condensed matter physics and solid stete physics ..crystal
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Module #20 Module #20 Module #20 Module #20
Physical Basis for Deformation II Physical Basis for Deformation II
Influence of Temperature
SUGGESTED READING*
• DIETER: Ch. 4, pp. 124132, 135137, and 139144
• HERTZBERG: Ch. 3
• COURTNEY: Ch 4 • COURTNEY: Ch. 4
*Thi li d h d d ll f h h I h b bl d
Prof. M.L. Weaver
*This list does not mean that you need to read all of these chapters. It has been assembled to
provide you with suggested reading from that you may be using OR referring to in your course.
Most of these chapters cover similar material. Any “required” reading will be noted separately.
Critical Resolved Shear Stress
cos cos
Schmid Factor
CRSS YS YS
m t o  ì o = =
F
A
o
ì

• t
CRSS
is the resolved shear
stress needed to induce slip
Slip plane
_
via dislocation motion.
Th ti li t h
A
s
p p
normal
Slip
direction
s
A
• The active slip system has
the largest Schmid factor.
F
Prof. M.L. Weaver
What factors influence What factors influence tt
CRSS CRSS
or or oo
ys ys
??
►Temperature
– Increase T ÷decrease t
CRSS
.
►Strain rate
D d
– Decrease c ÷decrease t
CRSS
.
• Defect content
c
• Defect content
– Impurity level
• Decrease impurity content ÷decrease t
CRSS
. ec ease pu ty co te t ÷dec ease t
CRSS
– Dislocation density
Prof. M.L. Weaver
• Decrease µ
±
÷decrease t
CRSS
.
tt
CRSS CRSS
as a function of T and as a function of T and cc
··
2
c =
1 2
c c = >
CRSS
t
*
Figure 4.2:
Schematic illustrating the variation of
I II III
*
t
µ
t
τ
CRSS
with temperature and strain rate.
[Adapted from Courtney, p. 143]
~0.7 T
mp
~0.25 T
mp
T
µ
In general, single crystals exhibit 3stage temperature
and strain rate dependence of t
CRSS
Not all crystals exhibit 3 stages;
d d t l t t
Prof. M.L. Weaver
depends on crystal structure.
Seeger, 1954 Seeger, 1954
In general, the flow stresses in crystals can be
separated into two components:
1. Athermal component t
μ
which is proportional to the
shear modulus (μ);
2. Thermal component t
*
that is more sensitive to
temperature and strain rate.
*
CRSS µ
t t t = +
t
μ
is the athermal component
of flow stress (t
μ
= F(T))
t* is the thermally dependent
component of flow stress
( * + as T )
Prof. M.L. Weaver
(t* + as T )
The components of The components of tt
internal internal
• t
μ
(athermal)
– Arises from the stress to move dislocations in the presence of
longrange internal stress fields long range internal stress fields.
– Longrange stress fields exist over large distances in comparison
to atomic dimensions. Examples of such barriers include the
stress fields of other dislocations. stress fields of other dislocations.
• t
*
(thermally dependent)
– Represents resistance to dislocation motion due to “shortrange
barriers” (i.e., atomic scale barriers). These barriers are so small
that temperature induced thermal vibrations can affect the Peierls
t (i l tti f i ti ) E l f h b i i l d stress (i.e., lattice friction). Examples of such barriers include
dislocation kinks and segregated/interacting impurity atoms.
– This component decreases as T increases and goes to zero at
t t h h t b t l b i ff ti i
Prof. M.L. Weaver
temperatures where shortrange obstacles become ineffective in
restriction dislocations.
Thermally activated flow Thermally activated flow
• Very important in creep Can play a significant role in deformation • Very important in creep. Can play a significant role in deformation
above 0 K. Can reduce stress necessary to overcome obstacles.
C id ff ti t hi h d i di l ti ti Th • Consider an effective stress which drives dislocation motion. The
effective stress, t
eff
, is:
ff l d l
t t t = ÷
• The internal stress, t
internal
, is the stress that resists dislocation motion.
i t f t t
eff applied internal
t t t
• t
internal
consists of two terms:
– (1) a resistance to longrange obstacles (>10 atom diameters) that cannot
be overcome by thermal fluctuation, t
μ
;
– (2) a resistance to shortrange obstacles (~10 atom diameters) that can be
overcome by thermal fluctuations, t
*
.
