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Analysis of the von Karman equations by group methods
Inr. J. Non-Lmear Mwhonics. Printed I” Great Bntarn.
Vol.
20. No.
ANALYSIS
4. pp.
201-209.
1985
0020-7462/85 13.00 + 0.00 Pcrgamon Press Ltd
OF THE VON KARMAN EQUATIONS GROUP METHODS
BY
K. A. AMES-~and W. F. AMEs$§ TDepartment of Mathematics, Iowa State University, Ames, Ia 50011, U.S.A.; and &School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. (Received 10 August 1984: receiwdfor
publication 7 March 1985)
Abstract-One of the systems ofequations approximating the large deflection of plates consists of two coupled non-linear fourth order partial differential equations, known as the von Karman equations. The full symmetry group for the steady equations is a finitely generated Lie group with ten parameters. For the time-dependent system the full symmetry group is an infinite parameter Lie group. Several subgroups of the full group are used to generate exact solutions of the time-independent and the timedependent systems.Theseinclude thedilatation group (similar solutions), rotation group, screw group and others. Physical implications and applications are discussed.
1. INTRODUCTION
Perhaps the most widely applicable method for determining analytic solutions of partial differential equations utilizes the underlying (Lie) group structure. The mathematical foundations for the determination of the full group for a system of differential equations can be found in Ames [l] and Bluman and Cole [2], and the general theory is found in Ovsiannikov [3]. The determination of the full group requires extremely lengthy calculations. Detailed calculations can be found in Ames [ 1 ] and Ovsiannikov [3 ] and for the Navier Stokes equations in Boisvert [4] (see also [5]). Here we give the results of those calculations for the von Karman equations (see equation 1.1) of non-linear elasticity in the form of the infinitesimal generators of the full group. These have also been obtained in an independent study by Schwartz [6] using an algebraic program package which uses REDUCE [7]. In Russia such an algebraic programming system, for this purpose, is available under the name CINO (see [3], p. 57). MACZYMA [8] has also been used for this purpose on the other problems. Our goal is to obtain explicit invariant solutions to the system of partial differential equations, due to von Karman, A’F = E[w& - w,,wy,,]
(1.1)
A2w= ; + ;
[Fyywx,
+ Fxxwyy- 2Fxywcyl
by employing various subgroups of the full group admitted by these equations. To apply this procedure we choose a one (or more) parameter subgroup and calculate the general form of the subgroup invariants. We then require the equations to be invariant under this group. As a result a set of simultaneous algebraic equations arises and their solution, possibly involving arbitrary parameters, leads to a more specific form of the invariants. Substitution of these invariants into (1.1) results in a representation of the system from which solutions to the original system can be constructed. A preliminary effort in this direction appeared in (91. 2. THE 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 [lo]. Let us consider a rectangular elastic plate under the combined action of a uniform lateral load and a
§Research supported by U.S. Army Grant DAAG-29-81-K-0042. NI.a
20:4-A
K. A. AMES and W. F. AMES
202
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 R of the ~1’ 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-l)
1.
(2.2)
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 A2 is the biharmonic operator, i.e.,
a4 a4 ax2ay2 + ay4’
A2zg+2---
Determination of the stress function F allows us to calculate the stresses in the middle surface of the plate by means of the relations a2F. 6X=2’
cy =
i-3
d2F
a2F
(2.3)
Tf,y = -axay.
2;
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 (2.1) and (2.2), we shall be interested in the two cases q = 0 and 4 = - pa2w/ar2. 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 (2.1) and (2.2) directly, we introduce the new variables, Z, J, 7, F and 0 which are defined by Y =
x = mX,
my,
t = pi,
F = bF,
w = CR.
