Theoretical predictions of dynamic necking formability of ductile metallic sheets with evolving plastic anisotropy and tension-compression asymmetry

The rate of deformation tensor d is the sum of an elastic part d^e and a plastic part d^p:

where the relationship between the elastic part and the rate of the stress is: where \(\overset\triangledown\sigma\) is an objective derivative of the Cauchy stress tensor, and C is the fourth-order isotropic elastic tensor defined as: with 1 being the unit second-order tensor and \(\mathbf {I}^{\prime }\) the unit deviatoric fourth-order tensor. Moreover G and K are the shear and bulk moduli, respectively.

Assuming an associated plastic flow rule, the plastic part of the rate of deformation tensor is defined as:

where \(\dot {\lambda }\) is the rate of the plastic multiplier and \(\bar {\sigma }\) is the effective stress [15] given by:

where k is the tension-compression asymmetry parameter, see Eq. 10a, and Σ_i (i = 1,...,3) are the principal values of the transformed stress tensor Σ defined as:

with L being an orthotropic, deviatoric, and symmetric fourth-order tensor, and s the deviatoric part of the Cauchy stress tensor σ. In the Cartesian coordinate system (X₁,X₂,X₃) associated to the orthotropy axes of the material, the deviatoric part of the Cauchy stress tensor is represented by a 6-dimensional vector s = (s₁₁,s₂₂,s₃₃,s₁₂,s₂₃,s₁₃) and the fourth-order orthotropic tensor L is represented by a 6 × 6 matrix:

with the components of L being the anisotropy coefficients, see Eq. 10b. Moreover, the parameter \(\tilde {m}\) in Eq. 5 is defined as:

with Φ_i (i = 1,...,3) expressed as follows:

The evolution of plastic anisotropy and tension-compression asymmetry with accumulated plastic strain is modeled using the methodology developed by Plunkett et al. [48]. Firstly, the values of the parameters k and L_ij (i,j = 1,...,6), see Eqs. 5 and 7, are identified for selected levels of effective plastic strain \(\bar {\varepsilon }^{p,1}<\bar {\varepsilon }^{p,2}<...<\bar {\varepsilon }^{p,m}\). Secondly, a linear interpolation is used to obtain the values of the tension-compression and anisotropy coefficients corresponding to any given level of effective plastic strain \(\bar {\varepsilon }^{p,n}<\bar {\varepsilon }^{p}<\bar {\varepsilon }^{p,n+1}\) (n = 1,...,m − 1): where the interpolation parameter α is: such that \(\alpha \left (\bar {\varepsilon }^{p,n}\right )=1\) and \(\alpha \left (\bar {\varepsilon }^{p,n+1}\right )=0\).

The yield function is defined as:

where σ_Y is the tensile yield stress in the rolling direction of the material X₁, given in the present work by the following relationship:

with A₀, A₁ and A₂ being material parameters. Notice that viscous and thermal effects on the yield stress evolution are neglected. Moreover, the effective plastic strain is \(\bar {\varepsilon }^{p}={{\int \limits }_{0}^{t}}\dot {\bar {\varepsilon }}^{p}\mathrm{d}\tau\), where t refers to time, and \(\dot {\bar {\varepsilon }}^{p}\) is the effective plastic strain rate given by:

where \(\lvert \mathbf {b}\rvert =\sqrt {{b_{1}^{2}}+{b_{2}^{2}}+{b_{3}^{2}}}\), and b₁, b₂ and b₃ are the principal values of the transformed rate of deformation tensor defined as:

where \(\mathbf {H}:\mathbf {G}=\mathbf {I}^{\prime }\) with \(\mathbf {G}=\mathbf {L}:\mathbf {I}^{\prime }\).

