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On the behavior of shock waves in deformable dielectric materials
ht. I. Engq %I. Vol. 21, No. IO, pp 1145-1155. 1983 Printed in Great Britain
0
002Cb7225/83 $3.00+ .oO 1983 Pergamon Press Ltd.
ON THE BEHAVIOR OF SHOCK WAVES IN DEFORMABLE DIELECTRIC MATERIALS
Universitt
B. COLLET Pierre et Marie Curie (Paris VI), Laboratoire de Mecanique Thtorique Tour 66,4 place Jussieu, 75230 Paris Cedex 05, France
(Communicated
associt
au C.N.R.S.,
by G. A. MAUGIN)
Abstract-In this paper, the behavior of plane shock waves propagating in elastic dielectrics is examined. The differential equation governing the amplitude of wave and formulae concerning the polarization changes across the shock are deduced. Some particular situations are studied in detail.
I. INTRODUCTION THERE
exists an important class of natural or synthetic solid deformable materials which exhibit electromechanical coupling effects. Such solids are said to be piezoelectric, electrostrictive, . . . There also exists a quite common sub-class of piezoelectric materials which possess the remarkable property to exhibit a spontaneous electrical polarization. Materials of this sub-class are said to be ferroelectric[ I, 21. Some of these deformable dielectric materials selected for their intrinsic properties (exceptionally high piezoelectric coupling coefficient, high dielectric constant, good stability in time and temperature, very low loses,. . .) are used in various devices[4-61 such as transducers, filters, resonators, delay-lines, acoustic elements, shockactuated power supplies,. . . . In recent decades significant advances have been made in the experimental and theoretical study of the formation and propagation of shock waves in solids[7-lo]. Much of this work has been directed to the use of shock waves as a convenient method for the determination of the equation of state of a solid under dynamic conditions and to the development of mathematical techniques on the basis of which the propagation of large-amplitude waves can be analyzed in certain special cases. In particular, as a result of these researches, some substantial progress in the understanding of thermal effects and rate effects on the propagation of shock waves has been obtained from both the theoretical and experimental point of view. Advances have also been made in the analysis of the propagation, under certain circumstances, of shock waves in deformable materials of solid type in interaction with electromagnetic fields[ll-IS]. By their very nature, these studies are in general complex, but this, evidently, does not mean that it is not possible to extract from past and actual works in this area useful informations which provide a better physical knowledge of electromechanical effects and electrical conduction effects on the response of deformable electromagnetic materials when subjected to dynamic loads. The purpose of this paper is to examine the behavior of plane shock waves in nonlinear elastic dielectric media. We assume that the stress and the electric field depend on the deformation gradient and the polarization per unit mass. The constitutive relations proposed include the particular behavior of deformable ferroelectric materials whose relaxation properties can be neglected. We derive the equations both for the propagation velocity and the amplitude of the shock wave as functions of the jumps in certain quantities across, and the state of the material immediately ahead of the wave front; we also give explicit formulae for the changes in the polarization per unit mass across the discontinuity surface, what should be of interest to experimentalists. After discussing the implications of the results, we consider special cases: combined mode, normal mode and axial mode. For the first mode each component of the polarization per unit mass which is parallel to it. Finally, the analysis is specialized to the case of weak shocks. UP.5 Vol.
21. No. 10-A
1145
B.COLLET
1146
2.BASIC EQUATIONS AND CONSTITUTIVE ASSUMPTIONS
Let (X, Y, Z) be the rectangular Cartesian coordinates of material point of a homogeneous nonlinear elastic dielectric material in a fixed reference configuration K0 with mass density p,,. We assume that the “generalized” motion of the body may be described by the set of scalar functions x(.,t), ~~,(.,t) and xn2(.,f) such that (x = x(X, t), Y, Z) and (r, =x=,(X, t), nTT2 = xn2(X, t), 0) give, respectively, the location and the polarization per unit mass of the material point (X, Y, Z) at time t. In addition, we assume that at each point of the body the electric field is of the form (E, = 6(X, t), 6 = 6(X, t), 0). In the framework of quasi-electrostatics the relevant global balance equations, in the absence of body force of non electrostatic origin, are[16-181
(2.1)
POUdX = CCX,,t) - x(X,, t),
0 = wx,, t) -
wx,, t),
(2.2)
0 = c%,cxp, t) - %2(X,,t),
(2.3
where v=
a,x, F = axx,
9 = E, + poF-‘a,.
(2.4)
Equations (2.1)-(2.3) must hold at all times t and for all X,, X, in Ko. u = u(X, t), F = F(X, t) > 0, C = Z(X, t), 9 = 9(X, t), represent the velocity of the solid continuum, the deformation gradient, the one-dimensional stress, the electric displacement in the X direction, respectively. Equation (2.1) is the one-dimensional statement of the law of balance of linear momentum, (2.2) is the one-dimensional Gauss law in the absence of free charges and (2.3) is the one-dimensional form of the Faraday law of quasi-electrostatics. We assume a functional dependence for C, E, and cCR2 of the form
C = %F,~T,,v), EI
=
-J%(E
TI,
7~21,
TI,
~2).
(2.5)
1 $2
=
82E
For later use, in view of (2.4), and (2.9, it is more convenient to introduce the constitutive relation ,. 22= 9(F,
(2.6)
~2).
T,,
Furthermore, we assume that the response functions 2, 6 and g2 are of class C2. Thus the following material coefficients defined by a=d(F,~l,~2)=~,
a$
aC
b,=&(F,n,,lr3=~,
1
”
.
2(F, T,, 9~2)= $,
2 A
d2 = ci,(F, 7~,,7~~)= g,
*
‘9~ d, = d,(F, r,, n2) = g, c = c(F, PI, ~2)= aF,
e=
x3
A
b2=b2(F,~,,~J=~,
I
(2.7) 2
I A
A
f, =j,(F,~T,,TT~)= 2,
f2 = fz(F,
T,,
4
=
2, 2
are of class C’. We further assume that NE
TI,
7~2) >
0,
&(F,
r,,
~2)
>
0,
f2(F,
7~19 ~2)
>
0.
