definition 30.31 The engineering moduli

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:954 · p. 1017

ground object -- no derivation owed

Rests on

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 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.125 Multiple integral ¶
    - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
  - 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 ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - 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 8.18 Liouville ¶
      - proves proof ch:06-complex-analysis@proof-13 ¶
    - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - proves proof ch:03-linear-algebra-representations@proof-26 ¶
    - 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 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - 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 ¶

Supports

depends_on phenomenon 30.70 Hertzian contact ¶
- depends_on definition A.698 The Hertz contact problem ¶
  - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶
    - depends_on theorem A.702 Hertz's solution ¶
      - depends_on example A.703 The numbers of the chapter, recomputed ¶
      - depends_on remark A.704 $E^{*}$ in two roles, and one trap ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
  - depends_on proposition A.701 The two-body compliance rule ¶
    - depends_on theorem A.702 Hertz's solution ¶ ↺
  - depends_on remark A.699 The third condition is not decoration ¶
  - depends_on theorem A.702 Hertz's solution ¶ ↺
- depends_on example A.703 The numbers of the chapter, recomputed ¶ ↺
- depends_on example 30.71 Contact stresses are enormous ¶
depends_on phenomenon 30.60 Coulomb's law of torsion ¶
- depends_on example 30.61 A torsion fibre as a force sensor ¶
- depends_on phenomenon 34.3 The stiffness of a twisted wire is geometric ¶
- depends_on remark 30.62 Saint-Venant's theory: non-circular sections warp ¶
- tests experiment Coulomb: the torsion pendulum ¶
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 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 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 ¶
depends_on proposition 30.51 Moment and curvature ¶
- 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 lemma A.681 Strains under the Kirchhoff hypothesis ¶
    - depends_on proposition A.684 The plate energy ¶ ↺
- depends_on definition 30.63 Flexural rigidity ¶ ↺
- 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 ¶
depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
- depends_on definition A.698 The Hertz contact problem ¶ ↺
- depends_on lemma A.697 Surface displacement under a distributed pressure ¶
  - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶ ↺
  - depends_on proposition A.701 The two-body compliance rule ¶ ↺

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
`cites`	→	Mémoire sur l'équilibre et le mouvement des corps élastiques	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:962
`cites`	→	A Course of Lectures on Natural Philosophy and the Mechanical Arts	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:960
`depends_on`	→	Pressure and deviatoric stress	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:969
`depends_on`	→	eq:elast-hooke-isotropic	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:969
`depends_on`	←	Hertzian contact	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2477
`depends_on`	←	Coulomb's law of torsion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2186
`depends_on`	←	Conversion among the moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1036
`depends_on`	←	Moment and curvature	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1895
`depends_on`	←	The third cubic constant is measured, and it is not small	declared	appendices/A-long-proofs.tex:32962
`depends_on`	←	Boussinesq's point-force solution, quoted	declared	appendices/A-long-proofs.tex:33755