I have the equilibrium equation in elasticity for a static case.i.e Div T=0. For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is it done in that case? I am aware of getting the weak form using the vector equation as such, but what I need are two separate weak form for the following equations.
For your first equation you obtain the weak form integrating by parts: $$\int_\Omega\left[ \frac{\partial T_{11}}{\partial x}+\frac{\partial T_{21}}{\partial y}\right]\phi(x,y)\,dxdy=0$$
as follows
$$\int_\Omega\left[ \frac{\partial\phi}{\partial x}T_{11}+\frac{\partial\phi}{\partial y}T_{21}\right]\,dxdy=\int_\Omega\left[ \frac{\partial T_{11}\phi}{\partial x}+\frac{\partial T_{21}\phi}{\partial y}\right]\,dxdy$$
According to Green's theorem, the right hand side of the previous equation equals to: $$\int_{\partial\Omega} T_{11}\phi\,dy-\int_{\partial\Omega}T_{21}\phi\,dx$$ Since $n_x=\frac{dy}{dl}$ and $n_y=-\frac{dx}{dl}$ you can write:
$$\int_\Omega\left[ \frac{\partial\phi}{\partial x}T_{11}+\frac{\partial\phi}{\partial y}T_{21}\right]\,dxdy=\int_{\partial\Omega} \left[T_{11}\phi n_x+T_{21}\phi n_y\right]\,dl$$
You can follow the same procedure with the other equation.
