PSI - Issue 42

Jaynandan Kumar et al. / Procedia Structural Integrity 42 (2022) 806–812 Jaynandan Kumar, Anshul Faye / Structural Integrity Procedia 00 (2019) 000–000

808

3

2.2. Phase-field Model

Phase-field method is widely used to model the fracture and crack propagation in linear and non-linear solids. The free energy function ψ with degradation is defined as

ψ ( F , A m , A m ′ ; d ) = g ( d ) ψ 0 ( F , A m , A m ′ )

(5)

where, g ( d ) = (1 − d ) 2 + ϵ is the degradation function, where ϵ is the auxiliary parameter to avoid the numerical singularity. Therefore, the governing equation of the displacement field derived from equation (5) is expressed as

J div( σ ) + ρ 0 ¯ γ = 0

(6)

where, σ = g ( d ) σ 0 and ¯ γ is the body force.

d − l 2 ∆ d − (1 − d ) H = 0

(7)

where, H is defined as

ψ iso 0

ψ ani 0

(8)

H =

+

g iso

g ani

c / l

c / l

2.3. Finite Element Formulation

The constitutive model along with the phase-field model discussed section 2.1 and 2.2 is implemented in FEAP –Version 8.6.1j using user element subroutine. Four noded tetrahedral elements with full integration are used. The standard form of residual vector and sti ff ness matrix for an element is shown in the equations (9)-(13).

R φ

e = f int − f ext

(9)

f int =

e γ e d V

B e h

B T i

σ e − N i ρ

(10)

where, R φ is the residual vector associated with the deformation. f int is defined in equation (10) and f ext comes from the forces at the boundary. B is the standard strain matrix which is the derivative of shape function with respect to spatial coordinates and σ is the Cauchy stress.

e =

B e h N

e d V = 0

e − (1 − d e ) H e + ∇

i d

R d

T N i le 2 ∇ d

(11)