Solve Plane Elasticity w/ Airy Stress Function

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Discussion Overview

The discussion revolves around the challenges of solving plane elasticity problems using the method of Airy stress functions, particularly when shear is applied to two opposite sides of a body. Participants explore the implications of boundary conditions and the nature of solutions available for different loading scenarios.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes successfully solving uniaxial or biaxial uniform tension problems with a second-order polynomial but struggles when shear is applied to two opposite sides, questioning the existence of a solution.
  • Another participant suggests that the lack of "nice" polynomial solutions may indicate the need for numerical techniques, such as finite element or boundary element methods, to address elasticity problems.
  • A participant expresses confusion over why uniaxial tension allows for polynomial solutions while uniform shear does not, indicating a need for further clarification on the underlying mechanics.
  • Another participant points out that the body may not be in equilibrium under the shear conditions described unless kinematic boundary conditions are applied, suggesting that this scenario may fall outside traditional mechanics of deformable solids.

Areas of Agreement / Disagreement

Participants express differing views on the nature of solutions available for elasticity problems under various loading conditions. There is no consensus on whether the issue lies in the formulation of boundary conditions or the inherent nature of the problems being discussed.

Contextual Notes

Participants note the potential for solutions to be non-polynomial and the implications of equilibrium conditions on the analysis of the problem. The discussion highlights the complexity of applying boundary conditions in the context of shear loading.

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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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.
 
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!
 
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. .
 

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