Convex analysis of a vectorial function

Question

In the context of equilibrium equations on structural concrete study, I deal with an system (3 non linear equations) that relates an entry of variables $(\epsilon_0,\kappa_x,\kappa_y)$ (which represents the deformation of a section) with an output $(N,M_x,M_y)$ (force and moments on it). Let $f$ be the function such that $(N,M_x,M_y)=f(\epsilon_0,\kappa_x,\kappa_y)$. I do not have the explicit relation of $f$, but I have a conjecture: looking at many examples, the volume enclosed by the surface defined by the points $(N,M_x,M_y)$ is convex. The points on the surface define the ultimate state of equilibrium of a section loaded with an axial force $N$ and two perpendicular moments $M_x$ and $M_y$, and every point enclosed by the surface represents an equilibrium outside the ultimate state of breaking the section. The system has the form \begin{equation*} N=R_{cc}\left(\epsilon_0,\kappa_x,\kappa_y\right)+\sum_{i=1}^nA_{si}\sigma_{si}\left(\epsilon_0,\kappa_x,\kappa_y\right)\\M_x=M_{cc,x}\left(\epsilon_0,\kappa_x,\kappa_y\right)+\sum_{i=1}^nA_{si}\sigma_{si}\left(\epsilon_0,\kappa_x,\kappa_y\right)Y_{si}\\ M_y=M_{cc,y}\left(\epsilon_0,\kappa_x,\kappa_y\right)+\sum_{i=1}^nA_{si}\sigma_{si}\left(\epsilon_0,\kappa_x,\kappa_y\right)X_{si} \end{equation*} where $R_{cc}$, $R_{cc,x}$ and $R_{cc,y}$ are the force and moments resisted by the compressed concrete (which depends on the deformation); $A_{si}$ is the area of the $i$-th steel bar, with its corrdinatinates $(X_{si},Y_{si})$ and stress $\sigma_{si}$ (which depends on the deformation). All dependences are not linear. What can be the hint to show that convexity? It seems a big challenge to show it since, in order to solve this system, I use, for example, Newton-Rhapson Method (in fact, the conjecture is based on plotting many points of the ultimate state of breaking, that is, it is based on visual fact).