proposition 30.19 The stress tensor is symmetric

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:601 · p. 1014

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 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 ¶
    - depends_on definition 13.3 Orthogonal coordinate system ¶
- proves proof ch:13-continuum-elasticity@proof-7 ¶
proves proof ch:13-continuum-elasticity@proof-8 ¶

Supports

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 lemma A.700 The displacement produced by an elliptic pressure ¶
      - depends_on proposition A.701 The two-body compliance rule ¶
      - 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 proposition A.682 The plane-stress law ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - 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 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 lemma A.681 Strains under the Kirchhoff hypothesis ¶
    - 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 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 ¶ ↺
- 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 ¶ ↺

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
`depends_on`	→	Traction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:605
`depends_on`	→	Cauchy: the stress tensor exists	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:605
`depends_on`	←	Pressure and deviatoric stress	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:644
`proves`	←	ch:13-continuum-elasticity@proof-8	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:608