91Ó°ÊÓ

The matrixis $p^{\left(1\right)}=\left(\begin{array}{ccc}-1& 1& 3\\ 1& 2& 0\\ 3& 0& 2\end{array}\right)$.

Stress is force per unit area, quantify the stress on a cube of material in multiple directions if given an arbitrary load. These data will be combined to generate the stress tensor, which is the second rank tensor, stress tensor.

The matrix is $p^{\left(1\right)}=\left(\begin{array}{ccc}-1& 1& 3\\ 1& 2& 0\\ 3& 0& 2\end{array}\right)$.

There are four stress tensors.

$$\begin{gathered} p^{\left(1\right)}=\left(\begin{array}{ccc}-1& 1& 3\\ 1& 2& 0\\ 3& 0& 2\end{array}\right) \\ {p}^{\left(2\right)}=\left(\begin{array}{ccc}1& 2& 2\\ 2& 3& 0\\ 2& 0& 3\end{array}\right) \\ {p}^{\left(3\right)}=\left(\begin{array}{ccc}-1& 2& 1\\ 1& 3& 0\\ 1& 0& 3\end{array}\right) \\ {p}^{\left(4\right)}=\left(\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right) \end{gathered}$$

Find the eigen values for $\left(\begin{array}{c}{P}_1^{\left(1\right)}\\ P_2^{\left(1\right)}\\ P_3^{\left(1\right)}\end{array}\right)$.

$$\begin{gathered} \left(\begin{array}{c}{P}_1^{\left(1\right)}\\ P_2^{\left(1\right)}\\ P_3^{\left(1\right)}\end{array}\right)=\left(\begin{array}{c}4\\ -3\\ 2\end{array}\right) \\ {V}_1^{\left(1\right)}=\left(\begin{array}{c}2\\ 1\\ 3\end{array}\right) \\ {V}_2^{\left(1\right)}=\left(\begin{array}{c}-5\\ 1\\ 3\end{array}\right) \\ {V}_3^{\left(1\right)}=\left(\begin{array}{c}0\\ -3\\ 1\end{array}\right) \end{gathered}$$

Find the eigen values for role="math" localid="1659255663231" $\left(\begin{array}{c}{P}_1^{\left(2\right)}\\ P_2^{\left(2\right)}\\ P_3^{\left(2\right)}\end{array}\right)$.

$$\begin{gathered} \left(\begin{array}{c}{P}_1^{\left(2\right)}\\ P_2^{\left(2\right)}\\ P_3^{\left(2\right)}\end{array}\right)=\left(\begin{array}{c}5\\ 3\\ -1\end{array}\right) \\ {V}_1^{\left(2\right)}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right) \\ {V}_2^{\left(2\right)}=\left(\begin{array}{c}0\\ -1\\ 1\end{array}\right) \\ {V}_3^{\left(2\right)}=\left(\begin{array}{c}-2\\ 1\\ 1\end{array}\right) \end{gathered}$$

Find the eigen values for $\left(\begin{array}{c}{P}_1^{\left(3\right)}\\ P_2^{\left(3\right)}\\ P_3^{\left(3\right)}\end{array}\right)$.

$$\begin{gathered} \left(\begin{array}{c}{P}_1^{\left(3\right)}\\ P_2^{\left(3\right)}\\ P_3^{\left(3\right)}\end{array}\right)=\left(\begin{array}{c}4\\ -3\\ 2\end{array}\right) \\ {V}_1^{\left(3\right)}=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right) \\ {V}_2^{\left(3\right)}=\left(\begin{array}{c}0\\ -1\\ 2\end{array}\right) \\ {V}_3^{\left(3\right)}=\left(\begin{array}{c}-5\\ 2\\ 1\end{array}\right) \end{gathered}$$

Find the eigen values for $\left(\begin{array}{c}{P}_1^{\left(4\right)}\\ P_2^{\left(4\right)}\\ P_3^{\left(4\right)}\end{array}\right)$.

$$\begin{gathered} \left(\begin{array}{c}{P}_1^{\left(4\right)}\\ P_2^{\left(4\right)}\\ P_3^{\left(4\right)}\end{array}\right)=\left(\begin{array}{c}-2\\ 2\\ -1\end{array}\right) \\ {V}_1^{\left(4\right)}=\left(\begin{array}{c}0\\ -1\\ 1\end{array}\right) \\ {V}_2^{\left(4\right)}=\left(\begin{array}{c}2\\ 1\\ 1\end{array}\right) \\ {V}_3^{\left(4\right)}=\left(\begin{array}{c}-1\\ 1\\ 1\end{array}\right) \end{gathered}$$

The basis spanned by their corresponding eigen vector, stress tensors are given below.

$$\begin{gathered} p{\text{'}}^{\left(1\right)}=\left(\begin{array}{ccc}4& 0& 0\\ 0& -3& 0\\ 0& 0& 2\end{array}\right) \\ p{\text{'}}^{\left(2\right)}=\left(\begin{array}{ccc}5& 0& 0\\ 0& 3& 0\\ 0& 0& -1\end{array}\right) \\ p{\text{'}}^{\left(3\right)}=\left(\begin{array}{ccc}4& 0& 0\\ 0& 3& 0\\ 0& 0& -2\end{array}\right) \\ p{\text{'}}^{\left(4\right)}=\left(\begin{array}{ccc}-2& 0& 0\\ 0& 2& 0\\ 0& 0& -1\end{array}\right) \end{gathered}$$