ZERO STIFFNESS IN NANO-CRYSTALLINE STRUCTURES
Seminar Report
Submitted in partial fulfillment of the requirements for the
award of the degree of
Bachelor of Technology In
Mechanical Engineering
by
ADIL IQBAL HASSAN PVB100512ME
Department of Mechanical Engineering
NATIONAL INSTITUTE OF TECHNOLOGY CALICUT
MAY 2015
CERTIFICATE
This is to certify that this report entitled ZERO STIFFNESS IN
NANO CRYSTALLINE STRUCTURES is a bonafide record of the Seminar
presented by ADIL IQBAL HASSAN PV (Roll No.: B100512ME), in partial
fulfillment of the requirements for the award of the degree of
Bachelor of Technology in
Mechanical Engineering from National Institute of Technology
Calicut.
Mr. A M Srinath Faculty-in-charge(ME4097 - Seminar)Dept. of
Mechanical Engineering
Dr. Allesu KanhirathinkalProfessor & HeadDept. of Mechanical
Engineering
Place : NIT CalicutDate: 3 May 2015
ABSTRACT
Dislocations are an important type of defects in a crystalline
solid. Movement of dislocation through the structure because of the
normal stress result in plasticly compromising the size of the
material. When the crystals of the size of nano size are
considered, the disturbances move to the surface on its own for
making themselves more fitting of their entropy. This movement from
dislocations are understood with a concept namely image force
concept. The plate bends and those dislocations may move along the
crystal without changing their energy. The neutral equilibrium in
mechanics is a concept similar to this. The seminar here gives us
an enlightment about the plastic deformations and what the concept
called zero stiffness concerning nano-crystals.
CONTENTS
Abbreviations iiSymbols iiiFigures iv
1 Introduction 1 1.1 Introduction 1 1.2 Concept of stiffness
2
2 Review of Literature 3 2.1 Dislocation 3 2.2 Image Force 2.3
Finite element methodology
3 Results and Discussion on the topic 3.1 Results and
discussion
4 Conclusions and Scope for Future Work 4.1 Conclusions
ABBREVIATIONS
Al Aluminium
FEM Finite Element Method
Ni Nickel
PN Peierls-Nabarro
ZSMS Zero Stiffness Material Structures
SYMBOLS
deformation
k Stiffness constant
F Force
G Shear modulus
b Burgers vector
v Poissons ratio
0_ size of the control volume ~ 70b
d Distance from surface
E Energy per unit length of dislocation line
L length of nanocrystal
FIGURES
1.1 Neutral equilibrium of structures in mechanics
1.2 Edge dislocations
1.3 Burgers vector
1.4 Screw dislocations
1.5 Mixed dislocations
2.1 Concept of Image Force
2.2 Image Force in nanocrystals
2.3 Eshelby bend
3.1 Unconstrained and constrained domains
3.2 Energy-distance curve
3.3 Contours plotted
INTRODUCTION
1.0INTRODUCTION
A system of particles is said to be in the state of static
equilibrium when every particle in the system is stationary and the
net force on each and every particle is nil. The necessary
conditions for a system of particles to be in the state of
mechanical equilibriua are:
(i) the total vector sum when all the externally applied
forcesare added is zero (ii) the net sum of moment of all the
externaly applied forces about any line has to be zero
When the potential energies are at local maxima, the system is
in a state of equilibrium called unstable equilibrium. If the
system is displaced from that state state for an arbitrarily small
distance, the forces on the system force it to move further
more.
If the potential energies are at local minima this can be a
stable equilibrium. So a response to very small deviatons in forces
tend to restore that equilibrium. If two or more stable equilibrium
states are possible for our system, an equilibria which has a
higher potential energy than the absolute minimum comprise a
metastable state.
The second of the derivative tests fails, one is forced to use
the first of the tests. Both the previous results may be still
possible, if not a third. This would be at a region where the
energy remains constant, where the the type of equilibrium can be
called a neutral or an indifferent or a marginally stable one. To
the smallest order, it may continue in the same state even if
displaced by a small amount.
