theorem A.688 The Kirchhoff plate equation

open in the book · appendices/A-long-proofs.tex:33429 · p. 3127

Rests on

depends_on lemma A.687 Kinetic energy ¶
- depends_on definition A.680 Kirchhoff kinematics ¶
  - 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 proposition 30.51 Moment and curvature ¶
    - depends_on definition 30.31 The engineering moduli ¶
      - depends_on definition 30.20 Pressure and deviatoric stress ¶
      - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
    - depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
    - proves proof ch:13-continuum-elasticity@proof-27 ¶
- proves proof app:A-long-proofs@proof-408 ¶
depends_on lemma 16.18 Fundamental lemma ¶
- depends_on definition 16.15 Variation; the first variation ¶
  - depends_on definition 16.11 Admissible class; functional ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on equation 7.177 eq:ana-functional-derivative ¶
- depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
- proves proof ch:14-calculus-of-variations@proof-8 ¶
depends_on proposition A.684 The plate energy ¶
- depends_on definition 30.63 Flexural rigidity ¶
  - depends_on proposition 30.33 Conversion among the moduli ¶
    - depends_on definition 30.31 The engineering moduli ¶ ↺
    - depends_on phenomenon 30.32 Poisson contraction ¶
      - depends_on phenomenon 30.28 Hooke's law ¶
      - depends_on proposition 30.30 Stability bounds ¶
      - proves proof ch:13-continuum-elasticity@proof-14 ¶
    - proves proof ch:13-continuum-elasticity@proof-15 ¶
  - depends_on proposition 30.51 Moment and curvature ¶ ↺
- depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶
  - depends_on definition A.680 Kirchhoff kinematics ¶ ↺
  - depends_on definition 30.6 Linear strain and infinitesimal rotation ¶ ↺
  - proves proof app:A-long-proofs@proof-404 ¶
- depends_on proposition A.682 The plane-stress law ¶
  - depends_on definition 30.63 Flexural rigidity ¶ ↺
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
  - depends_on proposition 30.33 Conversion among the moduli ¶ ↺
  - proves proof app:A-long-proofs@proof-405 ¶
- proves proof app:A-long-proofs@proof-406 ¶
proves proof app:A-long-proofs@proof-409 ¶

Supports

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 ¶

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`	→	Kinetic energy	declared	appendices/A-long-proofs.tex:33446
`depends_on`	→	Fundamental lemma	declared	appendices/A-long-proofs.tex:33446
`depends_on`	→	The plate energy	declared	appendices/A-long-proofs.tex:33446
`depends_on`	←	Poisson's count, and why it cannot be right	declared	appendices/A-long-proofs.tex:33563
`depends_on`	←	Kirchhoff's edge conditions and the corner force	declared	appendices/A-long-proofs.tex:33593
`proves`	←	app:A-long-proofs@proof-409	declared	appendices/A-long-proofs.tex:33450