nanoindentation - trinity college dublin · nanoindentation touch the surface of a material to...

9.11.2015 PY5020 Nanoscience 59

Nanoindentation

References

• Nanoindentation, 2nd Ed., Anthony C. Fischer-Cripps, Springer, 2010.

• Introduction to Contact Mechanics, 2nd Ed., Anthony C. Fischer-Cripps,

Springer, 2007.

• Contact Mechanics, Kenneth L. Johnson, ,1985

• M. R. VanLandingham, Review of instrumented indentation, J. Res. Natl.

Inst. Stand. Technol. 108, 249-265 (2003).

• W. C. Oliver, G. M. Pharr, Measurement of hardness and elastic modulus

by instrumented indentation: Advances in understanding and refinements

to methodology, J. Mater. Res., 19(1) (2004).


Nanoindentation

Touch the surface of a material to determine mechanical properties

Origins in Mohs’ hardness scale of 1822 based on scratching, numbered 1-10

Followed by Brinell, Rockwell, Vickers and Knoop indentation 1900-1935

Distinguishing features of nanoindentation:

• Displacement is continuously measured with resolution in nanometers

• Load is continuously measured in nanoNewton to milliNewtons

• Contact area is inferred from known geometry and properties of indenting

tip

• Small amplitude oscillatory response is continuously measured


Nanoindentation

Properties measured:

• Hardness

• Elastic modulus (..Poisson’s ratio, compliance tensor,..)

• Strain hardening exponent

• Fracture toughness (brittle)

• Viscoelastic properties (Storage and Loss Modulus, complex viscosity,…)

• Adhesion properties (interfacial fracture toughness, …)

• Isolated micro/nanoscopic defect and defect population behaviour

• Coupled parameters: Raman signal, electrical, etc.


Instrumentation

One dimensional simple harmonic

oscillator


Basic Measurement

Warning on notation differences from contact mechanics conventions:• For load (ie. force) nanoindentation

sometimes uses L instead of P• For displacement, nanoindentation

usually uses h instead of d

Berkovich 3-sided pyramid


Elastic-plastic Deformation

The elastic limit of deformation of most materials is a

few percent, then Hooke’s Law fails

Plasticity deals with behaviour beyond this limit.

Rigid plastic Elastic plastic Elastic strain

hardening plastic


Yield Criterion

The physical conditions for yield in metals are:

Transition from elastic to plastic behaviour in ductile materials is called the

yield point.

Under the simple tension test, this occurs at the yield stress 0

• Hydrostatic pressure does not cause yield in solids (empirical result)

• For an isotropic material, the yield condition must be independent of

coordinate system

Yield conditions are based on invariants of the stress deviator


Stress tensor decomposition

Define the mean normal stress

The stress tensor can be decomposed into a sum of a mean and deviation

A physical interpretation of this decomposition can be made when a body is

subject to an arbitrary stress:

0 0

0 0

0 0

xx xy xz m xx m xy xz

ij xy yy yz m xy yy m yz

xz yz zz m xz yz zz m

• The mean or hydrostatic component is associated with volume change

ie. Dilation or compression with shape self-similarity

• The deviatoric component is associated with shape change while preserving

volume

3

xx yy zz

m

Hydrostatic (H) Deviatoric (J)

By definition, a fluid is a body that, at rest, cannot sustain a

deviatoric component of stress (this happens at long times)

Total (I)


Decomposition of stress


Principle stresses and von Mises stress

In 3D, the following comprise the stress invariants. Ie. Quantities invariant under

coordinate transformation.

1 xx yy zzI 2 2 2

2 xx yy yy zz zz xx xy yz zxI

2 2 2

3 2xx yy zz xx yz yy zx zz xy xy yz zxI

von Mises Stress: The condition for plastic yield based on strain energy density

211 1 2

23 cos

3 3

II I

212 1 2

2 23 cos

3 3 3

II I

213 1 2

2 43 cos

3 3 3

II I

3

1 1 2 3

3 22

1 2

2 9 271arccos

3 2 3

I I I I

I I

The principle stresses in 3D are

1

2

3

0 0

0 0

0 0

ij

See Barber

2 2 2 2

1 2 1 2 2 3 3 13 ( ) ( ) ( )E I I


Von Mises (Distortion Energy) Criterion

The Von Mises condition states that yield occurs when the second invariant of the

stress deviator J reaches a critical value k2 (eg. see Dieter p. 76):

