PSI - Issue 42

Neha Duhan et al. / Procedia Structural Integrity 42 (2022) 863–870 Duhan et al./ Structural Integrity Procedia 00 (2019) 000 – 000

866

4

M

klij ij kl C   = (4) where, , k T is the temperature gradient w.r.t. index ‘ k ’, κ is thermal conductivity, klij C is the elastic tensor and ( ) M ij ij T I    = − is the mechanical strain tensor obtained from the total strain tensor ij  and thermal strain vector T  .

trac

temp

y

Ω

x

y

disp

hf

x

Fig. 1. Schematic representation of an arbitrary semiconductor domain with a dislocation (edge type) showing boundary conditions

The boundary conditions of the problem domain (semiconductor domain with edge type dislocation) are shown in Fig. 1 where ‘disp’ represents displacement, ‘trac’ represents traction, ‘temp’ represents temperature and ‘hf’ represents heat flux. For the solution of the boundary value problem of the heat equilibrium equation, the primary and secondary boundary conditions are defined respectively as follows,

(5)

0 temp T =

0 (6) Similarly, for the solution of the boundary value problem of the force equilibrium expression, the primary and secondary boundary conditions are defined respectively as follows, hf q =

(7)

disp u =

0

0 (8) The equilibrium equations in the variational form are known as the weak form of the governing equations. The virtual work method can be adopted to get the weak form of a differential equation. For this purpose, a test function ( T  or k u  ) is multiplied with the equilibrium equation and then the integration over the domain is set to zero. This results in the partial differential equation of one order less than the original differential equation. The benefit of doing this is to use the shape functions with low order polynomials. The weak form equations, from Eq. (1) and (2) after substituting Eq. (3) and (4), respectively, are given as, ( ) , , 0 k k JH T T Q d      − −  =      (9) ( ) , 0 M k klij ij k u C d       =      (10) With the help of integration by parts and divergence theorem, the weak form can be arranged to separate out the force terms on the right side as, ( ) , 0 , hf k JH k T T d T Q d T q d         = −     (11) trac t =