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.