A new temperature-dependent anisotropic constitutive model for predicting deformation and spring-back in warm deep drawing of automotive AA5086-H111 aluminium alloy sheet

Abstract

A new temperature-dependent anisotropic hardening and yield function based on non-associated flow rule (NAFR) which accurately describes the anisotropic properties of aluminium alloy sheet metal is presented in this paper. This new NAFR coupling yield function has been successfully implemented in commercial FEM software Abaqus via user material subroutine UMAT to model warm forming of AA5086-H111 aluminium alloy and good agreement was found in the predicted earing profiles and spring-back compared with experiments. The new NAFR coupling yield function has been shown be superior to the existing Hill48 (associated flow rule) yield function making it suitable for FEA of warm deep drawing and design of tooling for various automotive panels and components.

Appendices

Appendix A

σ₁,σ₂ and σ₃ are the three principal values of the Cauchy stress tensor:

$$ \left[\begin{array}{llll} \sigma_{11} &\sigma_{12} &\sigma_{13}\\ \sigma_{12} &\sigma_{22} &\sigma_{23}\\ \sigma_{13} &\sigma_{23} &\sigma_{33} \end{array}\right] $$

(34)

These principal values are the roots of the characteristic equation:

$$ P(\sigma_{k})=-{\sigma_{k}^{3}}+ 3H_{1}{\sigma_{k}^{2}}+ 3H_{2}\sigma_{k}+ 2H_{3}= 0 $$

(35)

The associated 1st, 2nd and 3rd invariants of σ_k:

$$\begin{array}{@{}rcl@{}} H_{1}\!&=&\!\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} \end{array} $$

(36a)

$$\begin{array}{@{}rcl@{}} H_{2}\!&=&\!\frac{\sigma_{23}^{2}+\sigma_{13}^{2}+\sigma_{12}^{2}-\sigma_{22}\sigma_{33}-\sigma_{11}\sigma_{33}-\sigma_{11}\sigma_{22}}{3} \end{array} $$

(36b)

$$\begin{array}{@{}rcl@{}} H_{3}\!&=&\!\frac{2\sigma_{12}\sigma_{13}\sigma_{23}+\sigma_{11}\sigma_{22}\sigma_{33}-\sigma_{11}\sigma_{23}^{2}\,-\,\sigma_{22}\sigma_{13}^{2}\,-\,\sigma_{33}\sigma_{12}^{2}}{2}\\ \end{array} $$

(36c)

Change the variable using:

$$ {\sigma}_{k}= \overline{\sigma}_{k} $$

(37)

The characteristic equation becomes:

$$ P(\overline{\sigma}_{k})=-\overline{\sigma}_{k}^{3}+ 3p\overline{\sigma}_{k}+ 2q = 0 $$

(38)

The associated invariants of $\overline {\sigma }_{k}$:

$$\begin{array}{@{}rcl@{}} p&=&({H_{1}^{2}}+H_{2}) \end{array} $$

(39a)

$$\begin{array}{@{}rcl@{}} q&=&\left( 2{H_{1}^{3}}+ 3H_{1}H_{2}+ 2H^{3}\right)/2 \end{array} $$

(39b)

$$\begin{array}{@{}rcl@{}} &&\theta=arccos\left[\frac{q}{p^{3/2}}\right] \end{array} $$

(39c)

Cardan’s solutions to Eq. 40a:

$$\begin{array}{@{}rcl@{}} &&\overline{\sigma}_{1}=z^{1/3}+\overline{z}^{1/3} \end{array} $$

(40a)

$$\begin{array}{@{}rcl@{}} &&\overline{\sigma}_{2}=\omega z^{1/3}+{\overline{\omega}} {\overline{z}}^{1/3} \end{array} $$

(40b)

$$\begin{array}{@{}rcl@{}} &&\overline{\sigma}_{3}=\overline{\omega} z^{1/3}+{\omega} {\overline{z}}^{1/3} \end{array} $$

(40c)

where z, w are complex number and $\overline {z}$, $\overline {w}$ are their conjugate quantities, they are given as:

$$\begin{array}{@{}rcl@{}} &&z=q+i\sqrt{p^{3}-q^{2}} \end{array} $$

(41a)

$$\begin{array}{@{}rcl@{}} &&\overline{z}=q-i\sqrt{p^{3}-q^{2}} \end{array} $$

(41b)

$$\begin{array}{@{}rcl@{}} &&{\omega=\text{cos}\left( -\frac{2\pi}{3}\right)+i\cdot\sin\left( -\frac{2\pi}{3}\right)} \end{array} $$

(41c)

$$\begin{array}{@{}rcl@{}} &&{\overline{\omega}=\text{cos}\left( -\frac{2\pi}{3}\right)-i\cdot \sin\left( -\frac{2\pi}{3}\right)} \end{array} $$

(41d)

The principal values of Cauchy stress tensor are computed as:

$$ \sigma_{1}=\overline{\sigma}_{1}+H_{1} \qquad \sigma_{2}=\overline{\sigma}_{2}+H_{1} \qquad \sigma_{3}=\overline{\sigma}_{3}+H_{1} $$

(42)

Appendix B

First-order deviator of the new coupling yield function required in the stress integration algorithm:

$$\begin{array}{@{}rcl@{}} \frac{\partial{\sigma}_{e}}{\partial\boldsymbol{\sigma}}&=& \frac{1}{(n + 1)(\overline{\sigma}^{n + 1})} \left( \frac{\partial F_{quad}}{\partial {\boldsymbol{\sigma}}} \cdot F_{non-quad}\right.\\ &&+ \left. \frac{\partial F_{non-quad}}{\partial {\boldsymbol{\sigma}}} \cdot F_{quad}\right) \end{array} $$

