- Email: [email protected]
Similarity solutions for plane waves in hyperelastic materials
0020-7225/92 $5.00 + 0.00 Copyright @ 1992 Pergamon Press plc
lnr. 1. Engns Sci. Vol. 30, No. 6, pp. 701-715, 1992 Printed in Great Britain. All rights reserved
SIMILARITY SOLUTIONS FOR PLANE WAVES IN HYPERELASTIC MATERIALS E. S. SUHUBi Faculty of Science, Istanbul Technical University, Ma&k 80626, Istanbul, Turkey
Ah&act-The general scheme which was developed for balance equations of arbitrary order is applied to determine isovector fields (infinitesimal generators) of symmetry groups (Lie groups) related to field equations describing the propagation of plane waves in heterogeneous and anisotropic hyperelastic materials. This approach yields in principle all classes of materials (strain energy functions) admitting possible symmetry groups. Homogeneous and isotropic cases are also treated. Ordinary differentiat equations which determine similarity solutions corresponding to some symmet~ groups are given.
1. INTRODUCTION
The theory which has been developed for a general system of balance equations of finite arbitrary order to determine its isovector fields and resulting similarity solutions fl] is applied directly here to the problem of propagation of plane waves in an hyperelastic solid of infinite extent. As is well known isovector fields correspond to infinitesimal generators of the symmetry group (Lie group) of the field equations such that the point transformations induced by the orbits of isovector fields leave the field equations invariant. In other words they transform a set of solution of the equations to another set of solution. A similarly solution is then defined as an invariant solution under a symmetry transformation generated by a particular isovector field. The most general infinitesimal symmetry transformation for propagating plane waves in infinite hyperelastic solids are deduced by solving the set of equations, which were reported in [l], determining isovector components. All we have to do is to introduce into these equations the special forms of the coefficient functions appearing in the balance equation ~rresponding to the present problem. The same set of equations leads also to relations which determine the admissible classes of strain energy functions corresponding to particular symmetry transformations. Once we prescribe a particular symmetry transformation admitted by a particular form of the strain energy function it is relatively easy to study the associated similarity solutions by deriving nonlinear ordinary differential equations in a similarity variable. These equations can very effectively be treated by numerical approaches. The whole procedure becomes somewhat simpler for isotropic and homogeneous materials. These cases have also been dealt with in the present work.
2.
PLANE
MOTION
IN
HYPERELASTIC
SOLIDS
We represent a plane motion of a body by the following equations in Cartesian coordinates X/&XK,t) =X,4&
+ &(X, t),
x,=x,
k, K = 1,2, 3
(2-l)
where xk and X, are spatial and material coordinates, respectively, and ZQare the components of the displacements vector. Without loss of generality we assume that the direction of propagation is parallel to X1=X axis. We further suppose that the material exhibits heterogeneity in the X,-direction. The equations of motion in material coordinates in the absence of body forces are
702
E. S. SUHUBi
where
=dI:
T,k
(2.3)
aPk
are non-vanishing components of the first Piola-Kirchhoff stress tensor TKk, pa(X) is the given density of the material in the reference configuration and Z = x(Pk, X) is the strain energy function of the hyperelastic solid [2]. Throughout this work the summation convention is adopted. We have also defined dUk
duk
(2.4)
vk=-.
pk=z,
at
It is easy to see that the Cauchy stress tensor is then given by t Ik -
T,k,
tak = Pa
&k,
1 +PI
a = 2, 3.
(2.5)
Hence the field equations can be written in the balance form as follows
(2.6) to determine pk(X, t) and v,(X, t). The balance equations (2.6), can be cast into the standard form axkl
-
axk2 -++k=o ax2
+
axI
which was treated in [l] with a slight modification x1=x,
3.
(2.7) in the notation if we define
x2 = t,
DETERMINATION
OF
THE
xk2 = -POvk,
ck=o.
ISOVECTOR
FIELD
(2.8)
We now specialize the general results given in [l] to the present case. Since the number of independent variables is IZ= 2, the number of dependent variables iV = 3 and the order of balance equations (2.6) in u is m + 1 = 2 or m = 1, we deduce immediately that (2.6) or (2.7) admits no generalized symmetries so that the Lie symmetries of the first order system in pk and vk are completely equivalent to those of the second order system in uk and the isovector field can be represented by the following expression v=~(X,t,u)~+W(X,t,“)%+wk(X,t,u)$+pk~+V,~
k
aPk
k
(3.1)
where Pk and vk are given by pk=~-~~pk-~vk+~p,-~pkp,-~vkp,
‘k=%-$pk
-$fvk
1
I +z
I
v,-$k&$kv,q
I
I
1
(3.2)
If we write the balance equations (2.7) in the form axko -=o, ax,
a =
1, 2,
k = 1, 2, 3
(3.3)
703
Similarity solutions for plane waves
= I&,&, uk, uk,J then the coefficient functions listed in (3.32) of [l] for an arbitrary system of balance equations become with a slight change in the indicial notation
with &
A obkbn= -A
baklm =
-
A abkml=
aZk@ aab] au,
A abckh
-A backlm
=
azk, =av,au,,
ah,
dab]
where square brackets denote antisymmetrization vk,
=
Uk,a
and for any function f =f(xa,
=pk,
c”kl
(I, b, c = 1, 2
k, 1, m = 1,2,3,
vk2
=
of bracketed %
vk),
=
(3.4)
indices,
@,
w2=1v
(3.5)
uk, v&l
af . v,----
(3.6)
avk
Similarly equations (3.41) and (3.44) of [l] which help to determine isovector field can be written in the present case as follows
components
of the
(3.7) A
ax,,, ki~vnz
=
Awkm
-
~~~~b~k~lvf~
lb)
where ilkl are unknown functions of x,, uk, v kb. Parentheses represent the symmetrization of the indices involved. It is then straightforward to see that the coefficients defined in (3.4) take the following explicit forms
a& A ,=,-&~+ck$ A
a& A 2ki Ikl=au,'
=-@,z+ckg I
I
av
A
a$
-dB”+ckt-$ llkf
-
AI2kt=$+&x6k,,
aP1
I
Ck all, Azm= A 2,kl=_POaVk_ aPl 'ax' A 12klm -02A
12lklm
=
where & is Kronecker’s Bk
=
-24
2llkIm
=
ck,
$
m
,
avk d -PO---au,
m122kl,,,
=
ax
(PO@)6kkl
-u2,2klm
=
PO -$
Sk,
m
(3.8)
delta, v(zkl)
=
ckg,
+
cktp,,
v@k,)
=
-ct,
$$
vk
-
&,vk
(3.9)
and we define the second order symmetric tensor C,, (elastisities) and the vector C, by
a2Z:
CkI(P,X) = ___
aPk aP,
’
G(P,
X) = ~
a22
ap, ax’
(3.10)
In order to determine the isovector field (3.1) we have to find the five unknown functions $, q and wk of X, 1 and II. We can easily see that these functions have to satisfy the following relations {cf. (3.41) and (3.44) of [l]}
or introducing (3.8) into the foregoing equations we have
(3.11) The third relation of (3.11) yields directly the unknown functions llkf (3.12) Recalling (3.9), and (3.2) we immediately
see that
Introducing these expression into (3.11)., we obtain (3.13) Thus, (3.13) immediately
gives @LO XI*
Furthermore,
Lo. dU[
since it is not physically feasible for the tensor of elastisities C,, to acquire an
705
Similarity solutions for plane waves
isotropic form even in isotropic materials we deduce that W x=0.
