Two rectangular columns of soft tissue are placed end$-$to$-$end and constrained between two rigid walls.

Each column is composed of the same incompressible material, with a strain$-$energy density function

given by

$W=\frac{C}{a}[e^{a(I_1-3)}-1]$

where $C$ and a are constants and

$I_1={}_x^2+{}_y^2+{}_z^2$

is the first strain invariant. At $t=0,$ both columns have the same length. For $t>0,$ column $1$ undergoes

growth in the $x-$direction only, as described by the growth stretch ratio $(t),$ while column $2$ does not

grow. Assume that no buckling occurs.

$($a$)$ Show that, at any time, ${}_{x1}^{**}+{}_{x2}^{**}=2,$ where ${}_{x1}^{**}$ and ${}_{x2}^{**}$ are the longitudinal stretch ratios of

columns $1$ and $2$ with respect to their zero$-$stress configurations.

$($b$)$ Derive expressions for Cauchy stresses ${}_{x1}$ and ${}_{x2}$ in columns $1$ and $2$ in terms of ${}_{x1}^{**}$ and ${}_{x2}^{**},$

respectively

$($c$)$ With $(t)$ given and using your answers from a$)$ and b$),$ find an equation to be solved for ${}_{x1}^{**}(t).$

$($d$)$ Consider the case where

$(t)=1+A(1-e^{-bt})$

For $A=0\mathrm{.}4,a=2$ and $b=1h{r}^{-1},$ solve the equation of part $($c$)$ using any method you wish. Plot the following

for $0t2hr$ :

$($i$)$ vs$.$ t;

$($ii$){}_{x1}=\frac{\frac{?}{\text{b}\text{a}\text{r}}(L{)}_1}{L_1}vs.$ t$,$ where $L_1$ and $\frac{?}{\text{b}\text{a}\text{r}}(L{)}_1$ are the actual lengths of column $1$ before and after growth,

respectively.

$($iii$)\frac{{}_{x1}}{C}$ vs$.$ t$.$