Table 1. Stress and strain equations... | Emerald Publishing

Table 1.

Stress and strain equations (TSHC and triaxial)

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	$τ_{z ϑ} = \frac{3 M_{T}}{2 π (r_{o}^{3} - r_{i}^{3})}$	$γ_{z ϑ} = \frac{2 Θ (r_{o}^{3} - r_{i}^{3})}{3 H (r_{o}^{2} - r_{i}^{2})}$
Major principal	$σ_{1} = \frac{σ_{z} + σ_{ϑ}}{2} + \sqrt{{(\frac{σ_{z} - σ_{ϑ}}{2})}^{2} + {(τ_{z ϑ})}^{2}}$	$ε_{1} = \frac{ε_{z} + ε_{ϑ}}{2} + \sqrt{{(\frac{ε_{z} - ε_{ϑ}}{2})}^{2} + {(\frac{γ_{z ϑ}}{2})}^{2}}$
Intermediate principal	σ₂ = σ_r	ε₂ = ε_r
Minor principal	$σ_{3} = \frac{σ_{z} + σ_{ϑ}}{2} - \sqrt{{(\frac{σ_{z} - σ_{ϑ}}{2})}^{2} + {(τ_{z ϑ})}^{2}}$	$ε_{3} = \frac{ε_{z} + ε_{θ}}{2} - \sqrt{{(\frac{ε_{z} - ε_{ϑ}}{2})}^{2} + {(\frac{γ_{z ϑ}}{2})}^{2}}$
Deviatoric stress/strain	$q = \frac{1}{\sqrt{2}} \sqrt{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}}$	$ε_{q} = \frac{\sqrt{2}}{3} \sqrt{{(ε_{1} - ε_{2})}^{2} + {(ε_{2} - ε_{3})}^{2} + {(ε_{3} - ε_{1})}^{2}}$
Mean effective stress	$p' = \frac{{σ′}_{1} + {σ′}_{2} + {σ′}_{3}}{3} = \frac{{σ′}_{z} + {σ′}_{r} + {σ′}_{ϑ}}{3}$	—
Intermediate principal stress ratio	$b = \frac{{σ′}_{2} - {σ′}_{3}}{{σ′}_{1} - {σ′}_{3}}$	—
Lode angle	$θ = \frac{180}{π} [\frac{π}{6} - {tan}^{- 1} \sqrt{3} \frac{{σ′}_{2} - {σ′}_{3}}{2 {σ′}_{1} - {σ′}_{2} - {σ′}_{3}}]$ $θ = \frac{180}{π} \tan^{- 1} \frac{1 - 2 b}{\sqrt{3}}$	—
Direction of the major principal stress relative to the vertical axis	$α = \frac{180}{π} \frac{1}{2} \tan^{- 1} (\frac{2 τ_{z ϑ}}{σ_{z} - σ_{ϑ}}) if σ_{z} > σ_{ϑ}$ α = 45° if σ_z = σ_ϑ $α = 90 - \frac{180}{π} \frac{1}{2} \tan^{- 1} (\frac{2 τ_{z ϑ}}{σ_{ϑ} - σ_{z}}) if σ_{z} < σ_{ϑ}$	—
Stress ratio	$η = \frac{q}{p'}$	—
Volumetric strain	—	$ε_{v} = - \frac{Δ V}{V}$
Dilatancy		$D = \frac{{\dot{ε}}_{v}}{{\dot{ε}}_{q}}$