Stress functions explained

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:

where

is the stress tensor, and the Beltrami-Michell compatibility equations:

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

Using index notation:

Engineering notation
\sigmax = 2\Phi \partial yy \partialz\partialz + 2\Phi \partial zz -2 \partialy\partialy 2\Phi \partial yz \partialy\partialz		\sigmaxy=- 2\Phi \partial xy - \partialz\partialz 2\Phi \partial zz + \partialx\partialy 2\Phi \partial yz + \partialx\partialz 2\Phi \partial zx \partialy\partialz
\sigmay = 2\Phi \partial xx + \partialz\partialz 2\Phi \partial zz -2 \partialx\partialx 2\Phi \partial zx \partialz\partialx		\sigmayz=- 2\Phi \partial yz - \partialx\partialx 2\Phi \partial xx + \partialy\partialz 2\Phi \partial zx + \partialy\partialx 2\Phi \partial xy \partialz\partialx
\sigmaz = 2\Phi \partial yy + \partialx\partialx 2\Phi \partial xx -2 \partialy\partialy 2\Phi \partial xy \partialx\partialy		\sigmazx=- 2\Phi \partial zx - \partialy\partialy 2\Phi \partial yy + \partialz\partialx 2\Phi \partial xy + \partialz\partialy 2\Phi \partial yz \partialx\partialy

where

is a symmetric but otherwise arbitrary second-rank tensor field that is at least twice differentiable, and is known as the Beltrami stress tensor. Its components are known as Beltrami stress functions. 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 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 is at least four times continuously differentiable.

Maxwell stress functions

The Maxwell stress functions are defined by assuming that the Beltrami stress tensor

is restricted to be of the form.^[2]

The stress tensor which automatically obeys the equilibrium equation may now be written as:^[2]

\sigmax = \partial2B \partialz2 + \partial2C \partialy2		\sigmayz=- \partial2A \partialy\partialz
\sigmay = \partial2C \partialx2 + \partial2A \partialz2		\sigmazx=- \partial2B \partialz\partialx
\sigmaz = \partial2A \partialy2 + \partial2B \partialx2		\sigmaxy=- \partial2C \partialx\partialy

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]

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

is usually represented by and the stresses are expressed as Where and are values of body forces in relevant direction.

In polar coordinates the expressions are:

Morera stress functions

The Morera stress functions are defined by assuming that the Beltrami stress tensor

tensor is restricted to be of the form

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]

\sigmax =-2 \partial2A \partialy\partialz		\sigmayz=- \partial2A + \partialx2 \partial2B + \partialy\partialx \partial2C \partialz\partialx
\sigmay =-2 \partial2B \partialz\partialx		\sigmazx=- \partial2B + \partialy2 \partial2C + \partialz\partialy \partial2A \partialx\partialy
\sigmaz =-2 \partial2C \partialx\partialy		\sigmaxy=- \partial2C + \partialz2 \partial2A + \partialx\partialz \partial2B \partialy\partialz

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.

Application to bar in torsion

For an elastic bar undergoing Saint-Venant torsion about the

-axis, the shear stresses can be expressed as

where the Prandtl stress function

satisfies

Where

is the shear modulus and 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 satisfied by the stress function (

, 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 for shear stress in torsion.^[5]

See also

• Elasticity (physics)
• Elastic modulus
• Infinitesimal strain theory
• Linear elasticity
• Solid mechanics
• Stress (mechanics)

References

• Book: Sadd, Martin H. . Elasticity - Theory, applications and numerics. . 2005 . Elsevier Butterworth-Heinemann . New York . 0-12-605811-3 . 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 . 326–350 . 10.1093/qjmam/11.3.326 . Oxford University Press .

Notes and References

1. Book: Sadd, Martin H. . Elasticity: Theory, Applications, and Numerics . Elsevier Science & Technology Books . 363 . 978-0-12-605811-6 . 2005 .
2. Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364
3. Knops (1958) p327
4. Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365
5. Book: Timoshenko, S. P. . Theory of Elasticity . Goodier . J. N. . McGraw-Hill . 1970 . 9780070858053 . 3rd . 295303.