(43)

where, the first-order deviator of quadric part is given by:

$$ \frac{\partial F_{quad}}{\partial {\boldsymbol{\sigma}}} = \left[\begin{array}{ll} \alpha_{3}(\sigma_{11}-\sigma_{22})-\alpha_{2}(\sigma_{33}-\sigma_{11})\\ \alpha_{1}(\sigma_{22}-\sigma_{33})-\alpha_{3}(\sigma_{11}-\sigma_{22})\\ \alpha_{2}(\sigma_{33}-\sigma_{11})-\alpha_{1}(\sigma_{22}-\sigma_{33})\\ 2\alpha_{4}(\sigma_{12})\\ 2\alpha_{5}(\sigma_{13})\\ 2\alpha_{6}(\sigma_{23}) \end{array}\right] $$

The first-order deviator of non-quadric part is given by:

$$ \frac{\partial F_{non-quad}}{\partial {\boldsymbol{\sigma}}} = \left[\begin{array}{lll} {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{11}} \\ {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{22}} \\ {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{33}} \\ {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{12}} \\ {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{13}} \\ {\sum}_{k_{1}= 1}^{3} {\sum}_{k_{2}= 1}^{3} \frac{\partial{F_{non-quad}}}{\partial \sigma_{k1}} \frac{\partial {\sigma_{k1}}}{\partial{H_{k2}}} \frac{\partial H_{k2}}{\partial \sigma_{23}} \end{array}\right] $$

where, the terms $\frac {\partial {F_{non-quad}}}{\partial \sigma _{k}}$ are given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial{F_{non-quad}}}{\partial \sigma_{1}} &=& \frac{1}{2}n (|\sigma_{1} - \sigma_{2} ||\sigma_{1} - \sigma_{2} |^{n-2}\\ &&- |\sigma_{3} - \sigma_{1} ||\sigma_{3} - \sigma_{1} |^{n-2}) \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} \frac{\partial{F_{non-quad}}}{\partial \sigma_{2}} &=& \frac{1}{2}n (|\sigma_{2} - \sigma_{3} ||\sigma_{2} - \sigma_{3} |^{n-2}\\ &&- |\sigma_{1} - \sigma_{2} ||\sigma_{1} - \sigma_{2} |^{n-2}) \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} \frac{\partial{F_{non-quad}}}{\partial \sigma_{3}} &= &\frac{1}{2}n (|\sigma_{3} - \sigma_{1} ||\sigma_{3} - \sigma_{1} |^{n-2}\\ &&- |\sigma_{2} - \sigma_{3} ||\sigma_{2} - \sigma_{3} |^{n-2}) \end{array} $$

(46)

The terms $\frac {\partial {\sigma _{k1}}}{\partial {H_{k2}}}$ are given by:

$$\begin{array}{@{}rcl@{}} &&\frac{\partial {\sigma_{k1}}}{\partial{H_{1}}} = \frac{\sigma_{k1}^{2}}{\sigma_{k1}^{2}-2H_{1}\sigma_{k1}-H_{2}} \end{array} $$

(49a)

$$\begin{array}{@{}rcl@{}} &&\frac{\partial {\sigma_{k1}}}{\partial{H_{2}}} = \frac{\sigma_{k1}}{\sigma_{k1}^{2}-2H_{1}\sigma_{k1}-H_{2}} \end{array} $$

(49b)

$$\begin{array}{@{}rcl@{}} &&\frac{\partial {\sigma_{k1}}}{\partial{H_{3}}} = \frac{2}{3(\sigma_{k1}^{2}-2H_{1}\sigma_{k1}-H_{2})} \end{array} $$

(49c)

The terms $\frac {H_{k2}}{\sigma _{ij}}$ are given by:

$$\begin{array}{@{}rcl@{}} &&\frac{H_{1}}{\sigma_{ij}} = \frac{1}{3} (i=j) \qquad \frac{H_{1}}{\sigma_{ij}} = 0(i \neq j) \end{array} $$

(50a)

$$\begin{array}{@{}rcl@{}} &&\frac{H_{2}}{\sigma_{11}} = -\frac{\sigma_{22}+\sigma_{33}}{3} \qquad \frac{H_{2}}{\sigma_{22}} = -\frac{\sigma_{11}+\sigma_{33}}{3} \end{array} $$

(50b)

$$\begin{array}{@{}rcl@{}} &&\frac{H_{2}}{\sigma_{33}} = -\frac{\sigma_{11}+\sigma_{22}}{3} \qquad \frac{H_{2}}{\sigma_{ij}} = \frac{2\sigma_{ij}}{3}(i \neq j) \end{array} $$

(50c)

$$\begin{array}{@{}rcl@{}} &&\frac{H_{3}}{\sigma_{11}} = \frac{\sigma_{22}\sigma_{33}-\sigma{23}^{2}}{2} \qquad \frac{H_{3}}{\sigma_{22}} = \frac{\sigma_{11}\sigma_{33}-\sigma{13}^{2}}{2} \end{array} $$

(50d)

$$\begin{array}{@{}rcl@{}} &&\frac{H_{3}}{\sigma_{33}} = \frac{\sigma_{11}\sigma_{22}-\sigma{12}^{2}}{2} \qquad \frac{H_{3}}{\sigma_{12}} \,=\, \sigma_{23}\sigma_{13} - \sigma_{33}\sigma_{12} \end{array} $$

(50e)

$$\begin{array}{@{}rcl@{}} &&\frac{H_{3}}{\sigma_{13}} = \sigma_{12}\sigma_{23} - \sigma_{22}\sigma_{13} \qquad \frac{H_{3}}{\sigma_{23}} \,=\, \sigma_{12}\sigma_{13} - \sigma_{11}\sigma_{23} \end{array} $$

(50f)