The choices b=;
and
D
c2 =(h*E)
as well as 0”’
m =
and
in the case q = -ppd’w/at’, lead to the dimensionless system (2.4)
A2w
_
a2FazwI a2F azw a$ a2 ax2 ay2
2 --a9
@W
axayaxay
(2Sa)
or Azw=
_$+$2+?2$-2-p
d2F d2w
axay axay
(2.5b)
Analysis of the von Karman equations by group methods
203
where the bars have been dropped. We shall investigate (2.4)-(2.5a) or (2.4)-(2.5b) using group analytical techniques with the aim of obtaining exact solutions of these systems. 3. FULL
The full (Lie) group of the time-dependent 14 infinitesimal operators
GROUP
von Karman equations (2.4)-(2.5b) is givenby
x1 = ;;
X6=-&;
x11 = fY-$
x2=;;
x,=g;
Xl2
x3=$;
X8 =
x-g;x13
X9 = Y$
X,=Zt&+x&+y~; aY
X 10
=mx& =hwYg
(3.1)
X14 = Mr)-&. a
=
tx--;
aw
Each of the first 11 generators is associated with a parameter independent of all the others. These generate a finite dimensional Lie algebra L1 i. The last three operators contain arbitrary functions of time, h(t), i = 1,2,3. 4. DILATATION
GROUP
In this section we investigate the action of the dilatation group (from X,)
z=
dx,
j = a’y,
(a > 0)
(4.1)
on the dimensionless time independent von Karman equations (2.4)-(2.5a). The invariants of this transformation group are ([l]) ? = XIY.
(4.2)
The von Karman equations are invariant under the dilatation group and
F = f(rl1;
w = h(q).
(4.3)
Substitution of these relations into equations (2.4)-(2.5a) leads to the following two coupled ordinary differential equations for f and h :
(4.4)
(4.5) Several exact solutions of the system (4.4)-(4.5) can be obtained. If we choose the deflection to be constant [i.e. h(v) = constant], then equation (4.5) is satisfied identically while (4.4) becomes
$
(1 + y12+ ?4)$ [
1
= 0.
(4.6)
204
Integration
K. A. AMES and W. F. AMES
of this equation gives the invariant solution
f(V) = -L i {(cl sin+,)ln(z2 &=O + (c2 cos &)arctan + ?arctan
- 2zcos & + 1)
[(z - cos &)/sin &I)
(4.7)
[(4~ + 2)/3] + c4,
where z = A, & = - 2x 13 + VIIand the ci(i = 1,. .4) are arbitrary constants. Another exact solution can be determined by letting h(q) = cf(q) for a constant c # 0. Combining equations (4.4) and (4.5) we find that f(q) must satisfy
Thus
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 equation y2a, + x20, = 2xytxy. In practice, one is usually more interested in these quantities than the deflection surface. Remark. Since the equations for the Airy stress function F and the deflection w are invariant under coordinate translations Z = x + b and JJ = y + d, it follows that the von Karman equations are also invariant under the group given by )I = X/y, f(f) and h(f). 5. ROTATION
GROUP
We now examine the properties of the von Karman equations under lne rotation group which, for any real a > 0, is defined by the transformation (from X4) ji- = xcosa - ysina;
J = xsina +ycosa;
ii, = w;
F = F.
.It is known [l] that the invariants of this group are q=x2+y2;
F =fhL
w = h(q).
(5.1)
Upon substituting (5.1) into equations (2.4)-(2.5a), we find that f and h must satisfy the two differential equations (5.2)
(5.3)
205
Analysis of the von Karman equations by group methods
Both of these equations can be integrated once to obtain (5.4)
(5.5) for constants A and B. Two particular choices of h lead to exact solutions of this system. With h = constant, equation (5.5) is satisfied identically if we take B = 0 while (5.4) reduces to a linear equation which has solutions of the form (singular at the origin),
f(v) =
cl +
ttl # 0)
c21nv + w + c4vlnv,
where the ci are constants. A second approach Equations (5.4)-(5.5) then lead to
is to set h(q) = cf(q) for a constant
c.
