PSI - Issue 42

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

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discussed. This section has sub-sections as: 2.1 Fundamental Equations, 2.2. Discrete equations from XFEM and 2.3. Peach-Koehler force. Section 3 has the explanation of the geometry, material properties, boundary conditions, mesh and the results of the problem. Finally, the entire work is concluded in section 4 of this paper.

Nomenclature k q

heat flux vector

JH Q

Joule heat stress tensor strain tensor

kl  ij 

thermal strain vector temperature gradient

T 

, k T

 thermal conductivity   N shape functions for displacement   T N shape functions for temperature   B derivatives of shape functions for   N   T B derivatives of shape functions for   T N W elastic strain energy  thermal expansion coefficient a lattice constant b Burgers vector ij c elastic constants

2. Problem Formulation This section has a description of all basic equations, including governing equilibrium equations, constitutive equations, boundary conditions and weak forms. The XFEM approximation for obtaining discrete form equations is also discussed here. The computation of the Peach-Koehler force is also deliberated in this section. 2.1. Fundamental equations The fundamental mathematical equations that define any problem are the differential equations. In mechanics, the differential equations are in the form of equilibrium equations. Fig. 1 shows domain Ω , where the heat and force equilibrium equations are the governing equations. The static heat governing equation when the internal heat is getting generated due to the Joule heat effect of the electric field application is written as,

0

, k k JH q Q − =

(1)

, k k q representing the differentiation of k q w.r.t.

where, k q is the heat flux vector with index ‘ k ’ following comma in

( ) . Q  = E E is the Joule heat defined as a function of electrical conductivity  and electric field

index ‘ k ’ and

JH

k

k

k E . The force equilibrium equation for a linear elastic body without the body force is as follows,

(2)

0

, kl l  =

where, , kl l  represents the differentiation of Cauchy stress tensor kl  w.r.t. index ‘ l ’. The constitutive equations define the material behavior by giving the relationship between stresses and strains and heat flux and temperature gradients, such as,

(3)

, k k T q  − =