example 30.35 From single crystals to the isotropic aggregate

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1153 · p. 1019

example -- no derivation owed by its kind

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depends_on proposition 30.34 Minimum of the potential energy ¶
- depends_on proposition 30.30 Stability bounds ¶
  - depends_on equation 30.23 eq:elast-energy-density ¶
  - depends_on theorem 5.116 Sylvester's law of inertia ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-53 ¶
  - proves proof ch:13-continuum-elasticity@proof-13 ¶
- depends_on proposition 30.23 Virtual work; the weak form ¶
  - depends_on definition 30.22 Boundary conditions of elastostatics ¶
    - depends_on definition 10.24 Well-posed problem ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on theorem 30.21 Cauchy's equation of motion ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - proves proof ch:13-continuum-elasticity@proof-9 ¶
  - depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-10 ¶
- proves proof ch:13-continuum-elasticity@proof-16 ¶
depends_on proposition 30.33 Conversion among the moduli ¶
- 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 theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
      - 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 definition 5.41 Adjoint ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-31 ¶
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
- depends_on phenomenon 30.32 Poisson contraction ¶
  - depends_on phenomenon 30.28 Hooke's law ¶
    - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
    - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
      - proves proof ch:13-continuum-elasticity@prooflink-1 ¶
    - proves proof ch:13-continuum-elasticity@proof-12 ¶
  - depends_on proposition 30.30 Stability bounds ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-14 ¶
- proves proof ch:13-continuum-elasticity@proof-15 ¶

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type	direction	node	provenance	where
`depends_on`	→	Minimum of the potential energy	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1191
`depends_on`	→	Conversion among the moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1191