Size Effect on the Ductile Fracture of the Aluminium Alloy 2024-T351

Ductile fracture is a common failure mechanism that causes significant issues in many engineering fields. Most of the metals used in civil engineering, transportation and military industry exhibit ductile behaviour to some degree. Better understanding of that behaviour allows further reduction of the weight of structures while maintaining their reliability. The fracture strains for different stress states can be obtained experimentally only for the sheet metal (even though not in all cases), while numerically for cylindrical, tubular and other specimens manufactured from bulk material, for which the cracks can often initiate below the surface, which is not trackable by optics. The experiments can be performed on macroscopic or miniature samples. Miniaturized ones have roots in the nuclear industry, where material is scarce or too dangerous to handle in large quantities. Thus, it is important to find a way to determine the properties of a small amount of material without risking excessive irradiation or the properties of a new material that is being developed. The number of possible miniature testing methods has gradually been extended and has begun to be utilized in various fields of application, providing information on stress–strain behaviour, fatigue, creep or fracture. Although miniature testing methods require less material, it should be noted that they also have limitations and disadvantages related mainly to the size effect and problematic sample preparation. On the other hand, such testing is suitable for estimating the remaining life of structures with the minimum amount of material needed. The repair process after the removal of the material is then either unnecessary or simple, unlike the extraction of standard specimens, which significantly disrupts the structure.

When the size of a material is reduced to a microscale level, the sample often consists of only a few grains in the deformation zone. Then, an individual grain plays a more significant role in the deformation process, which causes the material to behave differently from macroscale parts. This phenomenon is known as the size effect, which affects the properties of a material and makes conventional models and methods no longer applicable in the analysis of microscale specimens [1‐4]. The surface grains are less restricted than the inner grains, leading to lower hardening and lower resistance against deformation and rotation, resulting in lower flow stresses [5, 6]. It is also easier for voids to coalesce and cracks to grow throughout the thickness of microscale materials [2]. The decrease in flow stress with increasing miniaturization can be explained by a surface layer model. Based on that, Lai et al. [5] proposed a new constitutive model to express the stress–strain relationship. In this model, the internal layer of the sample is treated as a polycrystalline material, whereas the surface grains are treated as a single crystal. Then, the fracture strain and energy are much lower in positive stress triaxiality than in negative stress triaxiality regardless of the shape and grain size of the sample [6].

There are many issues in the testing of miniaturized specimens associated with specimen geometry, preparation methods or measuring devices [7, 8]. Therefore, Beerli et al. [9] suggested a semi-automatic miniature testing method to prevent damage to the specimens during handling. The test results can also be affected by the aforementioned size effect, which becomes more significant if the sample thickness was only six to ten times greater than the average grain size [10]. Moreover, the small dimensions of these specimens do not allow the use of standard extensometers for measurements. Thus, optics is used to record the strains [11, 12]. Finally, the specimens have a limited grip section and often require the development of special grippers [7, 13]. The most commonly used testing method for miniaturized specimens is a tension. The specimens generally have rectangular or cylindrical cross-sections. Regarding the device, it must be able to ensure a very slow feed rate because of a very small gauge length. The advantage of the tensile testing method is that it allows for the achievement of different stress states in the crack initiation location by varying the geometry and features of the specimens (Fig. 1), which can also be helpful in investigating the properties of functionally graded materials [14], for example. These tests can be combined with other miniature experiments, such as the Small Punch Test (SPT), to capture a wider range of stress states. SPT is a punching experiment in which a ball or pin is pressed through the centre of a small disk-shaped specimen [15]. Wang et al. [16] used the SPT to evaluate the fracture strain and fracture toughness of the reactor vessel steel. SPT can also be used to obtain experimental creep data, as conducted in a study of AlSi9Cu3 and AZ31 alloys by Andrés et al. [17]. Furthermore, SPT is also a valuable tool for fatigue analysis, as shown by Lewis et al. [18], who carried out experiments on additively manufactured materials. Moreover, the testing devices have been developed to allow elevated temperatures [13], in situ electron microscopy [19, 20], in situ optical microscopy [21, 22] or high strain rates [23]. Last but not least, Watanabe et al. [24] used the in situ miniature testing for subsequent crystal plasticity finite element analysis. A basic overview of the possible miniature specimen geometries is shown in Fig. 1.

