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Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
1st~J. Non-Linear Mechanics Vol. 5, pp 513-524. Pergamon Press 1970. Printed in Great Britain
GROWTH OF ACCELERATION WAVES IN AN UNSTRAINED NON-LINEAR ISOTROPIC ELASTIC MEDIUM B. L. JUNEIA* and G. A. NARIBOLI? Iowa State University, Ames, Iowa Abstract-A wave is taken as a singular surface across which second derivatives of the displacement vector suffer discontinuities. Use of compatibility conditions, equation of continuity and the equations of motion gives the relations between the acoustic tensor, amplitude of the dis~ntinui~ vector and the normal speed of propagation. Separability of modes and corresponding propagation speeds am then discussed. Use of compatibility conditions on the differentiated forms of the basic equations then leads to the basic equation that governs the growth of the amplitude for each mode. in an unstrained medium the shear wave, as in linear theory, does not grow; the dilatational mode may grow, depending on the elasticities. Comparison is made with known results.
1. ~~ODUC~ON CROSS effect is one of the most important evidences of non-linear elastic response of a medium. Most of the known constitutive laws show these effects, known as Kelvin and Poynting effects. These features are characteristic only of finite elasticity and are absent in a linear theory. Problems discussed so far pertain to deformation of bodies which are in a state of equilibrium. Recently, the phenomenon of wave propagation in non-linear elasticity has received fresh attention. For this paper a propagating wdve is defined as follows: There exists, inside the medium, a moving surface such that at least some of the field variables or their derivatives are discontinuous across it. The magnitude of the discontinuity is called the amplitude ofthe wave. The velocity ofthis surface normal to itselfis the speed of propagation. Our main interest is to discover how the discontinuities across such a singular surface grow as the surface moves. Taking the displacement vector niy let us assume its components to be continuous while its first derivatives are discontinuous. We call this a first order or velocity wave. The growth of such a wave indicates separation of material or its ‘fracture’. Though physically more important, its study is not undertaken here. If the displacement and its first derivatives are continuous while its second derivatives are discontinuous, we call this an acceleration or second-order wave. Our study in this work is limited to such an acceleration wave. The most important feature of waves in non-linear mechanics is their growth. Gas dynamics [l], magneto-gas-dynamics [2, 31 and hypoelasticity [4] provide some new features. Discontinuities grow indefinitely in these cases. The non-linearity of the basic equations is one of the main reasons for this. Also, non-linear elasticity may provide some results which may be distinctive features, such as cross effects. Truesdell’s recent work [5,6] in the field of wave propagation in non-linear elasticity is * Department of Mechanical Engineering, Indian Institute of Technology, Delhi, India. t Department oi Engineering Mechanics and Engineering Kesearch Institute, Iowa State University, Iowa, 513
514
B. L. JUNFJAand G.A.
NARIBOLI
of interest to our work. Let & be the amplitude of the discontinuity across a wave front with unit normal IZ~. Then in a number of physical problems, of which ours is a type, the relevant equations lead to relations of the form
Qij4j = Po@
(1-l)
Here G is the velocity of propagation of the wave front normal to itself, which will be the actual velocity if it is moving in a medium at rest and will be the relative velocity, if it is moving through a medium in a state of motion. The tensor Qij, called the acoustical tensor, is a function of the physical variables ahead of the wave front and the normal vector n,. It does not involve 5;. This equation determines the speeds of propagation. Truesdell now introduces the idea of principal waves ; these are the waves travelling along one of the principal directions of strain; the waves are called longitudinal or transverse depending on whether the motion is along, or perpendicular to, the principal directions. Thus there are nine different waves. By this artifice the velocities separate in an elegant way. His aim is, however, to obtain velocities and their relationships with elasticities of the material but not to study the growth of the wave. Here, we adopt a different viewpoint. The system of equations (1.1) are linear homogeneous in &. For the existence of a singular surface they must have a non-trivial solution. So their determinant must vanish. This gives a cubic in G2 ; so in our physical space of three dimensions there are only three speeds of propagation which are the roots of a cubic. We further assert that there is one mode of propagation, i.e. one linear combination of &, which is non-zero for a wave travelling with each one of these velocities. A simple device would determine both the speeds and the modes of propagation. Let us for the present assume Qij is symmetric. Its three principal values are given by a cubic
1Qij - QsijI = 0.
(1.2)
Let Q(‘@(a = 1, 2, 3) be three roots of this equation and Lp) be the three corresponding principal directions. Thus we have QijLp)
= Q@)@‘).
(1.3)
Multiplying (1.1) by L$@we obtain by use of (1.3) (1.4) Thus the velocities are given by (a)2 POG
=
Q'"'
L!“‘~. # 0. J 3
(1.5)
This determines both the velocities and the modes of propagation. In addition, it is easy to prove, when principal values Q(‘) are distinct, that Ly) are also distinct. The mode Ly)cj gives the discontinuity across the wave travelling with velocity (Q”‘/P,,)*. Since principal
Growth of acceleration
waves in an unstrained non-linear isotropic elastic medium
515
directions are orthogonal, we also have Lj25j = LSrj = 0, for this wave front. Thus once the principal directions and numerical value of the acoustic tensor are known, the real velocities of propagation together with separate modes of propagation are easily obtained. The restriction of symmetry of Qij is unnecessary. In fact in a general elastic medium with arbitrary initial strains, the condition may not be satisfied. However all the above arguments continue to hold only if LT are taken to be left eigen-vectors of Qil But such an analysis is then useful only if the principal values of Qij are real and positive; otherwise the velocities will be complex. The study of wave propagation by use of theory of singular surfaces has, we believe, little meaning when the velocities are complex. If the principal values of Qij depend on ni, the velocity of propagation will also depend on it. Different parts of wave surface will then move with different velocities. The wave propagation is then said to be anisotropic. The only methods of solution of anisotropic wave propagation are those given by Lighthill [7] and Ludwig [8] on one hand and by Weitzner [9] on the other, both by transform techniques. The former leads only to asymptotic estimates but in an explicit form, while the latter to the formal full solution but in terms of the roots of a quartic which are not always easy to obtain.* When one wants to study the growth of a wave, we believe our approach is more appropriate. The integration of the growth equation in the anisotropic case remains open, as far as we know, even for the linear case.* When, however, the equation for velocities of propagation is non-linear in amplitude (which is the case of first-order waves), the singular surface is a shock wave, the study of which will be made later. Lastly it is important to note that for a plane wave front, ni is constant in time. The study of the growth of a wave is then relatively simple. Thus, here we study the growth of second-order waves in an unstrained non-linear elastic medium. Even in the most general elastic medium the growth of shear waves is entirely the same as in classical linear elasticity. However, the growth equation for the dilatational mode is now non-linear. Its growth or decay depends on the elasticities of the material. 2. VELOCITIES
OF PROPAGATION
Let xi be a spatial Cartesian orthogonal system of coordinates with which we work throughout. The singular surface which we also call the ‘wavefront’ is described by xi = x,(ua; t) with t as time and #(a = 1,2) as a system of surface coordinates G is the velocity of surface normal to itself Greek indices denote surface coordinates and Latin indices the Cartesian coordinates with the exception of n and t. These letters, when used as suffixes, denote components normal and tangential to the wave front We frequently use the resolution of vector fi as J = .L
ff
fnni
=.hi
=f”f,
+f
LIxi, a
f*
=
fp
=fixi,fl
Safl
s=sf
=
g,figBy
%,axi,p =
67(I
(2.1)
* The anisotropic and even non-homogeneous cases have since been exhaustively studied. Refer to ‘On Some Aspects of Wave Propagation.’ Engineering Research Institute Report 57, Iowa State University, Ames, also to appear in Professor Seth’s 60th Birthday Volume, I.I.T., Madras, India.
