PSI - Issue 58

P.C. Sidharth et al. / Procedia Structural Integrity 58 (2024) 115–121 P.C. Sidharth and B.N. Rao / Structural Integrity Procedia 00 (2019) 000–000 3 existence of the crack. In this context, the symbol  denotes the phase-field order parameter, while Gc signifies the critical fracture energy release rate specific to the material. The function  characterizes the crack surface and is influenced by both the phase-field parameter  and its spatial fluctuations. The balance equations of the phase-field model are derived through the application of the Gauss divergence theorem, as follows. 117

1

 

  

 2 1      

   H  ε

   σ b

0

0

c G l 





(3)

l

In this particular model, a scalar auxiliary field denoted as  , ranging from 0 to 1, serves as the damage indicator. In this framework, a  value of zero signifies undamaged or intact material, whereas a  value of one corresponds to completely damaged material. To establish boundary conditions, Dirichlet-type conditions are utilized to prescribe displacements in the undamaged region, while Neumann-type conditions are employed to prescribe tractions in the damaged region. In this study, the spectral decomposition of strain energy into tensile and compressive parts proposed by Miehe et al. (2010) is used, along with isotropic constitutive equations, following the hybrid implementation of Ambati et al. (2015). 2. Methodology To solve phase-field fracture problems using the finite element method, the discretization strategy is to divide the domain into 4-noded quadrilateral elements. Weak forms for equations are derived, leading to the creation of residual and stiffness matrices. The displacement field is represented using standard linear finite element (LFE) shape functions, while exponential finite element (EFE) shape functions are used to approximate the phase-field parameter. The resulting system of nonlinear equations is solved iteratively through the Newton-Raphson method. The coupled equations of the phase-field model are then implemented and solved within MOOSE, which stands for Multiphysics Object-Oriented Simulation Environment. The Multiphysics Object-Oriented Simulation Environment, developed by Idaho National Laboratories (INL) in the USA, is a cutting-edge open-source, parallel finite element framework. This powerful simulation environment is equipped with the Jacobian-Free Newton-Krylov (JFNK) solver, which enhances computational efficiency and accuracy. The software allows a wide variety of physics modules within this framework, making it versatile for a multitude of applications. For specialized needs, MOOSE sports dedicated modules for tackling complex phenomena, including fracture analysis, phase field modeling, and XFEM simulations. This approach follows a staggered scheme where displacement and phase degrees of freedom are alternately calculated in subsequent load increments. This methodology allows for the efficient resolution of phase-field fracture problems. 2.1. Exponential finite element shape functions The motivation behind the construction of exponential finite element (EFE) shape functions (Kuhn et al. 2010) arises from the exponential nature of the analytical solution. To accurately capture this exponential behavior, the authors suggest the use of EFE shape functions instead of the conventional standard linear finite element (LFE) shape functions. A visualization of EFE shape functions is shown in Fig. 1.

(1 ') 

(1 ') 

 

 

  

  

  

  

exp

1

exp

1









(4)

4

4

( ', ) 1  

( ', )  

e

e

N

N

 



1

2

2       

2       

exp

1

exp

1

 

 

The exponential finite element (EFE) shape functions described in equation (4) are defined by a scalar parameter denoted as  . This parameter signifies the ratio of the element edge length, h , to the length scale parameter, l .