Crystal Basics

8/2/2019 Crystal Basics

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I. Basics of Crystal Plasticity

Plastic deformation

deformation which remains after load is removed

atomic rearrangements (change of neighbours)

Plastic deformation of crystals preserves lattice structure.

shearing of lattice planes against each other (slip)

plastic displacements are quantized in terms of multiple lattice vectors

At low temperatures / high stresses: Deformation of crystals occurs exclusively

by slip of lattice planes

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

n

r

sr


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On surface: slip can be seen in the form of slip steps along lines where

the slip planes intersect the surface

Example: Slip lines on the surface

of a polycrystalline Ni specimen. Slip

traces from two slip systems are visible

A

F

F

No shear stress if slip direction or slip plane normal are perpendicular

to the tensile axis

Maximum shear stress if slip plane and slip direction are under 45o

to the tensile axis.In single crystals: Slip starts on slip system with

highest resolved shear stress

Driving force for slip: Tensile stress

leads to resolved shear stress in slip system

coscosR =

nr

sr

AF/=R


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How do lattice planes slip?

Not: breaking all bonds simultaneously (this would imply a perfectly strong, but

also perfectly brittle material!)

Instead: slip involves motion of defects: Dislocations

Side view of the motion of an edge dislocation

Dislocation is characterized by slip vector (Burgers vector) and slip plane

Burgers vector: when you make a closed circuit around a dislocation,

there is a mismatch of one lattice vector. This is the Burgers vector


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Type of dislocation depends on orientation of the dislocation line

with respect to the Burgers vector.

Screw dislocation: line is parallel to Burgers vector

Edge dislocation: line is perpendicular to Burgers vector

Some top views of dislocations: (blue = atoms above slip plane

red = atoms below slip plane)

an edge dislocation a screw dislocation

A dislocation loop

Dislocation = boundary ofslipped area

(follow a vertical row of blue atoms

to see the displacement


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Stress required to move dislocations:

1) Stress required to displace dislocation by one Burgers vector (stress to

break bonds): Peierls stress

2) Stress required to move dislocation across obstacles

A dislocation pinned

by a localized obstacle

Introduction of dislocation obstacles: solution/precipitation hardening

3) Stress required to overcome dislocation interactions: work hardening

Dislocations: Internal stresses and dislocation interactions

Dislocations create lattice distorsions and stress and strain fields

Typically 5% of work done during deformation is stored as elastic energy

of the dislocation stress fields

Elastic energy per unit length of dislocation line:(Consequence: only dislocations with smallest Burgers vectors occur)

Stress fields decay like where ris the vertical distance

from the dislocation

2

L ~ GbE

rGb /~


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Stress fields lead to dislocation interactions

Estimating the magnitude of dislocation interactions:

Dislocation density

Mean dislocation spacing

Characteristic interaction stress

areaunit

pointsonintersectiofnumber

eunit volum

linesndislocatiooflength

==

/1=d

GbdGb ~/~


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Rate of deformation by slip (shearing rate ):

proportional to

dislocation density

Burgers vector modulus b

dislocation velocity v

Orowans equation:

Dislocation velocity depends on stress, temperature and presence

of dislocation obstacles

Often: thermally activated process with activation energy

Empirical relation for dislocation glide velocity:

whereNG = stress exponent for glide (typically ~10), T temperature

bvdt

d

=

dt

d

E

=kT

E

Gvv G

NG

exp0

Dislocation climb:

Dislocation can move perpendicular to its glide plane only if matter

is added or removed: climb motion

By climbing, dislocations can move past obstacles

Climb is possible if diffusion processes lead to transport of matter

Usually: vacancy diffusion

Rate of climb depends on concentration and diffusivity of

lattice vacancies : proportional to self-diffusion rate

climb velocity:

whereNC= stress exponent for climb (typically ~3),ESD activation energy

for self-diffusion (formation + migration energies of vacancies)

=kT

E

Gvv SD

NC

exp0


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Diffusional creep:

No motion of dislocations, deformation due to matter transport by

point-defect diffusion

Stress exponent is small (1 or 2)

Often grain boundary diffusion: activation energy is small)

Creep rate

=kT

E

G

D

ND

exp0

&&

Different deformation mechanisms are characterized by different

activation energies and stress exponents

Determination of stress exponents:

Experiments at constant

temperature but varying stress

Plot log(deformation rate)

vs. log(stress)

Determination of activation energies:

Experiments at constant stress

but varying temperature

Plot log(deformation rate)

vs 1/temperature


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Different deformation mechanisms operate in general simultaneously

But: Usually one mechanism (the fastest) prevails

Boundaries between different deformation

mechanisms: Equate respective rates

For rates in the form

we get for the boundary between two mechanisms i and j

hence

choose semi-logarithmic representation of stress vs. temperature

=kT

E

Gdt

d iN

i

i

i

exp0

&

=

kT

E

GkT

E

G

j

N

j

i

N

i

ji

expexp 00

&&

kT

EE

GNN

ji

ji

j

i

=

+

ln)(ln0

0

&

&

Result: Deformation mechanism maps (Example: Ni-base superalloy)

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Crystal Basics