PSI - Issue 42

Karlo Seleš et al. / Procedia Structural Integrity 42 (2022) 1721–1727 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

1723

3

( ) d . ,     



( )   =

 (0.4) It can be seen from Eq.1 that crack can appear in both tensile and compressive state, which is not physical. To prevent crack initiation and propagation in compressive stress state, an decomposition is introduced. Thus, the deformation energy is decomposed and only positive part is degraded as follows: ( ) ( ) ( ) ( ) e e e , g      + − = + ε ε ε (0.5) In this work, volumetric-deviatoric split is used proposed by [3] as: ( ) ( ) ( ) ( ) ( ) 2 2 1 e dev dev 2 2 2 1 e 2 : tr : , : tr . n n        + + − − = + + = + ε ε ε ε (0.6) With describing all parameters, Eq.1 can be written: ( ) ( ) ( ) ( ) 2 2 2 e e c , 1 d 2 d , l        + −    = − +  + +             ε ε ε (0.7) with ( ) ( ) 2 1 g   = − and ( ) 2 2 1 1 AT-2 2 , l l       = +      as proposed by [4,5]. This model is noted as AT2 model. As it can be seen from homogeneous solution, the phase-filed parameter evolves as soon as loading starts. To postponed the phase-field evolution threshold parameter is introduced where c 3 c 8 2 G l  = serves as energetic threshold (TH model). The length parameter for both models can be calculated as: ( ) ( ) c c AT-2 TH 2 2 max max 27 3 , . 256 4 2 G E G E l l   = = (0.8) where is visible that TH model has 5 times greater length scale parameter and allows coarser meshes. 2.2. Finite element implementation The described phase-field formulation is implemented into commercial finite element package ABAQUS. The FE model is based on weak form of the equations of the energy potential. The FE model can be written: (1.9) This decoupled equation system corresponds to staggered algorithmic approach. The basic idea is to solve decoupled system iteratively: first solve one field at iteration ii and then solve other field with estimated solution from iteration ii-1. This implementation is described in detail in authors previous work [16,17]. Using this staggered algorithmic approach, it is possible to implement commercial FE-s and couple them with phase-field formulation. This is accomplished with so-called three-layered system which consists of three sets of elements: two user element (one for PF calculation and one for residual controlling) and one commercial FE for displacement calculation. This implementation can utilize full potential of commercial Fe-s and gives a number of benefits: use of different interaction properties like contact or fluid-solid interaction, different solver like Newton Rapshon or Riks, automatic incrementation, element deletion, parallelization and many more. For more details visit [18]. 2.3. Experimental investigation Experimental investigation is conducted on polymethylmethacrylate (PMMA) which is also known as acrylic glass. The PMMA is transparent thermoplastic often used in replacing conventional glass. In the engineering, it is often used for experimental validation since it has very homogeneous microstructure and little or no defects. The investigation of fracture behavior is conducted on standard specimens. In the beginning, to confirm material parameters the uniaxial tension tests are conducted according to ASTM D638-14. The specimen is laser-cut from single plate and has square cross-section with dimensions 5.75 x 6 mm. The tests are conducted on servo hydraulic INSTRON device with constant loading rate of 1 mm/min. The mean results of uniaxial test are shown on engineering stress strain diagram (Fig 1) and in the Table 1.