True stress-strain identification accounting for anisotropy of sheet metals

Sheet metals for the automotive industry are subjected to continuous research efforts aiming at ever increasing mechanical performance. A remarkable feature of modern high strength sheet metals is their anisotropy, intrinsic in the technologic process of their production. When the effect of anisotropy on the mechanical response of a material cannot be neglected, specimens along different directions are usually tested, possibly under different stress states, to assess the flow curves and the deforming ratios for each direction and each loading mode. Such data are then used to calibrate many possible plastic anisotropy models available in the literature. In this work, the experimental procedures for determining the stress-strain curve and the anisotropic straining ratio are studied in detail, referring to representative tensile tests along the rolling direction of two anisotropic sheet metals, respectively PHS-1800 steel and 6181 aluminium alloy. Both alloys are ductile and exhibit remarkably long post-necking phases in tension, revealing that, in such cases, the standard procedures for the experimental derivation of the hardening curves and of the anisotropic strain ratios are limited to the very early phases of the material life and miss to cover the major part of the strain range up to failure. Different alternative procedures for the derivation of experimental data and for their postprocessing are considered and compared to each other, identifying a set of guidelines for achieving a good engineering accuracy up to failure in deriving both the stress-strain curves and the anisotropic strains ratios. The above analyses are made on the results of tensile tests at static, intermediate and high strain rate, confirming the generality of the identified procedure.


Introduction
Advanced high-strength sheet metals are often characterized by pronounced anisotropy resulting from the technological processes to which they are subjected, making their characterization more challenging.In the literature, numerous testing methodologies [1] and advanced material models [2] have been proposed.However, regardless the approach, the fundamental step is to obtain the material's flow curve through a classical tensile test on a dog-bone specimen.
Many researchers employ indirect approaches to obtain the material's flow curve, relying on the iterative reduction of target quantities representing the differences between simulation results and experimental data [3][4][5][6][7][8].One of the proposed inverse approaches is the Virtual Fields Method (VFM), which involves calibrating a predetermined material model to comply with the principle of virtual work [9][10][11].Macek et al. [12] proposed an hybrid indirect/direct approach, considering experimental data from classical tensile tests and a novel heterogeneous strain field test.
With highly anisotropic complex materials, indirect approaches could give not-unique solutions to the characterization problem due to the multitude of parameters to be calibrated.Consequently, researchers have also devoted significant efforts to direct approaches.They typically involve evaluating the true stress-strain curve of the material from a tensile test, based on the total axial load acting on the specimen and the corresponding minimum cross-section area.Indeed, such curve is identical to the flow curve before necking and, afterwards, it must be corrected in order to take into account the growing triaxiality.Bridgman [13] proposed the most well-known correction based on the curvature of the deforming specimen, while Mirone [14] proposed a material-independent corrective function applicable to both flat and cylindrical specimens.
The pre-necking true curve can be simply obtained using the conventional length-based approach, which is still commonly employed today [15].However, deriving the true curve after necking onset becomes challenging for anisotropic materials.Indeed, in isotropic or at least transversely isotropic materials, the specimen maintains its initial geometric proportions until fracture.Thus, by measuring the width or diameter of the specimen, the current cross-sectional area can be determined [16,17].However, this is not the case for anisotropic materials, making it more difficult to evaluate the actual cross-sectional area [16,18].Consequently, complex experimental setups such as 3D Digital Image Correlation (DIC) have been proposed to directly reconstruct the full 3D geometry of the specimen [19,20].
In this work, a relatively simple method is proposed to obtain the true curve of anisotropic materials from tensile tests on dog-bone specimens throughout their entire straining life until fracture.The novel procedure is based on optically analyzing the evolution of the thickness relative to the width before necking onset.Then, the obtained trend is assumed to apply in the post-necking phase as well, allowing for the derivation of the effective cross-section resistant area of the specimen and, consequently, the true stress and strain until fracture.The robustness of the proposed procedure is validated through optical analysis of the geometric proportions of the fractured specimens.Moreover, the proposed methodology is compared with other classical approaches, demonstrating its superior accuracy.

Experimental campaigns
The experimental campaigns considered in this work have been carried out on tensile specimens of two materials, the high strength steel PHS1800 and the aluminium alloy AL6181.All the specimens had the geometry shown in Figure 1, with thicknesses of 1.5 mm for the PHS1800 ones and 2.0 mm for the AL6181 ones.For both materials, the tensile tests have been carried out at the quasistatic rate of 3x10 -4 s -1 with an electromechanical testing machine, at the intermediate strain rate of 10 s -1 with a hydraulic testing machine, and at the high strain rate of 250 s -1 with the direct-tension Hopkinson bar developed by the authors [21].The load histories have been acquired with the load cells in the static and intermediate strain rate tests and with the three waves procedure in the high strain rate tests.They have been coupled with the images of the deforming specimens obtained with a standard high-resolution camera (static rate) or with a high-speed camera (intermediate and high strain rates) in order to employ the DIC technique and to obtain optical measurements of the gauge length L and of the width w of the specimens.

