1. MSE 2016 Qualify Exam
Candidate: Songyang Han
Committee Member: Dr. Martin Crimp,
Dislocations and Grain Boundary Effects
1. Introduction
The different applications of metals or alloys are based on the mechanical properties.
Evaluations of mechanical properties of metallic materials can be achieved by continuous
deformation to fracture under external stresses. Crack initiation and propagation are the keys
in understanding fracture and in reciprocal, help improve properties of materials. To understand
such mechanism, Griffith's criterion [1] and Irwin's modifications [2] are provided in
calculating the criteria and driving force of the crack propagation. Mathematical expressions
may provide decent answer on why and how crack propagates yet, the secret behind cracks as
stress concentrator is still in need of investigation. Dislocations are microstructure defects in
crystal lattice which can pile up before a boundary or an obstacle and act as stress concentrators
in crack propagations [3]. Therefore, understanding how dislocations occur, align and
segregate provides the backup knowledge in crack investigations.
Theories of dislocations have been well established since 1930’s in single crystals. In
reality, dislocations within one crystal is not the only reason that accounts for the deformation
in polycrystalline materials. The discontinuous interfaces between a certain grain and its
neighbouring grain boundaries can either encourage or hinder the dislocation generation and
propagation under a given stress. Some adjacent grain boundaries might enable the pile up of
dislocations to overcome the energy barrier to propagate within the new ones and, some may
stop the dislocations crossing the boundaries by absorbing or bouncing them back.
Understanding the interactions between dislocations and grain boundaries is a crucial point in

2. understanding or predicting the failure of real metallic materials. When studying grain
boundaries and relative dislocations, several attempts are taken in account: the source of stress;
the orientations of the interest grain itself and the criteria for dislocations to occur; the adjacent
grain orientations and energy capacities which can influence the dislocation motions around
boundaries; the energy accumulation, relaxation and the residual energy on boundaries. Some
topics can be observed from in-situ or regular deformation process in SEM, TEM etc. to
directly emphasis the boundary influence on dislocations, while others have to be achieved
from computational modelling.
Herein, a history on discovering dislocations from cracking and failure analysis is
reviewed to get a general idea on the relationship between microstructure defect and
macrostructure failure. Further mechanism of dislocations motions on polycrystalline materials
is provided with agreement of the experimental data.
2. Background and Journal Review
Dislocation is defined by G.I Talyor 1934 [4] in a deformation study. A model was
created in order to describe the equilibrium of atoms in the lattice and used to calculate the
stress distribution around the dislocation centre. Connections between material failure and
motions of crystal defects were thus built up by the model.
After Talyor’s definition, J. M. Burgers 1940 [5] defined edge and screw dislocations
as well as their motions during deformations. It was indicated that multiple dislocation lines
were initially formed from the defects on exterior surface and propagated within a single crystal
through similar paths and finally accumulated to form a slip band. It was also important to point
out that dislocation lines may stop at certain places on the boundaries since to overcome local
surface stresses need greater forces and the accumulation of those stopped lines will finally
cause permanent deformation of a crystal. However, what the criteria was needed for the
dislocation to pass through a boundary was unknown.

3. During the same time, other models on dislocations stress during deformation and
hardening were developed by R. Peierls 1940 [6] and J. S. Koehler 1941 [7]. Peierls created
continuum theory, found the width of dislocations was only a few atom-distance and calculated
the shear stress for a plane to slip in perfect lattice. Koehler refined the theory by solving the
shear stress around dislocation center and surface. These theories are widely used in further
studies, yet the theories are established on single crystals thus not responsible for deformations
on polycrystals.
Single crystal materials are rare in reality. Alloys are mostly used in industries since
their mechanical properties are better. The solute elements will form different microstructures
with solvent elements according to processing technics. Even pure metal materials are often
polycrystallines. These discrepancies or in consistencies in structures, orientations will strongly
influence the propagation of dislocations thus effect the mechanical properties of materials.
Understanding dislocations on polycrystalline materials and knowing the influence of grain
boundaries are essential in predicting the failure of real life materials.
B. Chalmers 1937 [8] briefly founded a linear relationship between the stress and the
angle between two merged tin single crystals (Fig 1). Larger load was needed only when the
angle between two crystal orientations got bigger. The effect of direction of the boundary plane
was not observed, yet they found the boundary was a transition zone connecting the two crystals
that prevented glide. To understand further about boundaries effects, precise measurement was
needed.

