1. A probabilistic twin nucleation model for HCP polycrystalline metals
by I. J. Beyerlein, and C. N. Tomé
Proceedings A
Volume 466(2121):
September 8, 2010
©2010 by The Royal Society

2. Schematic of the process of twin nucleation from a grain boundary: (a) interaction of a dislocation pile-up with an array of grain boundary defects (GBDs), (b) dissociation of GBDs into twinning partials, and (c) coalescence to form a twin nucleus.
Schematic of the process of twin nucleation from a grain boundary: (a) interaction of a dislocation pile-up with an array of grain boundary defects (GBDs), (b) dissociation of GBDs into twinning partials, and (c) coalescence to form a twin nucleus. (Figure courtesy of Dr Jian Wang.)‏
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

3. Schematic of the grain boundary surface of an equi-axed grain: each facet corresponds to a boundary that is shared by a neighbouring grain.
Schematic of the grain boundary surface of an equi-axed grain: each facet corresponds to a boundary that is shared by a neighbouring grain. The relevant area a* is assigned to the area of the largest facet.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

4. Partial grain boundary area of a spherical grain: the relevant area a
Partial grain boundary area of a spherical grain: the relevant area a* is assigned to the area of the spherical cap inscribed by a cone.
Partial grain boundary area of a spherical grain: the relevant area a* is assigned to the area of the spherical cap inscribed by a cone.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

5. Schematic of the multi-scale constitutive model for HCP metals that deform by slip and twinning.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

6. Comparison of the stress–strain predictions from the former model using a deterministic treatment of twinning with the measured stress–strain response at room and liquid nitrogen temperature.
Comparison of the stress–strain predictions from the former model using a deterministic treatment of twinning with the measured stress–strain response at room and liquid nitrogen temperature.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

7. Basal and prism pole figure of the initial texture of clock-rolled Zr measured by EBSD. Intensity lines are 0.5/1/2/3/4/5 m.r.o.
Basal and prism pole figure of the initial texture of clock-rolled Zr measured by EBSD. Intensity lines are 0.5/1/2/3/4/5 m.r.o. Black line, 0.5; red line, 1; green line, 2; dark blue line, 3; light blue line, 4; pink line, 5.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

8. (a) Distribution of grain areas of the approximately 9000 grains measured by EBSD. (b) Distribution of grain diameters using the circle area approximation.
(a) Distribution of grain areas of the approximately 9000 grains measured by EBSD. (b) Distribution of grain diameters using the circle area approximation.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

9. Comparison of the predicted and measured stress–strain response of Zr at liquid nitrogen (black) and room temperature (red).
Comparison of the predicted and measured stress–strain response of Zr at liquid nitrogen (black) and room temperature (red). The hypothetical case in which twinning was suppressed in the model is shown to elucidate the impact of twinning on plastic deformation behaviour.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

10. Comparison of (a) predicted and (b) EBSD measured textures in the form of basal (0002) pole figures at (i) 4%, (ii) 9%, (iii) 14%, and (iv) 19% strain.
Comparison of (a) predicted and (b) EBSD measured textures in the form of basal (0002) pole figures at (i) 4%, (ii) 9%, (iii) 14%, and (iv) 19% strain. The compression direction is along RD in the pole figure. Intensity lines are 0.5/1/2/3/4/5 m.r.o.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

11. (a) Evolution of twin volume fraction with strain and (b) evolution of the fraction of twinned grains with strain.
(a) Evolution of twin volume fraction with strain and (b) evolution of the fraction of twinned grains with strain. Solid line, model; unfilled circle, EBSD. The symbols correspond to measurements from EBSD taken from McCabe et al. (2009) and Capolungo et al. (2009c).
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

12. Predicted relative activities of each deformation mode during straining in the (a) parent matrix: solid line, prismatic slip; unfilled triangle on a solid line, tensile twinning.
Predicted relative activities of each deformation mode during straining in the (a) parent matrix: solid line, prismatic slip; unfilled triangle on a solid line, tensile twinning. (b) twinned domains: solid line, prismatic slip; unfilled square on a solid line, compressive twinning.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

13. Distribution of twin Schmid factors at (a) 5% strain from the model, (b) 10% strain from the model and (c) 10% strain from EBSD.
Distribution of twin Schmid factors at (a) 5% strain from the model, (b) 10% strain from the model and (c) 10% strain from EBSD.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

14. Predicted SC factors (equation (4
Predicted SC factors (equation (4.4)) at the time of twin nucleation for the population of twins present by 10% strain.
Predicted SC factors (equation (4.4)) at the time of twin nucleation for the population of twins present by 10% strain.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

15. The fraction of twinned grains (grains containing at least one twin) as a function of the highest twin Schmid factor m(1) in the grain at (a) 5% strain from the model, (b) 10% strain from the model and (c) 10% strain from EBSD. The factor m(1) is calculated...
The fraction of twinned grains (grains containing at least one twin) as a function of the highest twin Schmid factor m(1) in the grain at (a) 5% strain from the model, (b) 10% strain from the model and (c) 10% strain from EBSD. The factor m(1) is calculated based on the orientation of the parent grain and is used here as a measure of the suitability of crystallographic orientation for twinning.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

16. The fraction of twinned grains as a function of grain area in the grain at 10% strain: (a) model prediction and (b) EBSD measurements.
The fraction of twinned grains as a function of grain area in the grain at 10% strain: (a) model prediction and (b) EBSD measurements.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society

17. Number of twins per grain at 10% strain: (a) model prediction and (b) EBSD measurements.
I. J. Beyerlein, and C. N. Tomé Proc. R. Soc. A 2010;466:
©2010 by The Royal Society