Stress functions

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

\sigma _{ij,i}=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 of 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

It can be shown ^[1] that a complete solution to the equilibrium equations may be written as

\sigma =\nabla \times \Phi \times \nabla

Using index notation:

\sigma _{ij}=\varepsilon _{ikm}\varepsilon _{jln}\Phi _{kl,mn}

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

where $\Phi _{mn}$ is a symmetric but otherwise arbitrary second-rank tensor field that is at least twice differentiable, and is known as the Beltrami stress tensor.^[1] 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. And $\nabla$ is the Nabla operator. For the Beltrami stress tensor to satisfy the Beltrami-Michell compatibility equations in addition to the equilibrium equations, it is further required that $\Phi _{mn}$ is at least four times continuously differentiable.

Maxwell stress functions

The Maxwell stress functions are defined by assuming that the Beltrami stress tensor $\Phi _{mn}$ is restricted to be of the form.^[2]

\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:^[2]

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

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:^[3]

\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.

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.^[2] This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function $C$ is usually represented by $\varphi$ and the stresses are expressed as

\sigma _{x}={\frac {\partial ^{2}\varphi }{\partial y^{2}}}~;~~\sigma _{y}={\frac {\partial ^{2}\varphi }{\partial x^{2}}}~;~~\sigma _{xy}=-{\frac {\partial ^{2}\varphi }{\partial x\partial y}}-(f_{x}y+f_{y}x)

Where $f_{x}$ and $f_{y}$ are values of body forces in relevant direction.

In polar coordinates the expressions are:

\sigma _{rr}={\frac {1}{r}}{\frac {\partial \varphi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\varphi }{\partial \theta ^{2}}}~;~~\sigma _{\theta \theta }={\frac {\partial ^{2}\varphi }{\partial r^{2}}}~;~~\sigma _{r\theta }=\sigma _{\theta r}=-{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \varphi }{\partial \theta }}\right)

Morera stress functions

The Morera stress functions are defined by assuming that the Beltrami stress tensor $\Phi _{mn}$ tensor is restricted to be of the form ^[2]

\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:^[4]

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

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.^[4]

Notes

^ ^a ^b Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Elsevier Science & Technology Books. p. 363. ISBN 978-0-12-605811-6.
^ ^a ^b ^c ^d Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364
^ Knops (1958) p327
^ ^a ^b Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365

References

Sadd, Martin H. (2005). Elasticity - Theory, applications and numerics. New York: Elsevier Butterworth-Heinemann. ISBN 0-12-605811-3. OCLC 162576656.
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.

[Sadd05_363-1] Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Elsevier Science & Technology Books. p. 363. ISBN 978-0-12-605811-6.

[Sadd05_364-2] Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364

[3] Knops (1958) p327

[Sadd05_365-4] Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365

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