Defects in Solids: Geometrical Structure and Internal ...home.iitk.ac.in/~ayanrc/sota.pdf · 3-D crystalline solids in the present report. First, a brief historical review of the

State of The Art of

Defects in Solids:

Geometrical Structure and Internal Stress Field

by

Ayan Roychowdhury

Contents

1 Introduction 1

2 Elementary Defects in Solids 6

2.1 A Brief Historical Review of Dislocation Concept . . . . . . . . . . . . . . . . . . . . . 6

2.2 Crystal Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Crystal Disclination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Dislocations in Elastic Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Compatible Deformation: Compatibility Conditions . . . . . . . . . . . . . . . 14

2.4.2 Displacement Field of a Compatible Deformation . . . . . . . . . . . . . . . . . 16

2.4.3 Multiply-Connected Bodies: Weingarten’s Theorem . . . . . . . . . . . . . . . 17

2.4.4 Volterra Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Modelling of Crystal Defects by Volterra Dislocations . . . . . . . . . . . . . . . . . . 24

2.5.1 Kroner’s Continuized Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.2 Burgers Vector of a Dislocated Continuized Crystal . . . . . . . . . . . . . . . . 25

2.5.3 Frank vector of a Disclinated Continuized Crystal . . . . . . . . . . . . . . . . 28

2.6 Elastic Fields of Elementary Straight Volterra Dislocations . . . . . . . . . . . . . . . 29

2.6.1 Straight Edge Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6.2 Straight Screw Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.3 Straight Disclination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

i

2.6.4 Corrections for Simulating Crystal Dislocations . . . . . . . . . . . . . . . . . . 34

2.7 Presence of Disclinations in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Continuous Distributions of Defects 37

3.1 Continuous Distribution of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Continuous Distribution of Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Elastostatics Problem for Continuous Distribution of Defects . . . . . . . . . . . . . . 41

4 Differential Geometric Structure of a Solid Body: Theory of Inhomogeneities 43

4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Geometric Objects on the Material Manifold and Two Basic Identifications . . . . . . 45

4.3 Two Relevant Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Consequences of Zero Curvature: Necessity of a Constitutive Base for Defect Theories 52

4.5 Noll’s Inhomogeneity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5.1 Continuous Body, Local Configuration and Tangent Space . . . . . . . . . . . . 53

4.5.2 Simple Bodies, Material Isomorphisms and Material Symmetry . . . . . . . . . 55

4.5.3 Material Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5.4 Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5.5 Relative Riemannian Structure, Contortion . . . . . . . . . . . . . . . . . . . . 61

4.5.6 Contorted Aeolotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.7 Special Types of Materially Uniform Bodies . . . . . . . . . . . . . . . . . . . . 65

4.6 Extension of Noll’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Interfacial Defects and their Distribution 68

5.1 Modelling of Interfacial Defects by Arrays of Line Defects . . . . . . . . . . . . . . . . 69

5.2 Distribution of Interfacial Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ii

6 Conclusion and Open Questions 71

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Chapter 1

Introduction

The theme of this present report is defects in crystalline solids and their mathematical modeling.

Crystalline solids form a subclass of a large group of ordered media. They are ordinary 3-D contin-

uum, at least in mathematical idealization, having specific microstructures characteristic of the media

considered. For example, ferromagnetic materials, liquid helium, various phases (smectic, cholesteric,

nematic etc.) of liquid crystals and solids (amorphous or crystalline) — each have their own char-

acteristic parameters that depict their ordered microstructures. Note that the term ‘order’ has a

Figure 1.1: Subclasses of ordered media.

1

(a) Metals (b) Smectic phase of liquid crystals.

(c) Ferro-magnets. (d) Amorphous glass.

Figure 1.2: Examples of ordered media.

special connotation in this specific branch of condensed matter physics: Amorphous microstructure is

categorized as ‘ordered media’, though they are structureless or disordered in ordinary sense. In case

of such materials, ‘disorder’ in our sense means presence of localized well-defined structures which are

otherwise unlikely to occur in other regions of the media. It is the localization of disorders in that

ordered continuum which we are interested in, not in its nice everywhere-perfect orderly structure.

The reason is that defects play the role of driving mechanisms for the dissipative processes (plasticity,

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diffusion etc.) taking place inside the structure. They also cause strain-hardening in metals. Steel

would not have its characteristic strength if it had not localized microstructural defects present in

its b.c.c. or f.c.c. crystalline structure. Defects also act as sites of crystal nucleation and growth. A

major practical application of defects in liquid crystalline materials is liquid crystal display screens.

Probably, the most interesting example of defects in ordered media and their significant role to gov-

ern its behaviour is our own universe: elementary particles we all are composed of are nothing but

localized defects in certain fields in 4-D space-time. Another interesting example from physics is

electromagnetic wave— they are localized defects in electromagnetic fields.

(a) Grain boundaries in

metal.

(b) Domain wall in ferro-

magnets.

(c) Disclinations in liquid

crystals.

(d) Electro-magnetic wave. (e) LCD screen.

Figure 1.3: Defects in ordered media.

So to talk of defects in a particular media1, we have to specify first what would be a defect-free state

of that media. A defect-free state may possibly correspond to the validity of certain rule everywhere,

1To be more technical, defects are singularities in the order parameter field.

3

while constructing the orderly structure throughout the 3-D space. For example, when we construct

a defect-free b.c.c. crystalline solid, we always try to make it sure that each cubic building block has

been properly translated exactly by one lattice parameter length in the 3 mutually perpendicular

directions that lie along the 3 mutually perpendicular edges emanating from the vertices of the cubic

lattice. If, by chance, we do any mistake to maintain this rule of translation, we would end up with a

defect in that mess-up zone. The drama begins when there are many such defects present throughout

a crystalline solid body and we force to deform it by some external agency. Then these defects would

interact with each other and result in interesting structural properties. Similar thing happens in other

type of ordered media also.

In the last two passages when talking about defects, we misdid something which is very sus-

ceptible to overlooking and passing-through. We used the words ‘continuum’ and ‘crystal’ almost

synonymously though they have poles-apart interpretations. Continuum is just a mathematical ide-

alization of real materials made up of discrete particles. In making such idealization, what we do

usually is to bring two adjacent material particles infinitesimally close together and subsequently

scale down their mass accordingly so that the overall mass and volume conservation are maintained.

Obviously splitting up of particles is a must to do such things. Then, the operation of translation

of building blocks, as seen in the previous paragraph, or in certain cases, may be accompanied with

rotations by certain angle, calls for some group theoretic structure super-imposed on the continuum

idealization. So, to mathematically model a given ordered media, we associate a certain group struc-

ture characteristic of that media to a bounded region of 3D space, called continuum. Breaking (or

discontinuity) of such group structures would then characterize defects mathematically. We are saying

these things in very loose terms here, but we will make everything rigorous in subsequent chapters.

Another important point which we must mention in any talk of defects in ordered media is the

internal stress state. It is they and they only why we study defects at all for. Structural defects

surround themselves with strain fields and, consequently, stress fields. When present in many, defects

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interact with their respective stress fields. As stress is a vector valued field, they add up to large

Figure 1.4: Stress field of distribution of dislocations

values somewhere, while get canceled out in other places. The sites of large added up values are where

the defects interlock and show up ‘work-hardening’ as a result. The sites with low valued stresses

are vulnerable to external forcing agency and render plastic flow. Internal stress fields are sometimes

called as residual stresses because they exist without any presence of external forcing agency.

In the next chapter, we will talk specifically of elementary defects in crystalline solids.

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Chapter 2

Elementary Defects in Solids

Defects in 3-D solids (or any other ordered media) are classified according to their dimensionality.

Point defects or 0-dimensional defects are the well-known vacancies, substitutional atoms or intersti-

tials. Dislocations and disclinations constitute the line or 1-dimensional defects. Interfacial or 2-D

defects are also there in the forms of grain and twin boundaries. Among these, the 1-dimensional

defects are the most important ones and mostly studied. Point defects are also important and play

major role in diffusion phenomena, but we will exclude them from our present discussion. Interfacial

defects present in polycrystalline materials are the main dictators in plastic deformation. Inadequacy

of a proper general theory of arbitrary interfacial defects has led to prosper the research in linear

defects, because still now, to calculate the physical fields surrounded by interfacial defects, they are

mathematically modeled as arrays of linear defects. Hence, we will mainly focus on linear defects in

3-D crystalline solids in the present report. First, a brief historical review of the dislocation concept

(references: [1, 2]).

2.1 A Brief Historical Review of Dislocation Concept

The idea of dislocation as a material transporting agency was always there from general facts like

crawling of a caterpillar or a puckered carpet. In the late 19th century, when the actual atomic

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structure of materials had not yet arrived, few nobles (Burton [3], Larmor [4]) tried to explain

movement of materials internal to a solid body as localized modifications of the internal structure

and its subsequent transportation. They called them strain figures. On 1900, Larmor initiated the

idea of an internally strained ring which is formed by first twisting a metallic wire and then welding

the free ends together. Various elastic fields of dislocations in an isotropic continuum were studied

around the first decade of the next century by Weingarten [5], Timpe [6] and Volterra [7], though they

were still not named ‘dislocations’. They appeared in the theory as incompatibilities of an otherwise

continuous and everywhere differentiable strain field which produced multiple valued displacement

fields (or single valued displacement fields having finite and rigid body jumps across certain cut

surfaces). Volterra called them ‘distosioni’. The term ‘dislocation’ is due to Love [8]. These continuum

mechanical ideas were still not related to crystal plasticity. Around this time, Ewing [9] was the first

to introduce the idea that structural anomalies of this kind might cause mechanical hysteresis. Later,

in 1920’s, through the works of Prandtl [10] and Dehlinger [11], the idea of crystal dislocations as

motive behind plastic deformation started gaining grounds. Finally, the foundations were laid by

Orowan [12], Polanyi [13] and Taylor [14] in 1934. They produced the modern diagram of a crystal

dislocation and described how it glides along the slip plane to cause plastic flow. Taylor was the

first to point out the possible mathematical modelling of crystal dislocations by Volterra dislocations.

Burgers extended the two dimensional picture of Taylor (edge dislocation) to screw dislocations in

3-D. Later, Frank described its significance. Other important names to develop the theory further

are Shockley, Nemeyi, Nabarro, Seitz, Read, Koehler, Mott, Cohen, Cottrell, Weertman etc.

2.2 Crystal Dislocation

Following is the celebrated diagram given by Taylor that illustrated, for the first time, the plastic

deformation of a chunk of single crystal by formation and subsequent gliding of an edge dislocation.

Instead of sliding of the portion above the slip plane as a whole, the deformation proceeds by the

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Figure 2.1: Taylor’s model of plastic deformation by dislocation glide: (a, d) before, (b, e) during

and (c, f) after deformation.

formation of an edge dislocation as shown which requires much lesser shear stress. Taylor also

explained, by assuming that every crystal contains many such dislocations, why the experimental yield

strength of materials is much less than the theoretical yield strength. As evident from figure 2.2(a),

dislocations surround themselves with distortion fields. Taylor approximated these distortion fields

by that of Volterra dislocations, which we will discuss later. Burgers considered the possibility of

(a) Edge dislocation (b) Screw disloca-

tion

Figure 2.2: Edge and screw dislocation is a single crystal.

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presence of another type of atomic arrangements that can give rise to a dislocation, as shown in

figure 2.2(b). This is called a screw dislocation. The adjacent figure illustrates the formation of an

(a) True Burgers

vector.

(b) Local Burgers vector. (c) Local Burgers vector.

Figure 2.3: Burgers vector.

edge dislocation in a hexagonal triangular lattice.

Burgers Vector of a Dislocation

(a) True Burgers vector. (b) Local Burgers vector.

Figure 2.4: Burgers vector.

