Expression for change in 2 dependent variables Deformation $ \delta $  and stress $ \sigma$ depend on  parameters
$r$ and $t$  varying inversely as $ r^3 \, t $ and $ r \, t  $ respectively.
Find the expression how  $(r,t)$ depend on $ \delta $ and $ \sigma.$   
EDIT1:
May I now slightly modify the question?
Given $ \delta_1, \sigma_1 ,\delta_2,  \sigma_2,  r_1, t_1,  $
find
$ r_2 ,   t_2  .$
 A: Notice, we have $$\delta=\frac{C_1}{r^3t}\tag 1$$ $$\sigma=\frac{C_2}{rt}\implies r=\frac{C_2}{\sigma t}\tag 2$$ Substituting the value of $r$ in above eq (1), we get 
$$\delta=\frac{C_1}{\left(\frac{C_2}{\sigma t}\right)^3t}=\frac{C_1\sigma ^3t^2}{C_2^3}\iff t^2=\frac{\delta C_2^3}{C_1\sigma ^3} \iff \color{blue}{t=K_1\frac{\delta^{\frac{1}{2}}}{\sigma^{\frac{3}{2}}}=K_1\sqrt{\frac{\delta}{\sigma^3}}}$$  Where, $K_1$ is a new constant. 
Now, substituting the value of $t$ in eq(2), we get $$r=\frac{C_2}{\sigma K_1\frac{\delta^{\frac{1}{2}}}{\sigma^{\frac{3}{2}}} }=\frac{C_2\sigma^{\frac{1}{2}}}{\delta^{\frac{1}{2}}} $$ $$\implies \color{blue}{r=K_2\frac{\sigma^{\frac{1}{2}}}{\delta^{\frac{1}{2}}}=K_2\sqrt{\frac{\sigma}{\delta}}}$$
Where, $K_2$ is a new constant.
A: ANSWER TO ORIGINAL QUESTION:
From your description we directly get
$$\delta\cdot r^3t=c$$
$$\sigma\cdot rt=d$$
for some constants $c$ and $d$. Dividing the first equation by the second we get
$$\frac{\delta}{\sigma}r^2=\frac cd$$
Solving for $r$,
$$r=\sqrt{\frac{c\sigma}{d\delta}}$$
at least, if $r\ge 0$. Place a plus-or-minus sign in front if $r$ may be negative.
Substituting that into the second equation,
$$\sigma t\sqrt{\frac{c\sigma}{d\delta}}=d$$
or
$$t=\sqrt{\frac{d^3\delta}{c\sigma^3}}$$
We may combine the constants $c$ and $d$ to get new constants $e$ and $f$ to simplify to

$$r=e\sqrt{\frac{\sigma}{\delta}}$$
  $$t=f\sqrt{\frac{\delta}{\sigma^3}}$$

Checking that answer, we get
$$\delta\cdot r^3t=\delta\left(e\sqrt{\frac{\sigma}{\delta}}\right)^3\left(f\sqrt{\frac{\delta}{\sigma^3}}\right)=e^3f$$
$$\sigma\cdot rt=\sigma\left(e\sqrt{\frac{\sigma}{\delta}}\right)\left(f\sqrt{\frac{\delta}{\sigma^3}}\right)=ef$$
and the right-hand sides are constants, as desired.

ANSWER TO EDITED QUESTION:
You now want to find $r_1,t_1$ from $\delta_1,\delta_2,\sigma_1,\sigma_2,r_1,t_1$. From the original equations and the first situation for the "$1$'s" we get
$$c=\delta_1\cdot r_1^3t_1$$
$$d=\sigma_1\cdot r_1t_1$$
And using my first, unsimplified derived equations for $r$ and $t$ for the $2$'s,
$$\begin{align}
r_2 &= \sqrt{\frac{c\sigma_2}{d\delta_2}} \\
 &= \sqrt{\frac{(\delta_1\cdot r_1^3t_1)\sigma_2}{(\sigma_1\cdot r_1t_1)\delta_2}} \\
 &= r_1\sqrt{\frac{\delta_1\sigma_2}{\delta_2\sigma_1}}
\end{align}$$
and
$$\begin{align}
t_2 &= \sqrt{\frac{d^3\delta_2}{c\sigma_2^3}} \\
 &= \sqrt{\frac{(\sigma_1\cdot r_1t_1)^3\delta_2}{(\delta_1\cdot r_1^3t_1)\sigma_2^3}} \\
 &= t_1\sqrt{\frac{\delta_2\sigma_1^3}{\delta_1\sigma_2^3}}
\end{align}$$
There are several other ways to derive these equations, such as from scratch or from my first answer's final equations for $r$ and $t$.
