PSI - Issue 52

Tomislav Polančec et al. / Procedia Structural Integrity 52 (2024) 348 – 355 Tomislav Polančec, Tomislav Lesičar, Jakov Rako / Structural Integrity Procedia 00 (2019) 000 – 000

350

3

t

prescribed surface force vector

UMAT XFEM

user-material

u

displacement field

extended finite element method

Greek α

backstress tensor

decrease rate of kinematic hardening

k  

strain tensor

p ekv 

equivalent plastic strain

,  

Láme constants

poisson ratio





phase-field parameter

H

history field



energy density accumulation variable material dependent fatigue parameter

  c  e  p 

specific fracture energy elastic strain-energy density plastic strain-energy density

free energy functional of the body 

 σ

Cauchy stress tensor initial yield stress n -dimensional body

0 y 



2. The phase-field method As mentioned in the introduction, problem of tracking complex crack paths in PFM is facilitated by the diffusive band whose width is controlled by the length scale parameter l . PFM relies on the calculation of the PF variable, which gives smooth transition from fully intact ( ) 0  = to fully deteriorated material ( ) 1  = [23]. Within this paper, small strain theory, elastoplastic constitutive behaviour and brittle fracture is assumed. Therefore, the PF problem as a minimization of the internal energy potential  for a body  is defined as follows [22]: ( ) ( ) ( ) ( ) ( )   2 e p e p e e p e c , , d 2 d . g l              + −        = + + + +         (1) e  and p  denote elastic and plastic strain tensors, while ( ) g  is the quadratic degradation function which deteriorates the material stiffness ( ) ( ) 2 1 g   = − [26]. c  represents the energetic threshold as a constant specific fracture energy, characteristic to the threshold (TH) model. c G represents the critical value of the fracture energy density [12]. To ensure physical consistency, it is assumed that crack propagates only in the tensile loading state. Hence, the volumetric-deviatoric composition proposed in [27] is used. Accordingly, a decomposition of the elastic deformation energy ( ) ( ) ( ) e e e e e e       + − = + is implemented. The internal energy potential in tensile state is e p e p ( ) ( ) ( )       + + = + , while in the compressive state is e e ( ) ( )     − − = . After derivation procedure described in many references, such as [22], the obtained governing set of equations is: in ,  + =  σ b 0 (2) on ,  =  t σ n t (3) on , =  u u u (4)   2 1 in , l   −  + + =  H H (5) 0 on .    =  n (6)