A dislocation is characterized by its line position and its Burgers vector. Dislocation line, in case

of edge dislocation, is the edge of the extra atomic half plane inserted inside an otherwise perfect

single crystal, and in case of screw dislocation, the edge of the plane along which the two cut lips are

slided (see figure 2.2(b)) relative to each other. Now, if a closed circuit, surrounding the dislocation,

on the deformed crystal after insertion of the half plane, is mapped on the ideal perfect reference

crystal, as shown in the figure below, the circuit will not close on the perfect crystal and the closer

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failure, with a chosen direction, is called the true Burgers vector. The adjective true signifies that this

type of Burgers vector is a constant because its magnitude is always equal to the atomic spacing. On

the other hand, if we map a closed circuit on the perfect crystal to the deformed dislocated crystal,

provided it encircles the dislocation, the mapped circuit will still not close. Now the closer failure

will depend on position and circumference of the mapped circuit, and with a specifically chosen sign,

it is called the local Burgers vector, for the obvious reason.

Note that, Burgers vector of an edge dislocation is perpendicular to its line while, for a screw

dislocation, it is parallel.

2.3 Crystal Disclination

Dislocations arise due to the translational symmetries of a crystal, whereas disclinations are defects

due to its rotational symmetries. See the figures below for these type of rotational defects in a

hexagonal triangle lattice.

Figure 2.5: A perfect hexagonal triangle lattice with rotational symmetry angle 60.

Figure 2.5 shows a perfect crystal with hexagonal triangle lattices that have a 60 rotational

symmetry angle. Figure 2.6(a) illustrates the method of constructing, in that perfect crystal, a

negative wedge disclination of Frank vector −60 a: taking out a wedge shaped element of wedge

angle 60 and subsequent merging of the two cut lips. When, instead of taking out, a wedge of same

angle is inserted, we end up with a positive wedge disclination of Frank vector +60 a (figure 2.6(b)).

Here, a is an unit vector in the direction coming out of the plane of the page.

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(a) Formation of a minus 60 wedge disclination.

(b) Formation of a plus 60 wedge disclination.

Figure 2.6: Edge and screw dislocation is a single crystal.

Insertion or removal, and subsequent welding up, of such finite wedges induce huge amount of

strains in a disclinated crystal. Strain and stress fields are singular at the disclination line and

also of very large values far from the core. Solids cannot sustain such large values of straining and

attempts to introduce a disclination typically end up breaking the crystal. Hence, appearance of single

disclinations is very unlikely in solid crystals. However, nearby presence of two wedge disclinations of

opposite sign dramatically reduces down the severe straining of single disclinations. These are known

as disclination dipoles. See the figure below. Notice that in the disclination dipole of figure 2.7,

spacing between the two constituting disclinations is one lattice parameter, which is the nearmost

spacing possible in crystals. This is known as an ‘infinitesimal’ disclination dipole. Now, compare

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Figure 2.7: A disclination dipole composed of a +60 (top) and a −60 (bottom) wedge disclination

in a hexagonal triangle crystal.

(a) Removal of an atomic half plane. (b) After welding, a negative edge dislocation

forms.

Figure 2.8: Edge dislocation in hexagonal triangle crystal.

this lattice arrangement with that of figure 2.8(b), which is a negative edge dislocation of the same

hexagonal triangle lattice. They look similar, except the more number of extra atomic half planes

in figure 2.7. Hence, an ‘infinitesimal’ disclination dipole is topologically equivalent to a straight

edge dislocation. Disclination dipoles have been observed in real solid crystals, since their elastic

states are comparable with that of edge dislocations. We will again see this fact in case of continuum

disclinations later.

Theory of crystal disclinations was developed mostly by Anthony.

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Frank Vector of a Disclination

The angle of the inserted or the removed wedge, with a preferred sign, is known as the Frank vector

of the disclination. For wedge disclinations, this angular vector is parallel to the disclination line. For

twist disclinations, it is perpendicular to the disclination line.

2.4 Dislocations in Elastic Continua

As mentioned in the historical review, the idea of dislocation in elastic continuum appeared as the

possible presence of multiple-valued displacement field or incompatible deformation field. We will

first discuss what we mean by a compatible elastic deformation, what are the necessary and sufficient

conditions to ensure it and, then, the incompatible deformation. References are mainly [15, 16].

The story begins with the constitutive assumption of first grade elastic materials which states

that the strain energy density W (r) at the current location r of a representative material point is

given by W (r) = W (β(r), r), where β(r) is some second order tensor field that describes how a first

order neighbourhood about r is getting deformed or mapped from the current configuration to some

other configuration in the 3-dimensional physical space. This type of constitutive law restricts the

dimension of the state space to 9, with β as the independent state variable.

Principle of material frame indifference further restricts the constitutive law to a 6-dimensional

sub-domain of the 9-dimensional state space, namely the space of symmetric second order tensors.

So in all type of elastic deformation of first grade materials, existence of a symmetric second order

tensor (having 6 dimensions) ε(r) is always guaranteed. The rest 3-dimensions can be included into

a 3-dimensional vector quantity ω(r). The knowledge of ω(r) is, of course, necessary to completely

specify the mapped configuration of the first order material neighbourhood, though it apparently

does not affect the energetics of the body.

In this report, we will restrict ourselves to linear elasticity or small deformation theory only, which

means ||β|| << 1 (||β|| =√

tr (βTβ)).

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2.4.1 Compatible Deformation: Compatibility Conditions

By compatible elastic deformation, we mean that the ever-existing 6-dimensional ε and the 3-

dimensional ω are so obliging that they together constitute a 9-dimensional β = ε + ω, where ω

is a skew tensor having axial vector ω, and, further, β(r) comes from a 3-dimensional vector poten-

tial u(r), called the displacement, as

β = ∇u. (2.1)

Here, the gradient ∇ is taken with respect to the current configuration of the neighbourhood around

r. Thus, we can set

ε =1

2[∇u+∇uT ] (2.2)

and

ω =1

2[∇u−∇uT ] or ω =

1

2∇× u. (2.3)

Hence, compatible deformation means a certain type of conservative process where a three dimen-

sional vector potential u contains all the information regarding deformation. So we do not need the

full 9-dimensional state space of β; the 3-dimensional space of u suffices. This statement is equivalent

to the statement

∇× β = 0 (as ⇔ β = ∇u), (2.4)

or, in terms of ε and ω,

∇× ε+ (∇ · ω) I −∇ω = 0, with ε = symβ, ω = skwβ. (2.5)

Here, the second order tensor (∇× β) is the curl of the second order tensor field β, which is defined

as (∇× β)c = ∇× (βTc).

Equation (2.2) and (2.3) implies, respectively,

∇× (∇× ε)T = 0 (2.6)

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and

∇ · ω = 0, (2.7)

which are known as, repectively, the strain and rotation compatibility equations. By (2.7) and (2.5),

we get

∇× ε−∇ω = 0, with ε = symβ, ω = skwβ. (2.8)

Equations (2.6) and (2.7), individually, are both the necessary and sufficient conditions for the exis-

tence of vector potentials φ and ψ, distinct in general, that satisfies

ε =1

2[∇φ+∇φT ]

and

ω =1

2∇×ψ.

In order to ensure φ = ψ = u, we need to specify the condition (2.8) additionally. Otherwise, the

potentials φ or ψ would not be a valid displacement which satisfies (2.1). Thus, strain or rotation

compatibility conditions, by their own, does not infer compatibility. Specifying either of them along

with their mutual compatibility condition (2.8) ensures that the deformation is indeed compatible.

We can reformulate the above result in terms of another set of independent variables: ε, κ,

where κ = (∇ω)T. Validity of equation (2.3) (or rotation compatibility) implies that

trκ = 0. (2.9)

To ensure the existence of ω, which is a must to completely specify the mapped material neighbour-

hood as stated earlier, we need

∇× κT = 0, (2.10)

which comes from the very definition of κ. Finally, to ensure that the set ε, κ indeed form a

compatible pair to ensure the existence of u, we need additionally

∇× ε− κT = 0, (2.11)

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which comes from (2.5) and (2.9).

2.4.2 Displacement Field of a Compatible Deformation

All the compatibility conditions stated in the last subsection, if valid over a simply connected region,

ensures the existence of a unique displacement field, that specify the mapped configuration from

the current configuration of the whole material body. Hence, in this sense, the above compatibility

conditions are integrability conditions for the vector field u(r). Now we will prove this statement and

find the displacement field with given, say, the pair of basic variables ε, κ which are sufficiently

smooth and compatible. Given a path connecting a chosen fixed point r0 to any arbitrary point r

inside a simply connected region V , we can write

ω(r) = ω(r0) +

∫ r

r0

dω

= ω0 +

∫ r

r0

κT(r′) dr′ (2.12)

since κ = ∇ω and, similarly, since β = ∇u,

u(r) = u(r0) +

∫ r

r0

β(r′) dr′

= u0 +

∫ r

r0

[ε(r′) + ω(r′)] dr′

= u0 +

∫ r

r0

[ε(r′) dr′ + ω(r′)× dr′]

= u0 +

∫ r

r0

[ε(r′) dr′ + dω(r′)× r′ − dω(r′)× r′]

= u0 +

∫ r

r0

[ε(r′) dr′ − dω(r′)× (r− r′)+ dω(r′)× (r− r′)]

= u0 −[ω(r′)× (r− r′)

]r′=r

r′=r0

+

∫ r

r0

[ε(r′) dr′ − (r− r′)× dω(r′)]

= u0 + ω0 × (r− r0) +

∫ r

r0

[ε(r′) dr′ − (r− r′)× κT(r′)dr′]

or

u(r) = u0 + ω0 × (r− r0) +

∫ r

r0

U(r, r′)dr′, (2.13)

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where

U(r, r′) = ε(r′)− (r− r′)× κT(r′). (2.14)

Here, u0 and ω0 are known constants. The rotation and displacement fields given by the above

expressions are single valued if the integrals involved are path independent which requires vanishing

of the cyclic integral of their integrands taken over every closed curve C in the simply connected body.

Vanishing of the first cyclic integral, i.e.

∮Cκ(r′) dr′ =

∫S∇r′ × κT(r′) dS′ = 0 (2.15)

follows from Stoke’s theorem and equation (2.10), while vanishing of the second cyclic integral, i.e.

∮C

[ε(r′)− (r− r′)× κT(r′)] dr′ =

∫S∇r′ × [ε(r′)− (r− r′)× κT(r′)] dS′

=

∫S∇r′ × [ε(r′)n′ − (r− r′)× κT(r′) n′] dS′

= 0 (2.16)

also follows from Stoke’s theorem and equations (2.10),(2.11).

Hence, the displacement and rotation fields are uniquely determined if the (sufficiently smooth)

variables ε, κ satisfy the compatibility conditions over a simply connected region.

2.4.3 Multiply-Connected Bodies: Weingarten’s Theorem

When the current configuration V of the linearly elastic body is not simply connected, we can, no

longer, apply Stoke’s theorem to make the above cyclic integrals zero, because the classical form of

Stoke’s theorem we were using is valid over only simply connected regions. In figure 2.9, a doubly

connected toroidal region is shown, along with few mutually homotopic curves. Contour C0 is passing

through the fixed point r0 and is homotopic (i.e. smoothly deformable) to the contour C. Since both

of these contours are irreducible, i.e. cannot be shrunk to a point that lies on the doubly connected

region, cyclic integral along these contours cannot be reduced down to a surface integral using Stoke’s

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x y

z

S

C0

C

C1

C2

r0

r

A

P

Q Γ

Figure 2.9: A doubly connected toroidal region. C0 and C are mutually homotopic, but irreducible;

C1 and C2 are both mutually homotopic and reducible.

theorem. Hence, after putting r = r0 and taking the cyclic integral along C0 in equation (2.12), it

boils down to

[[ω]](r0) =

∮C0

κT(r′) dr′, (2.17)

where [[ω]](r0) means the jump in the value of ω on crossing the point r0 along the positive direc-

tion of the closed contour C0, as shown. It can be proved, after making a cut and rendering the

toroidal region simply connected, that the value of the above cyclic integral is actually indepen-

dent of the chosen contour, provided the contours chosen are mutually homotopic. The values of

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the integral for different sets of mutually homotopic irreducible contours are, of course, different1.

