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:

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

Read more about this topic: Stress Functions

Famous quotes containing the words stress and/or functions:

“Anyone who wishes to combine domestic responsibilities and paid employment with the least stress and most enjoyment might start by pondering this paradox: the first step to better functioning is to stop blaming herself for not functioning well enough.”
—Faye J. Crosby (20th century)

“If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
—Charles Baudelaire (1821–1867)