Adhesion Contact Deformation in Nanobridge Tests · Schematic diagram of mechanics analysis for the vertical nanobridge bending test. In the linear elastic analysis, the governing

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Electronic Supplementary Information

Adhesion Contact Deformation in Nanobridge Tests

Yao Gaoa,b, San-Qiang Shib and Tong-Yi Zhanga,*

a Shanghai Materials Genome Institute and Shanghai University Materials Genome Institute, Shanghai University,

99 Shangda Road, Shanghai, China

b Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong,

China

* To whom correspondence should be addressed: [email protected] and [email protected]

Electronic Supplementary Material (ESI) for Nanoscale.This journal is © The Royal Society of Chemistry 2017

mailto:[email protected]


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Contents

Electronic Supplementary Material .............................................................................................................. 1

1. Fabrication process flow for suspending gold nanobridges.................................................. 3

2. Nanowire bending test theory ..................................................................................................... 4

3. Details for the 14 adhesion contact models ............................................................................... 7

Model 1 ............................................................................................................................... 9

Model 2 ............................................................................................................................. 11

Model 3 ............................................................................................................................. 13

Model 4 ............................................................................................................................. 14

Model 5 ............................................................................................................................. 16

Model 6 ............................................................................................................................. 17

Model 7 ............................................................................................................................. 18

Model 8 ............................................................................................................................. 19

Model 9 ............................................................................................................................. 20

Model 10 ........................................................................................................................... 21

Model 11 ........................................................................................................................... 22

Model 12 ........................................................................................................................... 22

Model 13 ........................................................................................................................... 23

Model 14 ........................................................................................................................... 24

4. Fitting results of each contact model ........................................................................................ 25

5. Experimental results of 𝑭𝒑𝒖 − 𝑭𝒎𝒂𝒙 from nanobridge bending test. .......................... 26

6. Theoretical calculated loading-unloading force-deflection curves .................................... 27

7. Influence of 𝑵𝒓, 𝑳 and 𝑭𝒎𝒂𝒙 on ∆𝑬 under given experiment conditions .................... 28

8. Influence of 𝑬𝒖 on ∆𝑬 under different experiment conditions ....................................... 29

9. Influence of 𝑬𝒖 on ∆𝝆 and 𝝏𝑬𝝏𝝆 under given experiment conditions ......................... 30

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1. Fabrication process flow for suspending gold nanobridges

Figure S1 illustrates the fabrication process flow of gold nanobridge samples with precisely controlled

geometrical dimensions. At first, a silicon nitride layer (thickness (t): 100nm) together with a silicon dioxide layer

(t: 500nm) was thermally grown on a four-inch silicon wafer. A ~10nm titanium layer was used as the adhesion

layer. Gold NWs with the same thickness of ~90nm were then patterned by direct write electron beam

lithography (EBL) (JBX-6300FS, JEOL Ltd., Japan). After that, another silicon dioxide layer (t: 500nm) serving as

clamps was deposited by PECVD and patterned through the conventional photolithography method. Finally,

the silicon dioxide and titanium adhesion layers were etched away in certain regions by Reactive Ion Etching

(RIE) (Oxide RIE Etcher, Oxford) and the gold NWs became suspended and turned into purely gold NBs with

precisely controlled lengths and fixed boundary conditions.

Figure S1. Fabrication process flow of gold nanobridges with double-clamped ends

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2. Nanowire bending test theory

As illustrated in Figure S4, the neutral plane of bending is chosen as the (x, y)-plane and the center of the

nanobridge as the origin. The residual force 𝑁𝑟 is assumed to be completely released along the nanobridge

width direction and homogeneously distributed only along the nanobridge length direction and hence the

residual stress is expressed as 𝜎𝑟 = 𝑁𝑟 𝑡⁄ , with t the thickness of nanobridge. The loading position is with distance

a from the left end.

Figure S2. Schematic diagram of mechanics analysis for the vertical nanobridge bending test.

In the linear elastic analysis, the governing equation for beam bending in terms of moment balance is given

by:

𝐷𝑑2𝑢1𝑑𝑥2

− 𝑁𝑟(𝑢1 − 𝑢10) + 𝑃1 (𝑥 +

𝐿

2) +𝑀1 = 0 − 𝐿/2 ≤ 𝑥 ≤ 𝑎 (𝑆1𝑎)

𝐷𝑑2𝑢2𝑑𝑥2

− 𝑁𝑟(𝑢2 − 𝑢20) + 𝑃2 (

𝐿

2− 𝑥) + 𝑀2 = 0 𝑎 ≤ 𝑥 ≤

𝐿

2 (𝑆1𝑏)

where u is the displacement in the z-axis, 𝐿 is the bridge length, 𝑤 is the bridge width, 𝐷 = 𝐸𝑡3/12 is the

flexural rigidity, 𝐸 is the Young’s modulus of the nanobridge, and 𝑃1 and 𝑃2 are the generalized forces.

𝑃1 and 𝑃2 could be reduced based on force equilibrium given as below:

𝑃1 + 𝑃2 = 𝑄 (𝑆2𝑎)

𝑀1 −𝑀2 + 𝑃1𝐿 − 𝑄 (𝐿

2− 𝑎) + 𝑁𝑟(𝑢1 − 𝑢2) = 0 (𝑆2𝑏)

𝑃1 = 𝑄 (1

2−𝑎

𝐿) +

𝑀2 −𝑀1𝐿

+𝑁𝑟(𝑢2

0 − 𝑢10)

𝐿 (𝑆2𝑐)

𝑃2 = 𝑄 (1

2+𝑎

𝐿) +

𝑀1 −𝑀2𝐿

+𝑁𝑟(𝑢1

0 − 𝑢20)

𝐿 (𝑆2𝑑)

Solving equations (S1a) and (S1b), we have the following solutions:

𝑢1(𝑥) = 𝐴1 𝑠𝑖𝑛ℎ(𝑘𝑥) + 𝐵1 𝑐𝑜𝑠ℎ(𝑘𝑥) +1

𝑁𝑟[𝑃1 (

𝐿

2+ 𝑥) +𝑀1] + 𝑢1

0 − 𝐿/2 ≤ 𝑥 ≤ 𝑎 (𝑆3𝑎)

𝑢2(𝑥) = 𝐴2 𝑠𝑖𝑛ℎ(𝑘𝑥) + 𝐵2 𝑐𝑜𝑠ℎ(𝑘𝑥) +1

𝑁𝑟[𝑃2 (

𝐿

2− 𝑥) + 𝑀2] + 𝑢2

0 𝑎 ≤ 𝑥 ≤𝐿

2 (𝑆3𝑏)

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where

𝑘 = √𝑁𝑟𝐷 (𝑆4)

