PSI - Issue 28

Vera Petrova et al. / Procedia Structural Integrity 28 (2020) 608–618 Author name / Structural Integrity Procedia 00 (2019) 000–000

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formulation of the equations of the problem; therefore, it is convenient to represent the midpoint coordinates of the cracks in complex form as z k 0 = x k 0 + iy k 0 ( i is the imaginary unit). The FGC/TGO/H structure is cooled by Δ T , Δ T > 0 (this can be cooling from operating temperatures) and an additional tensile load p is applied parallel to the surface.

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Fig. 1. Geometry of the problem: (a) FGC/TGO/H with a system of cracks, and (b) coordinate systems connected with cracks.

2.2. Key considerations The problem of interacting cracks in functionally graded coatings under thermo-mechanical loading is studied in the frame of linear fracture mechanics. Numerous investigations (e.g. Jin and Batra (1996)) reveal that the distribution of stresses near the crack tips in functionally graded materials (FGMs) has the same inverse square root singularity as in homogeneous materials, if the material properties are continuous functions of a spatial variable. Hence, the stress intensity factors (SIFs) can be determined near the crack tips. For interacting cracks in an FGC, the mixed-mode fracture conditions is realized, that is, both stress intensity factors mode I ( K I ) and mode II ( K II ) are non-zero. Further, fracture analysis is possible by applying an appropriate fracture criterion for mixed-mode cracks and determining the direction of the crack propagation and the critical loads when this propagation starts. To obtain the critical loadings, it is important to know the fracture toughness near the crack tips (Kim et al. (2007), Feng and Jin (2012), Zhang et al. (2019)). Therefore, a rule should be established for determining fracture toughness for a functionally graded material. An experimental evaluation of fracture toughness and residual stresses in ceramic-metal functionally graded materials can be found in Tohgo et al. (2005), Tohgo et al. (2008) and Jin et al. (2009). The thermal and mechanical properties of the FGC are assumed to be continuous functions of the thickness coordinate. It is also expected that, like other material properties of functionally graded materials (FGMs), the fracture toughness is a continuous function of the same spatial coordinate. Models for material properties for FGMs, including a model for the fracture toughness, are presented in the next section. If the coating contains many cracks, fracture is expected to start from a weakest crack. The weakest crack can be determined by the highest value of SIFs (which is applicable for homogeneous materials, but not for FGMs), or by a smallest value of critical load at the crack. In determining critical loads the variation of the fracture toughness of the FGM is accounted. Thus the global critical load is determined as P cr = min p crn ( n = 1 , 2, …, N ). Due to the mismatch of the thermal expansion coefficients of the FGC, a change of operating temperature leads to the appearance of thermal residual stresses in the considered structure. Besides, as noted in Tohgo et al. (2008), microscopic and macroscopic residual stresses arise due to the mismatch of the thermal expansion coefficients between the ceramic and metal phases in the fabrication of FGMs. It is physically reasonable to assume that the non homogeneity of the functionally graded material is a consequence of the form of the corresponding inhomogeneous stress distributions on the surfaces of the cracks, see Afsar and Song (2010), Tohgo et al. (2008). Under this assumption, the properties of the FGM should vary slightly in the depth of the coating. These respective stresses that are applied to the crack faces are determined below.