516
B. L.
JUNBJA
and G. A. NARIBOLI
Finally, let ui be the displacement vector, tij and eij the stress and strain tensors. Here we use the Eulerian strain tensor eij, defined as 2eij = Ui,j + Uj,i - h!k,ih!k, j in terms Of the displacement vector Ui. In order not to digress we have discussed in Appendix I all the compatibility conditions needed in the problem and in Appendix 2 the constitutive laws. As to the discontinuities we assume
[Uil = O,
[Ui,
j]
O>
=
[ui, jkllfljnknl
=
Pi.
(2.2)
Here and throughout, a bracket denotes the jump in the quantity across the wavefront. The medium ahead is unstrained and at rest with the value of density p as constant po. We frequently take the jumps of product of quantities. If one of the quantities is continuous it can be taken out and if, in addition, it vanishes ahead, the whole jump vanishes. Also when both factors are zero ahead the jump of product equals the product of jumps. If wi is the vorticity vector then we have [mil = [ieijkvk, jl = -@t?ijktknj = -$eijknj~aXk,
c(
co,= i&G.
b-d = 0,
o = (Jwiwih
(2.3)
So, for such a discontinuity, vorticity is always tangential and it vanishes with the tangential component of ti. To interpret the normal component of
at
+ Pvi, i + P,ivi
=
O,
so zl + Pot”
=
0,
(2.4)
where ul = [PC1 ni* If we take the alternative form of the continuity equation : p = poJ(l
- 21 + 411 - 8111)
(2.5)
where I, II and III are invariants of eip the result obtained above still holds for the assumed type of the singular surface. We thus have the interpretation of <,, i.e. the normal component of tie Here, 5, > 0 denotes compression across the front and 5, < 0 denotes the expansion wave. We now consider Cauchy’s most general elastic medium defined, on the assumption of homogeneity and isotropy only, by the constitutive law tij =
U6ij +
beij +
Ceimemj,
(2.6)
where Q,b and c denote arbitrary functions of I, II and III. We require the stress to vanish with the strain which imposes the condition a(0, 0,O) = 0. We next introduce the following
Growth
notation
of acceleration waves in an unstrained non-linear isotropic elastic medium
517
: aa --a2a El= ar)JIJrr=o;'12--armI I,II,Ilf=O
a,=;I ;BQ=
b(0, 0,O)
c(O,0, 0).
y() =
(2.7)
I,1I,l11=0
The equations of motion are dui 4j,j
=
(2.8)
Pdt
This gives (% + 38,) Cl% + 3Mi
(2.9a)
= PoG2&
which can be rewritten as
The linear homogeneous system for ti is (2.9a) and it is rewritten in the forms (2.9b) for 5, = cinj and
with
poG2=ifio
with
&#O;
t”#O;
<“=O; t,=O;
Sr =&ni ti={aXi,c
(2.lOa) (2. lob)
The ease of the Green-elastic body is included in this with gLtand /$, given in terms of strain energy functions from (A.2.8) For the second-order theory, 01~= d and PO = 2~ from (A.2.9). 3. GROWTH
OF WAVES
To obtain the growth equation we differentiate the equation of motion (2.8) with respect to xk, multiply by yzk,and take the jumps. The result of differentiating acceleration term as given in Appendix (A.1.8) is po(G& - 2G $
-I- 2G25&.
(2.1 la)
After noting the earlier remarks, the terms from that of stress can be written as nkltij,
jkl
=
tat
Lkl
+
POEeij,
jkl>
%i
+
A+
(2Slb)
where
(2.114
518
B. L. JIJNEJA~~~ G.A. NARIBOLI
Note that we have now six elasticities a,, CC 2, c~rr, PO, /I1 and y,, which reduce to five only when a strain energy function exists. To evaluate the first two terms in (2.1 lb) we use (A.1.9). After some simplifications we obtain here two alternative forms as
Thus after rearrangement
we get
{PoG’- (~1 + PO))Lni + boG2- fBo)taxi,. =2p,G ‘k - 2p,G2t”5i - (~(1+ fro) t2ni - floCJ
(2.12)
The important point to note is that the left member of (2.11) is the same as the equation for velocities (2.9) which always enables us to obtain growth equations. For a shear wave,
or (2.13) It is the same as in classical linear elasticity. For a dilatational wave, poG2 = x1 + PO; 5i = &,ni =
SO muhiplying
by
where k =
3 _
0 (
all+ VI+ -_____
2
30
(2.14)
2(x, + BCJ
The corresponding equation in linear elasticity is obtained by setting k, = 0; putting 4 = l/p so that p -+ 0 implies growth, it reduces to
dp
dn
+
k,
+
pR = 0.
The integration is now straight-forward and can be carried for a general surface by use of R for a system of parallel surfaces. It is, however, enough to note the cases for plane, cylind-
519
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
rical and spherical cases. We obtain with S2 = 0, - 1/2r, - l/R respectively, P = p. - ken 3
, plane
- 2k,r
, cylindrical
p =Po(~)-koRln(&)
(2.15)
,spherical
For k. > 0 a compressive wave p. > 0 grows. The converse holds for k. < 0; the growth characteristics thus depend on the sign of k,. 4. HYPERELASTIC
MATERIAL
We now note the particular case of second-order theory, based on the assumption that stored energy exists. The constitutive law we now use is (A.2.10) : tij zz -p Fij
(2.16)
PO
The wave velocities remain the same. For growth we evaluate the stress term as below.
Ltij,jJ nk
=
k ([P>
jl
[Fi, J nk + [F,j, jl btkl nk) + nk[Fij,jkI.