Substituting the relationship in Eq. (S2) into Eq. (S3) yields

𝑢1(𝑥) − 𝑢10 = 𝐴1 𝑠𝑖𝑛ℎ(𝑘𝑥) + 𝐵1 𝑐𝑜𝑠ℎ(𝑘𝑥) +

𝐿

𝑁𝑟[𝑄 (

1

2−𝑎

𝐿) (1

2+𝑥

𝐿) + (

1

2−𝑥

𝐿)𝑀1𝐿+ (1

2+𝑥

𝐿)𝑀2𝐿]

+(𝑢20 − 𝑢1

0)(1

2+𝑥

𝐿) − 𝐿/2 ≤ 𝑥 ≤ 𝑎 (𝑆5𝑎)

𝑢2(𝑥) − 𝑢20 = 𝐴2 𝑠𝑖𝑛ℎ(𝑘𝑥) + 𝐵2 𝑐𝑜𝑠ℎ(𝑘𝑥) +

𝐿

𝑁𝑟[𝑄 (

1

2+𝑎

𝐿) (1

2−𝑥

𝐿) + (

1

2−𝑥

𝐿)𝑀1𝐿+ (1

2+𝑥

𝐿)𝑀2𝐿]

+(𝑢10 − 𝑢2

0)(1

2−𝑥

𝐿) 𝑎 ≤ 𝑥 ≤

𝐿

2 (𝑆5𝑏)

Applying displacement boundary conditions 𝑢1 (−𝐿

2) = 𝑢1

0 = 0, 𝑢2 (𝐿

2) = 𝑢2

0 = 0; we have:

𝐴1 𝑠𝑖𝑛ℎ (𝑘𝐿

2) − 𝐵1 𝑐𝑜𝑠ℎ (

𝑘𝐿

2) =

𝑀1𝑁𝑟 (𝑆6𝑎)

𝐴2 𝑠𝑖𝑛ℎ (𝑘𝐿

2) + 𝐵2 𝑐𝑜𝑠ℎ (

𝑘𝐿

2) = −

𝑀2𝑁𝑟 (𝑆6𝑏)

Then considering continuity requirements 𝑢1(𝑎) = 𝑢2(𝑎), 𝑢1′ (𝑎) = 𝑢2

′ (𝑎) , we have the following

equations:

(𝐴1 − 𝐴2) 𝑠𝑖𝑛ℎ(𝑘𝑎) + (𝐵1 − 𝐵2) 𝑐𝑜𝑠ℎ(𝑘𝑎) = 0 (𝑆7𝑎)

(𝐴1 − 𝐴2) 𝑐𝑜𝑠ℎ(𝑘𝑎) + (𝐵1 − 𝐵2) 𝑠𝑖𝑛ℎ(𝑘𝑎) = −𝑄

𝑁𝑟𝑘 (𝑆7𝑏)

By solving equation set (S6) and (S7), the coefficients of 𝐴1, 𝐴2, 𝐵1 and 𝐵2 are determined to be:

𝐴1 =

𝑘(𝑀1 −𝑀2) + 𝑄𝑠𝑖𝑛ℎ (𝑘 (𝑎 −𝐿2))

2𝑘𝑁𝑟 𝑠𝑖𝑛ℎ (𝑘𝐿2)

(𝑆8𝑎)

𝐵1 = −

𝑘(𝑀1 +𝑀2) − 𝑄𝑠𝑖𝑛ℎ (𝑘 (𝑎 −𝐿2))

2𝑘𝑁𝑟 𝑐𝑜𝑠ℎ (𝑘𝐿2)

(𝑆8𝑏)

𝐴2 =𝑘(𝑀1 −𝑀2) + 𝑄𝑠𝑖𝑛ℎ (𝑘𝑎 +

𝑘𝐿2)

2𝑘𝑁𝑟 𝑠𝑖𝑛ℎ (𝑘𝐿2)

(𝑆8𝑐)

𝐵2 = −𝑘(𝑀1 +𝑀2) + 𝑄𝑠𝑖𝑛ℎ (𝑘𝑎 +

𝑘𝐿2)

2𝑘𝑁𝑟 𝑐𝑜𝑠ℎ (𝑘𝐿2)

(𝑆8𝑑)

The substrate deformation was taken into account by using three generalized springs[1], in which the

bending moments of 𝑀1 and 𝑀2, indicated in Fig. 2(c), make the major contribution. For simplicity, only

the bending moment-induced substrate deformation is considered in the present work by using the

compliance 𝑆𝑀𝑀.[1] The slop of the deflection at the two ends should satisfy:

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𝜃1 (𝑥 = −𝐿

2) =

𝑑𝑢1𝑑𝑥|𝑥=−

𝐿2

= 𝑘𝐴1(𝑀1, 𝑀2) 𝑐𝑜𝑠ℎ (𝑘𝐿

2) − 𝑘𝐵1(𝑀1, 𝑀2) 𝑠𝑖𝑛ℎ (

𝑘𝐿

2)

+1

𝑁𝑟[(1

2−𝑎

𝐿)𝑄 +

𝑀2 −𝑀1𝐿

] = −𝑆𝑀𝑀 ∗ 𝑀1 (𝑆9𝑎)

𝜃2 (𝑥 =𝐿

2) =

𝑑𝑢2𝑑𝑥|𝑥=𝐿2

= 𝑘𝐴2(𝑀1, 𝑀2) 𝑐𝑜𝑠ℎ (𝑘𝐿

2) + 𝑘𝐵2(𝑀1, 𝑀2) 𝑠𝑖𝑛ℎ (

𝑘𝐿

2)

−1

𝑁𝑟[(1

2+𝑎

𝐿)𝑄 +

𝑀1 −𝑀2𝐿

] = 𝑆𝑀𝑀 ∗ 𝑀2 (𝑆9𝑏)

Solving equation set S9 results in:

𝑀1

=𝑄𝑁𝑟[𝑆𝑀𝑀(−2𝑎 + 𝐿)] − 2𝑄 + 2𝑘𝑁𝑟[𝐴2 + 𝐴1(−1 + 𝐿𝑁𝑟𝑆𝑀𝑀)] 𝑐𝑜𝑠ℎ (

𝑘𝐿2) + 2𝑘𝑁𝑟[𝐵2 − 𝐵1(−1 + 𝐿𝑁𝑟𝑆𝑀𝑀)] 𝑠𝑖𝑛ℎ (

𝑘𝑙2)

2𝑁𝑟𝑆𝑀𝑀(2 − 𝑁𝑟𝑆𝑀𝑀𝐿) (𝑆10𝑎)

𝑀2

=𝑄𝑁𝑟[𝑆𝑀𝑀(2𝑎 + 𝐿)] − 2𝑄 − 2𝑘𝑁𝑟[𝐴1 + 𝐴2(−1 + 𝐿𝑁𝑟𝑆𝑀𝑀)] 𝑐𝑜𝑠ℎ (

𝑘𝐿2) + 2𝑘𝑁𝑟[𝐵1 − 𝐵2(−1 + 𝐿𝑁𝑟𝑆𝑀𝑀)] 𝑠𝑖𝑛ℎ (

𝑘𝑙2)

