Relating matrix stress to local stress on a hard microstructural inclusion for understanding cleavage fracture in high strength steel

Int J Fract https://doi.org/10.1007/s10704-021-00587-y (0123456789().,-volV) ( 01234567 89().,-volV) ORIGINAL PAPER Relating matrix stress to local stress on a hard microstructural inclusion for understanding cleavage fracture in high strength steel Quanxin Jiang . V. M. Bertolo . V. A. Popovich . J. Sietsma . Carey L. Walters Received: 26 May 2021 / Accepted: 12 August 2021 Ó The Author(s) 2021 Abstract Macroscale cleavage fracture toughness of high strength steels is strongly related to the fracture of hard microstructural inclusions. Therefore, an accurate determination of the local stress on these inclusions based on the matrix stress is necessary for the statistical modelling of macroscale cleavage fracture. This paper presents analytical equations to quantitatively estimate the stress of the microstructural inclusions from the far-field stress of the matrix. The analytical equations account for the inclusion shape, the inclusion orientation, the far-field stress state and matrix material properties. Finite element modelling of a representative volume element containing a hard inclusion shows that the equations provide an accurate representation of the local stress state. The equations are implemented into a multi-barrier model and compared with CTOD experiments with two different levels of constraint. Keywords Cleavage fracture  High strength steel  Microstructure  Hard inclusion Q. Jiang (&)  V. M. Bertolo  V. A. Popovich  J. Sietsma  C. L. Walters Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Delft, The Netherlands e-mail: C. L. Walters Structural Dynamics, TNO, Delft, The Netherlands Abbreviations a0 Initial crack length of SENB specimen B Thickness of SENB specimen ci Constant values used in equations (i is number or descriptive characters) Einclu Young’s modulus of inclusion Ematrix Young’s modulus of matrix fa Stress concentration factor of inclusion f(D) Distribution density of the grain major axis mm K Ia Crack arrest parameter of grain boundary K Hardening parameter of matrix L Length of SENB specimen nL Hardening exponent of matrix N Amount of potential cracking nuclei per unit volume Pf Fracture probability Principal semi-axes of spheroidal inclusion Ri (i = 1, 2, 3) S Span of SENB specimen in the three-point bending test W Width of SENB specimen ep Von Mises equivalent plastic strain Remote plastic strain ep;matrix Threshold value of remote plastic strain ep;th g Remote stress triaxiality defined by the ratio of the hydrostatic stress to the equivalent Von Mises stress h Angle between inclusion’s principal semiaxis R1 and the remote first principal stress r1 123 Q. Jiang et al. r1;inclu r1;matrix req;matrix rcH ri ry s sp Representative inclusion stress Remote first principal stress Remote Von Mises equivalent stress Critical stress for hard inclusion Principal stress (i = 1, 2, 3) Yield strength Shear stress Remote maximum shear stress 1 Introduction Mechanical integrity assessment of steel structures frequently requires knowledge of their resistance to catastrophic failure by fast, unstable crack growth, expressed as fracture toughness. Ferritic steels exhibit a transition from ductile fracture modes at higher temperatures to brittle fracture at lower temperatures. Toughness at lower temperatures and the transition temperature region are related to transgranular quasicleavage fracture, which will be called cleavage in this paper. Many material requirements, for example Charpy test results, are related to the prevention of cleavage. The need for more accurate cleavage modelling is particularly acute for a new generation of high- and very high-strength steels (yield strength of 500 to 1000 MPa) because they generally have lower toughness, and therefore, a lower safety margin. Furthermore, these classes of steels obtain their favorable properties through their complex, multiphase microstructures, which complicates microstructural modelling of cleavage-driven failure. As a highly localized phenomenon, cleavage fracture exhibits strong sensitivity to material characteristics at the microstructural level, dependent on material and structure fabrication, and it is coupled with a constraint effect originating from the macroscopic stress state. It is generally accepted that the micromechanism of cleavage fracture can be described by three critical events: particle fracture, propagation of a particle-size crack and the propagation of a grain-size crack (Lin et al. 1986; Martı́nMeizoso et al. 1994; Pineau 2008; Chen et al. 2010; Pineau et al. 2016; Namegawa et al. 2019). A summary of the models describing micromechanisms of cleavage fracture can be found in Jiang et al. (2019). Table 1 shows a schematic representation of these 123 three critical events and the corresponding parameters to define cleavage criteria. As the first in the chain of events that leads to cleavage, fracture of the hard particle requires special attention. Second-phase particles are particles which do not belong to the matrix phases. They are present because of the alloying elements that are added for hardenability, yield strength, and other properties. Carbides, brittle inclusions, and M-A constituent are examples of second phase particles that are widely reported as being detrimental for cleavage fracture in steels (Ray et al. 2012; Chen and Cao 2015; Jia et al. 2017; Tankoua et al. 2018). Although steels also have soft inclusions like MnS, they mostly affect the ductile failure mode and are not the focus in this paper. It has been observed that larger inclusions and inclusion clusters act as weakest features in the microstructure, allowing brittle crack initiation and propagation (Ghosh et al. 2013; Popovich and Richardson 2015; Pallaspuro 2018; Bertolo et al. 2020). The probability that cracks initiate in a given particle depends on the particle size, shape, volume fraction and orientation of the elongated particles with respect to the applied stress (Lindley et al. 1970; Ray et al. 1995; Bordet et al. 2005; Chen et al. 2010; Miao and Knott 2016). Therefore, it is important to be able to estimate the local stresses on hard inclusions based on the global loading in order to be able to capture the first stage of cleavage fracture, especially in high-strength steels. Studies on the stress distributions within or around inclusions have been performed extensively (Huang 1972; Lee and Smivri 1981; Wilner 1988; Ramakrishnan 1996; Lee and Mear 1999; Lauke and Schüller 2002; Huang and Li 2005; Gao 2008). These works contributed to a good understanding of the stress distribution within or around a hard inclusion embedded in an elastic–plastic matrix. It is found that there is a critical aspect ratio at which interface debonding changes to particle fracture, and the remote stress triaxiality has a significant effect on this transition (Lee and Mear 1999). However, methods that can directly determine the local stress on a hard (...truncated)