theorem 30.18 Cauchy: the stress tensor exists

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:551 · p. 1013

Rests on

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 7.20 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 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 axiom 7.1 Completeness of $\R$ ¶ ↺
- 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 ¶

Supports

depends_on definition 30.24 The elasticity tensor ¶
- depends_on definition 30.37 Voigt notation ¶
  - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶
- depends_on phenomenon 30.28 Hooke's law ¶
  - depends_on phenomenon 30.32 Poisson contraction ¶
    - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on definition 30.63 Flexural rigidity ¶
      - 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 remark 30.36 How the moduli are actually measured ¶
  - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
- depends_on proposition 30.25 Twenty-one constants ¶
  - depends_on definition 30.37 Voigt notation ¶ ↺
  - depends_on proposition 30.38 The Cauchy relations ¶
    - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶ ↺
  - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
- depends_on proposition 30.38 The Cauchy relations ¶ ↺
- depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶ ↺
- depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶ ↺
depends_on definition 31.1 Fluid ¶
- depends_on definition 31.21 Ideal fluid ¶
  - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on lemma A.712 The force is a far-field integral ¶
    - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on remark 31.89 The Mach number and the regimes it separates ¶
      - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on proposition A.797 The outer limit loses a boundary condition ¶
    - depends_on proposition 31.80 Rayleigh's circulation criterion ¶
      - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶
    - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
    - depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
    - depends_on theorem 31.24 Bernoulli ¶
      - depends_on corollary A.799 The impressed pressure gradient ¶
      - depends_on lemma 31.67 Blasius' force formula ¶
      - depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
      - depends_on theorem A.707 D'Alembert's paradox ¶
      - depends_on theorem 31.74 The interface dispersion relation ¶
    - depends_on theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
      - depends_on remark 31.66 Only three things make circulation ¶
      - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
    - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on definition A.741 The statistical setting ¶
      - depends_on definition A.760 The convecting layer ¶
      - depends_on definition 31.43 Creeping flow ¶
      - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
      - depends_on phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on proposition A.761 The Boussinesq equations ¶
      - depends_on proposition 31.40 Circular Couette flow ¶
      - depends_on proposition 31.39 Plane Couette flow ¶
      - depends_on remark 31.48 What is actually proved ¶
      - depends_on remark 31.38 Whose equations, and the Stokes hypothesis ¶
      - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
      - … 3 more
    - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
      - depends_on phenomenon 31.93 The Mach cone ¶
      - depends_on remark 31.92 Entropy selects the physical branch ¶
    - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
- depends_on definition 31.2 Knudsen number ¶
  - depends_on remark 31.3 When the continuum picture is allowed ¶
- depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
  - depends_on remark 31.35 Non-Newtonian fluids are out of scope ¶
  - depends_on remark 31.33 Reading the two viscosities ¶
  - depends_on theorem 31.36 The Newtonian constitutive relation ¶
    - depends_on theorem A.718 Stokes' drag law with its coefficient ¶
    - depends_on theorem 31.37 The Navier–Stokes equations ¶ ↺
- depends_on proposition 31.16 Hydrostatic balance ¶
  - depends_on example 31.19 The barometric distribution ¶
  - depends_on phenomenon 49.8 The white-dwarf luminosity function has an edge ¶
  - depends_on phenomenon 31.18 Archimedes' principle ¶
    - depends_on example 31.46 Terminal velocity, and Millikan's droplets ¶
  - depends_on proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere ¶
    - depends_on phenomenon 52.12 The mass–luminosity relation ¶
      - depends_on phenomenon 52.17 The Hertzsprung–Russell diagram ¶
    - depends_on proposition 49.3 Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter ¶
      - depends_on theorem 49.7 The Chandrasekhar mass ¶
    - depends_on theorem 52.2 Virial theorem for a self-gravitating sphere ¶
      - depends_on phenomenon 52.4 The Sun has shone for billions of years ¶
      - depends_on proposition 52.18 Schönberg–Chandrasekhar limit ¶
depends_on proposition 30.19 The stress tensor is symmetric ¶
- depends_on definition 30.20 Pressure and deviatoric stress ¶
  - depends_on definition 30.31 The engineering moduli ¶
    - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on definition A.698 The Hertz contact problem ¶
      - 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 proposition 30.51 Moment and curvature ¶
      - depends_on definition A.680 Kirchhoff kinematics ¶
      - depends_on definition 30.63 Flexural rigidity ¶ ↺
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on theorem 30.56 The beam equation ¶
    - 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 definition 30.79 Maxwell viscoelasticity ¶
    - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
  - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
    - depends_on phenomenon 30.73 Yield and brittle fracture ¶
    - depends_on remark A.705 The dangerous stress is below the surface ¶
    - depends_on remark 30.75 Two criteria, one uncited ¶
  - depends_on remark 30.75 Two criteria, one uncited ¶ ↺
depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶ ↺
depends_on remark 30.72 Stress concentration ¶
- depends_on proposition 30.76 Griffith's criterion ¶
  - depends_on example 30.77 Why glass is weak, and thin glass is not ¶
  - depends_on phenomenon 30.73 Yield and brittle fracture ¶ ↺
depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶ ↺
depends_on theorem 30.21 Cauchy's equation of motion ¶
- depends_on definition 30.22 Boundary conditions of elastostatics ¶
  - 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 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 ¶
- depends_on proposition 30.23 Virtual work; the weak form ¶ ↺
- depends_on proposition 31.16 Hydrostatic balance ¶ ↺
- depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶ ↺
- depends_on theorem 30.40 Navier–Cauchy equations ¶
  - depends_on phenomenon 30.43 Two bulk wave speeds ¶
    - depends_on phenomenon 30.47 The three seismic phases ¶ ↺
    - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
  - depends_on proposition 30.41 Energy balance and its flux ¶
    - depends_on theorem 30.42 Uniqueness in elastodynamics ¶
  - depends_on proposition 30.50 Love waves need a slow surface layer ¶ ↺
  - depends_on proposition 30.48 Rayleigh's secular equation ¶ ↺
  - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶ ↺
- depends_on theorem 31.22 Euler's equations of motion ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Traction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:560
`depends_on`	→	Tensor under orthogonal transformations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:560
`depends_on`	←	The elasticity tensor	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:778
`depends_on`	←	Fluid	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:60
`depends_on`	←	The stress tensor is symmetric	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:605
`depends_on`	←	The maximum shear stress, and its indifference to pressure	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2629
`depends_on`	←	Stress concentration	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2577
`depends_on`	←	Boussinesq's point-force solution, quoted	declared	appendices/A-long-proofs.tex:33755
`depends_on`	←	Cauchy's equation of motion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:668
`proves`	←	ch:13-continuum-elasticity@proof-7	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:563