Table 2. Torsional shear hollow... | Emerald Publishing

Table 2.

Torsional shear hollow cylinder: stresses, strains and non-uniformities criteria

Parameter definition	Stresses	Strains
Vertical	$σ_{z} = \frac{W}{π (r_{o}^{2} - r_{i}^{2})} + \frac{p_{o} r_{o}^{2} - p_{i} r_{i}^{2}}{r_{o}^{2} - r_{i}^{2}}$	$ε_{z} = - \frac{Δ H}{H}$
Radial	$σ_{r} = \frac{p_{o} r_{o} + p_{i} r_{i}}{r_{o} + r_{i}}$	$ε_{r} = - \frac{u_{o} - u_{i}}{r_{o} - r_{i}}$
Circumferential	$σ_{θ} = \frac{p_{o} r_{o} - p_{i} r_{i}}{r_{o} - r_{i}}$	$ε_{θ} = - \frac{u_{o} + u_{i}}{r_{o} + r_{i}}$
Shear	$τ_{z θ} = \frac{3 M_{T}}{2 π (r_{o}^{3} - r_{i}^{3})}$	$γ_{θ z} = \frac{2 θ (r_{o}^{3} - r_{i}^{3})}{3 H (r_{o}^{2} - r_{i}^{2})}$
Major principal	$σ_{1} = \frac{σ_{z} + σ_{θ}}{2} + \sqrt{{(\frac{σ_{z} - σ_{θ}}{2})}^{2} + {(τ_{z θ})}^{2}}$	$ε_{1} = \frac{ε_{z} + ε_{θ}}{2} + \sqrt{{(\frac{ε_{z} - ε_{θ}}{2})}^{2} + {(\frac{γ_{θ z}}{2})}^{2}}$
Intermediate principal	$σ_{2} = σ_{r}$	$ε_{2} = ε_{r}$
Minor principal	$σ_{3} = \frac{σ_{z} + σ_{θ}}{2} - \sqrt{{(\frac{σ_{z} - σ_{θ}}{2})}^{2} + {(τ_{z θ})}^{2}}$	$ε_{3} = \frac{ε_{z} + ε_{θ}}{2} - \sqrt{{(\frac{ε_{z} - ε_{θ}}{2})}^{2} + {(\frac{γ_{θ z}}{2})}^{2}}$
Deviatoric stress triaxial	$q_{13} = σ_{1} - σ_{3}$
Deviatoric stress octahedral/Octahedral shear strain	$q = \frac{1}{\sqrt{2}} \sqrt{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}}$	$γ_{oct} = \frac{2}{3} \sqrt{{(ε_{1} - ε_{2})}^{2} + {(ε_{2} - ε_{3})}^{2} + {(ε_{3} - ε_{1})}^{2}}$
Mean effective stress	$p' = \frac{{σ'}_{1} + {σ'}_{2} + {σ'}_{3}}{3} = \frac{{σ'}_{z} + {σ'}_{r} + {σ'}_{θ}}{3}$
Intermediate principal stress ratio	$b = \frac{{σ'}_{2} - {σ'}_{3}}{{σ'}_{1} - {σ'}_{3}}$
Principal stress angle	$α = \frac{1}{2} \arctan (\frac{2 τ_{z θ}}{σ_{z} - σ_{θ}})$
Stress ratio	$η = \frac{q}{p^{'}}$
Volumetric strain		$ε_{vol} = - \frac{Δ V}{V}$