W ==o, Hence
(3.14)
w = w(t)*
(1,= 9(X)*
Consequently relations (3.2) are simplified to a great extent to (3.15) where primes denote ordinary differentiation with respect to X and overdots ordinary differentiation with respect to f. Introducing (3.14), (3.15) and (3.12) into (3.11)1,2 and arranging the resulting expressions we obtain
where we employ the relation
due to the definition (3.10). Obviously (3.18) is a completely symmetric third order tensor. Therefore considering the antisymmetric part of (3.16) with respect to indices k and 1 we arrive at the relation C
m”
(3.19)
Now in order to satify (3.16)r identically we have to equate the coefficients of the second and first degree in vk to zero. This yields a2uk -=o,
2sf
aul au,
= rj&.,
the first of which gives ok
=
hk,(X,
t)uf
+
Ak(x,
t)
(3.20)
and the second determines the structure of the functions &, as (3.21) Inserting (3.20) into (3.19) we see that A should satisfy Ckm&nl)
=
A(km&nl
which implies that the symmetric part of A should commute with an arbitrary symmetric matrix
706
E. S. SUHUBi
C,, whence we can easily deduce that ACkljhas to be an isotropic tensor, i.e. &kf) = f&t-K r)&,. However write
(3.21) restricts the admissible form of 4,. Taking the symmetric
part of (3.21) we
which leads to
or &(X7 t) = &(X) +; r&t). We can now denote Akl by
=: [ MX) +; ll(r)]& where we introduced the vector a to represent permutation symbol. Thus (3.20) becomes c,k = {[&o(x) + i
+@)]ak!
+ e/&l%z(X)
the antisymmetric
+ ekl&rn(X)}&
+ I\-k&%
part of &,. ek/m is the
(3.22)
+.
Let us now employ (3.22) in (3.16)2. We obtain
Since this expression is linear in u we immediately
deduce that
a% (;l;tS,, +k
a%
+ em&)
=
0.
a&
If the material is not linear this equation is satisfied if and only if Ah&,, +emnro:=O
or
h&=0
or
An=ao,
ru;=O
or
Ek=izk
(3.23)
where a, and Q&are constant. Hence wk=[(u~+~ll)Sk,+ek,~n,]q+Ak
(3.24)
and the expressions (3.16)1,2 become
-([( ackm
aPi
-
Ckt[ 2(#’ - $I+
2 #]+$$Cp- (ekmrClm + elmrCk&, = 0.
Similarity solutions for plane waves
707
It now follows from (3.25)r that $ =0
or
q(t) = b1t2 + 2&t + b3
(3.26)
where bl, bz and b3 are arbitrary constants. Next we recall the de~nition (3.10) and note the relations
to transform (3.25),,z respectively,
into
It is then easy to verify that
or
c,,[(b’+ ($)‘I =o.
(3.28)
Thus (3.28) yields
c#f+
($.-f)‘=o
which can easily be integrated to
#4X)= --& [c&j(X) 0
+ c2] =
“lR$+ c2 0
(3.29)
where c1 and c2 are arbitrary constants and R,(X) is defined by the following indefinite integral of the prescribed function pO(X)
Ro(X)= /PO(X) dx.
(3.30)
If we employ the relation (3.29) and
in (3.27) we can see at once that these expressions can be written as
where we defined (3.32) ES 30:6-B
708
E. S. SUHUBi
Of course the functions 1/, and $.Iare now given by (3.26) and (3.29), respectively. we deduce from (3.32) that
a2A +3%, ax
Since 3 = 0
(3.33)
at2- ax at2ap, or if we note (3.31)2 $2 ap,
a3A,
ap, ap, ax at2
_0 *
Hence for strictly nonlinear materials we have
a3A
---==() ax at2
(3.34)
and in view of (3.33)
a%
dt2-Therefore
differentiating
0.
(3.35)
(3.31)r tw’ice with respect to t and making use of (3.35) we find that
a’l\,=O at4
(3.36)
.
Thus (3.34) and (3.36) have the solution Ak =
a
dkt3 + ;fktz + A,(X)t + pk(X)
(3.37)
where dk and fk are arbitrary constants and n,(X) and &X) are arbitrary functions. This completes the determination of the general isovector field. Collecting the results recorded in (3.24), (3.26), (3.29) and (3.37) we can now list all the relevant components of the isovector field which are none other than the infinitesimal generators of the associated symmetry (Lie) group:
q = blt2 + 2b,t f b3 co, = [(a, + b2 + b, t)cS, + ektma,]ut + i dkt3 +
kfkt2 +A,(X)t +pk(X).
