# Newton's law of cooling style problem with variable cooling coefficient

• Apr 17th 2012, 04:55 AM
xlxc34
Newton's law of cooling style problem with variable cooling coefficient
Hi, I'm developing a code for analysing fatigue of polymers. The equation looks similar to Newton's law of cooling:

dn/dt=(n0-n)Kb ------ n is the damage variable, n0 is the resistance to failure (analagous to the ambient temperature in Newton's law) and Kb is the bond rupture rate, which is dependent on the temperature and the amount of stress that the polymer is experiencing.

In the current implementation of the code, Kb is integrated with time to obtain an 'average' bond rupture rate which can be used in the solution of the above equation.

n(t)=n0*(1-exp(integral(-Kb(t))))

This solution assumes that Kb repeats itself (ie high bond rupture have an equal likelihood of occuring at any time in the time history), because it should be obvious from the first equation that high stresses will cause more damage (bigger increase in n) if they occur at the start than at the end of the polymers life. I'd like to improve the code so that it can be used to analyse time histories that do not repeat themselves. Kb cannot be described mathmatically (it is a piecewise linear time history), and I cannot solve the equation by time stepping the first equation as the time penalty isn't acceptable (I'm currently using MatLab so if it's possible to carry on using vectorised operations I'd like to, although I do have a simple time stepping equations for checking my answers). n0 takes the value of 1.5820 and n is 0 initially and 1 when failure occurs, so the answer can't be too far out if you just solve for integral (Kb(t))=1 but I'd like to get the correct answer if possible! Any ideas? Thanks very much, Pete
• Apr 18th 2012, 01:46 AM
xlxc34
Re: Newton's law of cooling style problem with variable cooling coefficient