2𝑁𝑟𝑆𝑀𝑀(2 − 𝑁𝑟𝑆𝑀𝑀𝐿) (𝑆10𝑏)

Then, the expression for 𝑀1+ 𝑀2 and 𝑀1 −𝑀2 are obtained as follows:

𝑀1+𝑀2 = 𝑄 ∙𝑐𝑜𝑠ℎ(𝑘𝑎) − 𝑐𝑜𝑠ℎ (

𝑘𝐿2)

𝑘𝑠𝑖𝑛ℎ (𝑘𝐿2) + 𝑁𝑟𝑆𝑀𝑀 𝑐𝑜𝑠ℎ (

𝑘𝐿2) (S11a)

𝑀1−𝑀2 = 𝑄 ∙−𝐿𝑠𝑖𝑛ℎ(𝑘𝑎) + 2𝑎 𝑠𝑖𝑛ℎ (

𝑘𝐿2)

𝑘𝐿𝑐𝑜𝑠ℎ (𝑘𝐿2) + (−2 + 𝐿𝑁𝑟𝑆𝑀𝑀) 𝑠𝑖𝑛ℎ (

𝑘𝐿2) (S11b)

Substituting above equations into either equation in (S5), then the expression of position 𝑎 dependent

compliance 𝑢(𝑎)/𝑄 could be obtained.

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3. Details for the 14 adhesion contact models

Effective Young’s modulus 𝐸 Dupre energy of adhesion Γ

Poisson ratio 𝑣 Contact area 𝐴

Yield strength 𝑌 Penetration depth 𝛿

Effective contact curvature 𝑅 Adhesion force 𝐹𝑠

External load 𝐹 Pull-off force 𝐹𝑝𝑢

Contact load 𝑃 Truncated contact radius 𝑎′

Contact pressure 𝑝𝑚 Intermolecular distance 𝑧

Contact radius 𝑎

Subscribe

𝑒 In the elastic contact regime

𝑐 The initial point of plastic yielding

𝑒𝑝 In the elastic-plastic contact regime

𝑝 The initial point of full plastic regime

𝑓𝑝 In the full plastic contact regime

𝑟𝑒𝑠 The residual values after unloading

𝑚𝑎𝑥 The values corresponds to the maximum loading point

𝑢𝑛 In the unloading process

As we mentioned previously, only several models provided complete equations: the TN[2] model, the MP[3]

model, the Jackson and Green (JG) model[4, 5] and the Rathbone[6] model. Besides all these models, the Big-Alabo[7]

model only lacks the equation for 𝑅𝑟𝑒𝑠 and the equations for 𝑅𝑟𝑒𝑠 provided in Brake[8] model exhibited great

error in the fitting; therefore, equations from other models[6, 9] for 𝑅𝑟𝑒𝑠 was used for these two models. For the

models developed by the Etsion group: Chang et al.[10] developed one expression for the critical penetration depth

corresponds to the inception of plastic deformation (𝛿𝑐) and has been widely adopted by the following works.

Kogut and Etsion (KE)[11, 12] developed the load-interference equations in the elastic-plastic region without[12]

and with[11] the adhesion force respectively. Etsion et al.[13] provided another load-interference equation and an

expression for 𝑅𝑟𝑒𝑠. Brizmer et al.[14, 15] provided equations for 𝛿𝑐, and the load-interference equations for elastic-

plastic contact for both slip and full stick contact conditions. The slip contact assumed no tangential stresses in

the contact area and the full stick contact assumed that contacting points of the sphere and the flat were prevented

from further relative displacement. Zait et al.[16] provided equations for the unloading process for both stick and

slip conditions. Since no single model in Etsion’s group provided complete equations, several different models

needed to be combined for the data fitting. As we mentioned before, only TN[2], MP[3] and KE[11] models

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considered the influence of adhesion force, in order to employ other models for data fitting, the adhesion force

in models developed by Etsion group followed the KE model[11], and for the JG[4], Brake[8], Rathbone[6] and Big-

Alabo[7] models, the adhesion forces were treated as a constant of 2𝜋𝑅Γ, and the pull-off force equals 2𝜋𝑅𝑟𝑒𝑠Γ.

Moreover, in the Brake[8] model, a different definition for effective hardness is adopted and in order to

differentiate the influence of this new hardness definition, both these hardness equations were used for the data

fitting. Based on these basic contact models, total 14 compositions were constructed for the fitting the detailed

equations for each model were provided in the supporting information.

Table S1. Composition of each model used for data fitting.

M

odel

No.

Initial

point of

yielding

Elastic-plastic&

full-plastic Region

Initial

point of full-

plasticity

Unloading

1 Johnson[17] TN[2, 18] Johnson[1

7] TN[2, 18]

2 Brizmer[14,

15] Brizmer[14, 15]

Brizmer[1

4, 15] Zait[16]+Hertz

3 Johnson[17] Brake[8] Brake[8] Rathbone[6]+He

rtz

4 JG[4, 5] JG[4, 5] JG[4, 5] JG[4, 5] + Hertz

5 Johnson[17] Big-Alabo[7] Stronge

[15]

Rathbone[6]+He

rtz

6 MP[3] MP[3]+Majumdar[

19]

Johnson[1

7] MP[3]

7 Johnson[17] Rathbone[6] N/A Rathbone[6]+He

rtz

8 Johnson[17] Brake[8] Brake[8] SK[9] + Hertz

9 Brizmer[14,

15] Brizmer[14, 15]

Brizmer[1

4, 15] EKK[13]

10 CEB[10, 20] KE[12] KE[12] EKK[13]

11 CEB[10, 20] EKK[13] EKK[13] EKK[13]

12 Johnson[17] Brake et al.[8] Brake[8] Rathbone[6]+He

rtz

13 Johnson[17] Big-Alabo[7] Stronge15 SK[9] + Hertz

14 Johnson[17] Brake[8] Brake[8] SK[9] + Hertz

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Model 1

I. Elastic region: [JKR][17, 21]

𝐹𝑒 = 4𝐸𝑎𝑒

3

3𝑅− √8𝜋Γ𝐸𝑎𝑒

3 (𝑆12)

𝛿𝑒 =𝑎𝑒2

𝑅−√

2Γ𝜋𝑎𝑒𝐸

(𝑆13)

II. Initial point of yield: [Johnson][17]

The yield of most ductile material is usually taken to be governed by von Mises’ shear strain-energy criterion:

𝐽2 ≡1

6{(𝜎𝑧 − 𝜎𝑟)

2 + (𝜎𝑧 − 𝜎𝜃)2 + (𝜎𝜃 − 𝜎𝑟)

2} = 𝑘2 =𝑌2

3 (𝑆14)

or by Tresca’s maximum shear stress criterion:

max{|𝜎𝑧 − 𝜎𝑟|, |𝜎𝑧 − 𝜎𝜃|, |𝜎𝜃 − 𝜎𝑟|} = 2𝑘 = 𝑌 (𝑆15)

In which 𝜎𝑧 , 𝜎𝑟 , 𝜎𝜃 are all principal stresses and 𝑘 and Y denote the values of the yield stress of the softer

material in simple shear and simple tension (or compression) respectively.

