mae 493n_503t_lec 2

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“Tribology in Mechanical Engineering”

MAE 493N/593T

Dr. Konstantinos

A.

Sierros

West Virginia University

Mechanical & Aerospace Engineering

ESB Annex

263

[email protected]

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Course material

You can read and download course material from the following blog;

http://wvumechanicsonline.blogspot.com

e‐handouts will be also uploaded in the blog

Please print responsibly!

http://wvumechanicsonline.blogspot.com/

http://wvumechanicsonline.blogspot.com/

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Surfaces

• No real surface can possess perfect geometry

• Surface roughness and errors in the shape (waviness)

Roughness

Waviness

Total profile

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Surfaces

Before polishing After polishing

http://www.leicester‐ils.co.uk/electropolishing.html

• Stainless steel

surface

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Roughness and waviness

• Waviness

(b):

Undulations

with

relatively

long

wavelengths

(mm

scale)• Roughness (c): Variations with much shorter wavelengths

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Contact between surfaces

•

When two surfaces contact each other, the

surface asperities (tips of the

surface roughness) must first carry the load

http://www.theshortspan.com/features/friction.htm

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Contact between surfaces

• Geometric texture

of

surfaces

depend

on;

a.Production route

b.Nature of underlying substrate

Atomic lattice resolution of a mica

surface. Height image. Image size 6.6 nm x

6.6 nm,

z‐range

0.4

nm

http://www.azonano.com/details.asp?ArticleID=2076

• Atomically smooth surfaces

• If 2 mica (clean) surfaces touch each other;

apparent contact area ≈

true contact area

• Squeezing 2 small

rubber

surfaces

together;

apparent contact area ≈

true contact area

(elastic deformation of asperities)

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How do we examine surfaces?• Optical microscopy (up to 1000x mag)

•

Scanning Electron Microscopy ‐

SEM (high

magnification up

to

300000x

and

better field of depth)

•

Transmission Electron Microscopy – TEM (even

higher magnification, 750000x)

•

Atomic

Force

Microscopy

– 3D

surface

topography and true atomic resolution

in high vacuum

•

Profilometry – 3D surface topography, longer

scans and

5 nm

vertical

resolution

diamond stylus

surface

Stylus

movement

MEMS –

Sandia Labs

http://upload.wikimedia.org/wikipedia/commons/d/d4/STILO4.jpg







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Statistical nature of surfaces

• No

single

numerical

parameter

can

adequately

describe

surface

geometry• Centre line average roughness (Ra

) and Root mean square roughness (Rq

)

are the simplest parameters

dx y

L

R

L

∫ =0

1α

Length

Coordinate in the surface

Height of surface measured above

mean level

( At mean level: area of material above = area of the voids below)

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Making smooth surfaces and using Ra

Decreasing roughness

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Issues with average roughness (Ra

)

• Different

topographies

may

have

the

same

Ra value!

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How do we overcome this problem?

• Using

the

root

mean

square

roughness

value

(Rq

)

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

= ∫

L

q dx L R y0

21

In this

case;

y becomes

y2

as

compared

to

Ra

equation

and this squared term gives greater significance to surface variation

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Abbott bearing area curve

• We

should

be

able

to

estimate

the

bearing

area•

Bearing area

is the proportion of the nominal contact area between two

opposing surfaces that are in true contact

It

is

reallybearing

length!

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Statistical treatment of surfaces

•

We first define Φ(z) which is the probability distribution function* of

the z axis

•

For Φ(z) ‐

Plot the number of times a particular value of z occurs in the

data vs. the value of z and scale the most close data fitting curve so the

area under

it

is

unity;

*A mathematical function that is used to model the frequencies and probabilities of

events over time.

