PSI - Issue 42

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

1147

3

b

a

e d Fig. 1. (a) FGC/TGO/H with a system of cracks; (b) global ( x , y ) and local ( x k , y k ) coordinate systems; (c) three edge cracks, (d) three edge cracks and an internal crack; (e) variation of fracture toughness in an FGC. 2.2. Material properties for FGCs and residual stresses The presented method is applicable to different material combinations. However, due to the application of FGCs for TBCs, only (ceramic/metal)/metal coatings are considered in this study. Thus, the material composition varies with the y -coordinate from ceramic at the top of the FGC to metal in the substrate, Fig. 1e. Consequently, the properties of the FGC also vary continuously with the thickness coordinate y and can be mathematically described by a continuous function. In previous works (Petrova and Schmauder (2020a and b)), an exponential form of Young's modulus and thermal expansion coefficient was used, and in Petrova and Schmauder (2018) – a linear model. Poisson’s ratio was assumed to be constant and equal to the value of the homogeneous substrate. Along with the change in these mechanical properties, the change in fracture toughness was also taken into account, see Fig. 1e. In the present study, the thermal and mechanical properties as well as the fracture toughness ( K Ic ) of the FGC are modeled by functions based on the RoM. The RoM with its various modifications has long been used for conventional composites. In contrast to conventional composites, the volume fraction of one material in another in FGMs varies, so the effective properties of FGMs depend on this volume fraction. The FGC consists of two constituents, ceramic on the top and metal as a homogeneous substrate, with their volume fractions V c and V m , respectively, determined by a power law function (1) with a power coefficient λ as the grading parameter for the FGC. This parameter can be set to different values to realize different volume fractions (as desired) and profile shapes. The coefficient of thermal expansion and Young's modulus of the FGC are assumed to take the following forms (see Noda et al. (2004)): c     n   n    0 1 Im sin m V y n y h h          z x  h         ,     1 V y   c V y m

   

    

    V y E V y E m

 / 1 / 1 

      

  V y E

 / 1



  2/3



       2/3



 





c m c m E E E V y  

  y

  E y E

tm m

m

tc c

c

,

, (2)

t 



 





  V y E

 / 1



c









 

c m c E E E V y   m

m V y



m

c

c

here ν is Poisson’s ratio. The fracture toughness ( K I c ) for FGCs also changes with the coordinate y and can be determined using one of the models, see Noda et al. (2004), Jin and Bartra (1996). In the present work, the second function in Eq. (2) is used for K I c , where E should be replaced by K Ic . The method of superposition is used to solve this problem, so the loads at infinity are reduced to the corresponding