On the convexity bound of the generalized Drucker’s yield function CB2001 for orthotropic sheets

Abstract

Drucker’s sixth-order yield function for isotropic materials (J Appl Mech 16:349–357, 1949) was extended by Cazacu and Barlat (Math Mech Solids 6:613–630, 2001) for modeling orthotropic sheet metals via a generalization of the two stress invariants according to the theory of representation. The constant c in the original Drucker’s isotropic yield function was found by Dodd and Naruse (Int J Mech Sci 31:511–519, 1989) to be bound between –27/8 and 9/4 per the convexity requirement. In many subsequent modeling applications of orthotropic sheets, the same bound is also tacitly assumed for the constant c used in this class of the generalized Drucker’s yield function CB2001. No actual proof has, however, been presented in the literature that such a bound is indeed absolutely necessary if not sufficient to guarantee the convexity of the orthotropic CB2001 yield function. In this study, the validity of assuming such a convexity bound on the adjustable constant c is examined using a recently proposed numerical convexity certification algorithm. Representative orthotropic CB2001 yield functions whose material parameters have been calibrated and reported in the literature for some 12 FCC, BCC, and HCP sheet metals are evaluated. It is found that a single convexity bound between $-27/8$ and 9/4 on the constant c does not hold at all for any of those yield functions. This is in contrast to another class of generalized Drucker’s yield function based on linearly transformed stresses where the original convexity bound on the constant c does still hold.

Appendices

Appendix A The compact polynomial form of the generalized Drucker’s yield function CB2001

Recall that the generalized Drucker’s yield function $\varPhi _{6cb}(\sigma _x,\sigma _y,\tau _{xy})$ of Eq. (4) can be written into a complete homogeneous sixth-order polynomial in plane stress $\varPhi _{6}(\sigma _x,\sigma _y,\tau _{xy})$ of a compact form Eq. (3). Here, we list the 16 polynomial coefficients ($A_1,\ldots ,A_{16}$) in terms of the 11 material constants ($a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$, $b_3$, $b_4$, $b_5$, $b_{10},c$) appearing in $\varPhi _{6cb}(\sigma _x,\sigma _y,\tau _{xy})$ with $c^*=c/27$:

$$\begin{aligned} \begin{aligned} 8A_1&=a_1^3+3 a_1^2 a_3+3 a_1 a_3^2+a_3^3-c^*(8 b_1^2 +16 b_1 b_2 +8 b_2^2), \\ 4A_2/3&=-a_1^3-2 a_1^2 a_3-a_1 a_3^2 +c^*(8 b_1^2 +8 b_1 b_2), \\ 8A_3/3&=5 a_1^3 + a_1^2 a_2 + 6 a_1^2 a_3 + 2 a_1 a_2 a_3 + a_1 a_3^2 + a_2 a_3^2 -c^*( 24 b_1^2 - 16 b_1 b_4 - 16 b_2 b_4), \\ 2A_4/3&=-5 a_1^3 - 3 a_1^2 a_2 - 3 a_1^2 a_3 - 3 a_1 a_2 a_3 - c^*(4 b_1 b_3 + 4 b_2 b_3 + 40 b_1 b_4 + 4 b_2 b_4), \\ 8A_5/3&=5 a_1^3 + 6 a_1^2 a_2 + a_1 a_2^2 + a_1^2 a_3 + 2 a_1 a_2 a_3 + a_2^2 a_3 + c^*( 16 b_1 b_3 + 16 b_1 b_4 - 24 b_4^2), \\ 4A_6/3&=-a_1^3 - 2 a_1^2 a_2 - a_1 a_2^2 + c^*( 8 b_3 b_4 + 8 b4^2), \\ 8A_7&=a_1^3 + 3 a_1^2 a_2 + 3 a_1 a_2^2 + a_2^3 -c^*( 8 b_3^2 + 16 b_3 b_4 + 8 b_4^2), \\ 4A_8/9&=a_1^2 a_4 + 2 a_1 a_3 a_4 + a_3^2 a_4 -c^*( 16 b_1 b_{10}+ 16 b_2 b_{10} - 8 b_1 b_5 - 8 b_2 b_5), \\ A_9/9&=-a_1^2 a_4 - a_1 a_3 a_4 + c^*(12 b_1 b_{10} - 8 b_1 b_5 - 2 b_2 b_5), \\ 2A_{10}/9&=3 a_1^2 a_4 + a_1 a_2 a_4 + a_1 a_3 a_4 + a_2 a_3 a_4 + c^*(24 b_4b_{10}+12 b_1 b_5 -12 b_4 b_5), \\ A_{11}/9&=-a_1^2 a_4 - a_1 a_2 a_4 - 4 b_3b_{10} -c^*( 4 b_4b_{10} - 2 b_3 b_5 - 8 b_4 b_5), \\ 4A_{12}/9&=a_1^2 a_4 + 2 a_1 a_2 a_4 + a_2^2 a_4 - c^*(8 b_3 b_5 + 8 b_4 b_5), \\ 2A_{13}/27&=a_1 a_4^2 + a_3 a_4^2 - c^*(24 b_{10}^2 - 24 b_5 b_{10} + 6 b_5^2), \\ A_{14}/27&=- a_1 a_4^2 - c^*(12b_5 b_{10} - 6 b_5^2), \\ 2A_{15}/27&= a_1 a_4^2 + a_2 a_4^2 - c^* (18 b_5^2), \\ A_{16}/3&=27 a_4^3. \end{aligned} \end{aligned}$$

