Math 2.0

Mechanics of deformable solids Definition

Ball–James theory of microstructure

Builds on Martensitic transformationSpecial orthogonal groupYoung measureRank-one convexity

Definition

Let Ω⊂ℝ3\Omega \subset \R^3 be a bounded domain, the reference configuration of a crystal in the austenite phase. Let φ(F,θ)\varphi(F, \theta) be a free energy density, defined for 3×33 \times 3 matrices FF with det⁡F>0\det F > 0 and temperatures θ\theta. Let SO(3)SO(3) be the special orthogonal group and let 𝒫⊂SO(3)\mathcal{P} \subset SO(3) be the point group of the austenite lattice. The Ball–James theory of a martensitic transformation assumes the following.

  1. Frame indifference: φ(RF,θ)=φ(F,θ)\varphi(RF, \theta) = \varphi(F, \theta) for all R∈SO(3)R \in SO(3).
  2. Material symmetry: φ(FQ,θ)=φ(F,θ)\varphi(FQ, \theta) = \varphi(F, \theta) for all Q∈𝒫Q \in \mathcal{P}.
  3. Energy wells: below the transformation temperature, the minimum of φ(⋅,θ)\varphi(\cdot, \theta) is attained exactly on K=SO(3)U1∪⋯∪SO(3)UNK = SO(3)U_1 \cup \cdots \cup SO(3)U_N. Here U1U_1 is a symmetric positive definite matrix, and U1,…,UNU_1, \ldots, U_N are the distinct matrices QU1QTQ U_1 Q^T with Q∈𝒫Q \in \mathcal{P}. These matrices are the variants of martensite.

Let yky_k be a minimising sequence of the energy ∫Ωφ(∇y,θ) dx\int_\Omega \varphi(\nabla y, \theta)\, dx under given boundary conditions. A microstructure is described by the gradient Young measure (νx)x∈Ω(\nu_x)_{x \in \Omega} generated by ∇yk\nabla y_k, which is supported in KK.

Two matrices AA and BB are compatible if A−B=a⊗nA - B = a \otimes n for some vectors aa and n≠0n \neq 0, that is, if A−BA - B has rank at most one. Two variants UiU_i and UjU_j form a twin if RUj−Ui=a⊗nR U_j - U_i = a \otimes n for some R∈SO(3)R \in SO(3), a∈ℝ3a \in \R^3 and unit vector nn.

Remarks

Example: a simple laminate. Let RUj−Ui=a⊗nR U_j - U_i = a \otimes n and 0<λ<10 < \lambda < 1. Take deformations whose gradients alternate between UiU_i and RUjR U_j in layers normal to nn, with volume fractions 1−λ1 - \lambda and λ\lambda. They are continuous, because the gradients jump only across planes normal to nn. As the layers get finer, these deformations approach the affine map with gradient (1−λ)Ui+λRUj(1 - \lambda) U_i + \lambda R U_j, while their energy stays at the minimum.

Non-example: two distinct matrices in the same well, A=U1A = U_1 and B=RU1B = R U_1 with R∈SO(3)R \in SO(3), R≠IR \neq I, are never rank-one connected, so no laminate can be made from one variant alone.

The compatibility condition A−B=a⊗nA - B = a \otimes n is the Hadamard jump condition for a continuous deformation whose gradient jumps across a plane with normal nn. It is linked to rank-one convexity of the energy.

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Identity matrix, object, no evidence levelIdentity matrixobjectMartensitic transformation, concept, no evidence levelMartensitic transform…conceptMatrix, definition, no evidence levelMatrixdefinitionRank-one convexity, concept, no evidence levelRank-one convexityconceptVector space, definition, no evidence levelVector spacedefinitionYoung measure, concept, no evidence levelYoung measureconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectBall–James theory of microstructure, definition, no evidence levelBall–James theory of …definition
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  1. Identity matrix object
  2. Martensitic transformation concept
  3. Matrix definition
  4. Rank-one convexity concept
  5. Vector space definition
  6. Young measure concept
  7. Square matrix definition
  8. Transpose definition
  9. Linear map definition
  10. Orthogonal matrix definition
  11. Multilinear map definition
  12. Orthogonal group object
  13. Alternating multilinear map definition
  14. Determinant definition
  15. Special orthogonal group object
  16. Ball–James theory of microstructure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Diffusionless_transformation
About this page
TagE0RYNA6FKY
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Kinddefinition
WritingW0 stub
MSC74N15, 74N05, 49J45
DefinesBall–James theory of microstructure
UsesMartensitic transformation, Special orthogonal group, Young measure, Rank-one convexity
Changed3 November 2026

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