$[$A$]$ is a second order tensor $($matrix$).$ Express the following expressions in the index notation:

$[A],[A][A{]}^T,[A][A]$

Consider a cantilever beam of thickness $"$t$",$ depth $"2$c$"$ and length $"$L$"(L>>c,L>>t,2D$

plane stress problem$),$ subjected to a uniform pressure $($force per unit area ${}_0)$ on the top

surface as shown in the figure. Following solution states are proposed for this problem. It is

required to:

${}_{11}=\frac{{}_0{x}_2}{2c}\{\frac{x_2^2}{c^2}-\frac{3x_1^2}{2c^2}-\frac{3}{5}\}$

${}_{22}=\frac{{}_0}{2}\{-1+\frac{3x_2}{2c}-\frac{x_2^3}{2c^3}\}$

${}_{12}=-\frac{3{}_0{x}_1}{4c}\{1-\frac{x_2^2}{c^2}\}$

$($a$)$ Check if the above solution states satisfy the equilibrium, stress boundary and geometry

compatibility conditions for this problem.

$($b$)$ In your undergraduate mechanics of materials class you had learned under simple beam

theory that the bending stress and shear stress have the following forms

${}_{11}==\frac{M{x}_2}{\text{I}}$ and ${}_{12}==\frac{VQ}{It},$ where I is the area moment of inertia, $Q$ is the first

moment of the part of the cross$-$sectional area, $M$ and $V$ are bending moment and shear force

at the cross section. Check if the solution state corresponds to the simple beam theory

formulations.