To show this, we make a cut along the surface A, as shown, to render the body simply connected

and choose a curve Γ in that surface that connects the two intersection points P and Q of the

two mutually homotopic curves C0 and C with A. Now, the green patch of area, S, in that sim-

ply connected region has become an open surface, bounded by the closed curve ‘P toQ along Γ →

travel contour C along its positive direction, with starting from and ending atQ→ Q toP along Γ

→ travel contour C0 along its negative direction, with starting from and ending atP ’. On this open

surface, with the mentioned boundary, we can apply Stoke’s theorem, because the region has now be-

come simply connected. The integrand and, hence, the integral will vanish because of equation (2.10).

The proof follows from this point after breaking off the boundary curve into the four distinct trav-

elling paths and noting the fact that, out of them, two were actually identical except their opposite

travelling directions. Hence, they together make zero contribution, and we are done. Hence,

[[ω]](r ∈ C) = Ω(C), (2.18)

where

Ω(C) =

∮CκT(r′) dr′ =

0, if C is reducible;

a non-zero constant vector, if C is irreducible.

Ω is known as the Frank vector.

Along the exactly same line, we can also show, from equation (2.13), that, in a multiply-connected

body, after following any irreducible contour C, the displacement vector u also suffers a jump of an

amount

[[u]](r ∈ C) = b(C) + Ω(C)× r, (2.19)

where

b(C) =

∮C

[ε(r′) + r′ × κT(r′)] dr′ =

0, if C is reducible;

a non-zero constant vector, if C is irreducible.

.

1for all reducible contours, like C1, C2, the integral will be zero, by Stoke’s theorem.

19

The displacement jump is not a constant, but a rigid body displacement.

So, at every point of a doubly connected domain, displacement field has multiple (two) values. It

is no longer a function now in ordinary sense. We can make it an ordinary function by a trick: We cut

the doubly connected body by surfaces like A to render it simply connected and then say that, over

that simply connected body, the displacement field is single valued having discontinuity of amount

[[u]] across the cutting surface A.

Weingarten extended the above results to bodies with arbitrary connectivity:

S1

S2S3

S4

Figure 2.10: A 5-tuply connected region. The four cutting surfaces S1, S2, S3 and S4 will render it

simply connected.

Weingarten’s Theorem. Let V be an n-tuply connected region and let S1, S2, . . . , Sn−1 be

n − 1 non-intersecting cutting surfaces (with chosen orientations) that render V simply connected.

Assume that the fields ε and κ are single valued and sufficiently smooth in V , with trκ = 0,

∇×κT = 0 and ∇×ε−κT = 0. Then the jumps of the rotation and the displacement vectors across

any surface Sk, k = 1, . . . , n− 1, from its negative to positive orientation, will be

[[ω]](r ∈ C) = Ω(C) (2.20)

and

[[u]](r ∈ C) = b(C) + Ω(C)× r, (2.21)

20

where

Ω(C) =

∮CκT(r′) dr′ =

0, if C is reducible;

Ω, a non-zero constant, if C is irreducible and encircles one hole;

n Ω, if C is irreducible and encircles n holes.

and

b(C) =

∮C

[ε(r′)+r′×κT(r′)] dr′ =

0, if C is reducible;

b, a non-zero constant, if C is irreducible;

n b, if C is irreducible and encircles n holes.

.

Here, r is the position vector of a current point on Sk and C is any closed contour in V .

2.4.4 Volterra Dislocations

Volterra dislocations are nothing but the deformed states of a smoothly strained hollow cylinder (a

doubly connected body). They have multiple valued displacement fields, or in other words, single

valued displacement fields having discontinuities of the aforementioned type. The famous Volterra

construction of elementary dislocations is shown in the figure [17] below.

An infinitely long cylinder is first rendered simply connected by cutting it through a half plane,

as shown, and then the two lips of the cut are forcibly given a relative rigid body displacement. The

six possible elementary relative rigid displacements are shown in the same figure. They are called

elementary Volterra dislocations. In fact, the last three states of deformation are now called discli-

nations, and the first three have been retained with the name dislocations. After giving the relative

displacements, the two cut lips are welded together to make the body again doubly connected and

the external forcing agency is removed. The strain field, thus produced, present without any exter-

nal forcing, will be smooth everywhere, according to the discussion of the last subsection. We will

calculate the strain fields later. We will see, then, that the continuous strain fields have singularities

along the longitudinal axis of the cylinder. We call this axis, the dislocation line and the relative rigid

21

(a) u+(r)− u−(r) = b1e1 (b) u+(r)− u−(r) = b2e2 (c) u+(r)− u−(r) = b3e3

(d) u+(r)−u−(r) = Ω1e1×r (e) u+(r)−u−(r) = Ω2e2×r (f) u+(r) − u−(r) =

Ω3e3 × r

Figure 2.11: Elementary Volterra Dislocations.

displacement, the Burgers vector of the corresponding Volterra dislocation. Value of the strain field

of a dislocated cylinder is very high near the cylinder axis, and has a singularity on the axis as just

mentioned. Volterra, and the other founders of the theory, tactfully discarded the possibility of ap-

pearance of such high strains inside the actual material body by the multiply connected construction,

or removing the cylinder core.

Lines of Volterra dislocations just discussed are straight. They can be closed loops also, lying

solely inside the body. These are called dislocation loops of Volterra type. They are constructed

by making finite cuts inside a material body by open surfaces and displacing the two lips of the cut

22

rigidly, followed by an welding. Materials required to fill any produced gaps are supplied or any

produced overlaps are also removed. The closed boundary curve of the open surface will be, then, the

dislocation loop. The strain field will be smooth everywhere, except at the dislocation loop where it

will be singular. Very close to the loop, it has large values. To restrict ourselves within the realm

of linear elasticity, we typically remove the dislocation core (of toroidal shape); hence, rendering the

body doubly connected.

The strain and stress fields of Volterra dislocations are called, respectively, the eigen strain and

the eigen stress because of their existence without any external forcing fields. They are invariably

known as internal strain and stress fields also.

Somigliana Dislocations. The stringent condition of continuity of strain field across the cut-

ting surface, when replaced by somewhat relaxed condition of equilibrium of tractions (t) at the two

lips of the cutting surface, which is actually sufficient to maintain the continuity of the body, what

we get is the Somigliana dislocation. In this case, the displacement jump is not rigid body type but

arbitrary,

[[u(r)]] = f(r),

with

[[t]](r) = 0.

Calculation of strain field of Somigliana dislocation is difficult, so we will not discuss them. These are

being considered recently, in anisotropic elastic continuum, as models for crystal dislocations. But

we will stick to Volterra model only.

23

2.5 Modelling of Crystal Defects by Volterra Dislocations

2.5.1 Kroner’s Continuized Crystal

To model crystal defects by defects in elastic continuum, we need to make a continuum idealization

of the defective crystal. Kroner [18] discussed a nice continuization process that preserve the inherent

symmetry groups of the material in the continuized crystal. A limiting process is done in course

Figure 2.12: Two stages in the continuization of a crystal with dislocations.

of which all atoms are scaled down such that total mass in a given volume remains fixed, and, at

the same time, each dislocation is also scaled down so that its Burgers vector still remains equal to

the lattice spacing of the current stage. It has been shown in [18] that while, macroscopically, two

adjacent dislocations tend to get infinitesimally close together, they are actually getting far and far

away from each other intrinsically. Thus, we end up with a continuum with microstructure, where

dislocations still have discrete existence intrinsically, but are smeared out macroscopically. Hence,

with respect to the reference frame of the external physical space into which the crystal in embedded,

we can use macroscopic elasticity theory and also can talk of densities of dislocations when they are

present in many (see next chapter). Another assumption is small deformation, of course to restrict

ourselves in the regime of linear elasticity.

24

2.5.2 Burgers Vector of a Dislocated Continuized Crystal

Taylor’s model of a crystal edge dislocation by an elementary Volterra dislocation has been shown

in the following figure. The Burgers circuit is also shown. Figure 2.13(a) illustrates a stress-free

(a) (b)

cC

pq

P

Q

bS−

S+S+

S−

S+

K0 K

Figure 2.13: Modelling of a crystal edge dislocation by Volterra dislocation. The true Burgers vector

b is shown along with the Burgers circuit.

reference configuration K0 and figure 2.13(b), a dislocated configuration K with relative rigid body

translation b of the two cut lips S− and S+, of an isotropic linearly elastic solid. If χ : K0 → K

denotes the deformation from the reference state to the current dislocated state, then we can represent

the position vector x of any representative material point in the dislocated configuration as a mapping

from the corresponding position X of the same material point in the reference configuration by the

relation

x = χ(X) = X + u(X), (2.22)

where u(X) is the displacement function w.r..t. the reference configuration. If the mapping χ is a

diffeomorphism, its inverse χ−1 : K →K0 will exist which satisfies

X = χ−1(x) = x− u(x), (2.23)

25

where u(x) is the displacement function w.r..t. the current configuration.

If we take the gradient of equation (2.22) with respect to X, we get

F(X) = ∇Xχ(X) = 1 + β(X), (2.24)

where F = ∇Xχ is the deformation gradient tensor which maps vectors from configuration K0 to

configuration K. Similarly, taking gradient of equation (2.23) with respect to x yields

F−1(x) = ∇xχ−1(x) = 1− β(x), (2.25)

where F−1 is the inverse deformation gradient that maps vectors from configuration K to K0.

Now, as expected, the closed circuit c in the dislocated configuration circulating the dislocation

line, when mapped to the reference configuration, will not close. The closer failure

b =

∮c

F−1(x)dx =

∮c(1− β(x)) dx = −

∮c

du(x) = u(p)− u(q), (2.26)

is a vector in the reference configuration and is constant, because we constructed the dislocated state

K from the reference state K0 by a rigid displacement b of the two cut lips S+ and S−. b is called

the true Burgers vector of the dislocation. If the position vector of our representative material point

(p or q) is x, we can rewrite the above definition as

b = u+(x)− u−(x). (2.27)

Above equation tells us that the displacement field u has a constant jump across the cut surface s.

Again, as from the construction,

χ+(X) = χ−(X + b), (2.28)

⇒ u+(X)− u−(X + b) = b.

Neglecting the second and higher order terms after expanding u−(X + b), the above equation boils

down to

⇒ u+(X)− u−(X)− β−(X) b = b,

26

⇒ b? = u+(X)− u−(X) = F−(X) b (2.29)

where b? = u+(X)−u−(X) is nothing but the local Burgers vector. We can also show, by replacing

X with X − b in (2.28), that

⇒ b? = F+(X) b. (2.30)

The last two relations express the deformation-dependent nature of local Burgers vector.

In the realm of linear elasticity, F(X) ≈ 1, i.e. b? ≈ b.

b has to be necessarily a member of the translational symmetry group of the body.

Note. Few comments about the above analysis:

1. For convenience and also for a natural formulation, we take the reference configuration K0 as

a strain (hence, stress) free configuration. Hence, to test whether a doubly connected body has

any dislocation or not, we render it simply connected by cutting along an arbitrary surface, such

as s in the above example, thus, taking it to the stress free state. If, in this state, the two lips of

the cut are found to be rigidly displaced (by an amount b) relative to each other, then we say

that the original doubly connected body was dislocated, the Burgers vector of the dislocation

being b. Kondo [19] called the deformation that brings a body to its natural stress-free state

an elastic deformation, because it has been a standard practice in elasticity theory to assume

that stress is always elastic in origin.

2. The cutting surface used to make the body simply connected is arbitrary. So, the displacement

field, actually, has a jump everywhere inside the doubly connected body, as such the displace-

ment is no more a function in the classical sense. We can adopt two view-point about this

fact:

(a) either the displacement field, in classical sense, does not exist on the doubly connected,

(b) or, the displacement field defined over the doubly connected body is multi-valued,

27

(c) or, the displacement field, defined over the simply connected stress free state, has a constant

rigid body jump across some arbitrary cut surface.

