# Determination of the Fracture Locus of a Cor-Ten Steel at Low and High Triaxiality Ranges
## 1 Introduction  
The failure criteria for materials play a very important role in the application of solid mechanics. Commonly, the failure of a metallic component is considered the onset of yielding [1]; thus, modelling its failure does not require particularly complex methodologies beyond performing a tensile test on the material. However, for applications in which plastic deformation may be tolerated or even desired (as is the case for manufacturing technologies involving cold-working or energy absorption applications), a more complex failure criterion must be devised to take into account the non-linear behavior of the material. This is the case for Cor-Ten steels, also called weathering steels, a class of steels whose application is growing, particularly in the fields of architecture and crash barriers. These steels are known mainly for their good mechanical and corrosion resistance properties but also forthe appealing color of their oxide, which makes them suitable for architectural purposes. It has been extensively shown in the literature that crack formation in the plastic field is governed by at least two quantities, namely, the equivalent plastic strain \(\varepsilon_{ps,eq}\) and the stress triaxiality \(\eta\)[2]; more complex models also involve the Lode parameter to fully characterize the stress state in the material [3]. Stress triaxiality (see Equation (1)) is defined as the ratio between the hydrostatic stress, namely, the average value of the principal stresses (\(\sigma_{1},\,\sigma_{2}\) and \(\sigma_{3}\)), and von Mises equivalent stress, \(\sigma_{VM}\):  
\[\eta=\frac{1}{3}\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{\sigma_{VM}}. \tag{1}\]  
Regarding the Lode parameter, there is a little bit of confusion in the literature since three quantities are used interchangeably: the normalized Lode angle \(\overline{\theta}\), the Lode angle parameter \(\xi\), and the Lode parameter \(L\)[4]. These quantities are all related to one to another, as shown in Equations (2)-(4):  
\[L=\frac{2\sigma_{2}-\sigma_{1}-\sigma_{3}}{\sigma_{1}-\sigma_{3}}, \tag{2}\]  
\[\xi=\frac{L(L+3)(L-3)}{\left(L^{2}+3\right)^{3/2}}, \tag{3}\]  
\[\overline{\theta}=1-\frac{2}{\pi}\text{arccos}\xi. \tag{4}\]  
In the present study, the normalized Lode angle \(\overline{\theta}\) was chosen since it is the most common in the literature and thus easier to compare. In order to successfully model the failure and damage of materials subjected to plastic deformation, it is necessary to determine the combination of the aforementioned parameters that leads to crack formation [3,5]. This combination or, better still, relation is called the fracture locus. The determination of the fracture locus is not trivial because one needs to induce different values of stress triaxiality and Lode angle parameters in the tested specimens to cover the whole range of values that could be of interest regarding the material during its application. Moreover, these quantities can be found only by means of numerical methods, which must be suitably supported by experimental testing. [PERSON] and [PERSON] were pioneers in suggesting a more complex trend for the fracture locus of metallic materials based on their work on an aluminum alloy, thus tackling the common assumption that the fracture locus can be completely described by a monotonic decreasing function [2]. During the extrapolation of the fracture locus, these researchers found three distinct trends:  
1. In the range of negative triaxiality (from \(-1/3\) to \(0\)), the fracture locus can be expressed by an exponential decreasing function. 2. In the range of low stress triaxiality (from \(0\) to \(0.4\)), the fracture locus is best fit by a parabolic function and reaches a local maximum. 3. In the range of high stress triaxiality (from \(0.4\) to \(0.95\)), the fracture locus starts to decrease again according to an exponential function. 4.