## Abstract

This paper investigates the influence of stress state on the equivalent plastic fracture strain in 2024-T351 aluminum alloy. Eighteen unique stress states at failure—with triaxialities ranging from 0.388 (compressive) to −0.891 (tensile) and Lode parameters ranging from −0.978 to 1.000—are explored through mechanical experiments on 2024-T351 aluminum specimens with various geometries under multiple loading conditions. These include tension tests of plane stress (thin), plane strain (thick), and axisymmetric specimens, as well as pure shear and combined axial–torsional loading on thin-walled tubes. Using a hybrid numerical–experimental approach, the dependence of fracture strain on stress triaxiality and Lode parameter is quantified for each experiment. Fracture strains are measured using three-dimensional digital image correlation. Equivalent plastic fracture strain for 2024-T351 generally increases with stress triaxiality (moving toward a more compressive state). Fracture strain decreases modestly as the Lode parameter decreases from 1 to 0, although there is a significant increase in ductility as the Lode parameter decreases below −0.8. Compression–torsion tests extend the data’s stress-space coverage well into the compressive triaxiality region (up to 0.388) and to Lode parameters approaching −1. This experimental program provides the most extensive set of ductile fracture data from a single 12.7-mm-thick 2024-T351 aluminum plate in the literature. These data can be used to calibrate ductile fracture models used in finite element simulations of dynamic events such as bird strikes, automotive collisions, and debris containment.

## 1 Introduction

Blade-off and rotor-burst events in gas turbine engines, although rare, pose a significant threat to the safety of passengers and crew onboard commercial aircraft. To mitigate this hazard, the Federal Aviation Administration (FAA) requires that commercial jet engines be outfitted with a containment system that prevents the engine case from being penetrated in the event of a compressor or turbine blade being released from the disk. Further, for certification of new, derivative, and modified jet engine designs, the FAA requires commercial engine manufacturers to demonstrate through full-scale destructive testing that the most critical blade (i.e., the fan blade) is contained within the case when separated from the disk while the engine is running at full-rated thrust (Title 14 of the Code of Federal Regulations (CFR) 33.94). These full-scale blade-off experiments are both difficult to conduct and very expensive, motivating the need for predictive numerical simulations to lessen the likelihood of failing a certification test (economically disastrous for the manufacturer) or experiencing an inflight failure (disastrous for the passengers and crew).

The impact of high-speed (60–275 m/s) engine fragments with the fan case during a blade-release or rotor-burst event is notoriously difficult to model and simulate [1]. In the case of metals, the complex impact-penetration physics involves high strain rates, steep temperature gradients, large plastic deformations, progressive damage, and ultimately ductile fracture, all of which must be taken into account in the analysis. Notably, the ductile fracture model must be capable of accounting for multiple potential failure modes (e.g., plugging, petaling, and mixed-mode failure), which are intimately tied to the impact conditions (e.g., projectile and target material properties, geometry, relative orientation, and impact speed) and subsequent state of stress at impact. For instance, in ballistic impact tests, thin-sheet failure (upon the impact of a rounded projectile) is often biaxial-tension-dominated and characterized by radial tearing of the impact zone into petals (petaling), whereas thick-plate failure (upon the impact of a blunt projectile) is often shear-dominated and characterized by the formation of a coin-like plug (plugging) [1]. In other words, different impact conditions lead to different states of stress at impact and thus different failure modes, all of which must be accurately captured by the ductile fracture model.

Most damage evolution and ductile fracture models for metals are inspired by theories of microscopic void nucleation, growth, and coalescence, first set forth in the pioneering work of McClintock [2] and Rice and Tracey [3]. McClintock [2] was the first to recognize the role of voids in ductile fracture; using a microscopic void growth model, he demonstrated that the fracture strain depends on the full state of stress rather than just the maximum principal stress. Rice and Tracey [3], again using a microscopic void growth model, demonstrated that the dilatational amplification (volumetric growth) of a spherical void increases exponentially with the far-field mean stress. Based on these seminal ideas, Gurson [4] developed a physics-based plasticity model where micro-scale porosity is incorporated into the continuum constitutive model via an internal variable, the void volume fraction. Gurson’s plasticity model was later modified by Tvergaard and Needleman [5] to account for ductile fracture (via microvoid coalescence) through the dependence of the yield function on a critical void volume fraction, thereby coupling damage and plasticity. Other modifications to the Gurson plasticity model to account for ductile fracture can be found, for instance, in Refs. [6–8].

