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=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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