proposition 30.74 The maximum shear stress, and its indifference to pressure

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:2618 · p. 1036

Rests on

depends_on definition 30.20 Pressure and deviatoric stress ¶
- 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 7.125 Multiple 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 ¶
    - 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 ¶
      - 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 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - 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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - proves proof ch:03-linear-algebra-representations@proof-13 ¶
  - proves proof ch:03-linear-algebra-representations@proof-31 ¶
depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
proves proof ch:13-continuum-elasticity@proof-36 ¶

Supports

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 ¶

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`	→	Mémoire sur l'écoulement des corps solides soumis à de fortes pressions	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:2628
`depends_on`	→	Pressure and deviatoric stress	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2629
`depends_on`	→	Cauchy: the stress tensor exists	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2629
`depends_on`	←	Yield and brittle fracture	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2599
`depends_on`	←	The dangerous stress is below the surface	declared	appendices/A-long-proofs.tex:34192
`depends_on`	←	Two criteria, one uncited	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2684
`proves`	←	ch:13-continuum-elasticity@proof-36	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2632