# Ductile Fracture Behavior of Mild and High-Tensile Strength Shipbuilding Steeds
## 1 Introduction  
Ductile fracture resistance is an important structural performance requirement for the steel components of naval structures. As discussed by [PERSON] et al. [1], computational tools are capable of implementing a range of elastic-plastic constitutive relations and ductile fracture initiation models for analyzing the structural problems relevant to ship steels that involve considerable strain beyond moderate plasticity, such as an analysis of the plastic zone ahead of the crack tip and, on a larger scale, the plasticity generated due to forming [2, 3] or extreme loads. Therefore, there is a demand for ductile fracture models that can be calibrated and implemented easily in finite element analysis but represent the micro-mechanisms of the failure process as accurately as possible. Fracture initiation in ductile metals depends strongly on the stress state. Earlier studies on micro-void-based fracture mechanisms highlighted the role of the hydrostatic pressure (or the first invariant of the stress tensor) on void growth [4, 5]. Experiments on notched round bars have been conducted to achieve a range of axisymmetric tension stress states. The failure initiation results have been correlated through stress triaxiality (hydrostatic stress normalized with equivalent von Mises stress) and equivalent plastic strain [6, 7], and micro-mechanism based models by [PERSON] [4] and [PERSON] [5] have been verified, particularly for high-stress triaxialities. A parallel development is the porous plasticity model by [PERSON] [8], which was later enhanced by [PERSON] [9], and [PERSON] and [PERSON] [10]. Porous plasticity models incorporate the void volume fraction into the constitutive behavior of the material, which evolves with void nucleation, growth, and coalescence mechanisms in a phenomenological manner [10]. On the other hand, [PERSON] and [PERSON] [11] proposed a purely empirical model using a stress triaxiality sensitivity factor similar to the [PERSON] and [PERSON] model and included the strain rate effect andthermal softening terms in a multiplicative format. [PERSON] and [PERSON] [12] proposed an alternative model based on the maximum principal tensile stress to predict fracture initiation in bulk metal forming at a low and negative stress triaxiality. [PERSON] et al. [13] modified the Cockcroft-Latham (CL) model slightly by normalizing the maximum principal tensile stress with the von Mises equivalent stress. CL model incorporates both the hydrostatic pressure (stress triaxiality) sensitivity and the deviatoric stress state sensitivity (so-called Lode angle sensitivity) in an implicit manner. [PERSON] [14], who first interpreted the experimental results of [PERSON] and [PERSON] [15] correctly, reported the strong dependence of ductile fracture on the deviatoric stress state or the third invariant of the deviatoric stress tensor, i.e., the tendency of one of the principal stresses to dominate over the other two. The test results by [PERSON] and [PERSON] [15] indicated that the equivalent plastic strain to fracture does not decrease monotonically with increasing stress triaxiality, particularly at a low stress triaxiality range bounded between pure shear and uniaxial tension, and between uniaxial tension and equi-biaxial tension. [PERSON] [14] emphasized that a second measure of the stress state, which is the Lode parameter representing the third invariant of the deviatoric stress tensor, is needed to discriminate between axisymmetric and shear-dominated stress states. Some other test results, such as those by [PERSON] and [PERSON] [16] and [PERSON] [17], support [PERSON]'s arguments. Consequently, [PERSON] [18] proposed a damage plasticity model for ductile fracture, which is dependent on the hydrostatic pressure and the Lode angle. In addition, [PERSON] [19] developed a Gurson-type model that included the void shearing mechanism because of the Lode angle effect.