- Email: [email protected]
On group analysis of the von Kármán equations
,~o,,,;,,cc,r Anolwir, Theq. Printed in Great Bntain.
0362-546X/82/080545 SO3XMYO 0 1982 Pergammon Press L[d
Melhodr & Aj~phcorion. Vol. 6. No. 8, pp. 845-853. 1982.
ON GROUP ANALYSIS OF THE VON KARMAN EQUATIONS KAREN A. AMES Iowa State University, Ames. Iowa 50011. U.S.A.
and W. F. AMES* School of Mathematics.
Georgia Institute of Technology. Atlanta, Georgia 30332. (Received for publicarion
16March 1982)
Key words and phrases: Group analysis, von Kiirmtin equations,
solutions,
U.S.A
nonlinear, exact solutions, invariant
Lie algebra. 0. INTRODUCTION
MOTIVATEDby a growing interest in group methods for differential equations ((1, 6, 71). we shall examine the properties of the von KgrmGn equations of nonlinear elasticity under three transformation groups. Our goal is to obtain explicit invariant solutions to these equations. In constructing such solutions, we also obtain the generators of the Lie algebra associated with this set of equations. Specificially, we shall be interested in determining invariant solutions to the system of partial differential equations a2w a2w -2’
I
ax ay-
(-4) 03)
by employing one parameter continuous transformation groups referred to in the literature as the dilatation, rotation and spiral groups. The idea of using such groups to generate solutions to differential equations was that of Morgan (see [l]) who built upon the classical theory of Lie. To apply this method, we choose a one parameter group of transformations and thereby prescribe a general form for the invariants. We then require the equations to be invariant under this group and, as a result, obtain a set of simultaneous algebraic equations whose solution (possibly in terms of an arbitrary parameter) leads to the specific form of the invariants. Substitution of these invariants into (A) and (B) results in a representation of the system. from which solutions to the original set of equations can be constructed. We shall make these ideas more precise in sections 2-5 where we consider the von KBrman equations under the action of three groups of transformations. In section 1 we give a brief discussion of the equations we intend to study and in section 6 we summarize our results by indicating the generators of the Lie algebra associated with these equations. We remark here that our aim in this paper is not to prove existence or uniqueness theorems * Research supported by U.S. Army Grant DAAG-29-XI-K-W32 845
846
AMES
K. A. AMES and W. F.
for the von Karman equations (e.g. see 141) but rather to exhibit group properties solutions of these equations. 1. THE
and exact
EQUATIONS
The investigation of large deflections of plates rests on the solution of two coupled nonlinear partial differential equations known as the von Karman equations [5]. Let us consider a rectangular elastic plate under the combined action of a uniform lateral load and a tensile force in the middle plane of the plate. Denote by w the deflection of the plate away from its equilibrium position in a region D of the xy plane and by F the Airy stress function. Assuming that there are no body forces in the plane of the plate and that the lateral load is perpendicular to the plate, then w and F satisfy the equations 2
ah -2-. ax
a% ay2
1'
(1.1)
Here E is the modulus of elasticity, D is the flexural rigidity, h* is the thickness of the plate, q is the lateral load intensity and A* is the biharmonic operator, i.e.
Determination of the stress function Fallows us to calculate the stresses in the middle surface of the plate by means of the relations a2F
OX=-’ ay2 ’
a2F qy=-. ax2 f
t1.v= --
d2F axay .
(1.3)
From the function w, which defines the deflection surface of the plate, we can obtain the bending and shearing stresses. In our group analysis of equations (1.1) and (1.2), we shall be interested in the two cases q = 0 and q = -p d’wld. The first of these cases represents the situation in which there is no lateral loading while the second describes the vibration of a plate whose deflections are large in comparison with its thickness. Rather than treat (1.1) and (1.2) directly, we introduce the new variables 2, j, f, E and V+which are defined by x = m.f,
y =m)i,
F=bF’,
t =pr,
The choices b =j$
and
c’=---
as well as m=D’”
and
D
(h*E)
p2=p
w =c*.
