Continuum Mechanics

Problem $6$

Consider the simple symmetric two$-$dimensional tensor $S$ with components

$[S]=\left[\begin{array}{cc}4& -1\\ -1& 4\end{array}\right]$

expressed in a cartesian reference frame $e_i.$

a$)$ Find the eigenvalues, ${}_i$ and eigenvectors, $v_i$ of $S.$ Show your work!

b$)$ Find the rotation tensor $Q$ that maps from the original reference frame of $S$ to the eigenspace. That

is solve for $Q_{ij}=v_i*e_j.$

Show that your matrix $Q$ is correct by computing $S^{*}=QS{Q}^{TT}.$ Here $S^{*}$ should be a spherical tensor

containing the previously found eigenvalues.

c$)$ Compute $A=ln(S)$ the operation of which is defined as

$ln(S)={\displaystyle \underset{i=1}{\overset{3}{}}}ln({}_i)v_{i}ox{v}_i$

with ${}_i$ the eigenvalues of $S$ and $v_i$ the corresponding eigenvectors.