Mechanical properties of
solid bulk materials and
thin films
Prof. Dr. Frank Richter
Institut für Physik
TU Chemnitz
[email protected]
tel. +371-531-38046
A Lecture Series for the Teaching Programme of the
International Research Training Group
“Materials and Concepts for Advanced Interconnects”
August 2010
2
Chapter 3. Inelastic Behaviour
1.
Introductory Remarks
1.1.
1.2.
1.3.
Importance of mechanical properties
Empirical: Stress-strain curves / HOOKE´s law
Hardness
2.
2.1.
2.1.1.
2.1.2.
2.2.
2.3.
Elastic Behaviour
Stress, strain and elastic moduli
Elongation and compression
Shear deformation
Interatomic forces and mechanical properties
Anisotropy of elastic behaviour
3.
3.1.
3.2.
3.3.
3.4.
3.4.1
3.4.2.
Inelastic Behaviour
Overview
Some relevant properties of the stress tensor
Failure criteria
Fracture
Ductile fracture
Brittle fracture
3.5.
3.5.1.
3.5.2.
3.5.3.
3.6.
Plastic deformation
Basic mechanisms
Dislocation interactions and hardening
Creep
Phase transformation and other mechanisms
4.
4.1.
4.2.
4.3.
4.4.
Mechanics of thin films
Introduction
The biaxial stress state
Manifestations of film stresses
Sources of film stresses
5.
5.1.
5.2.
5.3.
5.4.
Determination of mechanical properties: Overview
Macroscopic mechanical methods
Dynamic methods
Measurement of strain and intrinsic stresses
Indentation techniques
6.
6.1.
6.2.
6.3.
Determination of mechanical properties by nanoindentation
State-of-the-art indentation technique
The Image Load method
Examples
3
3.
Inelastic Behaviour
3.1. Overview1
In the following we deal with failure of materials. Thereby, we restrict ourselves to
such loading conditions how they appear in our practice, e.g. during a mechanical
testing procedure. We do not deal wit “exotic deformation”, for instance using high
deformation velocity.
A material may be subject to complex loading including
- compression/tension,
- shear,
- bending,
- torsion,
- (hydrostatic) pressure,
or - which is often the case - a combinations of them.
If the load is sufficiently severe the material may undergo failure, i.e. it may
- fracture,
- experience plastic deformation (yield), or
- undergo phase transformation.
Each failure mechanism is connected with a certain "picture" of the stress state in
the material. For instance
-
tensile stress

fracture,
-
hydrostatic pressure

phase transformation,
-
shear stress

plastic deformation.
However this “picture” is not always that simple since usually several stress components exist together and a given material can undergo different failure modes,
depending on the existing complex stress state as well as on the particular structure
of the material (i.e. existing defects and their spatial distribution).
In the following we will quantify the certain "picture" of the stress state mentioned
above by defining failure criteria, i.e. finding
certain stress components or
combinations of stress components
which quantitatively characterise whether a certain material shows a particular failure or not.
Before we can do so, we have to deal with the properties of the stress tensor in
more detail.
1
(sketches on the blackboard)
4
3.2. Some relevant properties of the stress tensor2
We have seen above (section 2.3.): At any point of a stressed material exist three
principal axes which are oriented perpendicular to each other. In the planes perpendicular to the principal axes only normal stresses 1, 2 and 3 exist (so called
principal stresses). Shear stresses are zero in those planes.
In other words: In the particular CARTHESIAN co-ordinate system formed by the
principal axes the stress tensor has diagonal form:
The three principal stresses include the extreme normal stress values, i.e. the maximum and minimum normal stress values. Without affecting the general validity we
can assume that
σ1 ≤ σ2 ≤ σ3 .
(1)
Analysis shows that similarly three principal shear stresses exist in any point of a
material which also include the extreme values:
1 ≤ 2 ≤ 3
(2)
The principal shear stresses act on planes which are rotated about the principal axes
by 45° with respect to the planes of the “normal stress cube”.
Figure (after [C]): a) The cube formed by the planes which carry the principal
stresses (“normal stress cube”) and b) – as an example - the principal shear stress 1
acting on planes corresponding to a 45° rotation about the principal axis (1):
2
Note: Complete derivation is given at the blackboard as well as in provisional form in
Appendix 3.
5
The magnitude of the principal shear stress vectors depends on the principal stress
magnitudes as follows:
(3)
From eq. (1) it follows that the maximum of the principal shear stress values is
max = ½ (σ3 – σ1) = |2|
The 12 planes carrying the principal
shear stresses form a rhombic dodecahedron:
(4)
...which should not be intermixed with
the (pentagon) dodecahedron, one of
the PLATONic solids:
While the three principal (normal) stresses are not accompanied with shear stresses,
the three principal shear stresses have normal stresses acting in the same planes
(“45° planes”). These are:
(5)
We can use several crystal planes (crystal directions) to describe the stress state in
the material by the σ´s and ´s in those planes, e.g.