1.2 STIFFNESS
Stiffness is defined as the rigidity of an object, ability of
the object to withstand external applied forces. Neutral
equilibrium, which is seen rarely in the theory of elastic
stability, are said to be present if even in the case of large
displacements also the magnitude of the load maintaining the
equilibrium remains constant. These structures may be able to
change their shape without the help of extra external load
themselves like they behave as if they are mechanisms.For elastic
bodies with a single degree of freedom the stiffnessis defined
as
Generally, deflection (or movement) of a very small element
(considered a point) can be there along two or more degrees of
freedom in an elastic body (max: six at one point). say, a point on
a horizontal beam could undergo both a vertical displacement and a
rotation w.r.t its undeformed axis. For M degrees of freedom an M x
M matrix can be used to describe the stiffness at that point. The
diagonal terms in the matrix denotes stiffnesses (or the
direct-related stiffnesses) along the degree of freedom while the
off-diagonal terms denotes the coupling stiffnesses between two
different DOFs (either at the same or different points) or else the
same degree of freedom at two distinct points. In industry, a
specific term called influence coefficient is often used , inorder
to refer coupling stiffness.
REVIEW OF LITERATURE2.1 DISLOCATION A perfect or an ideal
crystal is a myth; there is nothing like that in universe. No
arrangements of atoms follow perfect crystalline pattern in real
materials. Nevertheless , most of the materials used in engineering
are crystalline to a very good extent. There may be basic physical
reasons for these things. The recommended structures for solids at
lower temperatures are those structures which minimise the internal
energy. The low energy configurations of atoms are mostly
crystalline because the regular pattern of crystal lattices reccur
for whatever local configurations favorable for bonding. There is
also a primary physical reason for the crystasl being imperfect. If
a perfectly crystalline structure is energetically preferred , over
the limits of lower temperatures, the atoms will be comparatively
immobile in the solids and it hence, is, difficult to discard the
imperfections that are brought into the crystal during its use,
growth or processing .The truth that original materials will not be
ideal is critical to this branch of engineering. Their properties
would be decided by their composition and crystal structure alone,
moreover would be very restricted in values and their variety if
the materials were perfect crystals . The possibility of making
these imperfect crystals allows scientists to make material
properties to the varied combos that modern engineering
applications need. Wecan continously seethat the most relevant
features of the material microstructure are the defects in crystals
modified to control their nature.The dislocations are defects which
occur through a line; crystallographic registry is lost the lines
through the crystal along which . Its main role in the
microstructure of materials is monitoring of subsequent plastic
deformation of the crystalline solids and the yield strength at
normal temperatures. These dislocations take part in growth of
these crystals and also in the respective structures and their
interfaces in between these crystal structures. The electrical
defects seen in semiconductors as well as optical materials but
they are always undesirable but almost .Volterra in the nineteenth
century introduced the concept of a dislocation in a solid. But
their importance to the deformation of crystals were recognized
until much later . Until late 1930s the idea of dislocations as the
source of plastic deformation did not appear . It has been possible
since the 1950's to identify and analyse dislocations directly by
techniques like these namely x-ray topography and transmission
electron microscopy . Dislocations are studied almost exclusively
in Materials Science eventhough they influence many aspects of
physical behavior .
Figure 1.2
We first make a planar section part way through it, as shown by
the shaded region in the figure to create edge dislocations in this
body . Then we make fix the region of the body under the wedge, and
to the body above the cut introduce a force that is about to move
it in the direction of this cut, as shown in Fig. 1.2. the slips
over the lower by the vector distance b or the upper part slides,
the slips below by the vector b, which is a relative displacement
between the two sections of our cut. The plane of the cut, is
called plane of slip where the slip occurs. This cut is constrained
at its end and alsofinite , so materials are accumulated there.
There is a linear discontinuity in the material at the end of the
cut, or equivalently, the boundary of the surface region of the
slipDimensions of edge dislocations are comparatively simple to
visualise if the crystal structure is a simple cubic crystal . The
distribution of atoms around the line of dislocation is more
complex in crystal structures in real life. In principle, the
Burgers of any crystalline dislocation can be a lattice vector.
Stating an example, it may be possible geometrically for any edge
dislocation being the stopping of any number of lattice surfaces.
Originally, the Burgers vector is constantly equal to almost the
shortest lattice vector inside the crystal lattice. This is because
energy gradient of the dislocation line, known as line energy, or,
in a just variable condition, with the square of the magnitude of
b, the line tension of the dislocation, is further more .
Figure 1.3
By determining the slip that would be required to make it we can
always find the Burgers vector, b, of any dislocation , but often
this is an inconvenient.A geometric construction known as Burgers
circuit uses an even simpler method . First choose a direction for
the dislocation line to make the Burgers circuit, and then by
taking each (unit) steps along the lattice vectors, make a
clockwise closed circuit in the perfect crystal . The Burgers
vector, b, of the dislocation is the vector (from the beginning
position) which is needed to finish the circuit, and measure the
total displacement with respect to a virtual observer completeing a
loop around this dislocation which is normally enclosed in an ideal
crystal.