2

2J k

1 0 2 3; 0 2 2

06 2k

To relate this physical condition to the stress in a simple tension test consisting

of uniaxial tension (assume we are aligned to principal directions for simplicity):

2 2 2

0 1 2 2 3 3 1

1( ) ( ) ( )

2

where2 2 2

2 1 2 2 3 3 1

1( ) ( ) ( )

6J

0 3k

2 2 2 2 2 2

0

1( ) ( ) ( ) 6( )

2xx yy yy zz zz xx xy yz zx

Or for an arbitrary coordinate system this becomes:

“The uniaxial yield stress or yield stress in tension”

For an arbitrary stress, in terms of this uniaxial yield stress we have:


Distortion Strain Energy

1 1( )( )

2 2t xx xxU Pdu A e dx

For an elemental cube under tension

Elastic strain energy by work-energy theorem

Elastic strain energy density (per unit volume) is

1

2t tU Ph

1 1( )( ) ( )

2 2xx xx xx xxe Adx e dV

22

0

1 1 1( )

2 2 2

xxxx xx xxU e e E

E

2

2

0

1 1 1( )

2 2 2

xy

xy xy xyU e e

For shear:

0

1( )

2ij ijU e

2 2 2

0

1 1( ) ( ) ( )

2 2xx yy zz xx yy yy zz zz xx xy yz zxU

E E

In general:

(NB. Poisson effect does not appear as no extra lateral force applied)

(use simple case of P~h)

Peak load, displace.

Applying Hooke’s law


Distortion Strain Energy

In terms of principle stresses2 2 2

0 1 2 3 1 2 2 3 3 1

1( ) 2 ( )

2U

E

2

0 1 2

12 (1 )

2U I I

E In terms of (total) stress invariants

2

3 3(1 2 )b

EK

221

0 1 2

1( 3 )

18 6b

IU I I

K

2 2 2 2

0 1 2 1 2 3 1 2 2 3 3 1

1 1( 3 )

6 6distortionU I I

Bulk modulus,

Slide 29

2 2 2

1 2 2 3 3 1

1( ) ( ) ( )

12

or

Example: Uniaxial tension

1 0 2 3; 0 2

0 0

12

12distortionU

2 2 2

0 1 2 2 3 3 1 2

1( ) ( ) ( ) 3

2J

hydrostatic deviatoric

Ie. same as the von Mises criterion


Hardness

For indentation, the original motivation was to

measure plastic properties through a

parameter called hardness

Hardness is the resistance to penetration by the

indenter that leaves permanent deformation.

0H C

For hardness defined as proj

P

A

It is found (Tabor) that

The constant C is the constraint factor and has a

value of:

• ~3 for high ratio materials eg. metals

• ~1.5 for low ratio materials eg. polymers0E

0E


Hertz (spherical) inelastic contact

Maximum shear stress in the contact stress

field of two solids of revolution occurs

below the surface on the axis of symmetry

Solid lines:

Hertz contact

, ,r z Are principle stresses

on this axis

r On the z-axis, so critical principle stress

difference for von Mises stress equation is z r

1

2 2

0 0

1(1 ) 1 tan

2(1 )

r z a

p p a z z a

2 2

0

1

(1 )

z

p z a

0

4

(1 )

ap

R

z r Is max value of at 00.62 p 0.48a

0 02.8 1.60p k Von Mises

yield (Slide 70)

3 2 3 3 23 3

0 0

(1.6)

12 (1 ) 12 (1 )yield yield

R RP p

( 0.3)

where

and

Slide 56 along z-axis:


Sharp Indentation ProbesGenerally nanoindentation is performed with sharp pyramidal indenters

a

d

Ratio of contact radius with

depth of penetration is constant

giving geometric similarity

Scale invariant deformation

with constant strain

Value of strain is about 8% for

Vickers and Berkovich tips

Singular geometry at the tip is important

practical matter


Instrumented IndentationBulychev, Alekhin, Shorshorov (1970’s), Oliver, Hutchings and Pethica (1983)

Tabor, essential physics of unloading of elastic-plastic materials after

indentation:

1. Plastic deformation is not recovered: All is elastic

2. Under conical indentation, residual impression elastically recovers

depth, but diameter remains unchanged.

