Statistics of Mechanics 

or

Statistical modeling of plastic 

yielding at small length scales

J. R. Morris,1,2 H. Bei,1 E. P. George,1,2 G. M. Pharr2,1

1 MS&T Division, Oak Ridge National Lab, Oak Ridge, TN

2 Dept. of Materials Science & Engineering, 

University of Tennessee

Sponsor: Materials Science and Engineering Division, 

Basic Energy Sciences, Department of Energy
2 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Other projects:

• Last CMSN meeting: metallic glasses & interfaces

• Traditional CMSN topic: solid-liquid interfaces, crystal 

nucleation (JCP 2010)

• H2 storage in carbon structures (JPCC 2010)

• Hydrogenated planar structures and interfaces

(PRB 2009)
3 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

“Conventional” length scale effects in nanoindentation

Nix & Gao

J. Mech. Phys. Solids, 1998

•H0 = hardness due to pre-existing 

dislocations

•h

* = characteristic length scale arising 

from geometry, pre-existing dislocations
4 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

What happens as indenter becomes small?

(Focus on spherically tipped indenters)

•a large indenter samples many defects (bulk limit)

•a small indenter likely samples no defects (theoretical limit)

•Intermediate size is much more variable in response.

m-2101210131014

L m1.00.30.1

R

P

L 

1



highly stressed zone
5 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

•For very small indentors, there is a sharp 

transition from elastic to plastic behavior.

•Usually assumed to be due to homogeneous 

formation of dislocation loops

max = 16.3 GPa

R = 178 nm; E = 322 GPa;  =0.30

~ G/8

Hertzian Analysis

P

•

R a

0.48a 45°

h



max 0.31pm 0.31 6PEr

2



3R2













1/ 3

DFT: theo = 15 GPa

(Ogata et al, PRB, 2004)

Ppop-in = 0.4 mN

R

Electropolished (100) Mo
6 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Not necessarily “homogeneous” dislocation nucleation:

Depends on processing!

max= 5 GPa

(~G/14)

Highly annealed vs. pre-strained (100 Ni)

Spherical indenter

R = 580 nm

100 indentations

max= 2.8 GPa
7 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

New nanoindentation size effect:

Pop-in stress varies with both indenter size and dislocation 

density

P

Resolved shear stress 

distribution under indenter 

Large variability 

in pop-in stress

Indenter radius (μm)

Maximum shear stress at pop

-in

Theoretical strength ~ G/8

Annealed (100) Mo (pure)

Spherical indenters:

R = 0.115 - 700 m

Loading rate: P/P = 0.05 s-1

36 indentations each 

•
8 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

A statistical model for pop-in statistics:

• Assume a defect density ρ (per volume).

• Assume the defects are perfectly 

randomly distributed.

• Assign a pop-in stress τpop-in associated 

with each defect.

– For simplicity, assume this is identical for each 

defect.

• What is the probability that a given load P

samples does not affect any defects?

– Determined by the volume where τ> τpop-in

– Poisson statistics: Prob(no defect) = exp(- ρV)

P
9 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Scaling of volume in Hertzian contact theory

• Elastic solution from 

Hertzian contact theory

• Length scale set by 

contact radius a

• Stress scale is set by 

mean pressure or 

equivalently by the peak 

shear stress under the 

indenter.

• V/a3is a function of 

τpop-in/ τmax

τpop-in/ τmax

Volume/(contact radius)

3

τ >τpop-in

a
10 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Statistical model of defect-driven pop-in:

90%

50%

10%

Simplest model:

• Assume a completely random distribution 

of defects.

• Only 2 parameters: 

• Defect activation stress = 0.52 GPa

• Defect density = 2x1016 / m3

• Fit shown at right.

• Red line indicates a 50% cumulative 

probability of pop-in.

• Green, black lines show 10% and 90% 

values.

