remark A.676 The third cubic constant is measured, and it is not small

Rests on

• depends_on definition 30.31 The engineering moduli ¶
  • depends_on definition 30.20 Pressure and deviatoric stress ¶
    • depends_on proposition 30.19 The stress tensor is symmetric ¶
      • depends_on definition 30.17 Traction ¶
        • depends_on definition 7.126 Line and surface integrals ¶
        • depends_on definition 30.2 Representative volume element; continuum field ¶
      • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
        • depends_on definition 30.17 Traction ¶ ↺
        • depends_on definition 13.5 Tensor under orthogonal transformations ¶
        • proves proof ch:13-continuum-elasticity@proof-7 ¶
      • proves proof ch:13-continuum-elasticity@proof-8 ¶
    • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
        • depends_on theorem 8.19 Fundamental theorem of algebra ¶
        • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
        • proves proof ch:03-linear-algebra-representations@proof-27 ¶
      • depends_on definition 5.41 Adjoint ¶
        • depends_on definition 5.18 Inner product ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on theorem 5.40 Rank–nullity ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on lemma 5.38 Exchange and completion ¶
        • proves proof ch:03-linear-algebra-representations@proof-13 ¶
      • proves proof ch:03-linear-algebra-representations@proof-31 ¶
  • depends_on equation 30.21 eq:elast-hooke-isotropic ¶
• depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  • depends_on definition 30.24 The elasticity tensor ¶
    • depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
      • depends_on definition 13.91 Symmetric and antisymmetric parts ¶
      • depends_on equation 30.2 eq:elast-displacement ¶
    • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
  • depends_on proposition A.673 Cubic tensors of rank four ¶
    • depends_on definition A.672 The octahedral rotation group ¶
      • depends_on definition 14.26 The rotation group ¶
      • depends_on definition 5.131 Isotropic Cartesian tensor ¶
        • depends_on definition 5.32 Levi–Civita symbol; cross product ¶
        • depends_on equation 5.162 eq:rep-special-orthogonal ¶
    • depends_on lemma A.671 The three products are independent ¶
      • depends_on equation 5.168 eq:rep-isotropic-4 ¶
      • depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
        • depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
        • depends_on lemma 5.132 Parity constraint ¶
        • depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
        • proves proof ch:03-linear-algebra-representations@proof-58 ¶
      • proves proof app:A-long-proofs@proof-401 ¶
    • proves proof app:A-long-proofs@proof-402 ¶
  • depends_on proposition 30.25 Twenty-one constants ¶
    • depends_on definition 30.24 The elasticity tensor ¶ ↺
    • depends_on proposition 7.105 Clairaut–Schwarz ¶
      • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on equation 7.7 eq:ana-limit-left ¶
        • depends_on equation 7.5 eq:ana-limit-right ¶
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-64 ¶
    • proves proof ch:13-continuum-elasticity@proof-11 ¶
  • proves proof app:A-long-proofs@proof-403 ¶

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