(5.6) where the constant D depends upon A,B, and c. For q # 0, (5.6) integrates to give f=
&2Dr/li2 -t
K,
with K a constant of integration. It is interesting to observe that if we set dh/dq = df /dq in (5.4) or df /dq = - dh/dq in (5.5), we obtain in both cases an equation of the form (5.7) where g = df/dq and A= l/4 or g = dh/dq and i = l/2. For c = 0, (5.7) is an equation of the Emden-Fowler type about which there is considerable literature (e.g. see [ll I). Because analytic solutions of this class of equations are known only in a few cases, 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 (5.7) can be transformed into 2 dy
+
q-2y2
=
0
(5.8)
dr12
by setting y(q) = qIg(q). One established fact about equation (5.8) is that all of its solutions are oscillatory on (0, co), by which we mean that for any q1 > 0 there exists q2 > q1 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. 6. SPIRAL
GROUP
As we shall see, the von Karman equations also remain invariant under a third group of transformations, the spiral group, which has the form (a linear combination of X2, X3, Xs and Xl4 (with f3(t)constant))
z=
x +
y1a,
J =
Y +
Y24
F=e’=‘F,
E=@w,
(6.1)
for real a > 0 and yl, y2, a, /I constants. The invariants of this group are rl =
‘l’zx
-
YlY,
f(v)= ~2lnF- ax
h(v) = 72lnw-BY.
(6.2)
206
K. A. AMES and W. F. AMES
Substitution of (6.1) into equations (2.4)-(2.5a) leads to the two invariance conditions u = 2p and fl = a + /I which together imply that a = /I = 0. Thus, (6.2) becomes v =
YZX -
f(rt)
YlY7
= ~2 In F,
NV)
=
(6.3)
72 In w
and the von Karman equations are reduced to a fourth order ordinary differential equation which both f and h must satisfy, namely
c.+“) + 46g'g"' + 6c(g')2g" + 3b(g”)2 + d(g’)4 =
(6.4)
(I
where 4
r’:
r:
c=y,+Q+2-$ r3
b = y: + 27: + z;
&&I+?!+23 r:
v+
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 invariantsf(q) and h(q) satisfy the same differential equation. Consider equation (6.4) with y1 = y2 = 1 and u(q) = g’(q). We then have u”’ + 4~” + 624%’+ 3(~‘)~ + u4 = 0.
(6.5)
This equation is linearized by raising the order. The transformation y’ = uy
gives rise to the sequence y” = y[u’ + u2] y”’ = y[u” + 3UU’+ u”] y@“)= y [u”’ + 4uu” + 6u*u’ + 3(u’)* + u’]. Consequently,
the truth of (6.5) implies that
whereupon the solution for u is u = y’/y =
3A3q2 A3q3
+
+ A2q2
2A29 + Al +
Alfj
+
A0
(6.6) =
3~* + 2Ay + B q3 + Aq2 + Brj + c
Whereupon,
g(v) = In [q3+ A?’ + Bq + C] + D.
(6.7)
Since both f and h must satisfy (5.4), it follows that f(q) = h(q) = g(q) is an exact solution with four arbitrary constants A, B, C and D. Hence F = w = G[q3 + A?’ + BY/ + C],
G = eD,
constitute a set of exact solutions to the von Karman equations.
‘l=x-y
Analysis
of the von Karman
equations
by group
207
members
Remark. For the time independent
problem other linear combinations of X2, X3, X,, X5 (without a@), X6, X8, X9, Xiz, Xi3, Xi4 (withfi(t) = constants) can be used to generate solutions. Some of these may be the same as can be ascertained by examination of the commutator table and the splitting-nonsplitting analysis. That is not shown here. 7. THE
TIME
DEPENDENT
EQUATIONS
Equations (2.4) and (2.5b) will be studied here using the dilatation group invariants are i =
This transformation and h:
xl Jt;
w = 45, tt).
