Three-dimensional (3D) X-ray microscopy with submicron resolution has been used to make spatially resolved measurements of lattice curvature and elastic strain over two-dimensional slices in thin deformed Si plates. The techniques and capabilities associated with white-beam 3D X-ray microscopy are discussed, and both theoretical and experimental considerations associated with the measurement of Nye dislocation density tensors in deformed materials are presented. The ability to determine the local geometrically necessary dislocation (GND) density in the form of a dislocation density tensor, with micron spatial resolution over mesoscopic length scales, is demonstrated. Results are shown for the special case of an elastically bent (dislocation free) thin Si plate and for a similar thin Si plate that was bent plastically, above the brittle-to-ductile transition temperature, to introduce dislocations. Within the uncertainties of the measurements, the known result that GND density is zero for elastic bending is obtained, and well-defined GND distributions are observed in the plastically deformed Si plate. The direct and absolute connection between experimental measurements of GND density and multiscale modeling and computer simulations of deformation microstructures is discussed to highlight the importance of submicron-resolution 3D X-ray microscopy for mesoscale characterization of material defects and to achieve a fundamental understanding of deformation in ductile materials.

1.
Hirth
,
J. P.
, and
Lothe
,
J.
, 1982,
Theory of Dislocations
,
Wiley
,
New York
.
2.
Nabarro
,
F. R. N.
, 1987,
Theory of Crystal Dislocations
,
Dover
,
New York
.
3.
Jourdan
,
C.
, and
Gastaldi
,
J.
, 1977, “
X-Ray Topographic Investigation of Dislocations in Ti Single Crystals Grown by Recrystallization
,”
Phys. Status Solidi A
0031-8965,
43
, pp.
425
435
.
4.
Larson
,
B. C.
,
Yang
,
W.
,
Ice
,
G. E.
,
Budai
,
J. D.
, and
Tischler
,
J. Z.
, 2002, “
Three-Dimensional X-Ray Structural Microscopy With Submicrometre Resolution
,”
Nature (London)
0028-0836,
415
, pp.
887
890
.
5.
Larson
,
B. C.
,
El-Azab
,
A.
,
Yang
,
W.
,
Tischler
,
J. Z.
,
Liu
,
W.
, and
Ice
,
G. E.
, 2007, “
Experimental Characterization of the Mesoscale Dislocation Density Tensor
,”
Philos. Mag.
1478-6435,
87
, pp.
1327
1347
.
6.
Yang
,
W.
,
Larson
,
B. C.
,
Tischler
,
J. Z.
,
Ice
,
G. E.
,
Budai
,
J. D.
, and
Liu
,
W.
, 2004, “
Differential-Aperture X-Ray Structural Microscopy: A Submicron-Resolution Three-Dimensional Probe of Local Microstructure and Strain
,”
Micron
0968-4328,
35
, pp.
431
439
.
7.
Yang
,
W.
,
Larson
,
B. C.
,
Pharr
,
G. M.
,
Ice
,
G. E.
,
Budai
,
J. D.
,
Tischler
,
J. Z.
, and
Liu
,
W. J.
, 2004, “
Deformation Microstructure Under Microindents in Single-Crystal Cu Using Three-Dimensional X-Ray Structural Microscopy
,”
J. Mater. Res.
0884-2914,
19
, pp.
66
72
.
8.
Levine
,
L. E.
,
Larson
,
B. C.
,
Yang
,
W.
,
Kassner
,
M. E.
,
Tischler
,
J. Z.
,
Delos-Reyes
,
M. A.
,
Fields
,
R. J.
, and
Liu
,
W.
, 2006, “
X-Ray Microbeam Measurements of Individual Dislocation Cell Elastic Strains in Deformed Single-Crystal Copper
,”
Nat. Mater.
1476-1122,
5
, pp.
619
622
.
9.
Larson
,
B. C.
, and
Lengeler
,
B.
, 2004, “
High Resolution Three-Dimensional X-Ray Microscopy
,”
MRS Bull.
0883-7694,
29
, pp.
152
156
.
10.
Ice
,
G. E.
, and
Larson
,
B. C.
, 2004, “
Three-Dimensional X-Ray Structural Microscopy Using Polychromatic Microbeams
,”
MRS Bull.
0883-7694,
29
, pp.
170
176
.
11.
Poulsen
,
H. F.
,
Jensen
,
D. J.
, and
Vaughan
,
G. B. M.
, 2004, “
Three-Dimensional X-Ray Diffraction Microscopy Using High-Energy X-Rays
,”
MRS Bull.
0883-7694,
29
, pp.
166
169
.
12.
Poulsen
,
H. F.
, 2004,
Three-Dimensional X-Ray Diffraction Microscopy: Mapping Polycrystals and Their Dynamics
(
Springer
,
Berlin
, 2004).
13.
Yang
,
W.
,
Larson
,
B. C.
,
Ice
,
G. E.
,
Tischler
,
J. Z.
,
Budai
,
J. D.
,
Chung
,
K.-S.
