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The Effect of the Imposed Boundary Rate on the Formability of Strain Rate Sensitive Sheets Using the M-K Method

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Abstract

In spite of the fact that the experimental results indicate the significant effect of strain rate on forming limits of sheets, this effect is neglected in all theoretical methods of prediction of Forming Limit Diagrams (FLDs). The purpose of this paper is to modify the most renowned theoretical method of determination of FLDs (e.g., M-K model) so as to enable it to take into account the effect of strain rate. To achieve this aim, the traditional assumption of preexistence of an initial geometrical inhomogeneity in the sheet has been replaced with the assumption of a preexisting “material” inhomogeneity. It has been shown that using this assumption, the strain rate would not be omitted from equations; thus, it is possible to demonstrate its effect on FLDs. To validate the results, they are compared with some published experimental data. The good agreement between the theoretical and experimental results shows capabilities of the proposed method in predicting the effect of the imposed rate at the boundary (which is physically the effect of the punch speed difference in sheet forming) on FLDs.

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Abbreviations

\( f_{0} \) :

Initial imperfection factor

K :

Strength coefficient

m :

Strain rate sensitivity exponent

n :

Strain hardening exponent

\( r \) :

Normal anisotropy coefficient

t :

Material thickness

α :

In-plane principal stresses ratio \( (\upsigma_{2} /\upsigma_{1} ) \)

\( \dot{\bar{\upvarepsilon }} \) :

Rate of effective plastic strain

\( \bar{\upvarepsilon } \) :

Effective plastic strain

\( \upvarepsilon_{0} \) :

Pre-strain value

\( d\bar{\upvarepsilon } \) :

Effective plastic strain increment

\( d\upvarepsilon \) :

Strain increment tensor

\( d\upvarepsilon_{1} ,\,d\upvarepsilon_{2} ,\,d\upvarepsilon_{3} \) :

Strain increments in the material coordinates

\( d\upvarepsilon_{tt} ,\,d\upvarepsilon_{nn} ,\,d\upvarepsilon_{nt} \) :

Strain increments in the groove coordinates

θ:

Groove angle between the groove coordinates and the material coordinates

\( \upsigma_{1} ,\,\upsigma_{2} ,\,\upsigma_{3} \) :

Principal stress components

\( \upsigma_{nn} ,\,\upsigma_{nt} ,\,\upsigma_{tt} \) :

In-plane stress components in the groove coordinates

\( \bar{\upsigma }_{Y} \) :

Effective stress obtained from hardening law

\( \bar{\upsigma }_{y} \) :

Effective stress obtained from yield function

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Authors and Affiliations

  1. School of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran

    Ramin Hashemi & Karen Abrinia

  2. Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran

    Amir Ghazanfari & Ahmad Assempour

Authors
  1. Ramin Hashemi
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  2. Amir Ghazanfari
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  3. Karen Abrinia
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  4. Ahmad Assempour
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Correspondence to Ramin Hashemi.

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Hashemi, R., Ghazanfari, A., Abrinia, K. et al. The Effect of the Imposed Boundary Rate on the Formability of Strain Rate Sensitive Sheets Using the M-K Method. J. of Materi Eng and Perform 22, 2522–2527 (2013). https://doi.org/10.1007/s11665-013-0559-2

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