(2.8)
On the behavior of shock waves in deformable dielectric materials 3. GENERAL First,
we
PROPERTIES
1147
OF SHOCK WAVES
assume that the motion contains a plane shock wave moving in the X direction
with intrinsic velocity d V(t) = z s(t) > 0,
(3.1)
where (S(t), Y, Z) with - m < Y, Z < m are the material point at which the wave is to be found at time t. The jump I[fDin any variable f(X, t) across the shock wave is defined as
nfn=f--f’
(3.2)
where fz = limx+s=(t)f (X, t). Since U(t) > 0, f- and f’ are respectively, the limiting values of f immediately behind and just in front of the shock surface. Furthermore, [f] must obey the kinematical condition of compatibility [B]
$nm=ain+ unaxrn,
(3.3)
where s/St is the displacement derivative and a superimposed dot denotes the material derivative. As usual in the theory of shock waves, we assume that x is a continuous function everywhere, while F, u, 7~~and a2 and their derivatives suffer jump discontinuities across the shock. They were continuous everywhere else. The global laws (2-l)-(2.3) imply for all X# S(t) a,2
= pod, a,9 = 0,
axg2 = 0,
(3.4)
and across the shock, we have
~~uud, haI= pm, ua=o, uwa=o, usn= -
IWI = 0,
[a,sJ
(3.5)
= .
Clearly, the formulae (3.4),, imply that the electrical displacement in the X direction and the electric field in the Y direction are uniform in X; they may, of course, be expressed as functions of time alone. In view (3.3) with f = 9 and f = iZ2respectively and (3.5),, we see that
u&j= 0, u$n= 0.
(3.6)
Thus, we have the important results that not only 9 and ccP2 are continuous across the shock wave but so are their first (and all higher) order derivatives. The compatibility condition (3.3) with f = x implies that
nun= -uua,
(3.7)
and this relation together with (3.5)i yields the well-known result
u2= umpm,
(3.8)
for the intrinsic velocity of the shock. If we put f = u and f = F successively in (3.3) and combine the resulting relations with (3.5), and (3.7), we obtain
2U(G/St)uFn+uFn(S/St)u =u2[axFk(ilpo)[a,sj, which is the equation the amplitude [F]I of the shock wave must obey.
(3.9)
B.COLLET
1148
4,SHOCK AMPLITUDE EQUATION
In this section we use (3.9) to derive the differential equation governing the amplitude [F] of a shock wave propagating in an elastic dielectric body. To this end, we obtain explicit expressions for [&.Z] and sU/Zit. From (2.5), and (2.7)1,2,3 we have ax2 = a&F + b,axa,+ b2dXT2,
(4.1)
for all Xf S(t). From (2.5),, (2.6), (2.7)~ and (3.4)~~we obtain caxF + d,axn, + d2aXT2 = 0,
(4.2)
A solution of the system (4.2) is given by ax7h = -g,a,F, ax7b=
(4.3)
-g2axF,
where
g, = i,(E
~1,
~2)=
cf2- 4 4f2 - &f,’
g2 =
i2(E
rl,
572) =
cft- 4
d2f,
_
d,f2
(4.4)
This enables us to write (4.1) in the form a,%= ha,F,
(4.5)
where h = h(F,
rl,
i-r2)= a - b,g, - b2g2.
(4.6)
Since the jump in any product fg may be written in the form
nfd=f-ud+g+ufk
(4.7)
from (4.5), it follows that (4.8) Hence (3.9) and (4.8) imply that
(4.9)
Taking the displacement derivative of (3.8), we obtain (4.10) Now, by virtue of the definitions (2.7),-j we have
sx6t=a
wn
--+a-s+b;-6t SZ’
+
s;ln + b;!!!$ + b;m 6t +b;+, : SF+
-zi--a+-+b,--St
+‘fint;+ l,,;!$,
(4.11)
(4.12)
On the behavior
of shock waves in deformable
dielectric
materials
I149
and on taking the difference of (4.11) and (4.12), we find (4.13) It follows from (3.3), (4.3),,* and (4.7) that
&n=b,nhma- m(w)+, (4.14)
~Ud=W WUaxFPvuaw)+. From (2.7)4-9r(3.6)~ and (4.7) we obtain the system d;uti,n + d;ij7i2n = - c-@n - ucnP+ - ud,nir: - ud2n7j;,
(4.15) so that u?i,n= - g;iP]-
j,P+ - k,7j; - i,+;, (4.16)
where . I’=
k,
=
Udf;-uend2 . U4lf;-n4ldr d;f;-d;f; ’ 12= d;f;_d;f;
’
lId,lJf~ - Uf,lld; , kz= Ud,lJl - Uf,!dr d,fz -&f; dzf, -d;fi ’
/,J&~f+f#i d,f,-d;fT
(4.17)
I = IIdz~f I- Ufzlb T ’ ’ dzf, -d;f; ’
On substituting from (4.16) into (4.14) we obtain
$hn=-g;$uw~i,i+-k,ii:-h?i:uug,n(m+, (4.18)
fad=-n;$Fn-
j21s-+-k27;t-12ti:-
U[gJ(a,F)+.
The substitution of (4.18) into (4.13) with (4.3),.2, (4.6) and (4.7) yields $p1=+Fn+
u[h~(axF)++{[a~-j,b;-j2b;}~+ (4.19)
so that eqn (4.10) then gives
(4.20) +{l[bJ-
/lb;--
&b;},$.
1150
B.COLLET
Finally, on substituting from (4.20) into (4.9), we find that the amplitude [F] of shock wave propagating in an elastic dielectric body obeys the equation
(h- - pl#)
(4.21)
@*-kJxFk
where 1 I{Uall-j,b;-j~~~}~+3Uhil(axF)+ A* = - 2(h- - pJJ’){ (4.22)
Clearly, the growth equation is very complicated and as in other shock-wave theories it is not possible to deduce any useful information from it without adopting additional assumptions[8, 111.In the following section we shall consider the implications of (4.21) under certain particular circumstances. 5,SOMEPROPERTIESOFTHE
SHOCKTRANSITION
As a direct consequence of the description proposed, it is evident that the jump [Fj in the deformation gradient and the jumps [a,] and [TJ in the components of the polarization per unit mass in the X and Y directions, respectively cannot be independent. Indeed, their relationship is given by (3.&, i.e. &(F-, &(F-,
n;, T;)- i&F+,T:, n,‘) = 0,
(5.1)
r:, P:) = 0.