In contrast to the micromechanics-based Gurson-type models, continuum-scale phenomenological ductile fracture models that are inspired by—but do not explicitly model—void nucleation, growth, and coalescence have proven popular and effective, particularly in commercial finite element codes for structural-level analyses. Perhaps, the most famous phenomenological ductile fracture model is that of Johnson and Cook [9], which provides a mathematical relationship between the equivalent plastic strain at fracture (a common measure of ductility) and the stress triaxiality (or normalized mean stress, defined as the ratio of the mean (hydrostatic) stress to the equivalent (von Mises) stress). The Johnson–Cook model was empirically motivated by the seminal ductile fracture experiments of Hancock and Mackenzie [10] and Mackenzie, Hancock, and Brown [11], who showed that the equivalent plastic strain at fracture decreases exponentially with increasing stress triaxiality. This key result, obtained from tensile tests on notched axisymmetric specimens and Bridgman’s analysis of the stress state at the necking localization, is consistent with the analytical microvoid models of McClintock [2] and Rice and Tracey [3].

Building on the work of Hancock and Mackenzie [10,11], Refs. [12–17] experimentally investigate the effect of stress triaxiality on ductility over a broader range of stress states using different specimen geometries and loading conditions. However, more recent experimental studies have conclusively shown that triaxiality alone is insufficient to model the ductile fracture of metals over the full range of potential stress states. This limitation was first addressed by Wierzbicki et al. [18] and later confirmed through the experimental work of Barsoum and Faleskog [19], who found that the ductility (plastic strain at fracture) depends on the three-dimensional state of stress, which can be characterized by supplementing the triaxiality with the Lode parameter. Using scanning electron microscopy, Barsoum and Faleskog [19] also noted different rupture mechanisms under different loading conditions: growth and internal necking of voids at high (tension-dominated) triaxialities and internal void shearing at low (compression-dominated) triaxialities. Recent ductile fracture models incorporating both stress triaxiality and Lode parameter dependence can be found in Refs. [20–27]. Recent experimental work investigating this dual dependence can be found in Refs. [28–34].

This paper presents results from an experimental test program designed to investigate the stress-state-dependent ductile fracture behavior of a single 12.7-mm-thick 2024-T351 aluminum plate. The program is designed specifically to study the effect of stress state (characterized by stress triaxiality and Lode parameter) on the equivalent plastic fracture strain (ductility) of the material. These experiments include tension tests on specimens with several different geometries (thin/plane stress, thick/plane strain, and round/axisymmetric), pure shear (torsion) tests, and combined loading (tension–torsion and compression–torsion) tests. This experimental program provides the most extensive set of ductile fracture data from a single 12.7-mm-thick 2024-T351 aluminum plate in the literature. Digital image correlation (DIC) is used to measure the full-field strain on the surface of the specimens, allowing for the resolution of deformation gradients and strains in localizations (such as necking and shear bands). The results presented in this paper (a) illustrate how the equivalent plastic strain is influenced by the state of stress for 2024-T351 aluminum and (b) can be used to calibrate state-of-the-art tabulated and parameterized ductile fracture models used in explicit finite element simulations of impact events such as bird strikes, blast-related impacts, automotive collisions, debris containment, and metal forming.

## 2 Preliminary Continuum Kinetics

*p*= −

*I*

_{1}/3 is the hydrostatic pressure. The principal invariants of the deviatoric stress tensor are

Stress triaxiality (taken here to be the normalized hydrostatic pressure, following the definition used in LS-DYNA) and Lode parameter (represented here as the normalized third invariant of the deviatoric stress) are defined as

Alternative representations of the latter include the Lode angle parameter and normalized Lode angle parameter, as discussed in the study by Lou et al. [35]. Definition (5) implies that the triaxiality is −0.33 in a uniaxial tension test, 0 in a pure shear test, and 0.33 in a uniaxial compression test—differing by a minus sign from the alternative definition of triaxiality that employs a mean stress rather than hydrostatic stress.