4. After Chalmers, Cottrell 1949 [9] discussed in a literature that grain boundaries would
act as a barrier for dislocations since they had a higher yield stress than that within single
crystals. Dislocations would be bounced back and piled up near the boundary until the stress
accumulated was large enough to transfer themselves to the next boundary.
Another study called grain-boundary theory was done by E. O. Hall 1951 [10] and N.
J. Petch 1953 [11] that yield point of a crystal was found dependent on the grain size. More
rapidly Luders band would form on crystals with larger grains. They plotted the yield stress
versus the diameter and found the equation which described the inverse
root relationship between yield stress and the grain size. It was also pointed out that the yield
strength of grain boundary was not too high otherwise the grain would not deform at all, and
the grain boundary was actually continuous although it was too thin to observe with the
technology at hand.
With the help of the results from Chalmers and Cottrell, T. Kawada 1951 [12, 13]
provided the stress-strain curve of zinc bicrystal. Kawada pointed out that the difference in
stress was again accountable for the mutual interaction of neighbouring boundaries which
agreed with Chalmers result. Later, J.J. Gilman 1953 [14] did similar deformation experiments
on symmetrical zinc bicrystals and found the stress-strain curve was almost identical with

5. single crystals (Fig 2). The rotations in Y and Z directions were justified to have little effect on
the strengthening of the bicrystals but in Z directions, though it was hard to see how
dislocations flew into the grain boundary and caused such rotation. It was also mentioned that
the interactions of the two crystals with the grain boundary were independent and the
interaction happened only within a few microns around the boundary.
J. D. Livingston and B. Chalmers in 1957 [15] came up with a general criteria for a
slip transferring across the boundary and activating on the neighbouring crystal. Confinements
were made that in order for the bicrystal to deform simultaneously with continuity (also see
Fig 2), total strain component on both crystals should be same. They calculate the stress of the
pile-up dislocations on the boundary and figured out the slip systems with Highest N1i will be
favourable in the activation of a slip in adjacent crystal (Fig 3). They also pointed out that N1i
was mostly helpful in finding out slip planes in the neighbouring crystal, while modifications
were needed on their criteria since the existence of some secondary slip planes in some bicrystal
systems were not predicted by their model.

6. J. J. Hauser and B. Chalmers in 1961 [16] made complementary research based on the
theory in 1957. They tried to simulate the real-life deformation of polycrystals by the
deformation of total surrounded bicrystal (Fig 4). They clarified that dislocation pile-ups were
the reason for the existence of slip bands of the secondary slip systems in adjacent crystals.
Important factors controlling the stress-strain curve of the bicrystal were summarized:
1) The Schmid factor which control the resolved shear stress on the slip systems is the
preliminary element that influence the stress-strain curve
2) The incompatibility of a bicrystal is the strongest factor affecting the curve. As
crystals becoming more and more incompatible due to rotation during deformation, huge
amount of stress accumulated on the grain boundary cause the activation of the secondary slip
system

7. 3) The type of dislocations reactions with the primary system can be a reason for
hardening by creating Lomer–Cottrell locks, the type of secondary slip system also indicate the
interaction between crystals
4) The amount of multiple slip layers at the grain boundary and how they spread is
the final component representing the magnitude of the interaction within bicrystals.
Other researcher’s [17-20] used this theory to discuss the continuity of the slip and the
slip band crossing the grain boundary in pure aluminum, sliver and iron bicrystals and found
that low angle boundaries were easy for slip band crossing boundaries with continuity.
Numerous researchers studied the interactions between dislocations and boundaries,
especially on low angle tilt boundaries and high angle twist boundaries, since real life material
were not always perfectly compatible. The grain boundary dislocations motions were
visualized in TEM. The differences in Burgers Vectors in lattice and boundaries were also
calculated and found high angle boundaries were difficult for dislocations to transfer [21-26].
J. P. Hirth in 1972 [27] summarized the mechanism that Grain boundaries could be both sinks
and sources for lattice dislocations in a review that the lattice dislocations could dissociate into
partial to reduce the energy when they encountered the grain boundaries (Fig 5). Dislocations
absorbed into the boundaries could also act as dislocation sources by generating dislocations
on new grains, leaving residual dislocations within the boundaries to release stress
concentrations.

8. During the same period, M. J. Marcinkowski et al. and W. Bollmann in 1970 [28, 29]
talked about residual Burgers vector was also a preliminary factor to analyse slip
transmission(Fig 6). Residual Burgers vector is the burgers vector left in the grain boundary
when a new dislocation is generated into another grain. Original Burgers vector was just the
deduction of both burgers vectors in adjacent grains, while later in real cases, orientation of the
lattice should also be considered in a crystal coordinate system to make sure the consistency of
the system. Residual Burgers vector is required to be minimized for slip to transfer between
grain boundaries. L. C. Lim in 1984 [30] indicated that the slip system would be preferable to
be activated even the Schmid factor was not the highest among all possible systems.
Another criteria was proposed by Z. Shen et al [31, 32] since they found that N factor
in a pure stress system alone could not account for their operative system. They combined the
shear stress with a new geometric system and came up with the equation
or (Fig 7). According to their theory, slip transfer would be more

9. favourable when both the angle α between burgers vector in adjoining planes and the angle β
between two lines of intersection with the interface of two slip planes were minimized.