(5.2)
T;, T;) - &(F+,
Equations (5.1) and (5.2) regarded as functions of (F-, TT;, T;, F+, TTT:,T;) are assumed to be solvable and to yield 7; and a; as functions of (F-, F+, P:, TT;). In other words, we assume that there exist functions ii, (.,.,.,.) and ijz (.,.,.,.) such that
Umll= %(UFD, F’, TTT:, d), (5.3)
b2n= %W.F', r:, 6), where we have used the relations F- = F+ +[Fn, IT; = ?r: + [n,] and T; = nt + UT& We now define the function e(.,.,.,.) such that
(5.4) From (2.7),_3,(5.3),, and (5.4) we have (5.5) On taking the displacement derivative of (5.3),,2and employing (4.3),.*we obtain
(5.6)
~~uFn+~++~?i;+~?if+(~-R;~-g2~)U(a,F) -$iT2n= I 2
I
2
On the behavior of shock waves in deformable dielectric materials
1151
Since (4.18),,*and (5.6) hold for arbitrary shock waves, we have aii, _
86 _ _ -
g’T
a[F]
a+,_ +arl
ati, _/
I,
z=
-
f$=
_k
a*2 __
a[F]
g2,
ae2_ fanI
. -11,
aF’-
_k
” (5.7) -j2,
a+2__l 21 aa,‘-
*.
These relations should be of interest to experimentalists concerned with the study of polarization changes produced by shock waves. They show how the jumps Up,] and 174 are influenced by the various parameters. In particular, they give the derivatives of jumps UT,] and [TJ in terms of the jumps in certain material coefficients and the values of material coefficients prevailing in the material just before the arrival of the shock. Thus, by (4.6), (5.5) and (5.7),,5, we find
1
?r:,7r:)=a-b;g; - big; @IFI, F+,
= h(F-, r;, 7~;).
(5.8)
We assume that there exist functions 7j,(.,.,.) and ij*(.,.,.) such that rr~= +,(F, 9,821, (5.9) 7~2 =
7?2(E
9,821,
and it follows from (3.4),, and (4.3),,2that a+, -#=
-g,,
$=
-g,.
(5.10)
We now define the function 2 (.,.,.) by X = %F, %a,)
=
%F,
ii,(F,
9,
$21,
CiF,
9,
%2h
(5.11)
and from (2.7),_,, (4.6) and (5.10),,2we have h(F9
9 9
gJ=aS_a$ atafi, aeai;, ~-~+an~+;i;;~=a-b,g,-b2g2=h. 2 I
(5.12)
2
Thus we see that i simply is the stress-strain modulus at constant longitudinal material electric displacement and transverse material electric field. Henceforth, we shall assume that h(F, %%,) > 0.
(5.13)
We now consider the implications of (4.21) on the evolutionary behavior of certain types of shocks. Consider a compression shock propagating in a material which is initially in compression or unstrained so that
[F]I
F+
We assume that the stress-strain law (5.11) is concave (5.13) a*w,
93
aF2
a21
<
0
for
all
(5.14)
from below, and, hence, by virtue of
FG 1,
(5.15)
B. COLLET
1152
Clearly 2 (.,.,.) is assumed to be in C* class. It follows from (3.8), (5.12)-(5.15) that h- > p#,
(5.16)
[h] > 0.
From (4.21) and (5.16), it is clear that if IF! is the amplitude of a compressive shock wave propagating in an elastic dielectric, which is initially in compression and for which (5.13) and (5.15) hold, then at any instant [a#] 5 A* *;
(5.17)
lI[F]lZ 0.
It is clear from (5.17) that whether the amplitude of a shock wave grows or decays depends on the relative values of the jump [a#] in the gradient of the deformation gradient or strain gradient across the shock and the parameter h* defined by (4.22). In accord with the classical nomenclature we call A* the critical jump in strain gradient. 6. PARTICULAR
CASES
As consequence of the very general modelization the foregoing results are obviously complicated and it is not easy to extract “simple” informations without introducing some additional assumptions. We now examine three particular situations: (i) Combined mode We assume that the material considered may be characterized by the constitutive equations c = QF, Tl, %), 1 E, = ME n,), ^ $2 = %‘,(F, ~2).
(6.1)
It follows from (2.5), and (6.1), that we have . 9 = 9(F, a,).
(6.2)
It is clear from (2.7) that in material of this type d:: = f, = 0,
(6.3)
so that, by (4.4) (4.17) and (5.17), we find
aii,
C-
i$q= -d;’
_@I_ aF+
aii, Ud,B a6 _-_ an+ I -
d;'
d;'
_.
an:- ’
!!c& -kJ,
-=-g,
aua af2 a=+-I
_ug
(6.4)
aiiz
--
-
2
o
3
32=-y. 2
2
(ii) Normal mode If we assume that TT,= E, = 0 and that the behavior of the material is described by the constitutive equations
On the behavior
of shock waves in deformable
dielectric
materials
1153
it is clear that we have dii2 _ -- e-_ dlFD fi’
*_
aF+ -
_u
_E& _y
fi ’
2
(6.6) 2
(iii) Axial model’ If we assume that 7r2= SET2 = 0 and that the behavior of the material is defined by the constitutive equations
C = i
A 9 = WE 7~11,
it is obvious that we obtain
aii, -=
-- c-
a[F]
d;'
ucn
acl _ aF+ -
d;’
aii,
---x. aa+-
6dJl
(6.8)
I
In three above-examined situations (6.4),,s, (6.6), and (6.8)r determine the jump in the polarization per unit mass across the shock surface. In view of (2.8)2,3we have aii,
aiiz
-= -sgnc-, sgn a[Fj
= - sgn e-, sgn a[Fl
(6.9)
and, consequently, the jump in polarization per unit mass T,(T~) will increase or decrease across a compression shock according as c- > 0 (e- > 0) or c- < 0 (e- < 0). Now the critical jump in strain gradient (4.22) should be replaced by: (i) Combined mode
hc=a-c$-eb, I
(6.11)
f2
(ii) Normal mode
where (6.13)
hN=a-e?, 2
(iii) Axial mode
“=
* -2(h;-p&J
2
([IIh*l+c+[~D)~+3Uh,l(a,F)+
tNote that this situation is equivalent to that described in[l I]. However, analysis of induced polarization or depolarization phenomena as observed waves.
