definition 30.6 Linear strain and infinitesimal rotation

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:189 · p. 1009

ground object -- no derivation owed

Rests on

depends_on definition 13.91 Symmetric and antisymmetric parts ¶
depends_on equation 30.2 eq:elast-displacement ¶

Supports

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 remark A.690 Poisson's count, and why it cannot be right ¶
    - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on remark A.692 The corner force is real, and you can feel it ¶
- depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶
  - depends_on proposition A.684 The plate energy ¶
    - depends_on lemma A.685 The Gaussian-curvature term is a null Lagrangian ¶
      - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶
    - depends_on theorem A.688 The Kirchhoff plate equation ¶ ↺
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 30.11 Green strain ¶
- depends_on proposition 30.12 The Green strain measures length exactly ¶
  - depends_on example 30.13 A rigid rotation forges a strain of $-\theta^{2}/2$ ¶
depends_on example 30.13 A rigid rotation forges a strain of $-\theta^{2}/2$ ¶ ↺
depends_on example 30.10 Simple shear and pure shear are the same strain ¶
depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶ ↺
depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
- depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
depends_on proposition 30.8 The trace is the fractional volume change ¶
depends_on proposition 30.9 Principal strains ¶
- depends_on example 30.10 Simple shear and pure shear are the same strain ¶ ↺
depends_on proposition 30.7 Strain is a Cartesian tensor of rank two ¶

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`	→	Symmetric and antisymmetric parts	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:205
`depends_on`	→	eq:elast-displacement	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:205
`depends_on`	←	Kirchhoff kinematics	declared	appendices/A-long-proofs.tex:33097
`depends_on`	←	The elasticity tensor	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:778
`depends_on`	←	Green strain	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:331
`depends_on`	←	A rigid rotation forges a strain of $-\theta^{2}/2$	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:384
`depends_on`	←	Simple shear and pure shear are the same strain	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:313
`depends_on`	←	Strains under the Kirchhoff hypothesis	declared	appendices/A-long-proofs.tex:33112
`depends_on`	←	Saint-Venant compatibility is necessary	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:402
`depends_on`	←	The trace is the fractional volume change	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:250
`depends_on`	←	Principal strains	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:273
`depends_on`	←	Strain is a Cartesian tensor of rank two	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:215