21=-L,E2 -
El
[m3]2 + O([m,]‘).
(4.25)
Y. B. FU and N. H. SCOUR
1388
On substituting (4.25) into (4.23) and (4.6), we obtain
(
=)b31* HJ%=f*+ ~‘+6_51
[?I = &
0
+ot[m3l”h
(4.26)
+Wm31").
(4.27)
(5;+&)b314
1
By the second law of thermod~amics
(2.15) this shock wave may propagate
only if (4.28)
CASE iii(b):
[mzJ # 0 [m3] = 0.
Equation (4.8) is satisfied trivially, whilst (4.7) gives PG
= 51+
53b4 + O([m212).
(4.29)
From (4.6) we obtain
hl
=
&
By the second law of thermodynamics
mh.13
f
(4.30)
om*l”).
(2.15) this shock wave may propagate
only if (4.31)
Mm31 2 0.
None of the other shocks considered in this section has a squared wave speed possessing a linear term in the shock amplitude and none has a third order, rather than fourth order, entropy jump. This then is the only shock of the first category, rather than the second, that arises here. Summarising the results obtained so far, we have shown that when 8 = n/2, two shock waves may propagate, one being polarised along the X+&s and the other along the X,-axis; when 8 = 0, because of the material symmetry, a shock wave polarised in any direction perpendicular to n may propagate; when 8 # 0, 3t/2, two shock waves may propagate, one being polarised nearly along the X3-direction (in the sense that the component of the shock amplitude along the X,-axis is of second order in the component along the X,-axis), the other being polarised purely along the X,-axis. For easy reference in the following discussion, we refer to these five shock waves respectively as shock 1, . . . , shock 5. Our analysis has revealed that shocks 1-4 al1 belong to the second category in which the entropy jumps are of fourth order and the existence conditions (deduced from the second law of thermodynamics) for these shock waves all have the same form, namely 5 =L0, where 5 takes in turn the values &($), &($), Es(O) and jump across shock 5, on the other hand, is of third order and & + E:/(E2 51). Th e entropy the existence condition is of a different form: &[mJ L 0. Recalling the results of the previous section, we see that the acceleration wave counterparts of shock 1 to shock 4 always travel with constant amplitudes, whereas the acceleration wave counterpart of shock 5 may grow or decay. We shall show in the next section that shock 5 is the only transverse shock wave discussed here which resembles a dilatational shock wave in its evolutionary behaviour.
5. THE
EVOLUTION
LAWS
In the previous section, we found the types of shock waves which the materials under consideration here may transmit in any direction. In this section, we shall derive an asymptotic evolution law for each of the five shocks. First, we review briefly the theory of simple waves. Since according to (4.6), the entropy jump is of third order or higher in the shock amplitude, we shall work with the isentropic assumption within the order of approximation considered in
Acceleration waves and shock waves
1389
the rest of this paper. Then from (2.10) and (2.9) we have C22m2,x +
C23m3,x
=
VZ,[,
(5.1)
C32m2*x
C33m3.X
=
~3,~
(5.2)
+
From (4.2) and (3.6), we have the following expressions for C,: pG2 = 5i+ PC23
=
253m2
2hm3
+
+
3&d
2hm2m3
+
+
Ed
+
4m2),
o(m’)?
pC33 = E2 + 2g4m2 + &rns + 3&m: + o(m’). Equations (5.1) and (5.2) together can be written as a matrix equation:
with the compatibility
(5.3)
relations m2,t = v~,~, m3,r = v3,x
(5.4) where
We now look for simple wave solutions of (5.4) of the form
v =V(#),
(5.5)
where the function 9 = $(X, t) is to be determined and the curves $J(X, t) = constant represent travelling wavelets on which V is constant. On inserting (5.5) into (5.4), we obtain (AI - A)V’ = 0,
(5.6)
where Adsf - $J,~/@,~, and a prime signifies differentiation with respect to 9. Equation (5.6) can easily be manipulated to give the following equations: i1’ - C22
(
-G
-C23 rnb A2- C33>{mj I = O’
(5.7)
v;+Am;=O, v;+Am;=O.
I The characteristic
(5.8)
equation of (5.7) is A4
-
(c22
+
c,,)n’
+
c22c33
-
c;,
=
0.
(5.9)
Treated as a quadratic in 3c2,the above equation has two roots given by
A:
=
;
(C22
+
n: =; (Czz +
c33
+
I@22
-
c33)2
+
4G3)P
(5.10)
c33
-
V(C22
-
c33)2
+
4CZ3).
(5.11)
Therefore, the governing equations (5.1) and (5.2) admit two pairs of simple wave solutions. In each of the simple wave regions, the three Riemann invariants are obtained by substituting the corresponding value of A into (5.7) and (5.8) and then integrating. We now use the above simple wave theory to construct shock wave solutions. As in Section 2, three cases are dealt with separately.
Y. B. FU and N. H. SCOTT
1390 CASE i:
8 = n/2.
With &(n/2) = &(x/2)
= 0, equations (5.3) reduce to
PC22 = pl(;) PC23
=
-2 m: + O(m”), + 3&(5) m; + 56(“)
X6( 3 m2m3 +O(m”), (5.12)
pC33=f2(~)+56(S)m~+357(S)m:+O(m4).
Equations (5.10) and (5.11) then give
CASE i(a):
A: = C33 + O(m’),
(5.13)
nz = C22 + O(m”).
(5.14)
L = fAr.
By substituting il= *A1 into (5.7) and (5.8) invariants:
and then integrating,
we obtain the Riemann
(5.15)
where higher order terms have been neglected, and the alternative choices of the “k” signs correspond to the outgoing and incoming simple waves respectively. We now show that such an outgoing simple wave can follow shock 1 (defined in case i(a) of the previous section, see also the passage below (4.31)). To this end, we have to show that Rf, i = 1, 2, 3, are constant. It then suffices to show that [Ri] = 0 since Rt+ = 0. Across shock 1, [m2] = 0 and so from (2.14), [v2] = 0 and it is obvious from (5.15) that [R,] = [R2] = 0. (Note that m : = 0, v: = 0). With the use of (4.11), we have from (2.14) that c (-) l+-$$jb312
b31= -j/m i
22
[m31G
(5.16)
i
The other desired result [R3] = 0 then follows by taking the jump of (5.15~) with the use of (5.16). In the simple wave region behind the shock, the constant values of Ri are R; = 0, which together with (5.15) yields m2 = v2 = 0,
(5.17) (5.18)
1391
Acceleration waves and shock waves
where f;,, defined by (5.19) is a dimensionless material constant the magnitude of which characterises the degree of material nonlinearity in any direction with 6 = ~12. On differentiating (5.18) with respect to X and using the compatibility relation t~~,~= Q~, we obtain (5.20) This reduced equation of motion governs the propagation of simple waves behind the shock wave. It is of the same form as [2, equation (5.4)]. Therefore, all of the results there can be transcribed
here with the material
constants
g1 and & there
Jd here. As an example, we consider the propagation 5C-j 72 X = 0 with the strain m,(O, t) prescribed by
replaced
by &(f)
of a shock wave which is initiated at
O+stshl’“/A,
(5.21)
otherwise. Here the parameter
and
n may take any positive value and A is chosen as (5.22)
so that the initial jumps in [m3] and [FFZ~,~] are independent [~llx=o
= h,
[m3,_&=,, = k + O&h’).
Rewriting [2, equations (5.18), (5.25)] with the appropriate we have (?)I+‘“+
of n and are given by
(1 +;)tlkhX(+!r=
substitutions
1,
(5.23) of material constants
(5.24)
where higher order error terms have been neglected. The leading order evolution law for the shock amplitude can be obtained by solving the algebraic equation (5.24) for a given value of n and (5.25) is the explicit leading order evolution law for the amplitude of the accompanying second order discontinuity. CASEi(b): A = &Aq. It can be similarly shown that the outgoing simple wave corresponding to this case can follow shock 2 [defined in case i(b) of the previous section]. Results (5.24) and (5.25) apply here if we replace m3 and fr respectively by m2 and c2, the latter being defined as 1;2 = 55(;)/51(
f).