(2.17)
After evaluating all terms as before, we obtain PoG2[i - (1 + P) enni - p5^i = 2p,G $
- (B + 2@ + 4poG25,) Ci - (A + 1 + 2,~L5~) ni + (n + P) {Wp5n,aXi,p)
+
(52:- 2ntn) ni}9
(2.18)
where A = (22 + 4~ - 61 - m) t;
The growth of shear waves is governed by (2.13), the same as before while the equation for dilatational waves is also of the same type with k,=
9-z 2 ;1 + 2pL’
(2.19)
Finally it is interesting to note that our conclusions are in complete agreement with those of Bland [12]. He considers, however, the generalized one-dimensional problem and obtains a condition for formation of compression shock. His conditions, obtained from thermodynamic as well as mechanical considerations, coincide with each other. He uses the Langrangian coordinate system. Our condition, p. > 0, a compressive wave, IS identical to this. The condition for compressive shock on the elasticities, obtained by him, may be stated as
- 1+2pk <0 3 O
*
(2.20)
B. L. JUNEJAand G. A. NAR~BOLI
520
The derivation of this equivalence is noted in the appendix (A.2.15). From this we see that the second-order modulus 1 must be quite large compared to the first-order elastic modulii, to obtain a dilatational shock. Clearly, in contrast to fluid dynamics, we expect in solid mechanics the dilatational shock on physical grounds. Also it is interesting to note that other second-order elasticities play no part in shock formation. 5. CONCLUSION
Non-linear elasticity does admit the growth of acceleration waves. However if it is an expansion wave which must grow indefinitely, one second-order elasticity has to be quite large. In the most general theory of elasticity, we obtain a condition between the elasticities for growth or decay of the wave. REFERENCES [ 1] T. Y. THOMAS,The growth and decay of sonic dis~ntinuit~es in ideal gases. J. Z%furh.Mech. 6,455469 (1957). [2] K. 0. FRIEDRICHS and H. KRANZER,Non-linear wave motion, notes on M. H. VIII. AEC Res. Devei. Rep. N.Y.O. 6486 (1958). [3] G. A. NARIBOLI,The propagation and growth of sonic discontinuities in magnetohydrodynamics. J. Math. Mech. 12, 141-148 (1963). [4] G. A. NARIBOLI,Growth and propagation of waves in hypoelastic media. J. Math. Anal. Appl. 18, 57-65 (1964).
.151~ C. TRUESDELL.General and exact theory of waves in finite elastic strain, Archs raf. Mech. Anal. 8, 263
(1961). I [6] C. TRUESDELL, Second-order
theory of wave propagation in isotropic elastic materials. Int. Symp. Secondorder Effects in Elasticitv. Plasticitv and Fluid Dynamics, Haifa, April (1962). [7] M. J. LIGHTHILL,Studies’on magnetohydrodynamic waves and other a&tropic wave motion. Phil. Trans.
R. Sot. Ser. A 252 397430 (1960). [8] D. Luuwrc, Examples of hehaviour of solutions of hyperbolic equations for large times. J. Math. Mech. 12,
557-566 (1963). (91 H. WEITZNER,On the Green’s function for two-dimensiona
magnetohydrodynamic waves. I, II, AEC Rex Bevel. Rep. N.Y.O. 2886,9489 (1960). [lo] T. Y. THOMAS,Extended compatibility conditions for the study of surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6, 311-322 (1957). [ll] T. Y. THOMAS,Decay of waves in elastic solids. J. rat. Mech. Anal. 6,759-768 (1957). [12] D. R. BLAND,Dilational waves and shocks in large displacement isentropic dynamic elasticity. J. Mech. Phys. Solids 12. 245-267 (1964). [13] A. C. ERINGEN,~o~-~~ear Theory ~~Contj~~o~s Media. McGt-aw-Hill, New York (1962). (Received 17 August 1965; revised 3 September 1969)
APPENDIX 1. COMPATIBILITY CONDITIONS Second-order conditions
These follow from Thomas’s paper [lo]. If [Z,J t+ = 0,
[Zl = 0,
[Z,
(ASS)
then we have
[Z,ijl =
a22 -ax,at
[I I
Cninj,
Z
-
GCn,
a22
[ 1= 2F-
CG'.
(A.1.2)
Now let ui he the displacement vector. The reference system xi is spatial throughout. Then we have, for a secondorder wave, with & as the amplitude vector [UJ = 0,
[u, jk]
[Uj,j]
= &njnb
=
0,
Eui,jkl njnk
=
&
=G’T,.
(A.1.3)
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
521
We now take the velocity vector vi defined in terms of ui as
In all later derivations we limit attention to the particular case when the medium ahead is unstrained and at rest. Also we assume that the velocity of propagation is constant and that the ‘delta time derivative’, the time derivative for an observer on the wave front moving with it, vanishes. In view of these assumptions, when we take the jump of the product of two quantities, if one is continuous we can take it out with its value ahead of the wave front. In addition, if it is zero ahead, the jump in the whole quantity vanishes. So we obtain [I+] = 0;
[qj]
=
-G&nj
k] =k +,,,,,]=k] =G2&.
(A.1.4)
Let eij be the strain tensor defined in terms of displacement vector ui as eij = +(u~,~+ uj,! - u,,~u~,,). Using the above results and assumptions, we get [eij] = 0;
[eij,J = fnd&nj + cjni).
(A.1.5)
Third-order condition
We give a short derivation of these on the same lines as given by Thomas The others follow from it or can be obtained similarly. Further, let (A.1.6)
[ZijJ n,njnk = D; we then have
[Z>ij,J = Dninjn, + g”8C,Aninjx.t,p+ nkhixj,0 f n,niui,& - Cb@(n+j, d~k,s + njxk..xi, p + nkxi,.xj, s)
a32
1 1 ax,ax,at
=
-G{Dn,nj
+ g”kY,An,xj,P + njxi,J - Cb”xi,_xj,s}
=
G*(Dn, + gJC&xi,,)
(A.1.7)
+ gninj
- 2Gni E.
St Note that we have assumed &,/St = 0; we also note that uqa = 1, 2) is the surface coordinate system, b, is the second fundamental form and 2n = g7b4 gives the mean curvature. Proof of (A.1.7) Let us denote, to start with [Z,ij] = L, = Lji = Cninj
[Zip1 4 = Mji; Mijnj Y Mj,nj = Ni Mi,n,nj = D = N,ni.
From the known first-order compatibility we obtain
[&I
= Mijn, + g”Lj, a%,8;
but the right member must be symmetric in indices i,i and k. From this condition we obtain M$rk + @“I+, ok, p = M+ni + S”Lj,. oxi.0 = Mtinj + g”BLki,pj,P Multiplying by n, and summing over the repeated index we get Mij = Njn, + P’L,,,~,,x,,,r
= N,n, + g@L,, ~n,x~,~
522
B.L.
Again we multiply
JUNFJA and G. A. NAIUBOLI
the above by nj and sum over the repeated
index, to get
N-I = Dn.I + g@L qp, .vvi.
B
= Dni f g’LBC,,,xi,B Thus we have Mij = Dninj + g”@C,Axi,Bnj + nixj,J
- Ct@~,,$l~,~
Using this and simplifying. we obtain (A.1.7). We write now two more relations which we shall need. Let L”i,jd
njW$ = C
Then we have
(A.l.lO) Lastly [ePq, A
we need
nk =+[u,,4rk1 4 + f[u,,,.kl nk - f[%,&d [%,d =
H&v,
+ tit,p,n,
nk
-
+t%qrl
s”%.,Aw,,~ + wd - 5pbaBx,,P~r,d + sap5q,Xq,Anpxr,p + wp.B) - L?%-+~,,~~
hpkl
%
+
APPENDIX Cauchy elastic medium Based only on the assumptions
- 5%vr
2. CONSTITUTIVE
of isotropy
and homogeneity,
(A.l.ll)
LAWS we have a stress-strain
relation
as
tij = aSij + beij + cei,,,e,,,) Here 4 b and c are arbitrary
functions
of the invariants
I = e,,;
I, II and III of strain tensor eij defined as, III = (eijl.