+d$#$
(6.14)
our approach seems to be better adapted to the in dielectric materials and produced by shock
B.COLLET
1154
where (6.15)
hA=o-+ I
Furthermore if the conditions ahead of the shock are steady so that F, ~~ (nz) are independent of time then
( W?+ 2(h
&,A
-
Pou
(6.16)
c.N.~
and for a compressive shock it follows from (5.16)1,2that
sgn0 bd
= - sw {(W)‘l.
(6.17)
Equation (6.17) states that the sign of the critical jump in strain gradient is the negative of the sign of the strain gradient ahead of the wave. Thus for a wave propagating into a region of decreasing strain, A$,,,* would be positive, opposite to the case of wave propagating into a region of increasing strain. Another special case of (4.21) arises when the region ahead of the wave is at rest in a uniform state. Here the critical jump in strain gradient vanishes and the shock will grow, decay or remain steady according as (&F)- is positive, negative or zero. This last result is analogous to that arising in purely elastic shock-wave theory.
l.WEAK SHOCKWAVES
In this section we determine some properties of shock waves in the limit F-+F+, i.e. we consider the case of a weak shock wave or shock wave of infinitesimal amplitude. For the sake simplicity we derive only expressions for [Z], IT,], [Ed, U and A* in the three particular cases examined in Section 6. Since (lljF]lG l), from (3.5)3,5,(5.11) and (5.12) we obtain
where cc
=
%F,
9,
821,
Z:N
=
%F,
821,
C,A =
%F, 9),
Hc.r.,,/,= a&N,,+
(7.2)
From (5.3),.2we have (7.3) and in view of (6.4), (6.6) and (6.8), we may write
am
_
-awn I =--IuFn=o=-2’ aii2
aii,
IFn=O
nF1=0
;;
(7.4)
so that for the (C,A) and (C,N) cases (7.5) By using similar arguments it may be shown that
ukN,.d =ffbv,~uFn+ 4um
1155
On the behavior of shock waves in deformable dielectric materials u 2C,N,A = u &%4 + “2-jkA IlFIi + o(UFll),
(7.6)
Jfc,A[F]I+o([FJD,[F]=
(hc.N,~-~,,u::,~)=~~;N.nl4to(lFID,[~~=
K&[F]+o([Fli)
2
where
u
;,C,N,A
&,A = &&&,A? KC,N
= fw&.N7
=
hfc,N.A/Po, &&,A h&,N
= b,ld,,
(7.7)
= b2/f2.
From (6.10), (6.12) and (6.14) it follows that in the limit [F]+O the critical jumps in strain gradient Af, A$ and AT\have the form
where
A&(‘)“‘= -
-AA*@)ns
{~+e+~}&_(f;!$}_&
(7.9)
_Il+c+$}g-[d;$}$.
,N,A depend entirely on the conditions prevailing ahead of the wave. If Note that A $$y, and A *C(“)“S the conditions acuniform ahead, A@&’ vanishes while A$:!& vanishes if these conditions are steady. REFERENCES
111J. GRINDLAY, An Introduction to The Phenomenologicai Theory of Ferroelectricity. Pergamon Press, Oxford (1970). 121M. E. LINES and A. M. GLASS, Principles and Applications of Ferroelectrics and Refuted Materials. Clarendon Press, Oxford (1977). 131E. DIEULESAINT and D. ROYER, Ondes Eiastiques dans 14sSoiides. Masson, Paris (1974). [41 R. HOLLAND and E. P. EerNISSE, Design of Resonant Piezoelectric Devices. M.I.T. Press, Cambridge, Mass. (1969). [51 P. C. LYSNE and L. C. BARTEL, J. Appl. Phys. 46, 222-229(1975). [61 P. C. LYSNE and C. M. PERCIVAL, J. Appl. Phys. 46, 1519-1525(1975). (71 W. J. MURRI, D. R. CURAN, C. F. PETERSON and R. C. CREWDSON, In Avd. High. Press. Res., Vol. 4, pp. t-163. Academic Press, New York (1974). Bl P. J. CHEN, In Handbuch der Physik (Edited by C. Truesdell), Band VI-3, pp. 303-402.Springer, Berlin (1973). [91 M. F. MCCARTHY,In Continuum Physics (Edited by A. C. Eringen), Vol. II, pp. 450-513. Academic Press, New York (1975). [lOI L. BRUN, In Mechanical Waves in Solids (Edited by J. Mandel and L. Brun), pp. 63-155. Springer-Verlag, Wien (1975). [ill P. J. CHEN and M. F. MCCARTHY,Institute Lombard0 di Science, Rendiconti 107, 715-727(1973). 1121P. J. CHEN, M. F. MCCARTHYand T. R. O’LEARY, Arch. Rat. Mech. Anal. 62, 189-207(1976). 1131P. J. CHEN and M. F. MCCARTHY,Arch. Rat. Meek Anal. 62, 353-366(1976). [I41 M. F. MCCARTHYand H. F. TIERSTEN, J. Appl. Phys. 48, 159-166(1977). 1151M. F. MCCARTHY,Arch. Mech. 33, 443-450(1981). 1161H. F. TIERSTEN, Int. J. Engng Sci. 9, 587-604(1971). [I71 B. COLLET and G. A. MAUGIN, C.R. Acad. Sci. Paris 2798, 379-382(1974). [181 G. A. MAUGIN and A. C. ERINGEN, .f. Mecanique 16, 101-147(1977). (Received 30 Norember 1982)
0
002Cb7225/83 $3.00+ .oO 1983 Pergamon Press Ltd.
