13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Crack-tip stress series The representation of crack-tip stress field at the vicinity of a crack-tip is generally performed with the so-called Williams crack-tip stress expansion [1, 2]: ( ) ( ) 2 , /2 1 1 , m m ij k ij k k m k r a f r σ θ θ ∞ − = =−∞ =∑∑ (1) This polar description establishes the separate nature of radial and angular dependencies. These dependencies also appear to be universal in the sense that they are the same for all fracture configurations. The particularity of each problem appears through an infinite set of specific coefficients – the first one of those being the Stress Intensity Factor (SIF). While SIF have been determined for numerous configurations, the evaluation of higher order coefficients has been less prolific (see [5] for a list of works on the subject). On the analytical point of view, just a few papers (for instance [6, 7]) have dealt with the subject. The authors have recently determined general closed form expressions for the following fracture configuration: Figure 1. Fracture configuration: (left) mode-I, (right) mode-II The procedure leading to coefficients exact definitions is described in the next subsections. 2.1. Complex solutions Complex analysis provides a convenient way to deal with bi-dimensional elasticity. Positions are defined with a single complex number and Lamé-Navier equations may be expressed by simple “complex” operators. The formalism has largely beneficiated from the seminal works of Kolosov, Muskhelishvili and coworkers [8, 9]. A popular presentation of the method is due to Westergaard [10] who expressed the stress state in term of a complex potential (“Westergaard stress function”). Westergaard’s initial work has then been improved, for instance by Sih [11]. Stress functions have been found for several fracture configurations [12, 13]. The last improvement in the domain has been provided by Sanford [14, 15]. With his “generalized Westergaard approach”, he has shown that the solution should involve two complex potentials. For mode-I, the stress state may be expressed with the potentials 1Z and 1Y according to: ( ) 1 11 1 2 1 1 1 Re Im ' Im ' 2Re Z x Z Y Y σ = − + + (2) ( ) 1 22 1 2 1 1 Re Im ' Im ' Z x Z Y σ = + + (3) ( ) 1 12 1 2 1 1 Im Re ' Re ' Y x Z Y σ =− − + (4) And for Mode-II with the potentials 2Z and 2Y : ( ) 2 11 2 2 2 2 2 Im Re ' Re ' 2Im Y x Y Z Z σ = + + + (5)
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