Ductile tearing of aluminium plates: experiments and modelling

This paper presents an experimental and numerical study on quasi-static ductile tearing of thin plates of the aluminium alloy AA6016 in three tempers. Depending on the temper, the main fracture mechanism in the plate tearing tests changes from grain boundary failure to coalescence of voids nucleated at the constituent particles. The experiments are complemented by nonlinear finite element simulations using an enriched Gurson–Tvergaard–Needleman (GTN) model to describe the material response. The onset of accelerated void growth is initiated either by incipient material softening (named the softening model) or by the occurrence of strain localization (named the localization model). It was found that strain localization takes place at a critical porosity fc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\text{ c }$$\end{document}, which depends on the current hydrostatic and deviatoric stress states. While the failure strain depends on the stress path, the critical porosity appears to be path independent. A third method is proposed (named the fc(T,L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\text{ c }(T,L)$$\end{document}model), where a critical porosity surface fc=fcT,L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\text{ c } = f_\text{ c }\left( T,L\right) $$\end{document} is used to determine when accelerated void growth starts. The surface is generated beforehand by solving for strain localization under proportional stress states defined by the stress triaxiality T and the Lode parameter L. By comparing the simulations to the experiments, it was found that the localization model performed well for a wide range of stress states. The softening model does not portray dependence on the Lode parameter and is therefore less versatile. The localization model and the fc(T,L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\text{ c }(T,L)$$\end{document} model gave similar predictions, but some minor differences were observed for two of the three tempers.


Introduction
Thin-plate structures made of steel or aluminium alloys are widely used in engineering applications. Such structures are highly susceptible to crack formation and propagation, which can substantially reduce their performance. While the present practice is to seek stronger materials, research indicates that high ductility inhibits crack propagation and is therefore often preferred over strength in applications where cracks can form (Granum et al. 2019;Elveli et al. 2022;Espeseth et al. 2022). Since strength and ductility are generally mutually exclusive properties (Ritchie 2011), one should carefully select the correct material based on the appli-cation. The ability to accurately describe ductile failure for a wide selection of materials is therefore vital in many design processes. When identifying the material failure parameters of a model, it is typical to employ tests that imitate the conditions (e.g., the stress history) of the application. Rarely, however, can these parameters be employed for other applications where the material is exposed to entirely different conditions. While sophisticated failure models that can perform well over a wide range of applications have been proposed (see, e.g., Li et al. 2011;Gruben et al. 2012;Mohr and Marcadet 2015), the added complexity due to a considerable increase in mechanical tests required for calibration makes them inappropriate for many industrial applications (Wierzbicki et al. 2005). As a remedy, a hybrid experimental-numerical approach that utilizes localization analyses has been proposed, which can substantially reduce the number of mechanical tests required in the calibration of the failure model (Bergo et al. 2020;Granum et al. 2021).
In ductile tearing of metal alloys, the governing failure mechanism is the nucleation of micron-sized voids that grow during plastic straining and eventually coalesce, resulting in local degradation and material separation. Micromechanically-based models that can describe these processes have found increasing interest in the last four decades. Perhaps the most widely used model was proposed by Gurson (1977), where a macroscopic yield condition able to capture material softening due to void growth was established based on the upper-bound limit analysis of a hollow sphere. The Gurson model is especially appealing due to its simple and closed-form formulation, where the only additional microstructural variable is the initial volume fraction of voids. The model was later extended by Chu and Needleman (1980) to account for void nucleation, improved by Tvergaard (1981Tvergaard ( , 1982 to achieve better agreement with unit cell calculations, and extended by Needleman and Tvergaard (1984) to account for rapid loss of load carrying capacity during void coalescence. Collectively, this model is known as the Gurson-Tvergaard-Needleman (GTN) model.
Various approaches to determine the onset of rapid void growth during the coalescence phase have been proposed in the literature. A practical and widely used criterion states that void coalescence initiates at a fixed critical porosity f c , which is normally regarded as a material constant (see, e.g., McClintock 1968;Sun et al. 1989;Chen et al. 2020). Directly monitoring the critical porosity in real materials proves very difficult in practice, and the validity of such a model has mainly been assessed numerically (see, e.g., Becker 1987;Koplik and Needleman 1988;Pardoen and Hutchinson 2000).
While a fixed f c is presumably acceptable for alloys with a well-defined microstructure and relatively low porosity, previous studies suggest that the onset of void coalescence also depends on the stress triaxiality (Brocks et al. 1995;Steglich and Brocks 1998;Pardoen and Hutchinson 2000;Kim et al. 2004;Tekoglu and Pardoen 2010;Lecarme et al. 2011;Tekoglu et al. 2012;Tekoglu 2015;Torki et al. 2017;Vishwakarma and Keralavarma 2019;Reboul et al. 2020). In recent years, the importance of the deviatoric stress state (represented by the Lode parameter) on the failure strain has been acknowledged (see, e.g., Bao and Wierzbicki 2004; Faleskog 2007, 2011;Roth and Mohr 2016), particularly in shear dominated problems at low stress triaxialities. Research on how the deviatoric stress state influences the porosity at void coalescence is more sparse. Nevertheless, cell calculations have given some indications that the porosity at failure is lower for generalized shear than for generalized tension and compression (Barsoum and Faleskog 2011;Vishwakarma and Keralavarma 2019;Chouksey and Keralavarma 2022). It should be noted that the fracture mechanism in shear-dominated problems is mainly internal shearing between voids, resulting in small elongated dimples with little void growth. Nahshon and Hutchinson (2008) and Xue (2008) suggested a phenomenological extension to the Gurson model which can account for softening due to void distortion and inter-void linking. This extension decreases the strains to failure in shear without modifying the porosity (which is now interpreted as a damaged parameter) at coalescence. More recently, Daehli et al. (2018) proposed a modification of the void evolution equation to enhance the predictions as compared to unit cell calculations.
Some degree of non-proportionality is always present in realistic loading conditions. Strongly non-proportional loading paths have been observed in front of a propagating crack (Bergo et al. 2021;Espeseth et al. 2022), but also in simpler material tests due to the formation of a diffuse or local neck (Bao and Wierzbicki 2004). Experiments show that the strain at failure strongly depends upon the loading path, where variations in the local stress state can lead to changes in the ductility (see, e.g., the overview by Thomas et al. 2016). Also, numerical studies using unit cells clearly demonstrate the importance of non-proportional loading paths on the predicted failure strain (see, e.g., Benzerga et al. 2012;Daehli et al. 2016;Tekoglu and Koçhan 2022). Although the influence of non-proportional loads on the failure strain has been extensively studied, the influence of non-proportional loads on the porosity at coalescence has received little attention. Becker (1987) concluded in his numerical study that the volume fraction is not as sensitive to the stress history as the failure strain, even though addressing this issue experimentally could be extremely challenging.
Large plastic deformation is frequently accompanied by the localization of strains into narrow bands. Using unit cell simulations, Tekoglu et al. (2015) studied strain localization and compared it to the void coalescence phase. While macroscopic localization occurs simultaneously with void coalescence for low to moderate stress triaxialities (T < 1), a clear separation exists between these two modes for higher stress triaxialities where macroscopic localization takes place before void coalescence. Consequently, the localization of plastic deformation into narrow bands of intense straining can in many cases be considered as a precursor to ductile failure. The condition for localization has been formulated in a quite general context by Rice (1976), where localization in an infinite homogeneous elastic-plastic solid is assumed to occur once a discontinuity in the deformation rates across a planer band becomes possible. Localization in a homogeneously deformed solid can arise due to instabilities in the constitutive equations which lead to the possibility of bifurcation modes, or it can be triggered by initial imperfections in a thin planar band where the stress and strain rates are allowed to take values different from their values outside the band. To trigger localization at reasonable stresses, a strain softening mechanism must be present (Rudnicki and Rice 1975), which can naturally be introduced into the constitute equations by considering a porous plasticity model, such as the Gurson model. While there is no need for additional parameters in the bifurcation analysis, the initial imperfection size must be specified in the imperfection band approach. The imperfection band approach, on the other hand, has been shown to predict lower strains at localization (Daehli et al. 2017) and is thus more conservative. Some recent studies adopting the imperfections band approach are given by, e.g., Nahshon and Hutchinson (2008), Gruben et al. (2017), Morin et al. (2018aMorin et al. ( , b, 2019, Daehli et al. (2017Daehli et al. ( , 2018Daehli et al. ( , 2022, Reddi et al. (2019), Bergo et al. (2020) and Granum et al. (2021), while the bifurcation analysis has been studied in, e.g., Besson et al. (2003), Haddag et al. (2009), Chalal andAbed-Meraim (2015) and Erice et al. (2020).
To employ localization analysis in actual structures, Doghri and Billardon (1995) computed Rice's condition for localization during the finite element (FE) simulation. Macro-cracks were assumed to start forming once the condition for localization was met, and the simulation was subsequently stopped. Becker (2002) combined the GTN model with failure criteria based on both Drucker's condition for material stability (Drucker 1959) and a simplified version of the bifurcation analysis by Rice (1976) in FE simulations of an expanding ring experiment. Both criteria were evaluated at each integration point and at every time step, where fracture in an element was imposed by setting the stress to zero once instability or localization occurred. Besson et al. (2003) used Rice's localization criterion in the post-processing of simulations on slant fracture to indicate when strain localization occurred on the global response. It was shown that macroscopic failure was slightly underestimated by the localization criterion. As the use of localization as a fracture indicator might be somewhat conservative, they discussed the possibility of modifying the constitutive equations instead, either by using the localization condition to initiate a model for accelerated void growth or to start nucleation on secondary particles. In a more recent work, Erice et al. (2020) adopted in situ strain localization analysis where the occurrence of bifurcation was taken as a fracture initiation indicator in FE simulations of sharp surface defects in a plane-strain material deformed under compression before reverse loading to tension was imposed. Following the proposal of Besson et al. (2003) and the work of Erice et al. (2020) and Bergo et al. (2021) investigated the possibility of predicting crack initiation and growth in pipeline steel using the GTN model where the onset of accelerated void growth was determined based on bifurcation analyses performed in situ at every material point. They compared this approach to the more commonly used model with a fixed critical porosity. The bifurcation enriched model was able to obtain similar or even better predictions over a wide range of different stress states as the onset of accelerated void growth became dependent on both the stress triaxiality and the Lode parameter.
Crack propagation in metal plates has received some attention in the literature in the last decades (see, e.g., Pardoen et al. 2004;Simonsen and Törnqvist 2004;El-Naaman and Nielsen 2013a;Nielsen and Gundlach 2017;Tekoglu and Nielsen 2019;, Andersen et al. 2020Çelik et al. 2021;Morgeneyer et al. 2021). The single edge notch tension (SENT) and double edge notch tension (DENT) specimens are frequently used to investigate mode I tearing where a pre-manufactured edge-crack is introduced to promote cracking . Three experimentally observed crack propagation mechanisms are often reported: cup-cup, cup-cone and slant fracture (El-Naaman and Nielsen 2013a). While slant fracture is mainly observed for high-strength metals, a cup-cup fracture mode is more common for plates made of a low-strength material where severe necking at the crack tip occurs (Pardoen et al. 2004). Nevertheless, it is not fully resolved why one failure mode is preferred over the other.
This paper presents an experimental and numerical study on quasi-static ductile tearing of thin plates made of the aluminium alloy AA6016 in three different tempers using single edge notch tension (SENT) and double edge notch tension (DENT) specimens. The different tempers result in materials with different yield stress, work hardening and fracture mechanism, while micro-structural features such as size, shape and distribution of constituent particles and the shape and size of the grains are assumed to remain unchanged. The experiments are complemented by nonlinear finite element simulations using an enriched GTN model to describe the material response. Three different approaches to determine the critical porosity f c that activates accelerated void growth will be compared: 1. In the softening model, accelerated void growth is activated by the onset of material softening. 2. In the localization model, a bifurcation analysis is performed in situ to determine the occurrence of strain localization and thus the onset of accelerated void growth. This is the same approach as suggested by Bergo et al. (2021). 3. In the f c (T, L) model, a critical porosity surface f c = f c (T, L) that depends on the stress triaxiality T and the Lode parameter L is used to determine the onset of accelerated void growth. The surface is generated by solving for strain localization under proportional stress states.
Three advantages occur with the extensions mentioned above: (1) f c is based on physical events; (2) there is no need to calibrate f c , which is not easily determined experimentally; (3) f c depends on the stress state and is not a material property, which previous studies have already confirmed. Notch tension tests with different notch radii are used to calibrate the parameters governing the work hardening behaviour and the nucleation rate of voids. This paper presents a novel comparison between the use of incipient material softening and strain localization to initiate accelerated void growth. In addition, the study provides a new method to estimate the onset of strain localization in porous materials in terms of a critical porosity surface. The paper is organized as follows. Section 2 presents the characteristics of the material. Results from experiments on notched tensile specimens, the SENT specimen and the DENT specimen are presented in Sect. 3. The enriched GTN model is introduced in Sect. 4, followed by the calibration procedure and a discussion about the predicted porosity at strain localization. Simulations of the SENT and DENT tests are presented in Sect. 5 and further discussed in Sect. 6. Some concluding remarks are given in Sect. 7.