If we go with the first view-point, then non-existence of any displacement function means that

the equation (2.1) is no longer valid (or β is not a conservative field) and, hence,

∇× β 6= 0. (2.31)

In terms of the variables ε, κ, this implies

∇× ε− κT 6= 0, (2.32)

provided equation (2.10) still holds true, i.e. the rotation field ω does exist. These two are

the incompatibility conditions necessary to have a dislocated body. These are also the non-

integrability conditions for a displacement field. We will return to this point again when we

discuss continuous distribution of dislocations.

3. Here, the model of edge dislocation has been given as an example. The same analysis is true

for screw dislocation also, provided the slip of the two cut lips is made in the direction of the

cylinder axis, or, in other words, the Burgers vector b is parallel to the dislocation line.

2.5.3 Frank vector of a Disclinated Continuized Crystal

Similar analysis, what we did in case of dislocations, will work here, except the fact that now the rigid

jump of displacement field across any arbitrary surface that render the disclinated doubly connected

body simply connected, will be given by Ω × x, where Ω is the angular rotation found between the

two cut lips, and x is the position vector of a material point on the cutting surface in the doubly

connected configuration. Since, [[ω]] = Ω is now non-zero at all points on the disclinated body, a

single-valued rotation field ω, thus, is undefined. Hence,

∇× κT 6= 0, (2.33)

28

together with (2.32).

Ω has to be necessarily a member of the rotational symmetry group of the material.

2.6 Elastic Fields of Elementary Straight Volterra Dislocations

In this section, we will formulate and solve the elastic boundary value problem of a straight dislocation

in an infinitely long cylinder, with the dislocation line lying on the cylinder axis. We will assume the

material to be isotropic and linearly elastic with Lame’s constants λ and µ, and Poisson’s ratio ν.

We will use both Cartesian coordinates (x1, x2, x3) and cylindrical coordinates (r, θ, z), which are

related by

x1 = r cos θ, x2 = r sin θ, x3 = z.

x1

x2

x3

O

R

r0(x1, 0+)

(x1, 0−)

Γ

Γ0

Figure 2.14: Cut along the strip x2 = 0, −R ≤ x1 ≤ −r0, used to define a single valued displacement

field around the straight edge dislocation lying along x3-axis.

2.6.1 Straight Edge Dislocation

The problem is of plane strain type, i.e.

u1 = u1(x1, x2), u2 = u2(x1, x2), u3 = 0.

29

Non-zero components of the infinitesimal strain tensor ε are

ε11 = u1,2, ε22 = u2,2 and ε12 =1

2(u1,2 + u2,1).

Non-zero stress components with this reduced number of non-zero strain components, for isotropic

linear elasticity, read

σ11 = λ (u1,1 + u2,2) + 2µu1,1, σ22 = λ (u1,1 + u2,2) + 2µu2,2, σ12 = µ (u1,1 + u2,2)

and σ33 = λ(u1,1 + u2,2).

Only two non-trivial equilibrium equations remain:

σ11,1 + σ12,2 = 0, σ12,1 + σ22,2 = 0.

Boundary conditions: traction is zero at both free surfaces, namely r = r0 and r = R.

Jump condition for displacement: u1(x1, 0+) − u1(x1, 0−) = −b, −R ≤ x1 ≤ −r0, where b is the

magnitude of the true Burgers vector.

The above classical boundary value problem has been solved by a combined method of potentials and

complex variables. The various elastic fields, thus found, are

u1 = − b

2π

[θ +

sin 2θ

4(1− ν)

], u2 =

b

8π(1− ν)[2(1− 2ν) ln r + cos 2θ]; (2.34)

εrr = εθθ =b(1− 2ν)

4π(1− ν)

sin θ

r, εrθ = − b

4π(1− ν)

cos θ

r, εzz = εrz = εθz = 0; (2.35)

σrr = σθθ =1

2νσzz =

µ b

2π(1− ν)

sin θ

r, σrθ = − µb

2π(1− ν)

cos θ

r, σrz = 0; (2.36)

in Cartesian coordinates,

σ11 =µ b

2π(1− ν)

y(3x21 + x22)

(x21 + x22)2, σ22 = − µ b

2π(1− ν)

y(x21 − x22)(x21 + x22)

2,

σ12 = − µ b

2π(1− ν)

x(x21 − x22)(x21 + x22)

2, σ33 =

µν b

π(1− ν)

y

x21 + x22. (2.37)

The strain energy density produced by the edge dislocation is

W =1

2σij εij =

µ b2

8π2(1− ν)21− 2ν sin2 θ

r2(2.38)

30

and, consequently, the strain energy stored per unit length of the edge dislocation is

w =

∫ 1

0dz

∫ R

r0

∫ 2π

0W r dr dθ =

µ b2

4π(1− ν)lnR

r0. (2.39)

2.6.2 Straight Screw Dislocation

Let us now consider the elastic boundary value problem of a straight screw dislocation. Displacement

field has components of the following form

uz = uz(r, θ), ur = uθ = 0,

with the jump condition

uz(r, π)− uz(r, −π) = −b, −R ≤ r ≤ −r0,

where b is the magnitude of the true Burgers vector. This condition is satisfied, if we assume the

displacement field to be

uz = − bθ2π, θ ∈ (−π, π]. (2.40)

From this, we calculate the non-zero components of the infinitesimal strain field

εθz = − b

4πr(2.41)

and the non-zero components of the stress field

σθz = − µ b

2πr, (2.42)

which satisfies the same equilibrium equations and boundary conditions as stated before in case of

edge dislocation.

The strain energy density is

W =µ b2

8π2r2(2.43)

and the strain energy stored per unit dislocation length is

w =µ b2

4πlnR

r0. (2.44)

31

2.6.3 Straight Disclination

Mixed Disclination

The elastic fields of a straight Volterra disclination of mixed type (superposition of states (d), (e) and

(f) of figure 5.1), with line coinciding with the x3-axis and Frank vector Ω = Ω1 e1 + Ω2 e2 + Ω3 e3,

in an isotropic and linearly elastic solid body of infinite extent has been calculated by deWit [17] and

are reproduced below. Strain and stress fields are, as expected, singular on the x3-axis. So, as usual,

we remove a cylindrical volume with axis x3, called the disclination core, to restrict ourselves in the

realm of linear elasticity.

u1 = − Ω1 x34π(1− ν)

[(1− 2ν) ln r+

x22r2

]+ Ω2 x3

[θ

2π+

x1 x24π(1− ν)r2

]−Ω3

[x2 θ

2π− 1− 2ν

4π(1− ν)x1(ln r− 1)

],

(2.45)

u2 = −Ω1 x3

[θ

2π− x1 x2

4π(1− ν)r2

]− Ω2 x3

4π(1− ν)

[(1− 2ν) ln r+

x21r2

]+ Ω3

[x1 θ

2π+

1− 2ν

4π(1− ν)x1(ln r− 1)

],

(2.46)

u3 = Ω1

[x2 θ

2π− 1− 2ν

4π(1− ν)x1(ln r − 1)

]− Ω2

[x1 θ

2π+

1− 2ν

4π(1− ν)x1(ln r − 1)

]; (2.47)

ε11 = − Ω1 x34π(1− ν)

[(1−2ν)

x1r2−2

x1 x22

r4

]− Ω2 x3

4π(1− ν)

[(1−2ν)

x2r2

+2x21 x2r4

]+

Ω3

4π(1− ν)

[(1−2ν) ln r+

x22r2

],

(2.48)

ε22 = − Ω1 x34π(1− ν)

[(1−2ν)

x1r2

+2x1 x

22

r4

]− Ω2 x3

4π(1− ν)

[(1−2ν)

x2r2−2

x21 x2r4

]+

Ω3

4π(1− ν)

[(1−2ν) ln r+

x21r2

],

(2.49)

ε33 = 0, (2.50)

ε12 =Ω1 x3

4π(1− ν)

[x2r2− 2

x21 x2r4

]+

Ω2 x34π(1− ν)

[x1r2− 2

x1 x22

r4

]− Ω3 x1 x2

4π(1− ν)r2, (2.51)

ε23 =Ω1 x1 x2

4π(1− ν)r2− Ω2

4π(1− ν)

[(1− 2ν) ln r +

x21r2

], (2.52)

ε31 =Ω2 x1 x2

4π(1− ν)r2− Ω1

4π(1− ν)

[(1− 2ν) ln r +

x22r2

]; (2.53)

32

σ11 = − µΩ1 x32π(1− ν)

[(1−2ν)

x1r2−2

x1 x22

r4

]− µΩ2 x3

2π(1− ν)

[x2r2

+2x21 x2r4

]+

µΩ3

2π(1− ν)

[ln r+

x22r2

+ν

1− 2ν

],

(2.54)

σ22 = − µΩ2 x32π(1− ν)

[(1−2ν)

x1r2

+2x1 x

22

r4

]− µΩ2 x3

2π(1− ν)

[x2r2−2

x21 x2r4

]+

µΩ3

2π(1− ν)

[ln r+

x22r2

+ν

1− 2ν

],

(2.55)

σ33 = − µν

π(1− ν)r2(Ω1x1 + Ω2x2) +

µΩ3

2π(1− ν)

[2ν ln r +

ν

1− 2ν

], (2.56)

σ12 = µ ε12, (2.57)

σ23 = µ ε23, (2.58)

σ31 = µ ε31. (2.59)

Wedge Disclination

Disclinations that appear in real crystals are of wedge type. So, we specifically mention here the

stress and strain fields of a positive wedge disclination of Frank vector Ωe3, placed on the x3-axis, in

an isotropic linearly elastic cylinder of radius R with the same axis as the disclination and of infinite

length. These have been taken from [20].

σrr =µΩ

2π(1− ν)lnR

r, (2.60)

σθθ =µΩ

2π(1− ν)[ln

R

r− 1], (2.61)

σzz =µν Ω

2π(1− ν)[2 ln

R

r− 1]; (2.62)

εrr =Ω

4π(1− ν)[(1− 2ν) ln

R

r+ ν], (2.63)

εθθ =Ω

4π(1− ν)[(1− 2ν) ln

R

r− 1 + ν], (2.64)

εzz = 0; (2.65)

33

in Cartesian coordinates, the stress components are

σ11 =µΩ

2π(1− ν)

[1

2ln

R2

x21 + x22− x22x21 + x22

], (2.66)

σ22 =µΩ

2π(1− ν)

[1

2ln

R2

x21 + x22− x21x21 + x22

], (2.67)

σ33 =µν Ω

2π(1− ν)

[ln

R2

x21 + x22− 1

], (2.68)

σ12 =µΩ

2π(1− ν)

x1 x2x21 + x22

. (2.69)

Infinitesimal Wedge Disclination Dipole

Stress field of an infinitesimal wedge disclination dipole, placed on the x2-axis with seperation dx2,

can be found [20] by differentiating the last four equations with respect to x2:

σ11 = − µΩ dx22π(1− ν)

y(3x21 + x22)

(x21 + x22)2, σ22 =

µΩ dx22π(1− ν)

y(x21 − x22)(x21 + x22)

2,

σ12 =µΩ dx2

2π(1− ν)

x(x21 − x22)(x21 + x22)

2, σ33 = −µν Ω dx2

π(1− ν)

y

x21 + x22. (2.70)

These are just the negative of stress field produced by a straight edge dislocation of burgers vector

Ω dx2. This again proves that an infinitesimal wedge disclination dipole is topologically equivalent to

an edge dislocation.

2.6.4 Corrections for Simulating Crystal Dislocations

Corrections and modifications needed to the above fields so that they can be successfully applied to

approximate crystal dislocations are:

1. Real crystals are highly anisotropic and are composed of periodic arrangements of constituting

atoms, ions or molecules. Modifications taking account of these facts are there in literature that

closely approximate the situations in real crystals.

2. Near the dislocation core, lattices are in highly strained state and linear elasticity breaks down.

Non-linear effects should be taken into considerations while modelling the near core zones.

34

However, far from the dislocation core, say about 10 to 12 times atomic distance, the above fields

(with modifications mentioned in the last point) almost precisely approximates the crystalline

fields.

3. The boundary condition at the inner surface of the cylinder, or at the surface of the cylindrical

core which we remove from the infinite medium to make the body doubly connected, is not

actually ‘traction free’ as we considered in the above formulation. Forces arising from the

dislocation core act on it and we need to modify further the above solution taking care of this

fact. These forces are calculated with the help of some combined atomistic-continuum model.

