 Solution

In the one-dimensional limit, $v=0$ and $u=u(x)$. The first component of the Navier-Cauchy equation reduces toFor a nondegenerate one-dimensional elastic material, $\lambda +2\mu \ne 0$, so $u^{\prime \prime }=0$ 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