Similar results of the 16 polynomial coefficients for both generalized Drucker’s yield functions $\varPhi _{6db}$ and $\varPhi _{6dc}$ in terms of the material constants $(c_1,c_2,c_3,c_6)$ and $(d_1,d_2,d_3,d_6)$ in the linearly transformed stresses along with the constant c have already been given in the appendix of [34].

Appendix B Transversely isotropic Drucker’s yield function

The Drucker’s yield function has been specified for transversely isotropic sheets in [11] using the generalized $J^T_2$ and $J^T_3$ with transverse isotropy. These two stress invariants can be written in terms of two principal stresses $(\sigma _1, \sigma _2)$ in plane stress as

$$\begin{aligned} \begin{aligned} J^T_2(\sigma _1, \sigma _2)&={h_1\over 6}(\sigma _1-\sigma _2)^2 +{h_2\over 6}(\sigma ^2_1 +\sigma ^2_2), \\ J^T_3(\sigma _1, \sigma _2)&={3g_2-g_1\over 27}(\sigma ^3_1+\sigma ^3_2) -{g_1\over 9}(\sigma ^2_1\sigma _2 +\sigma _1\sigma ^2_2), \end{aligned} \end{aligned}$$

where $h_1, h_2, g_1$, and $g_2$ are the four material constants. The 10 materials constants that appeared in the orthotropic stress invariants $J_{2a}$ and $J_{3a}$ of Eq. (5) are given in terms of these four material constants as

$$\begin{aligned} \begin{aligned} a_1&=h_1, \quad a_2=a_3=h_2, \quad a_4={2h_1+h_2\over 3}, \quad b_1 =b_2=g_1, \\ b_3&=b_4=3g_2-2g_1, \quad b_5=4g_1-3g_2, \quad b_{10}={-g_1+3g_2\over 2}. \end{aligned} \end{aligned}$$

Appendix C The relation between two formulations of the generalized Drucker’s yield function

The generalized Drucker’s yield function $\varPhi _{6db}$ based on one linearly transformed stress can be rewritten into the form of the generalized Drucker’s yield function $\varPhi _{6cb}$ by setting $J_{2a}=J_{2b}$ and $J_{3a}=J_{3b}$. The ten material constants ($a_1,\ldots ,b_{10}$) of the latter can be obtained in terms of the four material constants ($c_1,c_2,c_3,c_6$) of the former as

$$\begin{aligned} \begin{aligned} 3a_1&= c_3(c_1 + c_2 + 2c_3) - c_1c_2, \quad 3a_2 = c_1(2c_1 + c_2 + c_3) - c_2c_3, \\ 3a_3&= c_2(c_1 + 2c_2 + c_3) - c_1c_3, \quad a_4 = c^2_6, \\ 3b_1&= c_2^2 c_3 + 2c_2 c_3^2 + c_2^2 c_1 - c_1 c_3^2, \quad 3b_2 = c_3^2 c_1 + 2c_3 c_2^2 - c_2^2 c_1 + c_2 c_3^2, \\ 3b_3&= c_3^2 c_2 + 2c_3 c_1^2 + c_3^2 c_1 - c_2 c_1^2, \quad 3b_4 = c_1^2 c_3 + 2c_1 c_3^2 + c_1^2 c_2 - c_2 c_3^2, \\ b_5&= c_1 c_6^2, \quad 2b_{10} = c_6^2 (c_1 + c_2). \end{aligned} \end{aligned}$$

However, the generalized Drucker’s yield function $\varPhi _{6dc}$ based on two linearly transformed stresses may not be rewritten into the form of the generalized Drucker’s yield function $\varPhi _{6cb}$ as $J^3_{2a} \ne J^3_{2b} + J^3_{2c}$ and $J^2_{3a} \ne J^2_{3b} + J^2_{3c}$ in general.