Section 3.2
Solid Mechanics Part II
Kelly
46
3.2
The Stress Function Method
An effective way of dealing with many two dimensional problems is to introduce a new
“unknown”, the
Airy stress function
φ
, an idea brought to us by George Airy in 1862.
The stresses are written in terms of this new function and a new differential equation is
obtained, one which can be solved more easily than Navier’s equations.
3.2.1
The Airy Stress Function
The stress components are written in the form
y
x
x
y
xy
yy
xx
∂
∂
∂
−
=
∂
∂
=
∂
∂
=
σ
2
2
2
2
2
(3.2.1)
Note that, unlike stress and displacement, the Airy stress function has no obvious physical
meaning.
The reason for writing the stresses in the form 3.2.1 is that,
provided the body forces are
zero
, the equilibrium equations are automatically satisfied, which can be seen by
substituting Eqns. 3.2.1 into Eqns. 2.2.3 {
▲
Problem 1}.
On this point, the body forces,
for example gravitational forces, are generally very small compared to the effect of
typical surface forces in elastic materials and may be safely ignored (see Problem 2 of
§2.1).
When body forces are significant, Eqns. 3.2.1 can be amended and a solution
obtained using the Airy stress function, but this approach will not be followed here.
A
number of examples including non-zero body forces are examined later on, using a
different solution method.
3.2.2
The Biharmonic Equation
The Compatability Condition and Stress-Strain Law
In the previous section, it was shown how one needs to solve the equilibrium equations,
the stress-strain constitutive law, and the strain-displacement relations, resulting in the
differential equation for displacements, Eqn. 3.1.4. An alternative approach is to ignore
the displacements and attempt to solve for the
stresses and strains only
.
In other words,
the strain-displacement equations 3.1.2 are ignored.
However, if one is solving for the
strains but not the displacements, one must ensure that the compatibility equation 1.3.1 is
satisfied.
Eqns. 3.2.1 already ensures that the equilibrium equations are satisfied, so combine now
the two dimensional compatibility relation and the stress-strain relations 3.1.1 to get
{
▲
Problem 2}