(5.26)
1392
Y. B. FU and N. H. SCOTT
CASEii: 0 = 0. In this case, we have cl(O) = &(O), Z&(O)= &(O) = &(O) in addition to &(O) = Z&,(O) = 0, and equations (5.3) reduce to PC22 = fl(0) + 55(0)(3& + m:) + Wn”), PC23 = 2&(O)VQ
+ O(m”), (5.27)
PC33 = MO) + &(O)(& + 3m:) + Wn’). Substituting (5.27) into (5.10) and (5.11) then gives PA:
= 51(O)
+ 355(OW + W”),
(5.28)
pn$ = &(O) + fs(0)m3 + O(m”).
(5.29)
CASEii(a): A = *Ai. By integrating (5.7) and (5.8), we can show that the three Riemann invariants corresponding to this case are RI = m3/m2 = constant, R~ = v2 f
$!$ii&
(1 + $$j
m2)m2 = constant,
1 + $$j
R3 = v3 f qm(
m2)m3 = constant.
(5.30)
Using the same arguments as in case i(a), we can show that such an outgoing simple wave [which corresponds to the “+” sign in (5.30b,c)] can follow shock 3, which was defined in case ii of Section 2. In the simple wave region behind the shock, the constant values of R, vanish as in case i(a), so that we have v2 +
YEXG
v3 + V&@&(1
(1+m55(O)
m2)m2
+ &$$
=
0,
m2)m3 = 0.
1
Combining these two equations,
we obtain
v+jfij$i&(l+~m2)m=0,
(5.31) 1
where v = vm and m = vm, as before. On differentiating and using the compatibility relation v,~ = m,,, we arrive at m,t+VZZGG(l+~53m2)m,x=0,
(5.31) with respect to X
(5.32)
where 5‘3 =
Mw51(0).
(5.33)
Equation (5.32) is of the same form as (5.20). Therefore, the shock wave under consideration evolves in the same manner as the two shock waves we have just studied in case i. The corresponding evolution laws can be obtained immediately from (5.24) and (5.25) by appropriate substitutions of material constants. CASEii(b): A = fA2. It can be shown that the simple waves corresponding to this case are circularly polarised and that m is constant in the simple wave region. Since we are dealing with waves advancing into an unstrained region, we must have m = 0 so that such simple waves cannot exist.
Acceleration waves and shock waves
1393
CASEiii: 8 # 0, 3612. On
inserting
(5.3)
into
(5.10) and (5.11), we obtain
EZ
A: = G3
4 El _ f* m: + o(m2),
-
Ez il; = c** + 4 51 _ f2 m: + o(m”). CASEiii(a): 3c= f&. Using these values of I in (5.7) and (5.8), we integrate invariant
(5.7) to obtain the first Riemann
5’4 R,==~.m*+;~~m:--
m$=constant.
E2-
(5.36)
51
As before, we can show that the constant value of RI is zero and so we have m2 = -
E4
E2-
Substituting invariants:
this relation
into
(5.8) and then
integrating
54 5-5
R2 = v2 f m
2
R,=t.+&@&(l+i
(5.37)
rn$ + o(m$).
El
(p+E 2
gives the other
rn: = constant,
two Riemann
(5.38)
1
(if, 22
))m:)m,=constant.
(5.39)
1
It can be shown that each Riemann invariant vanishes as before, so that v2=
v3+a On differentiating
(
1+1
-VZZ&f& &+ 2 ( E2
(5.40)
25: Er2(52-El)
>m
: > m3=o*
(5.41)
(5.41) with respect to X, we obtain m3,r +
VZZ(1+ i C4m?)m33 =0,
(5.42)
where (5.43) Once again, we have in (5.42) an equation of the same form as (5.20). Therefore, results concerning the evolution of shock 4 can be obtained from the analogous results in case i(a) by appropriate substitutions of material constants. CASEiii(b): A = fA2. We shall show that the outgoing simple wave corresponding to this last case can follow shock 5 [defined in case iii(b) of Section 21. This shock wave is, among all of the shock waves which may propagate, the only one which resembles a dilatational shock wave in its evolutionary behaviour in the sense that it is much more strongly affected by material nonlinearity than those which we have just considered.
Y. B. FU and N. H. SCOTT
1394
Putting 3, = flZ2 in (5.7) and (5.8) and then integrating,
we obtain
= constant,
+ gii3 m2) m2
R,=v,fV'-&7j(l
(5.44)
= constant,
(5.45)
m2m3 = constant.
R,=vrfd%79& 1
4
As before, we can show that each Riemann invariant vanishes both before and after the passage of the shock wave. In addition, we have m3 = v3 = 0 and so these equations are satisfied trivially except for (5.45) which becomes (5.47) Differentiating
(5.47) with respect to X then gives m2.t + V&&%1 + G5m2)m2,x = 0,
(5.48)
55 = E3i51.
(5.49)
where The governing equation (5.48) is of the same form as [l, equation (5.9)] for dilatational shock waves. If the shock wave is initiated at X = 0 by prescribing m2 in the same form as (5.21), then by transcribing [l, equations (5.23), (5.30)] with appropriate substitutions of material constants, we obtain (5.50) [m2,x] = k(1 + t;&X)-l.
(5.51)
For a given n, the leading order evolution law for the shock amplitude can be obtained by solving the algebraic equation (5.50). Summarising the results obtained in this section, we have shown that each of shocks l-4 evolve in the same manner and their evolution laws have the typical forms (5.24) and (5.25) for the specific initial conditions of the form (5.21), and that shock 5 evolves like a dilatational shock wave according to the evolution laws (5.50) and (5.51). Comparing (5.24) with (5.50) and (5.25) with (5.51) shows that shock 1 to shock 4 are much less strongly affected by material nonlinearity than is shock 5 because of the presence of the factor h in the typical combination g,kh in (5.24). The linear theory would put cl and & equal to zero. Evolution laws (5.24) and (5.50) reveal that the effects of nonlinearity are cumulative and they become most pronounced for distances of travel of order (5,kh)-’ for shocks 1 to 4 and for smaller distances of travel of order (&k)-’ for shock 5.
6. THE LIMIT
OF INEXTENSIBILITY
In this section we consider the propagation of shock waves in an unstrained incompressible transversely isotropic elastic non-conductor which is also inextensible along the preferred direction e. First, it is not difficult to establish the following relations between the linear elastic constants (Ye, a2 and a3 used here and the more familiar constants Pi, pL and /l used, for example, by Spencer [7]: @l
=
PT,
(Y2=;B.
a3=2(pL-
pT)>
(6.1)
Acceleration waves and shock waves
1395
pL, pT and j3 are as in (3.16)-(3.18). The material being inextensible in the direction e implies that EL+ TV, and hence through (3.18) and (6.1) that cy2-f~. We now study the behaviour of each of the shock waves in this limit.
where
CASEi: e = n/2. The two shock waves which can propagate in this case are controlled by material constants &@/2), E&r/2), 5&r/2) and E&/2). F rom (4.3) we see that they are all independent of (~2. Therefore, shock 1 and shock 2 behave in the same way as before. CASEii: 8 = 0. From (4.3) we see that when a2 is large, cl(O) is still finite, but &(O) is of order (Ye. This means that & is of order (Ye.Replacing t1 by & in (5.24) and taking n = l/2 as an example, we have the leading order evolution law [m3] = h(1+ 3&I&x)-“3.
