A FE model of a standard cylindrical coupon 0.5 inch in diam. was developed. The geometry of the constant-area test section is completely uniform, no geometric imperfection was induced in order to “help” the necking process. A simplified axisymmetric approach was devised, based on the PLANE183 (high-order) element. The model represents the entire coupon, symmetry was not used.
The mesh contains 2146 base elements, with a max. element size of 1.27 mm, reduced to approx. 0.2 mm in axial direction near the coupon center, and has 10 elements in radial direction.
The traction is applied at both ends equally, to keep the center of the coupon in place (for easier post-processing). It is considered that the coupon ends are threaded, so only the axial displacement is enforced, leaving the radial DOF free. The instantaneous reaction forces are recorded on the loading surfaces, and the elongation is measured two diameters away from the center (as specified by the standard).
The material chosen for this model is S355 steel. A real test stress-strain dataset was converted from engineering to true values, using the method described in [1]. Particular care was taken to ensure smoothness and C1 continuity of the true stress-strain curve around the necking point. The data was finally converted into a discrete set of true plastic strain vs. true stress points, since the Multi-linear Isotropic Hardening model was used. The material definition goes up to a true plastic strain of 2.0, to avoid automatic extrapolation (i.e. perfect plasticity).
Several runs were made initially, to calibrate the material model — this was needed because the power-law extrapolation of the true stress-strain data beyond the necking point is empirical. The calculations showed that the necking occurred spontaneously, right in the middle of the coupon. The calibration could not be done beyond 20% total elongation, but the true stress at the experimental breaking point was determined. After that, it was decided to add the breaking of the coupon; thus, contact element pairs were inserted in the necking section, with a predetermined max. allowable normal stress (1150 MPa) for contact release. The final model uses implicit time integration, and has an effective strain rate of 0.001 s^-1.

The results show that:

• the FE model captures very well the large-scale features of the tensile test;
• with proper modeling, there is no need for artificial seeding of necking, it develops naturally;
• extreme effective strains occur at the necking site, even beyond 100%; therefore, careful material properties definition is critical for such simulations;
• it is possible to match the experimental results very closely, even beyond the critical necking point, using quite simple material definitions; for the particular case of S355 steel, even the typical plateau is well capture;
• with a simple, contact-based approach, the coupon breaking can only be represented approximately; the fracture is initiated at the neck axis (location of max. stress) and very quickly propagated radially outwards (this behavior is well confirmed by [2]); however, the resulting surfaces are symmetrical, slightly cupped — as opposed to the typically observed cup-and-cone fracture, which likely requires far more in-depth physical modeling.

Please note that the animations are not uniform in time, they are very much slowed down around the breaking of the coupon, to facilitate the observation of the fracture initiation & development, along with the evolution of the stress fields.

V. Tvergaard, A. Needleman, Analysis of the cup-cone fracture in a round tensile bar, Acta Metallurgica, Volume 32, Issue 1, 1984, Pages 157-169, ISSN 0001-6160, https://doi.org/10.1016/0001-6160(84)90213-X.

Jones, Cliff. (2019). JONES-DISSERTATION-2019. 10.13140/RG.2.2.25385.01129.