On group analysis of the von Kimin
847
equations
in the case q = -p d’wld?, lead to the dimensionless system (1.4) (15a) or A$,
a2w
= - -
a2Fa2w
a2Fa2w
ay2 ax2
ax2 ay2
+ __+--_*J!C!k
at2
axay axay
where the bars have been dropped. We shall investigate (1.4)-(1.5a) or (1.4)-(1.5b) group analytic techniques with the aim of obtaining exact solutions of these systems. 2. DILATATION
(1.5b) using
GROUP
In this section we investigate the effect of the dilatation group f = a”x,
9 = aSy,
*=a%
F = aYF,
(a > 0)
(2.1)
on the dimensionless von Karman equations (1.4)-( 1.5a). The invariants of this transformation group are (PI)
q = xy-“‘B, F = f( V)Yy’B, Upon transforming
w = h( rj)yb’@
(P f 0).
(2.2)
equation (1.4), we find that it is invariant if 4a-
y=2a+
2/I- y= 4p-
y= 24x+ 2/3- 26.
(2.3)
The second equation (1 Sa) yields the invariance condition 4~~-6=2~~+2~-6=4@--6=ti+2Q--y--6.
(2.4)
We thus conclude from (2.3) and (2.4) that the von Karman equations are invariant under the dilatation group if CY= /3 and y = 26 = 0. Then (2.2) become r] = xly;
F=f(r),
M’= h(q).
Substitution of these relations into equations (1.4)-( 1.5a) leads to the following two coupled ordinary differential equations for f and h:
(2.5) (2.6) Several exact solutions of the system (2.5)-(2.6) can be obtained. If we choose the deflection to be constant (i.e. h(q) = constant), then equation (2.6) is satisfied identically while (2.5) becomes
-$[(l
+ rjz + #]
= 0.
(2.7)
K. A. AMES and W. F. AMES
848
Integration
of this equation yields the invariant solution
f(v)
=$$o[c
cl sin f#Q In V/(2* - 22 cos & + 1)
+ (~2cos &) arctan (’ iiriV”‘)]
+ %arctan(y)
+ cd,
(2.8)
where z = vu, Qy = -2~r/3 + vn and the q(i = 1,. . . ,4) are arbitrary constants. Another exact solution can be determined by letting h(q) = Q(q) for a constant c # 0. Combining equations (2.5) and (2.6) we find that f(q) must satisfy
Thus,
and h(q) = sfr$
n-(Cz’2)+ cD
where B and D are constants. We note here that under the dilatation group the stresses acting in the middle plane of the plate are related by the equationy2uY + x’u, = Zryr,,. In practice, one is usually more interested in these quantities than in the deflection surface. Remark 2.1. Since the equations for the Airy stress function F and the deflection w are invariant under coordinate translations f = x + b and j = y -+ d, it follows that the von Karman equations are also invariant under the group given by fi = Z/y, f(rj) and h(q). 3. ROTATION
GROUP
We now examine the properties of the von Karman equations under the rotation which, for any real a > 0, is defined by the transformation f=xcosa
-ysina;
y=xsina
+ycosa;
i+=w;
group
E=F.
It is known [l] that the invariants of this group are rj=x2+y*;
F=f(v);
Upon substituting (3.1) into equations (1.4)-(1.5a), differential equations
w = h(q).
(3.1)
we find that f and h must satisfy the two
(3.2)
(3.3)
On group
analysis
of the von Kgrmb
equations
849
Both of these equations can be integrated once to obtain (3.4) (3.5) for constants A and B. Two particular choices of h lead to exact solutions of this system. With equation (3.5) is satisfied identically if we take B = 0 while (3.4) reduces to a linear equation which has solutions of the form (singular at the origin),
h = constant,
f(11)=~1+c2ln11+c3TI+cjrllnr,
(17 =# 0)
where the ci are constants. A second approach is to set h(q) = cf( 17)for a constant c. Equations (3.4)-(3.5) then lead to (3.6) where the constant D depends upon A, B, and c. For rl # 0, (3.6) integrates to give
f = -‘-2Drj”’ + K, with K a constant of integration. It is interesting to observe that if we set dhldq = dfldq in (3.4) or dfldq = -dhldy we obtain in both cases an equation of the form
in (3.9,
(3.7) where g = dfldq and A = t or g = dhldq and il = 1. For c = 0, (3.7) is an equation of the Emden-Fowler type about which there is considerable literature (e.g. see [3]). Because analytic solutions of this class of equations are known only in a few cases [2], the appearance of such an equation in the analysis is indicative of the difficulties one finds in attempting to find exact solutions of the von Karman equations. We note here that for c = 0, equation (3.7) can be transformed into (3.8) by setting y(r,$ = #g(q). One established fact about equation (3.8) is that all of its solutions are oscillatory on [0, CQ),by which we mean that for any nl > 0, there exists n2 > vi such that y(q2) = 0. This type of result gives us information about a subset of those solutions of the von Karman equations which are invariant under the rotation transformation group. 4. SPIRAL
GROUP
As we shall show, the von K&man equations remain invariant under a third group of transformations, the spiral group, which has the form X = X + yia,
Y=y
+Y&
E= ea”F,
w = e&w,
(4.1)
for real a > 0 and yl, y2, LY,p constants. The invariants of this group are rl=Y2x-YlY,
f(v) = n hF-
w,
h(v) = ~2Inw - PY.
