proposition 30.33 Conversion among the moduli

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1022 · p. 1018

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 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 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 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 ¶
      - 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 ch:13-continuum-elasticity@prooflink-1 ¶
  - proves proof ch:13-continuum-elasticity@proof-12 ¶
- 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 ¶
- proves proof ch:13-continuum-elasticity@proof-14 ¶
proves proof ch:13-continuum-elasticity@proof-15 ¶

Supports

depends_on definition 30.63 Flexural rigidity ¶
- depends_on proposition A.684 The plate energy ¶
  - depends_on lemma A.685 The Gaussian-curvature term is a null Lagrangian ¶
    - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶
  - depends_on theorem A.688 The Kirchhoff plate equation ¶
    - depends_on remark A.690 Poisson's count, and why it cannot be right ¶
    - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on remark A.692 The corner force is real, and you can feel it ¶
- depends_on proposition A.682 The plane-stress law ¶
  - depends_on definition A.689 Edge resultants ¶
    - depends_on remark A.690 Poisson's count, and why it cannot be right ¶ ↺
    - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶ ↺
  - depends_on proposition A.684 The plate energy ¶ ↺
  - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶
- depends_on proposition 30.66 Chladni scaling ¶
  - depends_on phenomenon 30.65 Chladni figures ¶
- depends_on remark A.704 $E^{*}$ in two roles, and one trap ¶
- depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶ ↺
- depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶ ↺
- depends_on remark 30.68 Shells carry load in a different way ¶
- depends_on theorem 30.64 The Kirchhoff plate equation ¶
  - depends_on phenomenon 30.65 Chladni figures ¶ ↺
  - depends_on proposition 30.66 Chladni scaling ¶ ↺
  - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
depends_on example 30.35 From single crystals to the isotropic aggregate ¶
depends_on proposition A.682 The plane-stress law ¶ ↺
depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
- depends_on example 30.45 Elastic waves in steel, and the distance to a source ¶
depends_on remark 30.36 How the moduli are actually measured ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The engineering moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1036
`depends_on`	→	Poisson contraction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1036
`depends_on`	←	Flexural rigidity	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2276
`depends_on`	←	From single crystals to the isotropic aggregate	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1191
`depends_on`	←	The plane-stress law	declared	appendices/A-long-proofs.tex:33171
`depends_on`	←	The speed ratio is fixed by Poisson's ratio	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1529
`depends_on`	←	How the moduli are actually measured	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1224
`proves`	←	ch:13-continuum-elasticity@proof-15	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1039