Materials Science and Tesing

Crystal plasticity

o Crystal structures, dislocations

o Plastic deformation

o Deformation of single crystals

o Deformation of polycrystals

o Texture and anisotropy polycrystalline

materials

Today’s topics

Crystal plasticity

Crystals

• Long range order of atoms

• The atoms’ position can be described with

translation.

Crystal structures

ҧ𝑟 = 𝑛𝑎1 +𝑚𝑎2

m, n - integer numbers

ҧ𝑟 - translation vector

𝑎1, 𝑎2 - basis vectors

Crystal structures

Lattice: periodic array of points in space

Unit cell: the smallest building unit in a lattice

Primitive lattice: volume defined by the tree basis vectors.

Unit cell only contains atoms at the corners.

Non-primitive lattice: The unit cell contains atoms not only

at the corners

Unit cell3 vectors

3 angles a1

a2

a3

α12

α23

α13

Crystal structures

cubic

tetragonal

orthorombic

hexagonal

monoclinic

triclinic

rhombohedral

Unit cell3 vectors

3 angles

a1

a2

a3

α12

α23

α13

Bravais lattices

Crystal structures

Primitive and non-primitive lattice:

Crystal structures

Miller indices

• Vectors

• Crystal planes

Crystal structures

Distance between the planes

BCC (ത111)

For cubic laticces:

Crystal structures

Face Centered Cubic (FCC)

Crystal structures

Body Centered Cubic (BCC)

Crystal structures

Hexagonal lattice

Crystal structures

Diamond lattice

Crystal structures

Close-packed planes in FCC

and hexagonal lattices

stacking sequence

FCC hexagonal

ABCABCABC ABABABABAB

Crystal structures

Ideal lattice: no defects

Real lattice: defects

• Vacancy, interstitial atom 0D Point defects

• Dislocation 1D

• Grain boundaries 2D

• Precipitations 3D

Lattice defects

Dislocations – linear (1D) defects

Burgers vector: The mismatch, when we

make a closed circuit around a dislocation.

Lattice defects

Dislocations – types

Edge dislocation

Screw dislocation

line is parallel to Burgers vector

line is perpendicular to Burgers vector

Slip plane

Lattice defects

Deformation of the lattice

Lattice defects

A transmission electron

micrograph of a titanium alloy in

which the dark lines are

dislocations

Lattice defects

Plastic deformation

• deformation which remains after load is removed

• atomic rearrangements (change of neighbors)

Elastic deformation

• after the load is removed, no deformation remains

• no rearrangements in the atomic order

Plastic deformation of crystals does not change the lattice structure.

Elastic and plastic deformation

Frenkel model

τ - shear stress

x – displacement

𝜏 = 𝐴 𝑠𝑖𝑛2𝜋𝑥

𝑎

𝜏𝑚𝑎𝑥 =𝐺

2𝜋

Hooke's law

G – shear modulus· 102-103 Real strength

Theoretical

strength

Plastic deformation

Plastic deformation – slip of dislocations

Slip plane

Deformation of crystals occurs by slip of lattice planes, motion of

dislocations

Orowan and Taylor

Plastic deformation

Plastic deformation – slip system

Slip system: slip plane + slip direction

Slip system is characterized by:

• slip plane normal

• slip direction, slip vector (a lattice vector in the slip direction)

often: slip planes are most densely packed lattice planes

slip directions are most densely packed lattice directions

Plastic deformation

Reaction of dislocations

Meeting of two dislocation:

Sum of the their vectors

annihilation

Plastic deformation

Cottrel-Lommer junction

Two dislocations combine (two different

slip planes), and the plain of the resulting

dislocation is not a slip plane.

This immobile dislocation will act as a

barrier for other dislocations

Plastic deformation

Driving force for slip:

Tensile stress leads to resolved shear

stress 𝜏𝑟 in slip system

Stress (pressure) 𝜎 =𝐹

𝐴

No shear stress: slip direction or slip plane

normal are perpendicular to the

tensile axis

Maximum shear stress: slip plane and slip direction are

under 45º to the tensile axis.

