Math 2.0

Mechanics of deformable solids Definition

Strain tensor

Builds on DeformationGradientTransposeSkew-symmetric matrix

Definition

Let B⊆ℝ3B \subseteq \R^3 be open and let u ⁣:B→ℝ3u \colon B \to \R^3 be the continuously differentiable displacement of a deformation of BB. Let ∇u\nabla u be the matrix whose rows are the gradients of the components of uu, and let (∇u)T(\nabla u)^T be its transpose. The (infinitesimal) strain tensor of uu is the symmetric matrix field below. ε(u)=12(∇u+(∇u)T),εij=12(∂ui∂Xj+∂uj∂Xi).\varepsilon(u) = \tfrac12 \left( \nabla u + (\nabla u)^T \right), \qquad \varepsilon_{ij} = \tfrac12 \left( \frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} \right).

Remarks

For u(X)=(aX1,0,0)u(X) = (a X_1, 0, 0) with a∈ℝa \in \R, one gets ε=diag⁡(a,0,0)\varepsilon = \operatorname{diag}(a, 0, 0): a relative extension aa along the first axis. For an infinitesimal rotation u(X)=WXu(X) = W X with WW a skew-symmetric matrix, one gets ε=12(W+WT)=0\varepsilon = \tfrac12 (W + W^T) = 0: such a displacement produces no strain. The skew part ω=12(∇u−(∇u)T)\omega = \tfrac12 (\nabla u - (\nabla u)^T) is the infinitesimal rotation tensor, and ∇u=ε+ω\nabla u = \varepsilon + \omega. The trace tr⁡ε=div⁡u\operatorname{tr} \varepsilon = \operatorname{div} u is the relative change of volume to first order. In finite strain theory the name strain tensor also refers to measures such as the Green–Lagrange tensor, whose linear part in ∇u\nabla u is ε\varepsilon.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Deformation, concept, no evidence levelDeformationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptSkew-symmetric matrix, concept, no evidence levelSkew-symmetric matrixconceptTranspose, concept, no evidence levelTransposeconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptGradient, concept, no evidence levelGradientconceptStrain tensor, definition, no evidence levelStrain tensordefinition
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. Deformation concept
  2. Euclidean space object
  3. Function concept
  4. Skew-symmetric matrix concept
  5. Transpose concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Gradient concept
  11. Strain tensor definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Strain_(mechanics)
About this page
TagPPVD6A7F9M
URIm2://PPVD6A7F9M
Kinddefinition
WritingW0 stub
MSC74A05
DefinesStrain tensor
UsesDeformation, Gradient, Transpose, Skew-symmetric matrix
Changed3 November 2026

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