Material and tempers
Aluminium plates of alloy AA6016 are considered in this study. The plates are approximately 1.5 mm thick and were provided in tempers T4, T6, and T7 by Hydro Aluminium Rolled Products, Bonn, Germany. Previous studies on this particular alloy include: (1) the effect of heat treatment on the structural response from blast loads (Granum et al. 2019); (2) an extensive test program (consisting of notch tension tests, plane-strain tension tests, in-plane simple shear tests and studies on steady-state crack propagation using two loading cases on a modified Arcan test) used to validate the performance of the modified Mohr-Coulomb (MMC) fracture model which had been calibrated by use of localization analyses (Granum et al. 2021); (3) a study on steadystate crack propagation using the SENT specimen and dynamic crack growth in low velocity impact problems (Espeseth et al. 2022).
The left-hand part of Fig. 1 shows the engineering stress to the engineering strain response extracted from The strain fields at the surface are calculated using 2D-DIC on the 0 • direction specimens with an element size of approximately 1 mm. Note that the colour bars have different ranges. The surface strain measured in an element outside of the local neck is included in the figure. The figures are taken from Espeseth et al. (2022) a flat dog-bone specimen for all tempers (Espeseth et al. 2022). The yield stress of the naturally aged temper T4 and the over-aged temper T7 is comparable, while the peak-aged temper T6 displays significantly higher yield stress. Moreover, temper T4 experiences a substantially higher work hardening than the two other tempers.
The right-hand part of Fig. 1 shows the local equivalent strain at the surface of the specimens, which has been calculated from 2D-DIC using eCorr (Fagerholt 2017). The equivalent strain is defined by ε eq = 4 3 ε 2 1 + ε 1 ε 2 + ε 2 2 where ε 1 and ε 2 are the principal logarithmic strains. It is noted that the resolution of the measurements is probably not high enough to exactly capture the maximum local strain. Consequently, the fields do not give an accurate measure of the failure strain, but they can be used to compare the different tempers as the element size is identical. By defining ductility as the reduction of the cross-section area (or amount of thinning) at fracture, we infer that temper T7 exhibits the highest ductility, followed by tempers T4 and T6 in that order. On the other hand, the strain at necking is the lowest for temper T7 and the highest for temper T4. This is also featured in the field plots of the surface strain as the strain outside of the necked region is higher for temper T4 than for temper T7.

Microstructure characterization
Samples from temper T4 were used to study the initial grain structure, grain size and distribution of constituent particles. The samples were cast in Epofix and further ground and polished down before micrographs were taken in a Reichert MeF3 A optical microscope using a 10 × objective. The micrograph of the constituent particles in Fig. 2 was taken under bright field illumination. The constituent particles are elongated and aligned in stringers along the rolling direction. The length of the particles ranges from less than a micron up to over 30 µm. By counting the dark pixels in Fig. 2, the area fraction of constituent particles was found to be approximately 0.0097.
The grain structure, shown in Fig. 3, was revealed under polarized light on samples that had been anodized for 2 mins using HBF 4 . It is seen that the material has a typical recrystallized grain structure. The grains are slightly elongated along the rolling and transverse directions with an average diameter of 41.7 µm. The average diameter was found by counting the number of grains over 12 lines with a length of 1 mm. There are on average 36 grains across the thickness of the plate. From the enhanced views in Fig. 3, it is seen that constituent particles are evenly distributed and mainly located inside the grains.

