John Peter Whitney ES240 Term Project Fall 2007
Green's Function of the Biharmonic Operator
is not Positive Definite
John “Peter” Whitney
ES 240 Fall 2007
January 11, 2008
John Peter Whitney ES240 Term Project Fall 2007
Green’s Function
For the following inhomogeneous linear differential equation,
Lu ( x) = f ( x),
The Green’s function is defined as
LG ( x, y ) = δ ( x − y ).
If the Green’s function is known, the solution can be formed by the integral
u ( x) = ∫ G ( x, y ) f ( y )dy.
For electrostatics, the Green’s function of the Laplacian represents the
electric field due to a unit point charge. For the biharmonic equation,
we can interpret it as the deflection field of a thin plate due to a point load.
John Peter Whitney ES240 Term Project Fall 2007
The Green’s function of the Laplacian is positive definite. Consider
∇ 2u = f .
This equation models the deflection of a thin elastic membrane. Since the
Laplacian is positive definite, a point load results in all deflected points
moving in the direction of the applied force. Hadamard (1908) proposed
that this might also be true of the biharmonic equation (which models a
thin rigid plate instead of an elastic membrane).
But this is not true!
Hadamard’s conjecture was disproven by counter example in 1949 by
Duffin (semi-infinite strip), and in 1951 by Garabedian, for a finite, but
sufficiently eccentric ellipse.
We will show this behavior by constructing the analytical solution for a
point-loaded elliptical plate. We will also make mention of a curious
feature of the eigenfunctions of the analogous eigenvalue problem, which
is related to the non-positive definiteness of the biharmonic operator.
John Peter Whitney ES240 Term Project Fall 2007
Elliptical Coordinates
x = c cosh u cosν
y = c sinh u sinν
(c,0)
(c,0)
boundary ellipse equation:
1
u1 = sinh −1  
c
picture from http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html
John Peter Whitney ES240 Term Project Fall 2007
Clamped Elliptical Plate, Point Load
vertical point
force, P, at (0,0)
y
Clamped boundary:
b
a
x
w=0
∂w
=0
∂u
scaled deflection:
2
a
c =   −1
b
wD
w=
P
John Peter Whitney ES240 Term Project Fall 2007
Thin Plate Equations
p
∇ ∇ w=
K
Eh 2
K=
12(1 −ν 2 )
2
2
particular solution (Cartesian)
particular solution (elliptical)
w = w p + wh
P
wp =
(
x 2 + y 2 ) log( x 2 + y 2 )
16πK
∞
Pc 2 

wp =
α 0 ( u ) + ∑ α n ( u ) cos(nν )

16πK 
n =1

(The equations for the alphas are quite complex, and are omitted here. See
reference given on next slide for details.)
John Peter Whitney ES240 Term Project Fall 2007
Homogeneous solution is composed of a sum of hyperbolic
cosines in u, and cosines in nu.
“Missing” odd terms are required if point load is not at origin
(not considered here).
∞
wh = K 0 + E0 cosh ( 2u ) + ∑ [ An cosh ( nu ) + En −2 cosh ( ( n − 2) u ) + En cosh ( ( n + 2) u ) ] cos(nν )
n=2
Constants are found by adding homogeneous solution to
particular solution, and ensuring the boundary conditions
are satisfied (details omitted).
Solution based on Saito (1959) “Bending of Elliptic Plates under a Concentrated
Load” JSME, Vol. 2, No. 6 (Warning, this paper contains errors in the
coefficients for the homogeneous solution).
John Peter Whitney ES240 Term Project Fall 2007
Verification in the Limiting Case: Circular Plate
clamped circular plate
0.02
0.018
series solution (50 terms)
exact solution
0.016
0.014
wD/P
0.012
0.01
0.008
0.006
0.004
0.002
0
0
Pa 2
w(r ) =
16πK
0.1
0.2
0.3
0.4
0.5
radius
 r2
r2
 r 
1 − 2 + 2 2 log 
a
 a 
 a
0.6
0.7
0.8
0.9
1
w(0) D
1
=
≈ 0.0199
P
16π
John Peter Whitney ES240 Term Project Fall 2007
5:1 Ellipse, Central Point Load
(deflection normalized by maximum deflection)
John Peter Whitney ES240 Term Project Fall 2007
Y=0 Cross-section
5:1 ellipse, 2000 points, 400 series terms
0.03
0.025
max wK/P = 0.0287
0.02
wK/P
0.015
0.01
0.005
0
-0.005
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
John Peter Whitney ES240 Term Project Fall 2007
Y=0 Cross-section
5:1 ellipse, 2000 points, 400 series terms
0.03
0.025
max wK/P = 0.0287
0.02
wK/P
0.015
0.01
0.005
0
-0.005
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
John Peter Whitney ES240 Term Project Fall 2007
Plate Pushes Back!
x 10
5:1 ellipse, 2000 points, 400 series terms
-4
3.5
3
2.5
2
wK/P
1.5
min wK/P = -4.65x10-5
1
“back deflection” is 0.162%
of max “forward deflection”
0.5
0
-0.5
-1
-1.5
-4
-3.5
-3
x
-2.5
-2
John Peter Whitney ES240 Term Project Fall 2007
Related Phenomenon: Eigenmode Inversion
∇ 2∇ 2φ = λφ
N = 2 mode
N = 1 mode!
plots from Sweers (2001)
John Peter Whitney ES240 Term Project Fall 2007
Questions?