Gotoh’s 1977 Yield Stress Function with Kinematic Hardening for Modeling Strength Differential Yielding of Orthotropic Sheet Metals

Abstract

When a sheet metal is subjected to both tensile and compressive stresses in a forming process, there is a need to formulate a yield stress function that can accurately account for its strength differential effect in anisotropic yielding. The earliest classical approach is to combine Hill’s 1948 quadratic yield stress function with Prager’s kinematic hardening concept. Consistent with the requirement that a polynomial stress function admits only even-order shear stress components for an orthotropic sheet metal, the resulting quadratic yield stress function in plane stress has up to five material parameters for on-axis yielding but only one material parameter for off-axis yielding. The latter feature limits its modeling capabilities in general sheet metal forming simulations. In this paper, we present a user-friendly approach of formulating a non-quadratic yield stress function with tension-compression asymmetry by combining Gotoh’s 1977 quartic yield stress function with kinematic hardening. The new fourth-order yield stress function in plane stress has up to a total of eleven material constants: seven for on-axis yielding and four for off-axis yielding. The nonlinear parameter identification by least-square minimization with positivity and convexity constraints on the yield stress function is detailed for various sheet metals exhibiting strength differential effects. The results show that the new Gotoh-Prager yield stress function has adequate capabilities for modeling both on-axis and off-axis asymmetric yielding of many orthotropic sheet metals investigated over the years.

1 Introduction

Plastic yielding and flow behavior of a metal may be noticeably different in tension and compression. One well-known example of such behavior is the so-called Bauschinger effect in metals deformed plastically upon a reversal of loading directions. The strength differential effect has also often been observed to be rather strong in some rolled sheet metals [1,2,3,4]. An advanced anisotropic yield stress function with tension-compression asymmetry is thus sought for accurately modeling those sheet metals as they are increasingly used in various engineering applications [5,6,7,8].

For an orthotropic sheet metal under a state of plane stress $\pmb {\sigma }=(\sigma _x,\sigma _y,\tau _{xy})$, the well-known Hill’s 1948 quadratic yield stress function [9] has the following compact form [10]

$$\begin{aligned} \varPhi _2(\pmb {\sigma })=\varPhi _2(\sigma _x,\sigma _y,\tau _{xy})=A_1\sigma _x^{2} +A_2\sigma _x\sigma _y+A_3\sigma _y^{2} +A_4\tau _{xy}^{2}, \end{aligned}$$

(1)

where ($A_1,A_2,A_3$, $A_4$) are its four non-dimensional material constants that are often determined using three uniaxial and one equal biaxial tensile yield stresses. Hill’s 1948 quadratic yield stress function $\varPhi _2(\pmb {\sigma })$ is however symmetric for tension and compression, i.e., $\varPhi _2(-\sigma _x,-\sigma _y,\tau _{xy})=\varPhi _2(\sigma _x,\sigma _y,\tau _{xy})$. To account for the strength differential effect in anisotropic yielding, the simplest approach has been to incorporate the back stress concept in Prager’s kinematic hardening model [11] into Hill’s quadratic yield stress function, namely [2, 12]

$$\begin{aligned} \varPhi _{hp}(\pmb {\sigma })=A_1(\sigma _x-\alpha _x)^{2} +A_2(\sigma _x-\alpha _x)(\sigma _y-\alpha _y)+A_3(\sigma _y-\alpha _y)^{2} +A_4\tau _{xy}^{2}, \end{aligned}$$

(2)

where $(\alpha _x,\alpha _y)$ are two additional material constants.¹ The so-called Hill-Prager yield stress function $\varPhi _{hp}(\pmb {\sigma })$ can model reasonably well the tension-compression asymmetric yielding of a sheet metal in on-axis or coaxial loading ($\tau _{xy}=0$) with a total of five material constants $(A_1, A_2, A_3,\alpha _x,\alpha _y)$. However, its ability to model strength differential effect of the sheet metal under off-axis or non-coaxial loading is very limited as there is only one material constant $A_4$ uniquely related to off-axis yield stresses.