II = t(eiiejj + eijeji);
Hyperelastic Here it is assumed that the stress tensor has a potential; it can be obtained on this assumption we can write the constitutive law [ 131 as tij = p(6i, PO
(A.2.1)
- 2ei,) g.
from a stored energy function.
(A.2.2)
Based
(A.2.3) 1”
Also we have p = p,,/(l Assuming
isotropy
and homogeneity
- 21 + 411 - 8111).
we have z = Z(I, II, III).
(A.2.4)
523
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
so az a1 ax aI1 S& - aIaejn az
az am
---faIIz+--=
awae,
Im
am
arr
s+Z , a+
z
-+c 2
-,
aejm
3
aei,
(A.2.5)
which defines Xi, Z, and X3. We further
note
-
aI
=amj;
z! =Ih,,
-
fj! =
emj;
emnenj
-
(A.2.6)
Iem, + II&,
Im
1”
Qejm
Now we use (A.2.6) in (A.2.5) and the result in (A.2.3) and also the Cayley-Hamilton
theorem
as (A.2.7)
eimemnemj- Ieimemj + lIeij - 11115,~= 0. After simplification are given by
we obtain the same as (A.2.1). Now a, b and c are not arbitrary
u = ;
(Z, + I& + II&
b = ;(-
the second-order
theory,
of the invariants,
but
(A.2.3)
- 2111X,)
C2 - IX, - 2C, - 21X2)
c = qz, PO To obtain
functions
(A.2.8)
+ 2X,).
we assume (A.2.9)
z = f(n + 2/J) I2 - 2pII + /I3 + mI II + nIII. We then obtain
tij
zz
PFij
PO with
Fij = (II + (3[ + m) 1’ + (m - n) II} 6, + {2~ + (m - n + 21) I} eij + (n - 4~) eime,*
(A.2.11)
Finally we give the relations between the elastic modulli employed by the present authors and Bland [12]. He uses the Lagrangian strain tensor which we denote by E, and its invariants by Ir, II, III, His constitutive law can now be obtained from X = )&I; However
these invariants
+ /+(I:
are related I, =
D = We expand
- 113 f aI; + &(I;
- 211,) + ~(3111, - 31&
+ I;).
(A.2.11)
to ours as
I - 411 + 12111 D
II - 6111
; IIE=_-----_. D
III
, III, = D
1 - 21 + 411 - 8111.
these to third order terms as I, = I + 21’ - 411 + 12111 - 121 II + 413 I;=I’-413-8111; II,
I;=13
= II + 6111 + 21 II
Using (A.2.12) in (A.2.11) and comparing
m = -41 for compressive
= I 11; III, = III.
with (A.2.9) we obtain
,%=I,;
His condition
; I&
/I=P~;
1=21+4p+a+P+v
- 12/l - 28 - 3v;
shock is Q. < 0
n = 12fl + 3v.
(A.2.12)
524
B.L.
JUNFJA
and G. A.
NARIBOLI
where
This is same as (2.9) of the paper
R&m&On considere une onde comme une surfacesinguliereau traversde laquelleles derivbessecondes du vecteur de d&placement subissent des discontinuites. L’emploi des conditions de compatibilite, de l%.quation de continuite et des equations du mouvement donne les relations entre le tenseur acoustique, l’a(plitude de discontinuite du vecteur et la vitesse normale de propagation. On discute alors d’une man&e nouvelle la possibilitt de separation des differents modes et des vitesses de propagation correspondantes. L’emploi des conditions de compatibilid dans la forme differentielle des equations fondamentales conduit a l’equation de base rbgissant la croissance de l’amplitude de chaque mode. Dans un milieu exempt de contraintes,l'onde de cisaillementne croitpas (de meme que dans la thtorie lineaire); le mode de dilatation peut croitre, suivant les valeurs des coefficients d’elasticite. Entin on compare avec des resultats deja connus. Zusammenfassung-Eine Welle wird als singulare Fliiche betrachtef an der die zweiten Ableitungen des Verschiebungsvektors unstetig sind. Anwendung der Vertrlglichkeitsbedingungen, Kontinuititsgleichung und der Bewegungsgleichungen ftihrt zu Beziehungen zwischen akkustischem Tensor, Amplitude des Diskontinuitiitsvektors und der Ausbreitungsgeschwindigkeit senkrecht zur Wellenfront. Die Miiglichkeit der Trennung in verschiedene Schwingungstypen und die zugehiirigen Ausbreitungsgeschwindigkeiten wird sodenn in einer neuartigen Weise diskutiert Die Anwendung der VertrBglichkeitsbedingungen auf die differenzierte Form der Grundgleichungen fiihrt zu der Gleichung die das Anwachsen der Amplitude fur jeden Schwingungstyp bestimmt. Wie in der linearen Theorie wiichst in einem unverformten Medium die Schubwelle nicht an; die Dehnungswelle kann anwachsen, dies hlngt von den elastischen Konstanten ab. Ein Vergleich mit schon bekannten Ergebnissen wird gemacht. AaaoTaqKR-BonHa OnpCLWleHa HaK OT#3IbHaU IlOBepXHOCTb B IIOnepeK KOTOP0i-i BTOpaR npOEI3BoAHafr BeKTOpa nepeuemenun fionycuaer cua.‘Iuu. np&IMCHCHLlC yCJIOBMfi COBMeCTIrMOCTI4, ypaBaKyCTWIeCKOr0 TCH3Opa, aMnJIllTyRb1 HCHMti HerIpepbIBHOCTI4 H ~BlUKeHIUJ AaeT 3aBHCIiMOCTb BeKTOpa CKa9Ka 12 HOpMaJlbHOfi CKOpOCTM paCnpOCTpaHeHrifl BOJIHbI. BCCJIeAyCTCS pa3J&eJDIMOCTb ffpEZMt?HRR yCJlOBl4H COBMeCTKMOCTI4 (POpM BOJIH 5% COOTBeTCTByIomHX CKOpOCT&i paCnpOCTpaHCHHH. $&JIFI ~PI+$CpOH~HaJTbHbIX ,$Of,M OCHOBHOrO ypaBHeHLlR IIOJIyYaeTCR OCHOBHOe ypaBHoHHC, OItHCbtBatomee ynenwreHrre aMnnr4TyAbr Kaxcqoi4 3onHbx. B rr3ojlapoBafrHot cpege BOJIH~I cpesa, TaKKre KaK B JIHHeZtHOZtTeOpKl4,HCyBCJK4WBaIOTCJr,LV'rnaTatViOHHar=l @OpMa BOJIHbI MOFKeT yBeJlIIWTbCH B 3aBHCRMOCTA OT yttpyrOCTI4. Pe3yJIbTaTbI CpaBHCHbI C A3BeCTHbIMK.
