Materials Science and Tesing · o Plastic deformation o Deformation of single crystals o...
Transcript of Materials Science and Tesing · o Plastic deformation o Deformation of single crystals o...
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