PSI - Issue 42

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

867

5

 



 

(12)

C d 

u t d

C I d 



 =



+





0

kl klij ij

k

kl klij T



trac

2.2. Discrete equations from XFEM The approximate field equations of temperature and displacement in FEM are defined with the help of polynomial shape functions. However, the XFEM uses enrichment functions in addition to standard shape function terms of FEM. The Heaviside and core enrichment functions are included to define the approximate field equations of XFEM for thermo-elastic formulation (Duhan et al. (2022)). These approximate equations, when substituted in the weak form Eq. (11) and (12), result in the discrete form as follows,

    

    

       



    T T B B d T  

 

  0 T

   =

 

 



(13)

T JH N Q d

N q d

−



 



hf

    

    



    B C B d U  klij

 

  B C I d  klij T

   =

 



 

(14)

N t d

 +



0

 



trac

where, T N and N are shape functions with their derivatives as T B and B defined for temperature and displacement fields, respectively. The terms on the left hand side of Eq. (13) and (14) are stiffness and primary variables and that on the right hand side are forces. These discrete equations are algebraic equations that are solved by performing numerical integration. 2.3. Peach-Koehler force The Peach-Koehler force is the force exerted on the dislocation due to the stresses except for self-stresses of the dislocation. This force can be defined as the energy change per unit change in the position of the dislocation. By knowing the Burgers vector and the stress tensor, this force can be calculated. Another approach to compute the Peach Koehler force is to use the analogy of J -integral. The expression of Peach-Koehler force based on the J -integral concept for thermo-elastic case (Duhan et al. (2021)) is as follows,

  

k u      x i

T

 

 

 

(15)

J

W

n d

d

=

  −

 +





i

il

kl

l

kk

x

i

where, W is the elastic part of the strain energy and α is the thermal expansion coefficient. More conveniently, the J -integral in the domain form is used to directly use the stresses, displacement gradients and temperature gradients obtained from FEM. The domain form expression of the J -integral (Duhan et al. (2021)) is as follows,

  

  

u



w

T



 

 

A 

k

(16)

J

W dA 

wdA

=

−

+





i

kl

il

kk

x

x

x





i

l

i

where, A is the area of the integration domain taken around dislocation in the form of an annular disk and w is the weight function with value 1 at the inner radius of the annular disk and 0 at its outer radius. 3. Numerical Example Si x Ge 1-x semiconductor alloy material with one free surface at the left is considered for modeling using XFEM when the edge dislocation is located at half of the domain length from the free surface and half of the domain height from the bottom surface. This is a 2D plane strain thermo-elastic problem incorporating the effect of electric field induced