Before we introduce the JKR model for elastic-plastic contact, we will first introduce the elastic-plastic

contact without adhesion, which is the Hertz model.

For pure Hertz contact, the principal stresses are given by the following equations[17]:

𝜎𝑟 = 𝜎𝜃 = −𝑝0(1 + 𝑣) {1 − (𝑧

𝑎) 𝑡𝑎𝑛−1 (

𝑎

𝑧)} +

1

2𝑝0 (1 +

𝑧2

𝑎2)

−1

(𝑆16)

𝜎𝑧 = −𝑝0 (1 +𝑧2

𝑎2)

−1

(𝑆17)

The maximum value of |𝜎𝑧 − 𝜎𝑟|, for 𝑣 = 0.43 (value of gold), is 0.56𝑝0 at a depth 𝑧 = 0.52𝑎.

Thus by the Tresca criterion the value of 𝑝0 for the initial point of yield is given by

0.56𝑝0 = 2𝑘 = 𝑌 (𝑆18)

𝑝0 = 3.57𝑘 = 1.79𝑌 (𝑆19)

Whilst by the von Mises criterion

|𝜎𝑧 − 𝜎𝑟| = 0.56𝑝0 = 𝑌 (𝑆20)

𝑝0 = 1.79𝑌 (𝑆21)

When taking into consideration of surface adhesion, for the first point of plastic yielding, the contact radius

𝑎𝑐 should satisfy the following equation:

2𝐸𝑎𝑐𝜋𝑅

− (2Γ𝐸

𝜋𝑎𝑐)

12= 𝑝0 = 1.79𝑌 (𝑆22)

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III. The elastic-plastic region: [TN][2, 18]

The applied force is:

𝐹𝑒𝑝 =4𝐸𝑎𝑐

3

3𝑅− (8Γ𝜋𝐸𝑎)

12𝑎𝑐 + 𝜋𝑝0(𝑎

2 − 𝑎𝑐2) (𝑆23)

The interference is:

𝛿𝑒𝑝 =𝑎2

𝑅− √

2Γ𝜋𝑎

𝐸 (𝑆24)

IV. Initial point of full plastic: [Johnson] [17]

𝑎𝑝 =60𝑅𝑌

𝐸 (𝑆25)

V. Unloading:[TN][18]

Ning[18] assumed a continuous change of contact radius at the unloading point, and with 𝐹∗ as the

maximum applied force, 𝑎∗ as the corresponding contact radius, we have:

𝑎∗3 =3𝑅𝑃1

∗

4𝐸=3𝑅𝑝𝑃1𝑟

∗

4𝐸 (𝑆26)

where 𝑃1∗ and 𝑃1𝑟

∗ are the equivalent Hertzian forces at the end of loading and at the beginning of

unloading, respectively:

𝑃1∗ = 𝐹𝑒

∗ + 2𝑃𝑐 ± √4𝑃𝑐𝐹𝑒∗ + 4𝑃𝑐𝑃𝑐 (𝑆27)

𝑃1𝑟∗ = 𝐹∗ + 2𝑃𝑐𝑟 ± √4𝑃𝑐𝑟𝐹

∗ + 4𝑃𝑐𝑟𝑃𝑐𝑟 (𝑆28)

where 𝑃𝑐 =3

2𝜋Γ𝑅, 𝑃𝑐𝑟 =

3

2𝜋Γ𝑅𝑟𝑒𝑠 and 𝐹𝑒

∗ is the elastic force according to JKR theory:

𝐹𝑒∗ =

4𝐸𝑎∗3

3𝑅− √8Γ𝜋𝐸𝑎∗3 (𝑆29)

And the final expression for 𝑅𝑝 is expressed as:

𝑅𝑟𝑒𝑠 =𝑅𝑃1

∗

𝐹∗ + √4𝑃𝑐𝑃1∗ (𝑆30)

𝐹𝑝𝑢 = −1.5𝜋𝑅𝑟𝑒𝑠Γ (𝑆31)

The elastic unloading stiffness is

𝑑𝐹

𝑑𝛿= 2𝐸𝑎 [

3√𝑃1𝑟 − 3√𝑃𝑐𝑟

3√𝑃1𝑟 − √𝑃𝑐𝑟] (𝑠32)

11

Nano Res.

Model 2

I. Elastic: [Hertz][22]

𝑃𝑒 =4

3𝐸√𝑅𝛿𝑒

3 (𝑆33)

𝛿𝑒 =𝑎𝑒2

𝑅 (𝑆34)

𝑎𝑒 = (3𝑅𝑃𝑒4𝐸

)

13 (𝑆35)

II. Initial Point of plastic yielding: [Brizmer et al.][14, 15] [linear hardening, stick]

𝛿𝑐𝑠𝑙𝑖𝑝

= [𝐶𝑣𝜋(1 − 𝑣2)

2(𝑌

𝐸)]

2

𝑅 (𝑆36)

𝐶𝑣 = 1.234 + 1.256𝑣 (𝑆37)

𝑃𝑐𝑠𝑙𝑖𝑝

=𝜋3

6𝐶𝑣3𝑌 (

𝑅(1 − 𝑣2)𝑌

𝐸)

2

(𝑆38)

𝐴𝑐𝑠𝑙𝑖𝑝

= 𝜋𝑅𝛿𝑐𝑠𝑙𝑖𝑝 (𝑆39)

𝛿𝑐𝑠𝑡𝑖𝑐𝑘


= 6.82𝑣 − 7.83(𝑣2 + 0.0586) (𝑆40)

𝑃𝑐𝑠𝑡𝑖𝑐𝑘


= 8.88𝑣 − 10.13(𝑣2 + 0.089) (𝑆41)

𝐴𝑐𝑠𝑡𝑖𝑐𝑘 = 𝜋𝑅𝛿𝑐

𝑠𝑡𝑖𝑐𝑘 (𝑆42)

III. Elastic-plastic regime: [Brizmer et al.][15] [linear hardening, stick]

𝐴𝑠𝑙𝑖𝑝∗ =

𝐴

𝐴𝑐= 𝛿𝑠𝑙𝑖𝑝

∗ (1 + exp(1

1 − (𝛿𝑠𝑙𝑖𝑝∗ )

𝛼)) (𝑆43)

𝑃𝑠𝑙𝑖𝑝∗ =

𝑃

𝑃𝑐= (𝛿𝑠𝑙𝑖𝑝

∗ )32 (1 − exp(

1

1 − (𝛿𝑠𝑙𝑖𝑝∗ )

𝛽)) (𝑆44)

𝛼 = 0.25 + 0.125𝑣 (𝑆45)

𝛽 = 0.174 + 0.08𝑣 (𝑆46)

For the case of v=0.45, where the critical interference in stick and slip are almost identical, the pressure

distributions in stick and in slip are almost identical.