In single crystals:

Slip starts on slip system with highest 𝜏𝑟 active slip system

Deformation of a single crystal

Macroscopic slip in a single crystals

Deformation of a single crystal

Slip systems in different lattices

active slip system: Slip starts on slip system with highest 𝜏𝑟

Characteristic slip systems

structureslip

directions

slip

planes

Nr. of slip

sys.

bcc <111> {110} 12

fcc <110> {111} 12

hex <11ത20> {0001} 3

Deformation of a single crystal

Single and multiple slip

Singe slip: The deformations starts in only one slip system

(position of the force F)

Multiple slip: Two or more slip system is active

fcc

4 active slip systems

highest stress

Deformation of a single crystal

Str

ess

deformation

stage

stage

stage

easy glide

Multiple

glide

only one slip

system operates

multiplication

and interaction

of dislocations

multiple cross

slip and climb

Deformation of a single crystal

http://www.doitpoms.ac.uk/tlplib/slip/videos.php

Deformation of a single crystal

Only one slip system is active.

The necessary shear stress for the deformation increases only slightly.

Deformation in macroscopic scale:

• A lot of dislocation is necessary.

• 1 dislocation causes a displacement of b.

If b = 2·10-8 m then for the displacement of 1 mm:

dislocation are necessary.

There are not so many dislocation in the material at the beginning.

How are they generated?

Stage I.

Deformation of a single crystal

Frank-Read sources

Deformation of a single crystal

Frank-Read sources

Simulation of frank-read source

Deformation of a single crystal

Stage II.

More and more dislocations are generated,

the shear stress is increasing.

The crystal starts to rotate in a more favorable position:

slip occurs on two or more slip system

The dislocation are not evenly distributed.

Deformation of a single crystal

Stage II.

Dislocation on two or more slip system.

Cottrel-Lommer junction:(The combination of two dislocations is not on a

slip plane.)

1. This immobile dislocation will act as

a barrier for other dislocations.

2. For further deformation new slip

systems become active.

3. Increase of the shear stress

Deformation of a single crystal

Stage III. Cross slip of dislocations.

The dislocation are piled up behind the immobile ones.

The stress is high enough for the screw dislocation to

bypass the immobile dislocation, and continue on a

neighboring slip plane.

Screw dislocations are more

mobile, they are not bounded

to only one slip plane

Deformation of a single crystal

Str

ess

deformation

stage

stage

stage

polycrystal

single crystal

Deformation of polycrystalline materials

Polycrystalline material consist differently

oriented crystals.

During the deformation the continuity of the body

remains. Grain boundaries do not rip apart, rather

they remain together during deformation.

Each grain’s shape is formed by the shape

of its adjacent neighbors

There are more active slip systems in

each grains (min. 5)

The stress increase is always

more intensive in polycrystalline

materials.polycrystalline specimen of copper

Slip lines in

differently

oriented

grains

Deformation of polycrystalline materials

Alteration of the grain structure of a polycrystalline metal as a result of plastic deformation.

(a) Before deformation the

grains are equiaxed.(b) The deformation has

produced elongated grains.

Deformation of polycrystaline materials

𝜀𝑚𝑎𝑐𝑟𝑜 = 𝜀𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑠

Macroscopic and microscopic deformation

The macroscopic deformation

(deformation of the body) is the

average of the crystals’

deformation

𝜀𝑐𝑟𝑦𝑠𝑡𝑎𝑙 = 𝑠𝑙𝑖𝑝

𝑠𝑦𝑠𝑡𝑒𝑚𝑠

𝑠ℎ𝑒𝑎𝑟 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛

The deformation of the crystal

is the sum of the slips on the

slip systems

F

F

Deformation of polycrystaline materials

Texture of polycrystalline materials

> deformation textures

> recrystallization textures

> vapor deposition textures

The texture is the distribution of crystals’

orientations of a polycrystalline material.

No texture: the orientations are fully random.

Texture: some orientation are

more preferred than others

weak, moderate or strong texture

Almost all produced material have texture, and can have

a great influence on material properties.

Origin of the texture:

• Deformation texture (sheets, wires)

• Texture originates from crystallization, or recrystallization (casting, heat treatment...)