F =f(i>vl);
rl = y/ Jt;
generator
X5. The
(7.1)
results in the following partial differential equations for the functions f
(7.2)
(7.3) +
a'fa2h + a? a2h a+ ap ap ar12
(
2
ay
azh
I
ajarlagarl’
If, at this point, we attempt to reduce equations (7.2) and (7.3) 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 action of the rotation group [ = ices a - qsin a; tj = @in a + ~~0s a.
As indicated previously, the invariants of this transformation
z=~2+q2;
f = u(z);
h
group are
= u(z).
(7.4)
Substitution of these relations into (7.2) and (7.3) leads to two coupled ordinary differential equations for u and t‘, namely
-$[z$q = -g-[z ($)‘I;
(7.5)
(7.6)
One set of solutions to this system is U(Z)= c and u(z) = aizln z + a21nz + a3z + a4 where c and the ai (i = 1,. . . ,4) are constants. It thus follows that the time dependent von Karman equations possess the exact solution
208
K. A. AMES and W. F. AMES
If we choose u(z) = u(z) in equations (7.5) and (7.6), then we obtain the following equation for u(z)
from which a first integral can be obtained. It follows that 1 ,dv =Gz z+Y,
(7.7)
where y is a constant of integration. For z # 0 we can solve for du/dz in (7.7) and obtain u(z) [and thus u(z)] after a quadrature. An alternative way of dealing with equations (7.2) and (7.3) is to look for solutions of the form f = g(z) and h = p(z) where z = i - II. Under such a transformation, equations (7.2) and (7.3) become linear equations
d4p -+cz dz4
1 zd2p 3 dp ~+~Z~=o.
(7.8)
The first of these equations can be readily integrated to give g(z) = c1z3 + c2z2 + c3z + c4 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
tXY
=
fix
=
cJy
=
1 6c1(x-Y) +
2c
-
r
2. 1
Jt
Equation (7.8) has two classical solutions pi = sin(z2/8) and pi = cos(z2/8) and a third solution of the form pi = j sin (t - Z)T-3’2dz, t = z2/8. (These solutions are due to our colleagues Frank Stallard and Thomas Morley.) 8. OTHER
POSSIBILITIES
The four groups considered (dilatation, rotation, translation and spiral) are not the only ones under which the von Karman equations remain invariant. To illustrate another (among the many) possibility consider X4 + &X1, i.e. the invariants are obtained by integrating
r2-x:+e; 1
r ax
I(x,y,t)=O
ay
(8.1)
where E is a constant. The invariants are r = (x2 + y2)“2
and
I=sin-‘r-&
r
Analysis of the von Karman equations by group members
Consequently,
209
solutions of the form X
sin-’ - r
Et
1
with a corresponding form for w can be sought. It should also be noted that these analyses yield invariant solutions 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 or energy conservation integrals. However, these solutions are interesting since they are often asymptotic limits of other solutions.
REFERENCES 1. W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Vol. II, Chapter 2. Academic Press, New York (1972). 2. G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations. Springer, New York (1974). 3. L. V. Ovsiannikov, Group Analysis of Differential Equations (translation editor W. F. Ames). Academic Press, New York (1982). (Russian Editor, Nauka 1978) 4. R. E. Boisvert, Group analysis of the Navier-Stokes equations, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA 30332 (1982). 5. R. E. Boisvert. W. F. Ames and U. N. Srivastava Groun properties and new solutions of Navier-Stokes equations. J. kngng Math. In press. 6. F. Schwarz, Personal communication. 7. F. Schwarz, A reduce package for determining Lie symmetries of ordinary and partial differential equations. Comp. Phys Commun. 27, 179-186 (1982). 8. P. Roseneau and J. L. Schwarzmeier, Similarity solutions of systems of partial differential equations using MACSYMA, Courant Inst. of Math. Sci. Report No. COO-3077-16O/MF-94 (1979). 9. K. A. Ames and W. F. Ames, On group analysis of the von Karman equations. Int. J. Nonlinear Anul. Theory Meth. Appl. 6, 845 (1982). 10. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. McGraw-Hill, New York (1959). 11. J. S. W. Wong, On the generalized Emden-Fowler equations, SIAM Rev. 17, 339 (1975).
