just answer e only!!!

Pure shear

In a right$-$handed Cartesian coordinate system $O{x}_1{x}_2{x}_3,$ a stress state called pure shear

may be represented by a stress tensor of the following form,

$([0,,0],[,0,0],[0,0,0])$

a$.$ Draw a cubic volume element with surfaces perpendicular$/$parallel to the three axial

directions widehat$(x_1),$widehat$(x_2),$widehat$(x_3),$ indicate on all surfaces of the volume element the non$-$zero

stress components corresponding to the state of pure shear shown above;

b$.$ Determine the eigenvalues and eigenvectors for this pure shear stress state;

c$.$ Form a new Cartesian coordinate system $O{x}_1^{\text{'}}{x}_2^{\text{'}}{x}_3^{\text{'}}$ made up from the three

eigenvectors you determined in b$,$ calculate the form of the stress tensor in the $O{x}_1^{\text{'}}$

$x_2^{\text{'}}{x}_3\text{'}$ system;

d$.$ Shown from your answer for c that the rotated stress tensor is in the form of a biaxial

stress;

e$.$ Draw the $O{x}_1^{\text{'}}{x}_2^{\text{'}}{x}_3^{\text{'}}$ and the $O{x}_1{x}_2{x}_3$ coordinate systems together. Does this

make physical sense to you?