ON THE BEHAVIOR OF SHOCK WAVES IN DEFORMABLE DIELECTRIC MATERIALS
Universitt
B. COLLET Pierre et Marie Curie (Paris VI), Laboratoire de Mecanique Thtorique Tour 66,4 place Jussieu, 75230 Paris Cedex 05, France
(Communicated
associt
au C.N.R.S.,
by G. A. MAUGIN)
Abstract-In this paper, the behavior of plane shock waves propagating in elastic dielectrics is examined. The differential equation governing the amplitude of wave and formulae concerning the polarization changes across the shock are deduced. Some particular situations are studied in detail.
I. INTRODUCTION THERE
exists an important class of natural or synthetic solid deformable materials which exhibit electromechanical coupling effects. Such solids are said to be piezoelectric, electrostrictive, . . . There also exists a quite common sub-class of piezoelectric materials which possess the remarkable property to exhibit a spontaneous electrical polarization. Materials of this sub-class are said to be ferroelectric[ I, 21. Some of these deformable dielectric materials selected for their intrinsic properties (exceptionally high piezoelectric coupling coefficient, high dielectric constant, good stability in time and temperature, very low loses,. . .) are used in various devices[4-61 such as transducers, filters, resonators, delay-lines, acoustic elements, shockactuated power supplies,. . . . In recent decades significant advances have been made in the experimental and theoretical study of the formation and propagation of shock waves in solids[7-lo]. Much of this work has been directed to the use of shock waves as a convenient method for the determination of the equation of state of a solid under dynamic conditions and to the development of mathematical techniques on the basis of which the propagation of large-amplitude waves can be analyzed in certain special cases. In particular, as a result of these researches, some substantial progress in the understanding of thermal effects and rate effects on the propagation of shock waves has been obtained from both the theoretical and experimental point of view. Advances have also been made in the analysis of the propagation, under certain circumstances, of shock waves in deformable materials of solid type in interaction with electromagnetic fields[ll-IS]. By their very nature, these studies are in general complex, but this, evidently, does not mean that it is not possible to extract from past and actual works in this area useful informations which provide a better physical knowledge of electromechanical effects and electrical conduction effects on the response of deformable electromagnetic materials when subjected to dynamic loads. The purpose of this paper is to examine the behavior of plane shock waves in nonlinear elastic dielectric media. We assume that the stress and the electric field depend on the deformation gradient and the polarization per unit mass. The constitutive relations proposed include the particular behavior of deformable ferroelectric materials whose relaxation properties can be neglected. We derive the equations both for the propagation velocity and the amplitude of the shock wave as functions of the jumps in certain quantities across, and the state of the material immediately ahead of the wave front; we also give explicit formulae for the changes in the polarization per unit mass across the discontinuity surface, what should be of interest to experimentalists. After discussing the implications of the results, we consider special cases: combined mode, normal mode and axial mode. For the first mode each component of the polarization per unit mass which is parallel to it. Finally, the analysis is specialized to the case of weak shocks. UP.5 Vol.
21. No. 10-A
1145
B.COLLET
1146
2.BASIC EQUATIONS AND CONSTITUTIVE ASSUMPTIONS
Let (X, Y, Z) be the rectangular Cartesian coordinates of material point of a homogeneous nonlinear elastic dielectric material in a fixed reference configuration K0 with mass density p,,. We assume that the “generalized” motion of the body may be described by the set of scalar functions x(.,t), ~~,(.,t) and xn2(.,f) such that (x = x(X, t), Y, Z) and (r, =x=,(X, t), nTT2 = xn2(X, t), 0) give, respectively, the location and the polarization per unit mass of the material point (X, Y, Z) at time t. In addition, we assume that at each point of the body the electric field is of the form (E, = 6(X, t), 6 = 6(X, t), 0). In the framework of quasi-electrostatics the relevant global balance equations, in the absence of body force of non electrostatic origin, are[16-181
(2.1)
POUdX = CCX,,t) - x(X,, t),
0 = wx,, t) -
wx,, t),
(2.2)
0 = c%,cxp, t) - %2(X,,t),
(2.3
where v=
a,x, F = axx,
9 = E, + poF-‘a,.
(2.4)
Equations (2.1)-(2.3) must hold at all times t and for all X,, X, in Ko. u = u(X, t), F = F(X, t) > 0, C = Z(X, t), 9 = 9(X, t), represent the velocity of the solid continuum, the deformation gradient, the one-dimensional stress, the electric displacement in the X direction, respectively. Equation (2.1) is the one-dimensional statement of the law of balance of linear momentum, (2.2) is the one-dimensional Gauss law in the absence of free charges and (2.3) is the one-dimensional form of the Faraday law of quasi-electrostatics. We assume a functional dependence for C, E, and cCR2 of the form
C = %F,~T,,v), EI
=
-J%(E
TI,
7~21,
TI,
~2).
(2.5)
1 $2
=
82E
For later use, in view of (2.4), and (2.9, it is more convenient to introduce the constitutive relation ,. 22= 9(F,
(2.6)
~2).
T,,
Furthermore, we assume that the response functions 2, 6 and g2 are of class C2. Thus the following material coefficients defined by a=d(F,~l,~2)=~,
a$
aC
b,=&(F,n,,lr3=~,
1
”
.
2(F, T,, 9~2)= $,
2 A
d2 = ci,(F, 7~,,7~~)= g,
*
‘9~ d, = d,(F, r,, n2) = g, c = c(F, PI, ~2)= aF,
e=
x3
A
b2=b2(F,~,,~J=~,
I
(2.7) 2
I A
A
f, =j,(F,~T,,TT~)= 2,
f2 = fz(F,
T,,
4
=
2, 2
are of class C’. We further assume that NE
TI,
7~2) >
0,
&(F,
r,,
~2)
>
0,
f2(F,
7~19 ~2)
>
0.
