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A. K. M. B. Rashid

Professor, Department of MME

BUET, Dhaka

MME131: Lecture 9

Imperfections in atomic arrangements Part 2: 1D – 3D Defects

Today’s Topics

Classifications and characteristics of 1D – 3D defects

1D defect – dislocations

2D defects – free surface, grain boundary, twin boundary

3D defects – porosity, cracks, inclusions

References:

1. Callister. Materials Science and Engineering: An Introduction

2. Askeland. The Science and Engineering of Materials

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Line defects, a.k.a. dislocations, are one-dimensional imperfections

in crystal structure where a row of atoms have a local structure that

differs from the surrounding crystal.

These type of defects are almost always present in a real crystals.

In a typical material, about 5 out of every 100 million atoms

(0.000005%) belongs to a line defect.

In a 10-cm3 chunk of material (about the size of a six-sided die),

there will be about 1017 atoms belonging to line defects!

Linear defects

Line defects have a dramatic impact on yielding (i.e., mechanical

deformation) of materials.

Characteristics of line defects

Intrinsic defect

Not equilibrium defects

Concentrations not given by Boltzmann factors

Given enough time and thermal energy, atoms will rearrange

to eliminate dislocations

Caused by processing conditions (how the material is made) and

by mechanical forces that act on the material

Line defects are identified by

Dislocation line — indicates position and orientation of dislocation

Burger’s vector — describes unit slip distance (magnitude and direction )

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Classification of dislocations

Edge dislocation - A dislocation introduced into the crystal

by adding an “extra half plane” of atoms

Screw dislocation - A dislocation produced by skewing a

crystal so that one atomic plane produces a spiral ramp

about the dislocation.

Mixed dislocation - A dislocation that contains partly edge

components and partly screw components.

Can be viewed as an extra half-plane of atoms inserted into

the structure, which terminates somewhere inside the crystal.

• The termination of this half-plane of atoms creates a defect line

(dislocation line) in the lattice (line DC in figure ).

Edge dislocations

edge dislocation

slip plane

• The edge dislocation is

designated by a

perpendicular sign, either

⊥ if the plane is above

the dislocation line, or T if

the plane is below the

dislocation line.

Deformation occurs in material

along the slip plane by the

movement of dislocations.

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Edge dislocations can be quantified using a vector called the Burger’s vector, b,

which represents the relative atomic displacement in the lattice due to the dislocation

To determine Burger’s vector:

• make a circuit from atom to

atom counting the same

number of atomic distances

in opposite directions.

• If the circuit encloses a

dislocation it will not close.

• The vector that closes the

loop is the Burgers vector b.

For edge dislocation, the Burger’s

vector is perpendicular to the

dislocation line

Burger’s vector

Screw dislocations

The perfect crystal (a) is cut and sheared one atom spacing, (b) and (c).

The left region of the crystal is then shifted/twisted one atomic distance upward

relative to the right side of the crystal.

The line along which shearing occurs is a screw dislocation.

A dislocation produced by skewing a crystal so that one atomic plane

produces a spiral ramp about the dislocation

Formed due to the application of a shear stress

dislocation line

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The Burger’s vector for a screw dislocation is constructed in the

same fashion as with the edge dislocation.

representation of defect line (OC ),

Burger’ s circuit and Burger’ s vector

in a screw dislocation

• For screw dislocation,

the Burger’s vector is

parallel to the

dislocation line

Mixed Dislocations

In this case, the Burger’s vector is

neither parallel nor perpendicular

to the dislocation line, but can be

resolved into edge and screw

components.

When a line defect has both an

edge and screw dislocation

component, a mixed dislocation

results.

The exact structure of dislocations in real crystals is usually more complicated.

Edge and screw dislocations are just extreme forms of the possible dislocation

structures. Most dislocations have mixed edge/screw character.

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An edge dislocation in MgO showing

the slip direction and Burgers vector

Dislocations in ceramic materials

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transmission electron micrograph of nickel showing dislocations (dark lines and loops)

tensile zone

compressive zone

The interatomic bonds are significantly distorted

only in the immediate vicinity of the dislocation line.

This area is called the dislocation core.

When an impurity atom is added to the structure, it positioned itself at the

compressive/tensile compressive zone depending on the stress filed created

by the impurity atom.

Lattice Strain

Edge dislocations introduce compressive, tensile, and shear lattice strains.

Screw dislocations introduce shear strain only.

Dislocations have strain fields arising from distortions at their cores.

