definition 30.24 The elasticity tensor

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:766 · p. 1015

ground object -- no derivation owed

Rests on

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 7.20 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 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 30.37 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 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 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 ¶ ↺

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`	→	Linear strain and infinitesimal rotation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:778
`depends_on`	→	Cauchy: the stress tensor exists	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:778
`depends_on`	←	Voigt notation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1240
`depends_on`	←	Hooke's law	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:868
`depends_on`	←	Twenty-one constants	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:788
`depends_on`	←	The Cauchy relations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1271
`depends_on`	←	Anharmonicity: what the quadratic energy also suppresses	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2776
`depends_on`	←	Three constants for a cubic crystal, two for an isotropic solid	declared	appendices/A-long-proofs.tex:32876