PSI - Issue 14

Manish Kumar et al. / Procedia Structural Integrity 14 (2019) 839–848 Manish Kumar et. al/ Structural Integrity Procedia 00 (2018) 000–000

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2. Mathematical Modelling In this section, the mathematical modelling of elasto-plastic-creep analysis using XFEM is discussed. Incorporation of relaxation and redistribution of stress state due to creep in the mathematical formulation is described. The methodology for calculation of ( ) C t -integral to perform crack growth along with data transfer and null step analysis is also explained in this section. 2.1. Governing equations An isotropic homogeneous continuum occupying volume V bounded by boundary  is considered as shown in Fig. 1. The continuum has a traction free discontinuity represented by the surface C  . U  and T  represent boundaries subjected to the prescribed displacement and traction respectively. The equilibrium equation of the continuum is written as, 0    σ b in V (1) The associated boundary conditions are,   σ n t  on T  (2) 0   σ n on C  (3) u = u  on U  (4) where,  is the gradient operator, σ is the Cauchy stress tensor, b is the body force, n is the unit normal vector on the T  boundary, u is the displacement, t  and u  are the prescribed traction and displacement on the boundary T  and U  respectively. Using the principle of virtual work, strong form of Eq. (1) is converted into weak form as,







(5)

u σ

u b

 u t 

δ :

δ  

δ

0

dV

dV

d





 

s

V

V



T

T 

V

C 



U 

Fig. 1 : A cracked domain along with boundary conditions