(6.2)
The decay to zero of the shock amplitude occurs at a length scale of order (cy2kh)-’ which tends to zero as CY~--*~. This implies that shock 3 cannot propagate in the limit of inextensibility. This conclusion can also be verified by the following consideration. When the material is inextensible, we have J1 = 0, which from (2.6) gives m2 = 0 when 8 = 0. Therefore, as far as plane wave propagation is concerned, the material behaves like a rigid body in the direction of inextensibility. For another interpretation, let us go back to the constitutive equations (4.4) and (4.5), which reduce to n21
=
Elu%2
+
5mm2m2
+
O(m”),
n31
=
NW3
+
‘w%3m2
+
O(m4)9
when 8 = 0. We see that when the material is nearly inextensible, it exhibits strong material nonlinearity (since the second order material constant &(O) is large). We know that the decay of the shock amplitude is due to the material nonlinearity and the existence of initial higher order discontinuities, so it is no surprise that the material being nearly inextensible results in rapid decay of the shock amplitude. It may be remarked that the above prediction for shock 3 yielded by the weak nonlinear theory is totally different from that given by the linear theory. According to the linear theory, the existence and the evolutionary behaviour of a shock wave travelling in the preferred direction e is not affected by taking the limit of inextensibility; shock waves may always propagate with constant amplitude. The constraint condition J1 = 0 is inactive in the linear theory since then Ji = m2 sin 8 cos 8 which vanishes whatever m2 when 8 = 0 or ~r/2. We have seen, however, that this constraint condition is not inactive in the weak nonlinear theory. CASEiii: 8 #O, ~12. When CY,is large, we have from equations (4.3) the following approximations: & = 2a2 sin2 e ~0s~8,
Lj3= 3ff2 sin e c0s3 8,
E4 = CU, sin e c0s3 8, 57 = (YZ~0~48, provided that sin 8 and cos 8 are both of order one. Hence from (5.43) and (5.49) we have c4 =
O(l),
s;5 =
O(l),
(6.3)
noting that the O(cu,) terms on the right hand side of (5.43) cancel with each other. Since shock 5 travels with a speed given by (4.29) which tends to infinity as cr2+m, this shock cannot actually propagate. This is also made clear by considering the constraint condition J1 = 0, that is, m2sinB+~m2c0se=0.
(6.4)
Since shock 5 requires m3 = 0, (6.4) can be satisfied only if m2 = 0 also. Thus m = 0 and shock 5 cannot propagate even in the nonlinear theory in the inextensible limit.
1396
Y. B. FU and N. H. SCOTT
As for shock 4, since both its velocity and & are finite in the limite LQ-PCQ, it can still propagate as may be verified by considering (6.4). Equations (6.4) implies that
which is indeed satisfied by shock 4, as can be seen from (5.37) (note that .&/(Ljl - e2) = l/2 cot 0). This result comes out as something of a surprise, since in general one would not expect that equation (6.5), which is derived from the inextensibility condition J1 = 0, should coincide in the limit CY~-,~ with condition (5.37), which is required by the simple wave solution. In conclusion, we have shown that in the limit of inextensibility, shock 3 and shock 5 no longer exist according to the weak nonlinear theory, though shock 3 would be permitted by the linear theory. The shock waves which may propagate in such a constrained material are therefore shock 1, shock 2 and shock 4 and their evolution laws may be obtained in a similar manner to their counterparts in Section 4. Acknowledgement+-‘Ibe support of a joint grant given to the first author (Y.B.F.) by the Chinese State Commission of Education and the British Council is gratefully acknowledged.
REFERENCES [l] Y. B. FU and N. H. SCOTT, The evolution law of one dimensional weak nonlinear shock waves in elastic non-conductors. Q. J. Mech. appl. Math. 42,23-39 (1989). [2] Y. B. FU and N. H. SCOTT, the evolutionary behaviour of plane transverse non-linear shock waves in unstrained inwmpre~ible isotropic elastic non-conductor. Wave Alotion. To appear. f3] W. D. COLLINS, One-dimensional non-tmear waves propagating in incompressible elastic materiais. Q. 1. Mech. Appl. Math. l9,259-328 (lQ66). [4] R. J. WATERSTON, One-dimensional evolutionary discontinuities in compressible elastic materials. J. Inst. Math. Appf. 4,58-77 (1968). [5] I. C. HOWARD, Finite simple waves in compressible transversely isotropic elastic solids. Q. J. Mech. Appl. Math. 19,329-341 (1966). [ti] W. A. GREEN, Wave propagation in strongly anisotropic elastic materials. Arch. Mech. 30, 297-307 (1978). [7] A. J. M. SPENCER, Constitutive theory for strongly anisotropic solids. In C~ff~~u~ Theory of the mechanics of F&e-Reinforced Composites (Edited by A. J. M. SPENCER). Springer, Vienna (1984). (Received 20 March 1989)
0020-7225/89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc
ACCELERATION WAVES AND SHOCK WAVES IN TRANSVERSELY ISOTROPIC ELASTIC NON-CONDUCTORS Y. B. FUt and N. H. SCOTT School of Mathematics,
University of East Anglia, Norwich NR4 7TJ, U.K.
Abstract-We consider the propagation of plane acceleration waves and weak nonlinear shock waves in an unstrained transversely isotropic elastic non-conductor which is incompressible but extensible along the preferred direction denoted by the unit vector e. It is shown that such a material may transmit two different shock waves in the direction of any unit vector n which is neither perpendicular nor parallel to the preferred direction e. One of these shock waves is polarised in the plane II spanned by e and n and carries an entropy jump of third order in the shock amplitude, whilst the other is polarised perpendicularly to II and carries merely a fourth order entropy jump. In the degenerate cases when the angle 0 between n and e is either n/2 or 0, the two different shock waves which can propagate both carry fourth order entropy jumps. In all of these cases, an asymptotic evolution law describing the decay of the shock amplitude is obtained from a reduced first order governing partial differential equation which is itself obtained by differentiating a certain Riemann invariant that is approximately constant under the isentropic assumption adopted here (in common with all studies of non-conductors). We consider also the limiting case in which the material is inextensible along the preferred direction whilst remaining incompressible. It is shown that when 9 = n/2, two shock waves can propagate as before, but when 19= 0, no shock wave can propagate, and when 0 + n/2, 0, only one shock wave can propagate. Shock waves which can propagate in such a constrained material are all of the second type. The shock wave travelling in the direction of inextensibility is allowed by the purely linear theory of such constrained materials but is forbidden by the weak nonlinear theory.
1. INTRODUCTION
Within the family of propagating singular surfaces of different orders, shock waves are unique in that their amplitudes do not obey a single exact evolution equation; their evolutionary behaviour is always coupled with that of the higher order discontinuities which accompany them. It is probably because of this that the global evolutionary behaviour of shock waves has been less studied in the literature than that of acceleration waves, most previous studies on shock waves having been confined to their general properties and their instantaneous growth and decay behaviour. Weak nonlinear shock waves in elastic non-conductors fall into two categories, the first consisting of those shock waves across which the entropy jump is of third order in the shock amplitude, and the second consisting of those across which it is of fourth order. In two previous papers [l, 21 we have studied these two categories separately, choosing a prototype for shock waves belonging to each of them, in order to show how an asymptotic evolutionary law for the shock amplitude may be derived. In [l], plane dilatational shock waves in elastic nonconductors were found to have an entropy jump of third order in the shock amplitude and so to belong to the first category, whilst in [2], plane transverse shock waves in unstrained incompressible elastic non-conductors were found to have a fourth order entropy jump and so to belong to the second category. Shock waves in the first category were found to be much more strongly affected by material nonlinearity. In the present paper, we consider an elastic non-conductor with more general constitutive law, namely, an unstrained incompressible transversely isotropic elastic material. In view of the increasing interest in engineering in this kind of material (which includes a continuum model for the fibre-reinforced composites), it is desirable to clarify which categories of shock waves this kind of material can transmit and how any such shock waves evolve. For a general review of the relevant literature we refer to [l, 21 but three further papers are mentioned here. Simple waves and shock waves in incompressible isotropic and transversely isotropic elastic materials have been studied previously by Collins [3]. Shock waves in t Present address: Department
of Mathematics,
University of Exeter, Exeter EX4 4QE, U.K. 1379
1380
Y.
