PSI - Issue 14

Pankaj Kumar et al. / Procedia Structural Integrity 14 (2019) 96–103 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

99

4

superposition of three Armstrong and Frederick model. Chaboche postulates that the total back stress is summation of three back stresses. The material coefficients used with ―Chaboche Model‖ are determined by experimentally obtained stabilized hysteresis loop. Stabilized loop is considered at half of the fatigue life. This model can be mathematically expressed as is

  3 1 i dx dx i 

(2)

3 2

p i x i d eq  

dx i

c i

d p



 

(3)

Isotropic hardening model is responsible for capturing hardening and softening behavior of the material and it is expressed as

  

        r p Q e bp 1

(4)

where b and Q are material constants. Here, Q is the saturated value of isotropic model once peak stress is achieved. The constant b determines the rate at which saturation is achieved and p is accumulated plastic strain.

3.2. Extended finite element method

In XFEM approximation of primary variable is enriched by additional functions within partition of unity framework. Two types of enrichment functions i.e. Heaviside function for crack surface and Asymptotic crack tip functions for crack front are used to model a crack. The nodal information for enrichment implementation is updated for these enrichment functions as crack propagates. Mathematical relation provided by Moës et al. (1999) for enriched displacement approximation at any point can be written as:

     

     

    

    

        j H H a x x j

      b   x x

n

4

  x

 

h



(5)

x u

u

N

 



  

j

j

j

j



j

1

1









 j n r

 j n A

Where, N j ( x ) is the Lagrange interpolation function; u h ( x ) the nodal displacement vector associated with continues part of the finite element solution. H ( x ), the Heaviside function across the crack surface; a j is the additional degree of freedom associated with the Heaviside function; ϕ α ( x ) is the asymptotic enrichment function taken from the Westergaard- William’s solution for disp lacement field at crack tip and b j ( α ) is enriched nodal degree of freedom associated with crack tip enrichment function. Heaviside function is used to model strong discontinuity across the crack surface, H ( x ), is given as:





, otherwise if x x n

1

0

  

  

 

H x

(6)



1



Where x is the sample Gauss point; x* is the point on the crack closest to x and n is the unit outward normal to the crack at x* . Kumar and Singh (2017) have applied the mathematical relation for function ϕ α ( x ) which is asymptotic crack tip enrichment function used for capturing the stress singularity near crack