GROWTH OF ACCELERATION WAVES IN AN UNSTRAINED NON-LINEAR ISOTROPIC ELASTIC MEDIUM B. L. JUNEIA* and G. A. NARIBOLI? Iowa State University, Ames, Iowa Abstract-A wave is taken as a singular surface across which second derivatives of the displacement vector suffer discontinuities. Use of compatibility conditions, equation of continuity and the equations of motion gives the relations between the acoustic tensor, amplitude of the dis~ntinui~ vector and the normal speed of propagation. Separability of modes and corresponding propagation speeds am then discussed. Use of compatibility conditions on the differentiated forms of the basic equations then leads to the basic equation that governs the growth of the amplitude for each mode. in an unstrained medium the shear wave, as in linear theory, does not grow; the dilatational mode may grow, depending on the elasticities. Comparison is made with known results.
1. ~~ODUC~ON CROSS effect is one of the most important evidences of non-linear elastic response of a medium. Most of the known constitutive laws show these effects, known as Kelvin and Poynting effects. These features are characteristic only of finite elasticity and are absent in a linear theory. Problems discussed so far pertain to deformation of bodies which are in a state of equilibrium. Recently, the phenomenon of wave propagation in non-linear elasticity has received fresh attention. For this paper a propagating wdve is defined as follows: There exists, inside the medium, a moving surface such that at least some of the field variables or their derivatives are discontinuous across it. The magnitude of the discontinuity is called the amplitude ofthe wave. The velocity ofthis surface normal to itselfis the speed of propagation. Our main interest is to discover how the discontinuities across such a singular surface grow as the surface moves. Taking the displacement vector niy let us assume its components to be continuous while its first derivatives are discontinuous. We call this a first order or velocity wave. The growth of such a wave indicates separation of material or its ‘fracture’. Though physically more important, its study is not undertaken here. If the displacement and its first derivatives are continuous while its second derivatives are discontinuous, we call this an acceleration or second-order wave. Our study in this work is limited to such an acceleration wave. The most important feature of waves in non-linear mechanics is their growth. Gas dynamics [l], magneto-gas-dynamics [2, 31 and hypoelasticity [4] provide some new features. Discontinuities grow indefinitely in these cases. The non-linearity of the basic equations is one of the main reasons for this. Also, non-linear elasticity may provide some results which may be distinctive features, such as cross effects. Truesdell’s recent work [5,6] in the field of wave propagation in non-linear elasticity is * Department of Mechanical Engineering, Indian Institute of Technology, Delhi, India. t Department oi Engineering Mechanics and Engineering Kesearch Institute, Iowa State University, Iowa, 513
514
B. L. JUNFJAand G.A.
NARIBOLI
of interest to our work. Let & be the amplitude of the discontinuity across a wave front with unit normal IZ~. Then in a number of physical problems, of which ours is a type, the relevant equations lead to relations of the form
Qij4j = Po@
(1-l)
Here G is the velocity of propagation of the wave front normal to itself, which will be the actual velocity if it is moving in a medium at rest and will be the relative velocity, if it is moving through a medium in a state of motion. The tensor Qij, called the acoustical tensor, is a function of the physical variables ahead of the wave front and the normal vector n,. It does not involve 5;. This equation determines the speeds of propagation. Truesdell now introduces the idea of principal waves ; these are the waves travelling along one of the principal directions of strain; the waves are called longitudinal or transverse depending on whether the motion is along, or perpendicular to, the principal directions. Thus there are nine different waves. By this artifice the velocities separate in an elegant way. His aim is, however, to obtain velocities and their relationships with elasticities of the material but not to study the growth of the wave. Here, we adopt a different viewpoint. The system of equations (1.1) are linear homogeneous in &. For the existence of a singular surface they must have a non-trivial solution. So their determinant must vanish. This gives a cubic in G2 ; so in our physical space of three dimensions there are only three speeds of propagation which are the roots of a cubic. We further assert that there is one mode of propagation, i.e. one linear combination of &, which is non-zero for a wave travelling with each one of these velocities. A simple device would determine both the speeds and the modes of propagation. Let us for the present assume Qij is symmetric. Its three principal values are given by a cubic
1Qij - QsijI = 0.
(1.2)
Let Q(‘@(a = 1, 2, 3) be three roots of this equation and Lp) be the three corresponding principal directions. Thus we have QijLp)
= Q@)@‘).
(1.3)
Multiplying (1.1) by L$@we obtain by use of (1.3) (1.4) Thus the velocities are given by (a)2 POG
=
Q'"'
L!“‘~. # 0. J 3
(1.5)
This determines both the velocities and the modes of propagation. In addition, it is easy to prove, when principal values Q(‘) are distinct, that Ly) are also distinct. The mode Ly)cj gives the discontinuity across the wave travelling with velocity (Q”‘/P,,)*. Since principal
Growth of acceleration
waves in an unstrained non-linear isotropic elastic medium
515
directions are orthogonal, we also have Lj25j = LSrj = 0, for this wave front. Thus once the principal directions and numerical value of the acoustic tensor are known, the real velocities of propagation together with separate modes of propagation are easily obtained. The restriction of symmetry of Qij is unnecessary. In fact in a general elastic medium with arbitrary initial strains, the condition may not be satisfied. However all the above arguments continue to hold only if LT are taken to be left eigen-vectors of Qil But such an analysis is then useful only if the principal values of Qij are real and positive; otherwise the velocities will be complex. The study of wave propagation by use of theory of singular surfaces has, we believe, little meaning when the velocities are complex. If the principal values of Qij depend on ni, the velocity of propagation will also depend on it. Different parts of wave surface will then move with different velocities. The wave propagation is then said to be anisotropic. The only methods of solution of anisotropic wave propagation are those given by Lighthill [7] and Ludwig [8] on one hand and by Weitzner [9] on the other, both by transform techniques. The former leads only to asymptotic estimates but in an explicit form, while the latter to the formal full solution but in terms of the roots of a quartic which are not always easy to obtain.* When one wants to study the growth of a wave, we believe our approach is more appropriate. The integration of the growth equation in the anisotropic case remains open, as far as we know, even for the linear case.* When, however, the equation for velocities of propagation is non-linear in amplitude (which is the case of first-order waves), the singular surface is a shock wave, the study of which will be made later. Lastly it is important to note that for a plane wave front, ni is constant in time. The study of the growth of a wave is then relatively simple. Thus, here we study the growth of second-order waves in an unstrained non-linear elastic medium. Even in the most general elastic medium the growth of shear waves is entirely the same as in classical linear elasticity. However, the growth equation for the dilatational mode is now non-linear. Its growth or decay depends on the elasticities of the material. 2. VELOCITIES
OF PROPAGATION
Let xi be a spatial Cartesian orthogonal system of coordinates with which we work throughout. The singular surface which we also call the ‘wavefront’ is described by xi = x,(ua; t) with t as time and #(a = 1,2) as a system of surface coordinates G is the velocity of surface normal to itself Greek indices denote surface coordinates and Latin indices the Cartesian coordinates with the exception of n and t. These letters, when used as suffixes, denote components normal and tangential to the wave front We frequently use the resolution of vector fi as J = .L
ff
fnni
=.hi
=f”f,
+f
LIxi, a
f*
=
fp
=fixi,fl
Safl
s=sf
=
g,figBy
%,axi,p =
67(I
(2.1)
* The anisotropic and even non-homogeneous cases have since been exhaustively studied. Refer to ‘On Some Aspects of Wave Propagation.’ Engineering Research Institute Report 57, Iowa State University, Ames, also to appear in Professor Seth’s 60th Birthday Volume, I.I.T., Madras, India.