## 3 Material, Specimen Design, and Test Plan

The material considered in this paper is a single 12.7-mm-thick 2024-T351 aluminum plate (Kaiser Aluminum, Spokane, WA). Vendor-reported chemical composition is provided in Table 1. After being cast and rolled, the plate was solution heat-treated, stretched a controlled amount for stress relief, and naturally aged [36]. The resulting microstructural morphology is described in the study by Seidt and Gilat [37].

Quasi-static ductile fracture behavior of the 12.7-mm 2024-T351 aluminum plate is investigated using three separate tensile test series (plane stress, axisymmetric, and plane strain), torsion (pure shear) tests, and a combined loading (tension–torsion and compression–torsion) test series. All coupon specimens for these tests are machined from the same plate, with each specimen’s longitudinal axis aligned with the rolling direction as shown in Fig. 1. Plane stress specimens are machined using wire electrical discharge machining (EDM), with the recast layer subsequently ground off. Axisymmetric, plane strain, and combined loading specimens are machined using traditional manufacturing techniques (e.g., mills and lathes). Global tolerances for the specimen designs are ±0.127 mm, while tolerances for critical sample dimensions (e.g., gage length, width, thickness, and/or diameter) are ±0.0635 mm. After polishing, the arithmetic mean surface roughness (Ra) is measured using a contact profilometer. Ra values from a representative set of plane stress specimens are found to be less than 0.8 *µ*m (ISO roughness grade N6).

The first tensile test series is conducted on thin (0.762 mm), flat, smooth, and notched specimens (Figs. 2(a)–2(d)). When loaded, these thin flat tensile specimens are in a state of plane stress (i.e., the third principal stress vanishes) at the center of the specimen. The second tensile test series is conducted on axisymmetric, smooth, notched specimens (Figs. 2(e)–2(j)). When loaded in tension, these specimens undergo a stress state where two of the three principal stresses are equal, which translates to a Lode parameter of 1.0. The third tensile test series consists of experiments on thick, smooth, notched specimens (Figs. 2(k)–2(m)). These specimens are sufficiently thick (25.4 mm) to approach a state of plane strain at the center of the specimen. The specimens for these three tensile test series are designed to (a) cover a broad range of negative (tensile) stress triaxialities (approximately −0.9 to −0.3) and (b) demonstrate that the equivalent plastic strain at fracture depends on both the triaxiality and Lode parameter. As an example of the latter, two axisymmetric specimens (SG6 and SG9) and two plane strain specimens (SG11 and SG13) are designed to have similar triaxialities but substantially different Lode parameters. All specimens are fabricated such that the direction of loading is parallel to the rolled direction of the plate.

A key takeaway is that this relatively simple combined axial–torsional loading experiment can be used to investigate ductile fracture over a diverse range of potential stress states. In principle, stress triaxialities ranging from $\sigma *=\u22120.33$ (uniaxial tension) to $\sigma *=0.33$ (uniaxial compression) are achievable, with pure shear $(\sigma *=0)$ in-between, as is the full range of Lode parameters (−1.0 to 1.0). In practice, however, when approaching uniaxial compression, the thin-walled tube may buckle or not maintain its alignment with the load. The axial-to-shear stress ratio *a* can be controlled (i.e., held nearly constant) using load and torque control on a biaxial servohydraulic load frame. This implies that the stress state is nearly constant for the duration of the experiment. In contrast, the geometry of the tension specimens shown in Figs. 2(a)–2(m) evolves significantly from the onset of loading to fracture (even for un-notched specimens after the onset of necking); thus, the stress state also evolves significantly during the specimen’s deformation history.