13. [1] Griffith, A. A. 1921, “The phenomena of rupture and flow in solids”, Philosophical
Transactions of the Royal Society of London, 221, 163–198
[2] Irwin G, 1957, “Analysis of stresses and strains near the end of a crack traversing a plate”,
Journal of Applied Mechanics. 24, 361–364
[3] Yokobori. T, 1965, “The Strength, Fracture and Fatigue of Materiats”, Noordhoff,
Groningen. 372
[4] G. I. Talyor, F. R. S., 1934, “The Mechanism of Plastic Deformation of Crystals. Part I.-
Therotical”, Proceedings of the Royal Society of London. Series A 145 (855): 362–387
[5] J. M. Burgers. 1940, “Geometrical Considerations Concerning The Structural Irregularities
To Be Assumed In A Crystal”, Proc. Phys. Soc. 52, 23-33
[6] R. Peierls. 1940, “The Size of a Dislocation”, Proc. Phys. Soc. 52, 34-37
[7] J. S. Koehler. 1941, “On the Dislocation Theory of Plastic Deformation”, Physical Review.
60, 397-410
[8] B. Chalmers. 1937, “The influence of the Difference of Orientation of Two Crystals on the
Mechanical Effect of their Boundary”, Proc. R. Soc. Lond. A. 162, 120-127
[9] A. H. Cottrell, A. T. Churchman. 1949b, Trans. A.I.M.E., 185, 87
[10] E. O. Hall. 1951, “The Deformation and Ageing of Mild Steel: III Discussion of Results”,
Proc. Phys. Soc. B 64 747
[11] N. J. Petch. 1953, “The Cleavage Strength of Polycrystals”, J. Iron Steel Inst. 174, 25-28
[12] T. Kawada. 1951, “On the Plastic Deformation of Zinc Bicrystal I”, J. Phys. Soc. Jap. 6, 362-
366
[13] T. Kawada. 1951, “On the PlasticDeformation of ZincBicrystalII”, J. Phys. Soc. Jap. 6, 485-
488
[14] J. J. Gilman. 1952, “Deformation of Symmetric Zinc Bicrystals”, Acta Metallurgica, 1, 426-
427
[15] J. D. Livingston, B. Chalmers. 1957, “Multiple Slip In Bicrystal Deformation”, Acta Metallurgia,
5, 322-327
[16] J. J. Hauser, B. Chalmers. 1961, “The Plastic Deformation of Bicrystals of FCC Metals”,
Acta Metallurgia, 9, 802-818
[17] T. J. Koppenaal. 1962, “Slip Lines across an F.C.C.-B.C.C. Interface”, Acta Metallurgica, 10,
684-685
[18] N. Kratochvilova, B. Sestak. 1964, “Slip Propagation Over the Grain Boundaries at Plastic
Deformation of Fe-3% Si Alloy”, Physica Status Solidi (B), 5, 343-355

14. [19] R. E. Hook, J.P. Hirth. 1967, “The Deformation Behavior of Isoaxial Bicrystals of Fe-3%Si”,
Acta Metallurgica, 15, 535-551
[20] R. E. Hook, J.P. Hirth. 1967, “The Deformation Behavior of Isoaxial Bicrystals of Fe-3%Si”,
Acta Metallurgica, 15, 1099-1110
[21] W. Bollmann, 1967, “On the Geometry of Grain and Phase Boundaries II. Applications of
General Theory”, Philosophical Magazine, 16, 383-399
[22] G. H. Bishop, B. Chalmers, 1968, “A Coincidence- Ledge- Dislocation Description of Grain
Boundaries”, Scripta Metallgurica, 2, 133-140
[23] A. H. Clauer, B. A. Wilcox, J. P. Hirth, 1970, “Dislocation Substructure Induced by Creep in
Molybdenum Single Crystals”, Acta Metallurgica, 18, 381-397
[24] T. Schober, R. W. Balluffi, 1971, “Extraneous Grain Boundary Dislocation in Low and High
angle (001) Twist Boundaries in Gold”, Philosophical Magazine, 24, 165-180
[25] G. H. Bishop, B. Chalmers, 1971, “Dislocation Structure and Contrast in High Angle Grain
Boundaries”, Philosophical Magazine, 24, 515-526
[26] G. R. Kegg,C. A. P. Horton, J. M. Silcock, 1972, “Grain Boundary Dislocations in Aluminium
Bicrystals after High- Temperature Deformation”, Philosophical Magazine, 27, 1041-1055
[27] J. P. Hirth, 1972, “The Influence of Grain Boundaries on Mechanical Properties”,
Metallurgical Transactions, 3, 3047-3067
[28] M. J. Marcinkowski, W. F. Tseng,1970, “DislocationBehavior at Tilt Boundaries of Infinite
Extent”, Metallurgical Transactions, 1, 3397-3401
[29] W. Bollmann, 1970, “Crystal Defects and Crystalline Interfaces” DOI, 10.1007/978-3-642-
49173-3, Springer-Verlag Berlin Heidelberg
[30] L. C. Lim, 1984, “Slip-Twin Interactions in Nickel at 573K at Large Strains”, Scripta
Metallurgica, 18, 1139-1142
[31] Z. Shen, R. H. Wagoner, W. A. T. Clark, 1986, “Dislocation Pile-up and Grain Boundary
Interactions in 304 Stainless Steel”, Scripta Metallurgica, 20, 921-926
[32] Z.Shen, R. H.Wagoner, W. A. T. Clark,1988, “Dislocationand Grain Boundary Interactions
in Metals”, Acta Metall, 36, 3231-3242