proposition A.684 The plate energy

open in the book · appendices/A-long-proofs.tex:33246 · p. 3125

Rests on

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 definition 30.20 Pressure and deviatoric stress ¶
      - depends_on proposition 30.19 The stress tensor is symmetric ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - 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 lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - 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 ¶
      - proves proof ch:13-continuum-elasticity@proof-13 ¶
    - 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 definition 30.31 The engineering moduli ¶ ↺
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-27 ¶
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 ¶
    - 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 30.6 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 ¶

Supports

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 ¶

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`	→	Flexural rigidity	declared	appendices/A-long-proofs.tex:33260
`depends_on`	→	Strains under the Kirchhoff hypothesis	declared	appendices/A-long-proofs.tex:33260
`depends_on`	→	The plane-stress law	declared	appendices/A-long-proofs.tex:33260
`depends_on`	←	The Gaussian-curvature term is a null Lagrangian	declared	appendices/A-long-proofs.tex:33344
`depends_on`	←	The Kirchhoff plate equation	declared	appendices/A-long-proofs.tex:33446
`proves`	←	app:A-long-proofs@proof-406	declared	appendices/A-long-proofs.tex:33264