*
internal µ
t t t = +
Prof. M.L. Weaver
( )
*
internal
eff applied
µ
µ
t t t t
= ÷ +
For thermal fluctuations to assist
dislocation motion, enough energy
Total stress required
to deform the
i l b li
(i.e., ΔH) must be supplied to
overcome the shortrange stress fields.
material by slip
t
applied
t

t
o
applied
t
µ
0
+
x {hkl}
Internal
resistance 0

resistance
to flow
ì
b ì ·
Prof. M.L. Weaver
Figure 820 Schematic representation of superimposed longrange and shortrange
stress fields. (Figure adapted from H. Conrad, Materials Science and Engineering, v. 6
(1970) pp. 265273 and Dieter, p. 312).
Overcoming short Overcoming shortrange barriers range barriers
• Apply a stress t > t
a
. The force on this dislocation is:
* *
F bL
where L
*
is the length of dislocation segment involved
in thermal fluctuation
F bL t =
in thermal fluctuation.
• Energy (ΔH) must be supplied to overcome the
superimposed shortrange barriers superimposed shortrange barriers.
• The work done by the applied stress during thermal
activation is: activation is:
( )
* *
o o
W x x b t = ÷
Prof. M.L. Weaver
Short Shortrange barriers range barriers – – cont’d cont’d
• The energy, ΔH, is the area beneath the force
distance curve between x
o
*
and x
o
. It is designated
as:
( )
*
* * * * *
o
x
H F x bd dx H v t t ( A = ÷ = A ÷
¸ ¸ }
where ΔH
*
is the activation energy at zero applied
( )
o
x
H F x bd dx H v t t ( A A
¸ ¸ }
stress.
• v
*
is known as the activation volume. It represents
the average area swept out by the dislocation during
some thermally activated event
Prof. M.L. Weaver
some thermally activated event.
Activation Volume Activation Volume
• The activation volume v
*
= L
*
bd
*
= A
*
b The activation volume v L bd A b.
A
*
is known as the activation area.
• The activation volume is important because it can be used • The activation volume is important because it can be used
to identify the mechanism controlling the thermally activated
process.
• This is done by comparing the values of ΔH and v
*
with
values predicted from specific dislocation models.
Climb mechanism v
*
= 1 b
3
PeierlsNabarro mechanism v
*
= 10 – 10
2
b
3
Cross slip mechanism v
*
= 10 – 10
2
b
3
± Intersection mechanism v
*
= 10
2
– 10
4
b
3
Prof. M.L. Weaver
Nonconservative motion of jogs v
*
= 10
2
– 10
4
b
3
tt
CRSS CRSS
as a function of T and as a function of T and cc
·
*
CRSS µ
t t t = +
[1]
2
c =
1 2
c c = >
CRSS
t
*
Figure 4.2:
Schematic illustrating the variation of
I II III
*
t
µ
t
τ
CRSS
with temperature and strain rate.
[Adapted from Courtney, p. 143].
~0.7 T
mp
~0.25 T
mp
µ
T
• Region I (T s 0.25T
mp
)
– t
CRSS
 with + T.
– t
CRSS
 with c.
– Athermal component of flow stress is large. Difficult for
dislocations to surmount short range barriers
c
Prof. M.L. Weaver
dislocations to surmount shortrange barriers.
tt
CRSS CRSS
as a function of T and as a function of T and cc
·
*
CRSS µ
t t t = +
[2]
2
c =
1 2
c c = >
CRSS
t
*
Figure 4.2:
Schematic illustrating the variation of
I II III
*
t
µ
t
τ
CRSS
with temperature and strain rate.
[Adapted from Courtney, p. 143].
~0.7 T
mp
~0.25 T
mp T
µ
• Region III (T > 0.7T
mp
)
– In this temperature range, diffusive processes become
important. Diffusion aids dislocation motion (i.e., makes it important. Diffusion aids dislocation motion (i.e., makes it
easier for dislocations to surmount barriers to their motion).
– Both t
μ
& t
*
+ as T .
+ 
Prof. M.L. Weaver
– t
CRSS
+ with  T.
– t
CRSS
 with  c. c
tt
CRSS CRSS
as a function of T and as a function of T and cc
·
*
CRSS µ
t t t = +
[3]
Figure 4.2:
Schematic illustrating the variation of
2
c =
1 2
c c = >
CRSS
t
*
τ
CRSS
with temperature and strain rate.
