PSI - Issue 42

Minghua Cao et al. / Procedia Structural Integrity 42 (2022) 777–784 Minghua Cao et al. / Structural Integrity Procedia 00 (2019) 000–000

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Table 1 Geometry of numerical models for surface-evolution analysis

Type of graphite inclusion ( μ m) ( μ m) External dimensions of matrix ( μ m) Spherical graphite 15

Model

Notation

A B C D

Spherical_H7.5

Ver_H4

Vermicular graphite

3 4 8

15

30×30×15

W4H8 W8H8

Flake graphite

The constitutive parameters of the graphite and matrix phases were selected based on our previous work (Palkanoglou et al., 2022). A damage model was employed to describe the constitutive behaviour of graphite to limit its response to the elastic region. The onset of damage was considered when the plastic-strain-based criterion was fulfilled. After initiation, the material stiffness degraded gradually to 0. The integral ductile damage criterion is given as (Hooputra et al., 2004) = ∫ � � � , �̇ � = 1 , (11) where is the state variable increasing monotonically. At every increment, the state variable increment should meet the condition ∆ = ∆ � � ( , �̇ ) ≥ 0 , (12) where ̅ � , ̅̇ � is the plastic strain at the onset of damage and ̅̇ is the plastic strain rate. Stress triaxiality is the main parameter used in this criterion, defined as = − , (13) where is the pressure level and is the von Mises (equivalent) stress. The finite element is removed from the numerical model if all the section points at an integration point cannot carry the load (damage-initiation point) (Collini et al., 2019). At the point that material’s stiffness is fully degraded, the material loses its load-carrying capacity. After the damage initiation, a linear damage evolution was considered (Fig. 4).

Fig. 4. Scheme of progressive damage evolution (Collini et al., 2019).

Periodic boundary conditions (PBCs) control the nodes on the corresponding surfaces with the same level of displacement but in opposite directions. PBCs are commonly used in finite-element simulations to analyse the material behaviour with the RVE at the microscale and mesoscale (Garoz et al., 2019). PBCs allow the use of a small domain to represent the corresponding infinitely large system to simulate the deformation of the RVE model, permitting distortion of boundary surfaces (Omairey et al., 2019). The PBCs for a pair of any two points and + on corresponding surfaces with distance between them are given by the following equations: ( + ) = ( ) + ̅ , (14) ( + ) = − ( ) , (15) where and are the displacement and traction at , respectively, and ̅ is the average infinitesimal strain over the volume (Drago and Pindera, 2007). Additionally, in this study, another set of fully fixed boundary conditions