σ: stress, ε: strain
a, b, c, d, $ε_2$, $ε_3$ are constants.
σ is a piecewise CONTINUOUS and SMOOTH function:
$σ(ε)= d\cdot \begin{cases} 1-(1-\frac{ε}{ε_2})^2 & 0<=ε<ε_2 \\ 1 & ε_2<=ε<=ε_3 \end{cases}$
Cross section always are planar, so $ε = a x + b y + c$
To get the axial force I must double integrate the $σ$ on area domain.
Now, I want to avoid double integration of $σ$ on area domain, and I want to use half of Green theorem $\int_S L(x,y)dy=\iint_A σ(x,y) dx dy$ where $L(x,y)=\int σ(x,y)dx$
But piecewise function L(x,y) IS NOT CONTINUOUS in boundary domain (nor the area domain).
So this is a forbidance to use Green theorem for piecewise functions?
I really need to use Green Theorem. Is there a workaround to make $L(x,y)$ continuous and valid for Green theorem?
Ok. My mistake. $L(x,y)=\int σ(x,y)dx+f(y)$
$f(y)$ must be chosen wisely, to make piecewise function continuous.
If this cannot give an analytical function $L(x,y)$ then the area domain must be splitted into 2 parts and Green theorem must be used on boundary of each of them.