13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- stress field near the crack-tip in an elastic-plastic material is given as: ( ) ( ) ( ) (2) 2 0 2 (1) (0) 0 0 θ σ θ σ θ σ σ σ ij t s ij t ij s ij r A A Ar A r − + − = (2) In Eq. (2), σij(θ) are stress components, σr, σθ and σrθ in a polar coordinate system with origin at the crack-tip; ( ) (0) θ σ ij , ( ) (1) θ σ ij and ( ) (2) θ σ ij are normalized angular stress functions. The dimensionless radius r is defined as /( / )0σ r r J = , where J is the J-integral at the crack-tip. Power t is an eigenvalue depending on the hardening exponent n of Ramberg-Osgood relation, and power s=−1/(n+1). The polynomial coefficient A0 is defined as [11], A0=(αε0In) −1/(n+1), where I n is a scaling integral only depending on n, see Refs. [1, 2]. Nikishkov [11] has proposed a computational algorithm to determine the values of normalized angular functions ( ) (0) θ σ ij , ( ) (1) θ σ ij , ( ) (2) θ σ ij , asymptotic power t, and scaling integral In. The three-term expansion in Eq. (2) for the crack-tip stress (displacement) fields is controlled by two parameters, the magnitude of the first term (J-integral) and a second parameter (A) controlling the second and third term. 2.2. Numerical method for determining of constraint parameter The application of J-A (A2) two-parameter approach depends on the determination of the load-related parameter J and constraint parameter A (A2). The solutions of J-integral (including analytical, numerical and approximate solutions) have been well established in the literature, such as the numerical solution of J suggested by Moran and Shih [17], which has been adopted in the commercial code ABAQUS [18] utilized in the present research. Figure 2. Typical FEA mesh, a/W = 0.3 Based on some stress component with corresponding finite element solutions at one or several locations (points) within the plastic zone, Yang et al. [9, 10] determine the A2 values by matching a three-term expansion on crack-tip stress field. It is called the “point match” method by some researchers. Also based on FEA results, Nikishkov et al. [12] suggest a “fitting method” to determine parameter A values through a three-term expansion expressed by Eq. (2). In the present work, the proposed least square fitting method is utilized to obtained numerical solutions of constraint parameter A. See refs. [12] and [13] for more details about the procedure of the fitting method. 3. Finite element analysis, results and discussion 3.1. Material model and properties The material model for finite element analyses carried out in the present work is the deformation theory of plasticity. Ramberg-Osgood power-law strain hardening relation is applied in finite element code ABAQUS [18], which is used in the present FEA.
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