Stress functions

In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces on the boundary are (using index notation) the equilibrium equation:

$\sigma _{ij,j}=0\,$

where $\sigma$ is the stress tensor, and the Beltrami-Michell compatibility equations:

$\sigma _{ij,kk}+{\frac {1}{1+\nu }}\sigma _{kk,ij}=0$

A general solution of these equations may be expressed in terms the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations.

Beltrami stress functions

In can be shown that a complete solution to the equlibrium equations may be written as

$\sigma _{ij}=\varepsilon _{ijm}\varepsilon _{ijn}\Phi _{mn}$

Engineering notation
$\sigma _{x}={\frac {\partial ^{2}\Phi _{yy}}{\partial z\partial z}}+{\frac {\partial ^{2}\Phi _{zz}}{\partial y\partial y}}-2{\frac {\partial ^{2}\Phi _{yz}}{\partial y\partial z}}$		$\sigma _{xy}=-{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial z}}-{\frac {\partial ^{2}\Phi _{zz}}{\partial x\partial y}}+{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial y}}+{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial z}}$
$\sigma _{y}={\frac {\partial ^{2}\Phi _{xx}}{\partial z\partial z}}+{\frac {\partial ^{2}\Phi _{zz}}{\partial x\partial x}}-2{\frac {\partial ^{2}\Phi _{zx}}{\partial z\partial x}}$		$\sigma _{yz}=-{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial x}}-{\frac {\partial ^{2}\Phi _{xx}}{\partial y\partial z}}+{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial x}}+{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial x}}$
$\sigma _{z}={\frac {\partial ^{2}\Phi _{yy}}{\partial x\partial x}}+{\frac {\partial ^{2}\Phi _{xx}}{\partial y\partial y}}-2{\frac {\partial ^{2}\Phi _{xy}}{\partial x\partial y}}$		$\sigma _{zx}=-{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial y}}-{\frac {\partial ^{2}\Phi _{yy}}{\partial z\partial x}}+{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial y}}+{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial y}}$

where $\Phi _{mn}$ is an arbitrary second-rank tensor field that is continuously differentiable at least four times, and is known as the Beltrami stress tensor. . Its components are known as Beltrami stress functions. $\varepsilon$ is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations.

Maxwell stress functions

Main article: Maxwell stress function

The Maxwell stress functions are defined by assuming that the the Beltrami stress tensor $\Phi _{mn}$ 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:

$\sigma _{x}={\frac {\partial ^{2}B}{\partial z^{2}}}+{\frac {\partial ^{2}C}{\partial y^{2}}}$		$\sigma _{xy}=-{\frac {\partial ^{2}C}{\partial x\partial y}}$
$\sigma _{y}={\frac {\partial ^{2}A}{\partial z^{2}}}+{\frac {\partial ^{2}C}{\partial x^{2}}}$		$\sigma _{yz}=-{\frac {\partial ^{2}A}{\partial y\partial z}}$
$\sigma _{z}={\frac {\partial ^{2}B}{\partial x^{2}}}+{\frac {\partial ^{2}A}{\partial y^{2}}}$		$\sigma _{zx}=-{\frac {\partial ^{2}B}{\partial z\partial x}}$

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:

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

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

Airy stress function

The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.

Morera stress functions

Main article: Morera stress function

The Morera stress functions are defined by assuming that the Beltrami stress tensor $\Phi _{mn}$ tensor is restricted to be of the form

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

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::

$\sigma _{x}=-2{\frac {\partial ^{2}fyz}{\partial y\partial z}}$		$\sigma _{xy}=-{\frac {\partial ^{2}A}{\partial z^{2}}}+{\frac {\partial ^{2}C}{\partial x\partial z}}+{\frac {\partial ^{2}B}{\partial y\partial z}}$
$\sigma _{x}=-2{\frac {\partial ^{2}fzx}{\partial z\partial x}}$		$\sigma _{xy}=-{\frac {\partial ^{2}C}{\partial x^{2}}}+{\frac {\partial ^{2}B}{\partial y\partial x}}+{\frac {\partial ^{2}A}{\partial z\partial x}}$
$\sigma _{x}=-2{\frac {\partial ^{2}fxy}{\partial x\partial y}}$		$\sigma _{xy}=-{\frac {\partial ^{2}B}{\partial y^{2}}}+{\frac {\partial ^{2}A}{\partial z\partial y}}+{\frac {\partial ^{2}C}{\partial x\partial y}}$

Prandtl stress function

The Prandtl stress function is a special case of the Morera stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.

References

• Sadd, Martin H. (2005). Elasticity - Theory, applications and numerics. New York: Elsevier Butterworth-Heinemann. ISBN 0-12-605811-3.
• Knops, R. J. (1958). "On the Variation of Poisson's Ratio in the Solution of Elastic Problems". The Quarterly Journal of Mechanics and Applied Mathematics. 11 (3). Oxford University Press: 326–350. doi:10.1093/qjmam/11.3.326. {{cite journal}}: Cite has empty unknown parameter: |1= (help)