The combined loading test specimens are shown in Figs. 2(n) and 2(o). The thin-walled tube specimens are fabricated such that the longitudinal axis of each specimen is parallel with the rolling direction of the plate. Two combined tension–torsion tests (denoted LR1 and LR2) and a pure shear/torsion experiment (denoted LR3) utilize the thin-walled tube specimen in Fig. 2(n) with a longer gage section and reduced wall thickness. The targeted axial-to-shear stress ratios in LR1, LR2, and LR3 are *a* = 2.587, 1.148, and 0, respectively. Two combined compression–torsion tests (denoted LR4 and LR5) utilize the thin-walled tube specimen in Fig. 2(o) with a shorter gage section and increased wall thickness to prevent buckling. The targeted ratios of axial stress to shear stress in LR4 and LR5 are *a* = −0.920 and *a* = −1.696, respectively. Note that precise triaxiality and Lode parameter target values, such as those shown, do not need to be achieved; failure surface creation can use the available fracture strains at any stress state.

## 4 Experimental and Numerical Procedures

### 4.1 Experimental Procedure.

For the experimental series described in Sec. 3, quasi-static mechanical testing is conducted at room temperature and ambient conditions using an Instron 1321 biaxial servohydraulic universal testing machine. Depending on the amount of load and torque required, the Instron load frame is equipped with either a Lebow 6467-107 load cell (with maximum load and torque capacities of 88.964 kN and 1.13 kN-m, respectively) or an Interface 1216CEW-2K load cell (with maximum load and torque capacities of 8.896 kN and 0.113 kN-m, respectively). An MTS FlexTest SE controller is used for both experimental control and digital data acquisition. MTS 647.02B-22 axial–torsional side-loading hydraulic wedge grips are used and fitted with either flat wedges (for thick and thin flat specimens) or V-notch wedges (for round and thin-walled tube specimens). At least three tests are conducted for each specimen design.

The tension test series SG1–SG13 is conducted using constant axial actuator velocity (i.e., displacement control; refer to Table 2) to achieve a nominal strain rate of 0.01 s^{−1} in each test. For the tension–torsion tests (LR1 and LR2) and compression–torsion tests (LR4 and LR5), axial load and torque control are used, with load and torque both increased linearly in time so that the axial stress to shear stress ratio (refer to Eqs. (6) and (7) and the discussion thereafter) is nearly constant for the duration of the experiment. Recall that the targeted axial-to-shear stress ratios in the LR1/LR2 tension–torsion tests are *a* = 2.587 and *a* = 1.148, respectively, and for the LR4/L5 compression–torsion tests, *a* = −0.920 and *a* = −1.696, respectively. The pure shear/torsion experiment LR3 is conducted with axial load control set to zero and constant rotational actuator velocity (i.e., angle control; refer to Table 2) to achieve a nominal strain rate of 0.01 s^{−1}.

*L*

_{VSG}(in

*px*) is calculated from the user-defined subset size

*L*

_{subset}(in

*px*), step size

*L*

_{step}(in

*px*), and strain filter

*L*

_{filter}as follows [38]:

The VSG length in physical spatial units (in this case mm) can be computed by multiplying *L*_{VSG} (in *px*) by the image scale or calibrated physical length of each pixel (in this case mm/*px*, Table 3). The physical length of each pixel is dependent on both the camera sensor resolution and the field of view (FOV). This naturally varies from setup to setup, even when identical camera hardware is used, since it is impractical to achieve the same pixel length for different camera setups.

In previous work, several low-strain rate tensile tests were conducted on the SG1 specimen geometry in order to calibrate the constitutive behavior of the material [37]. These test specimens were fabricated from the same material stock as the samples in this investigation using the same manufacturing methodology. DIC was not used to measure displacement and strain during this previous test series. Instead, a 4-mm gage length, physical extensometer was used to measure displacement only. These tests were included in this data set to bring deeper statistical significance to the equivalent plastic strain at failure for this particular case (SG1). This is done by creating an average Hencky strain (at the failure point) versus displacement (from a DIC-measured 4-mm virtual extensometer) curve from the tests in which DIC is used. Using the displacement at failure (measured using the 4-mm physical extensometer), the Hencky strain state at failure is interpolated (in some cases slightly extrapolated) from the DIC-created average Hencky strain versus displacement curve.

