Show all your work $($i$.$e$.$ show each step$)$

Calculate the deformation gradients for the following operations. As discussed in class, $(x,Y,Z)$ correspond to the original configuration and $(x,y,z)$ correspond to the new, updated configuration $(5$ points each$).$

a$.x=2xcos+3xsin$;$y=3Y+2xsin$

b$.x=2x+5Y$;$y=3x+$Ycos$$

For the following deformation gradients, perform polar decomposition and obtain the right stretch tensor $U,$ rotation matrix $R$ and left stretch tensor $V(10$ points each$)$

a$.,F=\left[\begin{array}{cc}1\mathrm{.}3& -0\mathrm{.}375\\ 0\mathrm{.}75& 0\mathrm{.}65\end{array}\right]$

b$.,F=\left[\begin{array}{cc}0\mathrm{.}8& 0\\ 0& 1\mathrm{.}24\end{array}\right]$

c$.,F=\left[\begin{array}{cc}0\mathrm{.}94& -1\\ 0\mathrm{.}46& 1\mathrm{.}24\end{array}\right]$

For each of the above deformation gradients in part $2,$ calculate the following:

a$.$ Strain tensor $lon$ directly from $F,$ using the small strain assumption $(5$ points each$)$

b$.$ Strain tensor $lon$ from the right stretch tensor $U.(5$ points each$)$

c$.$ Are these two strain tensors the same? Calculate the $\%$ difference in each term of the strain tensor for both methods. If they are difference, comment on why they are different using the $\%$ difference values $(10$ points each$)$