(4.2)
850
K. A. AWS
and W. F. AMES
Substitution of (4.1) into equations (1.4)-(1.5a) leads to the two invariance conditions 20 and p = (Y+ /3 which together imply that cr = fl= 0. Thus, (4.2) become f(q)
11= yzx - YIY,
= y2 In F.
h(n) = y1 In u’
and the von K&-man equations are reduced to a fourth order ordinary differential which both f and h must satisfy, namely a,@“) + 4bg’g”’ + 6c(g’)‘g” + 3b(g”)’ + d(g’)” = 0,
cr = (4.3)
equation (4.4)
where
We observe that under the action of the spiral group, the original set of partial differential equations is not only uncoupled but also transformed in such a way that the invariants f(n) and h(q) satisfy the same differential equation. Consider equation (4.4) with yt = ye = 1 and ~(77) = g’(q). We then have u”’ + 4uu” + 6u’u’ + 3(u)’ + u’ = 0. This equation has the nontrivial solution u = n -I. Thus, g(q) = In n + constant. Since both f and h must satisfy equation (4.4), it follows that f(n) = h(q) = In n is one exact solution. In fact, the functions c In n for c = 1,2,3 are all solutions of equation (4.4) in this case. Hence, F = (x - y)’ = w (c = 1,2,3) constitute a set of exact solutions to the von Karman equations. Of physical interest are the stress components in the middle plane of the plate. Under the action of the spiral group, it turns out that for yI = y? = 1. the normal stress components a,, uY and the shearing stress rXYare all equal. More specifically, uX= oy = txy = effX_Y)[frr+ (f’)Z].
5. ANALYSIS
OF THE
TIME-DEPENDENT
EQUATIONS
In this section, we consider the system of equations
(5.1) a?F a’w a2F a’w --7--7+7-7-2--
A’w = Application
ay ax
ax ay
of the dilatation group, .f = a*x;
jj = a$;
f=ast;
B = a%,,
F= a’F;
for any real a > 0. to (5.1) leads to the invariance conditions
4&-- y= 2a-c zg-
y= 4/3- y= 2LI+ y3-
2tj;
4a-6=2cu+2~-6=4/3-6=2~-6=L?,8+2~~-y-6. from which we deduce my= fi = s/2 and y = 6 = 0. In view of these conditions, invariants 77= yt-“‘“; c = X{-“E; IV= h( 5;. q)r”‘“; F = f( 5, n)P;
the group
On group
analysis
851
of the von KckmAn equations
reduce to c = xl%+; This.transformation and h:
7j = ylVt;
F=f(L
VI;
results in the following partial differential
w = h(L 7).
(5.2)
equations for the functions f
(5.3)
(5.4) If, at this point, we attempt to reduce equations (5.3) and (5.4) by re-applying the dilatation group, we discover that such a reduction is not possible. However, we can find solutions of this system by considering the rotation group 5=C cosa@= fsina As indicated previously,
qsina; + qcosa.
the invariants of this transformation z = f + $;
f = u(z);
group are
h = u(z).