Vol.
20. No.
ANALYSIS
4. pp.
201-209.
1985
0020-7462/85 13.00 + 0.00 Pcrgamon Press Ltd
OF THE VON KARMAN EQUATIONS GROUP METHODS
BY
K. A. AMES-~and W. F. AMEs$§ TDepartment of Mathematics, Iowa State University, Ames, Ia 50011, U.S.A.; and &School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. (Received 10 August 1984: receiwdfor
publication 7 March 1985)
Abstract-One of the systems ofequations approximating the large deflection of plates consists of two coupled non-linear fourth order partial differential equations, known as the von Karman equations. The full symmetry group for the steady equations is a finitely generated Lie group with ten parameters. For the time-dependent system the full symmetry group is an infinite parameter Lie group. Several subgroups of the full group are used to generate exact solutions of the time-independent and the timedependent systems.Theseinclude thedilatation group (similar solutions), rotation group, screw group and others. Physical implications and applications are discussed.
1. INTRODUCTION
Perhaps the most widely applicable method for determining analytic solutions of partial differential equations utilizes the underlying (Lie) group structure. The mathematical foundations for the determination of the full group for a system of differential equations can be found in Ames [l] and Bluman and Cole [2], and the general theory is found in Ovsiannikov [3]. The determination of the full group requires extremely lengthy calculations. Detailed calculations can be found in Ames [ 1 ] and Ovsiannikov [3 ] and for the Navier Stokes equations in Boisvert [4] (see also [5]). Here we give the results of those calculations for the von Karman equations (see equation 1.1) of non-linear elasticity in the form of the infinitesimal generators of the full group. These have also been obtained in an independent study by Schwartz [6] using an algebraic program package which uses REDUCE [7]. In Russia such an algebraic programming system, for this purpose, is available under the name CINO (see [3], p. 57). MACZYMA [8] has also been used for this purpose on the other problems. Our goal is to obtain explicit invariant solutions to the system of partial differential equations, due to von Karman, A’F = E[w& - w,,wy,,]
(1.1)
A2w= ; + ;
[Fyywx,
+ Fxxwyy- 2Fxywcyl
by employing various subgroups of the full group admitted by these equations. To apply this procedure we choose a one (or more) parameter subgroup and calculate the general form of the subgroup invariants. We then require the equations to be invariant under this group. As a result a set of simultaneous algebraic equations arises and their solution, possibly involving arbitrary parameters, leads to a more specific form of the invariants. Substitution of these invariants into (1.1) results in a representation of the system from which solutions to the original system can be constructed. A preliminary effort in this direction appeared in (91. 2. THE 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 [lo]. Let us consider a rectangular elastic plate under the combined action of a uniform lateral load and a
§Research supported by U.S. Army Grant DAAG-29-81-K-0042. NI.a
20:4-A
K. A. AMES and W. F. AMES
202
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 R of the ~1’ 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-l)
1.
(2.2)
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 A2 is the biharmonic operator, i.e.,
a4 a4 ax2ay2 + ay4’
A2zg+2---
Determination of the stress function F allows us to calculate the stresses in the middle surface of the plate by means of the relations a2F. 6X=2’
cy =
i-3
d2F
a2F
(2.3)
Tf,y = -axay.
2;
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 (2.1) and (2.2), we shall be interested in the two cases q = 0 and 4 = - pa2w/ar2. 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 (2.1) and (2.2) directly, we introduce the new variables, Z, J, 7, F and 0 which are defined by Y =
x = mX,
my,
t = pi,
F = bF,
w = CR.
