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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In the studies Galkiewicz (2015), Gao et al. (1998), based on the simulation, the maximum value of opening stress for the matrix material was determined to be the level of seven times the yield stress, i.e., 1470 MPa for high constraints. Once this value has been reached, a degradation process begins in the material. Therefore, this value is used for simulations in the cohesive zone for the matrix. A determination of the energy of separation for manganese sulfide and the work necessary to debond the inclusions from the matrix is difficult. Fractographic studies are helpful in this process. On this basis, it was assumed that MnS particles fracture at an overall strain level of approximately 1%, while the matrix can be totally damaged at an overall strain of 4% provided that the constraint level is high. To obtain such a result during simulations, the cohesive energy of the inclusion is assumed to be 240 J/m 2 . In the studies Neimitz (2008), Siegmund and Brocks (2000), it was demonstrated that the cohesive energy strongly depends on the geometry of the sample and constitutes only a small part of the energy dissipated in the fracture process (even less than 1%). Therefore, it was assumed that the cohesive energy (work of surface debonding) for the matrix material is equal to 2400 J/m 2 , i.e., 10 times more than for the inclusion. With such values, the elementary cell containing a circular inclusion behaves in a manner similar to the experimental results. However, it should be emphasized that the cohesive energy was of a secondary importance from the point of view of the study, as its value defines the moment of complete separation of the element, while the maximum stress determined the start of the degradation process. During the study, three different types of behavior of the elementary cell are observed (Fig. 4). The cell can break along the symmetry plane running through the inclusion and the matrix (uniform fracture). A complete detachment of the inclusion from the matrix and fracture of the matrix material along the symmetry plane (a complete debonding and breaking) can happen. Alternatively, the elementary cell can fracture along the plane of symmetry, which is accompanied by a partial detachment of the inclusion from the matrix. On a two-dimensional chart, where the behavior of a cell is a function of the cohesive stress of the inclusion material and the maximum stress on the inclusion-matrix interface, the areas of the specific behaviors form a characteristic pattern, demonstrated schematically in Fig. 4d. To make reading the chart easier, figures depicting the behavior of the cell are placed in each field. The areas are limited by two lines (AC and BC). On the abscissa, the value of σ maxB /σ maxC , the relation of the cohesive stress in the zone modeling the fracture of the inclusion to the corresponding value in the zone modeling the fracture of the matrix, is indicated. On the axis of ordinates, σ maxA /σ maxC , the value of the cohesive stress on the inclusion-matrix interface normalized by the value of the maximum stresses for the cohesive zone in the matrix material, is indicated. 6. Investigation of the effect of constraints on behavior of the elementary cell

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Fig 4. Behavior of the elementary cell: (a) uniform fracture, (b) complete debonding and breaking, (c) breaking with partial debonding, and (d) a schematic map of the occurrence of the elementary cell behaviors.

In Fig. 5, maps of the conditions of the elementary cell damage as a function of the constraints level are presented. The first column (Fig. 5a,d,g) contains the results for the highest constraints level (T = +0.5), the central column (Fig. 5b,e,h) contains the results for the high constraints level (T = 0.0), while the left column (Fig. 5c,f,i) contains the results for the low constraints level (T = – 0.1). To facilitate a comparison of the results in the figures located in the left and right columns, the results for the constraints level of T = 0.0 are marked with a dashed line. The influence of constraints is least evident for the circular inclusion. However, this influence is more pronounced for the inclusions having an elongated shape, especially when comparing the low constraints level with other cases.