2.7 Presence of Disclinations in Crystals

As already mentioned in previous sections, Burgers vector of a single dislocation in crystals is always

of length equal to one lattice parameter. While making the continuum approximation of a real crystal,

we split the constituting atoms into several parts and bring two nearby parts close together. This

procedure conserves the total mass and volume of the crystal we are experimenting with. Now, almost

all crystals appearing in nature have some translational and rotational symmetries. After making

the continuum approximation, the elements of its translational symmetry become vanishingly small

compared to other macroscopic dimensions. Since the strain and stress fields of single dislocations

written above are proportional to the magnitude of Burger’s vector, values of those fields in a con-

tinuized crystal are not very large and simultaneous presence of many such discrete dislocations is,

thus, physically possible in a real crystalline material, because the added up fields at places are of

comparable values that are found in experiments.

On contrary, disclinations have the Frank vector (an angle) as their characteristic feature and in

a real crystal, to maintain the continuity of material, it has to be equal to some symmetry angle of

the crystal. In the continuum approximation, as opposed to the translational symmetry, elements of

its rotational symmetry cannot be made small. Angles always remain finite. So in continuum model

35

of a real crystal, strain and stress fields of a single disclination remain finite and, as compared to

dislocations, of much larger values. Hence, in real crystals, physical presence of discrete single discli-

nations is questionable. However, they can present as dipoles, as already mentioned and explained

before. Presence of isolated disclinations in many in a finite volume of crystal is also not physically

possible.

These points will be raised again in the next chapter when we model real plastic deformation of a

single crystal by simultaneous presence of many discrete defects. Then we will talk of defect densities

and question of existence of a disclination density in real crystal will be dealt with.

36

Chapter 3

Continuous Distributions of Defects

3.1 Continuous Distribution of Dislocations

In plastic deformation of macroscopic crystalline solid bodies, many dislocations take part which were

either there from the beginning (due to manufacturing processes) or get generated by various methods

of dislocation multiplication. It has been found experimentally that total length of dislocations in

action during a typical plastic deformation process is about 108 − 1010 mm/mm3. So, to study the

combined effects (i.e. elastic fields etc.) of a bunch of dislocations is more practical than to study

discrete dislocations, because they always occur in many. To tackle this problem mathematically,

we make, here also, a continuum approximation so that we can apply theory of elasticity [21]. To

conserve the total dislocation content of a body in the continuization procedure [18], we split the

atoms into several parts and bring two neighbouring parts close to each other. This will decrease the

Burgers vectors while increase the number of dislocation content in a given volume, thus conserving

the net dislocation content inside the volume. In the limit, the Burgers vector becomes vanishingly

small and the number of dislocations becomes infinite. At this point, we say that the dislocations are

continuously distributed and we can talk of its density function.

To define the density function formally, we categorized all the dislocations appearing inside a

37

body into few basis elementary types, say the Volterra dislocations of type (a), (b), (c) etc. The i-th

type is specified by its true Burgers vector b(i) and its line direction ξ(i), chosen by some fixed rule.

If the number of i-th type dislocation, crossing an unit area perpendicular to ξ(i), is N (i), then we

define the local density of the i-th type dislocation, at the current placement x of some representative

material point, by the following tensor

α?(i) = N (i) ξ(i) ⊗ b(i) (3.1)

and the local dislocation density, due to all dislocations, will be

α? =∑(i)

α?(i). (3.2)

Let us take a small planar circuit c inside the current configuration of the body, around x, and let

the small oriented area element bounded by c be n ds, n being the unit normal. Then, number of

the i-th type dislocation crossing this small area will be N (i) ξ(i) · n ds, and the total Burgers vector

of this many number of i-th type dislocations will be (N (i) ξ(i) · n ds) b(i). Hence, the total Burgers

vector of all dislocations threading the small circuit c is

db =∑(i)

(N (i) ξ(i) · n ds) b(i) = α?Tn ds. (3.3)

When we detach the circuit form its surroundings to relax it from the constraints impressed upon it by

the local neighbours, or in other words, we take the small circuit to its local stress free configuration

K0, it will not close there, as we have seen before while considering single dislocations. The closer

failure will, obviously, be db, because it is the true Burgers vector of the dislocations crossing the

area n ds and it also belongs to the local stress free configuration K0. Note that, since n is a vector

belonging to the current dislocated configuration κ, it seems that the tensor α?T should map vectors

from K to K0, which it surely does, by definition (3.1).

If the local mapping of vectors from the current dislocated configuration to the reference configu-

ration is denoted by F−1, which is nothing but the inverse deformation gradient previously discussed,

38

then the vector

b(C) =

∮C

F−1dx =

∫s(curl F−1)Tnds, (3.4)

belonging to the local natural state of the dislocated neighbourhood, represents the true Burgers

vector of the dislocations threading the closed loop c, and after comparing equations (3.3) and (3.4),

we can see that

α? = curl F−1. (3.5)

Here, the closed curve C encloses an arbitrary open area s, which is composed of small infinitesimal

oriented plane area elements n ds, with planar boundary curve c.

There is another tensorial measure of dislocation density in literature, which is called the true

dislocation density, and is defined by

α = JFF−1curl F−1. (3.6)

This density is called true because this measure is invariant under arbitrary compatible deformation

of the dislocated configuration. The measure α? is not invariant in this sense. Invariance of measure

is desired because compatible deformation does not introduce any defects inside the body. However,

as we will be dealing with statics only, the current configuration does not change (i.e. we are given

with a dislocated body and we are to find its strain and stress fields) and we can work the local

dislocation density (3.5) only.

Recalling that F−1(x) = 1− β(x), where β(x) is the distortion tensor with respect to the current

dislocated configuration, we have

α?(x) = −curl β(x), (3.7)

which can be used as another definition of local dislocation density. Hence, alternatively, the following

equation also holds.

b = −∫s

(curl β)Tn ds. (3.8)

39

Now, as already seen for single dislocations, nonzero curl β introduces incompatibilities in the body,

which yields non-existence of any displacement field. Same is true for continuous distribution also.

The situation is slightly delicate here, because if we take local neighbourhoods of all material points

to their respective natural states, they do not, in general, fit together to form a continuous stress-free

configuration in our physical 3-D space, which is Euclidean. This fact also corresponds to non-

existence of any displacement field. However, since β exists, we can at least form such non-fitting

collections of local natural configurations. Mathematically, we can imagine some unreal space where

these local configurations would perfectly fit in. That space will be non-Euclidean. We will talk

about this in the next chapter.

Equation (3.7) implies that

divα? = 0, (3.9)

which means that a dislocation line cannot end inside the medium. It can only end on on other

defects like another dislocation, disclination, point defects or grain boundary, or at the free boundary

of the medium. This is called the conservation of dislocations.

3.2 Continuous Distribution of Disclinations

Although it was already pointed out that distribution of single disclinations is very unlikely to occur

in crystalline solids, as a field theory we can consider them and can study their elastic fields. This

study will also be helpful to find out elastic fields of single disclinations of mixed type, of arbitrary

orientation, and to extend the theory to generalized continuum (solids of higher grades, liquid crystals

etc.) where disclinations are found in abundance.

It was mentioned in the last chapter, while discussing continuum model of crystal disclinations,

that in presence of disclinations, curl κT 6= 0. We can consider this non-zero term as incompatibility

in ω(x) field and, hence, this may successfully serve our purpose as a definition of local disclination

40

density θ?.

θ? = curl κT. (3.10)

Since, when disclinations are present, we can no longer define ˜ω(x), β(x) also does not exist. Hence,

the definition (3.7) of local dislocation density need a modification in this case. Recall the following

compatibility equation in terms of the variables ε, κ:

curl ε+ (trκ) 1− κT = 0. (3.11)

We can use this equation as to modify the definition of local dislocation density:

α? = −curl ε− (tr κ) 1 + κT. (3.12)

Now, the conservation equations of disclinations and dislocations look like

divθ?T = 0 (3.13)

and

divα? − θ = 0, (3.14)

where θ is the axial vector of Skwθ?. These are implied by (3.10) and (3.12).

3.3 Elastostatics Problem for Continuous Distribution of Defects

To formulate the elastic boundary value problem for this case, we need to consider, additional to the

equations (3.10) and (3.12), the constitutive law for linear isotropic elasticity:

σ = 2µ ε+ λ (tr ε) I, (3.15)

where it has been assumed that stress is purely elastic in origin, and the following equilibrium equation

with zero body force:

div σ = 0. (3.16)

41

So the problem is:

Given some know distributions α? and θ? of dislocations and disclinations, respectively, find the

elastic fields ε, κ and σ from equations (3.10), (3.12), (3.15) and (3.16).

The elastic fields, as calculated by de Wit [22], are given below, with respect to some global

Cartesian coordinates in the current configuration.

εij(r) =1

8π

∫V ′

[R,knn

(εilk α

?jl(r′))(ij)

+

(Rijk1− ν − δij R,knn

)εkml α

?ml(r

′)

+R,nn θ?(ij)(r

′) +

(R,ij

1− ν − δij R,nn)θ(r′)

]dV ′, (3.17)

σij(r) =µ

4π

∫V ′

[R,knn

(εilk α

?jl(r′))(ij)

+1

1− ν

(Rijk − δij R,knn

)εkml α

?ml(r

′)

+R,nn θ?(ij)(r

′) +1

1− ν

(R,ij − δij R,nn

)θ(r′)

]dV ′. (3.18)

42

Chapter 4

Differential Geometric Structure of a

Solid Body: Theory of Inhomogeneities

4.1 Outline

Founder of the inhomogeneity theory was Kondo [23]. After him, Bilby [24], Kroner [25], Anthony,

Noll [26], Wang [27] and many more material scientists, physicists and mathematicians have con-

tributed heavily to develop this still active field. We give here a brief outline of the theory in physical

terms, which we will make rigorous later.

As previously mentioned many times, stress is considered to be purely elastic in origin and we

call the stress-free configuration of a material body, its natural configuration. In this chapter, two

important new concepts will be introduced: material uniformity and homogeneity. The latter is

almost synonymous to ‘defect free’.

A uniform material body means all of its points are made up of the same material. To decide

whether the materials at two different points of the body are same or not, we need a constitutive

law, for example, some energy-deformation relationship (recall W = W (β)) or stress-deformation

relationship. Now, given two material points, if there exists a linear transformation that carries the

43

local material neighbourhood of one material point to the local material neighbourhood of the other,

to bring the two neighbourhoods to the same state of deformation, so that their constitutive response

exactly match (energy or stress have the same value at the two points) at that state, we say that

materials at these two points are same. If we can do this for all points of the body, the body is called

materially uniform, otherwise non-uniform.

If after removal of external force fields, the body is still stressed, we say that defect (or defect

distribution) is present inside the body, or the body is inhomogeneous; otherwise homogeneous. In

other words, we cut the body into infinitesimal volume elements to make each element stress free

(see reference). Now, if it is possible to fit together all these stress free elements to make a connected

configuration in our physical space, we say that our original uncut body is homogeneous. If they

do not fit together, the uncut body is inhomogeneous. In the latter case, we need to apply surface

tractions on each of these stress free elements to deform them elastically so that, along with these

externally applied tractions, they perfectly fit together. After this, we weld the joining surfaces and

remove the external tractions from all the internal welded surfaces. This will recover our original

uncut body. Stress field due to the defect distribution will, obviously, be the stress field due to the

distribution of those externally applied surface tractions. Note that, here also, we need a constitutive

law in advance to bring each element to its natural state.

Differential geometric structure of the material body emerges, when we place those non-fitting

natural elements to an imaginary mathematical space, where they fit perfectly well together. The

connected configuration that the set of those natural elements would attain in that non-physical space,

can be given a structure of differential manifold (Kondo called this a ‘material manifold’) which is not

Euclidean1. Geometrical objects defined over that manifold, to describe its characteristic features,

will naturally have a correspondence to the non-fitting characteristics of those natural elements in

our physical space which we call defects. Hence, construction of that non-Euclidean manifold is also

1Because our own physical space is Euclidean.

