materials hall petch equaltion

1. Hall Petch Effect in Bulk
Nanostructured Materials
BY
Mahfooz Alam
M.Tech., Materials Engineering
17ETMM10

2. Contents
• Introduction
• Synthesis of nanostructured materials
• Hall-Petch studies in different materials
• Models for grain-size strengthening
• The Hall-Petch breakdown
• Conclusions
• Acknowledgement
• References
2

3. Introduction
• Bulk nanostructured materials are defined as bulk
solids with nanoscale or partly nanoscale
microstructures.
• Early in the century, when “microstructures” were
revealed primarily with the optical microscope, it was
recognized that refined microstructures, for example,
small grain sizes, often provided attractive properties
such as increased strength and toughness in structural
materials.
• Other properties of nanocrystalline materials, apart
from increased strength and hardness, include higher
electrical resistance, increased specific heat capacity,
3

4. Synthesis of Nanostructured
Materials
• Solid-state processing
Solid-state processes do not involve melting or
evaporating the material and are typically done at
relatively low temperatures. Examples of solid
state processes include mechanical alloying
using a high-energy ball mill and certain types
of severe plastic deformation processes.
• Liquid processing
Nanocrystalline metals can be produced by
rapid solidification from the liquid using a4

5. Ref. Nanostructured materials: basic concepts and microstructure H Gleiter
Fig 1. classification of nanostructured materials. The boundary regions of the 1st and 2nd family of NsM are
indicated in black to emphasize the different atomic arrangements in the crystallites and in the boundaries.
The chemical composition of the boundary regions and the crystallites is identical in the 1st family. In the 2nd
family, the boundaries are the regions where two crystals of different chemical composition are joined
together causing a steep concentration gradient.
5

6. The Hall–Petch Studies in
Different Materials
In the early 1950’s, Hall and Petch empirically
demonstrated that the initial yield point(σy) of
low carbon steels, was related to their grain size
(D) according to the now well-known Hall–Petch
relationship:
σy = σ0 +kD-1/2 (1)
where σ0 and k are chemistry and microstructure
dependent constants respectively.
6

7. Continue…
The summary of the BCC
transition metals’ grain-
size strengthening data is
shown in Fig. 2(a–g). In
each of these figures, plots
are results from the
various Hall–Petch studies
on the different metals as
well as best fits to the
entirety of the data using
Hall-Petch equation.
Ref. On the Hall-Petch relationship and substructural evolution in type 316L stainless steel
Author links open overlay panelB.P.KashyapK.Tangri
Hall-Petch plot of 316 L SS at 400°C different
strains range 0.002 to 0.34
7

8. Fig 2.Aggregated Hall–Petch data for each of the BCC metals as well as best fits to the data using
Hall-Petch equation . The closed points indicate Vickers and nanoindentation hardness
measurements that were divided by a Tabor factor of 3 while the open points indicate yield
strengths measured using compression or tension tests.
Ref. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals Z. C.
Cordero, B. E. Knight & C. A. Schuh
Continue…
8

9. Fig 3.Aggregated Hall–Petch data for each of the BCC metals as well as best fits to the
data using Hall-Petch equation . The closed points indicate Vickers and nanoindentation
hardness measurements that were divided by a Tabor factor of 3 while the open points
indicate yield strengths measured using compression or tension tests.
Ref.Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure
metals Z. C. Cordero, B. E. Knight & C. A. Schuh
Continue…
9

10. Fig 4.Aggregated Hall–Petch data for each of the FCC
metals as well as best fits to the data using Hall-Petch
equation . The closed points indicate Vickers and
nanoindentation hardness measurements that were
divided by a Tabor factor of 3 while the open points
indicate yield strengths measured using compression or
tension tests.
Ref. Six decades of the Hall–Petch effect – a survey of grain-size
strengthening studies on pure metals Z. C. Cordero, B. E. Knight & C. A.
Schuh
Continue…
10

11. Fig 5.Aggregated Hall–Petch data for each of the HCP metals as well as best fits to the data
using Hall-Petch equation . The closed points indicate Vickers and nanoindentation hardness
measurements that were divided by a Tabor factor of 3 while the open points indicate yield
strengths measured using compression or tension tests.
Ref. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals Z.
C. Cordero, B. E. Knight & C. A. Schuh
Continue…
11

12. Fig 6.Aggregated Hall–Petch data for each of the HCP metals as well as best fits to the data
using Hall-Petch equation . The closed points indicate Vickers and nanoindentation hardness
measurements that were divided by a Tabor factor of 3 while the open points indicate yield
strengths measured using compression or tension tests.
Ref. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals Z.
C. Cordero, B. E. Knight & C. A. Schuh
Continue…
12

13. Ref. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals
Z. C. Cordero, B. E. Knight & C. A. Schuh
13

14. Models for Grain-size
Strengthening
Most of these models (shown in Table 2) treat
the two terms on the right hand side of equation
σy = σ0 +kD-1/2 separately:
The physics associated with grain-size
strengthening is assumed to be encoded in k
and the grain-size exponent in the second term
 while σ0 is taken to account for all
strengthening effects unrelated to the grain
size.
14

15. Hall’s explanation of the grain-size strengthening term in equation
σy = σ0 +kD-1/2 relied on the concept of a dislocation pile-up against
a grain boundary.
According to his model, dislocations pile-up at grain boundaries in
one grain and macroscopic yielding occurs when dislocations are
emitted into the adjacent grain.
This is possible when the sum of the external stress and the stress
at the head of the dislocation pile-up is larger than some threshold
stress. The strength depends on grain size because the total pile-up
length is limited by the grain size, which therefore limits the stress
at the head of the pile-up.
As a result, the pile-up model predicts a linear relationship between
the yield stress and the reciprocal square root of the grain size.
Continue…
15

