Navier-Cauchy equation

In a homogeneous isotropic elastic body without body forces, static equilibrium of the displacement field is

μ \nabla^{2} u + (λ + μ) \nabla (\nabla \cdot u) = 0.

This is the static, homogeneous-isotropic displacement form of linear elasticity.

Uniform extension of a one-dimensional elastic body

For a one-dimensional body with endpoints displaced by zero and

Δ

, the static Navier-Cauchy equation reduces to

u^{''} = 0

and gives the uniform strain solution

u (x) = Δ x / L

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