In this study, we presented a user-friendly asymmetric quartic yield stress function with enhanced capabilities for modeling the strength differential effect in anisotropic yielding of sheet metals. Following the similar development of quadratic yield stress functions above, we proposed to add the two axial back stress components per kinematic hardening as two additional material constants to Gotoh’s 1977 quartic yield stress function [13,14,15]. A brief description of the new Gotoh-Prager quartic yield stress function with seven on-axis material constants and four off-axis material constants will be given first. A novel nonlinear optimization algorithm will then be detailed to simultaneously identify its total 11 material constants and guarantee the convexity of any as-calibrated Gotoh-Prager quartic yield stress function. Finally, the capabilities of the new yield stress function will be assessed and demonstrated through numerical results of modeling various orthotropic sheet metals with asymmetric yielding.

2 Quartic Yield Stress Function with Kinematic Hardening and Its Calibration

The new asymmetric yield stress function in terms of Gotoh’s quartic yield stress function with two axial back stress components $(\beta _x,\beta _y)$ has the following form:

$$\begin{aligned} \begin{aligned} \varPhi _{gp}(\pmb {\sigma }) & = B_1(\sigma _x-\beta _x)^4 +B_2(\sigma _x-\beta _x)^3(\sigma _y-\beta _y) +B_3(\sigma _x-\beta _x)^2(\sigma _y-\beta _y)^2 \\ & +B_4(\sigma _x-\beta _x)(\sigma _y-\beta _y)^3 +B_5(\sigma _y-\beta _y)^4 +B_6(\sigma _x-\beta _x)^2\tau ^2_{xy} \\ {} & +B_7(\sigma _x-\beta _x)(\sigma _y-\beta _y)\tau ^2_{xy} +B_8(\sigma _y-\beta _y)^2\tau ^2_{xy} +B_9\tau ^4_{xy}, \end{aligned} \end{aligned}$$

(3)

where ($B_1, B_2,...,B_9,\beta _x,\beta _y$) are its 11 material constants. As the new yield stress function $\varPhi _{gp}(\pmb {\sigma })$ has seven on-axis material constants ($B_1, B_2,B_3,B_4,B_5,\beta _x,\beta _y$) and four off-axis material constants ($B_6, B_7,B_8,B_9$), its modeling capabilities will be superior in comparison with the Hill-Prager yield stress function $\varPhi _{hp}(\pmb {\sigma })$.

To apply the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$ to model a sheet metal with asymmetric yielding, one needs to identify its material constants from available measured yield stress values and to ensure that the calibrated yield stress function is positive and convex. The second task is rather easy: if Gotoh’s quartic yield stress function itself (i.e., both back stress components $\beta _x$ and $\beta _y$ are zero in Eq. 3) is known to be positive and convex, then the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$ is also positive and convex.² Several practical methods have already been established for certifying the convexity of a calibrated Gotoh’s quartic yield stress function using either sufficient conditions only or sufficient and necessary conditions [14, 18, 19].

The task of identifying the material constants of the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$ involves numerically solving 11 nonlinear equations per the yield condition

$$\begin{aligned} \varPhi _{gp}(\pmb {\sigma }^{(1)})-\sigma ^4_{gp}=0,\; \varPhi _{gp}(\pmb {\sigma }^{(2)})-\sigma ^4_{gp}=0,\; ...,\; \varPhi _{gp}(\pmb {\sigma }^{(11)})-\sigma ^4_{gp}=0, \end{aligned}$$

(4)

where $(\pmb {\sigma }^{(1)},\pmb {\sigma }^{(2)},...,\pmb {\sigma }^{(11)})$ are measured yield stresses at 11 unique and suitable plane stress states and $\sigma _{gp}(\xi )$ is the flow strength at the current hardening state (in terms of a scalar internal state variable $\xi$) of the sheet metal appeared in the yield condition $\varPhi _{gp}(\pmb {\sigma })=\sigma ^4_{gp}(\xi )$. Ideally, the first seven yield stresses shall be from on-axis loading (uniaxial and biaxial) with $\tau _{xy}=0$ so one can solve the first seven equations in Eq. (4) for the seven on-axis material constants ($B_1, B_2,B_3,B_4,B_5,\beta _x,\beta _y$). Afterward, one can find the four off-axis material constants ($B_6,B_7,B_8,B_9$) from the remaining four equations in Eq. (4) where four off-axis yield stresses are used.

