13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- [ ] ( ) 1 22 2 3 4 22 2 3 4 ,0 1 7979 32295 22585 16337 2 4616 2 36928 2 147712 2 2363392 2 4,4 4517 19 4203 985 1 4616 64 147712 2363392 r r r r r a a a a a r r r r r a a a a σ σ∞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = ⋅ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (33) In any case, the epsilon algorithm (Subsection 3.3) is able to provide the required value for the different PA at any point. 4.1. Qualitative comparison The series expansion is performed at the crack-tip z=a. The radius of convergence is therefore equal to 2a. The series is not able to converge beyond this limit. In (Fig. 2), the exact complex solution, the truncated series representation with 9 terms, and the PA with 9 coefficients are compared. As expected, the series representation has only a good correspondence with the reference near the crack-tip and diverges beyond the radius of convergence. PA exhibits an excellent correspondence with the complex solution inside the whole convergence disk. The behavior outside the disk is also close to the reference. A spurious singular points appears at z=-2a. PA solution seems thus to have better convergence rates and a larger convergence domain. 4.2. Quantitative comparison The approximate representations with truncated series and PA (Eq. 31-33) are now compared in (Fig. 3). Within the convergence disk, the more terms hold the series, the better is the solution. Beyond the convergence radius series diverge. PA approximations have far better convergence rates inside most of the convergence disk. PA are also able to provide accurate results far beyond the series convergence limit. 5. Conclusion In this communication, Padé approximants have been used in order to improve the convergence behavior of crack-tip stress series. Starting from closed-form truncated series, it is possible to either determine exactly the coefficients in the PA or to estimate directly its value at any point. With the same amount of “information” (ie coefficients), the PA associated to a given truncated series provides better convergence rates inside the series convergence disk and is also able to converge outside the disk. Figure 2. Stress state representation for: (left) Westergaard exact solution, (center) truncated series with 9 terms, (right) Padé approximant [4,4] with 9 coefficients.
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