Notch tension tests
Notch tension tests were used to calibrate the material model in Sect. 4. These tests are favoured over the flat dog-bone specimens for calibration of the hardening parameters at large strains as the notch introduces a strain concentration that predetermines the location of the neck and the symmetry in the post-necking region is thus preserved (Mohr and Marcadet 2015). Only a short summary is given herein, and the reader is referred to Espeseth et al. (2022) for a thorough review of the test setup and results. Tension specimens with two different radii were considered. The force to elongation response of these specimens is shown by thick solid lines in Fig. 4, where NT3 and NT10 refer to specimens with a notch radius of 3.35 mm and 10 mm, respectively. The elongation was extracted from a virtual extensometer in eCorr (Fagerholt 2017) with an initial length of 15 mm. A simplified illustration of the specimen geometry and test setup is included at the bottom right corner of Fig. 4. The minimum width of the gauge area over the notch is 5 mm for both specimen types.
Temper T4 shows the highest elongation at failure. However, temper T7 experiences higher local strains inside the notch and thus more thinning. This appears in the plots as a more prolonged elongation subsequent to peak force. Temper T6 experiences the highest peak force, followed by temper T4, and has the lowest elongation at failure.

Single edge notch tension tests
Experiments on single edge notch tension (SENT) specimens are excellent for validating the finite element method's ability to predict both crack initiation and propagation. A summary is given below, while a thorough review of the test setup and results can be found in Espeseth et al. (2022). The force to slit opening displacement response for all tempers is shown as thin black solid lines in Fig. 5, where the inner and outer boundaries of the experimental data are enclosed by a grey shaded area. The slit opening displacement is defined as the change in distance between the two outer corners of the slit, as shown in the upper left figure. The crack length measured on the surface of one of the specimens for each temper is included as a solid black line in the lower section of the plot with an axis on the right-hand side. As the exact location of the crack front is indistinct, an error margin of ±1.0 mm is included as a grey-shaded area. The SENT tests were carried out with different orientations of the slit with respect to the rolling direction of the plates to check for any effects from anisotropy. Slit orientations are not distinguished in Fig. 5 as the influence of anisotropy on the ductile tearing behaviour was minor. See Espeseth et al. (2022) for a more detailed discussion.
While temper T6 experiences the highest peak force, the force level drops more rapidly compared to the other two tempers as this temper is less resistant to crack initiation and growth. Notice that the scale on the xaxis is different between the figures to better illustrate this response. Tempers T4 and T7 show similar peak force, but cracking occurs somewhat more rapidly with respect to the slit opening displacement in temper T7. Consequently, the force also drops more rapidly.

Double edge notch tension tests
Double edge notch tension (DENT) tests were carried out to extend the experimental program for crack propagation under quasi-static conditions. The specimen geometry and test rig are shown in Fig. 6, where the thickness of the specimen is 1.5 mm. The test rig is a modified version of the rig used in . Sharp slits were manufactured on both sides of the specimen from which two cracks propagate toward each other within the 90 mm wide ligament. The measured radius of the tip of the slits is 0.51 ± 0.01 mm, more than twice the value of the SENT specimen which was measured to be 0.19 ± 0.01 mm. A slit tip with a larger radius was chosen for the DENT specimen as it is easier to accurately represent in the finite element mesh. The specimen was fastened to the loading plates using 5 M12 bolts on both ends. The loading plates were pinned to two connection blocks, which again were connected to a connection plate that was attached to an Instron 5985 series universal testing machine. The top surfaces of the loading plates were in contact with the connection plate, which restrained rotation around the pins. The DENT specimen was loaded in tension, resulting in a Mode I loading at the crack tips.
The force was measured using a 100 kN load cell attached to the actuator during the experiments. A constant cross-head velocity of 0.25 mm/min was used in all tests to ensure a quasi-static loading condition. The tests were monitored using a Basler acA2440-75um camera with a Samyang 100 mm f/2.8 ED UMC macro lens at a frame rate of 1 Hz. The specimens were spraypainted with a speckle pattern to allow for full-field 2D-DIC measurements of the surface displacement. A virtual extensometer with an initial length of 50 mm, positioned centric to the slits, was used to extract a global measurement of the elongation.  The global force to elongation is shown as thin black solid lines in Fig. 7, where the inner and outer boundaries of the experimental data are enclosed by a shaded area. Three consecutive tests for each temper have been carried out. Tempers T4 and T7 display similar peak force at approximately 26.6 and 26.7 kN, respectively, whereas temper T6 shows a notably higher peak force at approximately 34.5 kN. Moreover, temper T6 reaches peak force at the lowest elongation equal to approximately 0.25 mm, compared to approximately 0.7 mm for temper T7 and approximately 1.0 mm for temper T4. A similar observation can be made for the SENT tests. Furthermore, both temper T4 and T7 show a distinct change in stiffness prior to peak force at approximately 20 to 22 kN as plastic yielding occurs throughout the ligament between the two slits. This shift is not observed for temper T6 as crack initiation and growth happen before or closely after plastic yielding occurs throughout the ligament.
In all experiments, the remaining width of the ligament on the surface of the specimen between the two edge cracks was measured. Between 40 and 60 images were used in each test, depending on the temper. The total crack length, defined as the initial ligament width (equal to 90 mm) minus the remaining ligament width, is included in the lower section of Fig. 7 with its axis on the right-hand side of the plot. As the crack paths are normally not straight (see Fig. 8 and the discussion below), one should understand that this measurement does not represent the exact length of the cracks, but is rather a measurement of the remaining cross-section between the two cracks. For all tempers, crack initi-

Fig. 6
Geometry and clamping of the specimen in the DENT test setup. All measurements are given in mm and are the same above the slits. The clamping of the specimen is identical on the lower half of the specimen but is removed in the figure to better show the specimen geometry ation occurs slightly before peak force. Initially, the cracks grow at a steady speed with respect to the elongation before the two crack fronts suddenly merge and the crack length increases rapidly. The cracks grow at a slower speed for temper T4, resulting in the largest elongation at failure. The cracks grow more slowly for temper T7 than for temper T6.
The crack path in the plane of the specimen for the three tempers is shown in Fig. 8. It was observed that the two edge cracks tend to grow at an oblique angle, i.e., deviates from a straight crack path in the plane of the specimen, for tempers T4 and T6. For temper T7, the crack path is more straight. As will be discussed in Sect. 3.4, the higher ductility in temper T7 leads to a cup-cup fracture profile. It is believed that this type of fracture profile preserves the symmetry and thus constrains the path of the crack. In contrast, tempers T4 and T6 display a slanted fracture profile. We believe that a slanted fracture profile induces a slight out-of-plane action (mode III loading) (El-Naaman and Nielsen 2013b) which favours an oblique crack path in the current test setup. Furthermore, tracking of additional markers attached to the rig, see Fig. 6, revealed small rotations around the pin and lateral displacement of the loading plates. The amount of skewness in the lateral and longitudinal displacement of the upper loading plates does not easily quantify as it is somewhat nonlinear with respect to the total elongation. Nevertheless, it is estimated that the skewness compared to the average longitudinal displacement of the loading plates is in the range of 5-10% for tempers T4 and T7, and 15-20% for temper T6. It was therefore assumed that the oblique crack path could be linked to slack in the test setup. It should be mentioned that  did not report on oblique crack paths in their study using a similar setup with plates in the aluminium alloy Al Global force and crack length as a function of elongation in the DENT tests. a-c correspond to tempers T4, T6 and T7, respectively. The force-elongation response from the experiments is shown as thin lines, enclosed by a shaded area. The measured crack length is included in the lower section of the plots with the axis on the right-hand side. The response from the simulations (discussed in Sect. 5.2) is shown by coloured lines, where the line style separates the different models. A circle (•), square ( ) or triangle ( ) indicates the point when the first element is eroded Fig. 8 Crack path for a temper T4, b temper T6 and c temper T7. The crack path is highlighted by coloured lines 1050 H14. This alloy has very little work hardening and experiences extensive thinning. Cup-cup fracture was therefore the dominant crack propagation mechanism observed, thus supporting the theory that this type of fracture profile constrains the path of the crack. In addition, Granum et al. (2021) reported on straight crack paths in modified Arcan tests of alloy AA6016 in temper T7, while tempers T4 and T6 displayed curved paths. From Fig. 7b, it is observed that one of the temper T6 tests failed at a lower elongation compared to the other two tests. The crack path of this test is included in Fig. 8d with the label "Outlier". Compared to the other two tests, the crack path is less oblique, which explains the earlier occurrence of failure.