However this is obviously not the end of the whole story. We still have to satisfy (3.31) under the special forms given in (3.38). We see that (3.32) can now be written as 12=tl\,+fi,
where (3.39)
A,(X, p) = [(% + bz -
+‘)Pk
+
ekd%P~
f
cl;]
~+$~+($‘+2b,-za,-c,)X aPk
and (3.31) becomes
t@l\l -_ apk
32A2=o %I
+
apk
ap,
709
Similarity solutions for plane waves
which are satisfied if and only if -
=
sx=
- 3% =o apk apI ’
dkPO(X),
fkdx)
from which we can easily find that
& =
[d,%(X)
+
gk]pk
+
A2
dx),
=
[fk&(X)
hk]pk
+
+
(3.40)
p(x)
where gk and hk are arbitrary constants and (Yand /3 are arbitrary functions. Due to (3.39) and (3.40) we conclude that the strain energy function Z has to satisfy simultaneously the following relations in order that the field equations admit the general symmetry group generated by (3.38) [bk
+
n;(x)]
gk
&I
+
2bJ
+
=
b2
-
[dkb(X)
#‘(x)]Pk
+
+
-
a(x)
ekl&dl
+
[e’(x)
gk]pk
+
+
2b2
-
%I
Y;(x)}
-
cl]z
gk
=
+
d’(x)
[h&(X)
E
+
hk]pk
+
p(x)
(3.41)
where $ and R0 are given by (3.29) and (3.30), respectively. Obviously, the right-hand sides of the above expresssions, even if they satisfy integrability conditions, give rise to only a linear contribution in p to Z. This, in turn, requires a prescribed initial stress distribution in the material. If we consider an initially stress free body, then we can assume without loss of generality that the right-hand sides of (3.41) vanish. To achieve this we only have to take dk=&=fk=hk=O,
a(X)
Therefore, in this case, I: should satisfy simultaneously partial differential equations given below + 2bJ
(blp, + A;) g
= /3(X) = 0.
(3.42)
two homogeneous
linear first order
= 0
(3.43)
For an entirely arbitrary Z (3.43),,2 are satisfied if and only if we take bl=O,
A; = 0
or
& = 1, = const.,
@=O
or
c1=c2=0,
a,=O, b2=0,
p; = 0
or
ak = 0,
pk = mk = conk
which results in a trivial isovector field given by
To find a solution of this system while preserving all the parameters and functions involved is next to impossible since the first equation implies that Z should be a homogeneous function of degree -2 in pk + (&J/b,. However, by some adjustments of the parameters which restrict somewhat the isovector field, one obtains quite a rich class of admissible function C. For instance, we may choose bI=O,
In this case the first equation
A; = 0
or
Ak = 1, = mnst.
of (3.43) is satisfied identically
(3.44)
whereas its second equation
710
E. S. $UHUBI
determines
uniquely the form of C. The isovector field corresponding 9 = 2b,t + b3,
@=-+&+cJ,
to this case become
+ e klm~ml~, + hit+ b&q
mk = [(a~,+ b&l
Of course several other possibilities may be explored. A detailed study of the system (3.43), the associated isovector fields and similarity solutions which they generated and their numerical treatment will be the subject of a future work. Here we satisfy ourselves only dealing with an arbitrarily chosen example. Let us suppose that (3.44) is satisfied. We further assume that Cl/(=o and write (3.43)* as [(a - (p’)pB -t ,u;] 2
+ # $$+
(b + Cp’)r:= 0
where we defined the constants a and b by a=a,+b,,
b = 2b, - 2a. - c, ,
The characteristic of this first order linear partial differential integration of the ordinary differential equations dI:
dx =_=(a - #‘lPk + cl; cp %
(b + Cp’)Z ’
equation
indices are suspended
the solutions of which lead to the characteristic
variables
=
clPk (uic,)-1
are determined
by the
k = 1, 2, 3
where the summation convention on repeated The above equations can also be written as
nk
(3.45)
for underscored
-
indices.
(3.46)
and the admissible I: is found to have the following form (3.47)
where we made use of (3.29).
4. SPECIAL
CASES
We now conside some special cases. (‘i) Homogeneous
materials
In this case 5: is independent
of X and pO is a constant. Hence (3.29) becomes # = c,X + E7,
(4.1)
Similarity solutions for plane waves
711
where E2= cz/pO, and (3.43) now reduces to (b1p, + A;) z
+ 2612 = 0
[(a0+ b2- c&k + ekl&?#l + pi] g Differentiating
k
+ 2@2- %)Z = 0.
these expressions with respect to X we obtain
or & = 1,x + hk, where lk, mk, hk and nk are constants. Therefore
pk=mkX+nk
(4.2)
Z has to satisfy the following equations
I(ao + bz - c&k + ek&&,& + mk] E + 2(bz - aO)Z = 0. aPk
(4.3)
This case has been investigated in some detail earlier by using a direct approach based on exterior calculus [3]. Let us note that for an arbitrary Z (4.3) is satisfied if and only if we take b1=0, In this case the corresponding C#J = 2uoX + E*,
bz=urJ,
ak = 0,
Cl = 24&-J.
isovector field is determined v= 2u,t + b3,
by
ok = &uk + lkXt + mkX + hkt + mk.
(ii) Isotropic materials If the material is isotropic, then I: has the form [2] 2 = qp,
4, X)
(4.4)
where P =p1,
Consequently
4*=P~+P~=PcYPp.
(4.5)
we have
ax aI: -=apI ap'
ax pmax -=-ah 4 a4'
Hence equations (3.43) take the following form
[(a0 + b* - C#J’)p+ pi] g
+ (a0 + 62 - r#9)qE + #;;+
(@’ + 262 - 2uo-- c,)Z
+[(~-~~)ea(lg+~~pb]p,=O
(4.6)
where eaB is the two-dimensional permutation symbol. Obviously these equations may have a solution Z(p, q, X) if and only if (4.7)
112
E. S. $UHUBi
Thus for physically meaningful results we have only to assume that A;= 0
or
A, = I, = const.,
a, = 0,
pL=O
or
~~=ItZ,=const.