(5.5)
Substitution of these relations into (5.3) and (5.4) lead to two coupled ordinary differential equations for u and v, namely
$[z’$] =-g[*(g)*]; _cJzS!$
=;$[z!L@c]
-&[2z$+z*3
(5.6) (5.7)
One set of solutions to this system is U(Z) = c and u(z) = alz In z + a2 In z + u3z + a, where ,4) are constants. It thus follows that the time dependent von Karman candtheai(i=l,... equations possess the exact solution
If we choose U(Z) = v(z) in equations (5.6) and (5.7), then we obtain the following equation for v(z)
from which a first integral can be obtained.
It follows that
K. A. AMES and W. F. AMES
852
where Y is a constant of integration. For z f 0 we can solve for duidz in (5.8) and obtain U(Z) (and thus u(z)) after a quadrature. An alternative way of dealing with equations (5.3) and (5.4) is to look for similarity solutions of the form f = g(z) and h = p(z) where z = < - n. Under such a transformation, equations (5.3) and (5.4) become
d4g
0.
-=
2
dr’
d4p z+GZ
1
2
d'p
3
Q
dz’+igzz=o.
The first of these equations can be readily integrated to give g(z) = c,.z3 + cZzZ+ c3z + cJ for constants ci(i = 1, . . . ,4). Information about the stress distribution in the middle surface of the plate can then be obtained since
implies that
6. LIE
ALGEBRA
Our results in sections 2, 3 and 4 lead to a characterization of a Lie algebra (not the full one) associated with the von Karman equations. We found that this system is invariant under the dilatation, rotation and spiral transformation group whose invariants are respectively 1. q=xiy, w = h(n); F =f(rl)l 2. rj = XI + y2, w = h(rl); F=f(r), h(n) = y2 In w. f(q) = ~1ln F, 3. 17= Y2X - YlY, Thus, a set of elementary
generators
of the Lie algebra for equations (1.4)-(1.5a) X,,vd-xL. ax
x2=-$ ,
Consequently,
the (infinitesimal)
generator
X = (al + a2x +
ay
are (6.1)
can be written as
a3y) i
+ (a4
+
a5y
-
w> +j-
(6.2)
where the ai(i = 1, . . . ,6) are constants. We remark here that the three groups we have considered are not necessarily the only ones under which the von K&-man equations remain invariant (an obvious addition is the group of translations). The choice of these particular transformation groups is guided by physical considerations. This is especially true in the time dependent case (section 5) for it is easy to verify that Newton’s laws of motion are invariant [8] under the Galilei-Newton group (space and time translations, rigid rotations and moving axes). Further these transformations leave unchanged the definitions of such material constants as viscosity, density etc. In addition Newtonian mechanics is invariant under the dilation group. Consequently, the von KArm&n equations are consistent with the framework of theoretical Newtonian mechanics.
On group analysis of the von Karrnan equations
853
It should also be noted at this point that our analysis yields invariant solutions of the problem consisting of the differential equations alone. It may not be possible to match these exact invariant solutions to general auxiliary conditions such as prescribed boundary values. However these solutions are interesting since they may be asymptotic limits of other solutions. As a final remark, we mention that the generators (6.1) and consequently (6.2) can be obtained by the infinitesimal transformation group theory of Lie which is treated in detail by Ovsiannikov [7]. This type of analysis is extremely lengthy even for equations that are less complex than those studied in this paper. A table of generators for 15 systems is given in Ovsiannikov. REFERENCES 1.
AMES W. F., Nonlinear
Partial Differential Equations in Engineering,
Volsl and II, Academic Press, New York
(1965 and 1972). 2. KAMKE E., Differentialgleichungen (Ltisungsmethoden und Ltisungen), Vol. I, Akad. Verlagsges. Leipzig (1956). 3. WONG J. S. W., On the generalized Emden-Fowler equation, SIAM Rev. 17, 339 (1975). 4. CIARLET P. G. & RABIER P., Les Equations de von Kbrmhn, Lecture Notes in Mathematics 826, Springer, Berlin (1980). 5. TIMOSHENKOS. P. & WOINOWSKY-KRIEGER S., Theory of Plates and Shells, McGraw-Hill, New York (1959). 6. BLUMAN G. W. & COLE J. D., Similarity Methods for Differential Equations, Springer, Berlin (1974). 7. OVSIANNIKOV L. V., Group Analysis of Differential Equations (Russian Edition, Nauka (1978)); (English Translation edited by W. F. Ames), Academic Press (1982). 8. BIRKHOFFG., Hydrodynamics, Princeton University Press, Princeton, N.J. (1960).