The choices b=;
and
D
c2 =(h*E)
as well as 0”’
m =
and
in the case q = -ppd’w/at’, lead to the dimensionless system (2.4)
A2w
_
a2FazwI a2F azw a$ a2 ax2 ay2
2 --a9
@W
axayaxay
(2Sa)
or Azw=
_$+$2+?2$-2-p
d2F d2w
axay axay
(2.5b)
Analysis of the von Karman equations by group methods
203
where the bars have been dropped. We shall investigate (2.4)-(2.5a) or (2.4)-(2.5b) using group analytical techniques with the aim of obtaining exact solutions of these systems. 3. FULL
The full (Lie) group of the time-dependent 14 infinitesimal operators
GROUP
von Karman equations (2.4)-(2.5b) is givenby
x1 = ;;
X6=-&;
x11 = fY-$
x2=;;
x,=g;
Xl2
x3=$;
X8 =
x-g;x13
X9 = Y$
X,=Zt&+x&+y~; aY
X 10
=mx& =hwYg
(3.1)
X14 = Mr)-&. a
=
tx--;
aw
Each of the first 11 generators is associated with a parameter independent of all the others. These generate a finite dimensional Lie algebra L1 i. The last three operators contain arbitrary functions of time, h(t), i = 1,2,3. 4. DILATATION
GROUP
In this section we investigate the action of the dilatation group (from X,)
z=
dx,
j = a’y,
(a > 0)
(4.1)
on the dimensionless time independent von Karman equations (2.4)-(2.5a). The invariants of this transformation group are ([l]) ? = XIY.
(4.2)
The von Karman equations are invariant under the dilatation group and
F = f(rl1;
w = h(q).
(4.3)
Substitution of these relations into equations (2.4)-(2.5a) leads to the following two coupled ordinary differential equations for f and h :
(4.4)
(4.5) Several exact solutions of the system (4.4)-(4.5) can be obtained. If we choose the deflection to be constant [i.e. h(v) = constant], then equation (4.5) is satisfied identically while (4.4) becomes
$
(1 + y12+ ?4)$ [
1
= 0.
(4.6)
204
Integration
K. A. AMES and W. F. AMES
of this equation gives the invariant solution
f(V) = -L i {(cl sin+,)ln(z2 &=O + (c2 cos &)arctan + ?arctan
- 2zcos & + 1)
[(z - cos &)/sin &I)
(4.7)
[(4~ + 2)/3] + c4,
where z = A, & = - 2x 13 + VIIand the ci(i = 1,. .4) are arbitrary constants. Another exact solution can be determined by letting h(q) = cf(q) for a constant c # 0. Combining equations (4.4) and (4.5) we find that f(q) must satisfy
Thus
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 equation y2a, + x20, = 2xytxy. In practice, one is usually more interested in these quantities than the deflection surface. Remark. Since the equations for the Airy stress function F and the deflection w are invariant under coordinate translations Z = x + b and JJ = y + d, it follows that the von Karman equations are also invariant under the group given by )I = X/y, f(f) and h(f). 5. ROTATION
GROUP
We now examine the properties of the von Karman equations under lne rotation group which, for any real a > 0, is defined by the transformation (from X4) ji- = xcosa - ysina;
J = xsina +ycosa;
ii, = w;
F = F.
.It is known [l] that the invariants of this group are q=x2+y2;
F =fhL
w = h(q).
(5.1)
Upon substituting (5.1) into equations (2.4)-(2.5a), we find that f and h must satisfy the two differential equations (5.2)
(5.3)
205
Analysis of the von Karman equations by group methods
Both of these equations can be integrated once to obtain (5.4)
(5.5) for constants A and B. Two particular choices of h lead to exact solutions of this system. With h = constant, equation (5.5) is satisfied identically if we take B = 0 while (5.4) reduces to a linear equation which has solutions of the form (singular at the origin),
f(v) =
cl +
ttl # 0)
c21nv + w + c4vlnv,
where the ci are constants. A second approach Equations (5.4)-(5.5) then lead to
is to set h(q) = cf(q) for a constant
c.