Moreover, the work conjugacy relation: yields:

The formulation of the constitutive behaviour is completed with the Kuhn–Tucker loading–unloading conditions:

and the consistency condition during plastic loading:

We investigate 3 materials widely used in different industrial sectors and engineering applications in the form of plates and shells: AZ31-Mg alloy, high purity α-titanium and OFHC copper. Sheet products of these materials are often subjected to dynamic forming operations to produce complex metal parts, see the introduction to this paper (Section “Introduction”). In addition, the three materials display very different yielding behaviour, see Fig. 1, increasing the scope of the analysis carried out in Section “Analysis and results”.

Table 1 shows the values of the initial density, the elastic constants and the parameters of the yield stress in the rolling direction [50]. We are aware that using a Voce-type hardening relation Eq. 13 to describe the yield stress of the materials investigated may be a crude assumption, particularly at high strain rates, when viscous and thermal effects –due to adiabatic heating– are known to affect dynamic formability limits [27, 29, 42]. Nevertheless, using a simple constitutive relation reduces the number of material parameters, which helps in the identification of the effects of anisotropy and tension-compression asymmetry on dynamic formability.

The values of the anisotropy coefficients L_ij (i,j = 1,...,6) and the tension-compression asymmetry parameter k for the initial yielding and for several individual levels of effective plastic strain are given in Table 2. Data are taken from Revil-Baudard et al. [50] and Cazacu et al. [16]. Recall from Section “Formulation” that the values of L_ij and k for effective plastic strains within the intervals reported in Table 2 are calculated using a linear interpolation function, see Eq. 10a. Moreover, in absence of additional data, we consider that L_ij and k do not evolve for effective plastic strains outside the range of values included in Table 2, so that for larger strains the values of L_ij and k for AZ31-Mg alloy, high purity α-titanium and OFHC copper are the same employed for \(\bar {\varepsilon }^{p}=0.1\), 0.3 and 0.12, respectively. Moreover, notice that, while OFHC copper is considered to be isotropic, high purity α-titanium and AZ31-Mg alloy display both evolving anisotropy and tension-compression asymmetry.

Figure 1 pictures plane stress theoretical yield loci for biaxial loading conditions, σ_I versus σ_II, where σ_I and σ_II are the components of the Cauchy stress tensor in the rolling and transverse directions, respectively. Results are shown for the three materials investigated in this work and different levels of effective plastic strain. In the case of AZ31-Mg alloy, Fig. 1a, the yield locus evolves from an triangular shape for low values of effective plastic strain (the triangular shape is caused by very pronounced tension-compression asymmetry) to an elliptical shape at large strains. On the other hand, notice that the evolution of the yield locus with the plastic strain is different for high purity α-titanium, see Fig. 1b, such that yield locus is initially ellipsoidal and it becomes triangular at large strains (increasing strength differential effect with plastic straining). Moreover, notice that both AZ31-Mg alloy and high purity α-titanium display plastic anisotropy so that the yield stress in the rolling and the transverse directions is different. Moreover, for OFHC copper, Fig. 1c, the predicted yield loci have an elliptical type shape, with the yield stress in compression being slightly greater than in tension (mild strength differential effect so that the yield loci do not become triangular). Recall that this material is taken to be isotropic and therefore the yield stress in the rolling and the transverse directions is the same. Appendix A provides a comparison with the plane stress yield loci calculated with von Mises [39] criterion, which illustrates the effect of anisotropy and tension-compression asymmetry on the mechanical behavior of AZ31-Mg alloy, high purity α-titanium and OFHC copper.

