PSI - Issue 52

Made Wiragunarsa et al. / Procedia Structural Integrity 52 (2024) 583–593 Wiragunarsa et al. / Structural Integrity Procedia 00 (2023) 000–000

585

3

The second governing equation is the rate of change of deformation gradient, as given in Equation 2. This equation was used by Lee et al. [2019] to simulate large-strain solid dynamics.

d F dt − ∇ 

p ⊗ I  = 0

1 ρ

(2)

Integrating Equation 2 with respect to time yields the value of the deformation gradient. Then, the displacement gradient ∇ u is calculated using Equation 3. In the linear elastic assumption, the strain can be obtained using Equation 4.

∇ u = F − I

(3)

1 2

T )

( ∇ u + ∇ u

(4)

ε =

2.2. Constitutive relation

Constitutive equation is a mathematical model of the real behaviour of materials. It relates the primary field vari ables to the secondary field variable. In this research, we relate the strain and stress tensors using Hooke’s law. The relation of the stress-strain can be expressed as follow

σ = C ε

(5)

In the two-dimensional plane strain problem, the relation is given by

  

E (1 − ν ) (1 + ν )(1 − 2 ν )    1 ν 1 − ν

  ε xx ε yy ε xy

  =

 

 

ν 1 − ν

0

σ xx σ yy σ xy

1 0

(6)

1 − 2 ν 2(1 − ν )

0 0

inwhich E , v , and G are Young’s modulus, Poisson’s ratio, and shear modulus of the material.

2.3. Fracture mechanics

The stress concentration factor can not be used to analyse a crack because the value tends to reach infinity. There fore, the stress intensity factor was introduced. The stress intensity factor, or simply K I , K II , and K III , depend on the crack mode that determines the intensity or state of stress at the crack tip. In this research, the problem is limited to mode I crack only. One of the most popular method to calculate stress intensity factor is using the J -integral method. In linear elastic fracture mechanics, the value of J -integral is similar to the rate of energy released per unit of crack translation in x k direction, as shown in Figure 1. It is defined as J k =  Γ ( Wn k − t j u j , k ) d Γ (7)