When there are more than 11 measured yield stresses available for a given sheet metal, one can use a nonlinear optimization algorithm to solve for the 11 material constants by minimizing a sum of squared errors in the yield condition. Here we proposed to add the sum-of-squares (SOS) convexity constraints [18] directly to the least-square minimization used for parameter identification. The SOS-convexity constraints are the algebraic sufficient conditions to ensure a calibrated Gotoh’s quartic yield stress function is indeed convex and they are given as $\pmb {G}_{3\times 3}\ge 0$ and $\pmb {G}_{6\times 6} \ge 0$ [18,19,20], where

$$\begin{aligned} \pmb {G}_{3\times 3}= \left( \begin{array}{ccc} 12 B_1 &{} 3 B_2 &{} 2 B_3 \\ 3 B_2 &{} 2 B_3 &{} 3 B_4 \\ 2 B_3 &{} 3 B_4 &{} 12 B_5 \\ \end{array} \right) , \; \pmb {G}_{6\times 6}= \left( \begin{array}{cccccc} 12 B_1 &{} 3 B_2 &{} 0 &{} 2 B_3 &{} 0 &{} 2 B_6 \\ 3 B_2 &{} 2 B_3 &{} 0 &{} 3 B_4 &{} 0 &{} B_7 \\ 0 &{} 0 &{} 2 B_6 &{} 0 &{} B_7 &{} 0 \\ 2 B_3 &{} 3 B_4 &{} 0 &{} 12 B_5 &{} 0 &{} 2 B_8 \\ 0 &{} 0 &{} B_7 &{} 0 &{} 2 B_8 &{} 0 \\ 2 B_6 &{} B_7 &{} 0 &{} 2 B_8 &{} 0 &{} 12 B_9 \\ \end{array} \right) . \end{aligned}$$

(5)

Consequently, calibration of the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$ for a given sheet metal with some measured yield stresses is to solve the following constrained optimization problem with respect to its 11 material constants:

$$\begin{aligned} \begin{aligned} \text {min}\sum _{k=1}^{K}w_k[\varPhi _{gp}(\pmb {\sigma }^{(k)})-\sigma ^4_p]^2,\; & \text { subject to }\; \pmb {G}_{3\times 3}\ge 0 \;\text { or }\; \pmb {G}_{6\times 6}\ge 0, \end{aligned} \end{aligned}$$

(6)

where the SOS-convexity condition $\pmb {G}_{3\times 3}\ge 0$ is to be used first for on-axis yield stresses (with $K_{\textrm{on}}\ge 7$) in identifying the seven on-axis material constants ($B_1, B_2,B_3,B_4,B_5,\beta _x,\beta _y$) and the SOS-convexity condition $\pmb {G}_{6\times 6}\ge 0$ is then to be used for off-axis yield stresses (with $K_{\textrm{off}}\ge 4$) in identifying the remaining four material constants ($B_6,B_7,B_8,B_9$).³ The weight coefficient $w_k$ is set between 0 and 1 for each measured yield stress depending on its accuracy and precision (unless it is explicitly changed, it is by default set to 1 in this study).

When there are less than 11 measured yield stresses available for a given sheet metal, one can impose some additional conditions (equations) of reduced plastic anisotropy [15] to the above-constrained optimization problem. Alternatively, one can first calibrate the Hill-Prager yield stress function $\varPhi _{hp}(\pmb {\sigma })$ of Eq. (2) using the similarly constrained optimization with six or more yield stress inputs ($11>K\ge 6$) with respect to its six material constants ($A_1,A_2,A_3,A_4,\alpha _x,\alpha _y$)

$$\begin{aligned} \begin{aligned} \text {min}\sum _{k=1}^{K}w_k[\varPhi _{hp}(\pmb {\sigma }^{(k)})-\sigma ^2_{hp}]^2,\; & \text { subject to }\; A_1>0, \;4A_1A_3>A_2^2,\; A_4>0, \end{aligned} \end{aligned}$$

