PSI - Issue 16

Jaroslaw Galkiewicz / Procedia Structural Integrity 16 (2019) 35–42 Jaroslaw Galkiewicz / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Model

The plane strain is assumed; therefore, the inclusion takes the shape of a cylinder of infinite length, as in the McClintock model (McClintock (1968)). Based on the observation of the distribution of the inclusions, it was assumed that the size of the elementary cell is 100 µm. In the case of circular inclusions, it was assumed that the diameter is 5 µm. The influence of the inclusion shape on the nucleation process was also analyzed, and there fore it was assumed that the inclusion takes the elliptic shape with a ratio of the half-axes of 1:3 (elongated inclusion) and 3:1 (flattened inclusion) (Fig. 2). Because of the double symmetry, only ¼ of the cell was modeled. The investigated cell is located near the crack tip. In numerous studies, the influence of geometric constraints (both in-plane and out-of-plane) on the behavior of the material in front the crack tip was emphasized (Larsson and Carlsson (1973), Betegon and Hancock (1991), Neimitz and Galkiewicz (2004), Neimitz and Dzioba (2015)). To ensure a high level of constraints, a load in the form of a displacement was applied as shown in Fig. 2. In the study Galkiewicz (2015), the dependence between the displacement components and the level of in-plane constraints was determined in the form:   1 2 0 / 0.214 / 0.169 U U T    , (2) where: T is the amplitude of the second term in the Williams’ asymptotic solution describing the stress field before the crack tip in the elastic material (Al-Ani and Hancock (1991)), and σ 0 is the yield strength. To investigate the influence of constraints, three levels of the T/σ 0 quotient equal to – 1.0, 0.0 and 0.5 were assumed.

b

a

Fig. 2. (a) numerical model and (b) shapes of the analyzed inclusions.

Both the inclusion and the matrix are modeled using 4-node, bilinear, plane strain elements CPE4I. In the model, three zones of cohesive elements were used (Fig. 2). In zone A, the interaction between the inclusion and matrix is modeled using a surface-based approach. Preliminarily, in this area it is assumed that fracture may occur according to the mixed mode of fracture (opening mode and sliding mode). Areas B and C are the planes of a potential crack growth in the inclusion and in the matrix, respectively. Both regions are modeled using an element-based approach. Due to symmetry, the nodes in zones B and C are subject to additional constraints. Before the simulation was started, the height of the cohesive elements in these zones was reduced to zero by moving the nodes in a vertical direction on one of the sides. Additionally, to avoid uncontrolled movement of nodes located in the symmetry plane, the displacements are tied with displacements of corresponding nodes connected with the matrix.

4. Properties of the cohesive zones

The most important characteristics of the cohesive element in the standard library of the ABAQUS software are shown in Fig. 3a. To describe the behavior of the cohesive element, the cohesive stress value and energy of separation for the two loading directions (corresponding to mode I and II) should be defined. For simplicity, it is assumed that the peak stress values and cohesive energy in zone A are equal for each direction. In the zones modeling the fracture of the inclusion and matrix (zones B and C), only the parameters characterizing the opening