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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direction. FGMs have been developed as ultrahigh temperature resistant materials for the aerospace industry, but they currently have a wide range of applications, e.g. in the automotive and aeronautic industries, in power engineering, as well as in the biomedical industry. For thermal barrier coatings, functionally graded materials consisting of ceramics on the top and metals underneath of the structure are used. At elevated temperatures and changes of temperatures in combination with mechanical loadings, different crack patterns are observed in functionally graded coatings (FGCs) (Rangaraj and Kokini (2003)). To analyze the complex fracture behavior in FGMs and the fracture process, one needs to know the distribution of fracture toughness in FGMs. The present paper deals with the problem of thermal fracture of a structure consisting of a functionally graded coating on a homogeneous substrate (FGC/H) subjected to thermal loading. The main focus is laid on the application of fracture criteria for FGMs and the determination of critical mechanical and thermal loads. In order to apply fracture criteria for an FGM, it is required to introduce a model how to determine the fracture toughness near the crack tips on the basis of available values of the fracture toughness of the constituent materials of FGCs. An experimental investigation of FGMs consisting of partially stabilized zirconia (PSZ) and stainless steel (Tohgo et al. (2007)) demonstrates the influence of microstructure on fracture toughness distribution, namely, the fracture toughness in the FGMs is higher in the FGM with a finer microstructure than in the FGM with a coarse (rough) microstructure; in addition, the fracture toughness in the FGMs is higher than in non-FGM. The fracture toughness distribution in FGMs is in good agreement with changes in the morphology and components of FGM, e.g. the fracture toughness increases with increasing steel content, Tohgo et al. (2005) and (2007). Theoretical models for estimating the fracture toughness of FGMs have been discussed in many papers, e.g. Jin and Batra (1996), Feng and Jin (2012), Zhang et al. (2019). It is physically reasonable to assume that the fracture toughness is a function of a spatial coordinate, like other material properties. In Zhang et al. 2019, an exponential function and power-law function were chosen to describe the fracture toughness of the FGM, and in Jin and Batra (1996), the rule of mixtures was used to determine the fracture toughness of a ceramic-metal FGM. These functions are convenient for implementation in FGM models, but, generally, in these cases, the fracture toughness can be overestimated, since the fracture toughness of a metal in bulk form is much higher than that of metal particles dispersed in a brittle matrix (Jin and Batra (1996), Zhang et al. (2019)). The present work is devoted to the study of the interaction of multiple cracks in FGCs with respect to the main characteristics of fracture, such as stress intensity factors and critical loads. The theoretical basis for this problem has been described in authors’ previous works (Petrova and Schmauder (2017, 2020)), but the main parts are repeated here for completeness. The fracture angles (the deviation of cracks from their initial direction of propagation) were investigated previously in Petrova and Schmauder (2017, 2020), so that here the main attention is paid on the critical loads for the pre-existing system of cracks, edge and internal cracks, in the FGC. This investigation can provide important knowledge for choosing appropriate material combinations and the gradation profile of the FGCs in order to optimize the fracture resistance of FGC/homogeneous structures operating under high temperature. 2. Statement of the problem 2.1. Geometry of the problem and loading Consider a structure consisting of a functionally graded layer of thickness h on a semi-infinite homogeneous substrate, as shown in Fig. 1a. For thermal barrier coatings (TBCs), the top of the layer should be made of ceramic, and the homogeneous substrate is made of metal. TBCs operate at high temperatures at which oxidation processes can occur, resulting in the formation of an oxide layer between the coating and the underlying alloy. That is, the structure under consideration is a functionally graded coating (FGC) with a material gradient perpendicular to the interface on a homogeneous substrate with the thermally grown oxide (TGO) layer between them. The structure will be called FGC/TGO/H. In the functionally graded coating a system of N cracks of length 2 a k ( k = 1, 2, …, N ) is located, which can be edge and/or internal cracks. The oxide layer is modeled as a weak layer and is represented by cracks located in this thin layer, Fig. 1a. The global coordinate system ( x, y ) is connected to the FGC surface, and the local coordinates ( x k , y k ) are referring to each crack with the x-axis on the crack lines as shown in Fig. 1b. Using these coordinate systems, the positions of the cracks are determined by their midpoint coordinates ( x k 0 , y k 0 ) and the inclination angles α k to the x -axis, or β k for edge cracks, β k = – α k , Fig. 1b. Further, the complex variables method will be used in the