PSI - Issue 79

Manish Singh Rajput et al. / Procedia Structural Integrity 79 (2026) 26–33

29

approximated as given below:

(6)

; d a T a u N u d N d T N T    ; i a ai

i

Now, the governing equations are solved iteratively using the Newton-Raphson scheme and the Taylor series expansion [12, 13]. 3. Model validation A three-point bending specimen made of a transversely isotropic material (BaTiO 3 ) under thermo-mechanical loading conditions (purely mechanical, thermo-mechanical heating, and thermo-mechanical cooling) has been taken for validation purposes. Here, the temperature (thermal load) is applied to the right edge until it reaches 350 K. In contrast, the left edge is maintained at 300 K, as shown in Fig. 2(a), which represents the dimensions and loading conditions of the validation specimen. The material characteristics of the validation model are presented in Table 1. In this validation study, the length scale parameter is considered to be 0.1 mm. The obtained load-displacement relation and the crack growth plots under thermo-mechanical loading conditions are found in good agreement with the reference results [17], as shown in Fig. 2(b) and Fig. 2(c), respectively.

Table 1. Material properties of BaTiO 3 [17] Material Elastic stiffness constants (GPa)

Density (kg/m³)

Specific heat (J kg -1 K -1 )

Thermal expansion coefficient (10 -6 K -1 )

Thermal conductivity (W K -1 m -1 )

Critical energy release rate (N/mm)

C 66 43

C 11

C 22

C 12

C p

k xx

k yy

c G

xx 

yy 

BaTiO 3

166

162

78

5700

421.4

15.7

6.4

3.2 3.2 0.2

Fig. 2. (a) Validation specimen geometry (All dimensions in mm) and loading conditions, Comparison of obtained results from the present thermo-mechanical PFM with reference results [17] under different thermo-mechanical loading environments: (b) load-displacement response, and (c) crack growth plots.