Consider an orthotropic solid with engineering moduli $E_{11}=140$GPa,$E_{22}=E_{33}=10$

GPa,$G_{12}=G_{13}=7$GPa,$G_{23}=3\mathrm{.}36$GPa,$v_{12}=v_{13}=0\mathrm{.}3,v_{23}=0\mathrm{.}49.$

a$)$ Find the elastic constants $C_{11},C_{22}$ and $3.$

b$)$ Consider the case where $v_{12}=v_{13}=v_{23}=0.$ Compare the values of $E_{11},E_{22},E_{33}$

with $C_{11},C_{22},C_{33}$ and comment on the results.

The body shown in the figure below is subjected to a $2$ D plane stress field.

a$)$ Express the stress components in tensor form.

b$)$ Compute the stress tensor in the rotated coordinate system $(x_1^{\text{'}},x_2^{\text{'}})$ given $=20.$

c$)$ Express the stress tensor in contracted notation. Considering $=45,$ what are the stresses $($in vector form$)$ in the rotated coordinate system $(x_1^{\text{'}},x_2^{\text{'}}).$

d$)$ What conclusions can you draw from the results obtained in part $($a$)$ and $($b$)?$ If $$ is the fiber orientation, for the given stress conditions, what fiber orientations would you recommend for the lamina? Explain why.