3-) (25 pts) Given the 2D displacement field $\mathbf{u}(X,Y) = \begin{Bmatrix} u(X,Y) \\ v(X,Y) \end{Bmatrix} = \begin{Bmatrix} 0.1XY + 1 \\ -0.02XY^2 \end{Bmatrix}$;
a- Calculate the strain tensor $[\mathcal{E}] = \begin{bmatrix} \mathcal{E}_x & \mathcal{E}_{xy} \\ \mathcal{E}_{xy} & \mathcal{E}_y \end{bmatrix}$
b- Calculate the strain tensor numerical value at point $X = 0.3$, $Y = 0.2$.
c- Using the numerical strain tensor, calculate the elongation strain $\mathcal{E}_\theta$ and the shear strain $\gamma_\theta$ where $\theta = 30^\circ$ is the angle between the X axis and the surface normal $\hat{\mathbf{N}}$