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The ductile fracture criteria can also be calibrated using butterfly specimens that usually have a small-scale process area. The butterfly specimen allows multiple stress states to be reached by varying the loading angle, as done by Mae et al. [25]. A similar case could be with a cruciform specimen [26].

Rice and Tracey [27] analysed the growth of the spherical void and proposed a ductile fracture criterion dependent on the stress triaxialitywhere \(\sigma_{1}\) is the maximum principal stress, \(\sigma_{2}\) is the middle principal stress, \(\sigma_{3}\) is the minimum principal stress and \(\overline{\sigma }\) is the equivalent stress. Decades later, the criteria have been extended by the dependence on the Lode angle \(\theta\) [28‐30]where \(J_{3}\) is the third invariant of deviatoric stress tensor. Today, machine learning has been incorporated [31, 32], such as Particle Swarm Optimization (PSO) for example [33], which can even replace the criterion itself [34]. Finally, the ductile fracture criteria can be coupled or uncoupled, influencing hardening or not [35]. Coupling the criteria with a hardening results in a weakening (softening) effect on the constitutive law in the scope of Continuum Damage Mechanics (CDM), as in this work. The material parameters of the aforementioned models are generally calibrated using the hybrid experimental–numerical approach, which has also been used in this work. Yao et al. [36] reported on machine learning to provide similar results compared to the hybrid experimental–numerical approach. However, models with a large number of material parameters presented a challenge for the machine learning technique [36]. Regardless of the calibration procedure, the models work with stress triaxiality and Lode angle. This space must be reasonably covered by fracture tests for a reliable calibration of the ductile fracture criteria. Basic fracture tests are highlighted in Fig. 2 on the plane of stress triaxiality and normalized Lode anglewhich can consequently be rewritten for the plane stress as

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In conclusion, miniature testing attracts growing interest that focuses on many aspects. This work aims on the calibration of the ductile fracture criterion using monotonic tensile tests, apart from the small punch test, carried out in a miniature testing device developed in order to show the size effect in ductile fracture for aluminium alloy 2024-T351. It is supplemented by a possible computational modelling in the scope of CDM to show the predictability.

The material model can be established after explaining the stress states and quantities. Also, the model is described within this section so that the computational results could be plotted together with experiments in the following parts.

First, the flow curve is estimated. The hardening law is coupled with damage to describe the equivalent stress–equivalent plastic strain relationship in the scope of CDM, obeying the von Mises yield criterion with associated flow rule. A simple power law is chosen as it has a minimum number of material parameters to be calibrated, yet enough for accurate description of the behaviour as shown later. Then, the equivalent stress iswhere \(K\) is the strength coefficient, \(\overline{\varepsilon }_{p}\) is the equivalent plastic strain, \(n\) is the strain hardening exponent and \(D\) is the damage parameter, for which a nonlinear accumulation is anticipated according to Xue [38], which was also used later by Papasidero et al. [39] for the same materialwhere \(\overline{\varepsilon }_{f} \left( {\eta ,\overline{\theta }} \right)\) is the fracture strain as a function of the stress state variables and \(m\) is the damage exponent. The damage parameter reaches the critical value when unity. The smooth specimen tensile test was numerically simulated to obtain the aforementioned material parameters related to plasticity and damage, together with elastic constants, Young’s modulus \(E\) and Poisson’s ratio \(\nu\). All, summarized in Table 1, were found using the trial and error method so that the simulated responses corresponded to those obtained experimentally. This was at the expense of a higher damage exponent than reported by Xue [38] and Papasidero et al. [39], who based it on the low-cycle fatigue and tension–torsion tests ended up with \(m = 1.73\) and \(m = 0.71\), respectively. The weakening exponent is sometimes introduced to the damage parameter in equation (5), which can then artificially decrease the value of the damage exponent. It was not the case in the present study, where a higher damage exponent was a matter of fit in the scope of the coupled damage used. Moreover, it cannot be directly compared with Papasidero et al. [39], who used the uncoupled approach, so the damage exponent did not influence the stress–strain curve contrary to the present analysis. All experiments were simulated within Abaqus/Standard, employing the USDFLD and UHARD user subroutines. Stress triaxiality and normalized Lode angle were defined as state variables within the USDFLD code, while the ductile fracture criterion, damage law and hardening rule were defined in UHARD.

