Stress Functions - Maxwell Stress Functions

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

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

The stress tensor which automatically obeys the equilibrium equation may now be written as:

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

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 for stress. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:

$\nabla^4 A+\nabla^4 B+\nabla^4 C=3\left( \frac{\partial^2 A}{\partial x^2}+ \frac{\partial^2 B}{\partial y^2}+ \frac{\partial^2 C}{\partial z^2}\right)/(2-\nu),$

These must also yield a stress tensor which obeys the specified boundary conditions.

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