Solve Plane Elasticity w/ Airy Stress Function

In summary: In the third case, the body is not in equilibrium (unless some kinematic boundary conditions are applied), because it can rotate anticlockwise. In other words, this problem falls outside of the scope of the mechanics of deformable solid bodies.
  • #1
popbatman
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Hello to everyone, I have a problem with the solution of plane elasticity problems with the method of Airy stress functions.

For instance I can solve a problem of uniaxial or biaxial uniform tension with a 2nd order polynomial, but if I add shear on only two opposite sides the problem seems to have no solution. Is it possible that I have to formulate the no shear boundary conditions (on the free shear sides) in a weak form? If so I cannot understand the mathematical reason for this. Someone can help me? thank you! (In the attached file a little sketch to clarify my question)
 

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  • #2
popbatman said:
Hello to everyone, I have a problem with the solution of plane elasticity problems with the method of Airy stress functions.

For instance I can solve a problem of uniaxial or biaxial uniform tension with a 2nd order polynomial, but if I add shear on only two opposite sides the problem seems to have no solution. Is it possible that I have to formulate the no shear boundary conditions (on the free shear sides) in a weak form? If so I cannot understand the mathematical reason for this. Someone can help me? thank you! (In the attached file a little sketch to clarify my question)
Is that no solution at all, or no "nice" solutions in the form of polynomials?

Often times, elasticity problems don't have "nice" solutions in terms of polynomials, but can be solved using numerical techniques. That's one reason why finite element and boundary element techniques are used.
 
  • #3
SteamKing said:
Is that no solution at all, or no "nice" solutions in the form of polynomials?

Often times, elasticity problems don't have "nice" solutions in terms of polynomials, but can be solved using numerical techniques. That's one reason why finite element and boundary element techniques are used.

I mean nice solution in polynomial form. What I cannot really realize is why simple uniaxial tension admits such a solution, while the application of uniform shear on two opposite sides does not!
 
  • #4
In the third case, the body is not in equilibrium (unless some kinematic boundary conditions are applied), because it can rotate anticlockwise. In other words, this problem falls outside of the scope of the mechanics of deformable solid bodies. .
 

1. What is the Airy stress function?

The Airy stress function is a mathematical function used in the analysis of plane elasticity problems. It is a solution to the two-dimensional biharmonic equation and is used to simplify the stress calculations in elastic materials.

2. How is the Airy stress function used to solve plane elasticity problems?

The Airy stress function is used to derive the stresses and displacements in a two-dimensional elastic material by satisfying the equations of equilibrium and compatibility. By substituting the Airy stress function into these equations, the unknown stresses and displacements can be solved for.

3. What are the advantages of using the Airy stress function method?

The Airy stress function method offers several advantages in solving plane elasticity problems. It allows for the calculation of stresses and displacements in complex shapes and loading conditions, and it reduces the complexity of the equations to be solved. It also provides a general solution that can be applied to various boundary conditions.

4. Are there any limitations to using the Airy stress function method?

While the Airy stress function method is a powerful tool, it does have limitations. It can only be used for two-dimensional problems, and it assumes that the material is linearly elastic and homogeneous. Additionally, it may not be suitable for problems with highly irregular boundaries or material properties.

5. What are some real-world applications of the Airy stress function method?

The Airy stress function method has many practical applications in different fields of engineering, such as structural analysis, aerospace engineering, and mechanical engineering. It can be used to analyze the stresses and deformations in buildings, bridges, aircraft, and other structures subject to various loading conditions.

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