(2.8)
On the behavior of shock waves in deformable dielectric materials 3. GENERAL First,
we
PROPERTIES
1147
OF SHOCK WAVES
assume that the motion contains a plane shock wave moving in the X direction
with intrinsic velocity d V(t) = z s(t) > 0,
(3.1)
where (S(t), Y, Z) with - m < Y, Z < m are the material point at which the wave is to be found at time t. The jump I[fDin any variable f(X, t) across the shock wave is defined as
nfn=f--f’
(3.2)
where fz = limx+s=(t)f (X, t). Since U(t) > 0, f- and f’ are respectively, the limiting values of f immediately behind and just in front of the shock surface. Furthermore, [f] must obey the kinematical condition of compatibility [B]
$nm=ain+ unaxrn,
(3.3)
where s/St is the displacement derivative and a superimposed dot denotes the material derivative. As usual in the theory of shock waves, we assume that x is a continuous function everywhere, while F, u, 7~~and a2 and their derivatives suffer jump discontinuities across the shock. They were continuous everywhere else. The global laws (2-l)-(2.3) imply for all X# S(t) a,2
= pod, a,9 = 0,
axg2 = 0,
(3.4)
and across the shock, we have
~~uud, haI= pm, ua=o, uwa=o, usn= -
IWI = 0,
[a,sJ
(3.5)
= .
Clearly, the formulae (3.4),, imply that the electrical displacement in the X direction and the electric field in the Y direction are uniform in X; they may, of course, be expressed as functions of time alone. In view (3.3) with f = 9 and f = iZ2respectively and (3.5),, we see that
u&j= 0, u$n= 0.
(3.6)
Thus, we have the important results that not only 9 and ccP2 are continuous across the shock wave but so are their first (and all higher) order derivatives. The compatibility condition (3.3) with f = x implies that
nun= -uua,
(3.7)
and this relation together with (3.5)i yields the well-known result
u2= umpm,
(3.8)
for the intrinsic velocity of the shock. If we put f = u and f = F successively in (3.3) and combine the resulting relations with (3.5), and (3.7), we obtain
2U(G/St)uFn+uFn(S/St)u =u2[axFk(ilpo)[a,sj, which is the equation the amplitude [F]I of the shock wave must obey.
(3.9)
B.COLLET
1148
4,SHOCK AMPLITUDE EQUATION
In this section we use (3.9) to derive the differential equation governing the amplitude [F] of a shock wave propagating in an elastic dielectric body. To this end, we obtain explicit expressions for [&.Z] and sU/Zit. From (2.5), and (2.7)1,2,3 we have ax2 = a&F + b,axa,+ b2dXT2,
(4.1)
for all Xf S(t). From (2.5),, (2.6), (2.7)~ and (3.4)~~we obtain caxF + d,axn, + d2aXT2 = 0,
(4.2)
A solution of the system (4.2) is given by ax7h = -g,a,F, ax7b=
(4.3)
-g2axF,
where
g, = i,(E
~1,
~2)=
cf2- 4 4f2 - &f,’
g2 =
i2(E
rl,
572) =
cft- 4
d2f,
_
d,f2
(4.4)
This enables us to write (4.1) in the form a,%= ha,F,
(4.5)
where h = h(F,
rl,
i-r2)= a - b,g, - b2g2.
(4.6)
Since the jump in any product fg may be written in the form
nfd=f-ud+g+ufk
(4.7)
from (4.5), it follows that (4.8) Hence (3.9) and (4.8) imply that
(4.9)
Taking the displacement derivative of (3.8), we obtain (4.10) Now, by virtue of the definitions (2.7),-j we have
sx6t=a
wn
--+a-s+b;-6t SZ’
+
s;ln + b;!!!$ + b;m 6t +b;+, : SF+
-zi--a+-+b,--St
+‘fint;+ l,,;!$,
(4.11)
(4.12)
On the behavior
of shock waves in deformable
dielectric
materials
I149
and on taking the difference of (4.11) and (4.12), we find (4.13) It follows from (3.3), (4.3),,* and (4.7) that
&n=b,nhma- m(w)+, (4.14)
~Ud=W WUaxFPvuaw)+. From (2.7)4-9r(3.6)~ and (4.7) we obtain the system d;uti,n + d;ij7i2n = - c-@n - ucnP+ - ud,nir: - ud2n7j;,
(4.15) so that u?i,n= - g;iP]-
j,P+ - k,7j; - i,+;, (4.16)
where . I’=
k,
=
Udf;-uend2 . U4lf;-n4ldr d;f;-d;f; ’ 12= d;f;_d;f;
’
lId,lJf~ - Uf,lld; , kz= Ud,lJl - Uf,!dr d,fz -&f; dzf, -d;fi ’
/,J&~f+f#i d,f,-d;fT
(4.17)
I = IIdz~f I- Ufzlb T ’ ’ dzf, -d;f; ’
On substituting from (4.16) into (4.14) we obtain
$hn=-g;$uw~i,i+-k,ii:-h?i:uug,n(m+, (4.18)
fad=-n;$Fn-
j21s-+-k27;t-12ti:-
U[gJ(a,F)+.
The substitution of (4.18) into (4.13) with (4.3),.2, (4.6) and (4.7) yields $p1=+Fn+
u[h~(axF)++{[a~-j,b;-j2b;}~+ (4.19)
so that eqn (4.10) then gives
(4.20) +{l[bJ-
/lb;--
&b;},$.
1150
B.COLLET
Finally, on substituting from (4.20) into (4.9), we find that the amplitude [F] of shock wave propagating in an elastic dielectric body obeys the equation
(h- - pl#)
(4.21)
@*-kJxFk
where 1 I{Uall-j,b;-j~~~}~+3Uhil(axF)+ A* = - 2(h- - pJJ’){ (4.22)
Clearly, the growth equation is very complicated and as in other shock-wave theories it is not possible to deduce any useful information from it without adopting additional assumptions[8, 111.In the following section we shall consider the implications of (4.21) under certain particular circumstances. 5,SOMEPROPERTIESOFTHE
SHOCKTRANSITION
As a direct consequence of the description proposed, it is evident that the jump [Fj in the deformation gradient and the jumps [a,] and [TJ in the components of the polarization per unit mass in the X and Y directions, respectively cannot be independent. Indeed, their relationship is given by (3.&, i.e. &(F-, &(F-,
n;, T;)- i&F+,T:, n,‘) = 0,
(5.1)
r:, P:) = 0.
