Problem $1$

For uniaxial loading, the strain energy density $U_0$ is given by

$U_0={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}d$

where $$ is the stress and $$ is the strain. For more complicated loadings, the equation for strain energy

density is given in the notes. Your job is to derive the equation using the approach that follows. We will

first extend Eq$.1$ to a general state of stress given by the $3$D stress tensor $$ and the $3$D strain tensor $$ :

$U_0={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$:$d$

Here $($ is the matrix dot product, which is defined for matrices A and $B$ with components $A_{ij}$ and $B_{ij}$

$($where

subscripts i and $j$ index rows and columns, respectively$)$ as

$A$:$B={\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}{A}_{ij}{B}_{ij}$

It can be shown $($you will not need to show it here$)$ that for a linear elastic material, Eq$.2$ can be written

$U_0=\frac{1}{2}$:$$

Using Eqs. $3$ and $4,$ derive the equation for strain energy density $U_0$ in terms of the components of the stress

tensor $($i$.$e$.,$ derive the equation $U_0()=$dots in the notes$).$ Explain the important steps of the derivation

using full sentences.

Note: It may be helpful to use the matrix form of the $3$D Hooke's Law: $=\frac{1+}{E}$

$u-\frac{?}{E}$

$utr()I.$