(a) Let the rectangular sheet of Fig. 8.1(a) be composed of an isotropic material with material constants $E, v, G$, and $\alpha$. In this case, it is known by measurement that this sheet has been uniformly stretched a distance $2 u_0$ in the $x$ direction and a distance $2 v_0$ in the $y$ direction, and that there has been a uniform temperature change $T_0$. That is, the values $u_0, v_0$, and $T_0$ are constants. It is also observed that the sheet is in a state of plane stress. Guess that the mathematical description of the displacement field throughout the sheet is
$$
u(x, y)=\frac{x}{a} u_0 \quad \text { and } \quad v(x, y)=\frac{y}{b} v_0
$$
and, on this basis, determine the tractions at edges $x=a$ and $y=b$, which, along with the temperature change, have stretched the sheet in the $x$ and $y$ directions.
(b) If $\left(u_0 / a\right)$ is not equal to $\left(v_0 / b\right)$, is there a value of $\alpha T_0$ for which the tractions at all four edges can be zero?