Math 2.0

Mechanics of deformable solids Definition

Principle of material frame-indifference

Builds on Constitutive equationCauchy stress tensorDeformation gradientPolar decomposition of the deformation gradientOrthogonal groupSpecial orthogonal group

Definition

A change of observer is a map x↦x∗=c(t)+Q(t) xx \mapsto x^* = c(t) + Q(t)\, x of ℝ3\R^3, where c(t)∈ℝ3c(t) \in \R^3. Here Q(t)Q(t) is a rotation, that is, an element of the special orthogonal group SO(3)SO(3). Under a change of observer the deformation gradient becomes F∗=QFF^* = Q F, and a frame-indifferent (objective) tensor field TT, such as the Cauchy stress tensor, becomes T∗=QTQTT^* = Q T Q^T. The principle of material frame-indifference states that a constitutive equation must keep the same form under every change of observer. For an elastic material with σ=σ^(F)\sigma = \hat\sigma(F) it reads σ^(QF)=Q σ^(F) QTfor all Q∈SO(3) and all F with det⁡F>0.\hat\sigma(Q F) = Q\, \hat\sigma(F)\, Q^T \quad \text{for all } Q \in SO(3) \text{ and all } F \text{ with } \det F > 0.

Remarks

The function σ^(F)=μ(FFT−I)\hat\sigma(F) = \mu (F F^T - I), with μ∈ℝ\mu \in \R, satisfies the principle: σ^(QF)=μ(QFFTQT−I)=Qσ^(F)QT\hat\sigma(QF) = \mu (Q F F^T Q^T - I) = Q \hat\sigma(F) Q^T, using QQT=IQ Q^T = I. The function σ^(F)=μ(F+FT−2I)\hat\sigma(F) = \mu (F + F^T - 2I) with μ≠0\mu \neq 0 does not: for F=IF = I it gives σ^(Q)=μ(Q+QT−2I)\hat\sigma(Q) = \mu(Q + Q^T - 2I), which is nonzero for a rotation Q≠IQ \neq I, while Qσ^(I)QT=0Q \hat\sigma(I) Q^T = 0. By the polar decomposition F=RUF = R U, the principle holds exactly when σ^(F)=R σ^(U) RT\hat\sigma(F) = R\, \hat\sigma(U)\, R^T for all FF, so the response is fixed by its values on the stretch tensors UU. Some authors allow QQ in the full orthogonal group, including reflections.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptCauchy–Green deformation tensor, concept, no evidence levelCauchy–Green deformat…conceptConstitutive equation, concept, no evidence levelConstitutive equationconceptDeformation gradient, concept, no evidence levelDeformation gradientconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionOrthogonal matrix, concept, no evidence levelOrthogonal matrixconceptPositive definite matrix, concept, no evidence levelPositive definite mat…conceptSquare root of a matrix, concept, no evidence levelSquare root of a matr…conceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSquare root of a positive semidefinite matrix, definition, no evidence levelSquare root of a posi…definitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectPolar decomposition, statement, no evidence levelPolar decompositionstatementPolar decomposition of the deformation gradient, statement, no evidence levelPolar decomposition o…statementPrinciple of material frame-indifference, definition, no evidence levelPrinciple of material…definition
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. Cauchy–Green deformation tensor concept
  3. Constitutive equation concept
  4. Deformation gradient concept
  5. Identity matrix object
  6. Matrix definition
  7. Orthogonal matrix concept
  8. Positive definite matrix concept
  9. Square root of a matrix concept
  10. Vector space definition
  11. Square matrix definition
  12. Transpose definition
  13. Linear map definition
  14. Conjugate transpose definition
  15. Multilinear map definition
  16. Orthogonal matrix definition
  17. Hermitian matrix definition
  18. Alternating multilinear map definition
  19. Orthogonal group object
  20. Unitary matrix definition
  21. Positive semidefinite matrix definition
  22. Determinant definition
  23. Square root of a positive semidefinite matrix definition
  24. Special orthogonal group object
  25. Polar decomposition statement
  26. Polar decomposition of the deformation gradient statement
  27. Principle of material frame-indifference definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Objective_stress_rate
About this page
TagAW9FV2ARXF
URIm2://AW9FV2ARXF
Kinddefinition
WritingW0 stub
MSC74A20
DefinesPrinciple of material frame-indifference
UsesConstitutive equation, Cauchy stress tensor, Deformation gradient, Polar decomposition of the deformation gradient, Orthogonal group, Special orthogonal group
Changed3 November 2026

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