13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 0G is the activation energy and mτ is the effective stress at 0K . From Eqs. (2) and (3), we have ( ) 1 1 0 0 ln 1 q p m kT G γ γ τ τ ∗ ⎧ ⎫ ⎡ ⎤ ⎪ ⎪ = ⎨ − ⎬ ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ & & . (4) For given 0G , 0γ&, p and q , this equation expresses the temperature and strain-rate dependence of the effective stress τ∗ . Now we calculate the effective slip stress mτ at 0K. According to the work of Nakatani et al.[23], the effective slip stress agrees with Rise’s theoretical prediction. We proceed to map conformally the infinite region outside the elliptical hole in the z x iy = + plane to the infinite region outside a circle of a radius R in the i ζ ξ η = + plane through a transformation function ( ) 1 2 c z ω ζ ζ ζ ⎛ ⎞ = = ⎜ + ⎟ ⎝ ⎠ (5) where 2 2 c a b = − and ( ) ( ) R a b a b = + − . The stress fields can be expressed as[ ] 2[ ( ) ()] 2[ ()/ () ( ) / ( )] y x z z σ σ ϕ ϕ ϕ ζ ω ζ ϕ ζ ω ζ ′ ′ ′ ′ ′ ′ + = + = + (6) [ ] 2 2 ( ) ( ) y x xy i z z z σ σ τ ϕ ψ ′′ ′ − + = + [ ] [ ]3 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) / ( ) ωζ ϕζωζ ϕζωζ ωζ ψζ ωζ ⎡ ⎤ ′′ ′ ′ ′′ ′ ′ ′ = − + ⎣ ⎦ (7) Using Muskhelishvili’s complex potential method [24], the complex potential functions in the i ζ ξ η = + plane can be easily calculated as 2 2 0 0 ( ) ln( ) ln( ) ln( / ) ln( / ) d d R R ϕζ γζζ γζζ γζ ζ γ ζ ζ = − − − − − + − 2 2 0 2 2 2 2 0 0 ( / ) ( / ) d d d a R a R R R γ γ ζ ζ ζ ζ ζ ζ + − − − (8) ( ) 2 ( / ) ( ) ( ) ( ) Rω ζ ψ ζ χ ζ ϕ ζ ω ζ ′ = − ′ (9) 2 2 0 0 ( ) ln( ) ln( ) ln( / ) ln( / ) d d R R χζγζζ γζζ γζ ζ γ ζ ζ = − − − − − + − 0 0 d d a a γ γ ζ ζ ζ ζ + − − − (10) where ( ) 1 4 1 b i μ γ π ν = − , ( ) 2 2 0 0 0 1 z z c c ζ = + − , 4 2 2 2 0 0 0 0 0 2 2 2 0 0 0 ( ) ( 1) ( 1) ( 1) R a R ζ ζ ζ ζ ζ ζ ζ + + = − − − , ( ) 2 2 1 d d d z z c c ζ = + − ,
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