CONTINUUM MECHANICS for ENGINEERS

′ ∂
∂ =
Y
f 1
y
(8.4-15)
where the prime following f represents differentiation with respect to x.
Defining q(x) = [f′(x)] –1 , Eq 8.4-15 may be integrated to obtain
Y = q( x) y+ g( x)
(8.4-16)
where function g(x) is an arbitrary function of integration.
Use of given functions X and Y, the incompressibility condition is satisfied
and the equilibrium equations may be written as follows:
[ ( ) ]
∂Π
= G f ′ f ′′+ ( yq′ + g′ ) yq′′ + g′′
∂ x
[ ( ) ]
∂Π
= G q yq′′ + g′′
∂ y
(8.4-17a)
(8.4-17b)
where, again, primes after a symbol represent differentiation with respect to
x. The second equation, 8.4-17b, is easily integrated to obtain an expression
for Π
Π
G
1 2
= qq′′ y + qg′′ y + M( x)
2
(8.4-18)
where M(x) is a function of integration. Differentiation of Eq 8.4-18 with
respect to x results in an equation that must be consistent with Eq 8.4-17a.
This comparison gives rise to the following three ordinary differential equations:
1
2
d
qq q q
dx ′′ ( )= ′′′
d
qq g q q g
dx ′′ ( )= ′′′+ ′ ′′
M′( x)= f′ f′′+ g′ g′′
(8.4-19a)
(8.4-19b)
(8.4-19c)
Eq 8.4-19a may be expanded and integrated to obtain q′′ = k q from which
the function q(x) is determined to within constants A and B:
2