Tunnel Face Stability & New CPT Applications - Geo-Engineering

....Appendix BBasic Equations of ElasticityGiven a polar coordinate system defined in figure B.1, with stresses σ ij and displacements u i ,the equations of equilibrium in radial and tangential directions are∂σ rr+ 1 ∂r r∂σ rθ+ 1 ∂r r∂σ rθ∂θ + σ rr − σ θθ= 0r(B.1)∂σ θθ∂θ + 2σ rθ= 0r(B.2)The deformations in radial and tangential directions are defined asǫ rr = ∂u r∂rǫ θθ = u rr + 1 rǫ rθ = 1 2∂u θ∂θ( ∂uθ∂r + 1 r∂u r∂θ − u )θr(B.3)(B.4)(B.5)and the volume strain ase = ǫ rr + ǫ θθ = ∂u r∂r + u rr + 1 ∂u θr ∂θThe stresses can now be expressed asσ rr = µ ((m − 1)e + 2ǫ rr )σ θθ = µ ((m − 1)e + 2ǫ θθ )σ rθ = 2µǫ rθ(B.6)(B.7)(B.8)(B.9). . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . ..θ . .r... ... .. ... ......x. .zFigure B.1: Element in polar coordinates191