Stress and strain evaluation in flat specimens
Several characterization approaches have been considered and compared, aiming at the evaluation of the true curve of the anisotropic metals at hand.By definition, the true stress   and strain   are obtained by eqs.( 1) and ( 2), in which  is the resistant cross-section area of the specimen and  is the total axial force applied to the specimen.The subscript "0" refer to the initial value of every variable.
Given the difficulty in evaluating the actual resistant cross-section area, different estimations of such variables can be done experimentally with diverse approaches.Given the gauge length of the specimen , the classical length-based one makes use of the expressions shown in eqs.( 3) and ( 4) to obtain the corresponding true stress  − and strain  − .It is well known that these expressions, based on the volume conservation, are rigorously valid only until the necking onset and then diverge from the actual values quite consistently for ductile materials.
If the material is isotropic, then it is possible to consider that the proportions of the resistant crosssection of the specimen remain constant during the test.Then, measuring the width  of the specimen, it is possible to derive the corresponding thickness .With this hypothesis, the width-based true stress  − and strain  − can be calculated as shown in eqs.( 5) and (6).
Depending on the material and on the specimen shape at hand, the local peak strain measured with the DIC in the outer surface of the necking section can be higher or lower than the actual mean strain in that section [16].However, considering them approximately equal, it is possible to obtain the DIC Peakbased true stress  − and strain  − as shown in eqs.(7) and (8).
The deviation of  − from the effective true strain directly impacts the accuracy of eqs.( 7) -( 8), and the magnitude of such deviations cannot be estimated a priori.
If the material is anisotropic, the proportions of the resistant cross-section do not remain constant.Then, the width-based approach is not valid during the entire test, while the length-based one is valid only before the necking inception (volume conservation still applies).Then, there are no options for the correct evaluation of the post-necking part of the true curve.Therefore, a new procedure is proposed here.The main new concept is to evaluate the anisotropic behaviour of the material in the pre-necking phase and to consider that it applies also to the post-necking one.In particular, thanks to the volume conservation, the trend of the evolving thickness can be evaluated in the pre-neck phase as shown in eq. ( 9).Then, such trend is prolonged all over the post-necking range up to failure.This hypothesis will be also checked analysing the broken samples and evaluating if the final proportions of the cross-sections are compatible with it.

Strain histories
The strain histories obtained with the different approaches are shown in Figure 2 for two representative static tests of the two materials (similar trends are found for all tests).For the PHS1800 (left part of Figure 2), the necking strain is roughly 0.05.It is possible to see that in the pre-necking stage all three estimations are essentially overlapped.The equality between the DIC Peak-based and the L-based approaches is trivial in this stage.On the other hand, the fact that also the w-based approach gives the same trend means that anisotropy has negligible effect on the equality  0 / 0 = t/w.In the post-necking stage, obviously the L-based approach loses accuracy while the other two give very similar estimations.Then, the pre-necking strain estimation can be done with all three approaches, while, for the reasonings explained earlier, no reliable prediction is present for the postnecking range.
The AL6181 (right part of Figure 2) presents a necking strain of 0.13.In this case, the w-based prenecking estimation is very different from the other two evidencing a consistent anisotropic behaviour.Therefore, the pre-necking strain evaluation must be done with the L-based approach.Again, no reliable post-necking estimation is present.
Given that both metals exhibit a long post-necking ductile behaviour, the above uncertainties apply for more than half of their straining life.

Evaluation of the post-necking straining life of anisotropic materials
The trend of the thickness reduction is obtained for every test with eq. ( 9).As already mentioned, such approach is valid only before the necking onset.However, through the optical analysis of the broken specimens (Figure 3), it is possible to evaluate the final proportions of the resistant cross-sections.Considering three representative tests for each material, in Figure 4 are presented (PHS1800 left and AL6181 right), in terms of / 0 vs w/w 0 , the results obtained with eq. ( 9) (scattered curves for the prenecking phase and solid curves for the post-necking phase), together with circular points representing the specimens' geometries at failure.The initial condition of every test is the point (1;1).The prenecking points of all tests of each material are found to be approximable with a power law according to eq. ( 10), with exponent  equal to 1.35 for PHS1800 and 1.8 for AL6181 (green dotted lines in Figure 4).  0 = (   0 )  (10) As expected, the post-necking data represented by the solid lines quit to be approximable by those power laws, since they are not representative of the actual behaviour of the material.The black dashed lines represent a hypothetical isotropic behaviour descripted by a power low according to eq. ( 10) with the exponent  equal to 1.It is easy to demonstrate that the exponent  is equal to the inverse of the classic Lankford coefficient   =   /  .
From Figure 4, it is very clear that the power laws approximating the pre-neck data, comply really well also with the fracture circular points for both materials.Then, such trends are considered to be valid also in the post-necking range.Therefore, with this hypothesis, it is possible to evaluate the actual resistant cross-section of the specimens during the entire test and, in turn, the actual true strain and stress values.
It is important to underline that, if the Lankford coefficient is directly derived in the pre-necking phase from the strains instead as the inverse of the exponent , more uncertainties arise, especially for very early necking materials.This concept is clearly shown in Figure 5, in which the Lankford coefficients calculated directly from the strains obtained with optical measurements (green scattered lines) and with local DIC (yellow scattered lines) are compared to the inverse of the obtained exponents (blue dotted lines).Indeed, at the beginning of the plastic straining, the classical Lankford coefficient is an intrinsically undetermined ratio   /  = 0/0.Therefore, in the case of the very early necking PHS1800 (left column of Figure 5), it is almost impossible to obtain useful data without considering the sections at fracture.For the AL6181 (right column of Figure 5) it is possible to obtain useful information in the pre-necking phase but it is possible to see that the DIC and length-based evaluations quite differ from each other due to measuring uncertainties.Then, the proposed power-law based approach demonstrates to be more robust than the classical strain-based one.