This technique is based on the superposition of a small perturbation on the homogeneous solution of the problem, so that if the perturbation grows faster than the background solution, the plastic flow is unstable and a non-homogeneous, neck-like deformation field can develop. We have used the same 2D approach employed in [43, 51, 63] so that the stress multiaxiality effects that develop inside the necked section have been approximated using Bridgman [11] correction. The fundamental solution of the problem is composed of 16 equations which are nondimensionalized, perturbed using the frozen coefficients method and linearized (note that in the case of N’souglo et al. [42] the number of equations was 17 because the energy balance was taken into account). Imposing that the determinant of the matrix of coefficients of the resulting system of dimensionless linear algebraic equations is equal to zero leads to a third order polynomial for the instantaneous growth rate of the perturbation η which depends on the fundamental solution and on the perturbation wavenumber ξ. The root of the polynomial with physical meaning is the one with the greatest positive real part [19], hereinafter denoted by η⁺. The wavenumber is related to the normalized perturbation wavelength \(\bar {L}= L^{0} / h^{0}\) as \(\bar {L} = \frac {2 \pi \xi ^{-1}}{h^{0}}\). Hereinafter, \(\bar {L}\) will be also referred to as necking wavelength. On the other hand, the instantaneous growth rate is used to compute the cumulative instability index \(\bar {I}={{\int \limits }_{0}^{t}} \eta ^{+} d \tau\) which tracks the history of the instantaneous grow rate of all the growing modes. Assuming that a perturbation mode turns into a necking mode when the cumulative instability index reaches a critical value which depends on the loading path and the strain rate, the linear stability analysis is used to construct forming limit diagrams which are compared with nonlinear two-zone model predictions and unit-cell finite element simulations. The procedure for calibration of the linear stability analysis is detailed in Appendix B, and the influence of the calibration procedure in the stability analysis predictions is shown in Appendix C.

The non-linear two-zone model [29] is an extension of the classical analysis of localized necking proposed by Marciniak and Kuczyński [37] to account for inertia effects. The model considers an imperfection in the form of a band of reduced thickness, perpendicular to the main straining direction, at the center of a unit-cell of length L⁰ (see Fig. 4 in N’souglo et al. [42]). Note that L⁰ also corresponds to the neck spacing. The initial thickness in the imperfection zone is h^A,0 = h⁰(1 −Δ), Δ being the normalized imperfection amplitude. The ratio between the initial length of the imperfection \(L_{x}^{A,0}\) and the initial unit-cell length is denoted by \(R=L_{x}^{A,0}/L^{0}\). In the present work, the value of this parameter was set to R = 0.28. This value was identified in the case of isotropic material [29] and was found to remain appropriate for orthotropic materials obeying the Hill [24] yield criterion [42]. The methodology to obtain necking strains with the two-zone model is described in Section 2.5 of Jacques [29]. Computations have been carried out for several normalized cell sizes, \(0.5 \leq \bar {L} \leq 6\) with \(\bar {L}=\frac {L^{0}}{h^{0}}\) and h⁰ = 2 mm, in order to identify the critical neck spacing and the corresponding necking strains (critical necking strains). The results presented in Section “Analysis and results” have been obtained with a normalized imperfection amplitude Δ = 0.2%. This value has been chosen arbitrarily. It would have been probably possible to improve the agreement with the experimental results (see Section “The influence of material behavior”) by adjusting the value of Δ for each material investigated. However, in order to illustrate the influence of the material behavior on formability, we have preferred to keep the same value of Δ for all calculations.

