phenomenon 30.28 Hooke's law

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:854 · p. 1016

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on definition 30.24 The elasticity tensor ¶
- depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
  - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
  - depends_on equation 30.2 eq:elast-displacement ¶
- 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 ¶
depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
- depends_on definition 13.5 Tensor under orthogonal transformations ¶ ↺
- depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on equation 5.19 eq:lin-leibniz-det ¶
    - proves proof ch:03-linear-algebra-representations@proof-8 ¶
  - depends_on lemma 5.132 Parity constraint ¶
    - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
      - depends_on equation 5.162 eq:rep-special-orthogonal ¶
    - depends_on equation 5.162 eq:rep-special-orthogonal ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-57 ¶
  - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - 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-56 ¶
  - proves proof ch:03-linear-algebra-representations@proof-58 ¶
- proves proof ch:13-continuum-elasticity@prooflink-1 ¶
proves proof ch:13-continuum-elasticity@proof-12 ¶

Supports

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 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 remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶

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
`cites`	→	Lectures de Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:858
`cites`	→	Leçons sur la théorie mathématique de l'élasticité des corps solides	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:865
`cites`	→	Theory of Elasticity	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:865
`depends_on`	→	The elasticity tensor	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:868
`depends_on`	→	Isotropic Cartesian tensors of rank four	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:868
`depends_on`	←	Poisson contraction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:986
`depends_on`	←	Anharmonicity: what the quadratic energy also suppresses	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2776
`proves`	←	ch:13-continuum-elasticity@proof-12	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:871