Final true stress-strain curves
As already pointed out, considering that the trend obtained with eq. ( 10) in the pre-necking phase of every test always applies until fracture, it is possible to evaluate the actual resistant cross-section of the specimen during the entire test and, in turn, the actual true strain and stress values as in eqs.( 11) and (12).It is worth noting that with  = 1, eqs. ( 11) and ( 12) degenerate to eqs. ( 5) and ( 6) which are valid for isotropic materials.
The obtained curves with the different approaches, called L-based, w-based, DIC-peak, and Expo, are shown, together with the fracture points, in Figure 6 for representative tests at different strain rates for both materials.Analysing such figure, it is possible to highlight the differences between the approaches both in the pre-necking and in the post-necking phase.
In the pre-necking phase, the L-based, DIC-peak, and Expo approaches give obviously the same curve, while the w-based differs from the formers depending on the magnitude of anisotropy (more evident for the AL6181).
In the post-necking phase, evidently the L-based stops to be representative of the actual behaviour of the material, being the corresponding true curve much shorter and lower than the others, not adequately representing the local behaviour of the material.On the other hand, the w-based and Expo approaches give different estimates of the true curve depending on the magnitude of the anisotropy: for the PHS1800 there is a noticeable difference but for the AL6181 the difference is very significant, with the fracture strain increased of over 40%.Moreover, the Expo curves nicely comply also with the fracture points, confirming once again the reliability of the approach.The DIC-peak curves, depending on the single test, are more or less scattered due to the speckle degradation during the tests but are quite similar to the Expo curves.As already pointed out earlier and better explained in [16], the accuracy of this approach cannot be predicted since it depends on the relationship between the evaluated DIC peak strain and the actual strain distribution in the resistant cross-section.Given the results, for these materials combined with the corresponding geometries of the specimens, there is a quite good agreement between such two strains.Then, the Expo approach is the only one effectively reliable in evaluating the postnecking part of the curve, significantly improving the characterization procedure.
It is useful to recall that the proposed approach regards only the evaluation of the true curve of the material, which still must be corrected to obtain the actual corresponding flow curve taking into account the triaxiality [13,14].Moreover, additional tests would be necessary to assess the full anisotropic behaviour of the materials in different directions and stress states in order to correctly calibrate an appropriate anisotropic model.However, the proposed approach gives a fundamental basepoint in order to correctly evaluate the tensile behaviour of a material.

Conclusions
In this work, a novel procedure for the evaluation of the true curve of an anisotropic material from a tensile test on a dog-bone specimen has been proposed.The novel procedure has been compared to other possible approaches, highlighting the differences, and demonstrating the better performance of the proposed one.The different characterization approaches have been applied to tensile tests at different strain rates on dog-bone specimens made of PHS1800 steel and AL6191 aluminium alloy.
The novel characterization approach is based on the evaluation of the anisotropy of the material at hand thanks to optical evaluation of the geometry of the specimen during the test.In particular, the evolution of the normalized thickness / 0 related to that of the normalized width / 0 , is derived from optical measurements of elongation and width before the necking onset, when it is rigorously valid.Then, the obtained trend is supposed to apply also in the post-necking phase, deriving consequently the effective cross-section resistant area of the specimen and, in turn, the true stress and strain until fracture.The adopted hypothesis has been checked analysing optically the shape of the minimum section of the fractured halves of the specimens, confirming that their / proportions comply with the pre-necking trend of anisotropy.It was demonstrated that the proposed methodology based on the evolution of the normalized thickness / 0 related to that of the normalized width / 0 , is much more effective and robust than the classical Lankford coefficient-based one, especially for low necking strain materials in which the latter gives almost no useful information about the material anisotropy.
Finally, the true curves obtained from the different tests with the proposed methodology are compared with the curves obtained with the other classical approaches, demonstrating that it is the only one delivering accurate results in the post-necking range, which for the steel and Al alloy at hand, respectively covers nearly 90% and 75% of their whole straining life.

Figure 2 .
Figure 2. L-based, w-based and DIC-Peak strain histories for representative tests on PHS1800 steel (left) and AL6181 alloy (right) at static rates.

Figure 5 .
Figure 5. Lankford coefficients from DIC strains, from L-w strains and from power laws of the t-vs.-wreduction

Figure 6 .
Figure 6.Comparison of experimental true curves according to different approaches.