B. FU and N. H. SCOlT
compressible isotropic and transversely isotropic elastic materials have been considered by Waterston [4] and simple waves in compressible transversely isotropic elastic materials have been considered by Howard [5]. The results presented here are new in two respects. Firstly, the above-mentioned studies are all confined to the particular case in which the direction of propagation is perpendicular to the preferred direction, whilst in our discussion the direction of propagation is arbitrary. Secondly, Collins’ [3] and Waterston’s [4] studies on shock waves are concerned with the stability problem, whilst our concern is the determination of the evolutionary behaviour of the shock amplitude. After setting up the basic equations in the first section, we devote the second section to the discussion of acceleration waves in incompressible transversely isotropic elastic materials. We have two purposes in including such a discussion here. Firstly, the propagation of acceleration waves in such materials does not seem to have been studied before. Green [6] has studied acceleration waves in a nearly inextensible tranversely isotropic compressible elastic material and has employed a perturbation procedure to investigate the limiting behaviour of the wave speeds when the direction of propagation n tends to a direction perpendicular to the preferred direction e. However, he did not study the evolutionary behaviour of these acceleration waves. (In the linear theory, of course, all such plane waves propagate with constant amplitude.) We show here that on introducing the constraint of incompressibility, whilst retaining a fully general nonlinear theory, we are still able to analyse the limiting behaviour of the wave speeds and amplitudes exactly. Our second reason for studying acceleration waves is the desire to compare the propagation of acceleration waves with that of shock waves in order to draw the following conclusion: along any direction of propagation n, if an acceleration wave travels with changing (constant) amplitude, then a shock wave with the same polarisation will carry a third (fourth) order entropy jump and its evolution will be strongly (weakly) affected by the material nonlinearity. In Section 3, we determine the types of shock waves an unstrained incompressible transversely isotropic elastic material may transmit by expanding the strain energy function in a Taylor series as far as quartic terms in the strain gradient and then using the reduced jump conditions (2.12)-(2.14). It is shown that this kind of material may transmit one of each category of shock wave along any direction n which is neither perpendicular nor parallel to the preferred direction e. The shock wave of the first category, carrying an entropy jump of third order in the shock amplitude, is polarised in the plane II spanned by e and n, whilst the shock wave of the other category is polarised perpendicularly to II and carries a fourth order entropy jump. In the degenerate cases when the angle 8 between e and n is either 0 or n/2, the shock waves which propagate are necessarily of the second category. In Section 4, we study the propagation of simple waves in order to derive for each shock wave a reduced first order governing equation by using the isentropic assumption to construct a Riemann invariant that is approximately constant. We then show how to obtain the evolution laws for these shock waves by transcribing our previous results [l, 21 making appropriate substitutions for the material constants. The final section deals with the propagation of shock waves in an unstrained incompressible transversely isotropic elastic non-conductor which is also inextensible along the preferred direction. Results are obtained for this material by taking the appropriate limits of the corresponding results in the previous two sections. It is shown that when 8 = ~r/2, two shock waves can propagate as before, but when 0 = 0, no shock wave can propagate, and when 8 # n/2, 0, only one shock wave can propagate. The shock waves which can propagate in such a constrained material all belong to the second category. It is of interest to note that the purely linear theory of such a constrained material allows a shock wave to propagate in the direction of inextensibility though this is forbidden by the weakly nonlinear theory.
2. BASIC
EQUATIONS
Let the preferred direction e of the material lie in the X,+X,-plane at an angle 0 with the X,-axis which is chosen as the direction of wave propagation. The most general form of a plane
Acceleration waves and shock waves
1381
transverse motion is then given by Xl
=x1,
x2=x2
+ U2(XI,
x3 =x3
t),
+ U3(XI,
o,
(2.1)
where xi, i = 1, 2, 3, is the position at time t of a particle which is at position X,, A = 1, 2, 3, in the undeformed reference configuration, both sets of coordinates being referred to a common Cartesian frame. The quantities Ui(X1, t), i = 2, 3, are the particle displacements. The corresponding deformation gradient tensor and the Green strain tensor are given by 100
F=
respectively,
m2 1 m3 0
0 1
E=;(FTF-I)=
9
(2.2)
where m2=mi+m:,
m3=u3,~,
m2=u2,~,
(2.3)
a comma signifying partial differentiation. For simplicity, we have written X for X1. It is clear from (2.2a) that the incompressibility condition det F = 1 is satisfied automatically by the plane transverse motion (2.1). The behaviour of an incompressible transversely isotropic elastic non-conductor is completely described by the specific internal energy function E, which depends on the deformation through the four deformation invariants and on the specific entropy q: E = a,
Z2,Jb
(2.4)
J2, rl),
where Zl = tr E, Using
Z2= i (I: - tr E’),
(2.2b) and noting that e has components
J1= e - Ee,
.I2= e - E2e.
(cos 8, sin 8,0), we have
J1 = m2 sin 13cos 8 + k m2 cOsze,
(2.6)
J2=~m~sin2~+~m2cos2~+~m2m2sin~cosB+$m4cos2~. Therefore,
(2.7)
we can write E = 4Z, JI,
where ZdgfZ1= -2Z2. The first Piola-Kirchhoff given by Q = p &ldFti -pFi!, which gives
J2,
s),
stress in an incompressible
d&
n”=p=ip
(2.8)
elastic material is
(i = 2,3),
(2.9)
density and p is an arbitrary pressure. The reduced equations of motion are
where p is the constant
jdil,X
=
(i = 2,3),
Pi.t,
where v, = u,,~ are the velocity components,
(2.10)
whilst the reduced energy equation is ?j = 0.
(2.11)
1382
Y. B. FU and N. H. SCOTT
Finally, the following reduced jump conditions must be satisfied across a shock wave: [%I = P[s] = $ (JrJi:+
-P”Advi19
(2.12)
(i = 2,3),
nil)[W],
(2.13)
-“N[mi],
(2.14)
[VI2 09
(2.15)
[vi] =
the last being a consequence of the second law of thermodynamics. Here U, is the shock velocity in the reference configuration. In equation (2.13), summation over the repeated subscripts 2 and 3 is implied.
3. ACCELERATION
WAVES
In this section we study the propagation of acceleration waves in an unstrained incompressible transversely isotropic elastic non-conductor. Following the standard procedure we first derive the propagation condition and then determine the evolution laws. The basic kinematical compatibility relation for any function G(X, t) appropriate to one-dimensional acceleration waves is
-&[Cl =[G,xl +&N [GA
(3.1)
where now UN denotes the referential propagation speed of the acceleration denotes the space derivative following the wave. Replacing G by v, gives [P?li,t]= [Vi,x] = - & [tit] = - &
aij
wave and 3/3X
(3.2)
where a,, defined by Uj = [tii],
(3.3)
is the acceleration wave amplitude vector. To obtain the propagation condition, we first take the jump of the equations of motion (2.10) and use (3.1) and (3.3): [nil,,] = -puNai* (3.4) Recalling (2.9), we have
[Jhrl = q~,Jl?
(3.5)
where d2&
dnil =----&=
cij
-
I and a superscript “+” signifies evaluation
’
d??li
am,
(3.6)
’
in the natural strain-free
state. On substituting (3.5)
into (3.4) and using (3.2), we arrive at (p@/S,, - CC)% = 0, which is the desired propagation
condition.
It is straightforward
(3.7) to show that
(3.8)
Acceleration waves and shock waves
1383
Equation (3.7) then implies that two acceleration waves can propagate propagation speeds and amplitude vectors given by PGx = G,
in any direction,
with
az+O,
a3 = 0,
(3.9)
a2 = 0,
a3#0.
(3.10)
and P&
= G3,
It is clear that the two acceleration waves defined by (3.9) and (3.10) are polarised along the X,-axis and the X,-axis, respectively. We now proceed to establish the relations between the material constants (a&/%)+ etc. used here and the more familiar linear elastic constants. From (2.5) to (2.9) we see that 1 a& -JrZ1 =-m2-tdl P
$
1
(sin 8 cos 19+ m, cos2 e)
1 aa de -11-31=dlm3+dJm3cosze+1 P
- m2 + z (m’ + 2mz)sin e cos 8 + m’m, cos’ 8 ,
(3.11)
- m3 cos2 8 + m2m3 sin 8 cos 8 + m’m, co? e .