16.  There are many variations on this same basic model
that are summarized in a review by Li and Chou, but
they all give a grain-size strengthening equation of
the same basic form
σy = σ0 + βGb1/2 D-1/2
(2)
 where b is the magnitude of the Burgers vector,
 β is a model dependent constant and depends on
crystal structure (being on average 0.18 for FCC, 0.42
for BCC and 0.9 for HCP metals).
 The pre-factor βGb1/2 is equivalent to k from
equation (1); the values of k/(Gb1/2) given in Table 1
Continue…
16

17. Ref. Six decades of the Hall–Petch effect – a survey of grain-size strengthening studies on pure metals
Z. C. Cordero, B. E. Knight & C. A. Schuh 17

18. Despite these successes, the pile-up model
nonetheless suffers from several major
deficiencies, the most important of which is the
lack of direct evidence relating the dislocation
pile-up length to the grain size.
 In addition, the pile-up mechanism cannot
account for k’s sensitivity to the average grain
boundary structure and chemistry, which was
demonstrated by Floreen and Westbrook in
experiments on sulphur-doped nickel.
These issues with the pile-up model motivated
the development of an alternative class of work-
Continue…
18

19. Ashby’s theory of grain-size strengthening is another work-hardening model, but it
differs from the grain boundary ledge model in that it tries to explain equation (1) by
reconciling two sets of observations instead of speculating about a specific mechanism.
 The first set of observations is that ρ increases linearly with plastic strain and
inverse grain size during the uniform deformation of polycrystalline iron,
vanadium, titanium, niobium and aluminium.
 The second set of observations is that even when a polycrystalline specimen is
subjected to macroscopically uniform plastic deformation, the individual grains still
exhibit non-homogeneous plastic flow. This non-uniform plastic flow is due to
compatibility constraints at grain boundaries, which result in the activation of
additional slip systems and the generation of excess dislocations near grain
boundaries.
Ashby showed how his model could explain the Hall–Petch equation by combining his
predictions about the strain and grain-size dependence of the dislocation density with
the Taylor hardening equation. Assuming for simplicity that the two groups of
dislocations do not interact, the total dislocation density is ρ = ρGN + ρSS. Inserting
this cumulative dislocation density into equation gives
σ= σ0 + αGb (ρGN + ρSS) (4)
•
Continue…
19

20. All of the models described thus far predict a
reciprocal square root dependence on grain size, in
agreement with the classic Hall–Petch equation.
However, several investigators have suggested
that the grain-size exponent should be something
other than −1/2. For example, Christman fitted
experimental grain-size strengthening data using a
power law of the form σ= σ0 + kDn where n, σ0
and k were all fitting parameters, and found that n
was −1/3 for the FCC metals and varied between
−1/2 and −0.9 for the BCC metals. Christman did
not provide a physical explanation for these
Continue…
20

21. The Hall–Petch Breakdown
Studies have shown that while there can be
extensive dislocation activity in samples with
grain sizes as small as 10 nm, dislocations do
not accumulate in nanocrystalline metals as
they do in coarse-grained samples.
Instead, in nanocrystalline metals, dislocations
emitted from one grain boundary are often
rapidly absorbed by other grain boundaries
after they traverse the grain interior. 21

22. Conclusions
 The Hall-Petch coefficient depends on the crystal structure of the materials, the
similarities between the frictional stress and single crystal flow stress and several
other trends.
 Some of the trends are accounted for in the dislocation pile up model of grain
size strengthening.
 But there is no direct evidence to support this model’s underlying assumption that
the grain size limits the size of dislocation pile-ups.
 Though the Hall–Petch equation has been investigated for over half a century –
and grain-size strengthening for even longer – there are still many exciting and
relatively unexplored avenues for future research.
 Work on these topics will benefit from recent developments in imaging,
mechanical characterization and processing
22

23. Acknowledgement
I would like to thank all the faculty of SEST for their
excellent classroom teaching.
I would like to thank dean of SEST to provide me the
facility that I have availed.
I am thankful to my classmates for their support in sharing
their knowledge.
 I would like to AICTE to provide me the GATE
scholarship that I have availed.
23

24. References
1. E. O. Hall: ‘The deformation and ageing of
mild steel: III discussion of results’, Proc. Phys.
Soc., Sect. B, 1951, 64, (9), 747–755.
2. N. J. Petch: ‘The cleavage strength of
polycrystals’, J. Iron Steel Inst., 1953, 174, (1),
25–28.
3. R. W. Armstrong: ‘The influence of polycrystal
grain size on several mechanical properties of
materials’, Metall. Mater. Trans., 1970, 1, (5),
1169–1176 24

25. References
12. T. G. Lindley and R. E. Smallman: ‘The plastic
deformation of polycrystalline vanadium at low
temperatures’, Acta Metall., 1963, 11, (5), 361–
371.
13. M. A. Sherman and R. F. Bunshah: ‘Yield
strength of fine-grained vanadium produced by
high-rate physical vapor deposition’, J. Nucl.
Mater., 1975, 57, (2), 151–154.
14. A. F. Jankowski, J. Go and J. P. Hayes:
‘Thermal stability and mechanical behavior of 25

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