516
B. L.
JUNBJA
and G. A. NARIBOLI
Finally, let ui be the displacement vector, tij and eij the stress and strain tensors. Here we use the Eulerian strain tensor eij, defined as 2eij = Ui,j + Uj,i - h!k,ih!k, j in terms Of the displacement vector Ui. In order not to digress we have discussed in Appendix I all the compatibility conditions needed in the problem and in Appendix 2 the constitutive laws. As to the discontinuities we assume
[Uil = O,
[Ui,
j]
O>
=
[ui, jkllfljnknl
=
Pi.
(2.2)
Here and throughout, a bracket denotes the jump in the quantity across the wavefront. The medium ahead is unstrained and at rest with the value of density p as constant po. We frequently take the jumps of product of quantities. If one of the quantities is continuous it can be taken out and if, in addition, it vanishes ahead, the whole jump vanishes. Also when both factors are zero ahead the jump of product equals the product of jumps. If wi is the vorticity vector then we have [mil = [ieijkvk, jl = -@t?ijktknj = -$eijknj~aXk,
c(
co,= i&G.
b-d = 0,
o = (Jwiwih
(2.3)
So, for such a discontinuity, vorticity is always tangential and it vanishes with the tangential component of ti. To interpret the normal component of
at
+ Pvi, i + P,ivi
=
O,
so zl + Pot”
=
0,
(2.4)
where ul = [PC1 ni* If we take the alternative form of the continuity equation : p = poJ(l
- 21 + 411 - 8111)
(2.5)
where I, II and III are invariants of eip the result obtained above still holds for the assumed type of the singular surface. We thus have the interpretation of <,, i.e. the normal component of tie Here, 5, > 0 denotes compression across the front and 5, < 0 denotes the expansion wave. We now consider Cauchy’s most general elastic medium defined, on the assumption of homogeneity and isotropy only, by the constitutive law tij =
U6ij +
beij +
Ceimemj,
(2.6)
where Q,b and c denote arbitrary functions of I, II and III. We require the stress to vanish with the strain which imposes the condition a(0, 0,O) = 0. We next introduce the following
Growth
notation
of acceleration waves in an unstrained non-linear isotropic elastic medium
517
: aa --a2a El= ar)JIJrr=o;'12--armI I,II,Ilf=O
a,=;I ;BQ=
b(0, 0,O)
c(O,0, 0).
y() =
(2.7)
I,1I,l11=0
The equations of motion are dui 4j,j
=
(2.8)
Pdt
This gives (% + 38,) Cl% + 3Mi
(2.9a)
= PoG2&
which can be rewritten as
The linear homogeneous system for ti is (2.9a) and it is rewritten in the forms (2.9b) for 5, = cinj and
with
poG2=ifio
with
&#O;
t”#O;
<“=O; t,=O;
Sr =&ni ti={aXi,c
(2.lOa) (2. lob)
The ease of the Green-elastic body is included in this with gLtand /$, given in terms of strain energy functions from (A.2.8) For the second-order theory, 01~= d and PO = 2~ from (A.2.9). 3. GROWTH
OF WAVES
To obtain the growth equation we differentiate the equation of motion (2.8) with respect to xk, multiply by yzk,and take the jumps. The result of differentiating acceleration term as given in Appendix (A.1.8) is po(G& - 2G $
-I- 2G25&.
(2.1 la)
After noting the earlier remarks, the terms from that of stress can be written as nkltij,
jkl
=
tat
Lkl
+
POEeij,
jkl>
%i
+
A+
(2Slb)
where
(2.114
518
B. L. JIJNEJA~~~ G.A. NARIBOLI
Note that we have now six elasticities a,, CC 2, c~rr, PO, /I1 and y,, which reduce to five only when a strain energy function exists. To evaluate the first two terms in (2.1 lb) we use (A.1.9). After some simplifications we obtain here two alternative forms as
Thus after rearrangement
we get
{PoG’- (~1 + PO))Lni + boG2- fBo)taxi,. =2p,G ‘k - 2p,G2t”5i - (~(1+ fro) t2ni - floCJ
(2.12)
The important point to note is that the left member of (2.11) is the same as the equation for velocities (2.9) which always enables us to obtain growth equations. For a shear wave,
or (2.13) It is the same as in classical linear elasticity. For a dilatational wave, poG2 = x1 + PO; 5i = &,ni =
SO muhiplying
by
where k =
3 _
0 (
all+ VI+ -_____
2
30
(2.14)
2(x, + BCJ
The corresponding equation in linear elasticity is obtained by setting k, = 0; putting 4 = l/p so that p -+ 0 implies growth, it reduces to
dp
dn
+
k,
+
pR = 0.
The integration is now straight-forward and can be carried for a general surface by use of R for a system of parallel surfaces. It is, however, enough to note the cases for plane, cylind-
519
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
rical and spherical cases. We obtain with S2 = 0, - 1/2r, - l/R respectively, P = p. - ken 3
, plane
- 2k,r
, cylindrical
p =Po(~)-koRln(&)
(2.15)
,spherical
For k. > 0 a compressive wave p. > 0 grows. The converse holds for k. < 0; the growth characteristics thus depend on the sign of k,. 4. HYPERELASTIC
MATERIAL
We now note the particular case of second-order theory, based on the assumption that stored energy exists. The constitutive law we now use is (A.2.10) : tij zz -p Fij
(2.16)
PO
The wave velocities remain the same. For growth we evaluate the stress term as below.
Ltij,jJ nk
=
k ([P>
jl
[Fi, J nk + [F,j, jl btkl nk) + nk[Fij,jkI.