(3.12)
Since for an unstrained material there should be zero stress whenever both m2 and m3 vanish, we must take
a&+
o
(aJ,>=* Linearisation
(3.13)
of (3.11) and (3.12) then yields (3.14) (3.15)
For incompressible transversely isotropic elastic materials, given, for example, by Spencer [7] become
the linear stress-strain
relations
x21 = pLm2 f Bm2 sin2 e cos28,
(3.16)
3631= yTm3 + (iuL - k-b3 cos2e,
(3.17)
where pL and j.+ represent shear moduli (measured along and transverse to the preferred direction, respectively) and /3 is related to the Young’s modulus EL in the preferred direction by (3.18) B=&.-~PC~L+PT Comparing (3.14) with (3.16) and (3.15) with (3.17) gives =
2(PL- PT)P
B-
(3.19)
On introducing (3.19) into (3.8), we find that the squared wave speeds (3.9) and (3.10) become pi7& = pL + p sin2
e ~0s~e,
PO& = PL cos’ 8 + pT sin2 8.
(3.20) (3.21)
The material being nearly inextensible corresponds to EL being large, and hence, by (3.18), to /.I being large. Equations (3.20) and (3.21) show that in the limit of inextensibility, the acceleration wave which is polarised in the X,-direction and which has velocity ON3can still propagate, whilst the other acceleration wave with velocity ON2may propagate only if 8 = 0 or ~12 since otherwise ON*tends to infinity. However, when 8 = 0, the constraint relation J1 = 0 reduces to m2 - 0, which means that no disturbance actually propagates along the direction of
Y. B. FLJ and N. H. SCOTT
1384
inextensibility, although ON2is finite. Therefore, the acceleration wave which is polarised along the X2-axis can propagate only if 8 = ~r/2 and we have the exceptional case of two waves capable of propagating in any direction orthogonal to the direction of inextensibility. In any other direction only one wave can propagate except that no wave propagates in the preferred direction. This seemingly singular nature of the inextensible material can be resolved by considering a material that is almost inextensible. When p is large but not infinite, it can be deduced from (3.20) that oNz takes a large value of order (~/~~i’2 if both ~12 - 8 and 8 are of order one; it takes a value comparable with the other wave speed UN3if either 3r/2 - 8 or 8 becomes of the motion must order (pL//I)1’2. However, although ON2is finite when 8 is of order (p,/p)ln, necessarily be of vanishingly small amplitude. This is made clear by considering (2.6) which can be rewritten as (mz + tan
e)2 + rn: = tar?
8 + 2J1/cos2
8.
(3.22)
This relation confines m2 and m3 to lie on a circle in the m2,m3-plane of radius (tan2 8 + 2Jr/cos2 @)rn which is of order (~~/~)1’2 when 0 is of order (~~/~)~‘2. (J1 is of order &/3 since J,/3 remains of order one in the limit /3---, m). This circle contracts to a point when 0 = 0 and when the material is inextensible. Green [6] has considered acceleration wave propagation in compressible inextensibie and almost inextensible elastic materials and has used a perturbation procedure to study the behaviour of the wave speeds in the limit of inextensibility. In the inextensible limit our wave speed UN3is the same as his wave speed U, given by [6, equation (2.16)]. But since in our analysis the material is also incompressible, his wave speed U2 has no counterpart in this paper. We now turn to the derivation of the evolution laws. First, differentiating (2.10) with respect to time and using (3.6) and (2.11), we have (3.23) where (3.24) Repeated use of (3.1) gives [mj,x]
=
&
(3.25)
ajt
N
[mj,*t]
=
-$$ +& N
[Vj,c*]*
(3.26)
N
On taking the jump of (3.23) and using (3.2), (3.25) and (3.26), we arrive at (3.27) CASE
i: ~0% = p&
= Cz2, a2 # 0, a3 = 0.
Taking i = 2 in (3.27) and noting (3.8), we obtain (3.28) which has a solution a2(0)
(3.29)
a2(x) = 1 + (~~,/2p@&~(0)X where ~(0) is the initial amplitude. Using the definition (3.24), the expression
for C-&
can be calculated
with the aid of
Accelerationwavesand shock waves (2.5)-(2.8)
1385
and is given by
&=psinI!ZcostI
3 1 &‘+I
c&j+ +
+3
(
$
+
2>
3 cos2 8
3
d2& +
+
+ sin’ e cos2 e 2 . ( aJ: > ( 1)I
(3.30)
It then follows from (3.29) and (3.30) that the acceleration wave which is polarised along the X2-axis propagates with changing amplitude unless 0 = 0 or 1r/2. CASEii: ~0% = PI!?:, = Cz3, a2 = 0, a3 f 0. For this acceleration
wave, we take i = 3 in (3.27) giving the evolution equation ,.
-+
C da,+xai=(). dX
(3.31)
2pl&
It can be shown with the aid of (2.5)-(2.8) that c”333- 0 and therefore the acceleration which is polarised along the X3-axis always propagates with constant amplitude.
4. THE SHOCK
VELOCITY
AND THE ENTROPY
wave
JUMP
This section is devoted to discovering the precise nature of the shock waves which an unstrained incompressible transversely isotropic elastic material may transmit. In view of (2.8), the dependence of the internal energy function E on m2 and m3 is through I, J, and J2. Assuming that the region ahead of the shock is unstrained, we can expand E about this strain-free state in a Taylor series as far as fourth powers of the small quantities: PE = Carl + (YZJ:+
~y3J2 + p8o(Tj
+
~0) +
@lZJl+ /32JlJ2 + 835: + ~11~
y2ZJ: + y3ZJ2+ y4J’: + ysJfJ2 + yhJg + 6,( 7 - qo)Jl+ * * *,
(4.1)
where 4, /I,, yj and f3r are material constants and e0 and rlo are respectively the equilibrium temperature and entropy. In (4.1) we do not include the linear term in J1 since it gives a term in the expression for nzl which does not vanish with m2 and m3 [see (3.13)]. Only the linear terms in the entropy excess r,r- q. have been included in (4.1) in anticipation of the result that the entropy jump is at least of third order in the small quantities for a non-conductor. On substituting (2.5) (2.6) and (2.7) into (4.1), we obtain
+ i &mf: + peo(q - rlo) + &r4), where 1 c1 = (Y~+ 2 cu3+ 2a2 sin’ 8 c0s2 8, 1 E2 = al + 2 a3 CO? 8, E3 = 3 sin
8 cos e( 2 ( a3 +
j3, + i p2) + p3 sin’ 8 ~08~8 + @2~0s~e},
g4 = sin 6 cos e[ i ( a3 + @r) + ( a2 + a @2)~~s2e), E5=y,+~y3+$y6+(~3+~l+$32)~0s2e+~2c0s4e + (2P2 + 2y2 + ys)sin2 8 cos’ 8 +
6f?3
sin’ 8 cos4 8 + 4y, sin4 8 COST8,
(4.2)
1386
Y.
B. Fu and N. H. SCQn
+ Cu,COS40 + 4 & + y2 Sm 0 C0s2 8 + k (($3 + y&in* 6 cos4 8, (” P2
(
57= Yl + cu, + /3*+ ; y3>cos* 8 + ; 2&, + ,62f ; y6>cos4 8.
(4.3)
We have omitted the final term of (4.1) from (4.2) even though it is of fourth order when 77- pinis of th ir d order. This is because when q - q. is of third order we need only expand the strain energy as far as third order terms and so our omission does not affect the final results. Similar remarks apply to other approximate expressions in this section. Using (4.2), we have from (2.9) that x*1 = 5,m* +
53mI + E4m: + 5,m;
JE31= $$?I3 + Zj4rn2m3
+ $,mmZ
+ o(m3),
(4.4)
+ &jf?&Yl3 + &?2~ + o(t?Z”).
