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.


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

= frac{partial^2B}{partial z^2}
+ frac{partial^2C}{partial y^2} sigma_{yz}
=-frac{partial^2A}{partial y partial z}
= frac{partial^2C}{partial x^2}
+ frac{partial^2A}{partial z^2} sigma_{zx}
= -frac{partial^2B}{partial z partial x}
= 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=3left(
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.

Read more about this topic:  Stress Functions

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