, and
Lowe
,
W. P.
, 2003, “
Spatially Resolved Poisson Strain and Anticlastic Curvature Measurements in Si Under Large Deflection Bending.
,”
Appl. Phys. Lett.
0003-6951,
82
, pp.
3856
3858
.
14.
Poulsen
,
H. F.
,
Lienert
,
U.
, and
Pantleon
,
W.
, 2005, “
Characterisation of Orientation Distributions of Individual Grains Within Deformed Metals
,”
Mat. Sci. Technol.
1001-0181,
21
, pp.
1397
1400
.
15.
Margulies
,
L.
,
Winther
,
G.
, and
Poulsen
,
H. F.
, 2001, “
In Situ Measurement of Grain Rotation During Deformation of Polycrystals
,”
Science
0036-8075,
292
, pp.
2392
2394
.
16.
Ice
,
G. E.
,
Larson
,
B. C.
,
Yang
,
W.
,
Budai
,
J. D.
,
Tischler
,
J. Z.
,
Pang
,
J. W. L.
,
Barabash
,
R. I.
, and
Liu
,
W.
, 2005, “
Polychromatic X-Ray Microdiffraction Studies of Mesoscale Structure and Dynamics
,”
J. Synchrotron Radiat.
0909-0495,
12
, pp.
155
162
.
17.
Nye
,
J. F.
, 1953, “
Some Geometric Relations in Dislocated Crystals
,”
Acta Metall.
0001-6160,
1
, pp.
153
162
.
18.
El-Azab
,
A.
, 2000, “
The Boundary Problem of Dislocation Dynamics
,”
Modell. Simul. Mater. Sci. Eng.
0965-0393,
8
, pp.
37
54
.
19.
El-Azab
,
A.
, 2000, “
Statistical Mechanics Treatment of the Evolution of Dislocation Distributions in Single Crystals
,”
Phys. Rev. B
0163-1829,
61
, pp.
11956
11966
.
20.
Limkumnerd
,
S.
, and
Sethna
,
J. P.
, 2006, “
Mesoscale Theory of Grains and Cells: Crystal Plasticity
,”
Phys. Rev. Lett.
0031-9007,
96
, p.
095503
.
21.
Limkumnerd
,
S.
, and
Sethna
,
J. P.
, 2007, “
Stress-Free States of Continuum Dislocation Fields: Rotations, Grain Boundaries, and the Nye Dislocation Density Tensor
,”
Phys. Rev. B
0163-1829,
75
, p.
224121
.
22.
Larson
,
B. C.
,
Yang
,
W.
,
Tischler
,
J. Z.
,
Ice
,
G. E.
,
Budai
,
J. D.
,
Liu
,
W.
, and
Weiland
,
H.
, 2004, “
Micron-Resolution 3-D Measurement of Local Orientations Near a Grain-Boundary in Plane-Strained Aluminum Using X-Ray Microbeams
,”
Int. J. Plast.
0749-6419,
20
, pp.
543
560
.
23.
Chung
,
J. S.
, and
Ice
,
G. E.
, 1999, “
Automated Indexing for Texture and Strain Measurement With Broad-Bandpass X-Ray Microbeams
,”
J. Appl. Phys.
0021-8979,
86
, pp.
5249
5255
.
24.
Weygand
,
D.
,
Friedman
,
L. H.
,
Van der Giessen
,
E.
, and
Needleman
,
A.
, 2002, “
Aspects of Boundary-Value Problem Solutions With Three-dimensional Dislocation Dynamics
,”
Modell. Simul. Mater. Sci. Eng.
0965-0393,
10
, pp.
437
468
.
25.
Weygand
,
D.
, and
Gumbsch
,
P.
, 2005, “
Study of Dislocation Reactions and Rearrangements Under Different Loading Conditions
,”
Mater. Sci. Eng., A
0921-5093,
400–401
, pp.
158
161
.
26.
El-Azab
,
A.
,
Deng
,
J.
, and
Tang
,
M.
, 2007, “
Statistical Characterization of Dislocation Ensembles
,”
Philos. Mag.
1478-6435,
87
, pp.
1201
1223
.
27.
El-Azab
,
A.
, 2006, “
Statistical Mechanics of Dislocation Systems
,”
Scr. Mater.
1359-6462,
54
, pp.
723
727
.
28.
Kosevich
,
A. M.
, 1979, “
Crystal Dislocations and the Theory of Elasticity
,”
Dislocations in Solids
,
F. R. N.
Nabarro
, ed.,
North-Holland
,
Amsterdam
, Vol.
1
, pp.
33
141
.
29.
Mura
,
T.
, 1968, “
Continuum Theory of Dislocations and Plasticity
,”
Mechanics of Generalized Continua
,
E.
Kröner
, ed.,
Springer-Verlag
,
New York
, pp.
269
278
.
30.
Kröner
,
E.
