proposition 30.30 Stability bounds

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:921 · p. 1017

Rests on

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 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 ¶

Supports

depends_on phenomenon 30.43 Two bulk wave speeds ¶
- depends_on phenomenon 30.47 The three seismic phases ¶
- depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
depends_on phenomenon 30.32 Poisson contraction ¶
- depends_on proposition 30.33 Conversion among the moduli ¶
  - 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 theorem A.688 The Kirchhoff plate equation ¶
    - depends_on proposition A.682 The plane-stress law ¶
      - depends_on definition A.689 Edge resultants ¶
      - 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 ¶
depends_on proposition 30.34 Minimum of the potential energy ¶
- depends_on example 30.35 From single crystals to the isotropic aggregate ¶ ↺
depends_on theorem 30.42 Uniqueness in elastodynamics ¶

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`	→	eq:elast-energy-density	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:928
`depends_on`	→	Sylvester's law of inertia	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:928
`depends_on`	←	Two bulk wave speeds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1461
`depends_on`	←	Poisson contraction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:986
`depends_on`	←	Minimum of the potential energy	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1073
`depends_on`	←	Uniqueness in elastodynamics	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1416
`proves`	←	ch:13-continuum-elasticity@proof-13	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:931