(4.5)
On inserting (4.2), (4.4) and (4.5) into (2.13), we arrive at 2&[77] = ; g3[m213 + 64[m21[m312 + ; &b214
+ h[m2]2[m312 +;
‘&b31”
+ o([m14>,
(4.(j)
and on substituting (4.4) and (4.5) into (2.12) with the use of (2.14), we obtain &‘[m,]
= 51[m2] + E3[m21* + 54[m312 + 55b21’ &b31
= [m31(g2 + 2$4b2]
+ 56[m21[m312 + o([m13)>
+ 56[m212 + t%[m31”) + o([m]“).
(4.7) (4. f-9
CASE i: 8 = z/2. In this case E3( 5) = g4( 5) = 0, and (4.7) and (4.8) reduce to
(4.9)
Pmm31=
rm31( E*(5)
+ &i(;)[m212
+ s,(;)rm31*}
+ ~([~I”).
(4.10)
These two equations are compatible only if either [m2] or [m3] vanishes. These two possibilities are now studied separately. CASE i(a): [mJ = 0, [m3] f 0. Equation (4.9) is satisfied trivially, whilst (4.10) gives (4.11) and from (4.6) we have [rl
The second law of thermodynamics this shock may propagate only if
= +&
e,( $314
+ WIm316)*
requires the entropy jump to be non-negative
e,(f)=Yi 2 0.
(4.12) (2.15) and so
(4.13)
Acceleration
1387
waves and shock waves
CASEi(b): [m2] # 0, [m3] = 0. In this case, expressions analogous to (4.11)-(4.13) Pug=
51(t)
[rll = &
are
+ e@m212+
e,(;)[m214
(4.14)
OW214),
(4.15)
+ o([m21”),
(4.16) CASEii: 13= 0. In this case &(O) = g4(0) = 0 as before. In addition, we have cl(O) = f2(0), &(O) = 56(O) = E,(O). Equations (4.7) and (4.8) therefore reduce to PGh21
=
b21(51@)
+ 5&9M2)
+ o(b13h
(4.17)
PW4
=
b31(51K9
+ MO)M2)
+ o(b13)7
(4.18)
from which we have PG
= 51(O)
+ EmM2
(4.19)
+ o([ml”),
while from (4.6) we obtain [VI=&
0
55(oM~1”
(4.20)
+ o(bl”>.
Equations (4.19) and (4.20) are both independent of the ratio [m2]/[m3]. This implies that the polarisation vector can be along any direction in the X2,X3-plane because of the symmetry. The second law of thermodynamics (2.15) requires 55(O)=n2+~3+B,+~~,+ul+~~3+a~6~0.
(4.21)
More generally, it can be shown that if the material is prestrained, two shock waves can propagate when 8 = 0. One of them is plane polarised and has the same character as the one discussed above. The other is circularly polarised and does not exist in an initially unstrained material. CASEiii: 8 # n/2,0. In this case, the only two possibilities are [m,] = 0 and [m,] # 0, with [m2] being always non-zero [as is clear from (4.7)]. To simplify the notation, we shall suppress the explicit dependence of the g’s on 8. CASEiii(a): [m2] # 0 [m3] # 0. Equations (4.7) and (4.8) yield Pv,
= 51 + 53[m2] + 54%
P%
+ ‘&[m212 + f6b312
= E2 + 254[m21 + t6b21”
+
+ o([m12),
57b31’ + o([m12)*
(4.22) (4.23)
These two expressions for pU$ are compatible only if (4.24) that is,
b21=-L,E2 -
El
[m3]2 + O([m,]‘).
(4.25)
Y. B. FU and N. H. SCOUR
1388
On substituting (4.25) into (4.23) and (4.6), we obtain
(
=)b31* HJ%=f*+ ~‘+6_51
[?I = &
0
+ot[m3l”h
(4.26)
+Wm31").
(4.27)
(5;+&)b314
1
By the second law of thermod~amics
(2.15) this shock wave may propagate
only if (4.28)
CASE iii(b):
[mzJ # 0 [m3] = 0.
Equation (4.8) is satisfied trivially, whilst (4.7) gives PG
= 51+
53b4 + O([m212).
(4.29)
From (4.6) we obtain
hl
=
&
By the second law of thermodynamics
mh.13
f
(4.30)
om*l”).
(2.15) this shock wave may propagate
only if (4.31)
Mm31 2 0.
None of the other shocks considered in this section has a squared wave speed possessing a linear term in the shock amplitude and none has a third order, rather than fourth order, entropy jump. This then is the only shock of the first category, rather than the second, that arises here. Summarising the results obtained so far, we have shown that when 8 = n/2, two shock waves may propagate, one being polarised along the X+&s and the other along the X,-axis; when 8 = 0, because of the material symmetry, a shock wave polarised in any direction perpendicular to n may propagate; when 8 # 0, 3t/2, two shock waves may propagate, one being polarised nearly along the X3-direction (in the sense that the component of the shock amplitude along the X,-axis is of second order in the component along the X,-axis), the other being polarised purely along the X,-axis. For easy reference in the following discussion, we refer to these five shock waves respectively as shock 1, . . . , shock 5. Our analysis has revealed that shocks 1-4 al1 belong to the second category in which the entropy jumps are of fourth order and the existence conditions (deduced from the second law of thermodynamics) for these shock waves all have the same form, namely 5 =L0, where 5 takes in turn the values &($), &($), Es(O) and jump across shock 5, on the other hand, is of third order and & + E:/(E2 51). Th e entropy the existence condition is of a different form: &[mJ L 0. Recalling the results of the previous section, we see that the acceleration wave counterparts of shock 1 to shock 4 always travel with constant amplitudes, whereas the acceleration wave counterpart of shock 5 may grow or decay. We shall show in the next section that shock 5 is the only transverse shock wave discussed here which resembles a dilatational shock wave in its evolutionary behaviour.
5. THE
EVOLUTION
LAWS
In the previous section, we found the types of shock waves which the materials under consideration here may transmit in any direction. In this section, we shall derive an asymptotic evolution law for each of the five shocks. First, we review briefly the theory of simple waves. Since according to (4.6), the entropy jump is of third order or higher in the shock amplitude, we shall work with the isentropic assumption within the order of approximation considered in
Acceleration waves and shock waves
1389
the rest of this paper. Then from (2.10) and (2.9) we have C22m2,x +
C23m3,x
=
VZ,[,
(5.1)
C32m2*x
C33m3.X
=
~3,~
(5.2)
+
From (4.2) and (3.6), we have the following expressions for C,: pG2 = 5i+ PC23
=
253m2
2hm3
+
+
3&d
2hm2m3
+
+
Ed
+
4m2),
o(m’)?
pC33 = E2 + 2g4m2 + &rns + 3&m: + o(m’). Equations (5.1) and (5.2) together can be written as a matrix equation:
with the compatibility
(5.3)
relations m2,t = v~,~, m3,r = v3,x
(5.4) where
We now look for simple wave solutions of (5.4) of the form
v =V(#),
(5.5)
where the function 9 = $(X, t) is to be determined and the curves $J(X, t) = constant represent travelling wavelets on which V is constant. On inserting (5.5) into (5.4), we obtain (AI - A)V’ = 0,
(5.6)
where Adsf - $J,~/@,~, and a prime signifies differentiation with respect to 9. Equation (5.6) can easily be manipulated to give the following equations: i1’ - C22
(
-G
-C23 rnb A2- C33>{mj I = O’
(5.7)
v;+Am;=O, v;+Am;=O.
I The characteristic
(5.8)
equation of (5.7) is A4
-
(c22
+
c,,)n’
+
c22c33
-
c;,
=
0.
(5.9)
Treated as a quadratic in 3c2,the above equation has two roots given by
A:
=
;
(C22
+
n: =; (Czz +
c33
+
I@22
-
c33)2
+
4G3)P
(5.10)
c33
-
V(C22
-
c33)2
+
4CZ3).
