Q$3.$ Consider a unit cube of material undergoing a homogenous motion specified by:

$x_1={}_1(t)x_1+(t)x_2,$

$x_2={}_2(t)x_2,$

$x_3={}_3(t)x_3$

where ${}_1(t),{}_2(t),{}_3(t)$ and $(t)$ are positive functions and ${}_1(0)={}_2(0)={}_3(0)=1$ and

$(0)=0.$

a$.)$ Find $F$ and sketch how a unit cube deforms in this deformation.

b$.)$ Calculate the velocity and acceleration in both the Lagrangian and Eulerian form.

c$.)$ Calculate L$,$ D and W$.$

d$.)$ Show that if the motion given by Eq$.(2)$ is isochoric, the following two conditions

have to be satisfied:

$\frac{i_1^{}{?}^{}}{{}_1}+\frac{{}_2^{}}{{}_2}+\frac{{}_3^{}}{{}_3}=0$ and ${}_1{}_2{}_3=1$