-
in the planes of the “principal cube” by the three principal stresses,
in the planes of the rhombic dodecahedron by the i (eq.(3)) and the σi (eq.(5)).
6
Another important set of planes for the description of the stress state by the respective σ´s and ´s is given by the regular octahedron defined by the principal axes
(figure from [F]):
Each of the eight planes of the octahedron has one and the same value of the normal stress, the octahedral normal stress, σoct:
(6)
Also, the shear stress is one and the same in each of the eight planes (octahedral
shear stress, oct). It terms of the principal stresses it has the form:
(7)
In a general Carthesian co-ordinate system (x, y z) oct appears as follows:
(7´)
From eq.(6) we see that the sum σa + σb + σc for any three rectangular co-ordinates
a, b, c is the same. This is an invariant of the stress tensor (the trace of the tensor),
and it is equal to the hydrostatic stress:
σhyd =
(8)
7
3.3. Failure criteria
In general, a failure criterion is a certain function of characteristic stresses which in
case of failure exceeds a certain critical stress crit. This critical stress is a materialspecific quantity.
In order to be independent of the specific co-ordinate system, failure criteria are
usually expressed in terms of the principal stresses 1, 2 and 3.
Hence, the general form of a failure criterion of isotropic3 materials is
f(1, 2, 3) = crit
(at failure)
(9)
As critical stresses are usually used:
for fracture:
the ultimate tensile strength, u,
for yield:
the yield strength, 0,
both obtained from uniaxial tensile tests (cf. section 1.2.):
 ultimate tensile strength
 yield stress
In the practically important cases which will be treated here, the function f(1, 2,
3) can be simply expressed as an effective stress value *:
f(1, 2, 3) = *,
(10)
therefore the failure criterion is
* = crit
3
(at failure).
(9´)
The condition of isotropy is already substantiated by the fact that the deduction is fully
based on the tensile strength σ0 without referring to particular crystallographic directions.
8
If * < crit, a safety factor X may be defined corresponding to
X = crit / *
(11)
This means that * could be increased by the factor X until failure occurs.
(1) Maximum normal stress fracture criterion
Failure in form of fracture is expected when the largest principal normal stress
reaches the ultimate tensile stress u:
* = Max (1, 2, 3) = u
(at failure)
(12)
If small cracks are already available in the material the material may break at much
lower stresses. This will be discussed in section 3.4..
(2) Maximum shear stress yield criterion (TRESCA criterion)
Generally, shear stresses cause movement of dislocations and are therefore connected to plastic deformation or yield. The simplest case is the glide of an edge
dislocation (picture from [G]):
Plastic deformation is expected when the largest shear stress reaches a material
specific critical value 0 which is called the yield stress in shear4. Since the largest
shear stress is always one of the principal shear stresses 1, 2 or 3 (see above), we
can in this case write the criterion (9) as
* = Max (1, 2, 3) = 0
(at failure).
(13)
Here, 1 = |σ2 – σ3|/2, etc. (cf. eq. (3), therefore criterion (13) is defined completely
in terms of the principal stresses 1, 2 and 3.
In principle, 0 could be measured by an especially designed shear stress experiment, e.g. using a thin-walled tube in torsion. However, the onset of plastic deformation is usually characterised by the so called yield strength σ0 which is determined by (and defined via) the uniaxial tensile test. In this test, at the onset of plastic deformation we have
σ1 = σ0 ,
4
cf. DOWLING p. 246
σ2 = σ3 = 0
(at failure)
(14)
9
Under these special conditions, eq. (3) becomes
yield stress in shear....0 = σ0/2.....yield stress/2.
(15)
Therefore we can conclude (TRESCA criterion) that plastic deformation occurs if the
maximum shear stress exceeds half of the yield strength σ0.
Note: Yield strength and yield stress in shear are correlated one-to-one. However,
from the viewpoint of relevant physical mechanisms it is appropriate to say that σ0
is an empirical quantity while 0 is reflecting the very physics of plastic deformation.
(3) Octahedral shear stress yield criterion (VON MISES criterion)
We define a failure criterion on the basis of the octahedral shear stress corresponding to eq. (7). Then, failure occurs if oct exceeds a material-specific value oct,0:
oct,0 =
(at failure)
(16)
When we apply eq. (16) to the case of the uniaxial tensile stress (cf. eq. (14)) we
get
oct,0 =
(at failure)
(17)
In this equation, oct,0  0.47σ0 which is by about 6% less than for the TRESCA criterion (cf. eq. (15). Which criterion describes the behaviour of a given material best
depends on the special case. The reader should keep in mind that the maximum
shear stress exists in one plane only while the octahedral shear stress exists in four5
different planes at the same time. Therefore, the latter provides some averaging
which might by an advantage in case of complex, anisotropic stress states.
In eq. (17), oct,0 is again the yield stress in shear as we have introduced above.