44

based on the constitutive assumption.

4.2 Geometric Objects on the Material Manifold and Two Basic

Identifications

The naturalization process is shown in the figure below. The additional purely imaginary ‘ideal state’

Figure 4.1: Naturalization process of an internally stressed body.

is sometimes used in modern defect theories to define plastic deformation (deformation taking the

ideal configuration to the natural configuration), but theory can be built up without mentioning its

existence at all. Note, again, that, in this diagram, deformation from the current strained state to

the ‘intermediate’ natural state has been referred to as ‘elastic distortion’. To see more clearly the

non-Euclidean nature of the imaginary connected manifold made up of these mutually disjoint pieces

of local natural configurations, residing in Euclidean space, consider the following illustration of a

thought experiment done on a dislocated crystal. An attempt has been made here to construct a

parallelogram around the edge dislocation, keeping in mind the fact that the natural parallel lines

inside a crystal are the crystallographic lines joining successive atoms. The ‘horizontal’ and ‘vertical’

crystallographic vectors, a and b respectively, have been ‘translated parallelly’ to construct the op-

posite sides of the parallelogram. The result is the open parallelogram in the dislocated configuration

which is unlikely to occur in our usual Euclidean space. Eli Cartan considered such possibilities first

45

Figure 4.2: Local Burgers vector and Cartan’s circuit.

which led him to discover a fundamental geometrical object of general differential manifolds. It is

called the Torsion tensor of the corresponding ‘affine connection’ on the manifold. Affine connection

Γ on a manifold defines a rule to translate vectors parallelly on it.

vi(x+ dx) = vi(x)− Γijk(x) vk(x) dxj . (4.1)

It has components of the form Γijk, in some global Cartesian coordinate system, and it is not a tensor.

A fundamental tensor that arises out of it is formed from its antisymmetric components Γi[jk], and

these form the torsion tensor S of Γ. To see the interpretation of the components Γi[jk], consider

the following diagram which illustrates the construction of an infinitesimal open parallelogram on

some non-Euclidean space. After getting parallelly translated along d2x(A), the infinitesimal vector

A

B

C

D

E

d1x(A)

d1x(B)

d2x(A) d2x(C)

Figure 4.3: Infinitesimal open parallelogram.

d1x(A) becomes

d1xi(B) = d1x

i(A)− Γijk(A) d1xk(A) d2x

j(A)

46

and, after getting parallelly translated along d1x(A), the infinitesimal vector d2x(A) becomes

d2xi(C) = d2x

i(A)− Γijk(A) d2xk(A) d1x

j(A).

Closer failure of the resulting open parallelogram is, hence, given by

d1xi(A) + d2x

i(C)− d2xi(A)− d1x

i(B) = −Γijk(d1x

j d2xk − d1x

k d2xj)(A)

= −Γijk(Skw d1x⊗ d2x

)jk(A)

= −Γi[jk](Skw d1x⊗ d2x

)jk(A),

where the first equality comes from the definition (4.1) of affine connection and last equality comes

from the fact that dot product of a symmetric and a skew tensor is zero. Axial vector of the tensor

(Skw d1x ⊗ d2x) represents the oriented area element n ds of the open parallelogram ABDECA,

and, thus, components Sijk ≡ Γi[jk] of the torsion tensor S measures the closer failure of open

parallelograms in non-Euclidean spaces.

Now recall the definition of dislocation density tensor:

db = α?T n ds

If we construct a third order tensor α? which is antisymmetric in the sense α?ijk = −α?ikj , and whose

second order axial tensor is α?, then we can identify α? with the torsion tensor of the dislocated

space:

α? = S. (4.2)

This first basic identification is a major cornerstone of differential geometric defect theory of ordered

media.

Next, consider another thought experiment, now performed on a dislocated and a disclinated

crystal. In figure 4.4(a), a lattice vector has been translated parallelly around an edge dislocation

in a cubic lattice. The result is what is expected and familiar: the vector returns to its original

orientation after its circum-navigation. We are accustomed to this fact in our 3D Euclidean physical

47

(a) around an edge dislocation (b) around a wedge disclination

Figure 4.4: Parallel displacement of a lattice vector.

space. Another aspect of non-Euclidean geometry of defective crystals emerges when we perform the

same experiment on a disclinated crystal. As can be seen in figure 4.4(b), after parallel transportation

along a closed loop around the disclination, the lattice vector does not return to its original orientation.

Like the previously seen closer failure of parallelograms, this fact is due to the existence of another

fundamental tensorial object made up of the affine connection and its covariant derivatives: the

Riemann-Christoffel curvature tensor R of the affine connection Γ. It is a fourth order tensor, with

components defined by

Rijkl ≡ 2(∂jΓ

ikl − Γikp Γpjl

)[jk]. (4.3)

Figure 4.5: Parallel transport of a vector around a closed loop in (a) a flat space and (b) in a curved

space.

This tensor measures the deviation in orientation δv of a vector v after parallel transport along

48

a closed loop according to the following relation (see figure 4.5):

δvi = Rijkl vl dSjk, (4.4)

where dSjk are the components of the skew symmetric area element tensor enclosed by the loop. The

second basic identification of the differential geometric defect theory is the following:

θ?ij = Rij = Rkikj . (4.5)

The justification of the last equality follows from a comparison of the figures 4.4(b) and 4.5(b). Here,

Rij are the components of the Ricci tensor, the only possible second order contraction of Rkilj .

4.3 Two Relevant Questions

After the above formal discussion on relevance of differential geometry in defect theories and the two

basic identifications of densities of elementary defects with two fundamental tensors of the material

manifold, the following two important questions naturally arise:

Q1. Can a continuous torsion tensor field exist on the material manifold of a solid?

or

Can continuous dislocation density field occur in solids?

Q2. Can a continuous Ricci curvature tensor field exist on the material manifold of a solid?

or

Can continuous disclination density field occur in solids?

Answers to these questions should certainly involve material symmetry arguments because the

domain of density field or the fundamental tensorial objects is nothing but Kroner’s continuized crystal

discussed previously. This is not a ordinary continuum, but possesses crystal structure through the

preservance of the crystal symmetries. Recall, in this context, that during the continuization process,

49

while the elements of translational symmetry group get reduced to infinitesimals, elements of the

rotational symmetry group remain finite.

A1. Continuous distribution of dislocation does exist in solids, both observationally and theoret-

ically. They have been found in abundance during plastic deformation and infinitesimal translational

symmetry elements also support this fact. Inside a local Burgers circuit, if we add one single dis-

Figure 4.6: Continuous distribution of dislocations.

location, the circuit opens up only infinitesimally in the reference crystal. Energy addition is also

infinitesimal.

A drawback of the first identification is that,though for crystals, which have discrete symmetry

groups, the torsion tensor of the material connection works well as a measure of density of dislocations,

it fails when the symmetry group is continuous e.g. polymers, because, in this case, the material

connection no longer remains unique as we will see later when we will discuss Noll’s inhomogeneity

theory.

A2. As had been pointed out in the last section of chapter 2, single disclinations cannot exist in

crystals because of their large elastic fields. Wherever they have been found, they present in dipoles.

Addition of a single discrete disclination inside of a Burgers circuit opens up the circuit finitely in the

reference crystal because the elements of rotational symmetry group remain finite, and, hence, require

finite amount of energy for each disclination addition. Crystalline solids cannot sustain such huge

amount of strain energy. As a consequence, continuous distribution of disclinations cannot occur in

crystalline solids,neither a continuous curvature tensor field in its associated material manifold. But if

50

Figure 4.7: Continuous distribution of disclinations.

the rotational symmetry group is continuous, continuous distribution of disclinations can exist e.g. in

polymers. Unfortunately this fact has still not a sound theoretical basis (involving material symmetry

arguments), like its dislocation counterpart. The only result that we have is due to Noll, which says

that for a first grade solid, to have a consistent theory of dislocations, material connection needs

necessarily to be curvature-free. Otherwise the structure of the solid would break down. Moreover, if

we recall the set of independent elastic variables ε, κ for general defect theories (i.e. simultaneous

presence of both dislocation and disclination), we can further argue that this 15 dimensional state

space (6 for symmetric ε and 9 for κ) actually make our problem indeterminate, because we need

only 9 dimensions for description of deformations of first grade materials. The ambiguity is: Do we

necessarily need higher grade constitutive models to support continuous distribution of disclinations?

The problem is still open.

In this context, the comment of K. K. Anthony [28] is worth to mention:

“A phenomenological, macroscopical continuum theory with Riemann-Christoffel curvature given

by a continuously varying disclination density tensor θ(x, t) makes no physical sense for the topo-

logical defect disclination. The characteristic structure of the disclination would be eliminated from

such a theory from the very beginning. In a continuum theory of the deformed crystal disclinations

have to be taken into account as singular objects...”

51

4.4 Consequences of Zero Curvature: Necessity of a Constitutive

Base for Defect Theories

Hence, to save our first grade constitutive law, we usually exclude disclinations from our theory.

Then, the curvature tensor should necessarily be zero, i.e.

Rijkl ≡ 2(∂jΓ

ikl − Γikp Γpjl

)[jk]

= 0.

Above is a partial differential equation for the unknown variables Γijk. A general solution, can be

shown, to be given by

Γijk = Aiα∂jAαk , (4.6)

where Aiα are components of some arbitrary matrix. Now, as the boundary or some other conditions,

the components Sijk are given, then, to find the values of Aiα’s, we have the equations

Aiα∂[jAαk] = Sijk. (4.7)

Equations (4.7) are insufficient to compute Aiα’s, and, as can be shown, we need supplementary

conditions. These conditions are supplied by the constitutive equations, equilibrium equations and few

additional kinematic relationships. This fact led Noll to base his inhomogeneity theory on constitutive

laws. We discuss it in the next section. The bottom-line is that given a constitutive law, kinematic

behaviour of the body gets fixed.

52

4.5 Noll’s Inhomogeneity Theory

In this section, we discuss the crux of Noll’s theory [26] of material inhomogeneities which cane out

in his celebrated 1958 paper. The theory is based on the assumption of first grade constitutive law,

and, as has been shown, to be structurally devoid of any possible occurrence of disclinations. This is

mainly a theory of dislocations, its theoretical origin and measure. A similar theory of disclination,

found on a solid base of constitutive laws, is still unavailable.

4.5.1 Continuous Body, Local Configuration and Tangent Space

A continuous body B of class Cp is, by definition, a three dimensional differential manifold of class

Cp with a global chart κ : B → E 2(E is the three dimensional Euclidean point space with translation

space V). Charts κ are called configurations of the body. Elements of B, denoted by X, Y etc., are

called material points.

Functions of type KX : TXB → V are called local configurations at the material point X. These

may, or may not be equal to the derivative mappings ∇κ|X : TXB → V induced from κ (they will be

equal for homogeneous bodies defined later).

At a particular material point X, there are infinitely many local configurations related through

local deformations, which are members of the three dimensional general linear group (= group of linear

isomorphisms from V to V) InvLin 3. Hence, GX = LKX , where L ∈ InvLin and KX ,GX ∈ CX ,

the set of local configurations at X. Thus, objects like GXK−1X are always some L ∈ InvLin.

Though the tangent space TXB, also denoted by TX , is a vector space, it is not an inner product

space, because the inner product ? : TX × TX → R, defined by hX ? fX = KX(hX) ·KX(fX) with

hX , fX ∈ TX , depends on the local configuration KX .

Note: In Noll’s paper, the local configurations at a material point X are defined as

2Manifold of class Cp means the mapping γ κ−1 : κ(B)→ γ(B), for all configurations κ and γ is of class Cp. The

mapping just mentioned is called a deformation from the configuration κ to the configuration γ.3Set of all invertible second order tensors.

53

(a) Configurations of B (b) Local configurations

Figure 4.8: Global and local configurations in Euclidean point space of a continuous body B.

equivalence classes—an alternate way to define tangent spaces of a manifold in modern

differential geometry. According to this definition, two configurations κ and γ are called

equivalent at some X ∈ B if ∇(κ γ−1)|γ(X) = 1 (= the identity element of InvLin).