*2projAdP

Edh

h

P

rheh

th

A

B

C

tP

2 2

1 2

*

1 2

1 11

E E E

projA Projected area

of contact

Determine Aproj with a shape function for

the tip for some depth: eg. orrhth

rh was chosen based off of TEM

replication data as best choice

* 12

proj

dPE

dhA t

proj

PH

A

NB:

h d

It was shown that for any smooth body of revolution, the initial unload slope is given


Elastic Flat Punch Unloading

Cylindrical flat punch elastic solution for rigid punch:

*2P aE h

2* * *2 2 2

projAdP aaE E E

dh

*2dP A

Edh

True for all axially symmetric

indenters (Oliver-Pharr)

t

e

PdP

dh hAlso from diagram

Slide 3821

2

v P

E ad

2

*

1 1

E E


Sharp Punch: Doerner & Nix

Doener and Nix (1986) key observation:

Initial slope of unloading curve for Berkovich

indenter is linear

Due to combination of indenter shape and

deformation geometry

Key abstraction: Can treat as unloading of

cylindrical flat punch (initially)

*2projAdP

Edh

Stage 1

Stage 2

Stage 3


Doerner Nix

a

h

r

ahrh

eh

r fph

e fph

ph

h

P

rh eh

p r fph h

a e fph h

th

A

B

DC

tP

a e fph h

Consider the imaginary flat punch of radius a

The linear unload slope implies unloading of

the flat punch to the surface and

p r fph h

224.5Berk

proj pA h

is value to be plugged into tip shape function.

Eg. * 12

24.5p

dPE

dhh

24.5

t

p

PH

h

Measure: Peak displacement ht, unload displacement hr


Oliver and Pharr

Sneddon: Load on axi-symmetric indenter of revolution indented into elastic

half space:mP h

h Is elastic displacement in surface

,m Are constants based on materials and

geometry, Eg. m=1 flat cyclinder, m=2

cones, m=1.5 spheres/paraboloids

Unloading of elastic-plastic materials: Under

indentation residual impression recovers only

depth, not diameter

*2projAdP

Edh

h

P

rheh

p rph h

a eph h

th

A

B

DC

tP


Oliver and Pharr

c sh h h

* 2proj

SE

A

( , , )t t tP h S

dPS

dh

Measure at the peak:

Characterize the projected contact area via

the tip shape function at the peak:

( )proj cA F h

At any point:

c t sh h h

So must determine the surface deflection at the contact perimeter..sh

where

To extract elastic modulus:

t

Problem of elastic contact

hf

ht-hf


Oliver and Pharr

This depends on indenter geometry. For cone (Sneddon):

2( )s t fh h h

From shape of elastic surface just

outside of contact region

( )zu r a

The quantity is is the elastic component of the displacement( )t fh h

From the elastic load vs. displacement relation for a cone indenter (Sneddon)

( ) 2 tt f

t

Ph h

S

ts

t

Ph

S 2

( 2)

Conical indenter


An atomic limit for nanoindentation

How do we extend deformation to the smallest of length scales?

Do the deformation mechanics scale?

What does an atomic scale indentation crater look like? Is it stable?


FIM – Field Evaporation

Indenter (tip) engineering by field evaporation

W He

Field ion microscope

Imaging : Voltage 4.5 kV

Field Evaporation : Voltage 5.2 kV

Single

Atom Tip


Uses of a field evaporation point probe

Low energy e- holography

Surface forcesSTM/AFM tip definition

Nanoindentation: geometric BCNanoforming: limits

Contacts in nanoelectronics

Additive: W, Na, Cr, Ni, etc.

Mesoscopic and atomic level control:

M ~ d / Rtip

X-ray flash radiography

Coherent LE e- sourceUltra-bright


Size and symmetry limits on indentation

Carrasco et. al., Phys. Rev. B 68 (2003)

50 nm

What are the size limits to fabrication?

Minimum size Surprising shapes

Do the laws of plasticity work at the nanoscale?

Cu 001 surface


Nanoindentation defect processes


More information

See: G.L.W. Cross, J. Phys. D., 39 p. R1-R24 (2006).

G. L. W. Cross, et. al., Nature Materials, 5p. 370-76 (2006).

Rowland, H. et. al., Science, 302 p. 720-23 (2008).

nanoindentation - trinity college dublin · nanoindentation touch the surface of a material to...

Documents