Hence (4.6) now reduces to
[(a0 + b2 - $‘)p + ,u;] g
+ (a. + b, - O.)qg
and the relevant isovector components c,&
(4.8)
become +
~2 II,
@= RI,
w1 =
+ c#I~:+ (@’ + 2b2 - 2a,, - c,)Z = 0
=
b, t2 + 2b,t + b3
’
(a0 + b2 + b,t)u, + A,(X)t + p,(X)
o, = (a0 + b2 + b,t)u,
+ e,ga,up + 1,t + m,,
Ly= 2, 3.
(4.9)
(iii) Homogeneous and isotropic materials It is easy to see that (4.8) and (4.9) reduce in this case to
[(a0 + b2 - c,)p + ml] g + (a0 + b2 - c,)q g + 2(b2 - a& 3P +l
cj = c,x + c2,
TJJ=
=0
(4.10)
blt2 + 2b,t + b3
o1 = (a. + b2 + blt)u, + f,Xt + m,X + h,t + n, (Y= 2, 3.
w,=(ao+b2+bIt)u,+e,paIug+f,t+m,,
Note that in isotropic cases (ii) and (iii) w, include terms e,P+ which cannot be controlled by the structure of the material since the parameter a, does not appear in differential equations satisfied by Z.
5.
SIMILARITY
SOLUTIONS
If we choose a particular isovector field the associated similarity solution is determined the additional constraint [l] #(X)
2 +l)(t) 2 - w,(X,
t, u) = 0.
by
(5.1)
Since wk is linear in II, equation (5.1) is a set of linear first order partial differential equations which specify the functional form of the vector u for a similarity solution. The solution to (5.1) is easily obtainable. Characteristic equations are dX -=----_
dt
@(X) VW
d&c
(5.2)
%(Xiu)
whence we first get the similarity variable (5.3) where Y(t) = exp
I
dt
q(t) .
(5.4)
Similarity solutions for plane waves
713
Then integration of the linear ordinary differential equations along a characteristic E = const. as follows du, 1 - eJk[X@,5), & ul dt - w(t) yields the functional form of the similarity solution as r+ = A,(X, 0&(E) +&(X,
0
(5.5)
where Akl and Ak are known functions of X and t. In~~ucing (5.5) into the field equations (2.6) together with (2.4) we end up in three nonlinear ordinary differential equations in the similarity variable f to determine the functions u,. * In order to reach explicit results we have of course have to consider a particular isovector field. Such an example is reported in [3j for homogeneous materials. We shall consider here a material whose strain energy function is given by (3.47). We take the associated &vector field as MO
cp=
+
~2
T+‘I=
2
R’
2b,t + b3,
mk
=
auk
+
pk(x)
0
where we asssumed that 1, = 0 for the sake of simplicity. Let us now define a set of new constants by
2bz
-_=(y
C2
-=
,
Cl
B,
(5.6)
$=y
Cl
2
so that
?@ = 2b,(t
f Y),
wk = auk + pk
, clPk Itk
=
R;(R,
+
/-J)‘Q”“-1
-
(R.
$“I
Z(p,X)
dx9
=
RE, z(a). cl(Ro + @)@‘=l)+l
(5.7)
Hence (5.2) becomes % (&-dXdx-= Ro+B
dt
duk
l+y
auk+pk
whence we deduce that the similarity variable is given by $=
(Ro+BY
W3)
t+Y
and uk SatiSfieS
on a characteristic dtsl, a R& 1 R; -=-dX clRo+/3Uk+~Ro+~pk
whose solution may be obtained straightforwardly as (5.9) Noting that X$ CVR;, --if, %f-R,+p
s= al
--
f t+Y
714
E. S. $UHUBi
we can easily verify that
x
I
R;‘(R,, + j-j)-‘“““-
’ pk dX +
$RX& + fi)-h
and in view of (5.7), .n, =
[email protected]~+c’i~k
+E
We see then immediately and we find
/R~(R,+/3)-(.ki)-'pk &x + (R~~+~~-("rl)~k - ~(R"+~)-("'cl)~~ dx. that the last three terms cancel each other in the above expression
Similarly we obtain auk
Vk=-=-
at
On the other hand
and the field equations
become R;(j?&
$
fi)-++Wrl
_
= (Ro+ B)-
!!$!!g
R,
k
o (t + Y)*
(S2rrk’+ 2&J;).
(5.11)
Since due to (3.45) 2a + b
----+I=4L2* Cl
Cl
we see that (5.11) take the form = E’( ‘!j2c + 2&J;) or we obtain the following ordinary differential
26,5(~)’ +(a0 k
equations
36, -I- c,) g - = k
c&~(&Y;:+ XJ;)
(5.12)
with 2 = %(%,
3t2r
f3)t
d&I)
=
2b2WX)
+
6%
+
b2)&(8-
(5.13)
Of course, the system (5.12) is in general highly nonlinear and is amenable only to a numerical treatment. The free parameters involved are imposed by the structure of the strain energy function. It is quite clear that this particular example and the ensuing similarity solution we have studied so far may not be identified immediately as the one corresponding to an easily realizable physical situation. However, we believe that it is still of interest since the resulting ordinary nonlinear differential equations can be treated very efficiently by numerical methods.
Similarity solutions for plane waves
715
In this capacity it may serve as a benchmark test for a larger numerical scheme devised to solve the nonlinear partial differential equations corresponding to a realistic case.
