PSI - Issue 71

Ritesh Kumar et al. / Procedia Structural Integrity 71 (2025) 364–371

365

Stresses in a FG disk for constant angular velocity were investigated using finite element method (FEM) (Allam et al., 2018; Çallioğlu et al., 2011; Kurşun & Topçu, 2013; X. Peng & Li, 2009; X. -L. Peng & Li, 2010; Sharma & Kaur, 2019). Investigated the thermoelastic stresses experienced by an FG annular rotating disk under both internal and external pressure (Rani & Singh, 2024). An optimization of the volume fraction of material has been identified to minimize stresses in an FG disk (Rahman & Ali, 2023). Elastic limit analysis of rotating disks was performed analytically (Kutsal & Coşkun, 2023) . Apart from these, several other methods have been employed by many researchers in analyzing the behaviour of structures: finite volume method (Gong et al., 2014), FEM, and direct numerical integration (Durodola & Attia, 2000), complementary functions method (Tutuncu & Temel, 2013), Galerkin’s, and Runge -Kutta methods (Shahriari & Safari, 2020). In the current study, for free-free boundary conditions, thermo-elastic analysis of orthotropic functionally graded was performed under varying thermal loadings. The temperature variation was given for constant and variable temperature change. Exponential gradation of the material property was considered. To check the validity of the proposed method, firstly its results are compared with benchmark solutions available in literature. It is interesting to note that studies pertaining to investigating the impact of material grading indices (β) on stresses for different temperature changes are scarce. For different indices (β), a difference in the magnitude of stresses (radial, tangential, and von Mises stress) was seen. The stress plots show the induced stresses along the radial coordinate which then can be compared with the yield strength of FGMs to check the yielding locations in a disk further it can be enhanced by tailoring the materials to change the location to improve the performance. 2. Material gradation and Mathematical formulations

In a functionally graded material (FGM), the material properties vary along a specific direction, which can be unidirectional, bidirectional, or multidirectional. To estimate these properties effectively, two primary approaches are used. The first approach involves determining the volume fraction of the matrix and reinforcement phases using models such as the rule of mixtures, the modified rule of mixtures, the Mori-Tanaka method, and the Halpin-Tsai model (Madan et al., 2024; Madan & Bhowmick, 2023). The second approach directly assumes material property variations based on predefined functions such as power-law, exponential, or sigmoid distributions. This study follows the latter approach, considering an exponential variation in material properties, including Young’s modulus, density, and thermal conductivity, without explicitly accounting for the volume fraction of each phase (X.-L. Peng & Li, 2012; Sondhi et al., 2023, 2024). Figure 1 shows the schematic representation of the FG disk.

Fig. 1. FG rotating disk



   

r b r a  −     − 

1 

−

r r o E E e = 

(1)

 



   

r b r a  −     − 

1 

−

o E E e = 

(2)





 



   

r b r a  −     − 



−

2

o e

= 

 

(3)

r

r

 



   

r b r a  −     − 

3 

−

r r o K K e = 

  (4) where and are the Young’s modulus variation in r and direction respectively, is the variation of density and is the variation of thermal conductivity in the radial direction. The material properties at the outer boundary ( r=b ) are given by , , and respectively 1 , 2 and 3 are material grading index of material properties (E, , ) respectively. 2.1 Thermal conduction problem