PSI - Issue 23

Vera Petrova et al. / Procedia Structural Integrity 23 (2019) 407–412 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction

Thermal barrier coatings (TBCs) have wide application in various engineering systems that operate at elevated temperatures, for example, in the aerospace and airplane industries, as well as in power engineering. TBCs are used to prevent melting of metal parts of structures, therefore, they are made of materials with low thermal conductivity. Ceramics possess these properties, so they are used for TBCs. However, due to the difference in the thermal expansion coefficients of ceramics and metals, high residual stresses arise at the interface between the ceramic coating and metal substrates, which leads to cracking and debonding along the interface and, thus, can lead to the deterioration of the entire structure. In order to increase the effectiveness and durability of engineering components, new materials and new concepts for designing the materials are challenging tasks for researchers, see Clarke et al. (2012). Functionally graded materials (FGMs) are one of these concepts, Miyamoto et al. (1999). The properties of FGMs are continuously varying mainly in one direction, which is achieved by changing the composition of the material and its structure, for example, different porosities. In spite of numerous investigations in this field, the fundamental aspects for modeling of fracture in FGMs under elevated temperature are still challenging tasks for researches. The general case of the geometry of the problem is shown in Fig. 1a. The upper layer of thickness h is made from an FGM, and the semi-infinite substrate consists of a homogeneous material. The functionally graded coating (FGC) and the homogeneous substrate (FGC/H) are perfectly bonded with the exception of an interface crack disposed at the interface between the two materials. The FGC contains arbitrary located cracks of length 2 a k ( k =1,…, N ), which can be internal and/or edge cracks. The coordinate systems are chosen as follows: the global coordinates ( x, y ) with the x axis located on the surface of the FGC/H structure and the local coordinates ( x k , y k ) connected with cracks and with centers at the midpoint of the k -th crack. The crack positions are determined explicitly by the midpoint coordinates z k 0 = x k 0 + iy k 0 and the inclination angles α k to the x -axis (or β k for edge cracks, β k = – α k ) (Fig. 1b). 2. Formulation of the problem

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Fig. 1. (a) An FGC/H structure with a system of edge and internal cracks; (b) coordinate systems connected with cracks; (c) schematic representation of three edge cracks and an internal crack.

The following loads are considered: tension parallel to this surface (in the x -direction), and cooling by ΔT (Fig. 1a). In the case of cooling of the FGC/H structure tensile residual stresses are observed as shown in the experimental investigations, Gilbert et al. (2008). Since the problem is linear, the results for each load can be superimposed. In the case of FGC/H under a heat flux the problem is solved in two steps, first, the thermal problem for the FGC/H structure with cracks, and then the thermo-elastic problem for the same geometry. The following assumptions are used for this problem:

 The thermal and mechanical properties of an FGC are continuous functions of the thickness coordinate y.