0362-546X/82/080545 SO3XMYO 0 1982 Pergammon Press L[d
Melhodr & Aj~phcorion. Vol. 6. No. 8, pp. 845-853. 1982.
ON GROUP ANALYSIS OF THE VON KARMAN EQUATIONS KAREN A. AMES Iowa State University, Ames. Iowa 50011. U.S.A.
and W. F. AMES* School of Mathematics.
Georgia Institute of Technology. Atlanta, Georgia 30332. (Received for publicarion
16March 1982)
Key words and phrases: Group analysis, von Kiirmtin equations,
solutions,
U.S.A
nonlinear, exact solutions, invariant
Lie algebra. 0. INTRODUCTION
MOTIVATEDby a growing interest in group methods for differential equations ((1, 6, 71). we shall examine the properties of the von KgrmGn equations of nonlinear elasticity under three transformation groups. Our goal is to obtain explicit invariant solutions to these equations. In constructing such solutions, we also obtain the generators of the Lie algebra associated with this set of equations. Specificially, we shall be interested in determining invariant solutions to the system of partial differential equations a2w a2w -2’
I
ax ay-
(-4) 03)
by employing one parameter continuous transformation groups referred to in the literature as the dilatation, rotation and spiral groups. The idea of using such groups to generate solutions to differential equations was that of Morgan (see [l]) who built upon the classical theory of Lie. To apply this method, we choose a one parameter group of transformations and thereby prescribe a general form for the invariants. We then require the equations to be invariant under this group and, as a result, obtain a set of simultaneous algebraic equations whose solution (possibly in terms of an arbitrary parameter) leads to the specific form of the invariants. Substitution of these invariants into (A) and (B) results in a representation of the system. from which solutions to the original set of equations can be constructed. We shall make these ideas more precise in sections 2-5 where we consider the von KBrman equations under the action of three groups of transformations. In section 1 we give a brief discussion of the equations we intend to study and in section 6 we summarize our results by indicating the generators of the Lie algebra associated with these equations. We remark here that our aim in this paper is not to prove existence or uniqueness theorems * Research supported by U.S. Army Grant DAAG-29-XI-K-W32 845
846
AMES
K. A. AMES and W. F.
for the von Karman equations (e.g. see 141) but rather to exhibit group properties solutions of these equations. 1. THE
and exact
EQUATIONS
The investigation of large deflections of plates rests on the solution of two coupled nonlinear partial differential equations known as the von Karman equations [5]. Let us consider a rectangular elastic plate under the combined action of a uniform lateral load and a tensile force in the middle plane of the plate. Denote by w the deflection of the plate away from its equilibrium position in a region D of the xy plane and by F the Airy stress function. Assuming that there are no body forces in the plane of the plate and that the lateral load is perpendicular to the plate, then w and F satisfy the equations 2
ah -2-. ax
a% ay2
1'
(1.1)
Here E is the modulus of elasticity, D is the flexural rigidity, h* is the thickness of the plate, q is the lateral load intensity and A* is the biharmonic operator, i.e.
Determination of the stress function Fallows us to calculate the stresses in the middle surface of the plate by means of the relations a2F
OX=-’ ay2 ’
a2F qy=-. ax2 f
t1.v= --
d2F axay .
(1.3)
From the function w, which defines the deflection surface of the plate, we can obtain the bending and shearing stresses. In our group analysis of equations (1.1) and (1.2), we shall be interested in the two cases q = 0 and q = -p d’wld. The first of these cases represents the situation in which there is no lateral loading while the second describes the vibration of a plate whose deflections are large in comparison with its thickness. Rather than treat (1.1) and (1.2) directly, we introduce the new variables 2, j, f, E and V+which are defined by x = m.f,
y =m)i,
F=bF’,
t =pr,
The choices b =j$
and
c’=---
as well as m=D’”
and
D
(h*E)
p2=p
w =c*.