Equivalent configurations at a certain X form classes: all equivalent configurations fall

in the same class. Each of such classes is called a local configuration. The set of all

local configurations (KX , GX etc.) at X is denoted by CX . Local configurations, as

the nomenclature suggests, perform tasks similar to their global brothers: they carry the

localities of B to the localities of E . Members of a locality of E are just the vectors in V.

The fact that two different vectors u and v ∈ V are the results of mappings, through two

different local configurations KX and GX respectively, from one single member uX of the

locality at X ∈ B is expressed as (GXK−1X )u = v, or K−1X u = G−1X v = uX . The member

uX , belonging to the locality at X ∈ B and getting mapped to u and v, is nothing but a

tangent vector at X ∈ B; and the locality at X, with members like uX (i.e. the tangent

vectors), is the tangent space TX at X ∈ B. Hence, according to this alternate definition,

tangent vectors at some X ∈ B are equivalence classes which are just the partitions of the

tangent space TX , the equivalence relation being (GXK−1X )u = v.

54

4.5.2 Simple Bodies, Material Isomorphisms and Material Symmetry

Physical characteristics of a material body are determined by its constitutive properties. The property

depends upon the context (strain energy, stress etc. for pure mechanical response). The mathematical

object constructed from all such constitutive properties (or response descriptors), for a particular

material body and for a particular type of response considered, is a set R.

A simple material body with respect to R is a body B of class Cp, endowed with a function G

which assigns to each material point X, another function GX : CX → R. Form of this function gets

severely restricted by the principle of material frame indifference.

Next comes the important question: Are materials at all points of the body same or not? Materials

at two points X and Y are same if there exists an invertible transformation ΦXY : TY → TX such that

GX(GX) = GY (GXΦXY ) holds for all GX ∈ CX . Note that GXΦXY : TY → V, i.e. GXΦXY ∈ CY .

If materials at all points of a body are same, then the body is called materially uniform.

(a) Material isomorphisms (b) Uniform reference

Figure 4.9: Material isomorphisms and uniform reference of a continuous body B.

The transformations ΦXY form a group gXY , elements of which (i.e. ΦXY ’s) are called material

isomorphisms (because the tangent spaces TY and TX are isomorphic to each other through the

isomorphisms ΦXY ). The group gXX is called the intrinsic isotropy group at the material point X.

55

Functions of type Φ : (X, Y )→ gXY are called material uniformities. A materially uniform body

always admits such functions.

Invertible functions K with values K(X) = CX are called references of B. K’s satisfying

Φ(X, Y ) = K(X)−1K(Y ) are called uniform references of B. If such a K happens to be equal

to ∇κ for any configuration κ (a very special case), then the body is called homogeneous.

Now we redefine the notions of material response and isotropy group with respect to a uniform

reference K. The obvious fact GX(LK(X)) = GY (LK(Y )) (which comes from the definition of ΦXY ),

for any L ∈ InvLin, ensures the existence of a function HK : InvLin → R, point-wise defined by

HK(L) = GX(LK(X)). Such a function is called the response function of the simple body relative to

the uniform reference K. Thus, once a uniform reference is chosen, the material response becomes

dependent only upon the local deformations—a familiar idea in classical continuum mechanics where

we define response functions for simple local bodies for a particular configuration as functions of

deformation gradients only.

Again, group properties of gXY and definition of uniform reference imply K(X)gXXK(X)−1 =

K(Y )gY Y K(Y )−1 which defines a function gK : V → V with point-wise values gK = K(X)gXXK(X)−1.

The group gK (a subgroup of InvLin) is called the isotropy group of the simple body relative to the

uniform reference K, or the symmetry group of the simple body in the uniform reference K. This defi-

nition is also consistent with our usual definition of material symmetry group in continuum mechanics:

gK = P ∈ InvLin |HK(L) = HK(LP), for all L ∈ InvLin.

The following result relates various functions, defined till now, having domain on different refer-

ences:

For any two uniform references K and K, any L and F ∈ InvLin and a function P : B → gK, we

have

1. K(X) = LP(X)K(X),

2. gK = L gK L−1 (known as Noll’s rule), and

56

3. HK(F) = HK(FL).

From classical continuum mechanics, we know that the law of mass conservation restricts gK to

be a subgroup (not necessarily proper) of the unimodular group U(3, R) for all uniform reference K.

We call a uniform reference undistorted if either gK ⊆ Orth or gK ⊇ Orth, i.e. gK is comparable

to the three dimensional orthogonal group over the real field, Orth (⊂ U(3, R)). The body is called

isotropic if there exists some undistorted uniform reference K for which gK ⊇ Orth. It is called

a solid if, for some undistorted K, gK ⊆ Orth. Hence, for an isotropic solid, there always exists

some K for which gK = Orth. The body is called a fluid if, for some K, gK = U(3, R). Since

LuL−1 ∈ U(3, R), with a u ∈ U(3, R) and for all L ∈ InvLin, Noll’s rule implies that a body is fluid

with respect to all uniform reference if it is with respect to any one. Again, since Orth ⊂ U(3, R), a

fluid is always isotropic and every uniform reference of a fluid is undistorted.

4.5.3 Material Connections

A tangent vector field4 c defined over a materially uniform body, is materially constant, if c(X) =

Φ(X, Y )c(Y ), or, in other words, K(X)c(X) = K(Y )c(Y ), i.e. Kc = c (∈ V) = constant.

The notion of a materially constant tangent vector field over a materially uniform body is anal-

ogous to the idea of a vector field over a manifold for which all its values at various points on the

manifold are parallel transports of each other. The latter fact is a manifestation of the zero curva-

ture of the manifold corresponding to the same affine connection with respect to which the parallel

transports have been performed. The fact that it is always possible to define a materially constant

tangent vector field on a materially uniform body B (as c = K−1c with any c ∈ V) instigates us to

define an affine connection on B with zero curvature. Such an affine connection is called the material

connection (we will denote it by Γ), and its curvature is zero by definition.

Formally speaking, if c (= K−1c) is a materially constant tangent vector field on B, then Γc =

4Functions, with prescribed smoothness, that assign to each X ∈ B, an element of TX .

57

O ( = the zero vector in the vector space IB of all intrinsic second order tensor fields5 over B) serves

both as a definition of the material connection Γ (associated with the uniform reference K) and as

the condition that its curvature is zero. The general form of a material connection, in terms of any

uniform reference K and any tangent vector field h, is given by Γh = K−1∇K(Kh)K. Point-wise,

(IX 3) Γh|X = K(X)−1[∇K(X)(Kh)]

∣∣∣∣X

K(X) = K−1X [∇KX(Kh)]

∣∣∣∣X

KX = K−1X [∇κ(Kh)]

∣∣∣∣X

KX , if

KX = K(X) and for some representative κ ∈ KX . Note that ∇κ(Kh)

∣∣∣∣X

= ∇(Kh κ−1)∣∣∣∣κ(X)

.

Note: From elementary differential geometry, we know that affine connection is not a

tensor and we can define infinitely many affine connections on a manifold (ofcourse, they

have to satisfy some conditions which emerges from the frame invarience requirements

of derivatives of tensors). The three important tensorial objects that can be formed

from affine connections are Cartan’s torsion tensor and Riemann-Christoffel curvature

tensor, for each of the connections, and the difference of any two connections (this tensor

has no name). Recalling that Γ : TB → IB and with the definitions Γhf = [Γ(f)]h

and [h, f] = h f − f h (the Lie-Jacobi bracket of two vector fields), we can define

the torsion tensor as a skew-symmetric bilinear map S : TB × TB → TB, given by

S(h, f) = Γhf − Γfh − [h, f] and the curvature tensor as another skew symetric bilinear

map R : TB × TB → IB, given by R(h, f) = [Γh, Γf] − Γ[h, f]. It is clear from their

definitions that torsion is a third order and curvature is a fourth order tensor. The

anonymous third order tensor Dh = Γh − Γh mentioned above is useful when we write

the torsion S and the curvature R for connection Γ in terms of the torsion S and the

curvature tensor R for connection Γ, respectively, as S(h, f) = S(h, f)− (Dhf − Dfh) and

R(h, f) = R(h, f)− [Γh, Df]− [Dh, Γf] + [Dh, Df] + D[h, f].

5Functions I , with prescribed smoothness, that assign to each X ∈ B, a linear isomorphism IX : TX → TX .

58

4.5.4 Inhomogeneity

As stated earlier, the mapping (local configuration) KX at X ∈ B may sometimes be equal to the

derivative mapping ∇κ∣∣X

induced by some global configuration κ. That means, in these cases,

the uniform reference K = ∇κ. Bodies for which such K exists are called homogeneous, otherwise

inhomogeneous. To determine whether a body is homogeneous or inhomogeneous, i.e. to know if

such a global configuration κ exists or not for a given K, a quantity has been ingeniously constructed

from K whose values tell us whether K = ∇κ is possible or not. The quantity is nothing but the

torsion S of the material connection Γ associated with the uniform reference K. Hence, torsion

S of the material connection Γ is invariably called the inhomogeneity of the material uniformity

Φ(X, Y ) = K(X)−1K(Y ), and the important result below easily follows:

If B is homogeneous, then it admits a material uniformity with zero inhomogeneity.

The converse is not true. A body with zero inhomogeneity is not homogeneous, but locally ho-

mogeneous, a state where the neighbourhood at each X ∈ B is homogeneous. This point requires

clarification.

Figure 4.10: Deformation gradient

59

To make the calculation of S easy, we define torsion S of any arbitrary configuration γ of B as

follows. Let F = (∇γ)K−1 ∈ L , where L = set of all invertible linear transformations V → V. Then

(Su)v = KS(K−1u, K−1v) = F−1[(

(∇KF)v)u −

((∇KF)u

)v] =

((∇γF−1)h

)k −

((∇γF−1)k

)h,

where u, v ∈ V and h = Fu, k = Fv. It is clear from this expression of S that it is a third order skew

symmetric tensor in the sense that (Su)v = −(Sv)u. The corresponding second order axial tensor α

is the ‘true dislocation density’ found in literatures of continuous distribution of defects. The proof

is given below.

Let us choose one orthonormal Cartesian basis e1, e2, e3 in the configuration γ(B) ∈ E . In this

basis, (Su)v = Smpqupvqem, which is also equal to

((∇γF−1)h

)k−

((∇γF−1)k

)h =

(F−1mn,i Fipup Fnqvq − F−1mn,i Fiqvq Fnpup

)em

= F−1mn,i

(Fip Fnq − Fiq Fnp

)upvq em,

according to the above definition. Thus, Smpq = F−1mn,i

(Fip Fnq − Fiq Fnp

), which is skew in its last

two indices and, hence,

αjm =1

2εjpqSmpq

=1

2F−1mn,i

(εjpqFip Fnq − εjpqFiq Fnp

)=

1

2F−1mn,i

(εjpqFip Fnq + εjqpFiq Fnp

)=

1

2F−1mn,i

(εjpqFip Fnq + εjpqFip Fnq

)= F−1mn,i

(εpqj Fip Fnq

).

Now, the determinant of F is given by JF = 16εinrεpqj FipFnqFrj . Hence,

6JF F−1jl = εinrεpqj FipFnq

(Frj F

−1jl

)= εpqj FipFnq

(εinr δrl

)= εpqj FipFnq εinl.

60

Since εinl εinl = 6, the above expression implies that

εpqj FipFnq = JF F−1jl εinl.

Thus, αjm = JF F−1jl

(εinl F

−1mn,i

)= JF F

−1jl

(CurlF−1

)lm

, or, α = JF F−1 Curl F−1.

For a homogeneous body, K = ∇γ. Hence, F = 1, whence S = 0 and α = 0. But a body with

α = 0 (or S = 0) is locally homogeneous only.

