Problem $6.(10$ pts$)$

In solid mechanics, Mohr's circle is a graphical representation of the transformation law for the Cauchy stress tensor. The points on the circle and their rotation can be represented by the deformation gradient matrix and its polar decomposition, respectively. The deformation gradient matrix $F$ can be decomposed into a rotation followed by a stretch component, or a stretch component followed by a rotation, i$.$e$.,F=RQ=PR$ where $R$ is a rotation matrix, and $Q$ and $P$ are the right and left stretch matrices. For the following relationships between positions in the reference configuration $\{x_1,x_2\}$ and deformed configuration $\{x_1,x_2\},$

$x_1=3x_1-x_2,x_2=3x_1+4x_2$

solve the following subproblems.

$($a$)$ Find the deformation gradient matrix $F=[F_{ij}]=[del\frac{x_i}{d}el{x}_j]$ and the singular value decomposition $($SVD$)$ of the matrix $F=U{V}^T.$

$($b$)$ Find a rotation matrix $R$ and rotation angle $$ for the matrix $F$ in subproblem $6($a$).$ Then, determine right and left stretch matrices $Q$ and $P.$