4.4 Variational Problems of Functionals with Functions of Several Variables
291
F +
n
i=1
(ϕ x i − p i )F p i = 0
(4.4.15)
where, p i = u x i .
If the boundary curve of the functional (4.4.13) is on the known surface expressed
by the implicit function, namely
Φ(x 1 , x 2 , . . . , x n , z) = 0
(4.4.16)
then the transversality condition is
F −
n
i=1
Φ x i
Φ z
+ p i
F p i = 0
(4.4.17)
Of course, Corollary 4.4.1 also includes the case of n = 1. Now giving an example
to illustrate its application.
Example 4.4.1 The membrane contact problem. Let there be a membrane, its
perimeter is fixed on a plane curve Γ 1 , the tension for the membrane is N, under
the action of lateral uniform load q it causes vertical deformation w, a part of the
membrane touches to a rigid flat plate of distance d apart from the bottom, it becomes
the planar film close to the flat plate, the perimeter of the membrane contacting the
flat plate domain is Γ 2 , see Fig. 4.5. Find the equilibrium equation of the membrane
and the condition on the perimeter Γ 2 . When Γ 1 is a circle which radius is equal to
R (namely the circular membrane), calculating the shape of the membrane.
Solution Let D be the domain between the membrane perimeter Γ 1 and Γ 2 , D 0 is
the contact domain surrounded by Γ 2 . The internal energy of the membrane before
the deformation is
U 0 =
¨
D+D 0
N dxdy
(1)
D
D 0
Γ 1
Γ 2
d
q
w(x, y)
Fig. 4.5 The membrane deformation under the uniform load q and contacting surface on the flat
plate