Solve the following problems

Let $,u$ and $T$ be differentiable scalar, vector and tensor fields. Show that

div$(Tu)=u*divT+tr(Tgradu)$

div$(T)=T^T$grad$+divT$

We will see in tomorrow's lecture, that surface traction vector $t(x,n),$ denoted as $t_n$ for simplicity, relative to an ideal cut plane with normal $n,$ is obtained from the stress tensor $$ as:

$t_n={}^{T}n$

where the tensor $$ has representation, in a local basis,

$=\left[\begin{array}{ccc}{}_{11}& {}_{12}& {}_{13}\\ {}_{21}& {}_{22}& {}_{23}\\ {}_{31}& {}_{32}& {}_{33}\end{array}\right]$

Compute the traction vector relative to the plane with normal $e_1=(1,0,0)$ and provide a mechanical interpretation of the components of this vector as normal and tangential stress components relative to that plane.

Compute the traction vector relative to the plane with normal $e_2=(0,1,0)$ and provide a mechanical interpretation of the components of this vector as normal and tangential stress components relative to that plane.

Compare the two traction vectors $t_{e_1}$ and $t_{e_2}$ and note what are their differences.

Prove that if $n$ and $m$ are two unit normals, then in order that

$t_n*m=t_m*n$

it must result that the stress tensor $$ is symmetric. With this in mind, think again of the last point to the question above, and make sense of the components $t_{e_1}*e_2$ and $t_{e_2}*e_1.$

$4.$ A cubic block with edges along the orthogonal unit vectors $e_1,e_2,e_3$ is said to be under "equal pure shear stresses" when it is subject to a Cauchy stress $$ with components

$=\left[\begin{array}{ccc}0& S& S\\ S& 0& S\\ S& S& 0\end{array}\right]$

in the $\{e_i\}$ basis, where $S$ is a constant. Compute the principal invariants of $.$ Show that two of the principal stresses are equal. Show that the third principal stress takes place along a diagonal of the cube.