Yield Strength as a Function of
Dislocation Density
Travis Grider, David Bahr
Characterization of Advanced Materials
Introduction
Nanoindentation is used on samples of brass
whose grain size and hardness is controlled in
order to test the procedure’s viability on samples
that have a moderate amount of dislocations.
Procedures and Methods
Before doing nanoindentation, the bulk properties of
brass need to be determined so they can be separated
from the nanoindentation data
.
Cold Rolling-plastic deformation which occurs
well below melting point
•Annealing-Heat treatment used for lowering
dislocation density
•Grinding/Polishing-removes surface damage to
produce reasonably flat, reflective surface
•Etching-reveals accurate, sharp definition of
true microstructure of material
•Heyn Intercept Method-Fast and accurate
technique for calculating average grain size
diameter
Once the grain size of a sample is determined, the
Vickers Hardness is found using a micro-hardness
indenter, which uses a square pyramidal indenter
tip.
Yield strength of a material
is a very useful statistic to
know, as it gives a good
upper limit for the stress the
given material can undergo
before plastic deformation.
Many metals’ yield strength
follows the Hall-Petch
relationship,
http://www.hardnesstesters.com/vickers.jpg
Vickers Tip
where k is a material-dependent constant and d is the
average grain size diameter. The Vickers hardness
number is in kgforce/mm2, force per contact area. This
must be converted to SI units and scaled so that it is force
per projected area. The surface area of a Vickers indent is
given by A=d2/2sin(136°/2), where d is the average
diagonal length of the indent. The projected area of this
area is just d2/2. Taking the ratio of contact to projected
area and multiplying by the gravitational constant gives
10.57, divided by 3 for geometric scaling gives 3.52 as the
multiplication factor converting from HV to σy.
Results
20 μm
10 μm
Grain size ~ 2 μm
Grain size and calculated yield stress are plotted and
a curve fit is applied using the Hall-Petch equation to
give experimental values for σo and k.
Grain Size ~ 6 μm
100 μm
Grain Size ~ 16 μm
Monday, July 27, 2009
τ = 3250 MPa
τ=440 MPa
τ=510 MPa
τ=360 MPa
τ=220 MPa
τ=190 MPa
Conclusions
Microstructure after etching
A slightly different sequence of sample prep is needed to
prepare a sample for nano-indentation
Normal order: Cold roll> Anneal> Grind,Polish> Etch>
photograph
Nano-indent order: Cold roll> Grind, Polish> Anneal>
Etch> indent
The sample is etched after annealing to minimize
mechanical damage to the surface as much as possible
•Nanoindentation is
first tested on a single
crystal of Iron – 3% Si
which has a perfect
crystalline structure
•Gives perfectly elastic
load-displacement
graphs as well as
obvious excursions and
yield points
•These perfect indents
give a guideline for
what to look for when
testing brass
•Different samples of
brass whose grain size
and hardness has been
controlled are tested
•Similar behavior is
seen in brass as in the
Iron, albeit not perfect
•Previously
electropolishing was
tried during sample
prep which can alter the
atomic structure, hand
polishing avoids this
•Elastic load-unload
curve for Iron sample is
plotted and curve-fitted
to Hertzian model
P=4/3E*R1/2δ3/2,
using known elastic
modulus to find tip
radius. E* is the elastic
modulus, R is the
indenter tip radius, and
δ is depth.
•This radius value is
then used with the
elastic load-unload
curve of brass to solve
for its ‘effective’ elastic
modulus.
Using nanoindentation, yield points can be
found in materials with moderate dislocation
densities while avoiding the use of
electropolishing.
Literature Used
1.Vander Voort, George. Metallography Principles and Practices. New York: McGraw-Hill,
1984.
2.Shackelford, James. Introduction to Materials Science for Engineers. New Jersey:
Pearson Prentice Hall, 2009.
3.Hull, D and D J Bacon. Introduction to Dislocations. New York: Pergamon Press Inc.,
1984.
Acknowledgements: I would like to thank Mohammed
Zbib and Julie Reid for their assistance throughout the
summer.
This work was supported by the National Science
Foundation’s REU program under grant number
DMR-0755055