Physical significance of higher order spatial derivatives of displacement (longitudnal loading)

Hi,

I have encountered fourth order space derivative in the PDE derived for
smart material.I am looking for the physical significance of higher
order spatial derivative of displacment in longitudnal loading. In the
Euler-Bernoulli beam theory (lateral loading), the fourth order space
derivative referes to the load. 

In longitudinal loading, the first order space derivative is strain i.e 
du/dx .What are the significane of higher order space derivatives d^2
u/dx^2 , d^3 u/dx^3 , d^4 u/dx^4?

Please share your expertise.

Thanks in advance.

Rakesh