Combining eqs. (16) and (17) we obtain
(at failure)
(18)
The right side of eq. (18) is the so called VON MISES stress
5
The eight planes of the octahedron include four pairs of planes with the two planes of one
pair being parallel to each other and thus being equivalent.
10
σM =
.
(19)
This quantity relates any general stress state characterised by the principal stresses
1, 2 and 3 to the uniaxial tensile test. In case of failure (here: yield) the von
Mises stress equals the yield strength
σM = σ0
(at failure),
(20)
no matter how the magnitudes of the principal stresses are. Only their combination
in form of eq. (19) matters.
As far as there is still no yield, σM is smaller than σ0. Then, the safety factor X corresponding to eq. (11) can be formulated as
X = σM / σ0 .
(21)
Eq. (19) gives σM in terms of the principal stresses. In a general CATHESIAN coordinate system the stress state includes also shear components and the VON MISES
stress is to be written as
σM =
(22)
where σx = σxx etc..
(4) Hydrostatic pressure criterion
Hydrostatic stress corresponding to eq. (8) does not cause plastic deformation, and
- if compressive - does not cause fracture. However, it may cause phase transformation.
Under strong hydrostatic pressure, many materials undergo transformation into
other phases, typically those phases which have a higher mass density.
In such cases, usually the magnitude of the hydrostatic pressure determines the
onset of phase transformation, i.e. the failure criterion has the form
σhyd,crit =
(23)
The conditions of phase transition are usually determined by experiments in a highpressure chamber. They are therefore described in form of material specific critical
values of the hydrostatic pressure, i.e. a hydrostatic stress that is the same in all
directions.
11
Despite the fact that critical hydrostatic pressures for phase transformations are
usually very high (e.g. for Si about 10 GPa = 105 atm.) such pressures can occur in
a material during measurement of indentation hardness (cf. section 3.6.).
3.4
Fracture
We have to discriminate ductile fracture and brittle fracture:
Ductile fracture: The propagation of the crack involves substantial plastic flow.
Plastic strain causes small microvoids to form in the material. These microvoids
grow and join together (i.e. they coalesce). Failure occurs when the walls of material between the growing voids finally break. The fracture surface formed is rough.
Brittle fracture: Brittle fracture is characterised by rapid crack propagation and
very little plastic deformation. It yields a relatively flat fracture surface. Usually,
brittle crack propagation corresponds to the successive and repeated breaking of
atomic bonds along specific crystallographic planes (cleavage).
Many materials show a transition from brittle (low T) to ductile fracture (high T):
However, the transition temperature is material specific. So, at room temperature
some materials are essentially brittle and others are essentially ductile.
3.4.1. Ductile fracture
Uniaxial tensile test: The tensile stress σx along the sample axis x is the maximum
principal stress (in fact the only principal stress ≠ 0). The maximum shear stress
appears in directions which are 45° inclined with respect to the tensile direction.
Considering eqs. (3), (4) as well as (14), (15) this maximum shear stress, (45°), is
half the tensile stress
(45°) = ½ σx.
(24)
12
When σx reaches the yield strength σ0, (45°) equals ½σ0 or – in other words, becomes equal to the yield stress in shear corresponding to eq. (15).
Dislocation movement also takes place in the 45° direction. The material transport
connected with this is responsible for the “necking” of the sample.
Voids and internal cracks in the
necked region of a polycrystalline
specimen of high-purity copper
(picture taken from [B])
In single crystals it happens that many edge dislocations move along one individual
lattice plane thus forming macroscopic slip steps at the sample surface which are
often arranged in so-called slip bands. Two nice examples from [H]:
slip band in a cadmium single crystal:
particularly large slip steps in an
aluminium single crystal:
Uniaxial compression also causes material failure due to plastic deformation. Analogue to the tensile test, a yield strength σ0 as well as an ultimate fracture strength
σu is obtained. It comes out6 that these critical values are practically the same in
tension and compression.
6
cf. DOWLING p. 243
13
example for uniaxial compression:
A composite x-ray topograph
showing slip bands in two
adjacent grains of an hexagonal Ih ice polycrystal
slowly strained under uniaxial compression at -6°C.
The bands formed through
slip on basal planes. The
images of each grain were
obtained from separate Laue
spots, therefore an image of
the grain boundary is present
in both. The dislocations in
the slip bands were nucleated at facets on the grain
boundaries (cf. arrows). The
dislocations starting at S
traversed the grain and pile
up at P.
from F. LIU, I. BAKER, AND
M. DUDLEY, Phil. Mag. A,
71 (1995), p.15.
Influence of confining pressure during uniaxial compression (from [D]):
14
The uniaxial compression test became increasingly popular in nanomechanics for
testing of tiny samples where a tensile test is hard to achieve. Little pillars are
formed by (for instance) focussed ion beam etching (FIB) and then loaded by a
blunt indenter. Examples from the internet [I]:
Cu pillars before (l.) and after load (r.)