[Adapted from Courtney, p. 143].
I II III
*
t
µ
t
T
µ
~0.7 T
mp
~0.25 T
mp
• Region II (0.25T
mp
< T < 0.7T
mp
)
– In this temperature range, t
μ
= F(T) & t
*
~ 0; t
CRSS
=
t t constant.
– Temperature is too low for diffusion to influence permanent
deformation, thus t ~ t
μ
= F(T).
Prof. M.L. Weaver
– Does crystal structure play a role?
Full caption from Figure 4.2 Full caption from Figure 4.2
Figure 4.2:
S h i ill i h i i f i h d Schematic illustrating the variation of τ
CRSS
with temperature and
strain rate. At high temperatures (Region III), τ
CRSS
is a strong
function of both variables. At intermediate temperatures τ
CRSS
is p
CRSS
independent of strain rate and temperature and is given by τ
μ
. At
lower temperatures, τ
CRSS
again increases with decreasing
temperature and increasing strain rate Thus for example at temperature and increasing strain rate. Thus, for example, at
temperature T
1
, τ
CRSS
can be considered to be the sum of the
athermal stress τ
μ
and a thermally dependent stress τ*. At the
t iti f R i I t R i II * ff ti l b transition from Region I to Region II, τ* effectively becomes
zero. [Adapted from Courtney, p. 143].
Prof. M.L. Weaver
KEY POINTS:
Temperature variation of Temperature variation of tt
CRSS CRSS
for various materials for various materials
KEY POINTS:
• FCC metals have low resistance to
plastic deformation (i.e., they are
weaker)
[Covalent
bonding]
weaker)
• BCC metals have much higher
resistances to plastic deformation
(i e they are generally stronger)
bonding]
[Metallic
bonding]
(i.e., they are generally stronger)
• T dependence of t
CRSS
:
–BCC – high; FCC – lower BCC high; FCC lower
• Impurities  t
CRSS
, sometimes greatly
(principle behind solid solution
hardening)
[Ionic
bonding]
hardening)
• Ionically bonded materials have low
t
CRSS
(Ex., NaCl, CsCl, etc.)
Figure 4.3:
Prof. M.L. Weaver
• Covalently bonded materials have
high t
CRSS
(Ex., TiC, diamond, etc.)
The temperature variation of τ
CRSS
for materials
with different structures and bonding
characteristics. [Adapted from Courtney, p. 144]
RECALL
TaylorOrowan relation
• Dislocation velocity
bv ¸ µ
±
=
y
also varies from
material to material.
• Could there also be
some relationship to some relationship to
crystal structure, elastic
properties, and/or
melting temperature? melting temperature?
• Think about it!
Figure from D. Hull and D.J. Bacon, Introduction
to Dislocations, 4
th
Edition, (Butterworth
Heinemann, Oxford, 2001) p. 51; originally
Prof. M.L. Weaver
Think about it!
Heinemann, Oxford, 2001) p. 51; originally
adapted from Haasen, Physical Metallurgy, 3
rd
Edition, (Cambridge University Press,
Cambridge, 1996) p. 283.
RECALL
TaylorOrowan relation
THERE ARE
bv ¸ µ
±
=
THERE ARE
TEMPERATURE EFFECTS
Yield stress (and CRSS) in
creases as T decreases.
Figure
m
78K
~ 44
m
298K
~ 35
Stress dependence of the velocity of edge
dislocations in 3.25% silicon iron at four
temperatures (after Stein and Low, J. Appl.
Phys. 31, 362, 1960). Scanned from E.W.
Prof. M.L. Weaver
m
298K
35
Billington and A. Tate, The Physics of
Deformation and Flow, McGrawHill, New
York, 1981, pages 418 and 420.
Thermal activation theory has
been used, in part, to explain
the decrease in strength
observed above 0.5T
mp
.
T
mp,Si
= 1414°C
T
mp,Ge
= 938°C
Prof. M.L. Weaver
Temperature dependence of hardness for Ge and Si. Originally from Trefilov and Mil’man,
Sov. Phys. Dokl., 8 (1964) 1240. Scanned from J. Gilman, Electronic Basis of the
Strength of Materials, (Cambridge University Press, Cambridge, 2003) p. 232.