(5.2)
T;, T;) - &(F+,
Equations (5.1) and (5.2) regarded as functions of (F-, TT;, T;, F+, TTT:,T;) are assumed to be solvable and to yield 7; and a; as functions of (F-, F+, P:, TT;). In other words, we assume that there exist functions ii, (.,.,.,.) and ijz (.,.,.,.) such that
Umll= %(UFD, F’, TTT:, d), (5.3)
b2n= %W.F', r:, 6), where we have used the relations F- = F+ +[Fn, IT; = ?r: + [n,] and T; = nt + UT& We now define the function e(.,.,.,.) such that
(5.4) From (2.7),_3,(5.3),, and (5.4) we have (5.5) On taking the displacement derivative of (5.3),,2and employing (4.3),.*we obtain
(5.6)
~~uFn+~++~?i;+~?if+(~-R;~-g2~)U(a,F) -$iT2n= I 2
I
2
On the behavior of shock waves in deformable dielectric materials
1151
Since (4.18),,*and (5.6) hold for arbitrary shock waves, we have aii, _
86 _ _ -
g’T
a[F]
a+,_ +arl
ati, _/
I,
z=
-
f$=
_k
a*2 __
a[F]
g2,
ae2_ fanI
. -11,
aF’-
_k
” (5.7) -j2,
a+2__l 21 aa,‘-
*.
These relations should be of interest to experimentalists concerned with the study of polarization changes produced by shock waves. They show how the jumps Up,] and 174 are influenced by the various parameters. In particular, they give the derivatives of jumps UT,] and [TJ in terms of the jumps in certain material coefficients and the values of material coefficients prevailing in the material just before the arrival of the shock. Thus, by (4.6), (5.5) and (5.7),,5, we find
1
?r:,7r:)=a-b;g; - big; @IFI, F+,
= h(F-, r;, 7~;).
(5.8)
We assume that there exist functions 7j,(.,.,.) and ij*(.,.,.) such that rr~= +,(F, 9,821, (5.9) 7~2 =
7?2(E
9,821,
and it follows from (3.4),, and (4.3),,2that a+, -#=
-g,,
$=
-g,.
(5.10)
We now define the function 2 (.,.,.) by X = %F, %a,)
=
%F,
ii,(F,
9,
$21,
CiF,
9,
%2h
(5.11)
and from (2.7),_,, (4.6) and (5.10),,2we have h(F9
9 9
gJ=aS_a$ atafi, aeai;, ~-~+an~+;i;;~=a-b,g,-b2g2=h. 2 I
(5.12)
2
Thus we see that i simply is the stress-strain modulus at constant longitudinal material electric displacement and transverse material electric field. Henceforth, we shall assume that h(F, %%,) > 0.
(5.13)
We now consider the implications of (4.21) on the evolutionary behavior of certain types of shocks. Consider a compression shock propagating in a material which is initially in compression or unstrained so that
[F]I
F+
We assume that the stress-strain law (5.11) is concave (5.13) a*w,
93
aF2
a21
<
0
for
all
(5.14)
from below, and, hence, by virtue of
FG 1,
(5.15)
B. COLLET
1152
Clearly 2 (.,.,.) is assumed to be in C* class. It follows from (3.8), (5.12)-(5.15) that h- > p#,
(5.16)
[h] > 0.
From (4.21) and (5.16), it is clear that if IF! is the amplitude of a compressive shock wave propagating in an elastic dielectric, which is initially in compression and for which (5.13) and (5.15) hold, then at any instant [a#] 5 A* *;
(5.17)
lI[F]lZ 0.
It is clear from (5.17) that whether the amplitude of a shock wave grows or decays depends on the relative values of the jump [a#] in the gradient of the deformation gradient or strain gradient across the shock and the parameter h* defined by (4.22). In accord with the classical nomenclature we call A* the critical jump in strain gradient. 6. PARTICULAR
CASES
As consequence of the very general modelization the foregoing results are obviously complicated and it is not easy to extract “simple” informations without introducing some additional assumptions. We now examine three particular situations: (i) Combined mode We assume that the material considered may be characterized by the constitutive equations c = QF, Tl, %), 1 E, = ME n,), ^ $2 = %‘,(F, ~2).
(6.1)
It follows from (2.5), and (6.1), that we have . 9 = 9(F, a,).
(6.2)
It is clear from (2.7) that in material of this type d:: = f, = 0,
(6.3)
so that, by (4.4) (4.17) and (5.17), we find
aii,
C-
i$q= -d;’
_@I_ aF+
aii, Ud,B a6 _-_ an+ I -
d;'
d;'
_.
an:- ’
!!c& -kJ,
-=-g,
aua af2 a=+-I
_ug
(6.4)
aiiz
--
-
2
o
3
32=-y. 2
2
(ii) Normal mode If we assume that TT,= E, = 0 and that the behavior of the material is described by the constitutive equations
On the behavior
of shock waves in deformable
dielectric
materials
1153
it is clear that we have dii2 _ -- e-_ dlFD fi’
*_
aF+ -
_u
_E& _y
fi ’
2
(6.6) 2
(iii) Axial model’ If we assume that 7r2= SET2 = 0 and that the behavior of the material is defined by the constitutive equations
C = i
A 9 = WE 7~11,
it is obvious that we obtain
aii, -=
-- c-
a[F]
d;'
ucn
acl _ aF+ -
d;’
aii,
---x. aa+-
6dJl
(6.8)
I
In three above-examined situations (6.4),,s, (6.6), and (6.8)r determine the jump in the polarization per unit mass across the shock surface. In view of (2.8)2,3we have aii,
aiiz
-= -sgnc-, sgn a[Fj
= - sgn e-, sgn a[Fl
(6.9)
and, consequently, the jump in polarization per unit mass T,(T~) will increase or decrease across a compression shock according as c- > 0 (e- > 0) or c- < 0 (e- < 0). Now the critical jump in strain gradient (4.22) should be replaced by: (i) Combined mode
hc=a-c$-eb, I
(6.11)
f2
(ii) Normal mode
where (6.13)
hN=a-e?, 2
(iii) Axial mode
“=
* -2(h;-p&J
2
([IIh*l+c+[~D)~+3Uh,l(a,F)+
tNote that this situation is equivalent to that described in[l I]. However, analysis of induced polarization or depolarization phenomena as observed waves.
