Plastic deformation – crystalline materials

 Plastic deformation/Inelastic deformation – permanent strains set in

the crystalline lattice.

 No change in the crystal structure/crystallinity under normal

conditions of pressure & temperature & strain rates.

 Plastically deformed single crystal – surface observations shows

steps/lines, appears as though a portion of the crystal has slipped
over the other.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation – crystalline materials

Plastic deformation in crystalline materials occurs by

slip (glide), which occurs on specific crystallographic planes and

directions and/or by

mechanical twinning.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Theoretical strength of a perfect crystal

Consider two plane of atoms in a perfect crystal (assuming no defects)

subjected to a shear stress,

initial mid-position final position

b = interatomic spacing If „u‟ is the displacement, then plastic deformation

should have occurred when u = b (slip distance)
d = interplanar spacing

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Theoretical strength of a perfect crystal

Assuming sinusoidal relationship between stress

and displacement,
th
2 u where, th = theoretical
th sin
b stress, (at u = b/4)

b/4 u
At small displacements, sinx ~ x,
and the shear strain, = u/d,

2 u u
th
b d
b
b
th ~ th
2 d 2 6

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Theoretical strength of a perfect crystal

 Although some of the sophisticated calculations of theoretical strength of a crystal

predict values in the order of /10 - /30, the experimental yield strengths of
materials yield values which are orders of magnitude less than the predicted
theoretical strength!

Material th (= /30) exp

109 Pa 106 Pa
Ag 1 0.37
Al 0.9 0.78
Cu 1.4 0.49

 This led to the postulation of the existence of defects in the materials (dislocations),
independently postulated by, G. I. Taylor, Orowan and Polanyi that “ it is the
movement of these defects which cause plastic deformation in materials”, and the
stress required to move these dislocations to cause the same displacement „b‟ is
quite small !

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Micheal Polanyi (1891 - 1976)

Diverse interests

Arts, humanities,
mathematics, science

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Hungarian – as usual

Born on 12th March 1891

Nobel prize: 5 (chemistry), 3 (medicine), 4 (physics), 1 (economics), 1 (peace)

http://www.hungarianhistory.com/nobel/nobel.htm
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Academic & professional career

Doctor Doctor of medicine (1913)

Doctorate in chemistry (1917)

Physical
chemist Kaiser Wilhelm Institut für
Faserstoffchemie (1920)

Manchester University (1933)

Philosopher Manchester University (1948 - 58)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Quotation from Polanyi

"Biologists speak of explaining living beings in terms of physics and chemistry,

but they never actually realise what this means. They assume that to explain
life in terms of a mechanism based on physics and chemistry is to explain it in
terms of physics and chemistry, and this is false. Thus misconceived, the claim
to explain life by physics and chemistry comes to stand for the claim of
explaining life by mechanical models, and this claim has much truth in it [since
models are `as if' conceptions]. ..."

"Though the claim to explain all living processes mechanically is absurd, all
life has a mechanical aspect which is truly explained by a mechanism. Now
suppose that this appears to be as much as we can achieve at this time. It
would be a sound policy then to restrict enquiry to the mechanical aspects of
life as if they explained life altogether. And consequently, scientists --- being
primarily concerned with the advancement of science --- may come so firmly
to uphold this fiction that they will regard it as `the scientific view' of life and
condemn anyone challenging this fallacy as an anti-scientific obscurantist.
..."

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Popular publications in philosophy

Personal Knowledge Towards a Post Critical

Philosophy. N.Y.:Harper & Row (Harper Torchbooks ed),
1964.

Science, Faith and Society. Chicago: University of

Chicago Press, 1964.

Science, Economics and Philosophy: Selected

Papers of Michael Polanyi. Edited with an
introduction by R.T. Allen. New Brunswick (USA)
and London: Transaction Publishers, 1997.
Micheal Polanyi
(1891-1976)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Driving force?

Adaptability & flexibility


 /?
Association & Perseverance &
Passion
interaction criticisms

Einstein
Fritz Haber
Eugene Wigner

Curiosity & ingenuity


Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Egon Orowan (1901 – 1989)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Hungarian – as usual

Born on 2nd August 1901

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 On crystal plasticity

“My own introduction to dislocations happened on a hot Saturday afternoon in

1928. Until less than a year before that, I studied electrical engineering; I was
more interested in physics but my father, a mechanical engineer, knew that
one could not make a living from Physics (this was before the Age of
Government Contracts). ….to be read aloud.

and he had to start his experiments !

“Becker‟s theory demanded complete brittleness at very low temperatures.”