Finally, the ductile fracture criterion is defined according to Ganjiani [40] aswhere \(\overline{\sigma }^{ref}\) is the reference equivalent stress, \(\eta^{ref}\) is the reference stress triaxiality, \(\overline{\theta }^{ref}\) is the reference normalized Lode angle and \(c_{\eta }\) and \(c_{{\overline{\theta }}}\) are the material parameters related to the stress triaxiality and normalized Lode angle, respectively. Equation (7) has to be transformed from stress space to strain space using a power law considered in equation (5), only without brackets with the damage aswhere \(\overline{\varepsilon }_{f}^{ref}\) is the reference fracture strain. Then, the fracture strain iswith cut-off stress triaxiality as

The remaining material parameters were calibrated using an evolutionary algorithm, namely PSO within MATLAB with 10000 particles in the swarm, which found a global minimum (as listed in Table 2) for the four miniaturized fracture tests. The goodness of fit is clear from the following section, where the corresponding experiments and results of the described calculations are explained.

The Ganjiani ductile fracture criterion was calibrated earlier toward four fracture tests (crack initiation points are highlighted in Figs. 7, 9, 11 and 13) with stress triaxiality and normalized Lode angle averaged with respect to equation (6) as followswhere \(\hat{\varepsilon }_{f}\) is the fracture strain for the respective fracture test. All respective calibration points are summarized in Table 3. These were obtained using previously presented numerical simulations.

The results for miniature specimens (Table 3) are compared in Fig. 15 with the results for standard-sized specimens taken from [42] for the same heat (these specimens were manufactured from the same block of material to eliminate the role of different microstructure) and for standard-sized specimens taken from [39] for the same alloy. Visual comparison reveals no significant size effect for the ductile fracture of 2024-T351, which is consistent with a similar study on a pipeline steel [47]. Only the miniature shear specimen (no such geometry was included in the macroscale batch – the geometry producing the closest stress state was the torsional notched tube) exhibited significantly higher fracture strain, making the fracture envelope higher in generalized shear compared to uniaxial tension condition (Fig. 16a). However, testing shear properties is globally problematic and needs more research [48], to determine whether the geometry of the specimen (sharp corners compared to notched tube), different boundary conditions (tension versus torsion) or other aspects (misalignment) is to blame. Finally, this observation is in correspondence with the results of Li et al. [49], who studied the nickel-based superalloy K418 on a similar microscale. It should be noted that there were no data on the uniaxial compression condition on the microscopic scale. Therefore, the fracture envelope naturally increased with decreasing normalized Lode angle in Fig. 16(a).

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The Ganjiani ductile fracture criterion based on the previous calibration on miniature specimens is given in Fig. 16(a). This fracture envelope has a similar shape as its macroscale counterpart (Fig. 16(b)) in the region where calibration points are present. The miniature-based fracture envelope along with the cut-off significantly deviates only in the region of normalized Lode angle of –1, where no calibration point is present as discussed above. This can be seen for the plane stress condition in Fig. 17, where the difference is within percent points up to the stress triaxiality of 1/3. It should also be noted that the results for macroscale specimens from [42] are based on an uncoupled approach compared to the coupled approach in this study. However, it should produce similar results regardless of the calibration procedure, confirming that there is no distinctive size effect on ductile fracture within this study in terms of the phenomenological ductile fracture criterion. Also, presented approach is intended for crack-free bodies and primarily for failure initiation, which does not compete with classical fracture mechanics.

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Miniature tensile testing becomes important in many engineering disciplines. Therefore, the detailed design of the apparatus for such experiments was developed. Subsequent modelling of elasticity and plasticity reveals a good conformity with the experimental results for the 2024-T351 aluminium alloy, including the addition of a small punch test and comparison with post-mortem specimens. Furthermore, the Ganjiani ductile fracture criterion is calibrated with the help of machine learning. However, artificial intelligence could hardly replace the fracture criteria themselves because of the lack of experimental data. On the other hand, neural networks could be directly trained using more calibrated ductile fracture criteria at once to incorporate the uncertainty of calibration, as some criteria may result in much better fit in terms of mean absolute percentage errors. Finally, the following analysis based on miniature and standard-sized specimens reveals no significant size effect on ductile fracture on a presented level. Therefore, such miniature specimens can be used to characterize the bulk ductile fracture behaviour of 2024-T351 without interrupting the structure with the extraction of a considerable amount of material needed for large-scale testing.