Au pillar on MgO
3.4.2. Brittle fracture
Introduction
In fracture of a material, two main mechanisms are involved:
•
plastic deformation, and
•
formation and propagation of cracks.
Thorough reasoning makes clear that
•
ductile fracture (= material transport due to plastic deformation → “necking”
→ actual stress exceeds σu), and
•
brittle fracture (without any plastic deformation)
are limiting cases which occur very seldom in pure form.
As we know, the tensile strength σ0 measured in the uniaxial test is usually by a
factor of 10 – 100 smaller than the theoretical tensile strength (cf. section 2.2.).
However, actual tensile strength in a technical system may be even more reduced
due to the propagation of cracks.
These cracks may exist before in form of small crack nuclei or flaws (e.g. at grain
boundaries) or they may be formed by stress-rising geometries (notches, sharp corners on hatches, etc.) which on their part can be due to bad design or just as an imperfection of the material.
15
Crack propagation depends also on certain material properties (toughness of the
material).
Investigation of fracture behaviour was pushed forward by catastrophic failures
during and after Word War II. From the first all-welded ships (Liberty series) totally 4700 have been built. More than 200 of them suffered catastrophic failure,
some splitting in two parts while lying at anchor in port, and over 1200 suffered
some sort of severe damage due to fractures. (figures from [J]):
Failure of Liberty ships
After the war, similar failures happened in civilian aircraft production. For instance, ill-placed rivet holes destroyed several airplanes. The figure below shows a
British Comet aircraft which was “peeled off” during flight (figure from [K]):
Cracks as stress raisers
First, a crack in a material gives rise to an increase of stress – no matter, what the
stress will do. For sake of simplicity we consider a crack with the shape of an elliptic hole having semimajor and semiminor axes c and d.
The smallest radius of curvature of an ellipse, ρ, is given by
ρ = d2/c
(25)
16
The body shall be exposed to a tensile stress S in y direction, i.e. parallel to the
minor axis of the ellipse. For the case c/d = 3 the following stress distribution is
obtained (figure from [C]):
In the special case considered here, the stress at the tip of the ellipse, σy(x = 0), is
increased by a factor of 7 in comparison to the remote stress (or gross stress) S. The
general expression for this increase of stress is
σy(x = 0) = S[1 + 2c/d]
(26)
σy(x = 0) = S[1 + 2 (c/ρ)1/2].
(27)
or with eq. (25)
If the crack is really narrow (slit-like, i.e. d >> c), then ρ → 0 and σy(x = 0) / S becomes infinite. Of course, this does not happen in nature:
17
The figure (from [C]) shows how a material accommodates the stress by blunting
the sharp tip to a small but nonzero radius δ which is called the crack-tip opening
displacement (CTOD):
ductile material:
by plastic deformation,
polymer:
by forming a special structure of elongated voids (grazing),
ceramic:
by forming a zone containing many tiny cracks.
In all cases, the maximum of σy is reduced and the growth of the crack is prevented
- provided the load is not too high.
The stress intensity factor
In the following we deal with the linear elastic fracture mechanics (LEFM), in contrast to elastic-plastic fracture mechanics. LEFM means that any plastically deformed zone be small and, hence, for the total behaviour of the system the unrestricted validity of HOOKE´s law can be assumed.
Fracture mechanics defines a quantity K called stress intensity factor which describes the severity of a stress situation with respect to cracking. K reflects the joint
influence of
- external (gross) stress S,
-
crack size, and
-
geometry.
A given material can resist the crack, i.e., the crack does not propagate and failure
does not occur as long as
K < Kc .
(28)
Here, Kc is a material specific critical value called fracture toughness.
Example: We consider a crack with a (half) length a in a wide plate of half width b
where a << b. The plate is elongated by a gross stress S in a direction perpendicular
to the crack. In this case, the stress intensity factor is give by
K = S a .
(29)
Hence, for a particular material having a fracture toughness Kc and a given crack
length a the stress Sc which just causes failure is
Sc = Kc /  a .
(30)
The following example (from [C]) refers to 1.5 mm thick plates made of the aluminium material 2014-T6 which were tested at -195 °C. (The reader should keep in
mind that brittle behaviour is most pronounced at low temperature!):
18
The critical tensile stress for failure in dependence on the crack length a follows
nicely a curve calculated from eq. (30): The smaller the crack the bigger the stress
which can be withstand. However, for very small crack lengths where large gross
stresses (which finally approach the yield strength) have to be applied the experimental values deviate from the theoretical curve. This is due to the fact that under
these circumstances the assumptions of the LEFM are not longer maintained.
Brittle vs. ductile behaviour:
Considering the full validity of the LEFM we calculate the crack length at for
which the failure stress acc. to eq. (30) equals the yield strength σ0 (cf. figure
above). Under these conditions we get from eq. (30)
at = 1/π (Kc/σ0)2.
(31)
We call this value transition crack length. Despite the fact that eq. (31) is valid for
a particular geometry (cf. figure) we can make the following general statements:
If the existing crack length a is....
