Solution
= Solution
In the one-dimensional limit, $v=0$ and $u=u(x)$. The first component of the <Navier-Cauchy equation> reduces to
$$
(\lambda+2\mu)u''(x)=0.
$$
For a <nondegenerate one-dimensional elastic material>, $\lambda+2\mu\ne0$, so $u''=0$ and $u(x)=Ax+B$. The <boundary conditions> $u(0)=0$ and $u(L)=\Delta$ give $B=0$ and $A=\Delta/L$. Therefore the <uniform extension of a one-dimensional elastic body> is
$$
\boxed{u(x)=\frac{\Delta}{L}x}.
$$