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Orowan’s confusion

Polanyi, Meissner and Schmid showed in 1930, before Orowan‟s equipment

and crystals were ready, “that these metals were almost as ductile in liquid air
as at room temperature.” This was odd, “because the papers of Polanyi and
Schmid contained the stereotyped remark that their metal crystals were drawn
from the melt and then broken into pieces of suitable lengths in liquid air. When
I asked Polanyi about this, he replied “Metal crystals broke in liquid air in those
days: today they don’t”.

This confusion however formed the basis of his Diplomarbeit in February

1929 and of his first publication !

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 His understanding….

One Saturday afternoon he had only one zinc crystal….

To his surprise, it extended with sharp jerks instead of flowing

smoothly. From this observation, often repeated, he drew a
surprising amount of information and was “led, almost unavoidably,
to the concept of dislocation.”

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Driving force

Adaptability & flexibility

Association & Perseverance &

Passion
interaction criticisms

Polanyi
G I Taylor
W L Bragg

Curiosity & ingenuity

Orowan joined the Massachusetts Institute of Technology in the summer of 1950.

 G I Taylor (1886 – 1975)

Dept. of MME, IIT Madras

 British – Scientist, mathematician

Born on 7th
March 1886

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Dislocations
 Dislocations are line defects that separate the sheared and
unsheared portions of the crystal.

 Edge and screw dislocations, mixed.

 Movement of dislocation will cause breakage and reformation of

atomic bonds causing plastic deformation.

 Dislocations can move by

 Glide (conservative motion, dV = 0)

 Climb (non-conservative motion, dV ≠ 0)
 Cross-slip

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Dislocation Motion: Glide

Glide is a motion of a dislocation in its own

slip plane.

All kinds of dislocations, edge, screw and

mixed can glide.

Courtesy: Dr. Rajesh Prasad, IIT Delhi

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Glide of
an Edge
Dislocation

Courtesy: Dr. Rajesh Prasad, IIT Delhi

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Glide of crss
an Edge
Dislocation

crss is critical
resolved shear
stress on the slip
plane in the
direction of b.

Courtesy: Dr. Rajesh Prasad, IIT Delhi

crss
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Glide of crss
an Edge
Dislocation

crss is critical
resolved shear
stress on the
slip plane in
the direction
of b.

Courtesy: Dr. Rajesh Prasad, IIT Delhi

crss
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Glide of crss
an Edge
Dislocation

crss is critical
resolved shear
stress on the
slip plane in
the direction
of b.

Courtesy: Dr. Rajesh Prasad, IIT Delhi

crss
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Glide of crss
an Edge
Dislocation

crss is critical
resolved shear
stress on the
slip plane in
the direction
of b.

Courtesy: Dr. Rajesh Prasad, IIT Delhi

crss
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Glide of crss
an Edge
Dislocation

A surface step
of b is created
if a dislocation
sweeps over the Surface step,
entire slip plane not a
dislocation !

Courtesy: Dr. Rajesh Prasad, IIT Delhi

crss
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 1 2 3 4 5 6 7 8 9

Burgers vector
b Slip plane

slip t no slip

boundary = edge dislocation

1 2 3 4 5 6 7 8 9
Dislocation Line:
A dislocation line is the boundary between slip and
no slip regions of a crystal

Burgers vector:
The magnitude and the direction of the slip is
represented by a vector b called the Burgers vector,
HRTEM image
Line vector
A unit vector t tangent to the dislocation
line is called a tangent vector or the line
vector.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

Courtesy: MPI Stuttgart
 Screw
dislocation

b || t

Materials Science
and Engineering,
Callister
Dept. of MME, IIT Madras
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Peierls-Nabarro stress

 Displacement „b‟ is obtained by a localized motion of atoms rather than

simultaneous shear of a perfect plane.

 Displacement associated with the dislocation is „spread‟ across several

atoms as you see in the previous slide (causing dislocation width).

 The stress required to move the dislocation along the slip plane is called the
„Peierls-Nabarro stress‟ and it is very small compared to the theoretical and
experimental strength (analogous to movement of a carpet)

• Dislocation width „ w ‟ is governed by the nature

2 w of atomic bonding and crystal structure.
f exp
b • For eg. in covalent materials, w is very narrow
(w ~ b), hence high frictional stresses.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Dislocation climb: non-conservative motion
 Edge and mixed dislocations have a defined slip plane and hence
can leave the slip plane only by absorption or emission of point
defects (vacancies).

 Dislocations become sources or sinks for vacancies, that changes

the volume of the crystal and hence dislocation climb is referred to
as non-conservative motion.

