Potentially Useful Equations σ α υ ∆ ∆ TE T= − 1 σ σ σ σ σ σ σ σ σ σ ij = 11 12 13 21 22 23 31 32 33 VkdP dx i ij P j = η σ σ σ σ σ θ n * * * * * cos = + + − 1 3 1 3 2 2 2 σ σ σ θ s = − 1 3 2 2 * * sin ′= ′+ ′ − ′ − ′ ′ λ λ λ λ λ θ 3 1 3 1 2 2 2 cos ′= ′ − ′ ′ γ λ λ θ 3 1 2 2 sin tan tan tan ′= = θ θ λ λ θ 3 1 3 1 SSσ σ µs o n S= + * ˙ exp ε σ σ = − ( ) − CQ RTo n 1 3 ˙ ε σ σ = ( ) − ( ) C TD do n 1 3 UCrCr= − + 1 2 12 P gdz lith z = ∫ ρ 0 φ = + VV Vffs φ φ = − ( ) o az exp ∆ v final initial initial V VV= − e l l l f i i = − e= + ( ) − sin sin φ θ φ 1 Sl l fi = = λ λ = S2 ′= λ λ 1 sin sin cos 2 2 θ θ θ = cos cos 2 1 2 2 θ θ = + sin cos 2 1 2 2 θ θ = − U U E dx i o i ij j = + UUUUUUE E EE E EE E Edx dx dx o o o 1 2 3 1 2 3 11 12 13 21 22 23 31 32 33 1 2 3 = + σ σ σ σ m fP * = + + − ( ) 1 2 3 3 3 σ σ 1 3 * * = + C Ko K C S K o o = + − = 1 1 2 sin sin φ φ ;