(7)

where the three inequalities here are necessary and sufficient conditions to ensure the strict convexity of Hill’s 1948 yield stress function $\varPhi _{2}(\pmb {\sigma })$ of Eq. (1) [10]. One can then use the calibrated Hill-Prager yield stress function $\varPhi _{hp}(\pmb {\sigma })$ to estimate the missing yield stresses needed for calibrating the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$. One additional benefit of having a calibrated Hill-Prager yield stress function $\varPhi _{hp}(\pmb {\sigma })$ on hand first is to provide the initial guesses for the nonlinear optimization problem of Eq. (6), namely (noting $\varPhi _{hp}^2(\pmb {\sigma }) \approx \varPhi _{gp}(\pmb {\sigma })$)

$$\begin{aligned} \begin{aligned} & B^{(0)}_1=A_1^2,\; B^{(0)}_2= 2 A_1 A_2,\; B^{(0)}_3= (A_2^2 + 2 A_1 A_3), \; B^{(0)}_4=2 A_2 A_3,\; B^{(0)}_5= A_3^2, \;\\ & B^{(0)}_6= 2 A_1 A_4, B^{(0)}_7= 2 A_2 A_4, B^{(0)}_8=2 A_3 A_4, B^{(0)}_9=A_4^2, \beta ^{(0)}_x = \alpha _x, \beta ^{(0)}_y = \alpha _y. \end{aligned} \end{aligned}$$

(8)

3 Modeling Results of Selected Sheet Metals

Here application examples of using the Gotoh-Prager yield stress function $\varPhi _{gp}(\pmb {\sigma })$ for describing tension-compression asymmetry in anisotropic yielding are presented for some nine HCP, FCC and BCC sheet metals, respectively. Experimental values of their yield stresses have already been reported in the literature [1,2,3,4,5,6,7] and their strength differential behaviors have motivated some very recent research on developing advanced asymmetric yield stress functions [5,6,7,8].

In the following, modeling results were grouped according to the availability of numbers and types of measured yield stress data for each of the nine representative sheet metals. The calibrated values of material constants of both Hill-Prager and Gotoh-Prager yield criteria are listed in Table 1. The unit for the flow strength $\sigma _{hp}$ or $\sigma _{gp}$ that appeared in Hill-Prager and Gotoh-Prager yield criteria is MPa. For the first four materials listed in Table 1, only on-axis material constants are given as there are no actual off-axis yield stresses reported for those materials in [1, 2].

Table 1 List of material constants of two yield stress functions

Fig. 1

a Biaxial yield surfaces of Mg-0.5%Th at a plastic strain of 1% as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; b Biaxial yield surfaces of Mg-4%Li at a plastic strain of 1% as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria. Solid and open circular symbols are reported [1] and calculated yield stresses, respectively, where $2\sigma _{cb}=\sigma _{c0}+2\sigma _{c45}+\sigma _{c90}$ was used. Results of Gotoh-Prager yield surfaces are omitted here as they are identical to Hill-Prager ones

HCP metals with only five on-axis yield stresses reported. The well-known 1968 study by Kelley and Hosford [1] reported in their tabulated data only five on-axis yield stresses for three HCP metals, namely ($\sigma _{t0}$, $\sigma _{t90}$, $\sigma _{c0}$, $\sigma _{c90}$, $\sigma _{tb}$) at three deformed states. These yield stresses happen to match the total number of on-axis material constants of a biaxial Hill-Prager yield stress function $\varPhi _{hp}(\sigma _x,\sigma _y,0)$. One might expect that the Hill-Prager yield stress function would be good enough for modeling their biaxial yielding behavior and $\varPhi _{gp}(\sigma _x,\sigma _y,0)$ = $\varPhi ^2_{hp}(\sigma _x,\sigma _y,0)$. This is indeed the case as shown in Fig. 1a, b for two magnesium alloys Mg-0.5%Th and Mg-4%Li: the five solid symbols in each figure correspond to the five reported yield stresses that are well predicted by the corresponding Hill-Prager yield criterion. The insertion of dashed-line yield surfaces given by the symmetric Hill’s 1948 yield function $\varPhi _{2}(\sigma _x,\sigma _y,0)$ (using the same $A_1,A_2,A_3$ as in $\varPhi _{hp}(\sigma _x,\sigma _y,0)$) in each figure is to highlight visually the asymmetric nature of the yield surfaces given by the Hill-Prager yield criterion.