Figure 1.4
Moving on to other dislocations, dislocations in normal crystals
dont have a purely edge behavior. Their Burger's vectors lie at
different inclinations to the direction of their lines. In farthest
cases, the Burger's vector is collateral to these dislocation
lines, which are the depiction of screw dislocations. Same as an
edge dislocation the energy per unit length of screw dislocations
are directly proportional to the squares of their Burger's vector.
So the smallest lattice vectors which are reconcilable with the
directions of their lines are normally the Burgers vector of the
screw dislocation. An edge dislocation differs from screw
dislocation in their geometry and by the way it deforms plasticly.
Qualitative the most relevant differences deals with their
direction of motion under external forces and their freedom of
movement relative to it. Unlike edge, screw dislocations glide in
any of the planes.The Burger's vector will be lying parallel to the
dislocation lines so both will be in any surface that is containing
these dislocation lines, and also screw dislocations could move
normal to its line in any directiion. In real materials
dislocations are most predominantly not pure edge or purely screw,
but mixed dislocations in nature where Burger's vectors lie at
angles midway through local directions of dislocation lines. Due to
the nature by which they interact with other microstructure
elements, ordinarily the dislocation lines are curved. Since these
dislocations bound an area that is slipped by their Burger's
vector, the Burger's vector is same at all points in their
dislocation lines. So the nature of the curved dislocations change
continuously all along their length. Hence often it is very useful
considering dislocations as borders of planes on which the slip
occurs and not as defects with specific local configuration of
atoms .
Figure 1.5
2.2 IMAGE FORCE
Dislocations in the wake of free surfaces feel attracting forces
towards the material surface, known as an image forces. The
dislocations would be still feeling a force directed to their
interfaces if these free surfaces are replaced by interfaces with
materials of lesser modulus of elasticity, and they will be lower
than that of free surfaces. Such may be brought under the class of
dislocations seen in semi infinite domain. But free nano-crystals
have two or more surfaces at a sp[ecific distance from theiur
dislocation lines and the formulae used for theoretical analysing
of semi infinite domains are no more valid. Image forces can be
resolved into two with respect to plane of slip. a) Glide component
parallel to it. b) climb component normal to it.
The literal term image force came due to hypothetically negative
dislocations which are assumed to be existed at the opposite side
of the free surface for calculating the original force. When the
materials having dislocation bond with a different material with
lesser elastic modulus, the dislocations would still be having a
force of attraction from the interface but lower in magnitude than
for free surfaces.
The force experienced by the dislocation become repulsive in
nature if the material across the interface is elastically harder,
(depends on elastic modulus and the exact configuration ). This can
lead to equilibrium positions of dislocations inside the crystal
(otherwise on free surfaces). Also, if a constrained surface
replaces the free surface , then the nature and magnitude n the
forvce experienced. The image force could be then extended till
such cases also, remembering that the making of image dislocations
maybe possible or not , also a crystal bounded by free surface
(where, the material with opposite dislocatiuon replaces the vacuum
region).
Such cases,having interfaces adjacent to the dislocations, could
be analyzed a dislocation in a semi infinite domain. When
dislocations are nearer to interfaces, deformations inside the
interface wont be ignored and is considered in the calculationof
the energy that the system possess. For free-standing
nano-crystals, superposition of all the image forcesgives the total
force experienced by the system and more than two surfaces are at
comparable distances from the line of dislocation.
Also,the deformation inside the domain becomes even more
relevant with diminishing size of the crystal,. This shows that the
standard theoretical formulae used in the analysis of semi-infinite
domains for free-standing nano-crystals, wont give exact results.
These image forces experienced by any dislocation are divided into
a) glide component which is collateral to its plane of slip b)
climb component normal to its plane of slip. The glide component of
image force exceeding PN force in large crystals, leads to the
exhaustion of their dislocation from areas close to the surfaces
but these forces lead to purely almost dislocation-free lattices in
vcase of nano-crystals. The dislocation climb is feasible only at
large temperatures so only then climb components are present.
FEM or the Finite element method is a utility at nanoscale are
highlighted through works of Benabbas, Zhang ,Bower , Rosenauer and
many other researchers. In their interactions with other dstress
fields andand in the case of nano-crystals FEM has been proved
useful. Belytschko with his co-workers especially inthe area of
interfaces(both type) and in the study of dislocations has made
very important contributions. With the help of various techniques
likeJ integrals and enrichment of finite element
spacedevelopedmethods applying to anisotropic and non linear
materials.Evenon a small scale or scale of a few lattices ,a
continuum approach depicts the dislocation behaviour well . (a)
distribution of the dislocations or other internal-stress fields,
(b) external forces, loading and boundary conditions, (c) geometry
of domain or (d) material distribution and systems with such
complex nature FEM prove to be a handful. But, no model is valid
when their dislocations are too close to their interface for models
based on linear elasticity.