(5.11)
Therefore, the governing equations (5.1) and (5.2) admit two pairs of simple wave solutions. In each of the simple wave regions, the three Riemann invariants are obtained by substituting the corresponding value of A into (5.7) and (5.8) and then integrating. We now use the above simple wave theory to construct shock wave solutions. As in Section 2, three cases are dealt with separately.
Y. B. FU and N. H. SCOTT
1390 CASE i:
8 = n/2.
With &(n/2) = &(x/2)
= 0, equations (5.3) reduce to
PC22 = pl(;) PC23
=
-2 m: + O(m”), + 3&(5) m; + 56(“)
X6( 3 m2m3 +O(m”), (5.12)
pC33=f2(~)+56(S)m~+357(S)m:+O(m4).
Equations (5.10) and (5.11) then give
CASE i(a):
A: = C33 + O(m’),
(5.13)
nz = C22 + O(m”).
(5.14)
L = fAr.
By substituting il= *A1 into (5.7) and (5.8) invariants:
and then integrating,
we obtain the Riemann
(5.15)
where higher order terms have been neglected, and the alternative choices of the “k” signs correspond to the outgoing and incoming simple waves respectively. We now show that such an outgoing simple wave can follow shock 1 (defined in case i(a) of the previous section, see also the passage below (4.31)). To this end, we have to show that Rf, i = 1, 2, 3, are constant. It then suffices to show that [Ri] = 0 since Rt+ = 0. Across shock 1, [m2] = 0 and so from (2.14), [v2] = 0 and it is obvious from (5.15) that [R,] = [R2] = 0. (Note that m : = 0, v: = 0). With the use of (4.11), we have from (2.14) that c (-) l+-$$jb312
b31= -j/m i
22
[m31G
(5.16)
i
The other desired result [R3] = 0 then follows by taking the jump of (5.15~) with the use of (5.16). In the simple wave region behind the shock, the constant values of Ri are R; = 0, which together with (5.15) yields m2 = v2 = 0,
(5.17) (5.18)
1391
Acceleration waves and shock waves
where f;,, defined by (5.19) is a dimensionless material constant the magnitude of which characterises the degree of material nonlinearity in any direction with 6 = ~12. On differentiating (5.18) with respect to X and using the compatibility relation t~~,~= Q~, we obtain (5.20) This reduced equation of motion governs the propagation of simple waves behind the shock wave. It is of the same form as [2, equation (5.4)]. Therefore, all of the results there can be transcribed
here with the material
constants
g1 and & there
Jd here. As an example, we consider the propagation 5C-j 72 X = 0 with the strain m,(O, t) prescribed by
replaced
by &(f)
of a shock wave which is initiated at
O+stshl’“/A,
(5.21)
otherwise. Here the parameter
and
n may take any positive value and A is chosen as (5.22)
so that the initial jumps in [m3] and [FFZ~,~] are independent [~llx=o
= h,
[m3,_&=,, = k + O&h’).
Rewriting [2, equations (5.18), (5.25)] with the appropriate we have (?)I+‘“+
of n and are given by
(1 +;)tlkhX(+!r=
substitutions
1,
(5.23) of material constants
(5.24)
where higher order error terms have been neglected. The leading order evolution law for the shock amplitude can be obtained by solving the algebraic equation (5.24) for a given value of n and (5.25) is the explicit leading order evolution law for the amplitude of the accompanying second order discontinuity. CASEi(b): A = &Aq. It can be similarly shown that the outgoing simple wave corresponding to this case can follow shock 2 [defined in case i(b) of the previous section]. Results (5.24) and (5.25) apply here if we replace m3 and fr respectively by m2 and c2, the latter being defined as 1;2 = 55(;)/51(
f).
(5.26)
1392
Y. B. FU and N. H. SCOTT
CASEii: 0 = 0. In this case, we have cl(O) = &(O), Z&(O)= &(O) = &(O) in addition to &(O) = Z&,(O) = 0, and equations (5.3) reduce to PC22 = fl(0) + 55(0)(3& + m:) + Wn”), PC23 = 2&(O)VQ
+ O(m”), (5.27)
PC33 = MO) + &(O)(& + 3m:) + Wn’). Substituting (5.27) into (5.10) and (5.11) then gives PA:
= 51(O)
+ 355(OW + W”),
(5.28)
pn$ = &(O) + fs(0)m3 + O(m”).
(5.29)
CASEii(a): A = *Ai. By integrating (5.7) and (5.8), we can show that the three Riemann invariants corresponding to this case are RI = m3/m2 = constant, R~ = v2 f
$!$ii&
(1 + $$j
m2)m2 = constant,
1 + $$j
R3 = v3 f qm(
m2)m3 = constant.
(5.30)
Using the same arguments as in case i(a), we can show that such an outgoing simple wave [which corresponds to the “+” sign in (5.30b,c)] can follow shock 3, which was defined in case ii of Section 2. In the simple wave region behind the shock, the constant values of R, vanish as in case i(a), so that we have v2 +
YEXG
v3 + V&@&(1
(1+m55(O)
m2)m2
+ &$$
=
0,
m2)m3 = 0.
1
Combining these two equations,
we obtain
v+jfij$i&(l+~m2)m=0,
(5.31) 1
where v = vm and m = vm, as before. On differentiating and using the compatibility relation v,~ = m,,, we arrive at m,t+VZZGG(l+~53m2)m,x=0,
(5.31) with respect to X
(5.32)
where 5‘3 =
Mw51(0).
(5.33)
Equation (5.32) is of the same form as (5.20). Therefore, the shock wave under consideration evolves in the same manner as the two shock waves we have just studied in case i. The corresponding evolution laws can be obtained immediately from (5.24) and (5.25) by appropriate substitutions of material constants. CASEii(b): A = fA2. It can be shown that the simple waves corresponding to this case are circularly polarised and that m is constant in the simple wave region. Since we are dealing with waves advancing into an unstrained region, we must have m = 0 so that such simple waves cannot exist.
Acceleration waves and shock waves
1393
CASEiii: 8 # 0, 3612. On
inserting
(5.3)
into
(5.10) and (5.11), we obtain
EZ
A: = G3
4 El _ f* m: + o(m2),
-
Ez il; = c** + 4 51 _ f2 m: + o(m”). CASEiii(a): 3c= f&. Using these values of I in (5.7) and (5.8), we integrate invariant
(5.7) to obtain the first Riemann
5’4 R,==~.m*+;~~m:--
m$=constant.
E2-
(5.36)
51
As before, we can show that the constant value of RI is zero and so we have m2 = -
E4
E2-
Substituting invariants:
this relation
into
(5.8) and then
integrating
54 5-5
R2 = v2 f m
2
R,=t.+&@&(l+i
(5.37)
rn$ + o(m$).
El
(p+E 2
gives the other
rn: = constant,
two Riemann
(5.38)
1
(if, 22
))m:)m,=constant.
(5.39)
1
It can be shown that each Riemann invariant vanishes as before, so that v2=
v3+a On differentiating
(
1+1
-VZZ&f& &+ 2 ( E2
(5.40)
25: Er2(52-El)
>m
: > m3=o*
(5.41)
(5.41) with respect to X, we obtain m3,r +
VZZ(1+ i C4m?)m33 =0,
(5.42)
where (5.43) Once again, we have in (5.42) an equation of the same form as (5.20). Therefore, results concerning the evolution of shock 4 can be obtained from the analogous results in case i(a) by appropriate substitutions of material constants. CASEiii(b): A = fA2. We shall show that the outgoing simple wave corresponding to this last case can follow shock 5 [defined in case iii(b) of Section 21. This shock wave is, among all of the shock waves which may propagate, the only one which resembles a dilatational shock wave in its evolutionary behaviour in the sense that it is much more strongly affected by material nonlinearity than those which we have just considered.