(5.6) where the constant D depends upon A,B, and c. For q # 0, (5.6) integrates to give f=
&2Dr/li2 -t
K,
with K a constant of integration. It is interesting to observe that if we set dh/dq = df /dq in (5.4) or df /dq = - dh/dq in (5.5), we obtain in both cases an equation of the form (5.7) where g = df/dq and A= l/4 or g = dh/dq and i = l/2. For c = 0, (5.7) is an equation of the Emden-Fowler type about which there is considerable literature (e.g. see [ll I). Because analytic solutions of this class of equations are known only in a few cases, 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 (5.7) can be transformed into 2 dy
+
q-2y2
=
0
(5.8)
dr12
by setting y(q) = qIg(q). One established fact about equation (5.8) is that all of its solutions are oscillatory on (0, co), by which we mean that for any q1 > 0 there exists q2 > q1 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. 6. SPIRAL
GROUP
As we shall see, the von Karman equations also remain invariant under a third group of transformations, the spiral group, which has the form (a linear combination of X2, X3, Xs and Xl4 (with f3(t)constant))
z=
x +
y1a,
J =
Y +
Y24
F=e’=‘F,
E=@w,
(6.1)
for real a > 0 and yl, y2, a, /I constants. The invariants of this group are rl =
‘l’zx
-
YlY,
f(v)= ~2lnF- ax
h(v) = 72lnw-BY.
(6.2)
206
K. A. AMES and W. F. AMES
Substitution of (6.1) into equations (2.4)-(2.5a) leads to the two invariance conditions u = 2p and fl = a + /I which together imply that a = /I = 0. Thus, (6.2) becomes v =
YZX -
f(rt)
YlY7
= ~2 In F,
NV)
=
(6.3)
72 In w
and the von Karman equations are reduced to a fourth order ordinary differential equation which both f and h must satisfy, namely
c.+“) + 46g'g"' + 6c(g')2g" + 3b(g”)2 + d(g’)4 =
(6.4)
(I
where 4
r’:
r:
c=y,+Q+2-$ r3
b = y: + 27: + z;
&&I+?!+23 r:
v+
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 invariantsf(q) and h(q) satisfy the same differential equation. Consider equation (6.4) with y1 = y2 = 1 and u(q) = g’(q). We then have u”’ + 4~” + 624%’+ 3(~‘)~ + u4 = 0.
(6.5)
This equation is linearized by raising the order. The transformation y’ = uy
gives rise to the sequence y” = y[u’ + u2] y”’ = y[u” + 3UU’+ u”] y@“)= y [u”’ + 4uu” + 6u*u’ + 3(u’)* + u’]. Consequently,
the truth of (6.5) implies that
whereupon the solution for u is u = y’/y =
3A3q2 A3q3
+
+ A2q2
2A29 + Al +
Alfj
+
A0
(6.6) =
3~* + 2Ay + B q3 + Aq2 + Brj + c
Whereupon,
g(v) = In [q3+ A?’ + Bq + C] + D.
(6.7)
Since both f and h must satisfy (5.4), it follows that f(q) = h(q) = g(q) is an exact solution with four arbitrary constants A, B, C and D. Hence F = w = G[q3 + A?’ + BY/ + C],
G = eD,
constitute a set of exact solutions to the von Karman equations.
‘l=x-y
Analysis
of the von Karman
equations
by group
207
members
Remark. For the time independent
problem other linear combinations of X2, X3, X,, X5 (without a@), X6, X8, X9, Xiz, Xi3, Xi4 (withfi(t) = constants) can be used to generate solutions. Some of these may be the same as can be ascertained by examination of the commutator table and the splitting-nonsplitting analysis. That is not shown here. 7. THE
TIME
DEPENDENT
EQUATIONS
Equations (2.4) and (2.5b) will be studied here using the dilatation group invariants are i =
This transformation and h:
xl Jt;
w = 45, tt).
F =f(i>vl);
rl = y/ Jt;
generator
X5. The
(7.1)
results in the following partial differential equations for the functions f
(7.2)
(7.3) +
a'fa2h + a? a2h a+ ap ap ar12
(
2
ay
azh
I
ajarlagarl’
If, at this point, we attempt to reduce equations (7.2) and (7.3) 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 action of the rotation group [ = ices a - qsin a; tj = @in a + ~~0s a.