•
a ≥ at → strength limited by brittle fracture; fracture mechanics applicable;
•
a ≤ at → strength most probably limited by yielding.
Now, we consider two hypothetical materials:
a) low σ0, high Kc

at large, and
b) high σ0, low Kc

at small.
19
The figure (from [C]) shows that cracks of moderate size are below at for material
(a) and, hence, do not affect the performance of this material which is limited by
yielding anyway. In contrast, for material (b) even relatively small cracks (flaws)
are bigger than the tiny at, hence the behaviour of this material is largely dominated
by brittle fracture.
Material of type b) was responsible for the failure in the Liberty ships and Comet
airliners: New high-strength materials (high σ0) happened to be sensitive against
the existence of cracks which are larger than the (quite small) at and, therefore, did
limit the strength of the material by brittle fracture under unfavourable circumstances.
Fig. b) shows another interesting fact: Many materials (glass, stone, ceramics, cast
metals) naturally contain a large concentration of small cracks or flaws (“intrinsic”
flaws of size ai). These materials have a high yield strength σ0 and, therefore, can
withstand large compressive stresses. In tension, however, these materials fail due
to brittle fracture which results in a relatively small ultimate tensile strength in tension, σut. This reduction in strength appears only in tension while under compression the flaws are simply closing. - This is in contrast to the ductile fracture discussed above where σu was the same under tension and compression.
Many people have at least heard about the theory of brittle fracture published by A.
A. GRIFFITH in the 1920ies which is one of the earliest works in fracture mechanics.
This theory is an energy method which essentially compares
•
the elastic strain energy which is released due to crack propagation, and
•
the energy of the surfaces which are newly formed during cracking.
This concept does not take into account plastic deformation. However, G. R. IRVIN
(who introduced the Kc concept) could show in the 1950ies that GRIFFITH´s approach can be applied even in case of mixed brittle-ductile behaviour provided the
plastic zone is small, i.e. LEFM applies.
20
Crack opening modes
There are three different basic modes of crack opening (crack surface displacement) which are schematically shown in the next figure (from [C]):
Mode I is the opening mode (crack surfaces move apart). Mode II is called sliding
mode and Mode III tearing mode. In the two latter cases the displacement occurs
within the plane of crack propagation, however, in case II the direction of displacement is (anti)parallel and in case III perpendicular to the direction of crack
propagation. According to the mode one uses KIc, etc. instead of Kc
Most practical cases are due to tension stresses and correspond to mode I.
3.5.
Plastic deformation
3.5.1. Basic mechanisms
Dislocations
A dislocation is a 1D defect in a crystal (line defect). There are two limiting cases
of a dislocation, the edge dislocation (left) and the screw dislocation (right)7:
In a gedanken experiment, the dislocation can be formed by making a cut into the
crystal. In case of the edge dislocation, the material is displaced perpendicular to
the cut and the gap filled by an additional crystal plane. In case of the screw dislocation, the material is shifted within the cut plane, parallel to the edge of the cut,
7
Figure taken from [C]
21
and then reconnected. In both cases the edge of the cut is the locus of the dislocation line as shown in the figure.
The displacement of the material – which has both a magnitude and a direction and
can therefore be considered as a vector, is the so-called BURGERS vector. This vector is shown in the next figure ([G]) using the screw dislocation as an example:
The figure also shows how the BURGERS vector can be determined by traversing a
path around the dislocation line. The BURGERS vector is independent of the path.
One gets one and the same vector for any closed path around the line.
The BURGERS vector is also unchanged when the dislocation line changes its direction. In the next figure (from [G]) we see a 90° turn of a dislocation line, thus
changing its character from a pure screw (at A) to a pure edge dislocation (at C):


The case b || dislocation line corresponds to screw character, while b  dislocation
line represents an edge dislocation. In the general case, the angle  between the
BURGERS vector and the dislocation line is 0° ≤  ≤ 90°.
22
Dislocations are very important for plastic deformation of solids. The finding, that
the theoretical shear strength according to
b =
Gb
.
2h
(2.2-19)
is by far not achieved in real solids, was not understood for a long time. It can,
however, be explained considering the fact that plastic deformation in crystalline
materials proceeds via formation and movement of dislocation (model of carpet
hump or carpet wave).
additional note: Screw dislocation and crystal growth
Screw dislocations play also a crucial role also in the growth of a crystal from the
vapour phase. Atoms arriving at a surface, for energetic reasons, do not like to settle on a flat surface but rather like to be taken up by a monoatomic step. Such step
is provided by a screw dislocation. Most importantly, this step is self reproducing:
The same is true for the sublimation of a crystal, i.e. its degradation by evaporation
of atoms from the surface. Experimentally, this case was demonstrated first, already in the 1960ies. The figure shows dislocation-induced lamellar step pattern at
the surface of a NaCl crystal after annealing at 400 °C, 90 min.(H. BETHGE et al.):
23
The next figure shows growth of SiC at a screw dislocation as imaged by STM
(from G. WAGNER, IKZ Berlin)
Formation and movement of dislocations
The displacement of material (called slip) due to movement of dislocations can
most easily be understood considering an edge dislocation (fig. from [B]):
This kind of movement of the edge dislocation is called glide. The dislocation
moves within a certain set of parallel lattice planes (representing a particular slip
plane) and – when emerging at the surface – form a tiny slip step which in the case
of a edge dislocation is just one lattice constant high. Many such elementary steps
may accumulate to form macroscopic slip bands as we have seen in section 3.4.1..
