13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- element for dissimilar material wedge problems. Mote [13], Bradford et al. [14] and Madenci et al. [15] established a special global element based on asymptotical solutions around a dissimilar material junction edge. By using the eigenfunction expansion method, Barut et al. [16] derived a special hybrid global element on the basis of exact analytical solutions of stress displacement fields under mechanical and thermal loads. In addition to the leading singular order term, a few other higher order terms were also used in constructing the special elements in Ref. [11-13, 16], which leads to more accurate numerical results The hybrid-stress finite element method developed more than 40 years ago by Pian is now well recognized as a powerful and easy-to-use tool for solving a variety of two-dimensional linear elasticity problems containing a single or multiple singular points. To the best of the author’s knowledge, the studies related to singular thermo-mechanical fields of inclusions by the hybrid-stress finite element method are absent. Moreover, a numerical solution of even a single irregular inclusion by the method could not be found in the literature, either. To predict singular stress fields around an inclusion corner under thermo-mechanical loads, a new ad hoc super inclusion corner element based on the numerical asymptotic solutions developed is proposed in this paper to study inclusion problems shown in Fig. 1. The validity and applicability of present approach are established through available solutions 2. The hybrid variational functionals for thermo-elasticity involving an inclusion Let a super n-sided polygonal element centered at the inclusion corner is taken as the complementary region (C-region) which contains inclusion domain 2Ω with outer boundary 2Γ and its surrounding matrix domain 1Ω with outer boundary 1Γ, as shown in Fig.2. Under appropriate continuity conditions, the stiffness matrix for a super inclusion element of dissimilar material wedge subjected to thermo-mechanical loads is written as [16]: { } 2 ( ) ( ) ( ) ( ) 1 2 ( ) ( ) 0 1 1 2 k k k m k T T m k m k T T t k c k m k T T k k dS dS dS λ λ λ λ Γ Γ λ Γ Π Π + = = ⎧ ⎫ = − − ⎨ ⎬ ⎩ ⎭ + + ∑ ∫ ∫ ∑ ∫ % σ n u σ n u σ n u (1) Figure 1 Local coordinate system near the inclusion corner-tip Figure 2 A super n-sided polygonal inclusion corner element 2α θ r 1α α 2Ω 1Ω x y o (2) u% (1) u% 2Ω 1Ω 2Γ 1Γ
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