On group analysis of the von Kimin
847
equations
in the case q = -p d’wld?, lead to the dimensionless system (1.4) (15a) or A$,
a2w
= - -
a2Fa2w
a2Fa2w
ay2 ax2
ax2 ay2
+ __+--_*J!C!k
at2
axay axay
where the bars have been dropped. We shall investigate (1.4)-(1.5a) or (1.4)-(1.5b) group analytic techniques with the aim of obtaining exact solutions of these systems. 2. DILATATION
(1.5b) using
GROUP
In this section we investigate the effect of the dilatation group f = a”x,
9 = aSy,
*=a%
F = aYF,
(a > 0)
(2.1)
on the dimensionless von Karman equations (1.4)-( 1.5a). The invariants of this transformation group are (PI)
q = xy-“‘B, F = f( V)Yy’B, Upon transforming
w = h( rj)yb’@
(P f 0).
(2.2)
equation (1.4), we find that it is invariant if 4a-
y=2a+
2/I- y= 4p-
y= 24x+ 2/3- 26.
(2.3)
The second equation (1 Sa) yields the invariance condition 4~~-6=2~~+2~-6=4@--6=ti+2Q--y--6.
(2.4)
We thus conclude from (2.3) and (2.4) that the von Karman equations are invariant under the dilatation group if CY= /3 and y = 26 = 0. Then (2.2) become r] = xly;
F=f(r),
M’= h(q).
Substitution of these relations into equations (1.4)-( 1.5a) leads to the following two coupled ordinary differential equations for f and h:
(2.5) (2.6) Several exact solutions of the system (2.5)-(2.6) can be obtained. If we choose the deflection to be constant (i.e. h(q) = constant), then equation (2.6) is satisfied identically while (2.5) becomes
-$[(l
+ rjz + #]
= 0.
(2.7)
K. A. AMES and W. F. AMES
848
Integration
of this equation yields the invariant solution
f(v)
=$$o[c
cl sin f#Q In V/(2* - 22 cos & + 1)
+ (~2cos &) arctan (’ iiriV”‘)]
+ %arctan(y)
+ cd,
(2.8)
where z = vu, Qy = -2~r/3 + vn and the q(i = 1,. . . ,4) are arbitrary constants. Another exact solution can be determined by letting h(q) = Q(q) for a constant c # 0. Combining equations (2.5) and (2.6) we find that f(q) must satisfy
Thus,
and h(q) = sfr$
n-(Cz’2)+ cD
where B and D are constants. We note here that under the dilatation group the stresses acting in the middle plane of the plate are related by the equationy2uY + x’u, = Zryr,,. In practice, one is usually more interested in these quantities than in the deflection surface. Remark 2.1. Since the equations for the Airy stress function F and the deflection w are invariant under coordinate translations f = x + b and j = y -+ d, it follows that the von Karman equations are also invariant under the group given by fi = Z/y, f(rj) and h(q). 3. ROTATION
GROUP
We now examine the properties of the von Karman equations under the rotation which, for any real a > 0, is defined by the transformation f=xcosa
-ysina;
y=xsina
+ycosa;
i+=w;
group
E=F.
It is known [l] that the invariants of this group are rj=x2+y*;
F=f(v);
Upon substituting (3.1) into equations (1.4)-(1.5a), differential equations
w = h(q).
(3.1)
we find that f and h must satisfy the two
(3.2)
(3.3)
On group
analysis
of the von Kgrmb
equations
849
Both of these equations can be integrated once to obtain (3.4) (3.5) for constants A and B. Two particular choices of h lead to exact solutions of this system. With equation (3.5) is satisfied identically if we take B = 0 while (3.4) reduces to a linear equation which has solutions of the form (singular at the origin),
h = constant,
f(11)=~1+c2ln11+c3TI+cjrllnr,
(17 =# 0)
where the ci are constants. A second approach is to set h(q) = cf( 17)for a constant c. Equations (3.4)-(3.5) then lead to (3.6) where the constant D depends upon A, B, and c. For rl # 0, (3.6) integrates to give
f = -‘-2Drj”’ + K, with K a constant of integration. It is interesting to observe that if we set dhldq = dfldq in (3.4) or dfldq = -dhldy we obtain in both cases an equation of the form
in (3.9,
(3.7) where g = dfldq and A = t or g = dhldq and il = 1. For c = 0, (3.7) is an equation of the Emden-Fowler type about which there is considerable literature (e.g. see [3]). Because analytic solutions of this class of equations are known only in a few cases [2], the appearance of such an equation in the analysis is indicative of the difficulties one finds in attempting to find exact solutions of the von Karman equations. We note here that for c = 0, equation (3.7) can be transformed into (3.8) by setting y(r,$ = #g(q). One established fact about equation (3.8) is that all of its solutions are oscillatory on [0, CQ),by which we mean that for any nl > 0, there exists n2 > vi such that y(q2) = 0. This type of result gives us information about a subset of those solutions of the von Karman equations which are invariant under the rotation transformation group. 4. SPIRAL
GROUP
As we shall show, the von K&man equations remain invariant under a third group of transformations, the spiral group, which has the form X = X + yia,
Y=y
+Y&
E= ea”F,
w = e&w,
(4.1)
for real a > 0 and yl, y2, LY,p constants. The invariants of this group are rl=Y2x-YlY,
f(v) = n hF-
w,
h(v) = ~2Inw - PY.
