A Phenomenological Model for the Hysteresis Behavior of Metal Sheets Subjected to Unloading/Reloading Cycles
Artikel i vetenskaplig tidskrift, 2011

The springback phenomenon is defined as elastic recovery of the stresses produced during the forming of a material. An accurate prediction of the springback puts high demands on the material modeling during the forming simulation, as well as during the unloading simulation. In classic plasticity theory, the unloading of a material after plastic deformation is assumed to be linearly elastic with the stiffness equal to Young's modulus. However, several experimental investigations have revealed that this is an incorrect assumption. The unloading and reloading stress-strain curves are in fact not even linear, but slightly curved, and the secant modulus of this nonlinear curve deviates from the initial Young's modulus. More precisely, the secant modulus is degraded with increased plastic straining of the material. The main purpose of the present work has been to formulate a constitutive model that can accurately predict the unloading of a material. The new model is based on the classic elastic-plastic framework, and works together with any yield criterion and hardening evolution law. To determine the parameters of the new model, two different tests have been performed: unloading/reloading tests of uniaxially stretched specimens, and vibrometric tests of prestrained sheet strips. The performance of the model has been evaluated in simulations of the springback of simple U-bends and a drawbead example. Four different steel grades have been studied in the present investigation.

stress yield function

elastic stiffness degradation





youngs modulus







steel sheet

hysteresis effect


Per-Anders Eggertsen

Chalmers, Tillämpad mekanik, Material- och beräkningsmekanik

Kjell Mattiasson

Chalmers, Tillämpad mekanik, Material- och beräkningsmekanik

J. Hertzman

Industrial Development Centre

Journal of Manufacturing Science and Engineering, Transactions of the ASME

1087-1357 (ISSN)

Vol. 133