Y. B. FU and N. H. SCOTT
1394
Putting 3, = flZ2 in (5.7) and (5.8) and then integrating,
we obtain
= constant,
+ gii3 m2) m2
R,=v,fV'-&7j(l
(5.44)
= constant,
(5.45)
m2m3 = constant.
R,=vrfd%79& 1
4
As before, we can show that each Riemann invariant vanishes both before and after the passage of the shock wave. In addition, we have m3 = v3 = 0 and so these equations are satisfied trivially except for (5.45) which becomes (5.47) Differentiating
(5.47) with respect to X then gives m2.t + V&&%1 + G5m2)m2,x = 0,
(5.48)
55 = E3i51.
(5.49)
where The governing equation (5.48) is of the same form as [l, equation (5.9)] for dilatational shock waves. If the shock wave is initiated at X = 0 by prescribing m2 in the same form as (5.21), then by transcribing [l, equations (5.23), (5.30)] with appropriate substitutions of material constants, we obtain (5.50) [m2,x] = k(1 + t;&X)-l.
(5.51)
For a given n, the leading order evolution law for the shock amplitude can be obtained by solving the algebraic equation (5.50). Summarising the results obtained in this section, we have shown that each of shocks l-4 evolve in the same manner and their evolution laws have the typical forms (5.24) and (5.25) for the specific initial conditions of the form (5.21), and that shock 5 evolves like a dilatational shock wave according to the evolution laws (5.50) and (5.51). Comparing (5.24) with (5.50) and (5.25) with (5.51) shows that shock 1 to shock 4 are much less strongly affected by material nonlinearity than is shock 5 because of the presence of the factor h in the typical combination g,kh in (5.24). The linear theory would put cl and & equal to zero. Evolution laws (5.24) and (5.50) reveal that the effects of nonlinearity are cumulative and they become most pronounced for distances of travel of order (5,kh)-’ for shocks 1 to 4 and for smaller distances of travel of order (&k)-’ for shock 5.
6. THE LIMIT
OF INEXTENSIBILITY
In this section we consider the propagation of shock waves in an unstrained incompressible transversely isotropic elastic non-conductor which is also inextensible along the preferred direction e. First, it is not difficult to establish the following relations between the linear elastic constants (Ye, a2 and a3 used here and the more familiar constants Pi, pL and /l used, for example, by Spencer [7]: @l
=
PT,
(Y2=;B.
a3=2(pL-
pT)>
(6.1)
Acceleration waves and shock waves
1395
pL, pT and j3 are as in (3.16)-(3.18). The material being inextensible in the direction e implies that EL+ TV, and hence through (3.18) and (6.1) that cy2-f~. We now study the behaviour of each of the shock waves in this limit.
where
CASEi: e = n/2. The two shock waves which can propagate in this case are controlled by material constants &@/2), E&r/2), 5&r/2) and E&/2). F rom (4.3) we see that they are all independent of (~2. Therefore, shock 1 and shock 2 behave in the same way as before. CASEii: 8 = 0. From (4.3) we see that when a2 is large, cl(O) is still finite, but &(O) is of order (Ye. This means that & is of order (Ye.Replacing t1 by & in (5.24) and taking n = l/2 as an example, we have the leading order evolution law [m3] = h(1+ 3&I&x)-“3.
(6.2)
The decay to zero of the shock amplitude occurs at a length scale of order (cy2kh)-’ which tends to zero as CY~--*~. This implies that shock 3 cannot propagate in the limit of inextensibility. This conclusion can also be verified by the following consideration. When the material is inextensible, we have J1 = 0, which from (2.6) gives m2 = 0 when 8 = 0. Therefore, as far as plane wave propagation is concerned, the material behaves like a rigid body in the direction of inextensibility. For another interpretation, let us go back to the constitutive equations (4.4) and (4.5), which reduce to n21
=
Elu%2
+
5mm2m2
+
O(m”),
n31
=
NW3
+
‘w%3m2
+
O(m4)9
when 8 = 0. We see that when the material is nearly inextensible, it exhibits strong material nonlinearity (since the second order material constant &(O) is large). We know that the decay of the shock amplitude is due to the material nonlinearity and the existence of initial higher order discontinuities, so it is no surprise that the material being nearly inextensible results in rapid decay of the shock amplitude. It may be remarked that the above prediction for shock 3 yielded by the weak nonlinear theory is totally different from that given by the linear theory. According to the linear theory, the existence and the evolutionary behaviour of a shock wave travelling in the preferred direction e is not affected by taking the limit of inextensibility; shock waves may always propagate with constant amplitude. The constraint condition J1 = 0 is inactive in the linear theory since then Ji = m2 sin 8 cos 8 which vanishes whatever m2 when 8 = 0 or ~r/2. We have seen, however, that this constraint condition is not inactive in the weak nonlinear theory. CASEiii: 8 #O, ~12. When CY,is large, we have from equations (4.3) the following approximations: & = 2a2 sin2 e ~0s~8,
Lj3= 3ff2 sin e c0s3 8,
E4 = CU, sin e c0s3 8, 57 = (YZ~0~48, provided that sin 8 and cos 8 are both of order one. Hence from (5.43) and (5.49) we have c4 =
O(l),
s;5 =
O(l),
(6.3)
noting that the O(cu,) terms on the right hand side of (5.43) cancel with each other. Since shock 5 travels with a speed given by (4.29) which tends to infinity as cr2+m, this shock cannot actually propagate. This is also made clear by considering the constraint condition J1 = 0, that is, m2sinB+~m2c0se=0.
(6.4)
Since shock 5 requires m3 = 0, (6.4) can be satisfied only if m2 = 0 also. Thus m = 0 and shock 5 cannot propagate even in the nonlinear theory in the inextensible limit.
1396
Y. B. FU and N. H. SCOTT
As for shock 4, since both its velocity and & are finite in the limite LQ-PCQ, it can still propagate as may be verified by considering (6.4). Equations (6.4) implies that
which is indeed satisfied by shock 4, as can be seen from (5.37) (note that .&/(Ljl - e2) = l/2 cot 0). This result comes out as something of a surprise, since in general one would not expect that equation (6.5), which is derived from the inextensibility condition J1 = 0, should coincide in the limit CY~-,~ with condition (5.37), which is required by the simple wave solution. In conclusion, we have shown that in the limit of inextensibility, shock 3 and shock 5 no longer exist according to the weak nonlinear theory, though shock 3 would be permitted by the linear theory. The shock waves which may propagate in such a constrained material are therefore shock 1, shock 2 and shock 4 and their evolution laws may be obtained in a similar manner to their counterparts in Section 4. Acknowledgement+-‘Ibe support of a joint grant given to the first author (Y.B.F.) by the Chinese State Commission of Education and the British Council is gratefully acknowledged.
REFERENCES [l] Y. B. FU and N. H. SCOTT, The evolution law of one dimensional weak nonlinear shock waves in elastic non-conductors. Q. J. Mech. appl. Math. 42,23-39 (1989). [2] Y. B. FU and N. H. SCOTT, the evolutionary behaviour of plane transverse non-linear shock waves in unstrained inwmpre~ible isotropic elastic non-conductor. Wave Alotion. To appear. f3] W. D. COLLINS, One-dimensional non-tmear waves propagating in incompressible elastic materiais. Q. 1. Mech. Appl. Math. l9,259-328 (lQ66). [4] R. J. WATERSTON, One-dimensional evolutionary discontinuities in compressible elastic materials. J. Inst. Math. Appf. 4,58-77 (1968). [5] I. C. HOWARD, Finite simple waves in compressible transversely isotropic elastic solids. Q. J. Mech. Appl. Math. 19,329-341 (1966). [ti] W. A. GREEN, Wave propagation in strongly anisotropic elastic materials. Arch. Mech. 30, 297-307 (1978). [7] A. J. M. SPENCER, Constitutive theory for strongly anisotropic solids. In C~ff~~u~ Theory of the mechanics of F&e-Reinforced Composites (Edited by A. J. M. SPENCER). Springer, Vienna (1984). (Received 20 March 1989)