Fracture investigation
Fracture zone profiles from the SENT tests are shown in Fig. 9a-c for tempers T4, T6 and T7, respectively. Both tempers T4 and T6 exhibit slant fracture at an angle of about ±45 • , commonly observed for high-strength aluminium alloys. Temper T7 shows a cup-cup fracture mode, which is favoured when extensive thinning of the fracture zone occurs (Pardoen et al. 2004). In order to evaluate the local fracture mechanisms, fractography analysis of the failed SENT specimen was performed using a Zeiss Gemini SUPRA 55VP FESEM operated at 15 kV. The images of the fracture surface in Fig. 9d-e were taken about 8 mm from the tip of the slit.
As for the global fracture mode, different local fracture mechanisms are also observed between the three tempers. The fracture surface for temper T6 in Fig. 9e consists mainly of intergranular facets, suggesting that material failure occurs at the grain boundaries. A high density of small dimples cover the facets. Some void growth is observed around what appears to be large and flat particles with sizes in the range above 10 µm. This particular AA6016 alloy has an extremely low Mg:Si ratio of only 0.265 (Granum et al. 2019). According to Remøe et al. (2017), an Mg:Si ratio lower than 1.73 will cause excess Si to form coarse Si particles at the grain boundaries which promote grain boundary failure. This is a plausible explanation for the observed facets seen on the fracture surface. Some intergranular facets can also be observed on the fracture surface for temper T4 in Fig. 9d. However, the main fracture mode appears to be the coalescence of larger voids around the constituent particles, resulting in large dimples on the fracture surface. The fracture surface for temper T7 in Fig. 9f consists primarily of large dimples with some densely packed areas of smaller dimples in the intervoid ligaments. Some large oversized dimples can also be observed. The size, shape and content of constituent particles are expected to be the same between the three tempers. It follows from this observation that the yield stress and the strain hardening response of this particular alloy play a vital role on the fracture mechanism occurring, both on a microscopic level (grain boundary failure versus coalescence of voids nucleated at constituent particles) and on a global level (slanted versus cup-cup fracture mode). In a study on void coalescence of laser drilled holes in sheet metals, Weck and Wilkinson (2008) observed that secondary voids were able to nucleate in a material with high yield stress before the larger laser drilled holes had grown sufficiently to be able to coalesce. It is believed that for temper T6, and partially for temper T4, the higher flow stress allows for the nucleation of secondary voids at the Si particles near the grain boundaries before the primary voids forming around the constituent particles have grown sufficiently to coalesce.

Stress-state parameters
In the following, important stress-state parameters used in this work will be outlined. The stress triaxiality T , which proves to be an important quantity for void growth (McClintock 1968;Rice and Tracey 1969), is defined as the ratio between the hydrostatic stress σ H and the von Mises equivalent stress σ vM , i.e., where σ H = 1 3 σ : I, σ is the Cauchy stress tensor, and I is the second-order identity tensor. The von Mises equivalent stress reads σ vM = 3 2 σ : σ , where the deviatoric stress tensor equals σ = σ−σ H I. The Lode parameter L, another important quantity in ductile failure analysis (Bao and Wierzbicki 2004;Barsoum and Faleskog 2007), defines the deviatoric loading state by where σ I ≥ σ II ≥ σ III are the ordered principal stresses. It follows that L = −1 and L = 1 define the peripheral deviatoric stress states of generalized tension and

Constitutive relations
A hypoelatic-plastic formulation of the Gurson-Tvergaard-Needleman (GTN) model is adopted, which is valid for small elastic deformations but allows for large rotations and plastic deformations. The rate-ofdeformation tensor D is additively split as where D e is the elastic and D p is the plastic parts of D. Assuming linear isotropic elasticity, the hypoelastic formulation relates the Jaumann rate of the Cauchy stress tensor σ ∇J to the elastic rate-of-deformation ten-sor D e according to the generalized Hooke's law where C is the fourth-order elasticity tensor, defined by Young's modulus E and Poisson's ratio ν. The deviatoric and volumetric parts of the elastic rateof-deformation tensor are denoted D e and tr (D e ) I, respectively, where D e = D e − 1 3 tr (D e ) I. The Gurson yield function (Gurson 1977), including the modification by Tvergaard (1981Tvergaard ( , 1982 and Tvergaard and Needleman (1984), reads where = 0 defines the periphery and < 0 the interior of the elastic domain. The three parameters q 1 , q 2 and q 3 were first introduced by Tvergaard (1981Tvergaard ( , 1982 and are here given standard values found in the literature, i.e., q 1 = 1.5, q 2 = 1.0 and q 3 = q 2 1 . As proposed by Daehli et al. (2017), a heuristic extension of the Gurson model in which the nonquadratic Hershey-Hosford equivalent stress (Hershey 1954;Hosford 1972) replaces the von Mises equivalent stress is adopted. The Hershey-Hosford equivalent stress in terms of the principal stresses reads The curvature of the yield surface in the deviatoric plane is controlled by the parameter a. We will assume that a = 8 based on polycrystal plasticity calculations by Hosford (1996). By assuming associative plastic flow, the plastic rate-of-deformation tensor D p reads whereλ is the plastic multiplier, which can be obtained from the consistency condition˙ = 0 for plastic flow (λ > 0), see, e.g., Erice et al. (2020). The work hardening is heuristically introduced into the matrix flow stress σ M by a three-term Voce hardening rule where σ 0 is the initial yield stress, and Q i and C i are hardening parameters. The equivalent plastic strain ratė p is given from the equivalence in plastic power (work conjugacy), defined as where f is the porosity. The equivalent plastic strain is accumulated according to the time integral p = ṗ dt. For the GTN model, an initial porosity f 0 must be specified. In addition, an evolution rule for the porosity is necessary. We will assume that the evolution of the porosity is defined as the sum of the void growth ratė f g and the void nucleation rateḟ n , viz.
While void growth is related to the volumetric plastic strain rate, void nucleation is assumed to increase linearly with the equivalent plastic strain p, controlled by the constant material parameter A n . The continuous nucleation model has been adopted as the constituent particles have different shapes and sizes, and is assumed to represent the average behaviour of the nucleation mechanisms (Zhang et al. 2000).
To include accelerated void growth, an effective porosity f * has been introduced into the yield function in Eq. 5, defined as ) where f F and f * u = q 1 + q 2 1 −q 3 q 3 denote the actual and effective porosity at failure, respectively. Accelerated void growth occurs once the porosity f is greater than the critical porosity f c . The critical porosity is normally taken as a fixed material parameter which defines the porosity at incipient void coalescence. In the following, two methods to determine f c during a simulation will be discussed. We have in this paper adopted a more relaxed definition of f c as these different methods to activate accelerated void growth do not necessarily always correspond with the occurrence of void coalescence.
In the softening model, accelerated void growth is assumed to start at the onset of material softening. The condition for material softening is defined by (Morin et al. 2018b) The equivalent stress to equivalent plastic strain response under proportional loading of a single material point is included as blue lines in Fig. 10. For comparison, the response where no accelerated void growth is assumed is shown as black dashed lines. The point of material softening, and thus accelerated void growth, is indicated by blue dots. After this point, the porosity grows rapidly. Total loss of load-carrying capacity occurs when f = f F . In the localization model, strain localization by loss of ellipticity (or bifurcation) is taken as an indicator for incipient accelerated void growth. strain localization in an elastic-plastic solid occurs when a discontinuity in the deformation rates across a planer band with unit normal n becomes possible, as illustrated in Fig. 11a. As  Fig. 10 An illustration of the equivalent stress to equivalent plastic strain response for the two different methods to activate accelerated void growth. The porosity related to void nucleation ( f n ) and the total porosity ( f ) are included on the right-hand axis. A proportional stress state with a fixed stress triaxiality T = 1.0 and a fixed Lode parameter L = −0.5 is used. Solid blue lines show the response from the softening model, while solid red lines show the response from the localization model. The point at which accelerated void growth occurs is indicated by circles (•). Dashed black lines (marked GT) show the response where no accelerated void growth is assumed (i.e., when f F = f * u ). The following material parameters are used: σ 0 = 100 MPa, Q 1 = 100 MPa, C 1 = 20, Q 2 = 100 MPa, C 2 = 5, f 0 = 0.0, A n = 0.01 and f F = 0.14 derived by Rice (1976) and Rice and Rudnicki (1980), the necessary condition for strain localization corresponds to the occurrence of vanishing eigenvalues of the acoustic tensor A (n), assuming plastic loading both outside and inside of the band, i.e., det A (n) = 0 The acoustic tensor A is defined by (see, e.g., Rice and Rudnicki 1980) where C ep is the elastic-plastic tangent modulus tensor and the tensor R is given by In the computational procedure that checks for strain localization at a given stress state, the unit vector n that minimizes the determinant in Eq. 13 must be selected. As suggested by Rudnicki and Rice (1975), the planar band normal can be assumed to be contained in the plane defined by the major principal stress direction e I and the minor principal stress direction e III , provided that the material behaviour is isotropic. Consequently, the critical orientation of the band can be determined by sweeping through the range of n ϕ , which is defined by 0 ≤ ϕ ≤ π/2, at suitable increments, as illustrated in Fig. 11b. In this work, a total of 200 bands linearly distributed between the minor and major principal stress directions were used to ensure a converged solution. Strain localization is normally only possible after material softening takes place (Rice 1976). To reduce the computational cost, we only checked the more relaxed condition for strain localization min det A n φ ≤ 0 once the condition in Eq. 12 is fulfilled. Material failure is immediately assumed if the porosity exceeds the 70, 000 0.3 2700 1.5 1 .0 0 .0 0 .14 Table 2 Initial yield stress, hardening parameters and nucleation rate found by calibration to the notch tension tests porosity at failure, that is f > f F , before strain localization is detected. The response using the localization model is illustrated by red lines in Fig. 10. The occurrence of accelerated void growth is delayed compared to the softening model. As will become apparent later, the response obtained with the two models will coincide for stress states close to generalized shear and will start to differ as the stress state approaches generalized tension.
The constitutive relations are completed by the Kuhn-Tucker conditions which distinguish between elastic loading/unloading and plastic loading. A stress-update algorithm is used to ensure that these constraints are fulfilled. The material model was implemented in Abaqus/Explicit 2022 using a user-defined material subroutine (VUMAT), where the temporal integration of the constitutive equations is carried out using the cutting plane algorithm (Ortiz and Simo 1986). The Jaumann update is not available by default (a VUMAT uses the Green-Naghdi objective stress rate), but has been enforced through the VUMAT subroutine before the update of the stress. The condition for material softening, Eq. 12, or strain localization, Eq. 13, is checked after the stress-update routine is completed. If the required condition is fulfilled, accelerated void growth, as described by Eq. 11, is activated by assigning f c equal to the current value of the porosity. An element erosion technique is used to represent crack propagation. The element is eroded when f = 0.99 f F to avoid numerical problems when the yield surface collapses due to softening.

