Calculation Of Indentation Contact Area And Strain Using Hertz Theory

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 Tabor, 1951

Hertz, 1882 Load-displacement relationships of quadratic surfaces in elastic contact

Mesarovic & Fleck, 1998

Johnson, 1985

Comprehensive work on Collection of macroindentation, load analytical, experimental displacement results, theory relationships, hardness

Comprehensive FEM studies. Elastic-plastic behavior, friction, hardening, mesh quality, numerical errors.

Hill, 1950

Sneddon, 1965

Oliver & Pharr, 1992

Flow line predictions of plastic deformation for wedge indenter

Analytical expressions of load-displacement using integral transforms, theory of elasticity

Nanoindentation, modulus from unloading stiffness

** 1980 Myer proposes new hardness definition to correct size effect in Brinell Hardness

Load HB (Myer)  Projected Area

Local Properties Radovic et al., 2004

Modulus from unloading segment Basis: Hertz theory Limitations: friction, ansotropy, contact area 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

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 •Each body regarded as elastic half space with elliptical contact shape

Load

Modulus Poisson’s Ratio

•Frictionless contact, only normal pressures transmitted

Separation of Surfaces

•Dimensions of contact small compared to size of body and curvature of surfaces

Curvature of Surface

Displacement Contact Area

Quadratic Surfaces R1 R1

y1 x1

z1 z2

R2

R2

x2

Each body traced out by quadratic surfaces

z

X

1 2 1 2 z1  x1 + y1 2R1 2R1 1 2 -1 2 z2  x2 y2   2R2 2R 2 Separation between bodies

Y

(

z1 - z2 

1

1

) (

+  2R1 2R2

X+ 2

1

1

)

Y +   2R1 2R2

Axisymmetric Bodies, Circular Contact

y2

1 2 z1 - z2  * r 2R

1 1 1 + *  R R1 R2

2

Linear Elastic Solids d ds 

a Y +r   X 

z

u



X

p(r ) 

po 2 2 1/ 2 a (a - r ) 2p po 2 2 2 2 ( ) uz r  a r r cos + )d ( * 4aE 0

∫

Displacement Boundary Condition:

4aE

2R

z1

ht

 a

  Pressure Distribution and Displacement:

ppo ( 2 2 ) 1 2 r  ht + * 2a - r *

+

R1

z2

+ *

R2 *

po  2E h t po  2aE*  pa  pR

*

3PR a  * 4E 3

3P po  2 2pa

Elastic Contact •

Assumptions: linear elastic, isotropic material, frictionless contact

a 2Ri hc - h

1 hc  he 2

2 c

4 * 1/ 2 3/ 2 P  E Ri he 3

a  Ri he

For Purely Elastic Contact, Spherical Indenter and Flat Surface

Ri +

a

*

hc



2

hc << Ri hC

Ri  R ht  he

Complete Load/Unload Indenter Surface

Fully Unloaded Surface

Fully Loaded Surface

Preloaded Surface

1 (1- n ) (1- n ) + *  E Es Ei 1 1 1 + *  R Ri Rs 2 s

ht

hc

Ri

a

4 * P  E hea 3 4 * *1/ 2 3/ 2 P  E R he 3

Rs

hp

he

3PR   a * 4E  3

*

*

a  R he

2 i

i  indenter s  sample Displacement Must Be Purely Elastic

 Unloading segment is assumed purely elastic

Inelastic Contact Assume unloading segment is purely elastic

he  ht - hp

ht

a  Ri he



ing

ad Lo

in g

* Ri  R

hp

ad

Load, P

1 1 1 + *  R Ri -Rs

Rs  finite

he

Un lo

3/ 2 4 * *1/ 2 P  E R ( ht - hp ) 3

Displacement, h

The assumption is valid for the case of a spherical indenter on a flat surface, 1/R s ≈ 0 

State of the Field Most analyses use Hertz theory as foundation

Characterize local mechanical properties: Yield point, modulus Characterize local anisotropy

Challenges Surface 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 Pathak, et al. 2008

Physically unrealizable modulus determined Completely Elastic

Post-Elastic

ht a a   * a Ri R

ht a a   * a Ri R

Indentation Stress-Strain • •

How to construct an indentation stress-stain curve? For each point (Pi , hti ) on the unloading curve, compute the regression to get effective radius and plastic displacement

      •

-1

 3  Pi  i1   N   N

2

 N i 2   ht Pi 3  i1  N  i  h   t   i1 

Compute the contact depth and contact radius

hc   •

1 4 *  E 3 h pi

  N  2 3 2 2 P  3   i   R*   i1N    2 3 P  i     i1

1 (ht + hp ) 2

a  2hc Ri - hc2

* 3 PR a3  4E *

Compute the indentation stress and strain using Hooke’s law

P  2 pa



4 a 4 he e  *   E 3p Ri 3p a

Indentation Maps Mesarovic & Fleck, 2000

Park & Pharr, 2004

Stress-Strain Behavior 3PR a  * 4E 3

a 2Ri hc - h Meyer Stress (GPa)

Using Finite Elements:

t  4 ht 3p a

*

t  4 a 3p Ri

2 c

1.40

Versus Meyer Stress

P pa 2

Modulus and Yield Strength are imposed Raw data is load and displacement How do SS curve differ with definitions of ‘a’ and strain??

1.20 1.00

E *  370 GPa

0.80 0.60 0.40 0.20

0.00 0.000

a  2hc Ri - hc2

E *  179 GPa

(

a  3PR

t 

t 

)

1/ 3

*

4E*

4ht 3pa

E *  560 GPa a  2hc Ri - hc2 4a t  3pRi

4ht 3pa

0.010

0.020

0.030

Indentation Strain

0.040

0.050

0.060



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

ht  0.63nm

ht  0.73nm 0.479

a 2.4a

0.425



a 2.4a

 

Conclusions Significant difficulties exist in determining local behavior with nanoindentation

Not all definitions of strain are equal Finite element modeling of indetation useful in critically examining Hertz’ theory and generating indentation Stress Strain curves

Draft Slide

n  12 he  2(ht - hc )

R1

XY

z1

c  z1 - z2 z2





R2

Spherical Geometry

Pure Elastic Behavior

ht  he, hp  0 1 hc  2 he