Courtesy
: PNAS

Dislocation is a
sink for
vacancy !
Absorption of defects ! Change in the volume of the crystal !
Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V
 Dislocation climb: non-conservative motion

Emission of defects !

Dislocation is a Change in the volume of the crystal !

source for
vacancy ! http://www.tf.uni-kiel.de/

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation in Single Crystals

Physics of single-crystal plasticity

 Established by Ewing and Rosenheim(1900), Polanyi (1922), Taylor

and others (1923 - 38), Schmid & Boas (1924 - 35), Bragg (1933)

Mathematical representation

 Initially proposed by Taylor in 1938, Bishop & Hill (1951)

 Further developments by Hill (1966), Kocks (1970), Hill and Rice

(1972) , Asaro and Rice (1977), Hill and Havner (1983)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic flow in single crystals

F • Consider a single crystal of some

orientation subjected to a uniaxial tension
as shown in Figure.
Outward
normal • At room temperature the major source for
plastic deformation is the dislocation motion
through the crystal lattice.

Slip direction • Dislocation motion occurs on fixed crystal

planes (“slip planes”) in fixed
As crystallographic directions (corresponding
Slip plane
to the Burgers vector of the dislocation that
carries the slip).
Ao
• The crystal structure of metals is not
altered by the plastic flow.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic flow in single crystals
F •The resolved shear stress is now the
ratio of the resolved shear force to the
area of the slip plane.
Outward
normal

Slip direction

As Fs F cos F cos cos

Slip plane RSS
As As Ao

Ao
RSS 1
m where, m
cos cos

1/m = Schmid – Boas factor

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic flow in single crystals
F • Plastic flow initiates when the resolved
shear stress reaches a critical value:

Outward Y m CRSS
normal

• where, CRSS is the critical resolved

shear stress, and Y is the yield stress
Slip direction of the material.

As • The value of CRSS depends on

Slip plane temperature, strain rate, initial
dislocation density and purity.
Ao

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Summary of Schmid-Boas Law

 Initial yield stress varies from sample to sample depending on,

among several factors, the position of the crystal lattice relative to
the loading axis.

 It is the shear stress resolved along the slip direction on the slip
plane that initiates plastic deformation.

 Yield will begin on a slip system when the shear stress on this
system first reaches a critical value (critical resolved shear stress,
crss), independent of the tensile stress or any other normal stress on
the lattice plane.

E. Schmid & W. Boas (1935), Kristallplastizität, Berlin: Springer.

E. Schmid & W. Boas (1950), Plasticity of Crystals, Hughes & Co., London.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Experimental evidence for Schmid’s law

“Soft orientation”,
with slip plane at
45°to tensile axis

The experimental evidence of

Schmid‟s Law is that there is a
critical resolved shear stress.

This is verified by measuring

“Hard orientation”, the yield stress of Mg single
crystals as a function of
with slip plane at
orientation.
~90°to tensile axis
Mg is hexagonal and slips
most readily on the basal plane
(all other crss are much larger).

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Influence of temperature & strain rate

 The figure exemplifies three regions of

CRSS as a function of temperature and strain
rate.
 The first and the third stages illustrating a
1 2 strong temperature dependence and strain
shear stress ( CRSS)

rate dependence of CRSS.

Critical resolved

 The CRSS can be expressed as

* *
CRSS a

I a II III a = athermal
component of stress
(temperature independent term)

* = thermal component of stress

0.25 Tm 0.7 Tm

Temperature (T)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Influence of temperature & strain rate
 The origins of both the temperature-
dependent and temperature-independent
components of stresses are directly related to
the microstructural features of the material –
2
specifically related to “ stresses required to
overcome the obstacles ”.
1
shear stress ( CRSS)
Critical resolved

*
Obstacles

I a II III

Short range Long range

0.25 Tm 0.7 Tm
(Thermal (Athermal
obstacles) obstacles)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Influence of temperature & strain rate
*
CRSS a

 Short range obstacles  Long range obstacles

(SROs) (LROs)
- 10 atomic spacings or less (field of
influence). - field of influence is more than 10 atomic
- Less energy required to overcome. spacings.
- Peierls-Nabarro energy barrier is a short - Very large energy is required to overcome
range obstacle. these obstacles.
-Randomly dispersed solute atoms act as - long range stress fields of other
short range obstacles. dislocations act as LROs.
- Small precipitates also act as short - Large precipitates act as LROs.
range obstacles. - these obstacles cannot be thermally
- these obstacles can be thermally activated.
activated.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Influence of temperature, solute additions &
bonding on CRSS – for various materials