IV. Initial point of full plastic: [Brizmer et al.][15] [linear hardening, stick]

𝛿𝑝 = 110𝛿𝑐 (𝑆47)

V. Unloading:

12

Nano Res.

[Zait et al.][16] [linear hardening, full stick]

For 1 ≤𝛿

𝛿𝑐≤ 150

𝛿𝑟𝑒𝑠𝛿𝑚𝑎𝑥

= (1 −1

(𝛿𝑚𝑎𝑥𝛿𝑐

)𝛼)

(

1 −1


)𝛽

)

(𝑆48)

𝛼 = 0.189𝑣 + 0.212 (𝑆49)

𝛽 = −6.758𝑣2 + 5.281𝑣 − 0.308 (𝑆50)

[Hertz] [22]

𝛿𝑟𝑒𝑠 = 𝛿𝑚𝑎𝑥 − (3𝑃𝑚𝑎𝑥

4𝐸√𝑅𝑟𝑒𝑠)

23

(𝑆51)

[DMT][23]

𝐹𝑝𝑢 = −2𝜋𝑅𝑟𝑒𝑠Γ (𝑆52)

VI. Adhesion: [KE][11] [elastic-perfect plastic, slip]

𝐹𝑠0 = 2𝜋𝑅Γ (𝑆53)

𝐹𝑠𝑃=𝐹𝑠𝐹𝑠0

∙

𝐹𝑠0𝑃𝑐𝑃𝑃𝑐

(𝑆54)

𝐹𝑠0𝑃𝑐=12

𝜋2∙Γ

𝑅𝐾𝐻(𝐸

𝐾𝐻)2

(𝑆55)

𝐾 = 0.454 + 0.41𝑣 (𝑆56)

For 0.005 ≤ 𝑧/𝛿𝑐 ≤ 0.5,

𝐹𝑠𝐹𝑠0

= 0.792(𝑧

𝛿𝑐)−0.321

(𝛿

𝛿𝑐)0.356

𝑓𝑜𝑟 1 ≤ 𝛿∗ ≤ 6 (𝑆57)

𝐹𝑠𝐹𝑠0

= 1.193(𝑧

𝛿𝑐)−0.332

(𝛿

𝛿𝑐)0.093

𝑓𝑜𝑟 6 ≤ 𝛿∗ ≤ 110 (𝑆58)

where 𝑧 is the equilibrium distance between two contacting materials, at which the repulsive force equals

the attractive force, according to ref.[24], for the case of Si-Au contact, the value for 𝑧 is around 0.1665 nm.

For 0.5 ≤ 𝑧/𝛿𝑐 ≤ 100,

𝐹𝑠𝐹𝑠0

= 0.961 +0.157𝑧𝛿𝑐

+0.261 ln (

𝛿𝛿𝑐)

𝑧𝛿𝑐

𝑓𝑜𝑟 1 ≤ 𝛿∗ ≤ 6 (𝑆59)

13

Nano Res.

𝐹𝑠𝐹𝑠0

= 1.756 − (0.516 −0.303𝑧𝛿𝑐

) ln (𝛿

𝛿𝑐) + 0.052 (ln (

𝛿

𝛿𝑐))2

𝑓𝑜𝑟 6 ≤ 𝛿∗ ≤ 110 (𝑆60)

Model 3

The effective hardness is chose to be:

𝐻 = 2.8𝑌 (𝑆61)

I. Elastic region: [Hertz] [22]

Same with model 2: Eq. (S33-S35).

II. The initial point of plastic yielding: [Johnson][17]

2𝐸𝑎𝑐𝜋𝑅

= 1.79𝑌 (𝑆62)

𝛿𝑐 =𝑎𝑐2

𝑅 (𝑆63)

III. For the elastic-plastic regime: [Brake et al.][8]

𝑃𝑒𝑝 = (2𝑃𝑐 − 2𝑃𝑝 + (𝛿𝑝 − 𝛿𝑐)(𝑃𝑐′ + 𝑃𝑝

′)) (𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

)

3

+ (−3𝑃𝑐 + 3𝑃𝑝 + (𝛿𝑝 − 𝛿𝑐)(−2𝑃𝑐′ − 𝑃𝑝

′)) (𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

)

2

+(𝛿𝑝 − 𝛿𝑐)𝑃𝑐′ (𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

) + 𝑃𝑐 (𝑆64)

𝑎𝑒𝑝 = (2𝑎𝑐 − 2𝑎𝑝 + (𝛿𝑝 − 𝛿𝑐)(𝑎𝑐′ + 𝑎𝑝

′ ))(𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

)

3

+ (−3𝑎𝑐 + 3𝑎𝑝 + (𝛿𝑝 − 𝛿𝑐)(−2𝑎𝑐′ − 𝑎𝑝

′ )) (𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

)

2

+(𝛿𝑝 − 𝛿𝑐)𝑎𝑐′ (𝛿 − 𝛿𝑐𝛿𝑝 − 𝛿𝑐

) + 𝑎𝑐 (𝑆65)

𝑎𝑐′ =

1

2√𝑅

𝛿𝑐 (𝑆66)

𝑃𝑐′ = 2𝐸√𝑅𝛿𝑐 (𝑆67)

IV. For the initiation point of full plastic regime: [Brake et al.][8]

𝛿𝑝 = (𝐻𝜋√𝑅

𝐸)

2

(𝑆68)

14

Nano Res.

𝑎𝑝 = √2𝑅𝛿𝑝 + 𝐶 (𝑆69)

𝐶 = (3𝜋𝑅𝐻

4𝐸)2

−2𝑅𝐻2𝛿𝑐𝑌2

(𝑆70)

𝑃𝑝 = 𝐻𝜋𝑎𝑝2 (𝑆71)

𝑎𝑝′ = 2𝑅(2𝑅𝛿𝑝 + 𝐶)

−12 (𝑆72)

𝑃𝑝′ = 2𝑅𝜋𝐻 (𝑆73)

V. For the full plastic regime: [Brake et al.][8]

𝑃𝑓𝑝 = 𝐻𝜋𝑎𝑓𝑝2 (𝑆74)

𝑎𝑓𝑝2 = 2𝑅𝛿𝑓𝑝 + 𝐶 (𝑆75)

VI. For the unloading:

[Rathbone][6]

𝑅𝑟𝑒𝑠 = 𝑅 [1 +(0.195 + 0.23𝑣)(𝛿 − 𝛿𝑐)

𝛿𝑐] (𝑆76)

[Hertz] [22]



23

(𝑆77)

[DMT] [23]


VII. Adhesion:

Same with model 6: Eq. (S126, S127).

Model 4

𝛿∗ =𝛿

𝛿𝑐 ; 𝑃∗ =

𝑃

𝑃𝑐 ; 𝐴∗ =

𝐴

𝐴𝑐 (𝑆79)

I. Elastic: [Hertz] [22] [𝟎 < 𝜹∗ < 𝟏. 𝟗 ]

𝐴𝑒∗ = 𝛿𝑒

∗ (𝑆80)

𝑃𝑒∗ = (𝛿𝑒

∗)3

2 (𝑆81)

II. Initial point of yielding: [JG][4]

𝛿𝑐 = (𝜋𝐶𝑌

2𝐸)2

𝑅 (𝑆82)

15

Nano Res.

