Page 292 - Introduction to Continuum Mechanics

276 Plane Strain

correspond to the strain components. To ensure the compatibility of the strain components,
we obtain the strain components in terms of <p from Hooke's law Eqs. (5.4.8) [and using
Eq. (5.163)]
•** ^

and substitute them into the compatibility equations, Eqs. (3.16.7) to (3.16.12). For plane
strain problems, the only compatibility equation that is not automatically satisfied is

Thus, we obtain the following equation governing the scalar function <p:

i.e.,

Any function <p which satisfies Eq. (5.16.8) generates a possible elastic solution. In par-
ticular, any third degree polynomial (generating a linear stress and strain field) may be utilized.
The stress function <p defined by Eqs. (5.16.5) and satisfying Eq. (5.16.8), is called the Airy
Stress Function.
We can also obtain from the Hooke's law [Eq. (5.16.6)], the compatibility equation
[Eq. (5.16.7)] and the equations of equilibrium [Eqs. (5.16.4)] the following : [See Prob. 5.77]

which may also be written as