Only a very few structures have been identified zero stiffness
structures so that we can make an exaple of.. Such structures show
mechanism-like properties because they are in equilibrium along
continuous paths in a configuration ( unlike normal structures that
dont undergo large deformations, they exist in a vast range and
configurations having same energy) .There are identified negative
stiffness materials also like buckling beams so the notion is that
both positive and negative stiffness structures together form
composite zero stiffness materials.
2.3 FINITE ELEMENT METHODOLOGYA demonstration of FEM is shown
here but not detailed. [18]. edge type dislocation for aluminium
(a0 4.04 A, plane of slip system is : h110 i{111} , Burger's vector
(b) is a0/2 , 2.86 A , G 26 . 18 GPa, and 0.348 ) by imposing
stress free or eigen strains,is simulated with the help of atoms in
half plane as shown in figure. By assuming that the propeties of
bulk materials can be used at a length scale of simulation,here an
isotropic condition of plane strain is considered here.Calculation
of the net energy of our system and the energy of the deformed
configuration is done using "ABAQUS" for different edge
dislocations(positions) along the domain and is henceforth
shown.
.
RESULTS3.1 Results and discussion
As the dislocation is placed very close to the free surface when
the both length of the plate L and the thickness is too small, the
energy possessed by the dislocation decreases (here we consider the
free lateral surface). From these observations we could easily
infer that the free surface attracts the dislocations and the
evolving of the much talked image force starts here. The image
force is determined from the slope of energy-distance curve.
If we consider some domains that are free to bend, the nature of
our energy contour becomes flat. From this we can say that even
though the system goes through a series of changes, the energy of
the system is unaffected (virtual) and our material could be called
a zero stiffness material structure. There are noted points which
denote the extend to which a zero stiffness material is represented
by our free bending domain. Looking through two plots a)the free
bending domain , as we know has lower energy (centre dislocations
34%) but the second point is somewhat surprising and says that
there is no more affectof the free surface. By the way our second
point was that the energy landscape has suddenly become flat. So
that also influences the (x stress contour plot very close to
dislocation position) restoring of left right mirror symmetry in
contour plot of x.
CONCLUSIONS
Some points worthy of comment are as follows.
The considerate system does not show zero stiffness if the
dislocation is positioned nearer to the free surface.of the crystal
. This may be similar to the range of configurations for some
zero-stiffness structures, showing zero stiffness; for instance, in
tensegrity the zero stiffness property exists only for a limited
range of configurations .
The range of configurations may not be truly continuous as in
the case of a zero-stiffness structure because the dislocation has
a minimum change in position with reference to theBurgers
vector.
The dislocation resides in a local energy minimum and has to
overcome a Peierls barrier(energy gap ) to reach the next position
(a Burgers vector apart). These Peierls oscillations (maxima and
minima of energy) have not been taken into account
If the plate is extremely long then there will be a region at
the centre of the domain which will show a flat energy landscape of
a trivial kind (as the free surfaces are far away).
The bending is of importance only when the thickness of the
plate is on the nanoscale. The scenario is different in case of
zero-stiffness structures, which are measured on the
macroscale.
Neglecting the core energy does not affect our conclusions,
beacause the core energy is a constant term which is always added
to the energy, except for a position of the dislocation nearer to
the surface, i.e. few Burgers vectors, which do not affect the
slope of the curves
REFERENCES
1. Materials analogue of zero-stiffness structures by Arun Kumar
and Anandh Subramanian, Department of Materials Science and
Engineering, Indian Institute of Technology, Kanpur, published on:
08 February 20112. Image forces on edge dislocations: a revisit of
the fundamental concept with special regard to Nano crystals by
Prasenjit Khanikar; Arun Kumar; Anandh Subramaniam, Department of
Materials Science and Engineering, Indian Institute of Technology
Kanpur.3. Materials Science by J.W. Morris, Jr4. Determination of
Image Forces in Nanocrystals using Finite Element Method by
Prasenjit Khanikar, Arun Kumar, Anandh Subramaniam , Department of
Material and Metallurgical Engineering, Indian Institute of
Technology Kanpur, India.