Fig. 2

a, b Biaxial yield surfaces of Zicaloy-2(J) at a plastic strain of 0.002% and $350^\circ$ as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; c Biaxial yield surfaces of the same material at the same conditions as given by the calibrated Gotoh-Prager (solid lines) and Gotoh 1977 (dashed lines) yield criteria. Solid symbols are reported yield stresses [2] while open circles are calculated ones in pure shear from the best-fit Hill-Prager yield stress function in b

Fig. 3

a, b Biaxial yield surfaces of Zicaloy-2(K) at a plastic strain of 0.002% and $25^\circ$ as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; c Biaxial yield surfaces of the same material at the same conditions as given by the calibrated Gotoh-Prager (solid lines) and Gotoh 1977 (dashed lines) yield criteria. Solid symbols are reported yield stresses [2] while open circles are calculated ones in pure shear from the best-fit Hill-Prager yield stress function in b

Fig. 4

a, b Biaxial yield surfaces of CP Ti as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; c Biaxial yield surfaces of the same material at the same conditions as given by the calibrated Gotoh-Prager (solid lines) and Gotoh 1977 (dashed lines) yield criteria. Solid symbols are reported yield stresses [5] while open circles are calculated ones in pure shear from the best-fit Hill-Prager yield stress function in b

HCP metals with six on-axis yield stresses reported. On the other hand, a total of six on-axis yield stresses ($\sigma _{t0}$, $\sigma _{t90}$, $\sigma _{c0}$, $\sigma _{c90}$, $\sigma _{tb}$, $\sigma _{cb}$) are provided by Shih and Lee [2] in their Table 1 for three HCP metals. If only the first five yield stresses are used to determine the material constants of the Hill-Prager yield stress function, its predicted equal biaxial compression yield stress $\sigma _{cb}$ will be way off from the measured one as shown in Figs. 2a and 3a. The yield surface defined by the least-square best-fit Hill-Prager yield stress function via Eq. (7) is much better as shown in Figs. 2b and 3b. Finally, the six reported on-axis yield stresses plus two calculated on-axis pure shear yield stresses $(\sigma _{ss0},\sigma _{ss90})$ from the best-fit biaxial $\varPhi _{hp}(\sigma _x,\sigma _y,0)$ were used to find the best-fit biaxial $\varPhi _{gp}(\sigma _x,\sigma _y,0)$ according to Eq. (6). The weights in the equation were set to be 0.1 instead of the default value of 1 for yield conditions based on the calculated pure shear yield stresses. The yield surfaces given by such calibrated Gotoh-Prager yield criteria are shown in Figs. 2c and 3c, respectively.

Fig. 5

a, b Biaxial yield surfaces of AZ31 as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; c Biaxial yield surfaces of the same material at the same conditions as given by the calibrated Gotoh-Prager (solid lines) and Gotoh 1977 (dashed lines) yield criteria. Solid symbols are reported yield stresses [6] while open circles are calculated ones in pure shear from the best-fit Hill-Prager yield stress function in b

Fig. 6

a Dependence of uniaxial tensile and compressive yield stresses $\sigma _{t\theta }$ and $\sigma _{c\theta }$ of CP Ti on the loading angle $\theta$ as given by the best-fit Hill-Prager (solid lines) yield criterion; b The same dependence as given by the best-fit Gotoh-Prager (solid lines) yield criterion. Solid symbols are reported yield stresses [5]

Fig. 7

a Dependence of uniaxial tensile and compressive yield stresses $\sigma _{t\theta }$ and $\sigma _{c\theta }$ of AZ31 on the loading angle $\theta$ as given by the best-fit Hill-Prager (solid lines) yield criterion; b The same dependence as given by the best-fit Gotoh-Prager (solid lines) yield criterion. Solid symbols are reported yield stresses [6]

