# Stress functions

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Equations describing elastic deformation

In [linear elasticity](/source/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](/source/Index_notation)) the equilibrium equation:

- σ i j , i = 0 {\displaystyle \sigma _{ij,i}=0\,}

where σ {\displaystyle \sigma } is the [stress tensor](/source/Stress_(physics)), and the Beltrami-Michell compatibility equations:

- σ i j , k k + 1 1 + ν σ k k , i j = 0 {\displaystyle \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

- σ = ∇ × Φ × ∇ {\displaystyle \sigma =\nabla \times \Phi \times \nabla }

**Using index notation:**

- σ i j = ε i k m ε j l n Φ k l , m n {\displaystyle \sigma _{ij}=\varepsilon _{ikm}\varepsilon _{jln}\Phi _{kl,mn}}

- Engineering notation σ x = ∂ 2 Φ y y ∂ z ∂ z + ∂ 2 Φ z z ∂ y ∂ y − 2 ∂ 2 Φ y z ∂ y ∂ z {\displaystyle \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}}} σ x y = − ∂ 2 Φ x y ∂ z ∂ z − ∂ 2 Φ z z ∂ x ∂ y + ∂ 2 Φ y z ∂ x ∂ z + ∂ 2 Φ z x ∂ y ∂ z {\displaystyle \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}}} σ y = ∂ 2 Φ x x ∂ z ∂ z + ∂ 2 Φ z z ∂ x ∂ x − 2 ∂ 2 Φ z x ∂ z ∂ x {\displaystyle \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}}} σ y z = − ∂ 2 Φ y z ∂ x ∂ x − ∂ 2 Φ x x ∂ y ∂ z + ∂ 2 Φ z x ∂ y ∂ x + ∂ 2 Φ x y ∂ z ∂ x {\displaystyle \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}}} σ z = ∂ 2 Φ y y ∂ x ∂ x + ∂ 2 Φ x x ∂ y ∂ y − 2 ∂ 2 Φ x y ∂ x ∂ y {\displaystyle \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}}} σ z x = − ∂ 2 Φ z x ∂ y ∂ y − ∂ 2 Φ y y ∂ z ∂ x + ∂ 2 Φ x y ∂ z ∂ y + ∂ 2 Φ y z ∂ x ∂ y {\displaystyle \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 Φ m n {\displaystyle \Phi _{mn}} is a [symmetric](/source/Symmetric_tensor) but otherwise arbitrary second-rank [tensor field](/source/Tensor_field) that is at least twice [differentiable](/source/Differentiable_function), and is known as the *Beltrami stress tensor*.[1] Its components are known as **Beltrami stress functions**. ε {\displaystyle \varepsilon } is the [Levi-Civita pseudotensor](/source/Levi-Civita_symbol), 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 ∇ {\displaystyle \nabla } is the [Nabla operator](/source/Del). For the Beltrami stress tensor to satisfy the Beltrami-Michell compatibility equations in addition to the equilibrium equations, it is further required that Φ m n {\displaystyle \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 Φ m n {\displaystyle \Phi _{mn}} is restricted to be of the form.[2]

- Φ i j = [ A 0 0 0 B 0 0 0 C ] {\displaystyle \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]

- σ x = ∂ 2 B ∂ z 2 + ∂ 2 C ∂ y 2 {\displaystyle \sigma _{x}={\frac {\partial ^{2}B}{\partial z^{2}}}+{\frac {\partial ^{2}C}{\partial y^{2}}}} σ y z = − ∂ 2 A ∂ y ∂ z {\displaystyle \sigma _{yz}=-{\frac {\partial ^{2}A}{\partial y\partial z}}} σ y = ∂ 2 C ∂ x 2 + ∂ 2 A ∂ z 2 {\displaystyle \sigma _{y}={\frac {\partial ^{2}C}{\partial x^{2}}}+{\frac {\partial ^{2}A}{\partial z^{2}}}} σ z x = − ∂ 2 B ∂ z ∂ x {\displaystyle \sigma _{zx}=-{\frac {\partial ^{2}B}{\partial z\partial x}}} σ z = ∂ 2 A ∂ y 2 + ∂ 2 B ∂ x 2 {\displaystyle \sigma _{z}={\frac {\partial ^{2}A}{\partial y^{2}}}+{\frac {\partial ^{2}B}{\partial x^{2}}}} σ x y = − ∂ 2 C ∂ x ∂ y {\displaystyle \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](/source/Beltrami%E2%80%93Michell_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]

- ∇ 4 A + ∇ 4 B + ∇ 4 C = 3 ( ∂ 2 A ∂ x 2 + ∂ 2 B ∂ y 2 + ∂ 2 C ∂ z 2 ) / ( 2 − ν ) , {\displaystyle \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 {\displaystyle C} is usually represented by φ {\displaystyle \varphi } and the stresses are expressed as

- σ x = ∂ 2 φ ∂ y 2 ; σ y = ∂ 2 φ ∂ x 2 ; σ x y = − ∂ 2 φ ∂ x ∂ y − ( f x y + f y x ) {\displaystyle \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 {\displaystyle f_{x}} and f y {\displaystyle f_{y}} are values of body forces in relevant direction.

