CONTINUUM MECHANICS for ENGINEERS

q( x)= Ae + Be
kx −kx
Eqs 8.4-19a and 8.4-19b may be combined along with the fact that
to obtain the function g(x)
g( x)= Ce + De
kx −kx
(8.4-20)
q′′ = k q
2
(8.4-21)
where C and D are constants. With functions q(x) and g(x) found it is now
possible to write functions X(x,y) and Y(x,y) as
X =
∫
dx
Ae + Be
kx −kx
( ) + +
Y = Ae + Be y Ce De
kx −kx kx −kx
(8.4-22a)
(8.4-22b)
In pursuit of the compressed rectangular rubber solution, constants C and
D are taken to be identically zero and constants A and B are set equal. Thus,
[ ]
−1 −1
kx
X = ( kA) tan ( e )+ C′
Y = 2Aycosh kx
(8.4-23a)
(8.4-23b)
where k, A, and C′ are constants that must be determined. The deformation
considered here has the rubber rhomboid deforming symmetrically about
the origin of both the referential and deformed coordinate system
(Figure 8.5a and b). Thus, set X = 0 when x = 0 to obtain constant C′, of
Eq 8.4-23a, to have the value 0.25π. Since the platens are fixed to the speci-
1
1
men, Y = y on planes x=± l the constant A is found to be cosh kl .
Finally, the constant k may be evaluated in terms of the deformed and
undeformed lengths l and L by
kL kl e kl ⎡ −1
π ⎤
= 2cosh(
) tan ( )−
⎣⎢ 4 ⎦⎥
(8.4-24)
To obtain an exact solution it must be shown that the barreled surface
initially at Y =±
H are stress free in the deformed configuration. Unfortunately,
this is not possible for this problem. Instead, a relaxed boundary
condition may be satisfied by enforcing the resultant force on the boundary
to be zero rather than have zero traction at every point.
2
[ ( ) ] −