PSI - Issue 68

Berkehan Tatli et al. / Procedia Structural Integrity 68 (2025) 1140–1146

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Tatli and Yalcinkaya / Structural Integrity Procedia 00 (2024) 000–000

3

where g 0 is the initial slip resistance, η ˆ α GND is the GND density on a slip system ˆ α . Incorporating the density of GNDs into the model using an intrinsic length scale parameter causes the material behavior to depend on the size and scale of deformations. This approach is employed to model the size-dependent material response in Section 3 (see Gu¨nay et al. (2024) for further details).

2.2. Park-Paulino-Roesler (PPR) Cohesive Zone Model

The Park-Paulino-Roesler (PPR), an intrinsic, mixed-mode, potential based cohesive zone model, is employed in this thesis to simulate the initiation and propagation of the cracks. The PPR model depends on four fundamental independent parameters in the normal and shearing fracture modes: cohesive strength, fracture energy, the shape of the softening curve, and the initial slope of the traction–separation relationship. The model formulation is omitted here for the sake of simplicity; for further details, please refer to Park et al. (2009). Figure 1a and 1b presents the traction-seperation laws in both normal and tangential directions defined for the numerical examples presented in Section3.

(a) Normal Traction-Separation Curve

(b) Tangential Traction-Separation Curve

Fig. 1: Traction-separation laws utilized in the numerical examples

2.3. Hydrogen Transport Model

Latticedi ff usion of the hydrogen is forced by a chemical potential gradient ∇ µ . Onsager coe ffi cients ( L i j ) relate the chemical potential gradient ( ∇ µ ) to the mass flux ( J i ), assuming that only fluxes between lattice sites are considered, through the following relation:

J L = − L LL ∇ µ L , where, L LL = D RT C L

(3)

where D is the coe ffi cient of di ff usion, R is the gas constant, T is temperature and C L is the lattice concentration of the hydrogen atoms. Equation of chemical potential of the hydrogen is given as follows, see Di Leo and Anand (2013) fore more details:

µ = µ 0 + RT ln

θ L 1 − θ L −

V H σ H

(4)

where σ H is the hydrostatic stress, µ H is the partial molar volume of hydrogen. The total flux of the hydrogen di ff usion can be readily obtained by substituting 4 into 3 and rearranging the 0 is the standard state chemical potential, and V