R2

P

R

Theoretical strength ~ G/8

Maximum shear stress at pop

-in

Indenter radius (μm)
12 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

209 μm

700 μm 130 μm

85 μm

64 μm

36 μm

17.5 μm

6.62

μm

3.75 μm

1.5 

μm 0.58 

μm

Can compare with full probability distribution:

τ

Cumulative probability

theo

τmax(GPa)

Decreasing

indenter radius
13 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Scaling theory:

•From before,the high-stress region 

has a scaled volume

V/a3 =f(τpop-in/ τmax)

•Define τ1⁄2 to be value of τmax when 

cumulative probability = 1⁄2 

– Theory predicts ρV=ln(2) at this stress.

• (ρV)/(ρa

3

)=ln(2)/(ρa

3

)=f(τpop-in/ τ 1⁄2)

•Plotting 1/a3 vs. 1/τ 1⁄2 should 

give a universal curve.

Decreasing

indenter radius

τtheo

700 μm 64 μm

Cumulative probability = 1⁄2 

Cumulative probability

τmax(GPa)

τ >τpop-in

a

τpop-in/τmax

Volume/(contact radius)

3
14 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Behavior of experiments closely matches 

predicted scaling law when τ1⁄2 < τtheo:

• Solid red curve shows fit to theory.

• Only two parameters in fit: the defect 

density, and the defect strength.

• τpop-in =0.52 GPa

• ρ = 0.02 / μm3

• Dashed lines examine sensitivity to 

dislocation density.

Theoretical limit

1/ τ1⁄2 (GPa)

-1

(contact radius)

-3 

(

μm)

-3
16 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Is the defect density reasonable?

• Defect density = ρdefect = 0.02/(μm)3 =2x1016/m3

– Characteristic distance between defects ~ ρdisl

-1/3 = 3.7 μm

– Corresponds to a dislocation density of 2x1011/m2

• For our annealed Mo single crystal: 

Dislocation density ρdisl =1011/m2 = 0.1/(μm)2

– Coarse estimation from x-ray line broadening

– Characteristic distance between dislocations ~ ρdisl

-1/2 = 3.2 μm

–  reasonable estimate of spacing between defects

• Or: ρdisl ~1011/m2 = 0.1/(μm)2 =100 nm/(μm)3

– 1 defect per 50 nm of dislocation length
17 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

What about “Wires & whiskers?”

Size effects without strain gradients

Uchic et al.

Science 2004
18 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

• Defect-free Mo pillars grown by directional solidification 

• Yield at the theoretical strength independent of size

Bei et al, Scripta Mater. (2007)

1. Pillars yield at ~9.2 GPa, independent of size

2. yis ~G/26 (in the range for theo)

3. Plastic deformation is unstable (work softening) 

Important Observations:

<100> Mo Pillar Compression

y = 9.2 GPa

9.2 GPa

9.2 GPa

9.3 GPa

Directional solidification of

Mo-NiAl eutectic
19 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Preliminary experiments show that pre-straining can indeed be 

used to vary the material length scale

Bei, Shim, Pharr, & George, Acta Mater. (2008)

0% pre-strain (as grown)

500-550 nm pillars

4% pre-strain 8% pre-strain 11% pre-strain

 = 0%

4%

8%

11%

P
20 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Effects of FIB damage on mechanical 

properties of nanopillars

Shim et al., Acta Mater. (2009)

Compressive load Nanoindentation behavior
21 Managed by UT-Battelle

for the U.S. Department of Energy Presentation_name

Conclusions:

• A new, exactly solvable model shows that statistics of defects in small 

volumes affect size-dependent nanoindenter pop-in.

• The same two parameters in the model describe the experimental results for 

all indenter sizes (700 μm down to 3.75 μm): 

 the defect strength and the defect density.

• A scaling theory agrees nicely with the experimental results.

• Crossover from bulk (determinative) behavior to stochastic pop-in occurs 

when the defect density ρ is on order of 1/Vstressed. This volume may be 

much greater than (contact radius)3.

• Similar effects seen experimentally in deformation of pristine and predeformed pillars. Small amount of FIB damage hides the effects.

On-going work:

• Same model applied to pillar compression

• Examination of distribution of defect strengths