Stress Functions - Morera Stress Functions

The Morera stress functions are defined by assuming that the Beltrami stress tensor tensor is restricted to be of the form

$\Phi_{ij}= \begin{bmatrix} 0&C&B\\ C&0&A\\ B&A&0 \end{bmatrix}$

The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami-Michell compatibility equations. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:

$\sigma_x = -2\frac{\partial^2 A}{\partial y \partial z}$		$\sigma_{yz} =-\frac{\partial^2 A}{\partial x^2} +\frac{\partial^2 B}{\partial y \partial x} +\frac{\partial^2 C}{\partial z \partial x}$
$\sigma_y = -2\frac{\partial^2 B}{\partial z \partial x}$		$\sigma_{zx} =-\frac{\partial^2 B}{\partial y^2} +\frac{\partial^2 C}{\partial z \partial y} +\frac{\partial^2 A}{\partial x \partial y}$
$\sigma_z = -2\frac{\partial^2 C}{\partial x \partial y}$		$\sigma_{xy} =-\frac{\partial^2 C}{\partial z^2} +\frac{\partial^2 A}{\partial x \partial z} +\frac{\partial^2 B}{\partial y \partial z}$

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