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For the one-dimensional reduction of isotropic linear elasticity, nondegeneracy means that the effective longitudinal modulus $\lambda +2\mu$ is nonzero. The static Navier-Cauchy equation then forces the displacement to be affine.

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

Let the displayed integrand be $F$. Since it has no explicit dependence on $u$ or $v$, the Euler-Lagrange equations for two fields areIts derivatives areThusUsing the Laplacian, gradient, and divergence, these combine into the static Navier-Cauchy equation