Parameter identification
The notch tension tests in Sect. 3.1 were used in the calibration of the work hardening parameters in Eq. 8 and the nucleation rate A n in Eq. 10 using the localization model. The same material parameters have also been used in the softening model as it will be compared to the localization model. The force to elongation response is shown in Fig. 4 for the localization model in solid lines and the softening model in dashed lines. The point at which accelerated void growth is initiated in the centre element is indicated by markers in the plot. Material constants common between the three tempers are listed in Table 1. The porosity at failure f F was set to 0.14, although it could be calibrated separately for each temper. The influence of f F will be discussed in Sect. 6.1. Materials that exhibit softening behaviour are particularly sensitive to changes in the element size. To circumvent this, all material parameters were calibrated to an element size of 0.15 mm, which was found to be sufficiently small to describe the physical mechanisms while large enough to make possible simulations of the SENT and DENT tests. It follows that a "computational cell approach" has been employed (Ruggieri et al. 1996). The calibrated initial yield stress, hardening parameters and nucleation rate are compiled in Table 2. A reversed modelling approach was used in the calibration process where the optimum parameters were selected based on a systematic trial-and-error approach using the optimization software LS-OPT. We refer to Espeseth et al. (2022) for more details on the calibration procedure and the finite element model used.
As shown in Fig. 4, the work hardening model in Eq. 8 can accurately reproduce the response of the Table 3 Critical porosity f c and porosity f n related to void nucleation (not including void growth) in the centre element at the onset of accelerated void growth for the two notch tension tests (marked in Fig. 4 notched specimens before the onset of accelerated void growth. For the localization model, accelerated void growth in the centre element is initiated at, or close to, the rear end of the experimental curve. Reduction of the force occurs after this point for both specimens, but the decrease is not as prominent for temper T7. Accelerated void growth is initiated too early in the softening model, which is expected as the material parameters were adopted from the localization model. The softening model appears the best for the NT3 specimen. This can be explained by a less negative value of the Lode parameter in the centre of the NT3 specimen compared to the NT10 specimen (Espeseth et al. 2022). As will be discussed later in this section, the softening model and the localization model normally coincide for stress states close to generalized shear, while the localization model predicts higher ductility as the stress state moves towards generalized tension. Consequently, these two models are more similar for the NT3 specimen. As the Gurson model is derived based on the physical mechanisms of void evolution in a material, it is interesting to check if the micromechanical quantities are within physical constraints. Table 3 shows the critical porosity f c and the porosity f n related to nucleation in the centre element at the onset of accelerated void growth (marked in Fig. 4). While f c is comparable for tempers T4 and T6, a significantly lower critical porosity is found for temper T7. Comparing the NT10 to the NT3 specimen, one can see that f c is increased by approximately 40-70% in the case of the localization model. This is not the case for the softening model, where f c remains about the same for both specimens. Again, this is clearly related to the lack of dependency on the Lode parameter portrayed by the softening model. The amount of nucleated voids is the highest for temper T6 and the lowest for temper T7. Compared to the area fraction of constituent particles in Fig. 2, the amount of nucleated voids seems reasonable for tem-per T7. For temper T6, f n is rather high due to the large value A n takes. The experimental observation that the crack predominantly grows along the grain boundaries for temper T6 does not necessarily mean that damage evolution and softening within the grains are unimportant. Indeed, the constituent particles were found to be reasonably uniformly distributed within the material. Some void growth is present for temper T6, which can be seen in the SEM images Fig. 9e. It is therefore not completely clear if the physical mechanism assumed in the Gurson model is justified for this particular temper. This depends on how dominant the growth of voids is on the response, which cannot be easily assessed from the experiments. It follows that we have interpreted the GTN model as a coupled damage model, where fracture is initiated due to an instability (the occurrence of strain localization or softening) instead of the actual mechanism of void coalescence. Petit et al. (2019) concluded in their study on a compact tension specimen that A n should depend on the flow stress, where a higher flow stress yields a higher nucleation rate (or higher A n ). The magnitude of A n for the different tempers of alloy AA6016 follows this trend. Figure 12 shows the critical plastic strain at incipient softening and strain localization for different proportional stress states with constant stress triaxiality and Lode parameter. The curves have been generated by solving the constitutive relation for a single element where a multi-point constraint (MPC) user subroutine has been used to drive the deformation of the boundary of the single element for a given stress state (Daehli et al. 2016). For illustrative purposes, an upper limit on f c has not been considered, thus assuming f F = f * u . Solid lines display the results from the localization model, while dashed lines correspond to the softening model. As shown in Fig. 12a-c, the critical plastic strain decreases exponentially with stress triaxiality. The response from the softening model is shown by  Fig. 12d-f, the localization model predicts a significant variation in the critical plastic strain with the Lode parameter, while the predictions from the softening model, on the other hand, are practically unaffected by this parameter. It should be mentioned that due to the non-quadratic equivalent stress applied in the GTN yield criterion, some insignificant variations in the critical plastic strain can be observed in a magnified plot. The point of minimum critical strain at localization occurs at different values of L depending on the stress triaxiality. In all cases, however, the plastic strain is the lowest for stress states close to generalized shear and the highest in generalized tension. One should also be aware of the fact that there are stress states where strain localization is not possible, and that strain localization usually does not occur for positive values of L. On the other hand, softening will occur, even for positive values of L. The localization model and softening model coincide at the point of minimum critical plastic strain.