(2.17)
After evaluating all terms as before, we obtain PoG2[i - (1 + P) enni - p5^i = 2p,G $
- (B + 2@ + 4poG25,) Ci - (A + 1 + 2,~L5~) ni + (n + P) {Wp5n,aXi,p)
+
(52:- 2ntn) ni}9
(2.18)
where A = (22 + 4~ - 61 - m) t;
The growth of shear waves is governed by (2.13), the same as before while the equation for dilatational waves is also of the same type with k,=
9-z 2 ;1 + 2pL’
(2.19)
Finally it is interesting to note that our conclusions are in complete agreement with those of Bland [12]. He considers, however, the generalized one-dimensional problem and obtains a condition for formation of compression shock. His conditions, obtained from thermodynamic as well as mechanical considerations, coincide with each other. He uses the Langrangian coordinate system. Our condition, p. > 0, a compressive wave, IS identical to this. The condition for compressive shock on the elasticities, obtained by him, may be stated as
- 1+2pk <0 3 O
*
(2.20)
B. L. JUNEJAand G. A. NAR~BOLI
520
The derivation of this equivalence is noted in the appendix (A.2.15). From this we see that the second-order modulus 1 must be quite large compared to the first-order elastic modulii, to obtain a dilatational shock. Clearly, in contrast to fluid dynamics, we expect in solid mechanics the dilatational shock on physical grounds. Also it is interesting to note that other second-order elasticities play no part in shock formation. 5. CONCLUSION
Non-linear elasticity does admit the growth of acceleration waves. However if it is an expansion wave which must grow indefinitely, one second-order elasticity has to be quite large. In the most general theory of elasticity, we obtain a condition between the elasticities for growth or decay of the wave. REFERENCES [ 1] T. Y. THOMAS,The growth and decay of sonic dis~ntinuit~es in ideal gases. J. Z%furh.Mech. 6,455469 (1957). [2] K. 0. FRIEDRICHS and H. KRANZER,Non-linear wave motion, notes on M. H. VIII. AEC Res. Devei. Rep. N.Y.O. 6486 (1958). [3] G. A. NARIBOLI,The propagation and growth of sonic discontinuities in magnetohydrodynamics. J. Math. Mech. 12, 141-148 (1963). [4] G. A. NARIBOLI,Growth and propagation of waves in hypoelastic media. J. Math. Anal. Appl. 18, 57-65 (1964).
.151~ C. TRUESDELL.General and exact theory of waves in finite elastic strain, Archs raf. Mech. Anal. 8, 263
(1961). I [6] C. TRUESDELL, Second-order
theory of wave propagation in isotropic elastic materials. Int. Symp. Secondorder Effects in Elasticitv. Plasticitv and Fluid Dynamics, Haifa, April (1962). [7] M. J. LIGHTHILL,Studies’on magnetohydrodynamic waves and other a&tropic wave motion. Phil. Trans.
R. Sot. Ser. A 252 397430 (1960). [8] D. Luuwrc, Examples of hehaviour of solutions of hyperbolic equations for large times. J. Math. Mech. 12,
557-566 (1963). (91 H. WEITZNER,On the Green’s function for two-dimensiona
magnetohydrodynamic waves. I, II, AEC Rex Bevel. Rep. N.Y.O. 2886,9489 (1960). [lo] T. Y. THOMAS,Extended compatibility conditions for the study of surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6, 311-322 (1957). [ll] T. Y. THOMAS,Decay of waves in elastic solids. J. rat. Mech. Anal. 6,759-768 (1957). [12] D. R. BLAND,Dilational waves and shocks in large displacement isentropic dynamic elasticity. J. Mech. Phys. Solids 12. 245-267 (1964). [13] A. C. ERINGEN,~o~-~~ear Theory ~~Contj~~o~s Media. McGt-aw-Hill, New York (1962). (Received 17 August 1965; revised 3 September 1969)
APPENDIX 1. COMPATIBILITY CONDITIONS Second-order conditions
These follow from Thomas’s paper [lo]. If [Z,J t+ = 0,
[Zl = 0,
[Z,
(ASS)
then we have
[Z,ijl =
a22 -ax,at
[I I
Cninj,
Z
-
GCn,
a22
[ 1= 2F-
CG'.
(A.1.2)
Now let ui he the displacement vector. The reference system xi is spatial throughout. Then we have, for a secondorder wave, with & as the amplitude vector [UJ = 0,
[u, jk]
[Uj,j]
= &njnb
=
0,
Eui,jkl njnk
=
&
=G’T,.
(A.1.3)
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
521
We now take the velocity vector vi defined in terms of ui as
In all later derivations we limit attention to the particular case when the medium ahead is unstrained and at rest. Also we assume that the velocity of propagation is constant and that the ‘delta time derivative’, the time derivative for an observer on the wave front moving with it, vanishes. In view of these assumptions, when we take the jump of the product of two quantities, if one is continuous we can take it out with its value ahead of the wave front. In addition, if it is zero ahead, the jump in the whole quantity vanishes. So we obtain [I+] = 0;
[qj]
=
-G&nj
k] =k +,,,,,]=k] =G2&.
(A.1.4)
Let eij be the strain tensor defined in terms of displacement vector ui as eij = +(u~,~+ uj,! - u,,~u~,,). Using the above results and assumptions, we get [eij] = 0;
[eij,J = fnd&nj + cjni).
(A.1.5)
Third-order condition
We give a short derivation of these on the same lines as given by Thomas The others follow from it or can be obtained similarly. Further, let (A.1.6)
[ZijJ n,njnk = D; we then have
[Z>ij,J = Dninjn, + g”8C,Aninjx.t,p+ nkhixj,0 f n,niui,& - Cb@(n+j, d~k,s + njxk..xi, p + nkxi,.xj, s)
a32
1 1 ax,ax,at
=
-G{Dn,nj
+ g”kY,An,xj,P + njxi,J - Cb”xi,_xj,s}
=
G*(Dn, + gJC&xi,,)
(A.1.7)
+ gninj
- 2Gni E.
St Note that we have assumed &,/St = 0; we also note that uqa = 1, 2) is the surface coordinate system, b, is the second fundamental form and 2n = g7b4 gives the mean curvature. Proof of (A.1.7) Let us denote, to start with [Z,ij] = L, = Lji = Cninj
[Zip1 4 = Mji; Mijnj Y Mj,nj = Ni Mi,n,nj = D = N,ni.
From the known first-order compatibility we obtain
[&I
= Mijn, + g”Lj, a%,8;
but the right member must be symmetric in indices i,i and k. From this condition we obtain M$rk + @“I+, ok, p = M+ni + S”Lj,. oxi.0 = Mtinj + g”BLki,pj,P Multiplying by n, and summing over the repeated index we get Mij = Njn, + P’L,,,~,,x,,,r
= N,n, + g@L,, ~n,x~,~
522
B.L.
Again we multiply
JUNFJA and G. A. NAIUBOLI
the above by nj and sum over the repeated
index, to get
N-I = Dn.I + g@L qp, .vvi.
B
= Dni f g’LBC,,,xi,B Thus we have Mij = Dninj + g”@C,Axi,Bnj + nixj,J
- Ct@~,,$l~,~
Using this and simplifying. we obtain (A.1.7). We write now two more relations which we shall need. Let L”i,jd
njW$ = C
Then we have
(A.l.lO) Lastly [ePq, A
we need
nk =+[u,,4rk1 4 + f[u,,,.kl nk - f[%,&d [%,d =
H&v,
+ tit,p,n,
nk
-
+t%qrl
s”%.,Aw,,~ + wd - 5pbaBx,,P~r,d + sap5q,Xq,Anpxr,p + wp.B) - L?%-+~,,~~
hpkl
%
+
APPENDIX Cauchy elastic medium Based only on the assumptions
- 5%vr
2. CONSTITUTIVE
of isotropy
and homogeneity,
(A.l.ll)
LAWS we have a stress-strain
relation
as
tij = aSij + beij + cei,,,e,,,) Here 4 b and c are arbitrary
functions
of the invariants
I = e,,;
I, II and III of strain tensor eij defined as, III = (eijl.
