This is the profile which I am expecting
I need to obtain equation of profile with the following conditions My 'y' axis is σ and 'x' axis is ϵ Given conditions are:
- It has to satisfy (fc,ϵp) and (fc/3,ϵf) Note: ϵp is known
- $$\int_{ϵp}^{ϵf} σ dϵ = G$$ Note: G is known
I tried solving it in this way:
I assumed the equation to be $$σ=Aϵe^{Bϵ}$$ It becomes complicated when I solve for ϵf and B. Is there a better way to obtain solution or a simple equation to fit a profile similar to the shown (*It should look like water flowing through spillway)
Edit:: fc is also a known value
Assuming that the model is $$σ=A\,ϵ\,e^{Bϵ}$$ the given conditions give as equations $$f_c=A\,ϵ_p\,e^{B\,ϵ_p}$$ $$\frac{f_c}3=A\,ϵ_f\,e^{B\,ϵ_f}$$ $$G=A\frac{ e^{B \,\epsilon_f} (B\, \epsilon_f-1)-e^{B\, \epsilon_p} (B \,\epsilon_p-1)}{B^2}$$ I suppose that $f_c,\epsilon_p,G$ are given and that you search for unknowns $A,B,\epsilon_f$.
This is not the most pleasant system to solve (I suppose that otherwise you would not have posted the problem) but a few things can be done.
Using the first and second equations, you can express $A$ and $\epsilon_f$ as function of $B$. This gives $$A=\frac{f_c }{\epsilon_p }e^{-B\, \epsilon_p }$$ $$\epsilon_f=\frac{1}{B}\,W\left(\frac{B \,\epsilon_p}{3} e^{B\, \epsilon_p }\right)$$ where $W(z)$ is Lambert function.
So, you are left with one ugly equation in $B$ (the third equation replacing $A$ and $\epsilon_f$ by their expressions) but numerical methods (such as Newton) can do the job. If you use methods requiring derivatives, I suggest you use numerical derivatives instead of the analytical ones.