Stress Functions - Airy Stress Function

The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function is usually represented by and the stresses are expressed as

$\sigma_x = \frac{\partial^2\varphi}{\partial y^2} ~;~~ \sigma_y = \frac{\partial^2\varphi}{\partial x^2} ~;~~ \sigma_{xy} = -\frac{\partial^2\varphi}{\partial x \partial y}$

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