proposition 30.51 Moment and curvature

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1879 · p. 1028

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 equation 30.21 eq:elast-hooke-isotropic ¶ ↺
proves proof ch:13-continuum-elasticity@proof-27 ¶

Supports

depends_on definition A.680 Kirchhoff kinematics ¶
- depends_on lemma A.687 Kinetic energy ¶
  - 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 lemma A.681 Strains under the Kirchhoff hypothesis ¶
  - 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 definition 30.63 Flexural rigidity ¶
- depends_on proposition A.684 The plate energy ¶ ↺
- 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 phenomenon 30.52 Euler buckling ¶
- depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
- depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
- depends_on remark 30.55 Beyond the critical load: the elastica ¶
- depends_on remark 30.68 Shells carry load in a different way ¶ ↺
depends_on theorem 30.56 The beam equation ¶
- depends_on example 30.58 A steel ruler ¶
- depends_on proposition 30.57 Bending waves are dispersive ¶
  - depends_on example 30.58 A steel ruler ¶ ↺
  - depends_on remark 30.59 Where Euler–Bernoulli fails ¶

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:1895
`depends_on`	→	eq:elast-hooke-isotropic	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1895
`depends_on`	←	Kirchhoff kinematics	declared	appendices/A-long-proofs.tex:33097
`depends_on`	←	Flexural rigidity	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2276
`depends_on`	←	Euler buckling	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1927
`depends_on`	←	The beam equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2054
`proves`	←	ch:13-continuum-elasticity@proof-27	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1898