PSI - Issue 42

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 42 (2022) 1651–1659 Tuncay Yalc¸inkaya et al. / Structural Integrity Procedia 00 (2019) 000–000

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When ∆ n attains the critical opening displacement, δ nc , the value of normal traction is noted as the normal cohesive strength ( σ max ). Similarly, when ∆ t attains the critical sliding displacement, δ tc , the value of normal traction is noted as the tangential cohesive strength ( τ max ). The energy constants are defined as Γ n =      α m   m , φ n < φ t − φ n   α m   m , φ n φ t , Γ t =      β n   n , φ t φ n − φ t   β n   n , φ t > φ n (11) with

β ( β − 1) λ 2 t 1 − βλ 2 t .

α ( α − 1) λ 2 n 1 − αλ 2 n ,

n =

(12)

m =

Here, λ n , λ t are initial slope indicators. The gradients of the PPR potential lead to the traction vector,

− α   1 −

+ φ t − φ n   (13)

Γ n δ n   Γ t δ t  

m   1 −

∆ n δ n  

α   m α +

∆ n δ n   m − 1

∆ n δ n  

α − 1   m α +

∆ n δ n  

m  

×   Γ t   1 − |

∆ t | δ t  

β   n β + |

∆ t | δ t   n

T n ( ∆ n , ∆ t ) =

− β   1 − |

+ φ n − φ t   ∆ t | δ t |

n   1 − |

∆ t | δ t  

β   n β + |

∆ t | δ t   n − 1

∆ t | δ t  

β − 1   n β + |

∆ t | δ t  

n  

×   Γ n   1 −

∆ n δ n  

α   m α +

∆ n δ n   m

T t ( ∆ n , ∆ t ) =

(14)

where T n is the normal cohesive traction, T t is the effective tangential cohesive traction. Cohesive zone model is implemented by a traction-separation relation defined by φ n , φ t , σ max , τ max , λ n , λ t , α and β . For more details about the model see Cerrone et al. (2014). In this work, the cohesive zone elements are inserted solely at the martensite/ferrite grain boundaries, through a script identifying the locations. Determination of cohesive zone parameters is a lengthy process composed of a ferrite/martensite inclusion problem, an indentation study focusing on the hardness at ferrite/martensite boundary as well as full RVE simulations. In addition to the numerical studies, literature presents certain approaches and values for the required parameters. Hosseini-Toudeshky et al. (2015) conducts nanohardness measurements and identifies the normal cohesive strength between the martensite and ferrite as σ max = 1100 MPa which is quite close to the numerically obtained value of σ max = 1160. Scheider and Brocks (2003) proposes that tangential cohesive strength can be taken as τ max = 350 MPa for the same interface, that is identified to be τ max = 385 here. Moreover, Hosseini Toudeshky et al. (2015) assumes that the normal and the tangential fracture energies to be equal, each of which is taken to be 400 N/m in this study. Remaining parameters that define the traction-separation relation are identified as α = 4 . 5, β = 2 . 7, λ n = 0 . 01 and λ t = 0 . 02 for the initial qualitative study. Details of the inclusion and indentation study are not presented here for the sake of compactness. See Table 3 for the summary of material parameters used for cohesive zone relations in the numerical examples.

σ max (MPa)

τ max (MPa)

φ n (N/m)

φ t (N/m)

α

β

λ n

λ t

1100

385

400

400

4.5

2.7

0.01

0.02

Table 3: Material properties of the cohesive element.

In the FE computations, bulk elements are modelled with 10-noded second-order tetrahedral elements (C3D10), while compatible 12-node triangular cohesive elements are utilized between the interfaces. 3D microstructures were obtained through Voronoi-based tessellation generator software Neper (Quey et al. (2011)). An in-house MATLAB script is developed to modify the input file. The script identifies the interfaces between martensite and ferrite grains by considering the element connectivity in the mesh and inserts corresponding cohesive elements. The modified input file is then run through ABAQUS together with the material subroutines for plasticity, user element subroutine for the cohesive zone model and user-defined field for the damage model. The results are presented and discussed in the next section.