Observe the following:

 Resistance to plastic deformation is

CRSS (MPa)

relatively low in FCC metals in comparison

to BCC transition metals.
 Increase in CRSS as a result of solute
additions/impurity atoms.
 Increase in CRSS of
covalently bonded
solids in comparison to ionic solids.
 Increase in CRSS at higher strain rates
– see NaCl.
T/Tm

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

Shear stress – shear strain curve for a single crystal
FCC crystals

Onset of multiple slip

Role of stacking
Stage - I Stage - II Stage - III fault energy

Linear
Easy glide hardening
region Decreasing
Shear stress

hardening
Single rate
slip
Max. work Exhausion
hardening hardening
Low
hardening
rate

Shear strain

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Tensile stress-strain behavior of Cu single crystals
 The critical resolved shear stress
Diehl, Jorg , Z. Metallkunde, 47, (1956)
( CRSS) is the same for all crystal
Multiple slip orientations.

[111]  Significant work hardening in other

[112] orientations – [111] crystal has the most
60
active slip system.
Stress
(MPa)

Single slip

[123]

Easy glide

Certain crystal orientations promote slip in

0.02 0.04 0.06 0.08 0.1 2 or more slip systems at the onset of
Strain plastic deformation

Crystal orientation dependence on multiple slip

 Constraints on sample deformation – Tensile
testing
Multiple slip due to testing constraints

(a) Tensile deformation of a single

crystal without constraints.
(b) Rotation of slip planes due to
constraints
Slip
planes
Given an orientation for single slip,
i.e. the resolved shear stress
reaches the critical value on one
system ahead of all others, then
one obtains a “pack-of-cards”
straining (as seen in figure on the
(a) (b) left).

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Rotation of the crystal lattice

n – slip plane normal.

s – slip direction.

o – initial orientation of
the slip plane with the
tensile axis.
- orientation of the slip
plane with the tensile axis
after rotation.

The slip direction rotates towards the tensile axis and the reverse happens
during compressive loading.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation of bicrystals
cos cos ?
Grain
boundary
y  The following conditions must be met at the
grain boundary for material continuity, failing
which voids/cracks appear at the grain
boundary ( requirement of cooperative
displacements at the boundary).
x
z 1 11
I II yy yy
1 11
zz zz
1 11
yz yz

where, yy and zz are the normal strains along y

and z- axes respectively, and yz the shear strain
along the x-plane (yz plane).

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation of bicrystals
In the literature, the deformation of bicrystals of different materials has been studied extensively.
Detailed investigations have been carried out, for instance,

on tin,
B. Chalmers. Proc R Soc A162 (1937), p. 120.

on zinc,
T. Kawada. J Phy Soc Jpn 6 (1951), p. 362.
J.J. Gilman. Acta Metall 1 (1953), p. 426.

on magnesium,
J.D. Mote and J.E. Dorn. Trans AIME 218 (1960), p. 491.

on copper,
C. Rey and A. Zaoui. Acta Metall 30 (1982), p. 523.

on aluminum,
K.T. Aust and N.K. Chen. Acta Metall 2 (1954), p. 632.
R. Clark and B. Chalmers. Acta Metall 2 (1954), p. 80.
J.D. Livingston and B. Chalmers. Acta Metall 5 (1957), p. 322.
J. D. Livingston and B. Chalmers. Acta Metall 6(1958),p. 216
R.L. Fleischer and B. Chalmers. Trans AIME 212 (1958), p. 265.
S. Miura and Y. Saeki. Acta Metall 26 (1978), p. 93.
C. Rey and A. Zaoui. Acta Metall 28 (1980), p. 687.
S. Sun, B.L. Adams and W. King. Phil Mag A 80 (2000), p. 9.
P. Yu and K.S. Havner. J Mech Phy Sol 49 (2001), p. 173.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation of bicrystals
 Livingston and Chalmers, pioneers in this field, concluded from their work on aluminum
bicrystals that, in order to keep the macroscopic plastic compatibility across the grain boundary,
bicrystals have to deform by multiple slip.
J.D. Livingston and B. Chalmers. Acta Metall 5 (1957), p. 322.

 Hook and Hirth suggested in their studies on Fe–3%Si bicrystals that the elastic incompatibility at
the grain boundaries results in the activation of secondary slip systems. These investigations built
their discussions essentially on intergranular incompatibility assuming otherwise homogeneous
behavior of the two abutting crystals,
R.E. Hook and J.P. Hirth. Acta Metall 15 (1967), p. 535.
R.E. Hook and J.P. Hirth. Acta Metall 15 (1967), p. 1099.