II = t(eiiejj + eijeji);
Hyperelastic Here it is assumed that the stress tensor has a potential; it can be obtained on this assumption we can write the constitutive law [ 131 as tij = p(6i, PO
(A.2.1)
- 2ei,) g.
from a stored energy function.
(A.2.2)
Based
(A.2.3) 1”
Also we have p = p,,/(l Assuming
isotropy
and homogeneity
- 21 + 411 - 8111).
we have z = Z(I, II, III).
(A.2.4)
523
Growth of acceleration waves in an unstrained non-linear isotropic elastic medium
so az a1 ax aI1 S& - aIaejn az
az am
---faIIz+--=
awae,
Im
am
arr
s+Z , a+
z
-+c 2
-,
aejm
3
aei,
(A.2.5)
which defines Xi, Z, and X3. We further
note
-
aI
=amj;
z! =Ih,,
-
fj! =
emj;
emnenj
-
(A.2.6)
Iem, + II&,
Im
1”
Qejm
Now we use (A.2.6) in (A.2.5) and the result in (A.2.3) and also the Cayley-Hamilton
theorem
as (A.2.7)
eimemnemj- Ieimemj + lIeij - 11115,~= 0. After simplification are given by
we obtain the same as (A.2.1). Now a, b and c are not arbitrary
u = ;
(Z, + I& + II&
b = ;(-
the second-order
theory,
of the invariants,
but
(A.2.3)
- 2111X,)
C2 - IX, - 2C, - 21X2)
c = qz, PO To obtain
functions
(A.2.8)
+ 2X,).
we assume (A.2.9)
z = f(n + 2/J) I2 - 2pII + /I3 + mI II + nIII. We then obtain
tij
zz
PFij
PO with
Fij = (II + (3[ + m) 1’ + (m - n) II} 6, + {2~ + (m - n + 21) I} eij + (n - 4~) eime,*
(A.2.11)
Finally we give the relations between the elastic modulli employed by the present authors and Bland [12]. He uses the Lagrangian strain tensor which we denote by E, and its invariants by Ir, II, III, His constitutive law can now be obtained from X = )&I; However
these invariants
+ /+(I:
are related I, =
D = We expand
- 113 f aI; + &(I;
- 211,) + ~(3111, - 31&
+ I;).
(A.2.11)
to ours as
I - 411 + 12111 D
II - 6111
; IIE=_-----_. D
III
, III, = D
1 - 21 + 411 - 8111.
these to third order terms as I, = I + 21’ - 411 + 12111 - 121 II + 413 I;=I’-413-8111; II,
I;=13
= II + 6111 + 21 II
Using (A.2.12) in (A.2.11) and comparing
m = -41 for compressive
= I 11; III, = III.
with (A.2.9) we obtain
,%=I,;
His condition
; I&
/I=P~;
1=21+4p+a+P+v
- 12/l - 28 - 3v;
shock is Q. < 0
n = 12fl + 3v.
(A.2.12)
524
B.L.
JUNFJA
and G. A.
NARIBOLI
where
This is same as (2.9) of the paper
R&m&On considere une onde comme une surfacesinguliereau traversde laquelleles derivbessecondes du vecteur de d&placement subissent des discontinuites. L’emploi des conditions de compatibilite, de l%.quation de continuite et des equations du mouvement donne les relations entre le tenseur acoustique, l’a(plitude de discontinuite du vecteur et la vitesse normale de propagation. On discute alors d’une man&e nouvelle la possibilitt de separation des differents modes et des vitesses de propagation correspondantes. L’emploi des conditions de compatibilid dans la forme differentielle des equations fondamentales conduit a l’equation de base rbgissant la croissance de l’amplitude de chaque mode. Dans un milieu exempt de contraintes,l'onde de cisaillementne croitpas (de meme que dans la thtorie lineaire); le mode de dilatation peut croitre, suivant les valeurs des coefficients d’elasticite. Entin on compare avec des resultats deja connus. Zusammenfassung-Eine Welle wird als singulare Fliiche betrachtef an der die zweiten Ableitungen des Verschiebungsvektors unstetig sind. Anwendung der Vertrlglichkeitsbedingungen, Kontinuititsgleichung und der Bewegungsgleichungen ftihrt zu Beziehungen zwischen akkustischem Tensor, Amplitude des Diskontinuitiitsvektors und der Ausbreitungsgeschwindigkeit senkrecht zur Wellenfront. Die Miiglichkeit der Trennung in verschiedene Schwingungstypen und die zugehiirigen Ausbreitungsgeschwindigkeiten wird sodenn in einer neuartigen Weise diskutiert Die Anwendung der VertrBglichkeitsbedingungen auf die differenzierte Form der Grundgleichungen fiihrt zu der Gleichung die das Anwachsen der Amplitude fur jeden Schwingungstyp bestimmt. Wie in der linearen Theorie wiichst in einem unverformten Medium die Schubwelle nicht an; die Dehnungswelle kann anwachsen, dies hlngt von den elastischen Konstanten ab. Ein Vergleich mit schon bekannten Ergebnissen wird gemacht. AaaoTaqKR-BonHa OnpCLWleHa HaK OT#3IbHaU IlOBepXHOCTb B IIOnepeK KOTOP0i-i BTOpaR npOEI3BoAHafr BeKTOpa nepeuemenun fionycuaer cua.‘Iuu. np&IMCHCHLlC yCJIOBMfi COBMeCTIrMOCTI4, ypaBaKyCTWIeCKOr0 TCH3Opa, aMnJIllTyRb1 HCHMti HerIpepbIBHOCTI4 H ~BlUKeHIUJ AaeT 3aBHCIiMOCTb BeKTOpa CKa9Ka 12 HOpMaJlbHOfi CKOpOCTM paCnpOCTpaHeHrifl BOJIHbI. BCCJIeAyCTCS pa3J&eJDIMOCTb ffpEZMt?HRR yCJlOBl4H COBMeCTKMOCTI4 (POpM BOJIH 5% COOTBeTCTByIomHX CKOpOCT&i paCnpOCTpaHCHHH. $&JIFI ~PI+$CpOH~HaJTbHbIX ,$Of,M OCHOBHOrO ypaBHeHLlR IIOJIyYaeTCR OCHOBHOe ypaBHoHHC, OItHCbtBatomee ynenwreHrre aMnnr4TyAbr Kaxcqoi4 3onHbx. B rr3ojlapoBafrHot cpege BOJIH~I cpesa, TaKKre KaK B JIHHeZtHOZtTeOpKl4,HCyBCJK4WBaIOTCJr,LV'rnaTatViOHHar=l @OpMa BOJIHbI MOFKeT yBeJlIIWTbCH B 3aBHCRMOCTA OT yttpyrOCTI4. Pe3yJIbTaTbI CpaBHCHbI C A3BeCTHbIMK.