∫

+∞

∞− =Φ 1)( dz z

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Statistical treatment of surfaces

•

For real surfaces, a ‘bell‐shaped’

curve is observed and Φ(z) is described by a

Gaussian distribution

(Large

number

of

randomly

occurring

events)

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Statistical treatment of surfaces•

In general, it is not always easy to judge if any given surface

deviates

from

Gaussian

distribution•

To further investigate we can plot the number of times a particular

value of z occurs in the data vs. the value of z, on normal probability axes

SD of

data

A truly random

(Gaussian) distribution

falls

on

a

straight

line

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Engineering surfaces

• In general, random surfaces in engineering are the exception and not

the rule

•

Most surfaces are tailored to our needs and surface patterns are

apparent in most cases (depending on magnification of observation)

Si nanostructure, 200 nm pitch

Fabricated by lithography and DRIE

Chang‐Hwan Choi, UCLA

Turning on a lathe

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Statistical treatment of surfaces• We can attempt to quantify the shape of the distribution curve

• We can

use

the

moments

of

the

distribution*

*A set of numbers that describe various properties of a distribution are called

moments of the distribution

∫

+∞

∞− Φ= dz z zm

n

n )(nth

moment

• For n=1, we generally choose the level from which z is measured

to give us

m1

=0

• For n=2, and taking into account that m1

=0 we arrive at the following;

q

Rm == σ 2

SD of the distribution

root mean square

roughness

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Skewness, Sk

dz z zSm

k )(1 3

33

3

Φ== ∫

+∞

∞−σ σ

3d

moment

• If Sk

=0, the distribution is symmetrical

∫ +∞

∞−

Φ= dz z zmn

n )(

Skewness Sk

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Skewness, Sk

• If Sk

=0 the distribution is symmetrical (a)

• If a process starts removing the peaks (b)

Sk

<0

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Kurtosis, K

dz z zK m

)(1 4

44

4 Φ== ∫ +∞

∞−σ σ

4th

moment

• A

Gaussian

distribution

has

K=3• Distributions with sharper peaks than Gaussian have K<3

∫ +∞

∞−

Φ= dz z zmn

n )(

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Considerations

(dealing only with height distributions)

• These statistical parameters are based on random data and they themselves are

subject to

random

statistical

variations

• Therefore, we may not be able to tell if a given value of Sk

or K represents a true

property of the surface or it has been influenced by the sampling process itself

• There is no information on the horizontal dimension and spatial

distribution!

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Surface height distribution

•

Based only on height distribution the two surfaces below give identical

roughness parameters

•

Their tribological response may be different since (a) exhibits

an ‘open’

texture

and (b) exhibits a ‘closed’

texture

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Surface ‘horizontal descriptors’

•

In order to take into account the horizontal dimension of the surface one should

consider possible parameters such as;

‐ Number of peaks per unit length of profile

‐ Number of crossing points per unit length with the mean level

• None

of

the

above

parameters

are

intrinsic properties

of

the

surface•

It is widely accepted that surfaces exhibit a form of self similarity with increasing

magnification

•

Therefore; Peak density, zero‐crossing density etc are all highly dependent on

the

sampling

interval

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Self ‐similarity

Autocorrelation functions (ACF)

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• Approach to incorporate information on spatial variations

• The ACF is

defined

as;

⎟

⎟⎟

⎠

⎞

⎜

⎜⎜

⎝

⎛

+×= ∫ ∞→

L

L

dx x z x z

L

t 0

2)()(

1lim

1)( τ

σ

ρ

Sampling lengthSD of z


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• When τ=0, ρ(τ)=1 and the curve

must decay

from

a value

of

unity

• As τ

goes to infinity, ρ(τ) becomes

asymptotic to zero

• The form of the curve decay provides

info on

the

spatial

distribution

of

roughness

Summary

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Summary

• No real engineering surface can possess perfect geometry

• There are

several

techniques

of

surface

examination

• Engineering surfaces are different compared to ‘real’

surfaces

• Surface roughness parameters

• Attempts to incorporate information on spatial variations

Out of class activity

You can attempt to solve the following problem;

Find the values of Ra, Rq, Sk, and K for a surface whose profile is given by

z(x)=αcos(πx/α)

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