PSI - Issue 33

7

Naoya Oie et al. / Procedia Structural Integrity 33 (2021) 586–597 Oie, N. / Structural Integrity Procedia 00 (2019) 000–000

592

� � ������� �� � � � � �7� The approximate solution using the least-squares method yielded ������� = 419.5 MPa, n = 0.149, and α = 0.00092. The relationship between the plastic strain and true stress obtained from fitting Swift's equation was read as the equivalent plastic strain and equivalent stress and used as the stress-strain relationship in the FEM analysis. To avoid the effects of element division in the calculation of Weibull stress, which will be discussed later, element division near the notch tip was standardized for FEM analysis of the eight types of specimens. In the vertical plane in the thickness direction, elements were divided into 30 µm long elements within a radius of 500 µm from the notch tip. The elements used were 8-node hexahedral isoparametric elements, and the software used was Abaqus2018. 3.2. Weibull stress In this study, the critical Weibull stresses defined by the Beremin model (1983), the Bordet model (2005), and the Yoshizu model (2014), which are based on the weakest link theory (1939), are used as evaluation models for the fracture toughness of materials in three-point bending tests. These models evaluate the brittle fracture of the material probabilistically, and the fracture probability of the specimen or the whole structure in all three models is obtained using equation (8). � � � � ��� �� � � � � � � �8� The Weibull stress at the initiation of brittle fracture is referred to as the critical Weibull stress, and the calculation method and equation numbers for Weibull stress in the Beremin, Bordet, and Yoshizu models are shown in Table 3 below. The value of β in equation (11) is set to 1.5 in this calculation. The derivation of these equations is omitted, but they are all transformed by defining the failure probability of a small volume and considering the weakest link mechanism for the overall failure probability.

Table 3. Formulae to calculate Weibull stress. model

formulae � � �� �� � � � � � � �� �� �� � � � � � �� �� � ��� �� � � � � � � � �� � � � � � �

Beremin

(9)

��

(10)

Bordet

� �

��

Yoshizu (11) , � , and � of each element obtained in the FEM analysis were used in the calculations of equations (9)-(11). The volume integrals in equations (9)-(11) consider the volume of each element to be a small volume. The quasi CTOD and Weibull stress were automatically calculated at each intermediate time step generated in the FEM analysis, and the critical Weibull stress was calculated by linear interpolation using the information of time steps adjacent to �������� obtained in the experiments shown in Chapter 2. Based on equations (9)-(11), the critical Weibull stresses, ���� for eight different specimens were calculated using the Beremin, Bordet, and Yoshizu models. The value of m in each model was approximated as follows: Beremin: m = 21, Bordet: m = 31, and Yoshizu: m = 31 using the maximum likelihood estimation procedure of m presented by Minami et al. (2006).