The simulations consider a unit cell with a sinusoidal spatial imperfection defined as \(h=h^{0}-\delta \left (1 + {\cos \limits } \left (\frac {2 \pi X}{L^{0}} \right ) \right )\), where δ is the amplitude of the imperfection. Exploiting the symmetry of the model, only one eight of the plate has been analyzed, with reference configuration (imperfection-free) defined by the domain \(0 \leq X \leq \frac {L^{0}}{2}\), \(0 \leq Y \leq \frac {L^{0}}{2}\) and \(0 \leq Z \leq \frac {h^{0}}{2}\). Recall that the origin of coordinates of the Lagrangian Cartesian coordinate system \(\left (X, Y, Z\right )\) is located at the center of mass of the whole cell. As in the two-zone model calculations, we consider several normalized cell sizes \(0.5 \leq \bar {L} \leq 6\) with \(\bar {L}=\frac {L^{0}}{h^{0}}\) and h⁰ = 2 mm fixed, and the normalized imperfection amplitude is \({\Delta } = \frac {2\delta }{h^{0}}= 0.2\%\). Note that the normalized cell size \(\bar {L}\) will be also called necking wavelength, to be consistent with the notation employed in the stability analysis and the two-zone model. The finite element calculations are performed with the finite element software ABAQUS under the initial and boundary conditions defined in equations \(\left (45\right )\) and \(\left (46\right )\) of N’souglo et al. [42], which are consistent with the stability analysis and the two-zone model. The unit-cell calculations have been carried out for selected loading paths χ = 0, 0.125, 0.25, 0.375, 0.5, 0.625 and 0.75. The major stretching direction coincides with the X axis. The finite element model is meshed with eight-node solid elements, with reduced integration and hourglass control (C3D8R in ABAQUS notation), with dimensions ≈ 25 × 25 × 50 μm³ (elements are initially shorter along the loading directions). The constitutive model presented in Section “Constitutive framework” has been implemented in ABAQUS/Standard [2] and ABAQUS/Explicit [1] via UMAT and VUMAT user subroutines, respectively, using the stress-update algorithm based on the numerical approximation of the yield function gradients developed by Hosseini and Rodríguez-Martínez [25]. The UMAT [2] has been used to obtain the low strain rate finite element results, 0.0001 s^− 1 and 100 s^− 1, reported in Figs. 3 and 13, and the VUMAT [1] has been used to obtain all the other finite element calculations reported in this paper for higher strain rates. Note that both ABAQUS/Standard and Explicit simulations take inertia effets into account. The difference is that the time integration of the equations of motion is based on the central difference scheme in ABAQUS/Explicit [1] and on the Hilbert-Hughes-Taylor (HHT) method in ABAQUS/Standard [2]. The use of the implicit HHT scheme leads to smaller computation times for low or moderate loading rates, while the explicit scheme is more efficient for highly dynamic loading conditions. Note also that ABAQUS/Explicit uses the Green-Naghdi rate to compute the objective derivative of the stress (see Eq. 2), while the stress rate in ABAQUS/Standard corresponds to the Jaumann derivative.

Figure 3 displays forming limit diagrams (critical major necking strain \(\varepsilon ^{c}_{xx}\) versus critical minor necking strain \(\varepsilon ^{c}_{yy}\)) predicted by the stability analysis, the nonlinear two-zone model and the unit-cell finite element calculations. The critical major necking strain is the logarithmic strain in the major loading direction when necking occurs and it is determined as in N’souglo et al. [42], so that the reader is referred to Section 6.1 therein for the procedure to obtain \(\varepsilon ^{c}_{xx}\), and no additional details are provided here for the sake of brevity. The critical minor necking strain \(\varepsilon ^{c}_{yy}\) is the logarithmic strain in the minor loading direction when \(\varepsilon ^{c}_{xx}\) is measured. The results correspond to AZ31-Mg alloy with material orientations ψ = 0^∘ and 90^∘, see Fig. 3a, high purity α-titanium with ψ = 0^∘ and 90^∘, see Fig. 3b, and OFHC copper, see Fig. 3c. A comparison is performed with calculations in which the material behavior is modeled with von Mises plasticity (i.e. imposing on the CPB06 criterion k = 1 and L_ij = δ_ij, with i,j = 1,...,6 and δ_ij being the Kronecker unit delta) and the same values of initial density, elastic constants and parameters of the yield stress (see Table 1). The results are obtained for an imposed strain rate of \(\dot {\varepsilon }^{0}_{xx}=100~\text {s}^{-1}\), that for all purposes can be considered quasi-static loading for the calculations performed in this work, as for this loading rate the effect of inertia on the necking formability of the three materials is negligible (for the specimen thickness considered), see Appendix D.

