phenomenon 30.70 Hertzian contact

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:2461 · p. 1034

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

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 proposition 30.23 Virtual work; the weak form ¶
- depends_on definition 30.22 Boundary conditions of elastostatics ¶
  - depends_on definition 10.24 Well-posed problem ¶
    - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on theorem 30.21 Cauchy's equation of motion ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
    - proves proof ch:13-continuum-elasticity@proof-9 ¶
- depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
- proves proof ch:13-continuum-elasticity@proof-10 ¶
proves proof ch:13-continuum-elasticity@proof-34 ¶
proves proof ch:13-continuum-elasticity@prooflink-3 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`cites`	→	Ueber die Berührung fester elastischer Körper	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:2474
`depends_on`	→	The engineering moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2477
`depends_on`	→	Virtual work; the weak form	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2477
`depends_on`	←	The Hertz contact problem	declared	appendices/A-long-proofs.tex:33844
`depends_on`	←	The numbers of the chapter, recomputed	declared	appendices/A-long-proofs.tex:34148
`depends_on`	←	Contact stresses are enormous	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2554
`proves`	←	ch:13-continuum-elasticity@proof-34	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2480
`proves`	←	ch:13-continuum-elasticity@prooflink-3	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2512