definition 30.22 Boundary conditions of elastostatics

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:694 · p. 1015

ground object -- no derivation owed

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - 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 definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - 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 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 definition 13.5 Tensor under orthogonal transformations ¶
    - depends_on definition 13.4 Orthogonal coordinate transformation ¶
      - depends_on definition 13.3 Orthogonal coordinate system ¶
  - proves proof ch:13-continuum-elasticity@proof-7 ¶
- proves proof ch:13-continuum-elasticity@proof-9 ¶

Supports

depends_on proposition 30.50 Love waves need a slow surface layer ¶
depends_on proposition 30.48 Rayleigh's secular equation ¶
- depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
  - depends_on phenomenon 30.47 The three seismic phases ¶
depends_on proposition 30.23 Virtual work; the weak form ¶
- 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 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`	→	Well-posed problem	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:703
`depends_on`	→	Cauchy's equation of motion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:703
`depends_on`	←	Love waves need a slow surface layer	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1827
`depends_on`	←	Rayleigh's secular equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1680
`depends_on`	←	Virtual work; the weak form	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:719