𝐶 = 1.295exp(0.736𝑣) (𝑆83)

𝑃𝑐 =4

3(𝑅

𝐸)2

(𝐶

2𝜋𝑌)

3

(𝑆84)

𝐴𝑐 = 𝜋3 (𝐶𝑌𝑅

2𝐸)2

(𝑆85)

III. Elastic-plastic regime: [JG][4, 5] [𝟏. 𝟗 < 𝜹∗; 𝒂

𝑹< 𝟎. 𝟒𝟏 ]

𝐴𝑒𝑝 = 𝜋𝑅𝛿𝑒𝑝 (𝛿𝑒𝑝

∗

1.9)

𝐵

(𝑆86)

𝐵 = 0.14 exp(23𝑒𝑦) (𝑆87)

𝑒𝑦 =𝑌

𝐸 (𝑆88)

𝑃𝑒𝑝 = 𝑃𝑐 {[exp (−1

4(𝛿𝑒𝑝

∗)512)] (𝛿𝑒𝑝

∗)32 +

4𝐻𝐺𝐶𝑌

[1 − exp(−1

25(𝛿𝑒𝑝

∗)59)] 𝛿𝑒𝑝

∗} (𝑆89)

𝐻𝐺𝑌= 2.84 − 0.92 (1 − cos (3.14

𝑎

𝑅)) (𝑆90)

IV. Full plastic regime: [JG][4]

𝐴𝑓𝑝∗ = 2𝛿𝑓𝑝

∗ (𝑆91)

𝑃𝑓𝑝∗ =

3𝐻

𝐶𝑌𝛿𝑓𝑝

∗ (𝑆92)

V. Unloading:

[JG][5]


= 1.02 [1 − (𝛿𝑚𝑎𝑥 + 5.9𝛿𝑐

6.9𝛿𝑐)−0.54

] (𝑆93)

[Hertz] [22]

𝑅𝑟𝑒𝑠 =1

(𝛿𝑚𝑎𝑥 − 𝛿𝑟𝑒𝑠)3(3𝑃𝑚𝑎𝑥4𝐸

)2

(𝑆94)

[DMT] [23]


I. Adhesion:


16

Nano Res.

Model 5

I. For the elastic region: [Hertz] [22]


II. At the point of initial yielding: [Big-Alabo et al.][7]


III. Elasto-plastic region: [Big-Alabo et al.][7]

Region I: Nonlinear elasto-plastic loading

𝐾ℎ =4

3𝐸𝑅

12 (𝑆96)

𝑃𝑒𝑝𝐼 = 𝐾ℎ(𝛿𝑒𝑝

𝐼 − 𝛿𝑐)32 + 𝐾ℎ𝛿𝑐

32 (𝑆97)

At the transition point from region I to region II:

𝛿𝑡𝑒𝑝3 − Λ1𝛿𝑡𝑒𝑝

2 + Λ2𝛿𝑡𝑒𝑝 − Λ3 = 0 (𝑆98)

Λ1 = 6𝛿𝑝 − 3𝛿𝑐 (𝑆99)

Λ2 = 9𝛿𝑝2 − 6𝛿𝑝𝛿𝑐 (𝑆100)

Λ3 = 4𝛿𝑐3 − 12𝛿𝑐

2𝛿𝑝 + 9𝛿𝑐𝛿𝑝2 + (

2𝑍

𝐾ℎ)2

(𝑆101)

𝑍 = 𝑃𝛿=𝛿𝑃 − 𝐾ℎ𝛿𝑐32 (𝑆102)

Region II: Linear elasto-plastic loading

𝑃𝑒𝑝𝐼𝐼 = 𝐾𝑙(𝛿𝑒𝑝

𝐼𝐼 − 𝛿𝑡𝑒𝑝) + 𝐾ℎ [(𝛿𝑡𝑒𝑝 − 𝛿𝑐)32 + 𝛿𝑐

32] (𝑆103)

𝐾𝑙 = 1.5𝐾ℎ(𝛿𝑡𝑒𝑝 − 𝛿𝑐)12 (𝑆104)

IV. At the initial point of fully plastic regime: [Big-Alabo et al.][7, 25]

𝛿𝑃 = 82.5𝛿𝑐 (𝑆105)

𝑎𝑝2 = 𝑅(2𝛿𝑃 − 𝛿𝑐) (𝑆106)

V. Fully plastic loading regime: [Big-Alabo et al.][7]

𝑃𝑓𝑝 = 𝜋𝐻𝑎𝑓𝑝2 (𝑆107)

𝐻 = 2.8𝑌 (𝑆108)

𝑎𝑓𝑝2 = 2𝑅𝛿𝑓𝑝 + 𝑐 (𝑆109)

𝑐 = 𝑎𝑝2 − 2𝑅𝛿𝑃 (𝑆110)

17

Nano Res.

VI. Unloading: [Rathbone et al.] [6] + [Hertz] [22] +[DMT] [23]


VII. Adhesion:


Model 6

I. Initial point of yield: [MP] [3]

For v=1/3, when

Γ∗ =16Γ𝐸2

9𝑅𝑌3≥ 6.25; Γ ≥ 0.00174 (𝑆111)

pure surface adhesion force could cause plastic yielding upon contact with a critical contact radius equals

a𝑐∗ =

4a𝑐𝐸

3𝑅𝑌= 4.9; a𝑐 = 2.34 (𝑆112)

II. Elastic-plastic:

[Studman et al.][26]

For v = 0.5

𝑝𝑚𝑌≈ 1 +

2

3ln𝐸𝑎

3𝑌𝑅 (𝑆113)

[Majumder et al.] [19]

𝛿 =𝑎2

2𝑅+𝑎𝑦2

2𝑅− √

𝜋𝑎𝑦Γ

2E (𝑆114)

III. Initial point of full plasticity: [Studman et al.][26]

𝑎𝑝 ≈60𝑅𝑌

𝐸 (𝑆115)

𝐹𝑝 = 𝐻𝜋𝑎𝑝2 − 2𝜋𝑅Γ (𝑆116)

IV. Full-plastic:

𝑝𝑚 = 𝐻 = 3𝑌 =𝐹 + 2𝜋ΓR

𝜋𝑎2 (𝑆117)

[Majumder et al.][19]

𝛿 =𝑎2

2𝑅+𝑎𝑦2

2𝑅− √

𝜋𝑎𝑦Γ

2E (𝑆118)

18

Nano Res.

