airy stress function in boundary conditions in 2D elasticity

[robinho]

Airy stress function in boundary conditions in 2D elasticity has a requirement that its x and y derivatives be related to the x and y boundary tractions Rx and Ry, i.e.: dAiry/dy=Rx, dAiry/dx=-Ry. Then dRx/dx=-dRy/dy? How to understand this identity?

[studiot]

51 minutes ago, robinho said:

Airy stress function in boundary conditions in 2D elasticity has a requirement that its x and y derivatives be related to the x and y boundary tractions Rx and Ry, i.e.: dAiry/dy=Rx, dAiry/dx=-Ry. Then dRx/dx=-dRy/dy? How to understand this identity?

Not sure if you want a physical or mathematical explanation.

The physical one is easier so I will start with that.

 

When you integrate the differential equations you get many or infinitely many solutions that differ by an arbitrary constant.

 

(actually since you should be using partial derivatives for this, you get an arbitrary function).

Physically what we want is for that arbtrary function to affect all parts of the region and its boundary equally.
Then its effect  may be discounted.

Finding a suitable function requires differentiating ande substituting the derivatives into the proposed stress function as you have indicated.
The only way for the arbitrary part to affect all parts equally is for it to a constant function or at most a linear function, whose derivatives are zero or constant.

 

 

 

[robinho]

Hi studiot, thanks so much for answering. I want a physical explanation. I don't think you answered directly why dRx/dx=-dRy/dy. If we take a differential element on the boundary and analyze force balance, can we get that?

[studiot]

Your question  -  A good one   - seems the sort of question that might (should) pop up in class when you have all the background of the course and its notation.

So it is a very good idea to give as much information as possible about where you are coming ffrom and where you want to go.

 

We normally like to place replies into the body of a thread, to benefit any member who might be interested. There is nothing confidential  abolout this.

 

The honest answer is that we are looking for solutions to Airy's equation. There is no pat answer as with say linear differential equations, each solution must be worked out for the specific situation. Airy's original was an inspired guess as are pretty well all other known solutions.

This is the drawback for this method and the reason it has never proceeded very far. Complex analysis and conformal mapping techniques have largely replaced it as an analytical technique.

 

Anyway here are a few pages from a book that is ideal for engineers studying stress analysis.

It works from an engineering point of view and explains, when it comes to it, that we 'guess' the form of the solution and back substitute to check it fits the equations of constitution and discusses this for several example cases.

Are you familiar with the idea of the equations of constitution and compatibility ?

For Constitution either the equations of equilibrium or the equations of motion may be employed, but these alone will not give enough equations to produce a solution.

That is where compatibility (or configuration or constraint) come in.

 

 

Advanced Strength and Applied Stress Analysis

R G Budynas

 

[robinho]

Hi studiot, thanks so much for answering.  Actually I already got my answer to this particular question. By the way, is there Michell problems in the book you scanned?