HCP metals with six on-axis and two off-axis yield stresses reported. A total of eight such yield stresses have been provided by Raemy et al. [5] for CP titanium and by Hu and Yoon [6] for magnesium alloy AZ31. The best-fit on-axis material constants of both yield criteria were identified in a similar way as described above. The resulting biaxial yield surfaces are given in Fig. 4 for CP Ti and Fig. 5 for AZ31, respectively. With both $\sigma _{t45}$ and $\sigma _{c45}$ yield stresses available, one obtained the best-fit $A_4$ of $\varPhi _{hp}(\pmb {\sigma })$ per Eq. (7). For the case of $\varPhi _{gp}(\pmb {\sigma })$, one first added to the measured two off-axis yield stresses $(\sigma _{t45},\sigma _{c45})$ four additional estimated off-axis yield stresses $2\sigma _{t225}=\sigma _{t0}+\sigma _{t45}$, $2\sigma _{t675}=\sigma _{t45}+\sigma _{t90}$, $2\sigma _{c225}=\sigma _{c0}+\sigma _{c45}$ and $2\sigma _{c675}=\sigma _{c45}+\sigma _{c90}$. The best-fit off-axis material constants of the Gotoh-Prager yield stress function were identified using these six off-axis yield stresses per Eq. (6). The weights for those four estimated yield stresses were set to 0.1 instead of the default value of 1. The resulting dependence of uniaxial yield stresses on the loading angle $\theta$ is given in Fig. 6 for CP Ti and Fig. 7 for AZ31, respectively.

FCC aluminum alloys with five on-axis and ten off-axis yield stresses reported. Such a set of yield stress data has been provided in [7] for the two aluminum alloy sheets AA2008-T4 [3] and AA2090-T3 [4]. As there are only five on-axis yield stresses available for both aluminum sheets, one obtained biaxial Gotoh-Prager yield function from biaxial Hill-Prager yield stress function as before to (i.e., getting $B_1,...,\beta _y$ per Eq. (8) directly). Then the best-fit four off-axis material constants $(B_6,B_7,B_8,B_9)$ were identified using those 10 reported off-axis yield stresses per Eq. (6). The resulting biaxial yield surfaces are given in Fig. 8a for AA2008-T4 and Fig. 8b for AA2090-T3, respectively. The resulting dependence of uniaxial yield stresses on the loading angle $\theta$ is given in Fig. 9 for AA2008-T4 and Fig. 10 for AA2090-T3, respectively.

Fig. 8

a Biaxial yield surfaces of AA2008-T4 as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; b Biaxial yield surfaces of AA2090-T3 as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria. Solid and open circular symbols are reported [7] and calculated yield stresses, respectively, where $2\sigma _{cb}=\sigma _{c0}+2\sigma _{c45}+\sigma _{c90}$ was used

Fig. 9

a Dependence of uniaxial tensile and compressive yield stresses $\sigma _{t\theta }$ and $\sigma _{c\theta }$ of AA2008-T4 on the loading angle $\theta$ as given by the best-fit Hill-Prager (solid lines) yield criterion; b The same dependence as given by the best-fit Gotoh-Prager (solid lines) yield criterion. Solid symbols are reported yield stresses [7]

Fig. 10

a Dependence of uniaxial tensile and compressive yield stresses $\sigma _{t\theta }$ and $\sigma _{c\theta }$ of AA2090-T3 on the loading angle $\theta$ as given by the best-fit Hill-Prager (solid lines) yield criterion; b The same dependence as given by the best-fit Gotoh-Prager (solid lines) yield criterion. Solid symbols are reported yield stresses [7]