REFERENCES [l] E. S. !$JHUBi, Znt. J. Engng Sci. 29, 133 (1991). [2] A. C. ER1NGE.N and E. S. SUHUBI, Elastodynamics, Vol. I-Finite Motions. Academic Press, New York (1974). [3] E. S. SUHUBI, Similarity solutions for one-dimensional nonlinear elastodynamics. In Proceedings of the Vth International Meeting Waves and Stability in Continuous Media (Edited by S. RIONERO), Series on Advances in Mathematics for Applied Sciences, Vol. 4, pp. 390-401. World Scientific, Singapore (1991). (Received 23 August 1991; accepted 9 October 1991)
ES 30:6-C
lnr. 1. Engns Sci. Vol. 30, No. 6, pp. 701-715, 1992 Printed in Great Britain. All rights reserved
SIMILARITY SOLUTIONS FOR PLANE WAVES IN HYPERELASTIC MATERIALS E. S. SUHUBi Faculty of Science, Istanbul Technical University, Ma&k 80626, Istanbul, Turkey
Ah&act-The general scheme which was developed for balance equations of arbitrary order is applied to determine isovector fields (infinitesimal generators) of symmetry groups (Lie groups) related to field equations describing the propagation of plane waves in heterogeneous and anisotropic hyperelastic materials. This approach yields in principle all classes of materials (strain energy functions) admitting possible symmetry groups. Homogeneous and isotropic cases are also treated. Ordinary differentiat equations which determine similarity solutions corresponding to some symmet~ groups are given.
1. INTRODUCTION
The theory which has been developed for a general system of balance equations of finite arbitrary order to determine its isovector fields and resulting similarity solutions fl] is applied directly here to the problem of propagation of plane waves in an hyperelastic solid of infinite extent. As is well known isovector fields correspond to infinitesimal generators of the symmetry group (Lie group) of the field equations such that the point transformations induced by the orbits of isovector fields leave the field equations invariant. In other words they transform a set of solution of the equations to another set of solution. A similarly solution is then defined as an invariant solution under a symmetry transformation generated by a particular isovector field. The most general infinitesimal symmetry transformation for propagating plane waves in infinite hyperelastic solids are deduced by solving the set of equations, which were reported in [l], determining isovector components. All we have to do is to introduce into these equations the special forms of the coefficient functions appearing in the balance equation ~rresponding to the present problem. The same set of equations leads also to relations which determine the admissible classes of strain energy functions corresponding to particular symmetry transformations. Once we prescribe a particular symmetry transformation admitted by a particular form of the strain energy function it is relatively easy to study the associated similarity solutions by deriving nonlinear ordinary differential equations in a similarity variable. These equations can very effectively be treated by numerical approaches. The whole procedure becomes somewhat simpler for isotropic and homogeneous materials. These cases have also been dealt with in the present work.
2.
PLANE
MOTION
IN
HYPERELASTIC
SOLIDS
We represent a plane motion of a body by the following equations in Cartesian coordinates X/&XK,t) =X,4&
+ &(X, t),
x,=x,
k, K = 1,2, 3
(2-l)
where xk and X, are spatial and material coordinates, respectively, and ZQare the components of the displacements vector. Without loss of generality we assume that the direction of propagation is parallel to X1=X axis. We further suppose that the material exhibits heterogeneity in the X,-direction. The equations of motion in material coordinates in the absence of body forces are
702
E. S. SUHUBi
where
=dI:
T,k
(2.3)
aPk
are non-vanishing components of the first Piola-Kirchhoff stress tensor TKk, pa(X) is the given density of the material in the reference configuration and Z = x(Pk, X) is the strain energy function of the hyperelastic solid [2]. Throughout this work the summation convention is adopted. We have also defined dUk
duk
(2.4)
vk=-.
pk=z,
at
It is easy to see that the Cauchy stress tensor is then given by t Ik -
T,k,
tak = Pa
&k,
1 +PI
a = 2, 3.
(2.5)
Hence the field equations can be written in the balance form as follows
(2.6) to determine pk(X, t) and v,(X, t). The balance equations (2.6), can be cast into the standard form axkl
-
axk2 -++k=o ax2
+
axI
which was treated in [l] with a slight modification x1=x,
3.
(2.7) in the notation if we define
x2 = t,
DETERMINATION
OF
THE
xk2 = -POvk,
ck=o.
ISOVECTOR
FIELD
(2.8)
We now specialize the general results given in [l] to the present case. Since the number of independent variables is IZ= 2, the number of dependent variables iV = 3 and the order of balance equations (2.6) in u is m + 1 = 2 or m = 1, we deduce immediately that (2.6) or (2.7) admits no generalized symmetries so that the Lie symmetries of the first order system in pk and vk are completely equivalent to those of the second order system in uk and the isovector field can be represented by the following expression v=~(X,t,u)~+W(X,t,“)%+wk(X,t,u)$+pk~+V,~
k
aPk
k
(3.1)
where Pk and vk are given by pk=~-~~pk-~vk+~p,-~pkp,-~vkp,
‘k=%-$pk
-$fvk
1
I +z
I
v,-$k&$kv,q
I
I
1
(3.2)
If we write the balance equations (2.7) in the form axko -=o, ax,
a =
1, 2,
k = 1, 2, 3
(3.3)
703
Similarity solutions for plane waves
= I&,&, uk, uk,J then the coefficient functions listed in (3.32) of [l] for an arbitrary system of balance equations become with a slight change in the indicial notation
with &
A obkbn= -A
baklm =
-
A abkml=
aZk@ aab] au,
A abckh
-A backlm
=
azk, =av,au,,
ah,
dab]
where square brackets denote antisymmetrization vk,
=
Uk,a
and for any function f =f(xa,
=pk,
c”kl
(I, b, c = 1, 2
k, 1, m = 1,2,3,
vk2
=
of bracketed %
vk),
=
(3.4)
indices,
@,
w2=1v
(3.5)
uk, v&l
af . v,----
(3.6)
avk
Similarly equations (3.41) and (3.44) of [l] which help to determine isovector field can be written in the present case as follows
components
of the
(3.7) A
ax,,, ki~vnz
=
Awkm
-
~~~~b~k~lvf~
lb)
where ilkl are unknown functions of x,, uk, v kb. Parentheses represent the symmetrization of the indices involved. It is then straightforward to see that the coefficients defined in (3.4) take the following explicit forms
a& A ,=,-&~+ck$ A
a& A 2ki Ikl=au,'
=-@,z+ckg I
I
av
A
a$
-dB”+ckt-$ llkf
-
AI2kt=$+&x6k,,
aP1
I
Ck all, Azm= A 2,kl=_POaVk_ aPl 'ax' A 12klm -02A
12lklm
=
where & is Kronecker’s Bk
=
-24
2llkIm
=
ck,
$
m
,
avk d -PO---au,
m122kl,,,
=
ax
(PO@)6kkl
-u2,2klm
=
PO -$
Sk,
m
(3.8)
delta, v(zkl)
=
ckg,
+
cktp,,
v@k,)
=
-ct,
$$
vk
-
&,vk
(3.9)
and we define the second order symmetric tensor C,, (elastisities) and the vector C, by
a2Z:
CkI(P,X) = ___
aPk aP,
’
G(P,
X) = ~
a22
ap, ax’
(3.10)
In order to determine the isovector field (3.1) we have to find the five unknown functions $, q and wk of X, 1 and II. We can easily see that these functions have to satisfy the following relations {cf. (3.41) and (3.44) of [l]}
or introducing (3.8) into the foregoing equations we have
(3.11) The third relation of (3.11) yields directly the unknown functions llkf (3.12) Recalling (3.9), and (3.2) we immediately
see that
Introducing these expression into (3.11)., we obtain (3.13) Thus, (3.13) immediately
gives @LO XI*
Furthermore,
Lo. dU[
since it is not physically feasible for the tensor of elastisities C,, to acquire an
705
Similarity solutions for plane waves
isotropic form even in isotropic materials we deduce that W x=0.
