13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) 2 22 2 2 2 2 Im Re ' Re ' Y x Y Z σ = − + (6) ( ) 2 12 2 2 2 2 Im ' Im ' Re x Y Z Z σ =− + + (7) The need for two complex potentials in the Westergaard approach appears coherent with the two complex potentials involved in the Kolosov-Muskhelishvili formalism. For the fracture configuration depicted in (Fig. 1), mode-I complex potentials have the following expressions: ( ) ( ) ( ) 1 22 1 22 2 2 1 , 2 z Z z Y z z a α σ σ ∞ ∞ − = = − (8) While for mode-II: ( ) ( ) 2 12 2 2 2 , 0 z Z z Y z z a σ∞ = = − (9) 2.2. Identification of Williams series coefficients In order to describe the stress-field in a planar cracked domain at the vicinity of the crack-tip, it is possible to use either Williams series (Eq. 1) or Sanford generalized Westergaard approach (Eq. 2-7). The stress state being unique, the two approaches must be equivalent. In [5], a procedure is described in order to transform the complex solution into the series one. Basically, complex potentials are expanded as closed-form convergent power series, these expression lead to complex stress series through (Eq. 2-7), and then, using the coordinates inter-relation i z a re θ = + , polar series may be retrieved. For the configuration depicted in (Fig. 1), mode-I coefficients are then: ( ) ( ) ( ) ( ) 1 1 22 2 1 1 1 3 2 2 2 1 2 ! , 0 2 ! 2 1 n n n n n a n n n a σ + ∞ + + − − = ≥ − (10) ( ) 1 22 2 1 4 a σ α ∞ − = (11) 1 0, ka otherwise = (12) And for mode-II : ( ) ( ) ( ) ( ) 2 12 2 1 1 1 3 2 2 2 1 2 ! , 0 2 ! 2 1 n n n n n a n n n a σ∞ + + − − = ≥ − (13) 2 0, ka otherwise = (14) If one considers the stress state along the radius associated with a given angle, it may be described by two polynomial series ofr : % ( ) %( ) %( ) , , , , 1 2 1 2 1 2 2 0 1 1 , m m ij m ij k m ij m ij k ij k k k k k k r a f r a f r r σ θ θ θ ∞ ∞ − + + = = = + ∑ ∑ (15) 3. Padé Approximants 3.1. Generalities When series are used so as to describe physical quantities, the question of the accuracy of such a
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