PSI - Issue 28

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Eda Gok et al. / Procedia Structural Integrity 28 (2020) 2043–2054 Gok et al./ Structural Integrity Procedia 00 (2019) 000–000

Fig. 2. Representation of bond-based PD model.

The micro potential energy is the energy in a single bond at a given point. It depends on the relative displacement vector η through the distance between the deformed points. This scalar valued function can be expressed as: ( , ) ( , ). w w        (3) Combining Eq. (2) and (3) and differentiating with respect to η shows that the PD force density vector, f is a function dependant on the reference position vector ξ , relative displacement vector η , bond constant of the material c , and the value of stretch s in the absence of thermal loading. This relation is given in Eq. (4). s, c     f(η, ξ) η + ξ (4) where c is called as the PD bond constant of the material and s is the stretch of the bond and may be calculated as: . s   η + ξ ξ ξ (5) The PD bond constant c can be obtained by equating the strain energy density for PD theory and the strain energy density for CCM. As e result of this, c may be obtained as: (6) where K is the bulk modulus of the material, and  is the radius of the horizon. PD formulation is implemented in ABAQUS using truss elements (T3D2) that represent the PD bonds as suggested by Macek and Silling, (2007). Mesh of truss elements are generated with an in-house code developed in MATLAB. Elastic modulus   t E and cross sectional area of truss elements ( ) t A are taken as in Macek and Silling (2007): 4 2 , , t t E c x A x     (7) where x  indicates the grid size. 2.2.1. PD Interface Failure Model Material failure can be considered in PD theory by breaking the bonds when the stretch between material points reaches a critical value. The material model in the original PD formulation Silling and Askari (2005) is brittle (see the black line in Fig. 3a). However, the interface failure problem requires a softening behaviour that is considered in traction separation laws in CZM. Therefore, the original PD material model is modified as a bilinear model (see the red line in Fig. 3a). In the modified formulation, the micro potential at the failure can be calculated as: 0 1 , w w w   (8) where 0 w and 1 w are given in Eq. (9) and Eq. (10): 4 18 , c K  