+d$#$
(6.14)
our approach seems to be better adapted to the in dielectric materials and produced by shock
B.COLLET
1154
where (6.15)
hA=o-+ I
Furthermore if the conditions ahead of the shock are steady so that F, ~~ (nz) are independent of time then
( W?+ 2(h
&,A
-
Pou
(6.16)
c.N.~
and for a compressive shock it follows from (5.16)1,2that
sgn0 bd
= - sw {(W)‘l.
(6.17)
Equation (6.17) states that the sign of the critical jump in strain gradient is the negative of the sign of the strain gradient ahead of the wave. Thus for a wave propagating into a region of decreasing strain, A$,,,* would be positive, opposite to the case of wave propagating into a region of increasing strain. Another special case of (4.21) arises when the region ahead of the wave is at rest in a uniform state. Here the critical jump in strain gradient vanishes and the shock will grow, decay or remain steady according as (&F)- is positive, negative or zero. This last result is analogous to that arising in purely elastic shock-wave theory.
l.WEAK SHOCKWAVES
In this section we determine some properties of shock waves in the limit F-+F+, i.e. we consider the case of a weak shock wave or shock wave of infinitesimal amplitude. For the sake simplicity we derive only expressions for [Z], IT,], [Ed, U and A* in the three particular cases examined in Section 6. Since (lljF]lG l), from (3.5)3,5,(5.11) and (5.12) we obtain
where cc
=
%F,
9,
821,
Z:N
=
%F,
821,
C,A =
%F, 9),
Hc.r.,,/,= a&N,,+
(7.2)
From (5.3),.2we have (7.3) and in view of (6.4), (6.6) and (6.8), we may write
am
_
-awn I =--IuFn=o=-2’ aii2
aii,
IFn=O
nF1=0
;;
(7.4)
so that for the (C,A) and (C,N) cases (7.5) By using similar arguments it may be shown that
ukN,.d =ffbv,~uFn+ 4um
1155
On the behavior of shock waves in deformable dielectric materials u 2C,N,A = u &%4 + “2-jkA IlFIi + o(UFll),
(7.6)
Jfc,A[F]I+o([FJD,[F]=
(hc.N,~-~,,u::,~)=~~;N.nl4to(lFID,[~~=
K&[F]+o([Fli)
2
where
u
;,C,N,A
&,A = &&&,A? KC,N
= fw&.N7
=
hfc,N.A/Po, &&,A h&,N
= b,ld,,
(7.7)
= b2/f2.
From (6.10), (6.12) and (6.14) it follows that in the limit [F]+O the critical jumps in strain gradient Af, A$ and AT\have the form
where
A&(‘)“‘= -
-AA*@)ns
{~+e+~}&_(f;!$}_&
(7.9)
_Il+c+$}g-[d;$}$.
,N,A depend entirely on the conditions prevailing ahead of the wave. If Note that A $$y, and A *C(“)“S the conditions acuniform ahead, A@&’ vanishes while A$:!& vanishes if these conditions are steady. REFERENCES
111J. GRINDLAY, An Introduction to The Phenomenologicai Theory of Ferroelectricity. Pergamon Press, Oxford (1970). 121M. E. LINES and A. M. GLASS, Principles and Applications of Ferroelectrics and Refuted Materials. Clarendon Press, Oxford (1977). 131E. DIEULESAINT and D. ROYER, Ondes Eiastiques dans 14sSoiides. Masson, Paris (1974). [41 R. HOLLAND and E. P. EerNISSE, Design of Resonant Piezoelectric Devices. M.I.T. Press, Cambridge, Mass. (1969). [51 P. C. LYSNE and L. C. BARTEL, J. Appl. Phys. 46, 222-229(1975). [61 P. C. LYSNE and C. M. PERCIVAL, J. Appl. Phys. 46, 1519-1525(1975). (71 W. J. MURRI, D. R. CURAN, C. F. PETERSON and R. C. CREWDSON, In Avd. High. Press. Res., Vol. 4, pp. t-163. Academic Press, New York (1974). Bl P. J. CHEN, In Handbuch der Physik (Edited by C. Truesdell), Band VI-3, pp. 303-402.Springer, Berlin (1973). [91 M. F. MCCARTHY,In Continuum Physics (Edited by A. C. Eringen), Vol. II, pp. 450-513. Academic Press, New York (1975). [lOI L. BRUN, In Mechanical Waves in Solids (Edited by J. Mandel and L. Brun), pp. 63-155. Springer-Verlag, Wien (1975). [ill P. J. CHEN and M. F. MCCARTHY,Institute Lombard0 di Science, Rendiconti 107, 715-727(1973). 1121P. J. CHEN, M. F. MCCARTHYand T. R. O’LEARY, Arch. Rat. Mech. Anal. 62, 189-207(1976). 1131P. J. CHEN and M. F. MCCARTHY,Arch. Rat. Meek Anal. 62, 353-366(1976). [I41 M. F. MCCARTHYand H. F. TIERSTEN, J. Appl. Phys. 48, 159-166(1977). 1151M. F. MCCARTHY,Arch. Mech. 33, 443-450(1981). 1161H. F. TIERSTEN, Int. J. Engng Sci. 9, 587-604(1971). [I71 B. COLLET and G. A. MAUGIN, C.R. Acad. Sci. Paris 2798, 379-382(1974). [181 G. A. MAUGIN and A. C. ERINGEN, .f. Mecanique 16, 101-147(1977). (Received 30 Norember 1982)