• Vapor depositions (PVD, CVD, …)

No texture: isotropic material direction independent properties

Texture: anisotropic material direction dependent properties

Texture of polycrystalline materials

Evolution of deformation textures

Because of the constrained deformation (neighboring crystals) of a crystal minimum 5

active slip system is necessary.

The number of active slip systems depends on the position of the crystal. e.g.

bcc crystal: 5 - 12 (maximum)

The less active slip system is necessary for the deformation the more preferred is the

position of the crystal.

Crystallographic slip results in lattice rotations relative to the external axes of a body.

Sheet rolling

Texture of polycrystalline materials

Representation of Texture

Pole figures and orientation distribution function

Describe the orientations in 3D space of thousands or millions

of individual grains. There ate two important tools:

• the pole figure, based on the stereographic projection.

• the Orientation Distribution Function (ODF), based on

the three Euler angles

http://aluminium.matter.org.uk/content/html/eng/0210-0010-swf.htm

Representation of the texture

http://aluminium.matter.org.uk/content/html/eng/0210-0010-swf.htm

Stereographic projection, pole figures

The unit cube is in the origin of the coordinate

systems and surrounded by the unit sphere.

1. To represent the cube faces:

2. intersection of the normal vector of each

cube face with the surface of the unit

sphere.

3. Only the intersections (1, 2, 3) on the

northern hemisphere are taken into account.

4. Connecting the points of intersection with

the south pole yields the intersecting points

(1', 2', 3') in the equatorial plane.

5. Poles of the respective cube faces.

Representation of the texture

Stereographic projection, pole figures

<100> poles on the stereographic projection (fcc unit cell)

Representation of the texture

Stereographic projection

Comparison of the <100>

and <111> pole figures

Representation of the texture

Pole figures of polycrystals

A set of poles can be plotted for each individual grain.

Pole figure showing

No texture

(no prefered orientation)

Pole figure showing

texture

(prefered orientation)

http://aluminium.matter.org.uk/content/media/flash/metallurgy/anisotropy/pole_figure_polycrystal.swf?targetFrame=Both

Representation of the texture

Inverse pole figures

„opposite” to the pole figure.

pole figure: how the specified crystallographic direction of grains are

distributed in the sample reference frame

inverse pole: figure shows how the selected direction in the sample

reference frame is distributed in the reference frame of the crystal

Partial inverse pole figure of

normal direction

http://www.resmat.com/resources/chapter7.htm

Complete inverse pole figure of

the normal direction

Representation of the texture

Anisotropy

Isotropy: properties are independent of the direction of measurement

Anisotropy: directionality of properties

The physical properties of single crystals depend on the crystallographic

direction in which measurements are taken.

e.g.: the elastic modulus, the electrical conductivity, the index of

refraction may have different values in the [100] and [111] directions

The anisotropy is associated with the variance of atomic or ionic spacing

with crystallographic direction.

Anisotropy in single crystals

The extent of anisotropic effects in crystalline materials are functions of

the symmetry of the crystal structure:

the degree of anisotropy increases with decreasing structural symmetry

e.g.: triclinic structures normally are highly anisotropic

Degree of anisotropy

Anisotropy in single crystals

Elastic modulus for some metals (GPa)

Metal [100] [110] [111]

Aluminum 63.7 72.6 76.1

Copper 66.7 130.3 191.1

Iron 125.0 210.5 272.7

Tungsten 384.6 384.6 384.6

F F =

L1 < L2

E1 > E2

Young modulus:

𝝈 = 𝑬𝜺

𝜎 =𝐹

𝐴− 𝑠𝑡𝑟𝑒𝑠𝑠 𝜀 =

∆𝐿

𝐿0− 𝑠𝑡𝑟𝑎𝑖𝑛

Hooke’s law:

Anisotropy in single crystals

No texture:

The crystallographic orientations of the crystals are totally random.

The polycristalline material behaves isotropic.

It represents the average of the directional properties

Texture:

Preferential crystallographic orientation

Certain properties contributes stronger in the macroscopical

ones.

Anisotropy and texture

Anisotropy in polycrystals

Mechanical anisotropy

Magentic anisotropy

Optical anisotropy

Dielectric anisotropy

Electrical anisotropy

…

Anisotropy in polycrystals