Effect of slide motion on springback in 2-D draw bending for AHSS

Abstract

The springback behavior of two advanced high strength steel (AHSS) grades, DP 780 and DP 980, after different forming conditions, was investigated. 2-D draw bending experiments were performed using a direct-drive digital servo-press in three operation modes, conventional (V-mode), relaxation mode (Stepwise and U-mode) and the attach-detach (A-D mode). Numerical simulations were conducted to in an attempt to reproduce the results and to perform parametric studies. The material behavior was captured using the homogeneous anisotropic hardening (HAH) distortional plasticity approach together with the chord elastic modulus model. In addition, the stress relaxation effects were implemented in the code by using a creep law in order to study the influence of a stepwise slide motion mode as well as holding at the bottom dead center. Both experimental and finite element (FE) simulation results demonstrate that detachment of tools from the work-piece was effective to reduce the springback while the effect of stress relaxation was insignificant. The numerical analysis was validated and successfully explained the importance of a forming path change on the final stress state. Based on the result of this study, a new method to reduce springback was introduced.

Appendix. The homogenous anisotropic hardening (HAH) model

The fluctuating component of the yield function in Eq. (2), Φ _h , can be expanded as:

$$ {\varPhi}_h={\left[{\mathrm{f}}_1{\hat{\boldsymbol{h}}}^s:\boldsymbol{s}-{\mathrm{f}}_1\left|{\hat{\boldsymbol{h}}}^s:\boldsymbol{s}\right|\left|{}^q+\right|{\mathrm{f}}_2{\hat{\boldsymbol{h}}}^s:\boldsymbol{s}-{\mathrm{f}}_2\left|{\hat{\boldsymbol{h}}}^s:\boldsymbol{s}\right|\Big|{}^q\right]}^{\frac{1}{q}} $$

(A-1)

The state variables, f₁ and f₂, depend on another sets of state variables, g _k :

${f}_k={\left[\frac{1}{g_k^q}-1\right]}^{\frac{1}{q}}$ for k = 1 and 2. (A - 2)

Based on the sign of double dot product, ${\hat{\boldsymbol{h}}}^s:\boldsymbol{s}$, the model gives the following relationships:

If ${\hat{\boldsymbol{h}}}^s:\boldsymbol{s}\ge 0$:

$$ \frac{d{g}_1}{d\epsilon }={k}_2\left({k}_3\frac{\sigma_0}{\sigma }-{g}_1\right) $$

(A-3)

$$ \frac{d{g}_2}{d\epsilon }=\frac{k_1\left({g}_3-{g}_2\right)}{g_2} $$

(A-4)

$$ \frac{d{g}_4}{d\epsilon }={k}_5\left({k}_4-{g}_4\right) $$

(A-5)

$$ \frac{d{\hat{\boldsymbol{h}}}^s}{d\epsilon }=k\left[\hat{\boldsymbol{s}}-\frac{8}{3}{\hat{\boldsymbol{h}}}^s\left(\hat{\boldsymbol{s}}:{\hat{\boldsymbol{h}}}^s\right)\right] $$

(A-6)

If ${\hat{\boldsymbol{h}}}^s:\boldsymbol{s}<0$:

$$ \frac{d{g}_1}{d\epsilon }=\frac{k_1\left({g}_4-{g}_1\right)}{g_1} $$

(A-7)

$$ \frac{d{g}_2}{d\epsilon }={k}_2\left({k}_3\frac{\sigma_0}{\sigma }-{g}_2\right) $$

(A-8)

$$ \frac{d{g}_3}{d\epsilon }={k}_5\left({k}_4-{g}_3\right) $$

(A-9)

$$ \frac{d{\hat{\boldsymbol{h}}}^s}{d\epsilon }=k\left[-\hat{\boldsymbol{s}}+\frac{8}{3}{\hat{\boldsymbol{h}}}^s\left(\hat{\boldsymbol{s}}:{\hat{\boldsymbol{h}}}^s\right)\right] $$

(A-10)

where σ ₀ and σ are initial and effective flow stress, respectively. g ₁ to g ₄ are state variables and k and k_1~5 are constants, which can be obtained from either loading-unloading-loading tension or forward-reverse simple shear experiments. The first coefficient, k, controls the rotation rate of the microstructure deviator, ${\hat{\boldsymbol{h}}}^s.$ Coefficients k_1~3 control the new flow stress and hardening rate after each load variation. If permanent softening is considered, coefficients k₄ and k₅ control this effect.