Porosity at strain localization
The surface plots in Fig. 13 show the porosity at strain localization for different proportional stress states with constant stress triaxiality and Lode parameter. The surfaces have been capped at f c = 0.14 to emphasize the constraints in the simulations. An exponential decrease in the critical porosity with increasing stress triaxiality is apparent, and the critical porosity is the highest for generalized tension (L = −1) and the lowest for stress states closer to generalized shear (L = 0). For stress states close to generalized tension, strain localization is not possible within the limit f c = 0.14 unless the stress triaxiality is higher than approximately 1.5. Moreover, strain localization will not occur when the Lode parameter increases slightly beyond the point of minimum critical porosity.  The porosity at strain localization for the three tempers is compared in Fig. 14 as a function of stress triaxiality at four constant values of the Lode parameter. We have now assumed no upper limit on the allowed porosity, i.e., f F = f * u , but the cap f = 0.14 is included as a dashed line for reference. Interestingly, there is essentially very little difference in the predicted porosity at strain localization between the three tempers down to L = −0.5. From Fig. 13 it is clear that this does not necessarily extend to higher values of the Lode parameter as strain localization at high stress triaxiality is not possible for temper T4 in these cases. A separate study revealed that materials with relatively high work hardening capacity compared to the initial yield stress, as for example temper T4, did not achieve strain local-ization for positive values of the Lode parameter even when the stress triaxiality became high.
The surfaces in Fig. 13 were constructed from proportional stress paths. Some non-proportional loading paths are considered in Fig. 15, marked by number 1 to 8 . We will only consider temper T4 for the sake of brevity, but the same findings apply to tempers T6 and T7. In Fig. 15a, all loading paths shown have a constant Lode parameter equal to L = −1, while the stress triaxiality is monotonically changed with respect to the plastic strain. The direction of loading is shown by arrows and a circle indicates when strain localization takes place. The dashed line shows the surface from Fig. 13a for the given constant value of the Lode parameter. Paths 1 and 2 show the evolution of the porosity for decreasing stress triaxiality, while path 3 considers an increasing stress triaxiality. Path 4 shows the response when the triaxiality is first decreased to T = 0.5 and then increased. In all cases, strain localization occurs once the porosity is about to intersect the surface from Fig. 13a. For the next example in Fig. 15b, the stress triaxiality is fixed at T = 0.8 while the Lode parameter is monotonically changed as a function of the plastic strain. Path 5 shows the evolution of the porosity for increasing values of the Lode parameter, paths 6 and 8 show the evolution for decreasing values, while path 7 shows the evolution for a combination of decreasing and subsequent increasing values of the Lode parameter. Similar to the previous examples, strain localization occurs once a critical porosity is obtained. However, the stress paths marked 8 never pass through the surface and strain localization occurs at a porosity above the f c -surface. The non-proportional loading paths discussed here are fairly simple. However, more complex loading paths (i.e., where T and L are cross coupled) yield the same result for our material: strain localization will occur if the porosity intersects, or is about to intersect, the f c -surface generated from proportional loading states. It should be emphasized that the load path independence does not necessarily transfer to the plastic strain. For example, the plastic strain at localization for load path 1 and 4 is 1.37 and 1.60, respectively, while the critical porosity is approximately the same.
As the localization model seemingly displays load path independence regarding the porosity at strain localization, simulations using a critical porosity surface f c = f c (T, L) that depends on the stress triaxiality and the Lode parameter should yield similar results. A third model is proposed, henceforward referred to as the f c (T, L) model, where a critical porosity surface f c = f c (T, L) is used to determine when strain localization occurs. The main advantage of this approach over the localization model is the decrease in computational cost. The surface is generated beforehand by solving for strain localization under proportional stress states, as previously discussed. Obviously, finding an appropriate algebraic expression that can accurately depict the surfaces in Fig. 13 is impractical. The data points will therefore be used directly, where the critical porosity can be determined by the use of 2D linear interpolation. If the porosity in a material point after the return mapping scheme has completed is higher than the critical porosity for the updated stress triaxiality and Lode parameter, accelerated void growth is initiated in the next time increment. Essentially, this model coincides with the original GTN model when f c is constant. It is emphasized that with this implementation, stress paths such as those labelled 8 in Fig. 15 would experience accelerated void growth once the porosity is above the f c -surface. However, this will not be an issue for the simulations performed in this paper and will therefore not be further discussed.
The predicted response from the notch tension tests using the f c (T, L) model is included in Fig. 4 as dotted lines. A square indicates when accelerated void growth is initiated in the centre element. The f c (T, L) model clearly resembles the response from the localization model quite well. For tempers T4 and T6, accelerated void growth is initiated in the centre element almost at the same time as for the localization model. For the NT3 specimen in temper T7, accelerated void growth is initiated earlier for the f c (T, L) model, and a deviation from the localization model in the softening region of the response curve is apparent.

Simulations of SENT and DENT tests
Finite element simulations of the SENT and DENT tests in Sect. 3 were carried out to investigate the differences between the softening and localization model and their ability to accurately predict crack initiation and growth. Simulations using the f c (T, L) model were compared to the results from the localization model to confirm that these models produce similar responses. All simulations were performed in Abaqus/Explicit 2022 using the user-subroutines described in Sect. 4.2 and the material parameters compiled in Tables 1 and 2.

