A probabilistic model for the cumulative probability of failure by cleavage fracture with a material related length scale is developed in this study. The model aims at describing the random nature of fracture in ferritic steels in the brittle-to-ductile transition temperature region. The model derives from use of an exponential function to describe the distribution of microstructural entities eligible to take part in the fracture initiation process, where also a dependence on effective plastic strain is incorporated. A nonlocal stress measure, calculated as the average stress in a spherical volume, drives the contribution to failure probability of an infinitesimal material volume. The radius of the spherical volume enters as the material length in this model. This length has a significant influence on failure probability predictions in geometries exposed to strong stress gradients as found ahead of cracks. The material length is associated with a fracture toughness threshold value. In a fracture application three model parameters need to be estimated based on testing; a parameter directly related to the mean fracture toughness, a parameter that primarily is related to crack-tip constraint effects and the material length parameter. The model is explored in a parametric study showing model features in concord with typical features found in toughness distributions from fracture mechanics testing in the transition region.

This study presents a large experimental investigation in the transition temperature region on a modified A508 steel. Tests were carried out on single-edge-notch-bend specimens with three different crack depth over specimen width ratios to capture the strong constraint effect on fracture toughness. Three test temperatures were considered, covering a range of 85 °C. All specimens failed by cleavage fracture prior to ductile tearing. A recently proposed probabilistic model for the cumulative failure by cleavage was applied to the comprehensive sets of experimental data. This modified weakest link model incorporates a length scale, which together with a threshold stress reduce the scatter in predicted toughness distributions as well as introduces a fracture toughness threshold value. Model parameters were estimated by a robust procedure, which is crucial in applications of probabilistic models to real structures. The conformity between predicted and experimental toughness distributions, respectively, were notable at all the test temperatures.

A probabilistic model for the cumulative probability of failure by cleavage fracture was applied to experimental results where cleavage fracture was preceded by ductile crack growth. The model, introduced by Kroon and Faleskog [Kroon M, Faleskog J. A probabilistic model for cleavage fracture with a length scale – influence of material parameters and constraint. Int J Fract 2002;118:99–118], includes a non-local stress with an associated material related length scale, and it also includes a strain measure to account for the number of nucleated cleavage initiation sites. The experiments were performed on single edge cracked bend test specimens with three different crack lengths at the temperature 85 °C, which is in the upper transition region for the steel in question. The ductile rupture process is modelled using the cell model for nonlinear fracture mechanics. The original cleavage fracture model had to be modified in order to account for the substantial number of cleavage initiators being consumed by the ductile process. With this modification, the model was able to accurately capture the experimental failure probability distribution.

In this and a companion study (Kroon, M., Faleskog, J., 2005. Micromechanics of cleavage fracture initiation in ferritic steels by carbide cracking. J. Mech. Phys. Solids 53, 171–196), the initiation of cleavage fracture in ferritic steels is studied. The initiation is modelled explicitly in the form of a microcrack, which nucleates in a brittle carbide and propagates into the surrounding ferrite. The carbide is modelled as an elastic cylinder and the ferrite as an elastic viscoplastic material. The crack growth is modelled using a cohesive surface, in which the tractions are governed by a modified exponential cohesive law. The advancing microcrack, which has nucleated in the carbide, may either continue into the ferrite or deflect into the interface between the carbide and the ferrite. Special attention is given to the influence of the mode mixity factor ββ, which is defined as the ratio between the shear and tensile strength of the interface between the carbide and the ferrite. Crack growth in the interface occurs in shear mode and is driven by a fibre loading mechanism. For mode mixity values β⩽0.2β⩽0.2, the crack deflects into the interface. The results indicate that crack growth in the interface can have a profound influence on the macroscopic fracture toughness of ferritic steels.