How Is Normal Strain Derived in Continuum Mechanics?

[nissan4l0]

http://imgur.com/SnHyP

What are the mathematical steps and assumptions to reach the conclusion that length(ab) ≈ dx + ∂u/∂x*dx ?

If you consider the the squares of the gradients to be negligible, you still have a square root and multiplication by the constant "2". What other assumptions do we make to derive the final equation?

 

[arildno]

You make a Taylor expansion of sqrt(1+e), where "e" is regarded to be small.
Then, you have, approximately, sqrt(1+e)=1+e/2.