(4.2)
850
K. A. AWS
and W. F. AMES
Substitution of (4.1) into equations (1.4)-(1.5a) leads to the two invariance conditions 20 and p = (Y+ /3 which together imply that cr = fl= 0. Thus, (4.2) become f(q)
11= yzx - YIY,
= y2 In F.
h(n) = y1 In u’
and the von K&-man equations are reduced to a fourth order ordinary differential which both f and h must satisfy, namely a,@“) + 4bg’g”’ + 6c(g’)‘g” + 3b(g”)’ + d(g’)” = 0,
cr = (4.3)
equation (4.4)
where
We observe that under the action of the spiral group, the original set of partial differential equations is not only uncoupled but also transformed in such a way that the invariants f(n) and h(q) satisfy the same differential equation. Consider equation (4.4) with yt = ye = 1 and ~(77) = g’(q). We then have u”’ + 4uu” + 6u’u’ + 3(u)’ + u’ = 0. This equation has the nontrivial solution u = n -I. Thus, g(q) = In n + constant. Since both f and h must satisfy equation (4.4), it follows that f(n) = h(q) = In n is one exact solution. In fact, the functions c In n for c = 1,2,3 are all solutions of equation (4.4) in this case. Hence, F = (x - y)’ = w (c = 1,2,3) constitute a set of exact solutions to the von Karman equations. Of physical interest are the stress components in the middle plane of the plate. Under the action of the spiral group, it turns out that for yI = y? = 1. the normal stress components a,, uY and the shearing stress rXYare all equal. More specifically, uX= oy = txy = effX_Y)[frr+ (f’)Z].
5. ANALYSIS
OF THE
TIME-DEPENDENT
EQUATIONS
In this section, we consider the system of equations
(5.1) a?F a’w a2F a’w --7--7+7-7-2--
A’w = Application
ay ax
ax ay
of the dilatation group, .f = a*x;
jj = a$;
f=ast;
B = a%,,
F= a’F;
for any real a > 0. to (5.1) leads to the invariance conditions
4&-- y= 2a-c zg-
y= 4/3- y= 2LI+ y3-
2tj;
4a-6=2cu+2~-6=4/3-6=2~-6=L?,8+2~~-y-6. from which we deduce my= fi = s/2 and y = 6 = 0. In view of these conditions, invariants 77= yt-“‘“; c = X{-“E; IV= h( 5;. q)r”‘“; F = f( 5, n)P;
the group
On group
analysis
851
of the von KckmAn equations
reduce to c = xl%+; This.transformation and h:
7j = ylVt;
F=f(L
VI;
results in the following partial differential
w = h(L 7).
(5.2)
equations for the functions f
(5.3)
(5.4) If, at this point, we attempt to reduce equations (5.3) and (5.4) by re-applying the dilatation group, we discover that such a reduction is not possible. However, we can find solutions of this system by considering the rotation group 5=C cosa@= fsina As indicated previously,
qsina; + qcosa.
the invariants of this transformation z = f + $;
f = u(z);
group are
h = u(z).
(5.5)
Substitution of these relations into (5.3) and (5.4) lead to two coupled ordinary differential equations for u and v, namely
$[z’$] =-g[*(g)*]; _cJzS!$
=;$[z!L@c]
-&[2z$+z*3
(5.6) (5.7)
One set of solutions to this system is U(Z) = c and u(z) = alz In z + a2 In z + u3z + a, where ,4) are constants. It thus follows that the time dependent von Karman candtheai(i=l,... equations possess the exact solution
If we choose U(Z) = v(z) in equations (5.6) and (5.7), then we obtain the following equation for v(z)
from which a first integral can be obtained.