### 4.2 Finite Element Simulations.

In each of the ductile fracture experiments, the stress and strain fields are inhomogeneous (non-uniform) in the specimen gage section, particularly after the onset of localization. Thus, the history of the state of stress (triaxiality and Lode parameter, Eq. (5)) and equivalent plastic strain are obtained using a hybrid experimental–numerical approach. For each experiment, measured data (e.g., force-displacement, torque-angle, and principal surface strains) are compared to the corresponding results from a parallel numerical simulation. Agreement between the simulation and the measured data implies that the simulation is adequately capturing the three-dimensional stresses and strains during the experiment. The simulation is then used to extract the history of the triaxiality, Lode parameter, and equivalent plastic strain at the site of fracture initiation.

Three-dimensional solid models of the 2024-T351 aluminum alloy specimen geometries (Fig. 2) are rendered in solidworks (v. 2019). Nominal dimensions in the computer-aided design drawings are modified in the 3D solid models to reflect the exact, measured dimensions of the as-machined, as-tested specimens. Finite element meshes (Fig. 3) are created using the grid-generation software HyperMesh (v. 2017.1, Altair, Troy, MI). Numerical simulations are performed in the commercial finite element software LS-DYNA (v. R10.1.0, Ansys/LST, Livermore, CA). Three-dimensional, constant-stress, eight-node, solid hex elements are used in the simulations. The finite element models of the specimens contain anywhere from 9640 elements (SG2) to 334,730 elements (SG11), with a characteristic mesh size of 0.15 mm and each element’s aspect ratio approaching 1:1:1 in the gage section. A mesh sensitivity analysis confirmed that plastic deformation is relatively insensitive to mesh size as long as the curvilinear features of the coupon are well-represented. In this case, the 0.15-mm mesh size is sufficiently small (five elements through the thickness of the plane stress samples and four elements through the wall thickness of the tension–torsion sample) to achieve post-yield convergence in each specimen of interest. To account for the tendency of under-integrated elements to hourglass, stiffness-based hourglass control is employed. Hourglass energies imposed by the hourglass control algorithm are acceptably low in all simulations.

The constitutive response of 2024-T351 aluminum is modeled using a publicly-available *MAT_224 material card calibrated by Park et al. [39] to an extensive set of experimental data for 2024-T351 aluminum. In *MAT_224, isotropic linear elastic deformation is characterized in a parametrized fashion by specifying two elastic constants, Young’s modulus (*E* = 70 GPa)_{,} and Poisson’s ratio (*ν* = 0.33). The (constant) mass density is *ρ* = 2600 kg/m^{3}. After yield, thermo-viscoplastic deformation is characterized through user-input hardening curves (equivalent true stress versus equivalent true plastic strain) at different temperatures and strain rates. Using these curves, a J2 (von Mises) yield surface is constructed numerically and used internally by the code. In this work, the *MAT_224 plasticity model is simplified by using only a single quasi-static (1 × 10^{−4} 1/s) room-temperature hardening curve (Fig. 4), rendering it rate-independent. This is a reasonable simplification, as the plastic deformation of 2024-T351 aluminum does not exhibit significant rate sensitivity at the low-strain rates used in the experimental test series [37,39].

To specify the boundary conditions in tests performed under displacement or angle control (all but LR1, LR2, LR4, and LR5), boundary node sets are created at the two DIC “boundary points” in the specimen grip section. Only the section of the specimen between these two points is modeled in the simulation (Figs. 2 and 3). The motion of the boundary node sets is specified according to the DIC-measured displacement history. In contrast, for tests performed under load and/or torque control (LR1, LR2, LR4, and LR5), the motion of the gripped sections is driven by load and/or torque signals from the physical experiment. In all cases, to numerically track the displacement and rotation in the specimen gage section, two nodes are selected to coincide with the two DIC virtual extensometer points.

## 5 Results and Discussion

Figures 5–8 illustrate comparisons—for representative plane stress (SG1), axisymmetric (SG6), plane strain (SG11), and combined loading (LR1) tests—between measured experimental data and the corresponding results from a parallel finite element simulation. Figure 5 presents a plot of the experimental and calculated axial loads and maximum and minimum principal strains at the location of eventual specimen fracture for the SG1 specimen. In this case, the simulated and experimentally measured results agree quite well. The numerical simulation tends to overestimate the measured force and underpredict the maximum/minimum principal strains. These deviations are likely due to the plastic deformation of 2024-T351 aluminum deviating from J2 flow theory, which does not consider tension–compression–shear yield asymmetry (important for this material) or plastic anisotropy (not important for this material) [37].

