PSI - Issue 79

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

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Consider a solid body subjected to a deformation field ' u ’, heat flux ‘ f q ’, and traction forces ‘ t ’ as shown in Fig. 1. The total energy functional can be written as follows [11]:     , e u e ij ij c d G d           (1) where, ' ' c G is the critical energy release rate and where   u e ij  represents the strain energy per unit volume associated with elastic deformation, and ‘ e ij ’ are the components of the elastic strain tensor and is defined as follows: e total thm ij ij ij   (2) where, thm ij T    represents the thermal strain tensor, and ‘β’ indicates th e materialthermal expansion coefficient. Nomenclature ζ Crack Surface Homogeneous elastic domain

t, q Solid domain Neumann boundary u, T Solid domain Dirichlet boundary C p Specific heat [Jkg -1 K -1 ] k Thermal conductivity [Wm -1 K -1 ] σ Cauchy stress tensor [Nm -2 ] G c Critical Energy release rate [Nm -1 ] b 0 Length scale parameter [m] ε Strain tensor [-] u Displacement field [m] Poisson’s ratio [ -] d Phase field variable H + History field variable [J] Density [kgm -3 ] N Nodal shape function R Residual force [N] F Body forces [N]

By applying the minimum potential energy condition and the first variational principle, Eq. (1) can be written as follows [12, 13]:   , 0, x e ij ij b i j d F x       (3)         2 0 0 2 2 , x u e u e c ij ij c j j b d G d G b x x                  (4) For a domain subjected to a thermal loading environment, the governing heat balance equation is expressed as follows [17]: (5) where ‘k’ represents the material thermal conductivity, ‘ ’ represents the density of the material, ‘ C p ’ represents the specific heat, q ’ represents the heat generation rate, and ‘j’ represents the dimension number. Now, using the finite element method, the field variables (displacement field, phase field variable, and temperature field) are 2 0, x    gen j j p T T k C x   t q x     