13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- i.e. by taking the function (6) and mapping the region of the physical plane (z=x+iy plane) onto the interior of the unit circle in the plane, the solution in closed form is found[9-11] !! "( ) = Hp 2# 2 i 1 1- " ln $"1 ( )- 1+" 1- " ( ) 1+" 2 ( ) ln $"" ( ) # $ % &% " " 2 1+" 2 ( ) ln 1+$2 ( )+ i 2 1+" 2 ( ) ln $"i $+i ' ( ) * + , - . % /%$ "a $"a (7) in which ! !a = !e !a H +2i 1!e !!a H 2!e !!a H ! !a = !e !a H !2i 1!e !!a H 2!e !!a H " # $ $ % $ $ (8) and (9) The corresponding stress intensity factor is determined as (10) The initiation of fault growth induced by stress drop ( - ) (the cohesive stress) is interpreted by the boundary condition as follows: (11) Boundary value problem (11) of eq. (1) can be solved similarly as above. And the corresponding stress intensity factor is determined as (12) Because the stresses over the breakdown zone are refer to (11), it means that there is no stress singularity at the fault tip, i.e. the total stress intensity factor must vanish: K total static =K !!! f static +K !b!! f static =0 (13) Substituting (5) and (12) into (13) determines the size R of the breakdown zone. Actually, R/ a 1; this and (13) offer a very simple expression for R. For explicitly, we here
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