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Max-Planck-Institut für

Eisenforschung GmbH

Düsseldorf/Germany

CPFEM model [12]

•Flow rule given by Kalidindi’s

constitutive model;

•Only 3 slip families are

introduced into the model :

Prismatic 1st order <a>, Basal

<a>, Pyramidal 1st order

<c+a>;

•The CPFE model used is purely

local formulation, and includes

only the changes in slip system

alignment across the boundary,

but no strengthening effect

from grain boundaries.

[1] Sutton A.P. and Balluffi R.W. , “Interfaces in Crystalline Materials.”, (1995).

[2] Randle V., J. Microscopy, 2005, 222, pp. 69-75.

[3] Livingston J.D. and Chalmers B., Acta Metall. (1957), 5(6), pp. 322-327.

[4] Shen Z. et al., Scripta Metallurgica (1986), 20(6), pp. 921–926.

[5] Luster J. et al., Metall. Mater. Trans. A26 (1995) p.1745-1756.

[6] Marcinkowski M.J. et al.., Metal. Trans. (1970), 1(12), pp. 3397-3401.

[7] Reid C.N., Pergamon Press, Oxford, United Kingdom, 1973.

[8] MTEX code : http://mtex-toolbox.github.io/

[9] Guo Y. et al., Acta Mater. (2014), 76, pp. 1-12.

[10] Kacher J. and Robertson I.M., Phil. Mag. (2014), 94(8), pp. 814-829.

[11] Lee T.C. et al. Scr. Metall. 23, 799 (1989).

[12] Zambaldi C. et al., J. Mater. Res. (2012), 27(01), pp. 356-367.

[13] Patriarca L. et al., Mater. Sci. & Eng. (2013), A588, pp. 308–317.

D. Mercier1 (d.mercier@mpie.de), C. Zambaldi1, P. Eisenlohr2, Y. Su2, M. Crimp2, T. R. Bieler2.

1Max-Planck-Institut für Eisenforschung, 40237 Düsseldorf, Germany

2Chemical Engineering and Materials Science, Michigan State University, East Lansing 48824 MI, USA

Microstructure Physics

and Alloy Design

Prof. Dr. D. Raabe

Theory and Simulation

Dr. Franz Roters

Motivation and Strategy

Spherical indentation and crystal plasticity modeling

near grain boundaries in alpha

‐

Ti.

Figure 4 : EBSD

orientation map with

IPF coloring scheme of

Ti–5Al–2.5Sn (wt.%)

sample.

Matlab Toolbox Bridge between experiments and modeling (for BCC, FCC and HCP materials, with 1 or 2 phases).

The development of pile-up topographies across GBs are affected by the geometrical compatibility of the 2 grains and

not inherent GB resistance to slip.

CPFE 3D models are necessary to accurately predict spatial distribution of shear per slip system.

Other functions to implement and other CPFEM models (µ-pillar compression or bending test of cantilever…).

Mercier D., Zambaldi, C. and Bieler T.R. (2014). “A Matlab toolbox to analyze slip transfer through grain

boundaries.” DOI: http://doi.org/10.5281/zenodo.11561

Gr. A

Gr. B

Plasticity of single crystal is well

understood... But, missing element to

predict polycrystal mechanics !

Micromechanics of grain

boundaries (GB) ?

1) Combination of EBSD and strain

experiments on bicrystal sample.

2) Analysis of slip transmission using

the Matlab Toolbox STABiX.

3) 3D Crystal Plasticity Finite

Element (CPFE) modeling

4) Model the slip transmission and

GB mechanic…

5 macroscopic degrees of freedom are

required to characterize a GB [1-2]:

3 for the rotation between the 2 crystals

(by EBSD or TEM characterizations);

2 for the orientation of the GB plane

defined by its normal n (inclination by

serial polishing or FIB sectioning and

trace by EBSD).

The rotation between the 2 crystals is defined by the

rotation angle and the rotation axis common to

both crystals uvw . Knowing Euler angles

of each crystal (from EBSD), a misorientation matrix

is calculated. Figure 2 : Possible strain transfer

across a GB [1].

Figure 1 : Schematic of a bicrystal.

Slip transmission parameters implemented in the toolbox:

What is a bicrystal ?

Quantification of slip transmission

Figure 3 : Geometrical

description of the slip

transfer.

Slip transmission parameter Function Ref.

(C-axis) Misorientation angle () = 1 2

[1]

factor from Livingston and Chamlers

= cos cos +cos cos [3]

factor from Shen et al. = cos cos [4]

parameter from Luster and Morris = cos cos [5]

Residual Burgers vector () = - [6]

Resolved Shear Stress () or

Schmid Factor (

) = :

with

= [7]

Generalized Schmid Factor () = [7]

100µm

Analysis of CPFE results

GB

Gr. B Gr. A

Figure 5 : AFM topography

of residual indent close to a

GB and profile of pile-up

surrounding the indent. Figure 6 : EBSD map GUI.

Figure 7 : Bicrystal GUI. Figure 8 : preCPFE bicrystal GUI.

Accumulated

basal shear

Accumulated

prism. 1 <a>

shear

Gr. B

Gr. A

Gr. B Gr. A

EBSD CPFEM

Validation on literature data

Conclusion

EBSD data are

required: average

grain orientations and

grain boundaries

positions.

Figure 10 : Calculated

topography from CPFEM.

Figure 11 : Local misorientation from

EBSD measurement and CPFEM results.

Figure 12 : Isosurfaces of

accumulated shear in the

bicrystal obtained by CPFEM.

https://github.com/stabix/stabix

Gr. A

Gr. B

IPF from

MTEX [8]

Figure 13 : Comparison of results obtained with the toolbox and

results from a) Kacher’s paper [10] and b) Patriarca’s paper [13].

a) b)

Bicrystal meshing Single crystal meshing

Gr. A

Gr. B

Indentation 2014

10 au 12 décembre 2014 - Strasbourg

Mainly prism.<a> to prism.<a> transfer for cp

α

-Tiour study, 9, 10.

Competition between prism.<a> to prism.<a> and prism.<a> to

basal<a> transfers for

α

-Ti alloy.

The strain transfer across grain boundaries (GB)

can be defined by 4 mechanisms :

Strain transfer across GBs

a) direct transfer and GB is

transparent to dislocations

(no strengthening effect);

b) direct transmission leaving

a residual dislocations;

c) indirect transmission

leaving a residual

boundary dislocations;

d) no transfer and the GB acts

as an impenetrable

boundary, which implies

stress accumulations,

localized rotations, pile-up

of dislocations…

STABiX workflow: from EBSD data to CPFEM model in 3 steps

1

2

3

Figure 9 : preCPFE single crystal GUI.

Transmission in cp-Ti

No transmission in cp-Ti

Strain transfer occurs at GB in cp-Ti for :

geometric consideration

m’ > 0.7 and low RBV

low critical resolved shear stress

Lee’s

criterion11

Ti alloy

Strain transfer

in cp-Ti

References