It follows that
K. A. AMES and W. F. AMES
852
where Y is a constant of integration. For z f 0 we can solve for duidz in (5.8) and obtain U(Z) (and thus u(z)) after a quadrature. An alternative way of dealing with equations (5.3) and (5.4) is to look for similarity solutions of the form f = g(z) and h = p(z) where z = < - n. Under such a transformation, equations (5.3) and (5.4) become
d4g
0.
-=
2
dr’
d4p z+GZ
1
2
d'p
3
Q
dz’+igzz=o.
The first of these equations can be readily integrated to give g(z) = c,.z3 + cZzZ+ c3z + cJ for constants ci(i = 1, . . . ,4). Information about the stress distribution in the middle surface of the plate can then be obtained since
implies that
6. LIE
ALGEBRA
Our results in sections 2, 3 and 4 lead to a characterization of a Lie algebra (not the full one) associated with the von Karman equations. We found that this system is invariant under the dilatation, rotation and spiral transformation group whose invariants are respectively 1. q=xiy, w = h(n); F =f(rl)l 2. rj = XI + y2, w = h(rl); F=f(r), h(n) = y2 In w. f(q) = ~1ln F, 3. 17= Y2X - YlY, Thus, a set of elementary
generators
of the Lie algebra for equations (1.4)-(1.5a) X,,vd-xL. ax
x2=-$ ,
Consequently,
the (infinitesimal)
generator
X = (al + a2x +
ay
are (6.1)
can be written as
a3y) i
+ (a4
+
a5y
-
w> +j-
(6.2)
where the ai(i = 1, . . . ,6) are constants. We remark here that the three groups we have considered are not necessarily the only ones under which the von K&-man equations remain invariant (an obvious addition is the group of translations). The choice of these particular transformation groups is guided by physical considerations. This is especially true in the time dependent case (section 5) for it is easy to verify that Newton’s laws of motion are invariant [8] under the Galilei-Newton group (space and time translations, rigid rotations and moving axes). Further these transformations leave unchanged the definitions of such material constants as viscosity, density etc. In addition Newtonian mechanics is invariant under the dilation group. Consequently, the von KArm&n equations are consistent with the framework of theoretical Newtonian mechanics.
On group analysis of the von Karrnan equations
853
It should also be noted at this point that our analysis yields invariant solutions of the problem consisting of the differential equations alone. It may not be possible to match these exact invariant solutions to general auxiliary conditions such as prescribed boundary values. However these solutions are interesting since they may be asymptotic limits of other solutions. As a final remark, we mention that the generators (6.1) and consequently (6.2) can be obtained by the infinitesimal transformation group theory of Lie which is treated in detail by Ovsiannikov [7]. This type of analysis is extremely lengthy even for equations that are less complex than those studied in this paper. A table of generators for 15 systems is given in Ovsiannikov. REFERENCES 1.
AMES W. F., Nonlinear
Partial Differential Equations in Engineering,
Volsl and II, Academic Press, New York
(1965 and 1972). 2. KAMKE E., Differentialgleichungen (Ltisungsmethoden und Ltisungen), Vol. I, Akad. Verlagsges. Leipzig (1956). 3. WONG J. S. W., On the generalized Emden-Fowler equation, SIAM Rev. 17, 339 (1975). 4. CIARLET P. G. & RABIER P., Les Equations de von Kbrmhn, Lecture Notes in Mathematics 826, Springer, Berlin (1980). 5. TIMOSHENKOS. P. & WOINOWSKY-KRIEGER S., Theory of Plates and Shells, McGraw-Hill, New York (1959). 6. BLUMAN G. W. & COLE J. D., Similarity Methods for Differential Equations, Springer, Berlin (1974). 7. OVSIANNIKOV L. V., Group Analysis of Differential Equations (Russian Edition, Nauka (1978)); (English Translation edited by W. F. Ames), Academic Press (1982). 8. BIRKHOFFG., Hydrodynamics, Princeton University Press, Princeton, N.J. (1960).