Consider a unit cube $[0,1{]}^3$ subject to the motion $x=(x,t)=A(t)x+b(t),$ where in the

right$-$hand orthonormal basis $e_1,e_2,e_3,$

$[A_{ij}]=([at+1,at,0],[0,1,0],[0,t,1]),\{b_i\}=([-t],[t],[0])$

Here, $x=x_i{e}_i$ is the material coordinates, $t0$ is the time, and $a0.$

Determine the inverse motion $(x,t).$

Determine the displacement $x-x$ in both Lagrangian and Eulerian coordinates, i$.$e$.,$

$U(x,t)$ and $u(x,t)$

Determine the velocity in both Lagrangian and Eulerian coordinate, i$.$e$.,V(x,t)$ and

$v(x,t).$

Determine the acceleration in both Lagrangian and Eulerian coordinates, i$.$e$.,A(x,t)$

and $a(x,t).$ Now let us consider the deformation at a fixed time $t=1,$ i$.$e$.,{}_1(x)=(x,1).$

Determine the deformation gradient $F$ of ${}_1.$

Determine the Right$-$Cauchy Green's deformation tensor C$.$

Determine the Cauchy deformation tensor c$.$

Determine the stretch in the direction $e_2$ in the reference configuration, i$.$e$.,{}_N$ for

$N=e_2.$

Determine the stretch in the direction $e_2$ in the deformed configuration, i$.$e$.,{}_n$ for

$n=e_2.$

Are the two stretches you have computed the same? Why or why not?

Repeat the previous three bullets for the direction $e_1.$

Determine the Lagrangian strain tensor $E.$

Determine the linearized small strain tensor $lon.$ Verify that the relationship detF$=1+Tr(lon)$ holds for this problem. Does this rela$-$

tionship hold for general deformations? What would a have to be for the deformation to

preserve the volume of the unit cube?

Compare the extensional strain $E_{11}$ and $lo{n}_{11}.$ What value of a limits the difference in this

strain to $1\%?$ Using this value for $a,$ what is the percent error in $lo{n}_{12}$ vs$.E_{12}?$

Compare this error with that of $lo{n}_{13}$ and $E_{13}.$ What is the large difference attributed to$?$