The same plot for the axisymmetric SG6 sample is presented in Fig. 6. In this case, the simulated maximum and minimum principal strain and load history agree quite well with the experimental data.

Comparison data from a plane strain sample (SG11) are presented in Fig. 7. For this case, the maximum and minimum principal strain histories agree. Simulated and experimental minimum principal strain histories for this case show strain values at fracture around −0.01, a very small value showing that the sample design led to the confinement of the material in the center of the sample and is indeed approaching plane strain conditions at that point. The main discrepancy in this comparison is the overestimation of the force by the simulation, believed to be due to tension–compression–shear yield asymmetry not captured by J2 flow theory [37].

Experimental and numerical results for a combined tension–torsion test are presented in Fig. 8. The case highlighted here is a tension–torsion (LR1) test where the axial stress is 2.587 times larger than the shear stress magnitude. Experimental strain magnitudes are slightly larger than their simulated counterparts; however, simulated and experimental axial load and torque histories generally exhibit good agreement.

*ɛ*

_{1}and

*ɛ*

_{2}at the site of fracture initiation are obtained from DIC virtual strain gauge measurements. The usual assumption of isochoric (volume-preserving) plastic deformations and small elastic strains imposes the constraint that the sum of the principal Hencky strains vanishes, i.e.,

*ɛ*

_{3}at the fracture site. The equivalent plastic strain at fracture is then computed using

Good agreement between simulation and experiment in surface strain histories at the eventual site of fracture (e.g., Fig. 7) is compelling evidence that the effective plastic strain calculated using Eq. (12) is reliable.

Values of equivalent plastic strain at fracture versus stress triaxiality for each test are presented in Fig. 11. As expected, the strain at fracture generally decreases with decreasing stress triaxiality (stress state trending toward hydrostatic tension) and increases with increasing stress triaxiality (stress state trending toward hydrostatic compression). There are some outliers to this general trend, however. First, there is a dip in the fracture strain value (0.240) at a stress triaxiality of zero, achieved with pure torsional tests on thin-walled tube samples. The nearest neighbor data points at positive stress triaxiality (compression–torsion test at a stress triaxiality of 0.210) and negative stress triaxiality (tension–torsion test at a stress triaxiality of −0.184) each have measured fracture strains substantially higher than those in pure shear (0.571 and 0.293, respectively). This trend has been observed previously by other researchers [14,28] and questioned by others [32–34].

Another interesting trend in Fig. 11 is found by comparing the plane stress and axisymmetric tension results. Plane stress and axisymmetric failure strain data at a stress triaxiality near −0.333 (uniaxial tension) agree quite well (0.297 failure strain for plane stress and 0.307 for axisymmetric). As stress triaxiality decreases from −0.333 (trending toward hydrostatic tension), the plane stress and axisymmetric fracture strain data diverge. The plane stress data form a lower “branch” and the axisymmetric data an upper “branch.” Plane stress data range from a stress triaxiality of −0.334 (SG1) to −0.592 (SG4-C), and their respective fracture strains are 0.297 and 0.095. Here, the stress state and failure strain values for SG4 are taken from elements and DIC strain field data at the center of the SG4 specimen notch (hence the label SG4-C).

When this ductile fracture test series was designed, it was assumed that the SG4 specimen would fail in the center of the gage section. However, after analysis associated with the material model calibration process, there is strong evidence that the SG4 specimen fails at the edge of the notch. The cameras used in the experiments are not fast enough to confirm fracture initiation and crack progression. In the experimental image just prior to fracture, the specimen is uncracked. In the subsequent image, the crack has formed and propagated entirely from one notch edge to the other. The specimen has dramatically different stress and strain state histories at the edge than it does at the center. The simulation predicts the stress triaxiality at the edge is −0.415 (compared to −0.592 at the center), and DIC-measured strain at fracture at the edge is 0.255 (compared to 0.095 at the center). The edge SG4 fracture strain (labeled SG4-E) agrees quite well with the SG2 fracture strain that has a similar stress state, providing more evidence that fracture occurs at the edge of SG4. Both center and edge stress state and fracture strain data for SG4 are included in Table 4, denoted SG4-E (edge) and SG4-C (center), respectively.