V. Unloading:

[Hertz] [22]:

𝑅𝑟𝑒𝑠 =4𝑎𝑓𝐸

3𝜋𝑝𝑚 (𝑆119)

[MP][3]:

Separation between the two solids could be either brittle (occur at the interface) or ductile (occur within the

softer of the two materials). Maugis et al. [3] divided the modes of separation into the following categories

[for unloading from the elastic-plastic region]:

𝜙𝑒𝑝 = √6𝜋𝑎∗

Γ ∗×[1 +

23ln (

𝐸𝑎3𝑌𝑅

)]

3 (𝑆120)

𝑎∗ =4𝑎𝐸

3𝑅𝑌 (𝑆121)

Γ ∗ =16Γ𝐸2

9𝑅𝑌3 (𝑆122)

1. When (2 − 0.105√𝑎

Γ ) ≤ 𝜙𝑒𝑝 ≤ 1:

𝐹𝑝𝑢 = 𝐹𝑚 = −√8𝜋𝐸Γ𝑎𝑓3 + 𝜋𝑎𝑓

2𝑝𝑚 (𝑆123)

2. When 𝜙𝑒𝑝 ≥ max [(9.54√𝑎

Γ ) , 1]:

𝐹𝑝𝑢 = 𝐹𝑏 = −1.5𝜋Γ𝑅𝑝 (𝑆124)

3. When 𝜙𝑒𝑝 ≤ min [ (2 − 0.105√𝑎

Γ ) , (9.54√

𝑎

Γ )] ∶

𝐹𝑝𝑢 = 𝐹𝑑 = −𝜋𝑎𝑓2𝐻 (𝑆125)

VI. Adhesion:

𝐹𝑠 = 2𝜋𝑅Γ (𝑆126)

𝐹 = 𝑃 − 𝐹𝑠 (𝑆127)

Model 7

II. Elastic region: [Hertz] [22]


III. Initial point of plastic yield: [Johnson][17]

19

Nano Res.


IV. Elastic-plastic region and full plastic region: [Rathbone et al.][6]

The contact pressure 𝑝 is expressed as:

𝑝 = 𝐷 arctan(𝑏𝛿) (𝑆128)

𝐷 = (1.22 + 0.69𝑣)𝑌 (𝑆129)

𝑏 =1

𝛿𝑐tan (

1.79

1.22 + 0.69𝑣) (𝑆130)

The contact radius 𝑎 is expressed as:

𝑎 = 𝑐𝛿 + 𝑑 (𝛿

𝛿𝑐)𝑧

(𝑆131)

𝑐 = 1.43 + 0.061(1 − 𝑣2)𝐸

𝑌 (𝑆132)

𝑑 = √2𝛿𝑐𝑅

3− 𝑐𝛿𝑐 (𝑆133)

𝑧 = 0.5 + 0.3𝑣2.56 (𝑆134)

Combining the expressions for pressure and the reduced contact radius gives the equation for the contact

force P:

𝑃 = 𝑝𝐴 = 𝑝𝜋𝑎2 (𝑆135)

IV. Unloading: [Rathbone et al.] [6] + [Hertz] [22] + [DMT] [23]


V. Adhesion:


Model 8

The effective hardness is calculated as: [Brake et al.][8]

𝐻 = (1

𝐻𝑆𝑅+2

𝐻𝐿𝑅)−1

(𝑆136)

SR means smaller radius, LR means larger radius.



II. The initial point of plastic yielding: [Johnson][17]

20

Nano Res.


III. Elastic-Plastic region: [Brake et al.][8]






VI. For the unloading:

[SK][9] [elastic-perfectly plastic]:


= (1 −𝛿𝑚𝑎𝑥𝛿𝑐

−13)(1 −

𝛿𝑚𝑎𝑥𝛿𝑐

−23) (𝑆137)

[Hertz] [22]



23

(𝑆138)

[DMT] [23]


VII. Adhesion:


Model 9



II. Initial Point of plastic yielding: [Brizmer et al.][14, 15]


= [𝐶𝑣𝜋(1 − 𝑣2)

2(𝑌

𝐸)]

2

𝑅 (𝑆140)

𝐶𝑣 = 1.234 + 1.256𝑣 (𝑆141)


=𝜋3

6𝐶𝑣3𝑌 (

𝑅(1 − 𝑣2)𝑌

𝐸)

2

(𝑆142)

𝐴𝑐𝑠𝑙𝑖𝑝

= 𝜋𝑅𝛿𝑐𝑠𝑙𝑖𝑝 (𝑆143)

III. Elastic-plastic regime: [Brizmer et al. 2006][15] [linear hardening, slip+stick]

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Nano Res.

𝐴𝑠𝑙𝑖𝑝∗ =

𝐴

𝐴𝑐= 𝛿𝑠𝑙𝑖𝑝

∗ (1 + exp(1

1 − (𝛿𝑠𝑙𝑖𝑝∗ )

𝛼)) (𝑆144)

𝑃𝑠𝑙𝑖𝑝∗ =

𝑃

𝑃𝑐= (𝛿𝑠𝑙𝑖𝑝

∗ )32(1 − exp(

1

1 − (𝛿𝑠𝑙𝑖𝑝∗ )

𝛽)) (𝑆145)

𝛼 = 0.25 + 0.125𝑣 (𝑆146)

𝛽 = 0.174 + 0.08𝑣 (𝑆147)

IV. Initial point of full plastic: [Brizmer et al.][15] [linear hardening, stick]

𝛿𝑝 = 110𝛿𝑐 (𝑆148)

IV. Unloading: [EKK][13] [linear hardening, slip] +[DMT] [23]


V. Adhesion: [KE][11] [elastic-perfect plastic, slip]

Same with mode l 2: Eq. (S53-S60).

Model 10

I. Elastic: [Hertz] [22]


II. Initial Point of plastic yielding:[CEB][10, 20]


III. Elastic-plastic regime: [KE][12] [slip, elastic-perfectly plastic]

𝛿∗ =𝛿

𝛿𝑐; 𝑎∗ =

𝑎

𝑎𝑐 ; 𝑃∗ =

𝑃

𝑃𝑐 (𝑆149)

(𝑎∗)2 = 0.93(𝛿∗)1.136, 𝑓𝑜𝑟 1 ≤ 𝛿∗ ≤ 6 (𝑆150)

(𝑎∗)2 = 0.94(𝛿∗)1.146, 𝑓𝑜𝑟 6 ≤ 𝛿∗ ≤ 110 (𝑆151)

𝑃∗ = 1.03(𝛿∗)1.425, 𝑓𝑜𝑟 1 ≤ 𝛿∗ ≤ 6 (𝑆152)

𝑃∗ = 1.40(𝛿∗)1.263, 𝑓𝑜𝑟 6 ≤ 𝛿∗ ≤ 110 (𝑆153)

IV. Unloading: [EKK][13] [linear hardening, slip] + [DMT] [23]




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Nano Res.