, 1981, “
Continuum Theory of Defects
,”
Physics of Defects
,
R.
Balian
,
M.
Kléman
, and
J.-P.
Poirier
, eds.,
North-Holland
,
Amsterdam
, pp.
217
315
.
31.
El-Azab
,
A.
,
Zaiser
,
M.
, and
Busso
,
E.
, 2007, “
Density-Based Modelling of Dislocations
,”
Philos. Mag.
1478-6435,
87
(
8–9
), pp.
1159
1160
.
32.
Eshelby
,
J. D.
, 1956, “
The Continuum Theory of Lattice Defects
,”
Solid State Phys.
0081-1947,
F.
Seitz
and
D.
Turnbull
, eds., volume
3
, pp.
79
144
.
33.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
,
2nd ed.
,
Martinus Nijhoff
,
Dordrecht, The Netherlands
.
34.
Kossecka
,
E.
, 1974, “
Mathematical Theory of Defects. Part I. Statics
,”
Arch. Mech.
0373-2029,
26
, pp.
995
1010
.
35.
Gurtin
,
M. E.
, 2002, “
A Gradient Theory of Single-Crystal Viscoplasticity That Accounts for Geometrically Necessary Dislocations
,”
J. Mech. Phys. Solids
0022-5096,
50
, pp.
5
32
.
36.
Needlemann
,
A.
, and
Sevillano
,
J. G.
, 2003, “
Preface to the Viewpoint Set on: Geometrically Necessary Dislocations and Size Dependent Plasticity
,”
Scr. Mater.
1359-6462,
48
, pp.
109
111
.
37.
Ungar
,
T.
,
Mughrabi
,
H.
,
Rönnpagel
,
D.
, and
Wilkens
,
M.
, 1984, “
X-Ray Line-Broadening Study of the Dislocation Cell Structure in Deformed (001)-Oriented Vopper Single Crystals
,”
Acta Metall.
0001-6160,
32
, pp.
333
342
.
38.
Acharya
,
A.
, and
Roy
,
A.
, 2006, “
Size Effects and Idealized Dislocation Microstructure at Small Scales: Predictions of a Phenomenological Model of Mesoscopic Field Dislocation Mechanics: Part I
,”
J. Mech. Phys. Solids
0022-5096,
54
, pp.
1687
1710
.
39.
Roy
,
A.
, and
Acharya
,
A.
, 2006, “
Size Effects and Idealized Dislocation Microstructure at Small Scales: Predictions of a Phenomenological Model of Mesoscopic Field Dislocation Mechanics: Part II
,”
J. Mech. Phys. Solids
0022-5096,
54
, pp.
1711
1743
.
40.
Zaiser
,
M.
,
Nikitas
,
N.
,
Hochrainer
,
T.
, and
Aifantis
,
E. C.
, 2007, “
Modelling Size Effects Using 3D Density-Based Dislocation Dynamics
,”
Philos. Mag.
1478-6435,
87
, pp.
1283
1306
.
41.
Cleveringa
,
H. H. M.
,
Van der Giessen
,
E.
, and
Needleman
,
A.
, 1999, “
A Discrete Dislocation Analysis of Bending
,”
Int. J. Plast.
0749-6419,
15
, pp.
837
868
.
42.
Ma
,
A.
,
Roters
,
F.
, and
Raabe
,
D.
, 2006, “
A Dislocation Density Based Constitutive Model for Crystal Plasticity FEM Including Geometrically Necessary Dislocations
,”
Acta Mater.
1359-6454,
54
, pp.
2169
2179
.
43.
Arsenlis
,
A.
, and
Parks
,
D. M.
, 1999, “
Crystallographic Aspects of Geometrically-Necessary and Statistically-Stored Dislocation Density
,”
Acta Mater.
1359-6454,
47
, pp.
1597
1611
.
44.
Jakobsen
,
B.
,
Poulsen
,
H. F.
,
Lienert
,
U.
, and
Pantleon
,
W.
, 2007, “
Direct Determination of Elastic Strains and Dislocation Densities in Individual Subgrains in Deformation Structures
,”
Acta Mater.
1359-6454,
55
, pp.
3421
3430
.
45.
Wert
,
J. A.
,
Huang
,
X.
,
Winther
,
G.
,
Pantleon
,
W.
, and
Poulsen
,
H. F.
, 2007, “
Revealing Deformation Microstructures
,”
Mater. Today
1369-7021,
10
, pp.
24
32
.
46.
Liu
,
W.
,
Ice
,
G. E.
,
Tischler
,
J. Z.
,
Khounsary
,
A.
,
Liu
,
C.
,
Assoufid
,
L.
, and
Macrander
,
A. T.
, 2005, “
Short Focal Length Kirkpatrick-Baez Mirrors for a Hard X-Ray Nanoprobe
,”
Rev. Sci. Instrum.
0034-6748,
76
, pp.
113701
113706
.
You do not currently have access to this content.