Strain drops radially with distance from dislocation core

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Slip – the motion of dislocation

When a shear stress is applied to the dislocation the atoms are displaced, causing the

dislocation to move one atomic distance in the slip direction. Continued movement of

the dislocation eventually creates a step, and the crystal is deformed.

Motion of caterpillar is analogous to the motion of a dislocation.

Slip is the movement of large numbers of dislocations to produce plastic deformation.

Slip allows deformation without breaking ductility

Though individual bonds must be broken for dislocation to move, new bonds are formed throughout the slip process

Analogy — caterpillars, carpets, worms

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Classes of surface defects

1. External surface

2. Grain boundary

3. Twin boundary

Surface defects

Surface defects – Imperfections that form a two-dimensional

plane within the crystal.

Surface atoms have unsatisfied atomic bonds, and higher surface

energies, g (J/m2 or, erg/cm2) than the bulk atoms.

To reduce surface free energy, material tends to minimize its surface

areas against the surface tension (e.g. liquid drop).

External Surfaces

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Polycrystalline material comprised of many small crystals or grains having

different crystallographic orientations.

Atomic mismatch occurs within the regions where grains meet.

These regions are called grain boundaries.

Grain Boundaries

(a) The atoms near the

boundaries of the three

grains do not have an

equilibrium spacing or

arrangement.

(b) Grains and grain

boundaries in a stainless

steel sample.

Segregation of impurities occurs at grain boundary.

Dislocations can usually not cross the grain boundary.

angle of misalignment

high-angle grain boundary

low-angle grain boundary

angle of misalignment

High angle grain boundaries cause greater mismatch along the

grain boundary and offer greater resistance to dislocation motion

Depending on

misalignments of atomic

planes between adjacent

grains we can distinguish

between the low and high

angle grain boundaries

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The effect of grain size on the yield strength of

steel at room temperature.

Hall-Petch equation

The relationship between

yield strength (sy) and

grain size (d) in a metallic

material

sy = s0 + K d-1/2

Finer the grains, better are

the mechanical properties

The yield strength of mild steel with an average grain size of 0.05 mm

is 20,000 psi. The yield stress of the same steel with a grain size of

0.007 mm is 40,000 psi. Assuming that the Hall-Petch equation is

valid, what will be the average grain size of the same steel with a yield

stress of 30,000 psi?

Example:

Design of a mild steel

SOLUTION

For a grain size of 0.05 mm the yield stress is

20 6.895 MPa = 137.9 MPa. (Note: 1,000 psi = 6.895 MPa).

Using the Hall-Petch equation

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For the grain size of 0.007 mm, the yield stress is 40 6.895 MPa

= 275.8 MPa. Therefore, again using the Hall-Petch equation:

Solving these two equations, we get

K = 18.43 MPa-mm1/2

σ0 = 55.5 MPa.

If we want a yield stress of 30,000 psi or 30 6.895 = 206.9 MPa,

the grain size will be 0.0148 mm.

Now we have the Hall-Petch equation as

σy = 55.5 + 18.43 d-1/2

Grain Size Measurement

N = 2G-1

N = number of observed grains per square inch

in area on photomicrograph taken at x100.

G = ASTM grain size number

ASTM grain size number (G) - A measure of the size of the

grains in a crystalline material obtained by counting the number

of grains per square inch using a magnification 100.

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SOLUTION

If we count 16 grains per square inch at magnification 250,

then at magnification 100 we must have:

N = (250/100)2 (16) = 100 grains/in2 = 2G-1

ln 100 = (G – 1) ln 2

4.605 = (G – 1)(0.693)

G = 7.64

Example:

Calculation of ASTM grain size number

Suppose we count 16 grains per square inch in a photomicrograph

taken at magnification 250. What is the ASTM grain size number?

Special grain boundaries with mirrored atomic positions across the boundary.

Produced by shear deformation of BCC/HCP materials (mechanical twin), or

during annealing following deformation (annealing twin) of FCC materials.

Twin Boundaries

Application of a stress to the perfect crystal (a) may cause a displacement of the atoms,

(b) causing the formation of a twin. Note that the crystal has deformed as a result of twinning.

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Twining gives rise to shape memory metals, which can recover their

original shape if heated to a high temperature.

Shape-memory alloys are twinned and when deformed they untwin.

At high temperature the alloy returns back to the original twin

configuration and restore the original shape.

A micrograph of twins within

a grain of brass (x250)

Pores affect optical, thermal, and mechanical properties

Cracks affect mechanical properties

Foreign inclusions affect electrical, mechanical, optical properties

Bulk or volume defects

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Next Class

MME131: Lecture 10

Diffusion in solids