The \(\varepsilon ^{c}_{xx}-\varepsilon ^{c}_{yy}\) curves display a concave-downward shape, with the formability increasing as the loading path moves away from plane strain tension. The results obtained with the stability analysis and the nonlinear two-zone model are in qualitative agreement with the finite element calculations –which are considered as the reference approach to validate the theoretical models– for the three materials investigated. For AZ31-Mg, see Fig. 3a, the formability for the von Mises material (black lines and markers) becomes greater than the results obtained with the CPB06 criterion as the loading path moves away from plane strain; the values of \(\varepsilon ^{c}_{xx}\) being slightly larger for ψ = 0^∘ (red lines and markers) than for ψ = 90^∘ (green lines and markers). By analogy with the results of N’souglo et al. [42] for materials modeled with Hill [24] criterion, the relative order of the \(\varepsilon ^{c}_{xx}-\varepsilon ^{c}_{yy}\) curves obtained with von Mises and CPB06 models depends on the Lankford coefficient when the material is considered anisotropic, the general trend being that if the Lankford coefficient is greater than 1 (as it is the case of AZ31-Mg for ψ = 0^∘ and 90^∘, such that \(r_{0^{\circ }}=1.40\) and \(r_{90^{\circ }}=1.88\) for \(\bar {\varepsilon }^{p}\geq 0.1\), see equation (17) in Cazacu et al. [15]) the von Mises material displays greater formability. In addition, Appendix A shows that the CPB06 criterion predicts greater curvature of the yield locus near equibiaxial tension than von Mises, which is assumed to decrease the material formability. The experimental formability results reported by Choi et al. [17] (orange markers) for AZ31B sheets obtained using the limiting dome height test (see Fig. 3b therein) are also included in Fig. 3a. Near plane strain tension, the experimental data are in good quantitative agreement with the finite element predictions, however, as the loading path moves away from plane strain, the agreement gets slightly worse. On the other hand, both stability analysis and two-zone model predictions overestimate the experimental results with the differences increasing as the loading path moves away from plane strain. Fig. 3b pictures the results corresponding to high purity α-titanium. As in the case of AZ31-Mg alloy, the calculations using von Mises criterion yield greater values for the critical major necking strain as the loading path moves away from plane strain tension, and the lower formability corresponds to the CPB06 criterion with orientation ψ = 90^∘. As mentioned before, this can be attributed to the fact that the anisotropic material displays a Lankford coefficient greater than 1 for ψ = 0^∘ and 90^∘ (namely, \(r_{0^{\circ }}=1.09\) and \(r_{90^{\circ }}=1.11\) for \(\bar {\varepsilon }^{p}\geq 0.3\)), and also because the curvature of the yield locus near equibiaxial tension is greater than for the von Mises material, see Appendix A. In Fig. 3b the experimental data reported in Fig. 7 of Stachowicz [57] for high purity α-titanium sheets are included. The finite element predictions are in very good agreement with the experiments. The two-zone model slightly underestimates the experimental data for all the loading paths and the linear stability analysis slightly underestimates/overestimates the experimental data for loading paths near plane strain/equibiaxial tension. We have also included in Fig. 3c the experiments reported in Fig. 10 of Melander [38] for OFHC copper sheets. The numerical simulations and analytical models show good overall qualitative agreement with the experimental results. Moreover, note that for OFHC copper both theoretical models and the finite element calculations show that the predictions using von Mises plasticity are above the results obtained for CPB06 as the loading path moves away from plane strain stretching and approaches equibiaxial tension. We have checked that taking k positive (see Table 2) the trend is reversed (results are not shown for the sake of brevity), and the critical major necking strain is smaller using von Mises plasticity, suggesting that for an imposed value of the initial major strain rate, the formability increases for a material that displays a larger yield stress in uniaxial tension than in uniaxial compression.

These results make apparent the influence of the yield criterion on the materials formability (for the same elastic constants and yield stress parameters), and the capacity of both stability analysis and two-zone model to provide predictions which are in good accord with experiments and consistent with finite element results.