W ==o, Hence
(3.14)
w = w(t)*
(1,= 9(X)*
Consequently relations (3.2) are simplified to a great extent to (3.15) where primes denote ordinary differentiation with respect to X and overdots ordinary differentiation with respect to f. Introducing (3.14), (3.15) and (3.12) into (3.11)1,2 and arranging the resulting expressions we obtain
where we employ the relation
due to the definition (3.10). Obviously (3.18) is a completely symmetric third order tensor. Therefore considering the antisymmetric part of (3.16) with respect to indices k and 1 we arrive at the relation C
m”
(3.19)
Now in order to satify (3.16)r identically we have to equate the coefficients of the second and first degree in vk to zero. This yields a2uk -=o,
2sf
aul au,
= rj&.,
the first of which gives ok
=
hk,(X,
t)uf
+
Ak(x,
t)
(3.20)
and the second determines the structure of the functions &, as (3.21) Inserting (3.20) into (3.19) we see that A should satisfy Ckm&nl)
=
A(km&nl
which implies that the symmetric part of A should commute with an arbitrary symmetric matrix
706
E. S. SUHUBi
C,, whence we can easily deduce that ACkljhas to be an isotropic tensor, i.e. &kf) = f&t-K r)&,. However write
(3.21) restricts the admissible form of 4,. Taking the symmetric
part of (3.21) we
which leads to
or &(X7 t) = &(X) +; r&t). We can now denote Akl by
=: [ MX) +; ll(r)]& where we introduced the vector a to represent permutation symbol. Thus (3.20) becomes c,k = {[&o(x) + i
+@)]ak!
+ e/&l%z(X)
the antisymmetric
+ ekl&rn(X)}&
+ I\-k&%
part of &,. ek/m is the
(3.22)
+.
Let us now employ (3.22) in (3.16)2. We obtain
Since this expression is linear in u we immediately
deduce that
a% (;l;tS,, +k
a%
+ em&)
=
0.
a&
If the material is not linear this equation is satisfied if and only if Ah&,, +emnro:=O
or
h&=0
or
An=ao,
ru;=O
or
Ek=izk
(3.23)
where a, and Q&are constant. Hence wk=[(u~+~ll)Sk,+ek,~n,]q+Ak
(3.24)
and the expressions (3.16)1,2 become
-([( ackm
aPi
-
Ckt[ 2(#’ - $I+
2 #]+$$Cp- (ekmrClm + elmrCk&, = 0.
Similarity solutions for plane waves
707
It now follows from (3.25)r that $ =0
or
q(t) = b1t2 + 2&t + b3
(3.26)
where bl, bz and b3 are arbitrary constants. Next we recall the de~nition (3.10) and note the relations
to transform (3.25),,z respectively,
into
It is then easy to verify that
or
c,,[(b’+ ($)‘I =o.
(3.28)
Thus (3.28) yields
c#f+
($.-f)‘=o
which can easily be integrated to
#4X)= --& [c&j(X) 0
+ c2] =
“lR$+ c2 0
(3.29)
where c1 and c2 are arbitrary constants and R,(X) is defined by the following indefinite integral of the prescribed function pO(X)
Ro(X)= /PO(X) dx.
(3.30)
If we employ the relation (3.29) and
in (3.27) we can see at once that these expressions can be written as
where we defined (3.32) ES 30:6-B
708
E. S. SUHUBi
Of course the functions 1/, and $.Iare now given by (3.26) and (3.29), respectively. we deduce from (3.32) that
a2A +3%, ax
Since 3 = 0
(3.33)
at2- ax at2ap, or if we note (3.31)2 $2 ap,
a3A,
ap, ap, ax at2
_0 *
Hence for strictly nonlinear materials we have
a3A
---==() ax at2
(3.34)
and in view of (3.33)
a%
dt2-Therefore
differentiating
0.
(3.35)
(3.31)r tw’ice with respect to t and making use of (3.35) we find that
a’l\,=O at4
(3.36)
.
Thus (3.34) and (3.36) have the solution Ak =
a
dkt3 + ;fktz + A,(X)t + pk(X)
(3.37)
where dk and fk are arbitrary constants and n,(X) and &X) are arbitrary functions. This completes the determination of the general isovector field. Collecting the results recorded in (3.24), (3.26), (3.29) and (3.37) we can now list all the relevant components of the isovector field which are none other than the infinitesimal generators of the associated symmetry (Lie) group:
q = blt2 + 2b,t f b3 co, = [(a, + b2 + b, t)cS, + ektma,]ut + i dkt3 +
kfkt2 +A,(X)t +pk(X).
