Math 2.0

Mechanics of deformable solids Definition

Linear elasticity

Builds on Strain tensorCauchy stress tensorElasticity tensorDivergenceLamé equations of elasticity

Definition

Let Ω⊆ℝ3\Omega \subseteq \R^3 be a bounded open set, the reference shape of a body. Let 𝖢\mathsf{C} be an elasticity tensor field on Ω\Omega, let ρ>0\rho > 0 be the mass density and let f ⁣:Ω→ℝ3f \colon \Omega \to \R^3 be a body force per unit volume. The equations of linear elasticity for a displacement field u ⁣:Ω×[0,T]→ℝ3u \colon \Omega \times [0, T] \to \R^3 are the following.

  1. Strain–displacement relation: the strain tensor is ε=12(∇u+(∇u)T)\varepsilon = \tfrac12 (\nabla u + (\nabla u)^T).
  2. Constitutive law: the Cauchy stress tensor is σ=𝖢:ε\sigma = \mathsf{C} : \varepsilon, that is σij=Cijklεkl\sigma_{ij} = C_{ijkl} \varepsilon_{kl} with summation over repeated indices.
  3. Equation of motion: ∇⋅σ+f=ρ ∂t2u\nabla \cdot \sigma + f = \rho \, \partial_t^2 u in Ω\Omega, where (∇⋅σ)i=∂jσij(\nabla \cdot \sigma)_i = \partial_j \sigma_{ij} is the row-wise divergence.

In the static case the right side of clause 3 is 00. A boundary value problem adds conditions on ∂Ω\partial \Omega. The displacement uu is prescribed on one part of the boundary. The traction σn\sigma n is prescribed on the rest, where nn is the outward unit normal.

Remarks

Example: let 𝖢\mathsf{C} be isotropic with Lamé parameters λ,μ\lambda, \mu, let f=0f = 0, and let u(x)=ax1e1u(x) = a x_1 e_1 for a small constant aa. The strain is ε=a e1⊗e1\varepsilon = a \, e_1 \otimes e_1 and the stress is constant, so ∇⋅σ=0\nabla \cdot \sigma = 0 and the static equations hold. This is a uniform stretch of the body.

Non-example: u(x)=ax12e1u(x) = a x_1^2 e_1 with a≠0a \ne 0 and f=0f = 0 does not solve the static isotropic equations, because ∇⋅σ=2a(λ+2μ)e1≠0\nabla \cdot \sigma = 2a(\lambda + 2\mu) e_1 \ne 0.

For an isotropic homogeneous material, substituting clauses 1 and 2 into clause 3 gives the Lamé equations for uu alone. The model is valid when ∣∇u∣\abs{\nabla u} is small, so that the reference and deformed shapes may be identified.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptElasticity tensor, concept, no evidence levelElasticity tensorconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLamé equations of elasticity, concept, no evidence levelLamé equations of ela…conceptStrain tensor, concept, no evidence levelStrain tensorconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptDivergence, concept, no evidence levelDivergenceconceptLinear elasticity, definition, no evidence levelLinear elasticitydefinition
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.
  • no evidence level
  1. Cauchy stress tensor concept
  2. Elasticity tensor concept
  3. Euclidean space object
  4. Function concept
  5. Lamé equations of elasticity concept
  6. Strain tensor concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Partial derivative concept
  10. Vector field concept
  11. Divergence concept
  12. Linear elasticity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_elasticity, Mathematical formulation
About this page
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URIm2://7FKWYYQPW7
Kinddefinition
WritingW0 stub
MSC74B05, 35Q74
DefinesLinear elasticity
UsesStrain tensor, Cauchy stress tensor, Elasticity tensor, Divergence, Lamé equations of elasticity
Changed3 November 2026

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