BCC Steel with seven on-axis inputs and three off-axis yield stresses reported. Such a set of yield stress data has been provided in [7] for a DP780 steel sheet. The seven on-axis yield stresses $(\sigma _{t0}, \sigma _{t90}, \sigma _{c0}, \sigma _{c90}, \sigma _{tb}, \sigma _{s0}, \sigma _{s90})$ provided happen to be sufficient in calibrating the on-axis material constants of both Hill-Prager and Gotoh-Prager yield functions. For the case of $\varPhi _{gp}(\pmb {\sigma })$, one first added to the three measured off-axis yield stresses $(\sigma _{t45},\sigma _{c45}, \sigma _{s45})$ four additional estimated off-axis yield stresses $2\sigma _{t225}=\sigma _{t0}+\sigma _{t45}$, $2\sigma _{t675}=\sigma _{t45}+\sigma _{t90}$, $2\sigma _{c225}=\sigma _{c0}+\sigma _{c45}$ and $2\sigma _{c675}=\sigma _{c45}+\sigma _{c90}$. The best-fit off-axis material constants of Gotoh-Prager yield stress function were identified using these seven off-axis yield stresses per Eq. (6). The weights for those four estimated yield stresses were set to 0.1 instead of the default value of 1. The resulting biaxial yield surfaces and dependence of uniaxial yield stresses on the loading angle $\theta$ are shown in Figs. 11 and 12, respectively.

Fig. 11

a, b Biaxial yield surfaces of DP780 steel as given by the calibrated Hill-Prager (solid lines) and Hill 1948 (dashed lines) yield criteria; c Biaxial yield surfaces of the same material at the same conditions as given by the calibrated Gotoh-Prager (solid lines) and Gotoh 1977 (dashed lines) yield criteria. Solid symbols are reported yield stresses at a plastic strain of 1.5% [7] while the open circle is the calculated equal biaxial compression yield stress $4\sigma _{cb}=\sigma _{c0}+2\sigma _{c45}+\sigma _{c90}$

Fig. 12

a Dependence of uniaxial tensile and compressive yield stresses $\sigma _{t\theta }$ and $\sigma _{c\theta }$ of DP780 steel on the loading angle $\theta$ as given by the best-fit Hill-Prager (solid lines) yield criterion; b The same dependence as given by the best-fit Gotoh-Prager (solid lines) yield criterion. Solid symbols are reported yield stresses at a plastic strain of 1.5% [7]

4 Discussion and Conclusions

In practice, slightly stronger SOS-convexity conditions $\pmb {G}_{3\times 3}> 0$ and $\pmb {G}_{6\times 6}> 0$ (both matrices are positive define) instead of $\pmb {G}_{3\times 3}\ge 0$ and $\pmb {G}_{6\times 6}\ge 0$ (both matrices are positive semi-define) were used here. One reason to do so is due to the fact that leading principal minors of those two matrices are somewhat simpler to obtain than their principal minors [18, 20]. The strong SOS-convexity conditions were actually implemented in numerical calculations given here, i.e., all eigenvalues of both $\pmb {G}_{3\times 3}$ and $\pmb {G}_{6\times 6}$ were required to be equal or larger than a small positive value (say, $10^{-4}$). This proves to be effective in avoiding any possible violation of convexity conditions when only five significant digits or up to four decimal points were kept for the calibrated material constants as given in Table 1.

As shown through numerical results for modeling tension-compression asymmetry of all representative sheet metals considered in the previous section, the SOS-convexity constrained least-square minimization per Eq. (6) was proved to be effective for parameter identification of the non-quadratic Gotoh-Prager yield function. This is similar to the use of any yield function based on a convex isotropic function of linearly transformed stresses (such as Yld2000-2d): any calibrated Gotoh-Prager yield function with a total of 11 material constants is also guaranteed to be convex automatically without a need for further certification. For sheet metals with purely isotropic-kinematic hardening behaviors with evolving $(\beta _x,\beta _y)$ but fixed $(B_1,...,B_9)$, the convexity of Gotoh-Prager yield function is preserved. On the other hand, for sheet metals with anisotropic-kinematic hardening behaviors with evolving $(\beta _x,\beta _y)$ and ($B_1,...,B_9$), the convexity of Gotoh-Prager yield function has to be verified continuously during the plastic deformation. This can be easily done in either the parameter identification stage or the post-calibration certification stage by using the algebraic SOS-convexity conditions $\pmb {G}_{3\times 3}> 0$ and $\pmb {G}_{6\times 6}> 0$.

In conclusion, the first effort in developing and applying the Gotoh-Prager yield function as presented here has shown that such a simpler asymmetric yield function can be an attractive alternative for modeling strength differential effects in sheet metals.