A unit solid cube is deformed from a reference configuration to a current configuration. The vertices $A,B,C,D,E,F,G,H$ in the reference configuration $($see figure$)$ are located at $a,b,c,d,e,f,g,h$ in the current configuration.

Consider that the block is subjected to the following deformation

$x_1=x_1+x_2,x_2=x_1+x_2,x_3=x_3$

where $>0$ is a constant.

$($a$)$ Sketch the shape of the solid in the current configuration when $=\frac{1}{2}.$

$[10$ marks$]$

$($b$)$ Compute the components $F_{ij}=$del$\frac{x_i}{d}el{x}_j$ of the deformation gradient $F.$ Is the deformation homogeneous? Is it isochoric? What is the range of allowable values for $?$

$($justify your answers to these questions$)$

$[20$ marks$]$

$($c$)$ Use Nanson's formula nda$=(detF)(F^{-1}{)}^{TT}$NdA to find out how the surface element NdA$=[0,1,0{]}^{TT}dA$ in the $CDHG$ face $($reference configuration$)$ is transformed into nda $($current configuration$).$

$[10$ marks$]$

$($d$)$ Compute the components of the right Cauchy$-$Green deformation tensor $C=F^{TT}F.$

$[10$ marks$]$

$($e$)$ Check that the vectors along $x_3$ and along the diagonals $AH$ and $DE$ are eigenvectors of $C$ and give their respective eigenvalue.

$[30$ marks$]$

$($f$)$ Find the components of a vector along the greatest stretch in the current configuration. Is it along a diagonal of the deformed solid?

$[20$ marks$]$