Model 11



II. Initial Point of plastic yielding: [CEB][10, 20]

𝛿𝑐 = (𝜋𝑀𝐻

2𝐸)2

𝑅 (𝑆154)

𝑀 = 0.454 + 0.41𝑣 (𝑆155)

𝑎𝑐 = (𝛿𝑐𝑅)12 (𝑆156)

𝑃𝑐 =2

3𝑀𝐻𝜋𝛿𝑐𝑅 (𝑆157)

III. Elastic-plastic regime: [EKK][13] [linear hardening, slip]

𝐹𝑜𝑟 1 ≤ 𝛿∗ ≤ 170

𝑃∗ =𝑃

𝑃𝑐= 1.32(𝛿∗ − 1)1.27 + 1 (𝑆158)

𝐴∗ =𝐴

𝐴𝑐= 1.19(𝛿∗ − 1)1.1 + 1 (𝑆159)

𝛿∗ =𝛿

𝛿𝑐 (𝑆160)

IV. Unloading: [EKK][13] [linear hardening, slip]

𝐹𝑜𝑟 1 ≤ 𝛿∗ ≤ 170


=

(

1 −1


)0.28

)

(

1 −1


)0.69

)

(𝑆161)

𝑅𝑟𝑒𝑠𝑅

= 1 + 1.275(𝐸

𝑌)−0.216

(𝛿

𝛿𝑐− 1) (𝑆162)

[DMT] [23]




Model 12

The effective hardness is calculated as: [Brake et al.][8]

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Nano Res.

𝐻 = (1

𝐻𝑆𝑅+2

𝐻𝐿𝑅)−1

= 0.303 (𝑆164)

SR means smaller radius, LR means larger radius.

I. Elastic regime: [Hertz] [22]


II. For the initial point of plastic yielding: [Johnson][17]


III. For the elastic-plastic regime: [Brake et al.][8]


IV. The initiation point of full plastic regime: [Brake et al.][8]


V. In the full plastic regime: [Brake et al.][8]


VI. Unloading: [Rathbone] [6]+ [Hertz] [22]+[DMT] [23]


VII. Adhesion:


Model 13



II. At the point of initial yielding: [Johnson][17]


III. Elasto-plastic region: [Big-Alabo et al.][7]


IV. At the initial point of fully plastic regime: [Big-Alabo et al.][7]


V. Fully plastic loading regime: [Big-Alabo et al.][7]


VI. Unloading: [SK][9]+ [Hertz] [22] + [DMT] [23]


24

Nano Res.

VI. Adhesion:


Model 14

The effective hardness is calculated as:

𝐻 = 2.8𝑌 = 0.616 (𝑆165)



II. The initial point of plastic yielding: [Brake et al.][8]


III. Elastic-Plastic region: [Brake et al.][8]






VI. For the unloading: [SK][9]+ [Hertz] [22]+[DMT] [23]


25

Nano Res.

4. Fitting results of each contact model

During the analysis, several additional parameters were involved: the effective Young’s modulus (𝐸), and

the effective Poisson’s ratio (𝑣) with the following definitions:

𝐸 = (1 − 𝑣𝑡𝑖𝑝

2

𝐸𝑡𝑖𝑝 −

1 − 𝑣𝑠2

𝐸𝑠𝑢)

−1

(S166)

𝑣 = 𝑣𝑠 (S167)

where 𝑣𝑡𝑖𝑝 and 𝐸𝑡𝑖𝑝 are the Poisson’s ratio and Young’s modulus of the AFM tip; and 𝑣𝑠, and 𝐸𝑢 are the

Poisson’s ratio and Young’s modulus for the tested sample. Considering the AFM tip was made of silicon,

the sample of our tests was gold, values for these parameters were set as 𝐸𝑡𝑖𝑝 = 165𝐺𝑃𝑎, 𝑣𝑡𝑖𝑝 = 0.22, and

𝑣𝑠 = 0.43. Values for 𝐸𝑢 could be set as different values based on the calculation purpose.

Table S2. Summary of fitting results of different models

Model

No.

Work

of

adhesion

Tabor

No.

STDE. Approach

1 0.015 0.151 3.79 Indentation

2 0.0042 0.063 4.12 Flattening

3 0.0057 0.077 4.23 N/A

4 0.0123 0.129 4.62 Flattening

5 0.0054 0.074 4.94 Indentation

6 0.0107 0.118 5 Indentation

7 0.0063 0.083 5.14 Flattening

8 0.0093 0.107 5.19 N/A

9 0.005 0.071 6.94 Flattening

10 0.005 0.071 7.17 Flattening

11 0.0046 0.067 7.34 Flattening

12 0.0052 0.073 8.26 N/A

13 0.0084 0.1 8.37 Indentation

14 0.0074 0.092 12.55 N/A

26

Nano Res.

5. Experimental results of 𝑭𝒑𝒖 − 𝑭𝒎𝒂𝒙 from nanobridge bending test.

Figure S3. The red solid triangles denote the averaged pull-off force values when 𝐹𝑚𝑎𝑥 equals 60nN, 90nN and 120nN.

27

Nano Res.

6. Theoretical calculated loading-unloading force-deflection curves

Figure S4. Theoretical calculated loading and unloading 𝐹𝑚𝑎𝑥 − 𝑑𝛿 curves for the pure contact tests based on the TN

adhesion contact model.

Figure S5. Theoretical predicted loading and unloading 𝐹𝑚𝑎𝑥 − 𝑑𝑢 (red and purple) and 𝐹𝑚𝑎𝑥 − 𝑑𝑢+𝛿 (blue and green)

curves based on the TN model and the Nanobridge Multi-point Bending Test Theory.

28

Nano Res.

7. Influence of 𝑵𝒓, 𝑳 and 𝑭𝒎𝒂𝒙 on ∆𝑬 under given experiment conditions

Figure S6. Plots of the theoretical predicted relationship between ∆𝐸 and experimental conditions: (black) the

maximum applied force 𝐹𝑚𝑎𝑥, (red) the suspended length of Au NB 𝐿 and (green) the work of adhesion Γ.

29

Nano Res.

8. Influence of 𝑬𝒖 on ∆𝑬 under different experiment conditions

Figure S7. (a) The curves are theoretical predicted relationships between Young’s modulus variation ∆E and samples’

Young’s moduli 𝐸𝑢 with different maximum applied load. (b) The curves are theoretical predicted relationships

between ∆E and 𝐸𝑢 with different suspended length. (c) The curves are theoretical predicted relationships between

∆E and 𝐸𝑢 with different residual force.

30

Nano Res.

9. Influence of 𝑬𝒖 on ∆𝝆 and 𝝏𝑬

𝝏𝝆 under given experiment conditions

Figure S8. The curves are theoretical predicted relationship between samples’ Young’s moduli 𝐸𝑢 and ∆𝜌 (black solid

line) and the relationship between 𝐸𝑢 and 𝜕𝐸

𝜕𝜌 (blue solid line) for the center-bending nanobridge test.

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Nano Res.

Reference

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[3] Maugis, D.; Pollock, H. Surface forces, deformation and adherence at metal microcontacts. Acta Metallurgica 1984,

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Address correspondence to Tong-Yi Zhang, email [email protected] and [email protected]



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