PSI - Issue 42

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

1722

2

1. Introduction Due to increasing computational power in last decade, a numerical modelling experienced strong take-off. A whole new set of different approaches for fracture and fatigue modelling is being proposed. The befits of numerical modelling in comparison with experimental investigation are still notable: the cost and time savings, ease of use are almost immeasurable compared to experiment. But even though numerical modeling is attractive, the accuracy of the novel mathematical approaches must be confirmed with experimental data usually on a homogeneous material samples. Since Francfort and Marigo [1] proposed the phase-field (PF) variational approach to brittle fracture two decades ago, a big interest started to form. The phase-filed method is diffusive continuum method commonly related to the system which consists of different phases. In the case of the fracture, or fatigue modelling the crack discontinuity is approximated by a diffusive band through the smooth transition. There are multiple reasons for its popularity but thermodynamical constituency and lack for any ad hoc criteria made PF method one of most popular methods for crack modeling. A significant number of PF methods for brittle fracture has been formulated [2 – 5]. Except the brittle fracture the ductile fracture is also formulated by Ambati et al. [6] and Aldakheel [7]. Nevertheless, the PF method has been proposed for different complex models both 2D [8] and 3D [9], homogeneous and heterogeneous microstructure. To model such complex micro- and macrostructure the validation is still required on simple homogenous samples. A number of different formulations were verified in accordance with standard benchmark results, yet few of them have been properly validated. Nguyen et al. [10] validated the brittle fracture and material parameters on a plaster material while Pham et. al [11] used polymethylmethacrylate (PMMA) polymer for validation. Even though validation is conducted by a few authors, the proposed formulation, consisting of both user finite elements and finite elements from commercial package has to be validated on simple samples so complex heterogeneous models can be properly investigated. To cover the first step in PF modeling using highly homogeneous PMMA polymer the validation of authors formulation has been caried out. Herein, the brief outline of the formulation will be described along with description of used standard samples and experimental parameters. The comparison between numerical and experimental data is given and short discussion is obtained at the end. 2. Materials and methods 2.1. Phase-field formulation According to Francfort and Marigo [1] variational approach as an upgrade of Griffith’s fracture theory, minimization of internal energy functional  is governing process as follows: ( ) b s / d d . c G      =  +  =  +    ε (0.1) where b  is body’s bulk energy and s  as dissipated energy induced by fracture. In the case of brittle fracture, material behavior is assumed as linear elastic where elastic strain energy density ( ) e  ε is given as ( ) ( ) ( ) 2 2 1 e 2 tr tr    = + ε ε ε ( ,   are Láme constants and ε is strain tensor). According to Griffith’s theory, (0.2) In order to account the loss of the stiffness induced by fracture propagation the bulk energy is regularized using monotonically decreasing degradation function ( )   0,1 g   as follows: ( ) ( ) ( ) , d d . b g        =  =     ε ε (0.3) Also, sharp crack is described by scalar function ( ) / x l x e  − = where l is length scale parameter, or width of diffusive crack. A functional ( )   is introduced as: material fail when reaching critical value of fracture energy density G c . In order to describe crack discontinuity crack density function ( ) ,     is introduced [2] where parameter  describes scalar damage field as follows: 0 intact state, 1 broken state.   = − = −