Even though edge dislocations are most efficient in this respect, other types of dislocations can also provide permanent deformation of the material due to their
movement. We will come back to this in the next section about dislocation interaction.
Most materials contain plenty of dislocations which can take part in plastic deformation when sufficient stress is applied. However, certain materials such as silicon
crystals can be obtained dislocation free. In this kind of material dislocations may
24
be formed by external stress fields. One mechanisms for generation of dislocations
is the FRANK-READ source (figure from [G]):
This source acts as follows: A dislocation is – driven by an appropriate shear stress
- moving from left to right. However, the dislocation is caught by two so-called
pinning sites8 A and B (1). When the shear stress is increased, the dislocation bows
(2) out until the angle  reaches 90° (3). This corresponds to the necessary shear
stress source. If the stress is slightly increased further, the dislocation will spontaneously pass through the configurations (4) and (5) and a dislocation loop is generated (6).
The dimension of the FRANK-READ source may give rise to a dependence of deformation behaviour on the size of the volume involved, e.g. in case of small samples of few µm.
Analysis shows that source depends on shear modulus G, magnitude of the BURGERS vector b and the distance L of the two pinning sites as follows:
source = Gb/L .
(32)
With b  2 Ǻ = 210-10 m and a typical L in the µm range we get source  10-4G
which is about three orders of magnitude smaller than the theoretical shear strength
given in eq.(2.2-20). This value corresponds with critical stress values observed in
the experiment.
Dislocations are governed by the symmetry of the crystal. They move only in certain slip planes and within the planes only in particular directions. The combination
of slip plane and slip direction is called slip system. Slip planes are usually densely
packed low-index crystal planes, as they exist for instance in fcc, bcc and hcp crystals. Most metals crystallise in those lattices and it is not surprising that many metals are quite ductile.
8
In a silicon crystal, pinning sites can for instance be represented by tiny SiO2 precipitates
which are frequently observed in crystals grown by the CZOCHRALSKI method.
25
The activation of a slip system depends on the
shear stress component9 σR resolved along the
slip direction
σR = σ cosθ cosλ
(33)
together with the critical resolved shear stress,
σR,c, of that slip system (fig. from [M]).
The following table (from [B]) shows slip systems observed in important crystal
structures:
9
σR is called the resolved shear stress. In the German literature, eq. (33) is written as
σR = σm with m = cosθ cosλ being the SCHMIDT-Faktor.
26

In the diamond-type lattice, the slip system 111 110 is activated (see the “fccline” in the table above). This means the glide dislocations lie in (111) planes and –
for one individual plane - have three different directions. One direction provides a
pure screw dislocation, the other two are so-called 60° dislocations which have an

angle of π/3 between b and the dislocation line. In the following, an infrared microscopy picture10 of a FRANK-READ source operating in single-crystal is shown:
The following picture (from [G]) gives the atomic structure of a 60° dislocation in
the diamond lattice. The additional lattice plane is drawn bold. s is the dislocation
line, b the BURGERS vector and  is the angle of 60° between s and b:
As can be seen from the table, all important slip systems are “manifold”, i.e. each
of them represents a number of equivalent single systems. This is particularly useful for plastic deformation of polycrystalline materials:
10
from W. C. DASH, J. Appl. Phys. 27, 1193 (1956)
27
Plastic deformation in polycrystalline Al (from [B]).
Slip planes are parallel within a
grain but are discontinuous across
grain boundaries.
Some grains show two families of
slip planes.
Another possibility of movement of dislocations is climb (for edge dislocations)
and cross slip (in case of screw dislocations), resp.. The enables the dislocation to
change their slip plane and, hence, to avoid obstacles in their path. We will show
the climb process as an example using the following figure from [G]:
Climb is essentially an interaction with point defects which can be supported by
hydrostatic pressure and in particular by high temperature. In the figure we see
from left to right an upward movement of the dislocation which is performed by
the emission of interstitials from or (which is the same) a condensation of vacancies at the “additional lattice half plane”.
3.5.2. Dislocation interactions and hardening
Completing the presentation about glide of edge dislocation above, we first add a
figure (from [B]) which illustrates how a screw dislocation is moving:
In this case, the BURGERS
vector and the dislocation
line are aligned and are
perpendicular to the direction of motion. With
progressing
movement
the slipped part of the
crystal increases.
