Calculation of Indentation

Contact Area and Strain

Using Hertz Theory

Brendan Donohue

Advisor: Prof Surya Kalidindi

Graduate Seminar

16 October 2009

Contributions to the Field

Hertz, 1882

Load-displacement

relationships of

quadratic surfaces in

elastic contact

Hill, 1950

Flow line

predictions of

plastic

deformation for

wedge indenter

Tabor, 1951

Comprehensive work on

macroindentation, load

displacement

relationships, hardness

Sneddon, 1965

Analytical expressions

of load-displacement

using integral

transforms, theory of

elasticity

Johnson, 1985

Collection of

analytical, experimental

results, theory

Oliver & Pharr, 1992

Nanoindentation, modulus

from unloading stiffness

Mesarovic &

Fleck, 1998

Comprehensive FEM studies.

Elastic-plastic

behavior, friction, hardening,

mesh quality, numerical

errors.

** 1980 Myer proposes new

hardness definition to correct

size effect in Brinell Hardness

Area Projected

Load(Myer)HB

Local Properties

Radovic et al., 2004

Pyrex Glass ALA4

Aluminum

4140 Steel

Displacement

(nm)

51.20 901.32 52.08 1452.7

6

53.43 962.36

Load (mN) 0.35 79.96 0.16 79.94 0.81 9.12

Modulus

(GPa)

62.33 61.208 77.38 74.02 216 187.55

Modulus from unloading segment

Basis: Hertz theory

Limitations: friction, ansotropy, contact area

Bell et al., 1992

Hertz Theory

Hertz Theory (1882): Provides a link between linear elasticity and

deflections of quadratic surfaces in contact

Linear

Elastic

Solids

Quadratic

Surfaces

Load

Modulus

Poisson’s Ratio

Separation of

Surfaces

Curvature of

Surface

Displacement

Contact Area

•Each body regarded as

elastic half space with

elliptical contact shape

•Frictionless contact, only

normal pressures transmitted

•Dimensions of contact small

compared to size of body and

curvature of surfaces

)

Quadratic Surfaces

z2

Axisymmetric Bodies, Circular Contact

1

R*

1

R1

+1

R2

1

2R* r

2

zx1

y1

z1

x2

y2

R1

R 1

R 2R 2

X

Y

1

2R1+

1

2R2

2+

1

2R1+

1

2R2

2

( ()X Yz1 - z2

z1 - z2

z1 1

2 R 1x1

2+

1

2 R 1y1

2

z2 -1

2 R 2x2

2-

1

2 R 2y2

2

Each body traced out by

quadratic surfaces

Separation between bodies

∫

Linear Elastic Solids

Pressure Distribution and Displacement:

po 2E

*ht

papo

2aE*

pR*

po 3P

2pa2

p r( ) a2- r

2( )

1/ 2

ppo

4aE*

2a2- r

2( ) +

1

2R*r

2 ht

Displacement Boundary Condition: a3

3PR*

4E*

po

a

+

+

a

r

ds

uz r( ) po

4aE* a

2- r

2+ r

2cos

2( )d

0

2p

Y

X

z2

ht

z1

u

a

R2

+

R1z

X

d

• Assumptions: linear elastic, isotropic material, frictionless contact

P4

3E

*Ri

1/ 2

he

3/ 2

a 2Ri hc-hc2

a

+

Ri

cha

For Purely Elastic Contact, Spherical

Indenter and Flat Surface

Ri he

hc2<< RihC

Ri R*

ht he

Elastic Contact

hehc

1

2

ht

hc

he

hpaRi

Rs

Indenter

Surface

Preloaded

SurfaceFully Loaded

Surface

Fully

Unloaded

Surface

1

E*

1-n s2

( )

Es

+1-n i

2( )

Ei

1

R*

1

Ri

+1

Rs

P 4

3E

*hea a

3

3PR *

4E*

P 4

3E

*R

*1/2he

3/ 2a R

*he

i indenter

s sample

Displacement Must Be Purely

Elastic

Unloading segment is assumed

purely elastic

Complete Load/Unload

Assume unloading segment is purely elastic

1

R*

1

Ri

+1

-Rs

Lo

ad, P

Displacement, h

Loading

Unl

oadi

ng

The assumption is valid for the case of a

spherical indenter on a flat surface, 1/Rs ≈ 0

he ht - hp

hp

ht

he

P 4

3E

*R

*1/2ht - hp( )

3/2

Ri R*

a Rihe

Inelastic Contact

finiteRs

State of the Field

Most analyses use Hertz theory as foundation

Characterize local mechanical properties: Yield point, modulus

Characterize local anisotropy

ChallengesSurface preparation

Friction

Valid Definition of Indentation Stress and Strain

Suggestions

Make use of finite elements

Impose magnitude of friction, hardening, flow stress, elastic modulus

Requires a sound definition of Indentation Stress and Strain

Strain Definition

Physically unrealizable modulus determined

ht

a

a

Ri

a

R*

ht

a

a

Ri

a

R*

Completely Elastic Post-Elastic

Pathak, et al. 2008

Indentation Stress-Strain • How to construct an indentation stress-stain curve?

• For each point on the unloading curve, compute the regression to get effective radius and plastic displacement

P

pa2

e

E*

4

3p

a

Ri

4

3p

he

a

( )i

ti hP ,

1

4

3E * R*

23

hpi

Pi23

2

i1

N

Pi23

i1

N

Pi23

i1

N

N

-1

htiPi

23

i1

N

hti

i1

N

• Compute the contact depth and contact radius

( )ptc hhh +2

1 22 cic hRha - *

*3

4

3

E

PRa

• Compute the indentation stress and strain using Hooke’s law

Indentation Maps

Mesarovic & Fleck, 2000

Park & Pharr, 2004

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.000 0.010 0.020 0.030 0.040 0.050 0.060

( )

a

h

EPRa

tt

p

3

4

43

3/1

*

*

Stress-Strain Behavior

a3

3PR*

4E*

a 2Ri hc-hc2

4

3p

ht

a

t a

Ri

t Using Finite Elements:

Modulus and Yield

Strength are imposed

Raw data is load and

displacement

How do SS curve differ

with definitions of ‘a’

and strain??

4

3pMeyer

Stress

(GPa)

Indentation Strain

Versus

Meyer

Stress2a

P

p

GPa560* E

i

t

cic

R

a

hRha

p

3

4

2 2

-

a

h

hRha

tt

cic

p

3

4

2 2

-

GPa370* E

GPa179* E

Stress Contours

ht (nm) aFEM (um) aQ (um) aE (um) Q (GPa) E (GPa)

0.63 0.089 0.10 0.088 0.426 0.552

0.73 0.108 0.111 0.093 0.429 0.608

0.4250.479

ht 0.73nm

ht 0.63nm

a

2.4a

a

2.4a

Conclusions

Not all definitions of strain are equal

Finite element modeling of indetation

useful in critically examining Hertz’

theory and generating indentation Stress

Strain curves

Significant difficulties exist in determining local

behavior with nanoindentation

c z1 - z2

R1

R2

z1

z2

XY

he 2(ht -hc)

n 12

Spherical Geometry

Pure Elastic Behavior

ht he, hp 0

hc 12 he

Draft Slide