Figure 4 shows forming limit diagrams obtained with unit-cell calculations, linear stability analysis and two-zone model for AZ31-Mg alloy and different initial major strain rates (5000 s^− 1, 10000 s^− 1, 20000 s^− 1 and 40000 s^− 1) for which the effect of inertia on the formability of the material is noticeable. The agreement between the predictions obtained with the analytical models and the finite element calculations is reasonably good for all loading paths and strain rates. Notice that, as the value of \(\dot {\varepsilon }^{0}_{xx}\) increases, the \(\varepsilon ^{c}_{xx}-\varepsilon ^{c}_{yy}\) curves are shifted upwards. Moreover, as the loading path moves away from plane strain tension, the von Mises criterion predicts greater formability than the CPB06 model (as in the results shown in Fig. 3a).

The evolution of the critical major necking strain with the imposed initial major strain rate for χ = 0 (plane strain tension) and χ = 0.5 is shown in Fig. 5. The \(\varepsilon ^{c}_{xx}-\dot {\varepsilon }^{0}_{xx}\) curves display a concave-upward shape, so that the necking strain is roughly constant for strain rates lower than ≈ 1000 s^− 1 (negligible inertia effects, see Appendix D), and then increases nonlinearly for greater strain rates. For plane strain tension, the unit cell finite element calculations predict that the increase of the necking strain is \({\Delta } \varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}^{40000~\text {s}^{-1}}= \frac {\varepsilon ^{c}_{xx}\rvert _{40000~\text {s}^{-1}} - \varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}}{\varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}}=223\%\) for the von Mises criterion, and 247% and 239% for the CPB06 criterion with orientations ψ = 0^∘ and 90^∘, respectively. Similar quantitative results are obtained with the stability analysis and the two-zone model. For χ = 0.5, the \(\varepsilon ^{c}_{xx}-\dot {\varepsilon }^{0}_{xx}\) curves are shifted upwards since necking is delayed as the loading path moves away from plane strain. The increase of the necking strain with the strain rate is \({\Delta } \varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}^{40000~\text {s}^{-1}}=88\%\) for the finite element calculations carried out with von Mises criterion, and 214% and 241% for the calculations performed with CPB06 criterion with ψ = 0^∘ and 90^∘, respectively. This shows that the influence of strain rate on necking strains is strongly dependent on the yield criteria considered. The results presented in Fig. 5 indicate that the stabilizing effect of inertia at high strain rates is enhanced for materials having low formability. The increase of the necking strain with the strain rate computed with the CPB06 model is of the same order as in the experiments of Xu et al. [62] (see Section “Introduction”). Moreover, note that for von Mises the finite element calculations show that the stabilizing effect of inertia strongly decreases as the loading path moves away from plane strain (the same trend was obtained in the unit-cell calculations performed by Rodríguez-Martínez et al. [52] and N’souglo et al. [42] for materials modeled with von Mises [39] and Hill [24] criteria); however, this trend is less evident for the calculations performed with CPB06, showing that the effect of inertia on neck retardation depends on the yield criterion.

Figure 6 shows the forming limit diagrams for high purity α-titanium and the same strain rates considered in the calculations performed for AZ31-Mg alloy, i.e. 5000 s^− 1, 10000 s^− 1, 20000 s^− 1 and 40000 s^− 1. The relative order of the \(\varepsilon ^{c}_{xx} - \varepsilon ^{c}_{yy}\) curves is independent of the strain rate, with the von Mises material displaying greater formability, being the results obtained with the CPB06 criterion for ψ = 0^∘ and 90^∘ very similar (see Section “The influence of material behavior”). While both theoretical models predictions find good correlation with the finite element results, the stability analysis generally overestimates the limit strains obtained from the unit-cell calculations, and the opposite trend is obtained with the two-zone model.

The results for the evolution of the critical major necking strain with the imposed initial major strain rate shown in Fig. 7 for χ = 0 and 0.5 display the ability of both theoretical models to capture the effect of inertia on neck retardation for high purity α-titanium (differences between theoretical model predictions and finite element calculations being less than 10%). For plane strain tension, the increase of the necking strain with the strain rate is \({\Delta } \varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}^{40000~\text {s}^{-1}}=245\%\) for von Mises criterion, and 212% and 255% for the finite element calculations performed with CPB06 model with ψ = 0^∘ and 90^∘, respectively. For χ = 0.5, the effect of inertia is significantly less, and the increases become 94% for von Mises, and 127% and 153% for CPB06 with ψ = 0^∘ and 90^∘, respectively. These results reinforce the idea that the stabilizing effect of inertia depends on both the loading path and the yield criterion.