4.5.5 Relative Riemannian Structure, Contortion

For a materially uniform body with uniform reference K, we can define the inner product, point-wise,

on TB as h ? f = (Kh)·(Kh), where h and f are two tangent vector fields. With this additional structure

of inner product (which is also a symmetric bilinear form on TB), the body manifold B becomes a

metric space. On this metric manifold B, we can choose one (out of many) very special connection

Γ which (a) has zero torsion (S = 0), and (b) satisfies the relation h(f ? l) = f ? (Γhl) + l ? (Γhf). The

last condition corresponds to ‘zero covariant derivative of the metric tensor’ in classical differential

geometry. The metric manifold B with this connection Γ is called a Riemannian manifold and the

associated structure (i.e. the inner product and the existence of the special connection Γ) is called

the Riemannian structure of B relative to the uniform reference K.

As a byproduct of this relative Riemannian structure, a tensorial object Dh = Γh − Γh emerges

which has important physical interpretation to be discussed later. The push forward D of Dh, with

values D(X) : V → L , through K, defined by Du = KDK−1u K−1 ∀u ∈ V, is called contortion.

It is skew-symmetric in the sense that Du = −(Du)T. In terms of the components Sijk in some

orthonormal basis e1, e2, e3 of V, the components Dijk of the third order tensor D are given by

Dijk =1

2

(Sijk − Sjik − Skij

),

from which it is clear that Dijk is skew with respect to its first two indices. The components κlq of

61

the second order axial tensor κ of D will, then, be

κlq =1

2εmplDmpq

=1

2

[1

2εmpl

(Smpq︸ ︷︷ ︸

εnpq αnm

− Spmq︸ ︷︷ ︸εnmq αnp

)− 1

2εmpl Sqmp︸ ︷︷ ︸αlq

]

=1

4

[εpml εpnq αnm − εmpl εnmq αnp

]− 1

2αlq

=1

4

[(δmn δlq − δmq δln)αnm − (δpn δlq − δpq δln)αnp

]− 1

2αlq

=1

2αmm δlq − αlq.

Hence, κ = −(α− 1

2tr(α)1

)= − Nye’s curvature tensor.

In general, the curvature R of Γ is not zero. We can define the push forward R of R through K

as

R(u, v) = K R(K−1u, K−1v) K−1 ∈ L , ∀u, v ∈ V,

which finally reduces down to

R(u, v) =((∇KD)v

)u−

((∇KD)u

)v + (Du)(Dv)− (DV)(Du)−D

((Su)v

).

Now, we try to write the components Rijkl of the fourth order tensor R in some basis e1, e2, e3

of V. First note that the first term((∇KD)v

)u in R(u, v) is equal to

(∇K(Du)

)v. Next, recalling

the relation

∇Kψ = ∇Gψ ((GK−1))

between the gradients of a field ψ on B relative to two uniform references K and G, if we take

62

G = ∇γ and ψ = Du, then

(∇K(Du)

)v =

[∇γ(Du)

((∇γ) K−1

)]v

=[∇γ(Du) F

]v with F = (∇γ) K−1 ∈ L

=[∇γ(Du)

]Fv

=(Dijl ul

),pFpk vk ei ⊗ ej

= Dijl, p Fpk ul vk ei ⊗ ej .

Hence,

Rijkl = Dijl, p Fpk −Dijk, p Fpl +DimlDmjk −DimkDmjl −Dijm Smkl.

It is clear from the above expression of Rijkl that

Rijkl = −Rjikl and Rijkl = −Rijlk,

which are the well-known anti-symmetries of the curvature tensor of a Riemannian manifold.

Components of the second order axial tensor E (Einstein tensor, as named by Kroner) of R are

given by

Epq =1

4εpij εqkl Rijkl

[after simplification]

=1

2εqkl

[κpl,r Frk − κpk,r Frl

]+

1

2εqklDprl κkr − κpmαqm.

4.5.6 Contorted Aeolotropy

When the deformation gradient F = (∇γ)K−1 ∈ L from the set of local configurations induced

by K to the global configuration γ has orthogonal values (i.e. F(X) = Q(X) ∈ Orth ∀X ∈ B),

interpretation of D becomes very easy. K for which there exists a configuration κ that satisfies

Q = (∇κ) K−1,

63

with Q(X) ∈ Orth, is called a state of contorted aeolotropy. D, in such case, is given by

(Du)v = −QT∇κ(Qv) Qu (4.8)

= −QT [∇K(Qv) (K(∇κ)−1

)] Qu

= −QT [∇K(Qv) Q−1] Qu

= −QT [∇K(Qv)] u

= −QT [(∇KQ) u] v (4.9)

⇒ Du = −QT (∇KQ) u.

This equation shows that the skew symmetric tensor −Du(X) is the rate of change of Q(X) in the

direction of u, as viewed from any local configuration belonging to K(X). Recalling the relation

(Su)v = F−1[(

(∇KF)v)u −

((∇KF)u

)v] and comparing it with equation (4.9), we can conclude

that while, for any general F ∈ InvLin, S captures the local behaviour of F (i.e. ∇KF), in the

special case of F = Q ∈ Orth, it is D that captures the local behaviour of F = Q. Hence, when K

is in a state of contorted aeolotropy, D becomes the measure of inhomogeneity. Additionally, the

curvature tensor R of the Riemannian connection Γ relative to this K also vanishes. The converse

of this statement is true only in a restricted sense: If the curvature of the Riemannian connection

relative to K vanishes, the K is locally a state of contorted aeolotropy.

We can write (4.8) in a basis e1, e2, e3 as

Dmpq uq vp em = −QimQjq Qip,j uq vp em

⇒ εmpn κnq = −QimQjq Qip,j

or, −QimQjq Qip,j = εmpn

[1

2αii δnq − αnq

]=

1

2εmpq Qsi (Curl QT)si − εmpnQsn (Curl QT)sq

[multiplying both sides withQ−1mt Q−1qr ⇒]

−Qtp,r =1

2QtmQrq εmpq Qsi (Curl QT)si −QtmQrq εmpnQsn (Curl QT)sq,

64

which means that Curl Q = 0⇒ ∇Q = 0. Hence, if a rotation field Q is curl free, it is constant.

Alternate proof: QTQ = 1 ⇒ QplQpk = δlk whence, upon differentiating, we get

Qpl,j Qpk + QplQpk,j = 0 ⇒ QplQpk,j is anti-symmetric w.r.t. k and l. Hence, there

exists some second order axial tensor A which satisfies εlmk Amj = QplQpk,j . Now,

Curl Q = 0⇒ εijkQpk,j = 0⇒

εijkQplQpk,j = 0, ⇒ εijk εlmk Amj = 0

⇒ (δil δjm − δim δjl)Amj = 0

⇒ δilAjj −Ail = 0

⇒ Ajj=0, ⇒ Ail = 0

∇Q = 0.

Again, it is straightforward to prove that ∇Q = 0⇒ Curl Q = 0. Hence, ∇Q = 0⇔ Curl Q = 0.

4.5.7 Special Types of Materially Uniform Bodies

In the penultimate section of Noll’s paper, material bodies with two types of isotropy groups have

been discussed: discrete and continuous.

If the isotropy group gK relative to some uniform reference K is discrete, then the continuous

function P : B → gK has to be a constant function. Thus, the relation K(X) = LP(X)K(X)

between two uniform references simplifies to K(X) = LK(X), with L =constant (absorbing P into

L), which implies K−1(X) K(Y ) = K−1(X) L−1 L K(Y ) = K−1(X) K(Y ), i.e., uniform references K

and K correspond to the same material uniformity Φ. Hence, the material uniformity Φ is unique

for discrete isotropy groups, and so is the material connection Γ associated with Φ. So, in case of

discrete isotropy groups, the torsion S becomes a true measure of inhomogeneity of the materially

uniform body.

65

But in case of continuous isotropy groups, the above reasoning that results in the identical unifor-

mities corresponding to two different uniform references breaks down and we, hence, have non-unique

material uniformity, which means, for continuous isotropy group, the torsion S is also non-unique.

Hence, S cannot be a measure of the inhomogeneity of the materially uniform body.

To find the measure of inhomogeneity for bodies with continuous symmetry groups, we relax the

severe restriction K(X) = LK(X), with L = constant, for discrete isotropy groups, to K(X) =

aQ(X)K(X), where a ∈ R and Q(X) ∈ Orth. Such a relation between uniform references is called

a similarity transformation. We construct a class U of uniform references related through similarity

transformations. The inner products relative to two uniform references K and K ∈ U , then, will

differ only by a constant factor a2. Hence, the Riemannian connection Γ is unique for all uniform

references belonging to U , because the definition condition h(f ? l) = f ? (Γhl) + l ? (Γhf) of Γ becomes

independent of the uniform reference with respect to which the inner product is defined (the constant

factor a2 will cancel out from both sides). Thus, the curvature R of the Riemannian connection

Γ becomes the measure for bodies whose uniform references belong to the class U . For example,

undistorted uniform references for an isotropic solid body always belong to U , and, hence, R measures

the deviation from homogeneity in this case. Further, since we already know that in case of a state

of contorted aeolotropy, the associated R is zero (i.e. the concerned body is homogeneous), we have

the following result:

If a uniform isotropic solid body has an undistorted state of contorted aeolotropy, it is

homogeneous.

In fact, the above result generalizes to all type of uniform isotropic bodies.

66

4.6 Extension of Noll’s Theory

A summary of the last section is given in the following figure. It can be seen that the starting

assumption of first grade constitutive law rules out, from the very beginning, the possibility of a non-

zero curvature tensor of the associated material connection. The open question, as already mentioned,

Figure 4.11: Summary of constitutive theory of dislocations given by Noll.

is:

Do we necessarily need higher grade constitutive theories to take non-zero curvature of the material

connection and, hence, a continuous distribution of disclinations into account?

A constitutive theory of disclinations may be a proper answer to this question.

67

Chapter 5

Interfacial Defects and their

Distribution

Interfacial defects present in solid bodies in form of grain and phase boundaries, twin boundaries

and surface cracks. They are abundantly found in polycrystals and are responsible for their plastic

(a) Grain bound-

ary.

(b) Twin bound-

ary.

Figure 5.1: Interfacial defects in crystals.

deformation and hardening. They have been modelled by arrays of line defects and the theoretically

computed stress fields have been matched with the experimentally found ones. But a serious drawback

of modelling by line defects is that it cannot be extended to cases when interfacial defects are present

in many. A theory of interfacial defects in its own (like point and line defects) is still missing.

68

5.1 Modelling of Interfacial Defects by Arrays of Line Defects

Read and Shockley [29] gave the dislocation modelling of low angle tilt boundaries for the first time.

His model successfully predicted the energy characteristics of a low angle tilt boundary and considered

to be a major footstep in defect theory. Drawback of this model was its intrinsic inadequacy to

get extended to high angle grain boundaries and twin boundaries. J. C. M. Li [20] then came up

with a disclination model where he replaced each edge dislocation in the dislocation model by a

wedge disclination dipole. This approach has been taken by many researchers [30, 31] to make more

accurate and appropriate models and to diversify its extent to complex interface arrangements. The

central idea behind disclination modelling is that since grain and twin boundaries are, structurally,

69

discontinuities in rotation field and since disclinations are rotational defects, disclination model would

be more natural and simpler.

5.2 Distribution of Interfacial Defects

Following is the picture of cross-section of a polycrystalline material under microscope. Presence of

Figure 5.2: Distribution of grain boundaries in a polycrystalline material.

multitude of grain boundaries is observable. It is evident that models with arrays of line defects will

simply not work here. We need a proper theory of interfacial defects in its own and a proper density

measure of its distribution. Algebraic topological methods used to discuss theory of topological defects

in ordered media [32] may be solicited. This field is still open.

70

Chapter 6

Conclusion and Open Questions

This report presents only a partial and brief overview of the existing theories of defects in solids.

Point defects and discussion of polymeric and amorphous solids have been excluded. But the follwing

open problems that emerge out from this short review are worth appreciating.

1. Necessity of higher grade theories in continuous distribution of disclinations: justification using

material symmetry arguments.

2. Consideration of piecewise smooth distortion maps to account for surface defects.

3. Density of surface defect distributions.

71

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