CONTINUUM MECHANICS for ENGINEERS

σ yy
⎡⎛
∂X
⎞ ⎛ ∂Y
⎞ ⎛ ∂X
⎞ ⎛ ∂Y
⎞ ⎤
= Π − G⎢⎜
⎝ ∂y
⎟ + ⎜
⎠ ⎝ ∂y
⎟ − ⎜
⎠ ⎝ ∂x
⎟ − ⎜
⎠ ⎝ ∂x
⎟ ⎥
⎣
⎢
⎠
⎦
⎥
1
2 2 2 2
2
σ xy G X ∂
=−
⎡ ∂X
∂Y
∂Y
⎤
⎢ + ⎥
⎣ ∂x
∂ y ∂x
∂x
⎦
(8.4-12b)
(8.4-12c)
where the results of Eq 8.4-11 have also been utilized. The nontrivial equilibrium
equations are associated with the x and y directions. Using the stress
components as defined in Eq 8.4-12 the equilibrium conditions are
∂
∂ =
∂
+
∂
∂
Π ⎡ X 2 Y 2 ⎤
G ⎢ X Y⎥
x ⎣ x ∂x
⎦
∂
∂ =
∂
+
∂
∂
Π ⎡ X 2 Y 2 ⎤
G ⎢ X Y⎥
y ⎣ y ∂y
⎦
The incompressibility condition may also be written as
∂ ∂ ∂ ∂
−
∂ ∂ ∂ ∂ =
X Y X Y
1
x y y x
(8.4-13a)
(8.4-13b)
(8.4-14)
Eqs 8.4-13 and 8.4-14 are three nonlinear partial differential equations that
must be satisfied to ensure equilibrium and the constraint of incompressibility.
To actually solve a problem, a solution must be found that satisfies the
appropriate boundary conditions. Often the way this is done is by assuming
a specific form for X(x,y) and Y(x,y) and demonstrating that the aforementioned
equations are satisfied. Following that, the boundary conditions are
defined and demonstrated, completing the solution. In a sense, this is similar
to the semi-inverse method discussed in Chapter Six. The assumed “guess”
of the functional form of X(x,y) and Y(x,y) requires some experience and or
familiarity with the problem at hand. Here, a function form will be assumed
in pursuit of the solution of a rectangular rubber specimen being compressed
in the x direction. The faces of specimen are perfectly attached to rigid
platens. In this case, assume that X = f(x). After a modest amount of work,
this will be shown to lead to a solution of a neo-Hookean material compressed
between rigid platens.
Substituting the assumed form of X into the incompressibility equation,
Eq 8.4-14, yields the ordinary differential equation