Single edge notch tension test
The finite element mesh and boundary conditions of the SENT specimen are shown in Fig. 16a. The attachment between the specimen and the machine was modelled using frictionless contact boundary conditions between the rigid pins (blue parts in Fig. 16a) and the specimen. The displacement of the upper pin was controlled by assigning a constant velocity, smoothly ramped up over the first 10% of the simulation time, and the lower pin was fixed. Time scaling was employed, where the amount of scaling was determined by a parametric study to ensure vanishing inertia effects. Symmetry was utilized at the mid-surface (the xz-plane), as highlighted in Fig. 16a, to reduce computation cost. The use of symmetry will decidedly prevent a slant fracture mode from occurring in the simulations. However, the mesh topology and an insufficient element size will not allow a slanted fracture profile to form even if no symmetry is applied. A more refined mesh and some sort of trigger in the model are needed in order to form a slant fracture profile in the SENT specimen. As a consequence, the influence of a slant fracture mode on tempers T4 and T6 will not be considered in this paper. Therefore, the results presented are only valid with the reservation that the element model cannot describe a slant fracture profile. The element size in the centre of the specimen where the crack propagates was set to 0.15 mm and gradually increased away from this region. The mesh consists of 41, 837 elements in total. The sharp slit has a radius of 0.2 mm and is shown in the enhanced view of Fig. 16a. Previous studies have shown that the orientation of the mesh strongly influences the predicted crack path when the element erosion technique is used to describe fracture without additional regularization techniques (Simonsen and Törnqvist 2004;Wu et al. 2016). In Granum et al. (2021) and Espeseth et al. (2022), the sweep meshing technique in Abaqus was employed to generate an irregular mesh orientation which gave rise to small variations in the crack path that are more in accordance with the experiments. As one of the main goals of this work, however, is to compare different models to activate accelerated void growth, a regular mesh throughout the cracked region is more convenient as this reduces uncertainties that may arise if the shape and size of the elements in the crack path vary a lot. However, this will result in a completely straight crack path.
The global force as a function of the slit opening displacement from the simulations is shown in Fig. 5 as coloured lines for all tempers. Predictions using the localization model are given by solid lines, while dashed lines correspond to the predicted response from the softening model. The predicted crack length at the surface of the specimen is included in the lower section of the plots, with the axis on the right-hand side. A marker indicates when the first element is eroded, which always occurs in the centre of the specimen. The overall force response in the simulation using the localization model is generally overestimated for temper T4 and underestimated for temper T7, while good overall predictions can be seen for temper T6. The crack propagates slightly faster for temper T6, as seen when comparing the crack length to the experiments, which results in a too rapid reduction of the force. The peak force is slightly overestimated in all tempers as crack initiation occurs at too large deformations. The delay in crack initiation becomes especially apparent for tempers T4 and T7 when we compare the predicted crack length to the experiments. For temper T7, however, a jump in the crack length occurs around a slit opening displacement of 2.1 mm, and the force is from this point below the experimental results. The softening model, on the other hand, underestimates the overall force level. Moreover, crack initiation occurs at a lower slit opening displacement compared to the localization model. It should be emphasized that the porosity at failure f F was selected based on previous experience from simulation of the SENT test where the localization model was used. A slight increase of f F would be in favour of the softening model. However, this would not be a remedy for the inadequate performance in the NT3 and NT10 specimens. A further discussion on the value of f F is given in Sect. 6.1. Although some discrepancy is observed between the simulations and the experiments, the prediction from the finite element model is still rather accurate regarding crack propagation. Interestingly, this also applies to temper T4 and T6 even though the model cannot reproduce the slant fracture mode. From this observation, it is conceivable that the fracture mode has a limited impact on the predicted global response of the SENT test.
To further understand the differences between the softening model and the localization model, we will consider the critical porosity along the crack path. As the porosity should be interpreted as a damage variable in the constitutive models, the variation along the crack path presented in the following does not necessarily reflect on the actual porosity in the experiments. We have only considered the porosity along the crack path in order to compare the different models. Figure 17 shows the critical porosity as a function of the distance from the tip of the slit for the elements near the centre of the specimen. The critical porosity calculated at strain localization (black lines) is higher than the critical porosity found at material softening (red lines). It is rather obvious that the localization model yields higher f c along the crack path as the softening model normally coincides with the point at minimum ductility predicted by the localization model, as shown in Fig. 12. Both models predict generally higher f c at material points in the vicinity of the slit. This is especially noticeable for the localization model. In fact, for temper T4, strain localization does not occur for the closest material point before the limit f = f F is reached. The same observations were made by Bergo et al. (2021) where it was argued that the failed elements could be split into elements that fail during crack initiation and elements that fail during crack propagation. Following the argument in their paper, the reason for the elevated porosity near the tip of the slit can be explained by considering the stress state in the material points. Figure 18 shows the porosity as a function of the Lode parameter and the stress triaxiality for all material  Fig. 18a and c   Fig. 18 Evolution of the porosity as a function of the Lode parameter and the stress triaxiality for elements that are eroded in the centre of the SENT specimen. Only elements within 10 mm from the tip of the slit are included. Vertical dashed lines correspond to a stress triaxiality of T = 1/ √ 3 and a Lode parameter of L = 0 (plane strain tension). The stress state in points marked by dots in Fig. 17 is highlighted by a green curve. Grey curves correspond to sections of the f c -surface (Fig. 13) where the stress triaxiality is fixed at 1.1 for tempers T4 and T6 and 0.75 for temper T7 points near the centre of the specimen within the first 10 mm from the notch. Dark colours indicate material points close to the slit, and the colour gradually changes to red for elements further away from the slit. Solid dots indicate incipient accelerated void growth, and the evolution of porosity after this point has not been included for clarity in the following discussion. A section of the f c -surface from Fig. 13 has been included in the figure as a reference. However, one should keep in mind that this is a three-dimensional surface and it can not be used to evaluate f c for all material points in the figure.
Material points in the vicinity of the slit undergo a stress state with a Lode parameter that is more negative compared to material points further away. It was shown in Sect. 4 that the localization model predicts substantially higher ductility for stress states in this region compared to the softening model. Consequently, the localization model predicts considerably higher critical porosity in this region. Material points further away from the slit are initially subjected to stress states between generalized shear (L = 0) and generalized compression (L = 1) before rapidly changing to a state of generalized tension (L = −1) as the crack approaches the considered material point and a through-thickness neck forms (Espeseth et al. 2022). It is clear from Fig. 18 that accelerated void growth is usually initiated during this transition of the stress state. While the porosity of tempers T4 and T6 approaches the f c -surface from the right-hand side, the porosity of temper T7 approaches the f c -surface from beneath. There are some material points along the crack in tempers T4 and T7 where f c is substantially higher compared to its neighbours. This appears as obvious spikes in Fig. 17a and c. The stress state of the two points marked with green dots in Fig. 17 is highlighted as a green curve in Fig. 18. The stress state in these material points is similar to its neighbouring points. However, strain localization is not detected as the stress path sweeps near the f c -surface. Consequently, deformation can continue in the stress region where the ductility is higher. It is assumed that the occurrence of these spikes in f c will slightly increase the global force. A consequence of these variations in f c can be seen in Fig. 5a, where periodic waves in the curves for the localization model (green solid line) appear after peak force. Different measures to remove these spikes have been investigated with little success. Increasing f F has some effects, but this measure will also increase the force. This issue is further discussed in Sect. 6.1. It has also been demonstrated that the use of smaller elements leads to a reduction in both the number and the height of the spikes in the SENT simulations. However, a more refined mesh makes an extensive numerical study of the DENT test impractical. Finite element analyses of another DENT test specimen with a finer mesh were provided by Nielsen and Felter (2019) with emphasis on the occurrence of slant fracture.
The predictions from the f c (T, L) model are included in Fig. 5 as dotted lines. The point of crack initiation agrees well with the localization model for all tempers. The two models are also almost indistinguishable from each other for temper T6, while the global force predicted by the f c (T, L) model is slightly higher for tempers T4 and T7. The reason for these discrepancies is not obvious, but the f c (T, L) model predicts a slightly higher critical porosity along the crack path for these two tempers compared to the localization model. It is clear from Fig. 18 that the porosity at strain localization in the SENT tests depends on both the stress triaxiality and the Lode parameter, but it is not obvious if reaching a critical porosity is a strict requirement for strain localization or if this just increases the likelihood of strain localization occurring. Indeed, some of the points at which strain localization occurs appear slightly outside of the f c -surface in Fig. 18. The strain localization analysis might be somewhat sensitive to numerical approximations in the VUMAT subroutine, which could be a possible explanation for the discrepancies between these two models. Also, the value for f c is calculated using linear interpolation on a convex surface which in general will yield a higher f c . However, effects from the interpolation are assumed to be minimal as a rather high density of data points was used.

Double edge notch tension test
The finite element mesh and boundary conditions of the DENT specimen are shown in Fig. 16b. The modelling approach is similar to the SENT specimen, where symmetry at the mid-surface (i.e., the xz-plane) was utilized and frictionless contact boundary conditions between the rigid pins (blue parts in Fig. 16b) and the specimen were used to model the attachment to the loading plates. The five lower pins were fixed, while the five upper pins were given a constant velocity, smoothly ramped up over the first 10% of the simulation time. Appropriate time scaling was used to decrease the computational cost while ensuring that inertia effects can be neglected. Regular shaped elements with a target size of 0.15 mm were used in the area between the two slits, and gradually increased outside of this region. The mesh consists of 197, 740 elements in total. The radius of the tip of the slits, shown in the enhanced view of Fig. 16b, is 0.5 mm.
The global force as a function of the elongation is shown in Fig. 7 as coloured lines for all tempers. Predictions using the localization model are given by solid lines, while dashed lines correspond to the predicted response from the softening model. The predicted crack length at the surface of the specimen is included in the lower section of the plots, with the axis on the righthand side. The localization model overestimates the peak force for all tempers. Crack initiation occurs earlier for the softening model, which results in a lower overall force compared to the localization model. Taking this into account, both the softening and localization models predict a force response that tends to drop too rapidly with elongation subsequent to peak force, especially for temper T6. The simulations predict a straight crack path due to the use of symmetric boundary conditions and loads. Consequently, the crack propagates a shorter distance compared to the oblique crack path seen in the experiments (see Fig. 8). A shorter path will result in a faster reduction of the cross-section area between the slits, and thus a more rapid drop of the global force with respect to the elongation. The influence of asymmetric loading will be discussed in Sect. 6.3.
The critical porosity along the crack path in the centre of the specimen is shown in Fig. 19, where predictions from the localization model and the softening model are shown by black and red curves, respectively. Similar to the SENT tests, the critical porosity is typically higher for the localization model, although the difference between these two models is almost indistinguishable for temper T7 at a 5 mm distance away from the slits. Material points closest to the slits show a higher f c for the localization model, which explains the delayed crack initiation and higher peak force. Some spikes in the f c curves can be seen for tempers T4 and T7 in the vicinity of the slits. Figure 20 shows the porosity as a function of the Lode parameter and the stress triaxiality for all material points on the righthand side from the centre line of the specimen. Dark colours indicate material points close to the slit, and the colour gradually changes to red for elements near the centre line. Solid dots indicate incipient accelerated void growth, and the evolution of porosity after this point has not been included. The stress triaxiality is generally slightly lower for the DENT test compared to the SENT test. Consequently, the critical porosity is slightly higher for the DENT specimen (Fig. 19) than for the SENT specimen (Fig. 17). Similar to the SENT tests, the higher f c near the tip of the slit is related to a stress state that evolves in the region of negative Lode parameter values. Moreover, some material points further away from the slit do not experience strain localization when the stress state changes due to the approaching crack. As for the SENT tests, this gives rise to the spikes seen in Fig. 19.
The predictions from the f c (T, L) model are included in Fig. 7 as dotted lines. The point of crack initiation is in agreement with the localization model for all tempers. The predicted global force for the two models corresponds well for tempers T6 and T7, while the force is slightly higher for temper T4 when the f c (T, L) model is used due to a higher f c along the crack path. and a Lode parameter of L = 0 (plane strain tension). Grey curves correspond to sections of the f c -surface (Fig. 13) where the stress triaxiality is fixed at 1.1 for tempers T4 and T6 and 0.75 for temper T7 6 Discussion 6.1 Influence from f F The actual porosity at failure f F cannot be accurately calibrated from the notch tension tests as the initiation and total failure of the specimen occur in quick succession. In addition, the release of stored elastic energy in the test machine might also influence the force-displacement curve after the initiation of failure occurs. The value f F = 0.14 was selected from previous experience and kept the same for all tempers. However, it could be calibrated separately for each temper using the results from the SENT test. A value of 0.25 was suggested by Tvergaard (1982) based on analytical estimates (Andersson 1977). However, other values have been proposed and Zhang and Niemi (1994) used an f F = 0.125 in their study. Bergo et al. (2021) found a f F = 0.35 when calibrating f F and A n to axisymmetric specimens for pipeline steel.
The force response of the SENT test for three different values of f F is shown in Fig. 21. Higher values of f F will restrain crack growth and consequently increase the force at a given displacement. This is especially prominent for tempers T4 and T6, while temper T7 is less affected by changes in f F . It appears that f F = 0.14 might be too large for temper T4 to accurately predict the correct global response. It should also be mentioned that the occurrence of abrupt variation in f c along the crack path, seen as spikes in Fig. 17, is greatly increased for low values of f F and completely eliminated in the case of f F = 0.20 for tempers T4 and T7. When a material point is strengthened due to an increase in f F , the porosity in neighbouring points is Results from the experiments are shown in shaded areas. The line styles separate the different values of f F allowed to grow slightly more before the Lode parameter changes toward a state of generalized tension. This might be the reason why Bergo et al. (2021) did not observe such spikes along the crack path for the Kahn test in their study.

