suppose that I have a 1-dimensional bar with length L. The potential energy of the bar is a function of the position along the bar and is indicated as $\phi$(x), which is considered to be uniform along the bar. Now what I am supposed to do is to superimpose another function to the $\phi$(x) of the bar showing a local reduction in the middle of the bar and call it $\phi^*$(x). I intend to Define a continuous explicit function for $\phi^*$(x) that has the following properties:
- $\phi^*$(0.5L) = 0.5 $\phi$(x) (x does not matter here, since the function is uniform).
- The function smoothly tend to $\phi$(x), as in x = 0.5L+ 0.2L and x = 0.5L-0.2L the value of $\phi^*$(x) is equal to $\phi$(x).
To clarify the above requirements, I have attached the desired graph of my function.
Any suggestion is appreciated. Click to see image
Since the new $\phi^*$ is a function of the position of the element, a sort of damage function should be introduced that acts like a reduction function as follows:
$\phi^*$ = $\phi$[1 - $\omega(x)$];
In order to fulfill the above requirements, $\omega(x)$ must a Gaussian distribution function (or normal distribution) and has the following form:
$\omega(x)$ = Exp[-((x - $\mu$)^2)/(2*$\sigma$^2)]/Sqrt[V]*$\sigma$
The above parameters are function of the element length and can be calculated readily.