As indicated previously, the invariants of this transformation
z=~2+q2;
f = u(z);
h
group are
= u(z).
(7.4)
Substitution of these relations into (7.2) and (7.3) leads to two coupled ordinary differential equations for u and t‘, namely
-$[z$q = -g-[z ($)‘I;
(7.5)
(7.6)
One set of solutions to this system is U(Z)= c and u(z) = aizln z + a21nz + a3z + a4 where c and the ai (i = 1,. . . ,4) are constants. It thus follows that the time dependent von Karman equations possess the exact solution
208
K. A. AMES and W. F. AMES
If we choose u(z) = u(z) in equations (7.5) and (7.6), then we obtain the following equation for u(z)
from which a first integral can be obtained. It follows that 1 ,dv =Gz z+Y,
(7.7)
where y is a constant of integration. For z # 0 we can solve for du/dz in (7.7) and obtain u(z) [and thus u(z)] after a quadrature. An alternative way of dealing with equations (7.2) and (7.3) is to look for solutions of the form f = g(z) and h = p(z) where z = i - II. Under such a transformation, equations (7.2) and (7.3) become linear equations
d4p -+cz dz4
1 zd2p 3 dp ~+~Z~=o.
(7.8)
The first of these equations can be readily integrated to give g(z) = c1z3 + c2z2 + c3z + c4 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
tXY
=
fix
=
cJy
=
1 6c1(x-Y) +
2c
-
r
2. 1
Jt
Equation (7.8) has two classical solutions pi = sin(z2/8) and pi = cos(z2/8) and a third solution of the form pi = j sin (t - Z)T-3’2dz, t = z2/8. (These solutions are due to our colleagues Frank Stallard and Thomas Morley.) 8. OTHER
POSSIBILITIES
The four groups considered (dilatation, rotation, translation and spiral) are not the only ones under which the von Karman equations remain invariant. To illustrate another (among the many) possibility consider X4 + &X1, i.e. the invariants are obtained by integrating
r2-x:+e; 1
r ax
I(x,y,t)=O
ay
(8.1)
where E is a constant. The invariants are r = (x2 + y2)“2
and
I=sin-‘r-&
r
Analysis of the von Karman equations by group members
Consequently,
209
solutions of the form X
sin-’ - r
Et
1
with a corresponding form for w can be sought. It should also be noted that these analyses yield invariant solutions 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 or energy conservation integrals. However, these solutions are interesting since they are often asymptotic limits of other solutions.
REFERENCES 1. W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Vol. II, Chapter 2. Academic Press, New York (1972). 2. G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations. Springer, New York (1974). 3. L. V. Ovsiannikov, Group Analysis of Differential Equations (translation editor W. F. Ames). Academic Press, New York (1982). (Russian Editor, Nauka 1978) 4. R. E. Boisvert, Group analysis of the Navier-Stokes equations, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA 30332 (1982). 5. R. E. Boisvert. W. F. Ames and U. N. Srivastava Groun properties and new solutions of Navier-Stokes equations. J. kngng Math. In press. 6. F. Schwarz, Personal communication. 7. F. Schwarz, A reduce package for determining Lie symmetries of ordinary and partial differential equations. Comp. Phys Commun. 27, 179-186 (1982). 8. P. Roseneau and J. L. Schwarzmeier, Similarity solutions of systems of partial differential equations using MACSYMA, Courant Inst. of Math. Sci. Report No. COO-3077-16O/MF-94 (1979). 9. K. A. Ames and W. F. Ames, On group analysis of the von Karman equations. Int. J. Nonlinear Anul. Theory Meth. Appl. 6, 845 (1982). 10. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. McGraw-Hill, New York (1959). 11. J. S. W. Wong, On the generalized Emden-Fowler equations, SIAM Rev. 17, 339 (1975).