Results using a fixed f c
The main feature of the softening and localization models presented in this paper is that f c is selected based on physical events and need not be predetermined by the user. However, it introduces additional complexity and the need for user-defined codes. The original Gurson-Tvergaard-Needleman model, which assumes that f c takes a fixed value, is available in several commercial finite element codes. However, one should be careful when applying the GTN model to other applications that undergo different stress states than the experiments used to calibrate the material parameters. In Fig. 22, results using a fixed f c are shown together with those obtained with the localization model. In this case, f c is given the same value as predicted by the localization model in the centre element of the NT10 specimen, as summarized in Table 3. The critical porosity used for each temper is also compiled in the caption of Fig. 22. The other material parameters remain unchanged. This way, the predicted response up to incipient accelerated void growth in the centre for the NT10 test coincides with the localization model. Clearly, a constant f c calibrated from the NT10 specimen yields unsatisfactory results for tempers T4 and T6. For temper T7, however, the response is better. As previously discussed, the value of the Lode parameter progresses towards L = 0 as the neck forms in the notched specimen in temper T7. Therefore, the stress state at accelerated void growth is not as different between the centre of the notch tension tests and the SENT tests. A phenomenological extension to account for shear softening due to void distortion and inter-void linking has been proposed by Nahshon and Hutchinson (2008) and Xue (2008). This extension would also decrease the ductility predicted by the GTN model for stress states closer to generalized shear but at the cost of introducing an additional material parameter.

Oblique crack path in the DENT tests
It has previously been argued that the discrepancy between the simulations and the experiments of the DENT specimen is partially because the simulations do not describe an oblique crack path. With a straight crack path, the cross-section decreases too quickly with respect to the elongation, which results in a more rapid reduction of the global force in the simulations compared to the experiments. To study possible effects from an oblique crack path, we imposed an asymmetry in the displacement of the loading plate. To this end, the five upper pins were given a small displacement in the xdirection, see Fig. 16b. In addition, the leftmost pin was given a slightly higher displacement in the z-direction than the rightmost pin. The displacement of the pins in between was linearly scaled with respect to their position. Together, these additional imposed displacements gave the best effect in provoking an oblique crack path. It should be mentioned that similar types of asymmetry in the displacement were measured in the experiments. However, the amount of asymmetric displacement was exaggerated by approximately 1.5 times in the simulations, where it ranges between 0.36 mm for temper  Table 3 and summarized in the following: f c = 0.038 for temper T4, f c = 0.034 for temper T6, and f c = 0.014 for temper T7 T7 and 0.71 mm for temper T4. The goal is not to exactly reproduce these asymmetric loadings, but to provoke an oblique crack path so that the effect on the force-elongation curve can be studied. Also, the FE model is still somewhat restrained due to a structured mesh and the use of symmetry over the xz-plane (see Fig. 16b). The effect of a slant fracture mode has not been taken into account in this study. As already discussed in Sect. 3.3, it is assumed that a cup-cup fracture mode will constrain the crack path as symmetry about the centre plane is preserved. For a slanted fracture mode, however, symmetry about the centre plane is no longer maintained. Seemingly, this favours an oblique crack path in the test setup.
The global force as a function of the elongation is shown in Fig. 23, where the predicted responses for symmetric and asymmetry loads are shown by solid and dashed lines, respectively. The localization model is used in both cases. A figure of the lower part of the failed specimen is included below their respective plot. Asymmetric loads in the simulations can induce similar crack paths as seen in the experiments. The crack path of temper T7 is somewhat exaggerated by the simulation, where a clear wave-shaped crack path is more prominent in the simulation than in the experiments. The force is in better accordance with the experiments for all tempers, although the higher peak force in the simulation of tempers T4 and T7 results in some discrepancy. The improvement is especially noticeable for temper T6 where the peak force and the decline in force after the peak is greatly reduced. The predicted crack length is also in better accordance with the experiments for all tempers, although it might be a bit too detained for temper T7.

Concluding remarks
This paper has presented experiments and simulations on ductile tearing using two types of plate tearing tests: the single edge notch tension (SENT) test and the double edge notch tension (DENT) test. The simulated material response is represented by an enriched Gurson-Tvergaard-Needleman model, in which three different methods to determine the onset of accelerated void growth are compared. These methods are described in Sect. 4 and named the softening model, the localization model and the f c (T, L) model. The main findings from this study are summarized as follows: • The SENT test setup appears to be highly reliable for all tempers, with little variation between repeated tests. In the DENT tests, the two edge cracks tend to grow at an oblique angle to each other, see Fig. 8. Temper T7 exhibits a less oblique crack path as a profound neck apparently restrains the crack path to remain fairly straight. • The fracture surface for temper T6, which is the least ductile of the considered tempers, consists of facets which suggests grain boundary fracture to be the main mechanism. Both temper T4 and T7 display larger dimples on the fracture surface, suggesting coalescence of voids nucleated at the constituent particles in the grain interior as the main fracture mode. As it is not completely clear if the physical mechanism assumed in the Gurson model is justified for temper T6, we have interpreted the GTN model as a coupled damage model, where accelerated void growth is initiated due to instabil- Pictures of the failed specimens are included below the plots together with predictions from the simulations ity (the occurrence of strain localization or softening) instead of void coalescence. • The predicted strain at material softening is not affected by the deviatoric stress state. On the other hand, the strain at localization greatly depends on both the deviatoric and hydrostatic stress states, where minimum ductility is predicted at stress states near generalized shear (L = 0). The highest strain where localization is still possible occurs for stress states at generalized tension (L = −1), whereas localization will not take place as the stress state approaches generalized compression (L = 1). Moreover, the strain at material softening coincides with the minimum strain at localization for a given stress triaxiality. The localization model is therefore more appropriate for a broader range of applications. • Studies on proportional stress states show that the porosity at strain localization depends both on the hydrostatic and deviatoric stress state, where the lowest critical porosity occurs for stress states closer to generalized shear (L = 0) and increases towards generalized tension (L = −1). While the failure strain depends on the stress path, the critical porosity appears to be path independent. • The localization model is able to represent the force response and initiation of failure of the notch tension tests it was calibrated to. Furthermore, the model describes crack initiation and growth in the SENT and DENT tests when combined with an element erosion technique. However, crack initiation appears to be slightly delayed compared to the experimental results, which can be explained by a stress state closer to generalized tension in the material points in the vicinity of the slit. Lastly, the finite element model of the SENT test predicts the crack propagation adequately, even though the fracture mode is not necessarily correctly captured. Conceivably, the fracture mode has a limited impact on the predicted global response as long as the crack path is not altered, e.g., as for the DENT test. However, this might not be the case for less ductile materials where plate thinning is not as pronounced. • The same material parameters are used in the softening model and the localization model. Consequently, some discrepancy between the experimental results of the notch tension tests and the predicted responses using the softening model is apparent. The softening model generally underestimates the forces in both the SENT and DENT tests. However, this could be ameliorated by increasing, e.g., the actual porosity at failure f F . Despite this improvement, the softening model is not suitable for a wide range of applications as it does not portray dependence on the Lode parameter.
• Although the f c (T, L) model generally coincides with the localization model, there were some minor differences observed in the overall force response for tempers T4 and T7.