It is also interesting to note the differences between the axisymmetric data (squares) and the plane strain data (triangles) in Fig. 11. There are two cases where an axisymmetric specimen and a plane strain specimen have similar stress triaxialities, yet very different fracture strains. The first case is SG6 and SG11. These samples have respective stress triaxialities of −0.535 and −0.558, yet fracture strains of 0.270 and 0.196. The second case is SG9 and SG13. Although their respective triaxialities are similar (−0.767 and −0.733), their failure strains are not (0.238 and 0.194). The discrepancy between axisymmetric and plane strain tension data illustrates that a ductile fracture model calibrated with stress triaxiality alone is insufficient to accurately describe the fracture behavior of this material.

The data in Fig. 12 (equivalent plastic fracture strain versus Lode parameter) can be used to explain (and model) the differing fracture strain values. Due to stress-state symmetry (off-axis principal stresses are equal, i.e., *σ*_{2} = *σ*_{3}), all axisymmetric tension test specimens have a Lode parameter of 1. That is not the case for the plane strain test specimens, having very different Lode parameters (ranging from 0.048 to 0.186). Figure 12 shows that strain at fracture is a minimum at Lode parameter near zero and increases dramatically as the Lode parameter decreased toward negative 1 (compression–torsion tests) and increases slightly as the Lode parameter increases toward positive 1. This trend is consistent with the relatively low fracture strain at pure shear (pure torsion tests). This test has a Lode parameter of zero. Adding compression to this stress state decreases the Lode parameter, greatly increasing the fracture strain; adding tension increases the Lode parameter, which only modestly increases the fracture strain. It is clear from Figs. 11 and 12 that both triaxiality and Lode parameter must be used to accurately model the fracture behavior of 2024-T351 aluminum.

Figure 13 is a depiction of every presented stress-state condition in the stress triaxiality versus Lode parameter stress-space. The specimens tested in this work are identified by the same markers used to depict average fracture strain in Figs. 11 and 12. The dashed line depicts the plane stress meridian; i.e., all stress states that fall on this line is in the plane stress condition. There are two interesting observations in Fig. 13. The first is that most of the data presented here are in the plane stress condition, illustrating how difficult it is to generate three-dimensional stress states in controlled mechanical tests. The second is that, aside from the plane stress meridian, the experimental data cover very little of the achievable stress space. This illustrates the need for further research to reliably impart three-dimensional stress states into thoughtfully designed specimens and loading conditions to explore these unpopulated regions. Recent work along this front is reported, for instance, by Spulak et al. [40].

## 6 Conclusions

The stress-state-dependent ductile fracture behavior of 2024-T351 aluminum is investigated using several different types of mechanical experiments. These tests include tension tests on several different specimen geometries, combined loading, and pure shear experiments. Tension tests are conducted on plane stress, axisymmetric, and plane strain specimens for a total of thirteen different sample geometries. The plane stress tension test series is conducted on thin specimens, one with a straight gage section and three with different notch radii. Tension tests are also conducted on axisymmetric specimens, one with a straight gage section and five others with different notch radii. The plane strain tension test series is conducted on thick specimens, one with a smooth gage section and two more with different notch radii. Specimen surface strains are measured experimentally with 3D DIC. Combined tension–torsion, compression–torsion, and pure shear experiments are also conducted. Stress-state histories in the thin-walled tube specimens are determined from load and torque cell measurements, while surface strains are measured with 3D DIC. The dependence of equivalent plastic fracture strain on stress triaxiality and Lode parameter is determined over a wide range of stress states. The data show that the triaxiality alone is insufficient to describe the ductile fracture behavior of 2024-T351: tests conducted with similar stress triaxiality, yet different Lode parameters, have significantly different equivalent plastic fracture strains.

## Acknowledgment

This work was funded by the Federal Aviation Administration (FAA) under Grant No. 2006G004 and Cooperative Agreement No. 692M151940011. Thanks to Don Altobelli, Chip Queitzsch, Bill Emmerling, and Dan Cordasco for their support and involvement.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.