By Dina M. Al-Sibai

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ASM 52, 54–80 (1960) 7. : A new model for Hydrogen-asisted cracking. Metall. Trans. 3, 437–451 (1972) 8. : The phenomenon of rupture and flow in solids. Philos. Trans. Roy. Soc. Lond. A221, 163–198 (1921) 9. : Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound. 9, 411–430 (2007) 10. : Energy minimizing brittle crack propagation. J. Elast. 52, 201–238 (1998/99) 11. : Modeling and numerical simulations of dendritic crystal growth. Physica D 63, 410–423 (1993) 12.

This chapter is a sort of review of papers [5, 6, 8–10]. H. jp © Springer Nature Singapore Pte Ltd. 2017 H. Itou et al. 1007/978-981-10-2633-1_4 35 36 H. Itou Fig. 1 An example of an illustration of the geometry A brief outline of this paper is as follows: In Sect. 2, three problems which we deal with are stated. Section 3 is a main part of this paper and convergent series expansions for each problems are shown. Finally, in Sect. 4, we summarize the results and mention future works. 1 Domain In the present paper, we restrict ourselves to two dimensional linearized elastostatic material denoted by Ω ⊂ R2 assuming a bounded domain with Lipschitz boundary and divided into two Lipschitz domains Ω (1) := Ω ∩ {x2 > 0} and Ω (2) = Ω ∩ {x2 < 0} by the x1 -axis.

44 H. Itou Proposition 5 ([10]) For slip state case, there exist complex numbers aˇ n , bn , cˇn satisfying the condition (8) and a constant vector c ∈ R3 such that for k = 1, 2 u(k) (r, θ) = ∞ e(−1) π r n+1−γˇ −Im aˇ n Q (k) 2b,n (θ) 2μ(k) (n + 1 − γˇ ) k n=0 ∞ r n+1 + n=0 ∞ − n=0 ∞ + n=0 μ(1) μ(2) (n + 1) Re [bn ] R(k) 2a,n (θ) ± f (κ (1) κ (2) − 1) Q (k) 2a,n (θ) k (m 1 + m 2 ) μ(k) m r n+1 d˜k (k) Im [bn ] S1,n (θ) μ(1) μ(2) (m 1 + m 2 )(n + 1) r n+1 Re cˇn k (n + 1) 2μ(k) m (k) (k) P 2a,n (θ) ∓ f Q 2a,n (θ) + F(x)c, where the upper and lower signs are taken when (10) is positive and negative, respectively, Q (k) 2b,n (θ) = (κ (k) + n + 1 − γˇ + e(−1) 2π ) sin n + 1 − γˇ θ − (n + 1 − γˇ ) sin n − 1 − γˇ θ k+1 (−κ (k) + n + 1 − γˇ + e(−1) 2π ) cos n + 1 − γˇ θ − (n + 1 − γˇ ) cos n − 1 − γˇ θ k+1 .