PSI - Issue 35

Deniz ÇelikbaŞ et al. / Procedia Structural Integrity 35 (2022) 269 – 278 D. C¸ elikbas¸ / Structural Integrity Procedia 00 (2021) 000–000

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In the SPH method, the volume of the plate is defined by nodes, which have the material properties of the plate. This numerical method is a griddles Lagrangian technique. Since the technique represents the volume by nodes, there is no need to model the voids to propagate the fractures. While SPH has some drawbacks on numerical instabilities and applying boundary conditions, it is still state-of-the-art modeling technique for large deformations Meng (2021).

2.2. Ceramic Material Model

In this study, ceramic tiles are modeled with (JH2) material model. This material model is developed by Johnson and Holmquist to model the mechanical behavior of brittle materials Johnson and Holmquist (1994). Later, JH2 material model implemented and validated in LS-DYNA by Cronin (2003). This material model accounts for the damaged and undamaged strength of the ceramic material. Figure 1 shows the intact and fractured strength curves for ceramic material. Therefore this material is appropriate to simulate the brittle nature of ceramic materials.

Fig. 1. Strength versus pressure curves for ceramic material Islam (2020).

The JH2 normalized material model can be expressed in Eq. (1) Livermore Software Technology Corporation (2018). σ ∗ = σ ∗ i − D ( σ ∗ i − σ ∗ f ) (1) where the superscript ∗ defines the normalized values. The strength parameters are normalized by the strength at Hugoniot elastic limit (HEL), the pressure parameters are normalized by the pressure at the HEL, and the strain rate is normalized by the reference strain rate. In Eq. (1), D designates the damage accumulated, and it can be computed from Eq. (2). D = ∆ ε p ε p f (2) where ∆ ε p represents the increase of plastic strain, and ε p f is the plastic strain to fracture defined in Eq. (3). ε p f = D 1 ( P ∗ + T ∗ ) D 2 (3) where D 1 and D 2 are material constants. P ∗ and T ∗ represents the pressure and maximum tensile pressure strength normalized by the pressure at HEL, respectively, as given in Eq. (4).

P P HEL

T P HEL

P ∗ =

T ∗ =

(4)

,

In Eq. (1), σ ∗ i represent the intact material strength given in Eq. (5), where A represents intact strength coe ffi cient, N represents intact strength exponent, and ˙ ε ∗ is the normalized strain rate. σ ∗ i = A ( P ∗ + T ∗ ) N (1 + C ln ˙ ε ∗ ) (5)