However this is obviously not the end of the whole story. We still have to satisfy (3.31) under the special forms given in (3.38). We see that (3.32) can now be written as 12=tl\,+fi,
where (3.39)
A,(X, p) = [(% + bz -
+‘)Pk
+
ekd%P~
f
cl;]
~+$~+($‘+2b,-za,-c,)X aPk
and (3.31) becomes
t@l\l -_ apk
32A2=o %I
+
apk
ap,
709
Similarity solutions for plane waves
which are satisfied if and only if -
=
sx=
- 3% =o apk apI ’
dkPO(X),
fkdx)
from which we can easily find that
& =
[d,%(X)
+
gk]pk
+
A2
dx),
=
[fk&(X)
hk]pk
+
+
(3.40)
p(x)
where gk and hk are arbitrary constants and (Yand /3 are arbitrary functions. Due to (3.39) and (3.40) we conclude that the strain energy function Z has to satisfy simultaneously the following relations in order that the field equations admit the general symmetry group generated by (3.38) [bk
+
n;(x)]
gk
&I
+
2bJ
+
=
b2
-
[dkb(X)
#‘(x)]Pk
+
+
-
a(x)
ekl&dl
+
[e’(x)
gk]pk
+
+
2b2
-
%I
Y;(x)}
-
cl]z
gk
=
+
d’(x)
[h&(X)
E
+
hk]pk
+
p(x)
(3.41)
where $ and R0 are given by (3.29) and (3.30), respectively. Obviously, the right-hand sides of the above expresssions, even if they satisfy integrability conditions, give rise to only a linear contribution in p to Z. This, in turn, requires a prescribed initial stress distribution in the material. If we consider an initially stress free body, then we can assume without loss of generality that the right-hand sides of (3.41) vanish. To achieve this we only have to take dk=&=fk=hk=O,
a(X)
Therefore, in this case, I: should satisfy simultaneously partial differential equations given below + 2bJ
(blp, + A;) g
= /3(X) = 0.
(3.42)
two homogeneous
linear first order
= 0
(3.43)
For an entirely arbitrary Z (3.43),,2 are satisfied if and only if we take bl=O,
A; = 0
or
& = 1, = const.,
@=O
or
c1=c2=0,
a,=O, b2=0,
p; = 0
or
ak = 0,
pk = mk = conk
which results in a trivial isovector field given by
To find a solution of this system while preserving all the parameters and functions involved is next to impossible since the first equation implies that Z should be a homogeneous function of degree -2 in pk + (&J/b,. However, by some adjustments of the parameters which restrict somewhat the isovector field, one obtains quite a rich class of admissible function C. For instance, we may choose bI=O,
In this case the first equation
A; = 0
or
Ak = 1, = mnst.
of (3.43) is satisfied identically
(3.44)
whereas its second equation
710
E. S. $UHUBI
determines
uniquely the form of C. The isovector field corresponding 9 = 2b,t + b3,
@=-+&+cJ,
to this case become
+ e klm~ml~, + hit+ b&q
mk = [(a~,+ b&l
Of course several other possibilities may be explored. A detailed study of the system (3.43), the associated isovector fields and similarity solutions which they generated and their numerical treatment will be the subject of a future work. Here we satisfy ourselves only dealing with an arbitrarily chosen example. Let us suppose that (3.44) is satisfied. We further assume that Cl/(=o and write (3.43)* as [(a - (p’)pB -t ,u;] 2
+ # $$+
(b + Cp’)r:= 0
where we defined the constants a and b by a=a,+b,,
b = 2b, - 2a. - c, ,
The characteristic of this first order linear partial differential integration of the ordinary differential equations dI:
dx =_=(a - #‘lPk + cl; cp %
(b + Cp’)Z ’
equation
indices are suspended
the solutions of which lead to the characteristic
variables
=
clPk (uic,)-1
are determined
by the
k = 1, 2, 3
where the summation convention on repeated The above equations can also be written as
nk
(3.45)
for underscored
-
indices.
(3.46)
and the admissible I: is found to have the following form (3.47)
where we made use of (3.29).
4. SPECIAL
CASES
We now conside some special cases. (‘i) Homogeneous
materials
In this case 5: is independent
of X and pO is a constant. Hence (3.29) becomes # = c,X + E7,
(4.1)
Similarity solutions for plane waves
711
where E2= cz/pO, and (3.43) now reduces to (b1p, + A;) z
+ 2612 = 0
[(a0+ b2- c&k + ekl&?#l + pi] g Differentiating
k
+ 2@2- %)Z = 0.
these expressions with respect to X we obtain
or & = 1,x + hk, where lk, mk, hk and nk are constants. Therefore
pk=mkX+nk
(4.2)
Z has to satisfy the following equations
I(ao + bz - c&k + ek&&,& + mk] E + 2(bz - aO)Z = 0. aPk
(4.3)
This case has been investigated in some detail earlier by using a direct approach based on exterior calculus [3]. Let us note that for an arbitrary Z (4.3) is satisfied if and only if we take b1=0, In this case the corresponding C#J = 2uoX + E*,
bz=urJ,
ak = 0,
Cl = 24&-J.
isovector field is determined v= 2u,t + b3,
by
ok = &uk + lkXt + mkX + hkt + mk.
(ii) Isotropic materials If the material is isotropic, then I: has the form [2] 2 = qp,
4, X)
(4.4)
where P =p1,
Consequently
4*=P~+P~=PcYPp.
(4.5)
we have
ax aI: -=apI ap'
ax pmax -=-ah 4 a4'
Hence equations (3.43) take the following form
[(a0 + b* - C#J’)p+ pi] g
+ (a0 + 62 - r#9)qE + #;;+
(@’ + 262 - 2uo-- c,)Z
+[(~-~~)ea(lg+~~pb]p,=O
(4.6)
where eaB is the two-dimensional permutation symbol. Obviously these equations may have a solution Z(p, q, X) if and only if (4.7)
112
E. S. $UHUBi
Thus for physically meaningful results we have only to assume that A;= 0
or
A, = I, = const.,
a, = 0,
pL=O
or
~~=ItZ,=const.
