PSI - Issue 42

Vera Petrova et al. / Procedia Structural Integrity 42 (2022) 1145–1152 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction Functionally graded materials (FGMs) are a special type of composites, the properties of which are continuously varying mainly in one direction, which is achieved by changing the composition of the material and its structure. FGMs have wide engineering applications, in particular, in thermal barrier coatings (TBCs), where ceramics are used on the coating top and then continuously vary up to metal in the substrate. Functionally graded coatings (FGCs) with a gradual compositional variation from heat resistant ceramics to fracture resistant metals have been proposed in order to reduce thermal residual stresses causing delamination and debonding of interfaces, enhancing coating toughness, and improving the long-term performance of TBCs, see Clarke et al. (2012). However, cracks can occur because of initial defects or microcracks that appear during manufacturing or operation. Therefore, the study of fracture of FGCs is important for a better understanding of the fracture resistance of graded coatings. The study of FGMs involves many challenging mechanical problems, including the prediction and measurement of the effective properties, thermal stress distribution, and fracture of FGMs. Modelling and evaluation of the effective properties of FGMs was considered in many works, see Zuiker (1995), Noda et al. (2004). The evaluation of fracture toughness of FGMs is also a very important problem for studying of FGCs fracture (Jin and Batra (1996), Tohgo et al. (2005), (2008), Zhang et al. (2019)). In Petrova and Schmauder (2019), (2020b), the analysis of the fracture parameters with respect to critical loads for a system of edge and internal cracks in FGCs showed the importance of considering the variation of fracture toughness through the thickness of FGCs, and accordingly the importance of using critical stresses as the main fracture characteristic for FGCs. In previous papers (Petrova et al. (2016), Petrova and Schmauder (2018, 2020a, 2020b, 2021)), different aspects of thermal fracture of FGC/H structures were investigated using an exponential model for material properties evaluation. In the present work, the problem for thermal fracture of FGC/H structures is considered with application of the functions based on the rule of mixtures (RoM) with corresponding gradation parameters. The RoMs, originally applied to conventional composites, has been successfully used in FGMs (Noda et al. (2004), Jin and Batra (1996)) and has more possibilities for optimizing FGMs in terms of improving fracture toughness. In the current paper, the general formulation of the problem is the same as in previous works, e.g. in Petrova and Schmauder, 2020b, so we only briefly repeat this formulation for completeness, see Sections 2.1 and 3. Section 2.2 presents the model for material properties. The results and their discussion are given in Section 4, and in Section 5 are conclusions. 2. Formulation of the problem 2.1. Geometry of the problem and loading The general geometry of the problem is shown in Fig. 1a, i.e. an FGC of thickness h , an underlying homogeneous substrate (H) and a layer of thermally grown oxide (TGO) that can form between them (FGC/TGO/H structures). For TBCs the top layer is made from ceramics, materials with low thermal conductivity, and the homogeneous substrate is made of metal. A FGC/TGO/H structure is cooled by ΔT , ΔT > 0 (e.g. due to cooling from operating temperatures) and a tensile load p is applied parallel to the surface. A pre-existing system of cracks (length 2 a k ) is located in the FGC. The global coordinate system ( x, y ) and the local coordinates ( x k , y k ), which are referred to each crack with the x k -axis on the crack lines, are shown in Fig. 1b. The position of cracks are determined explicitly by their midpoint coordinates ( x k 0 , y k 0 ) (or in the complex form z k 0 = x k 0 + iy k 0 , i is the imaginary unit) and the inclination angles α k to the x -axis, or β k for edge cracks, β k = – α k , Fig. 1b. As in the previous authors works (e.g. see Petrova and Schmauder (2017, 2018, 2020a, 2020b, 2021)), the following assumptions are used: the uncoupled, quasi-static thermo-elasticity theory is applied, that is, the temperature distribution is independent of the mechanical field; the thermal and mechanical properties of an FGC are continuous functions of the thickness coordinate y ; the non-homogeneity of the functionally graded material is revealed in the form of the corresponding inhomogeneous stress distributions on the surfaces of cracks (Afsar and Sekine (2000), Tohgo et al. (2008)).