In polar coordinates the expressions are:

- σ r r = 1 r ∂ φ ∂ r + 1 r 2 ∂ 2 φ ∂ θ 2 ; σ θ θ = ∂ 2 φ ∂ r 2 ; σ r θ = σ θ r = − ∂ ∂ r ( 1 r ∂ φ ∂ θ ) {\displaystyle \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 Φ m n {\displaystyle \Phi _{mn}} tensor is restricted to be of the form [2]

- Φ i j = [ 0 C B C 0 A B A 0 ] {\displaystyle \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]

- σ x = − 2 ∂ 2 A ∂ y ∂ z {\displaystyle \sigma _{x}=-2{\frac {\partial ^{2}A}{\partial y\partial z}}} σ y z = − ∂ 2 A ∂ x 2 + ∂ 2 B ∂ y ∂ x + ∂ 2 C ∂ z ∂ x {\displaystyle \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}}} σ y = − 2 ∂ 2 B ∂ z ∂ x {\displaystyle \sigma _{y}=-2{\frac {\partial ^{2}B}{\partial z\partial x}}} σ z x = − ∂ 2 B ∂ y 2 + ∂ 2 C ∂ z ∂ y + ∂ 2 A ∂ x ∂ y {\displaystyle \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}}} σ z = − 2 ∂ 2 C ∂ x ∂ y {\displaystyle \sigma _{z}=-2{\frac {\partial ^{2}C}{\partial x\partial y}}} σ x y = − ∂ 2 C ∂ z 2 + ∂ 2 A ∂ x ∂ z + ∂ 2 B ∂ y ∂ z {\displaystyle \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]

### Application to bar in torsion

For an elastic bar undergoing [Saint-Venant torsion](/source/Torsion_(mechanics)) about the z {\displaystyle z} -axis, the shear stresses can be expressed as

- τ x z = ∂ ϕ ∂ y ; τ y z = − ∂ ϕ ∂ x {\displaystyle \tau _{xz}={\frac {\partial \phi }{\partial y}}~;~~\tau _{yz}=-{\frac {\partial \phi }{\partial x}}}

where the Prandtl stress function ϕ {\displaystyle \phi } satisfies

- ∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ x 2 = − 2 G θ ′ {\displaystyle {\frac {\partial ^{2}\phi }{\partial y^{2}}}+{\frac {\partial ^{2}\phi }{\partial x^{2}}}=-2G\theta '}

Where G {\displaystyle G} is the [shear modulus](/source/Shear_modulus) and θ ′ {\displaystyle \theta '} is the rate of twist (change in angle per unit length). Applying the traction-free boundary condition at the outer surface of the bar leads to the result that the outer surface of the bar is a contour of the stress function. The shear stress in the bar acts along the contour of the stress function, and is proportional to the slope.

The [differential equation](/source/Differential_equation) satisfied by the stress function ( ∇ 2 ϕ = − k {\displaystyle \nabla ^{2}\phi =-k} , [Poisson's equation](/source/Poisson's_equation)) is the same as that governing the static deflected shape of an elastic membrane under uniform tension and pressure. This observation is the basis of the [membrane analogy](/source/Membrane_analogy) for shear stress in torsion.[5]

## See also

- [Elasticity (physics)](/source/Elasticity_(physics))

- [Elastic modulus](/source/Elastic_modulus)

- [Infinitesimal strain theory](/source/Infinitesimal_strain_theory)

- [Linear elasticity](/source/Linear_elasticity)

- [Solid mechanics](/source/Solid_mechanics)

- [Stress (mechanics)](/source/Stress_(mechanics))

## Notes

1. ^ [***a***](#cite_ref-Sadd05_363_1-0) [***b***](#cite_ref-Sadd05_363_1-1) Sadd, Martin H. (2005). [*Elasticity: Theory, Applications, and Numerics*](http://www.sciencedirect.com/science/book/9780126058116). Elsevier Science & Technology Books. p. 363. [ISBN](/source/ISBN_(identifier)) [978-0-12-605811-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-605811-6).

1. ^ [***a***](#cite_ref-Sadd05_364_2-0) [***b***](#cite_ref-Sadd05_364_2-1) [***c***](#cite_ref-Sadd05_364_2-2) [***d***](#cite_ref-Sadd05_364_2-3) Sadd, M. H. (2005) *Elasticity: Theory, Applications, and Numerics*, Elsevier, p. 364

1. **[^](#cite_ref-3)** Knops (1958) p327

1. ^ [***a***](#cite_ref-Sadd05_365_4-0) [***b***](#cite_ref-Sadd05_365_4-1) Sadd, M. H. (2005) *Elasticity: Theory, Applications, and Numerics*, Elsevier, p. 365

1. **[^](#cite_ref-Timoshenko70_295_5-0)** Timoshenko, S. P.; Goodier, J. N. (1970). *Theory of Elasticity* (3rd ed.). McGraw-Hill. pp. 295–303. [ISBN](/source/ISBN_(identifier)) [9780070858053](https://en.wikipedia.org/wiki/Special:BookSources/9780070858053).

## References

- Sadd, Martin H. (2005). *Elasticity - Theory, applications and numerics*. New York: Elsevier Butterworth-Heinemann. [ISBN](/source/ISBN_(identifier)) [0-12-605811-3](https://en.wikipedia.org/wiki/Special:BookSources/0-12-605811-3). [OCLC](/source/OCLC_(identifier)) [162576656](https://search.worldcat.org/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](/source/Doi_(identifier)):[10.1093/qjmam/11.3.326](https://doi.org/10.1093%2Fqjmam%2F11.3.326).

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