Figure 8 shows forming limit diagrams for OFHC copper and the same values of the initial major strain rate considered in Figs. 4 and 6, i.e., 5000 s^− 1, 10000 s^− 1, 20000 s^− 1 and 40000 s^− 1. Recall from Section “The influence of material behavior” that as the loading path moves away from plane strain the critical major necking strains corresponding to von Mises criterion are above the results obtained with CPB06 model because the yield stress in uniaxial compression is greater than in uniaxial tension (i.e. because k < 0, see Table 2). We have checked that, for the whole range of strain rates investigated, considering the same absolute values of the tension-compression parameter k, yet positive, yields critical major necking strains above the von Mises predictions. The reader is referred to the paper of N’souglo et al. [43] for specific analysis and discussion on the effect of the strength differential effect on the dynamic necking formability of ductile sheets subjected to plane strain stretching. The electrohydraulic forming experiments reported by Balanethiram and Daehn [6] for OFHC copper sheets tested at strain rates that were estimated to be ≈ 500 s^− 1 –this is a lower bound estimate obtained from a simple analytical model. Subsequently, more refined analyses of the deformation during conical-die electrohydraulic forming based on finite element computations revealed that strain rates larger than 10000 s^− 1 are achieved in some parts of the specimen [28]– are compared in Fig. 8c with the results obtained using both analytical approaches and the finite element calculations. The agreement between experimental data, two-zone model and finite element calculations is satisfactory, however, the stability analysis underestimates the formability obtained with the finite elements near plane strain tension for 40000 s^− 1. The evolution of the critical major necking strain with the imposed initial major strain rate for χ = 0 and 0.5 is pictured in Fig. 9a and b, respectively. These graphs make apparent the underestimation by the stability analysis of the stabilizing effect of inertia at high strain rates. A reason for this disagreement is that the stability analysis has been calibrated using the physical constants and yield stress parameters corresponding to AZ31-Mg, which has a mass density ≈ 5 times smaller than copper (see Table 1), so that the dependence of the critical instability index with the strain rate underestimates the effect that inertia has on necking in copper (we have checked that calibrating the set of coefficients of the critical instability index using the initial density, elastic constants and parameters of the yield stress corresponding to OFHC copper, see Table 3, the stability analysis predicts greater stabilizing effect of inertia at high strain rates; see also the results in Fig. 12 where the calibration with OFHC copper yields slightly greater values of critical major necking strain). Moreover, the increase of the necking strain with the strain rate computed with the finite element calculations for plane strain tension is \(\varepsilon ^{c}_{xx}\rvert _{100~\text {s}^{-1}}^{40000~\text {s}^{-1}}=352\%\) and 308% for the von Mises criterion and the CPB06 model, respectively. These values are similar to the experimental data reported by Balanethiram and Daehn [6] (see Table 1 therein) and they are greater than for AZ31-Mg alloy and high purity α-titanium (see the paragraphs above), which makes apparent that the contribution of inertia to neck retardation depends on the material. For χ = 0.5, the increase of the necking strain with the strain rate for both von Mises and CPB06 yield criteria is 126% and 191%; i.e., smaller than for plane strain tension.

Recall from Section “Materials investigated” that we have not considered viscous and thermal effects on the behavior of the materials. For specific insights into the role that thermal softening and strain rate sensitivity have on forming limit diagrams of anisotropic materials, the reader is referred to Section 6.4 of N’souglo et al. [42]. The general trend is that, while thermal softening promotes early necking localization, the strain rate sensitivity increases the formability of ductile metals. Furthermore, additional results are presented in Appendix E to shed light on the specific influence of tension-compression asymmetry on the formability.