28
In the following we consider the encounter of a screw dislocation and an edge
dislocation (pictures from [B]):
The two dislocations move towards each
other: The edge dislocation due to the horizontal component of the shear stress , the
screw dislocation due to the vertical component of .
After the dislocations met: In the upper
part, the extra half plane of the edge dislocation became identical to the screw dislocation plane and the screw dislocation line
lies in this plane.
In the lower part, however, the extra half
plane is missing and the screw dislocation
line has to continue in a neighbouring plane
 a horizontal jog is formed in the vertical
screw dislocation line.
The length of the jog equals the magnitude be of the BURGERS vector of the edge
dislocation. While the vertical part of the dislocation line is still a screw dislocation, its horizontal part is an edge dislocation!
As movement progresses, the edge dislocation also becomes jogged. The magnitude
of the jog is bs and the jog has screw dislocation character.
29
Effects of the jogs in both dislocations on their mobility:
vertical (originally screw) dislocation (figure right):
The vertical shear stress which causes the screw dislocation to move to the left tries to move the horizontal
(edge) jog in vertical direction.
As a consequence, the whole dislocation becomes immobilised!
In contrast, the existence of the screw-type jog in the
edge dislocation hampers their movement only slightly.
In conclusion, intersection of dislocations can immobilise them or at least reduce
their mobility. Since dislocation density (measured as dislocation length per volume, m/m3 = m-2) increases during plastic deformation, yield strength should get
larger. This effect is called work hardening or strain hardening.
The following figure (from [L] shows
the increase of yield strength with rising
dislocation density for single and polycrystalline materials (probably for Cu):
30
Analysis of stress-strain curves:
An ideal elastic, perfectly plastic behaviour corresponds to a flat curve beyond
yielding (left) while elastic, linear-hardening behaviour shows a rise following
yielding. However, the slope of the curve reduced from E (YOUNG´s modulus) to
δE with δ being the reduction factor.
However, even more realistic is the elastic, power-hardening relationship which
assumes the validity of the following expressions:
σ = E
(σ ≤ σ0)
σ = H1n1
(σ ≥ σ0)
(34)
with n1 being the strain hardening exponent and H1 an additional constant. The
graphic presentation of eq. (34) in linear (left) and log-log co-ordinates (right):
We see that H1 is the value of σ for  = 1. With the same scale on both axes (i.e. the
same length for one decade) the slope for σ ≤ σ0 is one and for σ ≥ σ0 n1.
This kind of model fits very well for metals with n1 values between 0.05 and 0.4.
Using both parts of eq. (34) for σ = σ0 and equate them yields
σ0 = E (H1/E)1/(1-n1)
(35)
We see that the yield strength, σ0, is not an independent constant, as any two of the
constants σ0, H1 and n1 may be used to calculate the remaining one.
The strain hardening exponent n1 is a useful and much used number (“like H”).
31
HALL-PETCH relation:
Not only dislocations formed during deformation of the sample but also defects
existing from the beginning may hamper the movement of dislocations and thus
increase yield strength. An example is the HALL-PETCH relationship, a strong dependence of yield strength on the grain size in polycrystalline material according to
σ = σ0 + ky/D1/2
(36)
with D being the grain size, σ0 the yield strength of the single crystal and ky a constant. The following figure (from [L]) shows an example for various steels:
Physics behind the HALL-PETCH relation:
-
grain boundaries as such are obstacles to dislocation movement,
-
favourable slip directions in one grain may be neighboured to unfavourable
slip direction in the adjoining grain,
-
for very small grains (< 10 nm) it is energetically unfavourable to have dislocations in the grain.
Concluding remark: In addition to the mechanisms discussed so far, i.e. large
dislocation densities or a structure comprising relatively small grains, several other
mechanisms are known how the yield strength and/or fracture toughness can be
increased by a particular “defective” structure which hampers the movement of
dislocations and/or the propagation of cracks:
Solid solution strengthening,
precipitation hardening,
nanocomposite structures.
These issues are not within the scope of this lecture.
32
3.5.3. Creep
Usually, elastic and plastic strains appear instantly upon the application of stress.
However, in certain cases further deformation occurs gradually with time which is
called creep strain or just creep.
Creep is generally connected with high temperatures, where “high” has to be considered in comparison with the melting temperature Tm of the material under consideration. Typically
T > (0.3 – 0.6) Tm
(at creep)
(37)
So, particularly polymers or biological material are prone to creep. A nice example
of creep from nature is the movement of a glacier.
In nanomechanical testing, even small amounts of creep may influence the accuracy of the measuring results, therefore one always has to keep an eye on this phenomenon.
A nice historic example from 191411 is given in the following picture showing the
creep of lead at 17°C under constant true stress (see mechanism in the right part):
Empirically, creep can be divided into three phases:
-
The primary (transient) phase,
-
the secondary (steady-state) phase, and
-
the tertiary (unstable) phase, which eventually leads to fracture.
