Modelling the torsion of thin metal wires by distortion gradient plasticity

A DCAMM seminar will be presented by

Associate Professor Lorenzo Bardella
University of Brescia, Italy


The torsion of thin metal wires is one of the few paradigmatic examples of the small-scale plasticity behaviour involving the “smaller being stronger” size effect, as firstly shown by the experimental results of Fleck, Muller, Ashby, and Hutchinson (1994), who observed, with diminishing specimen size, both an increase in strain hardening and a conspicuous strengthening. We model this problem (Bardella and Panteghini, 2015) by the phenomenological gradient plasticity (GP) theory of Gurtin (2004), which accounts for the dissipation due to the plastic spin and its energetic counterpart included in the  defect energy, a function of Nye’s dislocation density tensor α. To distinguish this phenomenological GP theory from the more common strain gradient plasticity (SGP) theories, overlooking the contribution of the plastic spin, we call the former distortion gradient plasticity (DGP). We consider both energetic and dissipative higher-order stresses, with related “energetic” and “dissipative” material length scales.

The DGP theory of Gurtin (2004) has been studied by Bardella (2010), who has shown that the plastic spin may play a fundamental role (even for small displacements and rotations). However, our previous studies are limited to the simple shear of a strip constrained between regions impenetrable to dislocations. In this investigation, to study the torsion problem, we have implemented a specific finite element, thus being able to obtain a numerically robust and accurate implicit algorithm. In the torsion problem the importance of the plastic spin readily emerges from the structure of α, which suggests to write the defect energy as a function of two invariants of α. This feature alone allows the prediction of some strengthening by a quadratic defect energy, not only strain hardening increase with diminishing size, as usual in SGP theories. We introduce two energetic length scales, each one related to a distinct invariant of α.We analyse both monotonic and cyclic loading, showing that each one of the two energetic length scales governs one of two specific features of the normalised cyclic twist-torque curve, that consist in its size and its shape.

We also discuss the role played by the dissipative length scale involved in the DGP modelling on the basis of the recent work of Fleck, Hutchinson, and Willis (2014, 2015) on SGP theory. We confirm their findings also for the DGP applied to the torsion problem. This result goes into the direction suggested by Hutchinson (2012) of avoiding finite higher-order stresses dependent on plastic strain rates.

Furthermore, we also consider less-than-quadratic defect energies. On the one hand, we show that a properly regularized logarithmic defect energy (see Forest and Guéninchault, 2013, and references therein) may help in representing the experimental results. On the other hand, we show that that choice of defect energy leads to an anomalous cyclic behaviour; noticeably, this problem is common to all the defect energies allowing the modelling of the same kind of energetic strengthening, including the defect energy written as a one-homogeneous function of α.
Danish pastry, coffee and tea will be served 15 minutes before the seminar starts.

All interested persons are invited


Tue 26 May 15
11:00 - 12:00




Room 105, Building 404
Technical University of Denmark