proposition 30.23 Virtual work; the weak form

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:706 · p. 1015

Rests on

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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - 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 definition 7.98 Functions of class $C^{1}$ ¶ ↺
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - 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 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-9 ¶
depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
proves proof ch:13-continuum-elasticity@proof-10 ¶

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 proposition 30.34 Minimum of the potential energy ¶
- depends_on example 30.35 From single crystals to the isotropic aggregate ¶
depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶

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`	→	Boundary conditions of elastostatics	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:719
`depends_on`	→	Cauchy's equation of motion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:719
`depends_on`	←	Hertzian contact	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2477
`depends_on`	←	Minimum of the potential energy	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1073
`depends_on`	←	Saint-Venant's principle, and its exact status	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2458
`proves`	←	ch:13-continuum-elasticity@proof-10	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:722