Math 2.0

Mechanics of deformable solids Linear elasticity Theorem

Papkovich–Neuber solution

Builds on Lamé equations of elasticityHarmonic functionPoisson's ratio

Statement

Let Ω⊆ℝ3\Omega \subseteq \R^3 be an open set. Consider a homogeneous isotropic linear elastic material with shear modulus μ>0\mu > 0 and Poisson's ratio ν∈(−1,1/2)\nu \in (-1, 1/2), and no body force. Let ψ ⁣:Ω→ℝ3\psi \colon \Omega \to \R^3 be a vector field and ϕ ⁣:Ω→ℝ\phi \colon \Omega \to \R a function, all of whose components are harmonic. Then the displacement u=12μ(4(1−ν)ψ−∇(x⋅ψ+ϕ))u = \frac{1}{2\mu} \Big( 4 (1 - \nu) \psi - \nabla (x \cdot \psi + \phi) \Big) solves the static Lamé equations μΔu+(λ+μ)∇(∇⋅u)=0\mu \Delta u + (\lambda + \mu) \nabla (\nabla \cdot u) = 0 in Ω\Omega, where λ=2μν/(1−2ν)\lambda = 2\mu\nu/(1 - 2\nu). Conversely, if Ω\Omega is convex, every solution uu of class C2C^2 of these equations on Ω\Omega has this form for some harmonic ψ\psi and ϕ\phi.

Remarks

The direct part is a computation: Δ(x⋅ψ)=2∇⋅ψ\Delta (x \cdot \psi) = 2 \nabla \cdot \psi for harmonic ψ\psi, so Δu=−1μ∇(∇⋅ψ)\Delta u = -\frac{1}{\mu} \nabla (\nabla \cdot \psi) and ∇⋅u=1−2νμ∇⋅ψ\nabla \cdot u = \frac{1 - 2\nu}{\mu} \nabla \cdot \psi. Since λ+μ=μ1−2ν\lambda + \mu = \frac{\mu}{1 - 2\nu}, the two terms of the Lamé operator cancel. Example: ψ=F8π(1−ν)∣x∣\psi = \frac{F}{8\pi(1 - \nu) \abs{x}} and ϕ=0\phi = 0 give the Kelvin solution for a point force on ℝ3∖{0}\R^3 \setminus \{0\}. The representation is not unique: for example, adding a constant to ϕ\phi does not change uu.

Going deeper

For a student

The representation is a standard tool for three-dimensional elastostatics. One chooses harmonic potentials adapted to the geometry, for example spherical harmonics for a sphere. The Boussinesq solution for a normal point load on a half-space can be written in this form. It plays for the Lamé equations a role similar to that of the Helmholtz decomposition for vector fields.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLamé equations of elasticity, concept, no evidence levelLamé equations of ela…conceptPoisson's ratio, concept, no evidence levelPoisson's ratioconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptPapkovich–Neuber solution, statement, no evidence levelPapkovich–Neuber solu…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Lamé equations of elasticity concept
  4. Poisson's ratio concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
  12. Divergence concept
  13. Laplacian concept
  14. Harmonic function concept
  15. Papkovich–Neuber solution statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Papkovich%E2%80%93Neuber_solution
About this page
TagGWGMRAD7HF
URIm2://GWGMRAD7HF
Kindstatement, theorem
WritingW0 stub
MSC74B05, 74G05, 35Q74
StatusThis copy of the site has no build data. Run m2 status GWGMRAD7HF.
AboutLamé equations of elasticity
UsesLamé equations of elasticity, Harmonic function, Poisson's ratio
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Changed3 November 2026

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