 Recent detailed studies by Rey and Zaoui on Cu bicrystals, revealed that the presence of grain
boundaries can also give rise to considerable intragranular heterogeneity. Rey and Zaoui found
that intragranular nonhomogeneity leads to internal stresses inside each of the two neighboring
crystals entailing activation of additional slip systems and considerable corresponding hardening
effects.
C. Rey and A. Zaoui. Acta Metall 28 (1980), p. 687.
C. Rey and A. Zaoui. Acta Metall 30 (1982), p. 523.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Room temperature tensile stress-strain curves for a single crystal,
bicrystal and a polycrystal

polycrystal What should be the condition for a

polycrystalline deformation ?
Load (lb)

Load (N)
bicrystal

Five independent strains

Single Niobium
crystal

polycrystal Five independent slip systems

Stress (MPa)

bicrystal Operation of five independent slip

Single
crystal systems in the vicinity of the grain
boundaries to maintain grain boundary
NaCl compatibility !
Strain
Gradient in dislocation density !

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 What is an independent slip system ?

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

Eg: Basal plane in a hexagonal close packed crystal
(HCP)
• Basal plane is a close packed plane and
a1 and a2 and a3 are close packed
directions (which constitute the slip
system) in which slip occurs.
a3

a2
• Number of geometrical slip systems is 3.
a1

• Number of independent slip systems is

only 2.

• Number of independent slip systems will

be always less than the geometrical slip
systems.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation of polycrystals

Plastic deformation of a polycrystal differs from that of

a single crystals in two respects:

(i) Grain boundaries act as obstacles to dislocation movement.

(ii) Individual grains in the polycrystal will possess a wide variety of

orientations.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Statistically stored & geometrically necessary
dislocations
 When a polycrystalline material is deformed, both statistically-stored and
geometrically-necessary dislocations are required to accommodate the deformation.
Hence the total dislocation density at a given strain, T, can be written as:
T = l + gb
where, l = density of statistically-stored dislocations associated with deformation of the
lattice - associated with dislocation multiplication and accumulation with plastic
strain.
gb = density of geometrically-necessary dislocations associated with
maintaining grain boundary compatibility during plastic deformation – associated with
plastic strain gradient in the crystal.

According to M.F. Ashby, the deformation of polycrystals can be divided into two parts:

a general, uniform deformation during which the statistically-stored dislocations accumulate

+
a local, non-uniform deformation during which the geometrically-necessary dislocations accumulate

M.F. Ashby, Phil. Mag.,

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V 21, 399 (1970)
 Statistically stored & geometrically necessary
dislocations
 The statistically-stored dislocation density ( l) is a characteristic of the properties of
the material; for instance: crystal structure, shear modulus, stacking fault energy.

 Geometrically-necessary dislocation density ( gb) is a characteristic of the particular

microstructure, for instance: the grain boundaries, second phase particles.

 Although statistically-stored and geometrically-necessary dislocations can interact

with each other, to a first approximation, the geometrically-necessary dislocation
density can be considered independent of the material.

 Thus, the total dislocation density is simply the sum of the statistically-stored and
geometrically-necessary dislocation densities. In bulk metals and their alloys, the
contributions from statistically stored dislocations to strain accumulations are much
greater than those due to geometrically necessary dislocations.

M.F. Ashby, Phil. Mag., 21, 399 (1970)

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Plastic deformation of polycrystals
Generation of geometrically necessary dislocations
during tensile deformation of a polycrystal.

 Part - II: Grains have been deformed according to

their orientations and Schmid factors as if they were
independent.
 Part - III: If the grains did not cohere then open-or-
doubly occupied wedges would appear during
I II deformation. The introduction of suitable dislocations
in part III allows restoration of material continuity
(Part IV).

It is plausible to place the geometrically necessary

dislocations at the grain boundaries where the
incompatibility stresses are higher and the
III IV statistically stored dislocations (not shown in fig) will
harden the crystals depending on their orientations.

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

To learn more about geometrically necessary
dislocations…..

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V

 Concept of geometrically necessary dislocations
(GNDs)
• What is a geometrically necessary dislocation?

• What does the concept of geometrically necessary dislocations tell us that we

would not otherwise know?

• Can geometrically necessary dislocations be identified experimentally, i.e.

what is testable about geometrically necessary dislocations?

Example-1
Example-2
Please refer to: Huajian Gao et al,Scripta Materialia, 48, (2003), 113-118

Dept. of MME, IIT Madras Dr. rer. nat. Ravi Kumar, N. V