definition 30.20 Pressure and deviatoric stress

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:632 · p. 1014

ground object -- no derivation owed

Rests on

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 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 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 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 ¶
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 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.56 Endomorphism ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - proves proof ch:03-linear-algebra-representations@proof-26 ¶
  - 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 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-31 ¶

Supports

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 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 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 lemma A.685 The Gaussian-curvature term is a null Lagrangian ¶
      - depends_on theorem A.688 The Kirchhoff plate equation ¶
    - depends_on proposition A.682 The plane-stress law ¶
      - depends_on definition A.689 Edge resultants ¶
      - depends_on proposition A.684 The plate energy ¶ ↺
      - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶
    - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on phenomenon 30.65 Chladni figures ¶
    - 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 phenomenon 30.65 Chladni figures ¶ ↺
      - depends_on proposition 30.66 Chladni scaling ¶ ↺
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
  - 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 theorem A.688 The Kirchhoff plate equation ¶ ↺
    - depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶
      - depends_on proposition A.684 The plate energy ¶ ↺
  - 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 example 30.58 A steel ruler ¶ ↺
      - depends_on remark 30.59 Where Euler–Bernoulli fails ¶
- 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.

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`	→	The stress tensor is symmetric	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:644
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:644
`depends_on`	←	The engineering moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:969
`depends_on`	←	Maxwell viscoelasticity	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2792
`depends_on`	←	The maximum shear stress, and its indifference to pressure	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2629
`depends_on`	←	Two criteria, one uncited	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2684