Hence (4.6) now reduces to
[(a0 + b2 - $‘)p + ,u;] g
+ (a. + b, - O.)qg
and the relevant isovector components c,&
(4.8)
become +
~2 II,
@= RI,
w1 =
+ c#I~:+ (@’ + 2b2 - 2a,, - c,)Z = 0
=
b, t2 + 2b,t + b3
’
(a0 + b2 + b,t)u, + A,(X)t + p,(X)
o, = (a0 + b2 + b,t)u,
+ e,ga,up + 1,t + m,,
Ly= 2, 3.
(4.9)
(iii) Homogeneous and isotropic materials It is easy to see that (4.8) and (4.9) reduce in this case to
[(a0 + b2 - c,)p + ml] g + (a0 + b2 - c,)q g + 2(b2 - a& 3P +l
cj = c,x + c2,
TJJ=
=0
(4.10)
blt2 + 2b,t + b3
o1 = (a. + b2 + blt)u, + f,Xt + m,X + h,t + n, (Y= 2, 3.
w,=(ao+b2+bIt)u,+e,paIug+f,t+m,,
Note that in isotropic cases (ii) and (iii) w, include terms e,P+ which cannot be controlled by the structure of the material since the parameter a, does not appear in differential equations satisfied by Z.
5.
SIMILARITY
SOLUTIONS
If we choose a particular isovector field the associated similarity solution is determined the additional constraint [l] #(X)
2 +l)(t) 2 - w,(X,
t, u) = 0.
by
(5.1)
Since wk is linear in II, equation (5.1) is a set of linear first order partial differential equations which specify the functional form of the vector u for a similarity solution. The solution to (5.1) is easily obtainable. Characteristic equations are dX -=----_
dt
@(X) VW
d&c
(5.2)
%(Xiu)
whence we first get the similarity variable (5.3) where Y(t) = exp
I
dt
q(t) .
(5.4)
Similarity solutions for plane waves
713
Then integration of the linear ordinary differential equations along a characteristic E = const. as follows du, 1 - eJk[X@,5), & ul dt - w(t) yields the functional form of the similarity solution as r+ = A,(X, 0&(E) +&(X,
0
(5.5)
where Akl and Ak are known functions of X and t. In~~ucing (5.5) into the field equations (2.6) together with (2.4) we end up in three nonlinear ordinary differential equations in the similarity variable f to determine the functions u,. * In order to reach explicit results we have of course have to consider a particular isovector field. Such an example is reported in [3j for homogeneous materials. We shall consider here a material whose strain energy function is given by (3.47). We take the associated &vector field as MO
cp=
+
~2
T+‘I=
2
R’
2b,t + b3,
mk
=
auk
+
pk(x)
0
where we asssumed that 1, = 0 for the sake of simplicity. Let us now define a set of new constants by
2bz
-_=(y
C2
-=
,
Cl
B,
(5.6)
$=y
Cl
2
so that
?@ = 2b,(t
f Y),
wk = auk + pk
, clPk Itk
=
R;(R,
+
/-J)‘Q”“-1
-
(R.
$“I
Z(p,X)
dx9
=
RE, z(a). cl(Ro + @)@‘=l)+l
(5.7)
Hence (5.2) becomes % (&-dXdx-= Ro+B
dt
duk
l+y
auk+pk
whence we deduce that the similarity variable is given by $=
(Ro+BY
W3)
t+Y
and uk SatiSfieS
on a characteristic dtsl, a R& 1 R; -=-dX clRo+/3Uk+~Ro+~pk
whose solution may be obtained straightforwardly as (5.9) Noting that X$ CVR;, --if, %f-R,+p
s= al
--
f t+Y
714
E. S. $UHUBi
we can easily verify that
x
I
R;‘(R,, + j-j)-‘“““-
’ pk dX +
$RX& + fi)-h
and in view of (5.7), .n, =
[email protected]~+c’i~k
+E
We see then immediately and we find
/R~(R,+/3)-(.ki)-'pk &x + (R~~+~~-("rl)~k - ~(R"+~)-("'cl)~~ dx. that the last three terms cancel each other in the above expression
Similarly we obtain auk
Vk=-=-
at
On the other hand
and the field equations
become R;(j?&
$
fi)-++Wrl
_
= (Ro+ B)-
!!$!!g
R,
k
o (t + Y)*
(S2rrk’+ 2&J;).
(5.11)
Since due to (3.45) 2a + b
----+I=4L2* Cl
Cl
we see that (5.11) take the form = E’( ‘!j2c + 2&J;) or we obtain the following ordinary differential
26,5(~)’ +(a0 k
equations
36, -I- c,) g - = k
c&~(&Y;:+ XJ;)
(5.12)
with 2 = %(%,
3t2r
f3)t
d&I)
=
2b2WX)
+
6%
+
b2)&(8-
(5.13)
Of course, the system (5.12) is in general highly nonlinear and is amenable only to a numerical treatment. The free parameters involved are imposed by the structure of the strain energy function. It is quite clear that this particular example and the ensuing similarity solution we have studied so far may not be identified immediately as the one corresponding to an easily realizable physical situation. However, we believe that it is still of interest since the resulting ordinary nonlinear differential equations can be treated very efficiently by numerical methods.
Similarity solutions for plane waves
715
In this capacity it may serve as a benchmark test for a larger numerical scheme devised to solve the nonlinear partial differential equations corresponding to a realistic case.
REFERENCES [l] E. S. !$JHUBi, Znt. J. Engng Sci. 29, 133 (1991). [2] A. C. ER1NGE.N and E. S. SUHUBI, Elastodynamics, Vol. I-Finite Motions. Academic Press, New York (1974). [3] E. S. SUHUBI, Similarity solutions for one-dimensional nonlinear elastodynamics. In Proceedings of the Vth International Meeting Waves and Stability in Continuous Media (Edited by S. RIONERO), Series on Advances in Mathematics for Applied Sciences, Vol. 4, pp. 390-401. World Scientific, Singapore (1991). (Received 23 August 1991; accepted 9 October 1991)
ES 30:6-C