Constitutive Equations Based on Non-associated Flow Rule for the Analysis of Forming of Anisotropic Sheet Metals

Abstract

In this study, an anisotropic constitutive model based on the non-associated flow rule was developed for anisotropic sheet metals. This model was defined in the quadratic form of the Hill’s anisotropic function under a general three-dimensional stress condition. The anisotropic parameters for the yield function were identified using the directional planar yield stresses, bulge yield stress and shear yield stress, while those for the plastic potential function were identified using the directional r-values. A full expression related to the non-associated flow rule was applied and the model was implemented into the finite element code ABAQUS. A static-implicit analysis and the solid element were applied. Capability of the developed model for predicting the anisotropic behavior of sheet metal was investigated by considering two different sheet metal forming processes: cylindrical cup drawing of AA2090-T3, A6061P-T6 and SPCE; and hole expansion forming test of A6016-O. Cup heights and through-thickness strain distributions obtained from the simulations were compared with the experimental data. Results demonstrate that the developed material model considering 3D condition can improve accuracy of predicting the anisotropic behaviors. Furthermore, the simple formulations are efficient and user-friendly for computational analyses and solving the common industrial sheet metal forming problems.

Appendix

For this multistage method, each plastic correction step is divided into k substeps. The non-linear equations for the sub-steps are

$$\begin{aligned} & \phi ^{{(0)}} (\bar{\varepsilon }^{{p(0)}} ) = \bar{\sigma }({\mathbf{\sigma }}^{{tri}} - d\lambda ^{{(0)}} \cdot {\mathbf{D}}^{{{e}}} {\mathbf{n}}^{{(0)}} ) - \sigma _{Y} (\bar{\varepsilon }^{p} + d\lambda ^{{(0)}} \cdot P^{{(0)}} ) - \Psi ^{{(0)}} = 0, \\ & \phi ^{{(1)}} (\bar{\varepsilon }^{{p(1)}} ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \bar{\sigma }({\mathbf{\sigma }}^{{tri}} - d\lambda ^{{(1)}} \cdot {\mathbf{D}}^{{{e}}} {\mathbf{n}}^{{(1)}} ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \sigma _{Y} (\bar{\varepsilon }^{p} + d\lambda ^{{(1)}} \cdot P^{{(1)}} ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \Psi ^{{(1)}} = 0, \\ & \phi ^{{(2)}} (\bar{\varepsilon }^{{p(2)}} ) = \bar{\sigma }({\mathbf{\sigma }}^{{tri}} - d\lambda ^{{(2)}} \cdot {\mathbf{D}}^{{{e}}} {\mathbf{n}}^{{(2)}} ) - \sigma _{Y} (\bar{\varepsilon }^{p} + d\lambda ^{{(2)}} \cdot P^{{(2)}} ) - \Psi ^{{(2)}} = 0, \\ & \vdots \\ & \phi ^{{(k)}} (\bar{\varepsilon }^{{p(k)}} ) = \bar{\sigma }({\mathbf{\sigma }}^{{tri}} - d\lambda ^{{(k)}} \cdot {\mathbf{D}}^{{{e}}} {\mathbf{n}}^{{(k)}} ) - \sigma _{Y} (\bar{\varepsilon }^{p} + d\lambda ^{{(k)}} \cdot P^{{(k)}} ) - \Psi ^{{(k)}} = 0 \\ \end{aligned}$$

with

$$\varPsi^{(0)} > \varPsi^{(1)} > \cdots > \varPsi^{(j)} > \cdots > \varPsi^{(k)} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} d\varPsi = \varPsi^{(j)} - \varPsi^{(j - 1)} < \sigma_{Y} ,$$

where ${{\Psi }}^{j}$ is a prescribed sub-potential residual value at the jth sub-step and it is adjustable in accordance with simulation case.

Each sub-step includes complete Newton–Raphson iterations. At the beginning of the jth sub-step, the initial direction ${\boldsymbol{n}}^{(j)}$ is estimated from the direction ${\boldsymbol{n}}^{(j - 1)}$ at the previous sub-step. The iteration is completed when the stress is driven onto the auxiliary yield surface. Finally, this procedure is finished when ${{\Psi }} = {{\Psi }}^{(k)} = 0$ and the plastic strain increment remains normal to the yield surface at the final stress $\varvec{\sigma}_{{{\text{n}} + 1}}$. The iteration procedure for the jth sub-step is given as:

$$\phi ^{{(j)}} (\bar{\varepsilon }^{{p(j)}} ) = \bar{\sigma }_{f} ({\mathbf{\sigma }}^{{(j)}} ) - \sigma _{Y} (\bar{\varepsilon }^{p} + d\bar{\varepsilon }^{{p(j)}} ) = \Psi ^{{(j)}} ,$$