11
E. M. DA C. ANDRADE: “The flow in metals under large constant stresses”, Proc. Royal
Soc., Series A, London, Vol. 90, pp.329-342
33
This is illustrated in the following picture from [C]:
The strain rate, i.e. the slope of the (t) curve in the steady-state region depends
strongly on the conditions of deformation, particularly on stress and temperature:
NaCl crystal at 737 °C (from [L]):
single-crystal TiO2 (from [C]):
34
Analysis shows that the dependence on σ is a power law
d/dt  σn
(at T = const)
(38)
with n being the stress exponent, while the dependence on T is an exponential one
with an energy of activation, Q
d/dt  exp(-Q/kBT)
(at σ = const.)
(39)
Combining both dependences one finds
(40)
with G being the shear modulus and A being a constant.
Careful analysis revealed that the activation energy Q for creep equals the activation energy for self diffusion, i.e. the diffusion of vacancies and self-interstitials:
Q = QSD
(41)
This is shown for a variety of materials in the following figure (from [L]):
The self diffusion supports the creep process by providing point defects that enable
dislocation to move by climb, thus overcoming obstacles by changing the glide
plane.
35
3.6.
Phase transformation and other mechanisms
When a material is loaded, it depends on the properties of that material as well as
on the very stress distribution which failure mode will occur first. Most materials
fail by plastic deformation, by cracking accompanied with more or less plastic deformation, or by brittle fracture.
However, in some cases these mechanisms may be unfavourable and, hence, other
mechanisms take place, for instance mechanical twining or phase transformation.
mechanical twinning:
Twinned crystals share some of the crystal lattice points in a symmetric manner. In
the simplest case they share one plane (contact twin). A more complicated case is
penetration twinning:
penetration (l) and contact twin (r.) from [G]:
This is a sequence
of twin planes
forming a “multiple contact twin”
or “set of twin lamellae” (from [L]):
real quartz twin:
36
Twin crystals can not only be formed during growth but also by stresses. In particular: - at low temperatures ( only few point defects available for climb), and
- in lattices with few glide systems (e.g. the hcp lattice – cf. table p. 8)
plastic deformation by dislocation movement may be prevented and deformation
twinning may occur.
Phase transformation
In addition to the usual crystal structure (diamond type, called Si-I), silicon exhibits
several phases which occur under high hydrostatic pressure.
important phases of silicon12:
Since we do not perform high-pressure experiments, this should be not important to
us. However, semiconductor silicon is dislocation-free, so plastic deformation is
unlikely. Also, cracking is not favoured because of the perfection of the crystals
(no flaws), and strong bonds.
Therefore it happens during nanoindentation of silicon crystals that stress states are
built up under the indenter whose hydrostatic pressure components acc. to
σhyd =
(23)
get high enough to form high-pressure phases. During unloading, re-transformation
takes place, however, the coherence of the single crystal lattice has been disturbed.
This has to be taken into account when thin films on Si shall be measured.
12
from Properties of Crystalline Silicon, 20th ed., edited by R. Hull, (INSPEC, London,
1999), pp. 104–107.
37
References
[A] W.D. NIX, 353 class notes 2005, Standford University.
[B] MELVIN M. EISENSTADT, Introduction to Mechanical Properties of Materials,
Macmillan, New York and London, 1971.
[C] NORMAN E. DOWLING, Mechanical behaviour of Materials, Prentice-Hall,
Upper Saddle River NJ, USA, 1999.
[D] ANTONY C. FISCHER-CRIPPS, Nanoindentation, 2nd ed., Springer, 2004.
[E] MATTHIAS HERRMANN: ”A short note about the calculation of elastic constants for loading cases associated with non-isotropic elastic behaviour”, Report, TU Chemnitz, Solid State Physics, 2007.
[F]
H.G. HAHN, Elastizitätstheorie, B.G. Teubner, Stuttgart 1985.
[G] CH. WEIßMANTEL, C. HAMANN: Grundlagen der Festkörperphysik, Deutscher
Vlg. der Wissenschaften, Berlin 1989.
[H] Cd from http://www.doitpoms.ac.uk/tlplib/miller_indices/images/cadmium%20slip.jpg
Al from http://www.univie.ac.at/hochleistungsmaterialien/mikrokrist/characterization.htm
[I]
Cu from http://www.tms.org/Meetings/Annual-08/images/AM08educ_clip_image002.jpg
Au from http://www.imechanica.org/node/679
[J]
http://www.fiu.edu/~thompsop/liberty/photos/fractures.html
[K]
http://chaos.ph.utexas.edu/%7Emarder/fracture/phystoday/how_things_break/how_things_break.html
[L] G. Gottstein, Physikalische Grundlagen der Materialkunde, Springer Vlg.,
2001.
[M] Yip-Wah Chung, Introduction to Materials Science and Engineering, CRC
Press, Boca Raton, FL, USA, 2007.
38
Appendix 3: Complete derivations from section 3.2.
39
40
41
42