Purpose
In order to avoid expensive 3-D finite element calculations for predicting the initial necking and post-necking behaviour during sheet metal forming, an internal length scale is incorporated directly into the constitutive relation. Thereby it is possible to account for thickness effect during necking in a 2-D plane stress analysis.
Application
The applicability of the approach has been demonstrated in a number of simple sheet metal forming processes. The figures below show the contours of the maximum principal strain for a deformation for an internal pressurized tube far into the instability region. The figure to the left shows the results from a conventional 2-D plane stress model; a physical unrealistic mesh dependent solution. The figure to the right shows the corresponding result obtained from an enhanced plane stress model. A result, which predict the true 3-D deformation state quite well.
Instability modes (click to enlarge)
Selected Publications:
Mikkelsen, L.P., Necking in rectangular tensile bars approximated by a 2-D gradient dependent plasticity model. Eur. J. Mech. A/Solids 18, 805-818, 1999.
Mikkelsen, L.P. and Tvergaard, V., A nonlocal two-dimensional analysis of instabilities in tubes under internal pressure. J. Mech. Phys. Solids 47, 953-969, 1999.
Mikkelsen, L.P. and Tvergaard, V., A 2-D Non-local Analysis of Hydroforming for Thin Sheets. Journal de Physique IV Pr8, 249-256, 1998.
Mikkelsen, L.P., Post-necking behaviour modelled by a gradient dependent plasticity theory. Int. J. Solids and Structures 34, 4531-4546, 1997.
Mikkelsen, L.P., On forming limits of thin metal sheets using length scales. Plasticity, Damage and Fracture at Macro, Micro and Nano Scales (eds. A.S. Khan and O. Lopez-Pamiers). NEAT Press, Maryland, USA, 424-426, 2002
Mikkelsen, L.P. and Tvergaard, V., Bending effects on instabilities of internally pressurised tubes modelled by a nonlocal membrane theory. In Thin-Walled Structures - Research and Development (eds. N.E. Shanmugam, J.Y. Richard and V. Thevendran). Elsevier, Singapore, 679-686, 1998. |