(28)

where

$${{\boldsymbol\upsigma}}^{(j)} { = }{{\boldsymbol\upsigma}}^{tri} - d\lambda^{(j)} \cdot {\boldsymbol{D}}_{e} {\boldsymbol{n}}^{(j)} ,$$

$$\sigma _{Y} ^{{(j)}} = \sigma _{Y} (\bar{\varepsilon }^{p} + d\bar{\varepsilon }^{{p(j)}} )\quad {\text{or}}\quad \sigma _{Y} ^{{(j)}} = \sigma _{Y} ^{n} (\bar{\varepsilon }^{p} ) + d\bar{\varepsilon }^{{p(j)}} \cdot h,$$

and

$$d\bar{\varepsilon }^{{p(j)}} = d\lambda ^{{(j)}} \cdot P^{{(j)}} .$$

The following residuals are defined as:

$$\begin{aligned} & R_{1} (\bar{\varepsilon }^{{p(j)}} ) = \phi ^{{(j)}} (\bar{\varepsilon }^{{p(j)}} ,\sigma _{Y} ^{{(j)}} ) = 0, \\ & R_{2} (\bar{\varepsilon }^{{p(j)}} ) = {\mathbf{D}}^{{e - 1}} ({\mathbf{\sigma }}^{{(j)}} - {\mathbf{\sigma }}^{{tri}} ) + d\lambda \cdot {\mathbf{n}} = 0, \\ & R_{3} (\bar{\varepsilon }^{{p(j)}} ) = h^{{ - 1}} (\sigma _{Y} ^{{(j)}} - \sigma _{Y} ^{n} ) - d\lambda ^{{(j)}} \cdot P^{{(j)}} = 0. \\ \end{aligned}$$

To perform Newton–Raphson iterations, the above three functions are linearized with respect to the plastic strain increment $d\bar{\varepsilon }^{p}$ at each iteration, giving

$$\begin{array}{*{20}l} {{\text{I}}.} \hfill & {R_{1} (d\lambda_{(k)}^{(i)} ) + {\boldsymbol{m}}_{(k)}^{(i)} {\boldsymbol{d\sigma }}_{(k)}^{(i)} - \sigma_{{Y_{(k)}^{(i)} }} = 0,} \hfill \\ {{\text{II}}.} \hfill & \begin{aligned} {\boldsymbol{d\sigma }}_{(k)}^{(i)} = - {\boldsymbol{E}}_{(k)}^{(i) - 1} (R_{2} (d\lambda_{(k)}^{(i)} ) + {\boldsymbol{m}}_{(k)}^{(i)} d\Delta \lambda_{(k)}^{(i)} ), \hfill \\ {\boldsymbol{E}}_{(k)}^{(i) - 1} = \left[ {{\boldsymbol{D}}^{e - 1} + d\lambda_{(k)}^{(i)} \frac{{\partial {\boldsymbol{n}}_{(k)}^{(i)} }}{{\partial {{\boldsymbol\upsigma}}_{(k)}^{(i)} }}} \right]^{ - 1} , \hfill \\ \end{aligned} \hfill \\ {{\text{III}}.} \hfill & {d\sigma_{{Y_{(k)}^{(i)} }} = \left( - R_{3} (d\lambda_{(k)}^{(i)} ) + d\Delta \lambda_{(k)}^{(i)} \cdot P_{(k)}^{(i)} \right) \cdot h^{(i)} .} \hfill \\ \end{array}$$

(29)

By substituting Eqs. (II) and (III) into Eq. (I) and rearranging, the following expression is obtained:

$$d\Delta \lambda_{(k)}^{(i)} { = }\frac{{R_{1} \left( {d \lambda_{(k)}^{(i)} } \right) - {\boldsymbol{m}}_{(k)}^{(i)} {\kern 1pt} {\kern 1pt} {\boldsymbol{\rm E}}_{(k)}^{(i) - 1} R_{2} \left( {d \lambda_{(k)}^{(i)} } \right) + R_{3} \left( {d \lambda_{(k)}^{(i)} } \right)h^{(i)} }}{{{\boldsymbol{m}}_{(k)}^{(i)} {\kern 1pt} {\kern 1pt} :{\boldsymbol{\rm E}}_{(k)}^{(i) - 1} :{\boldsymbol{n}}_{(k)}^{(i)} + H^{(i)} \cdot P_{(k)}^{(i)} }},$$

(30)

$d{{\Delta }}\lambda_{(k)}$ represents the incremental change in the factor $d\lambda_{(k)}$ at the kth iteration. The iterative change is fully defined by Eqs. (29) and (30). Thus, the variables can be updated as follows:

$$\begin{aligned} d\lambda _{{(k)}}^{{(i + 1)}} = d\lambda _{{(k)}}^{{(i)}} + d\Delta \lambda _{{(k)}}^{{(i)}} , \\ {\boldsymbol{\sigma }}_{{(k)}}^{{(i + 1)}} = {\boldsymbol{\sigma }}_{{(k)}}^{{(i)}} + {\boldsymbol{d\sigma }}_{{(k)}}^{{(i)}} , \\ \sigma _{{Y(k)}}^{{(i + 1)}} = \sigma _{{Y(k)}}^{{(i)}} + d\sigma _{{Y(k)}}^{{(i)}} , \\ d\bar{\varepsilon }^{p} = d\lambda _{{(k)}}^{{(i)}} \cdot P_{{(k)}}^{{(i)}} . \\ \end{aligned}$$

(31)