Bhushan B. Nanotribology and Nanomechanics: An Introduction

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14 Mechanical Properties of Nanostructures 777

Fig. 14.25. Bending stress

contours, vertical displace-

ment contours, and bending

stress contours after load-

ing trapezoidal Si nanobeam

= 200 nm, w

= 370 nm,

t = 255 nm,  = 6 µm,

E = 169 GPa, ν = 0.28) at

70 µN load [72]

stresses and this eﬀect is more pronounced in the case of semicircular asperity as

it exhibits a higher value of maximum bending stress than that in grooved asperity.

Figure 14.26 shows the stress contours obtained at a section of the beam from front

and from the side for both cases when we have a single semicircular asperity and

when four adjacent semicircular asperities are present. Trends are similar to that

observed earlier for a smooth nanobeam (Fig. 14.25).

14.4.3 Eﬀect of Roughness in the Transverse Direction and Scratches

We analyze semicircular asperities when placed along the transverse direction fol-

lowedbytheeﬀect of scratches on the maximum bending stresses in varying num-

bers and diﬀerent pitch [72]. In the analysis of semicircular transverse asperities

778 Bharat Bhushan

Number of asperities

10 234

Semicircular and grooved asperities

Longitudial direction

Maximum bending stress (GPa)

R = L = 25 nm

Compressive Semicircular

Tensile

Grooved

Bending stress contours

Single asperity

B-B

–1

–6

–10

–19

–21

GPa

–1

–6

–10

GPa

Four asperities

B-B

–6

–14

–20

–22

GPa

–6

–14

GPa

A-A

– 20 GPa

–19 GPa

13 GPa

14 GPa

B-B

A-A

B-B

Fig. 14.26. Eﬀect of lon-

gitudinal semicircular and

grooved asperities in dif-

ferent numbers on maxi-

mum bending stresses af-

ter loading trapezoidal Si

nanobeams (w

= 200 nm,

= 370 nm, t = 255 nm,

 = 6 µm, E = 169 GPa,

ν = 0.28, Load = 70 µN).

Bending stress contours ob-

tained in the beam with

semicircular single asperity

and four adjacent asperities

of R = 25 nm [72]

three cases were considered which included a single asperity, asperities throughout

the nanobeam surface separated by pitch equal to 50nm and pitch equal to 100nm.

In all of these cases, c value was kept equal to 0 nm. Figure 14.27 shows that the

14 Mechanical Properties of Nanostructures 779

Single

asperity

Semicircular asperities

Transverse direction

Maximum bending stress(GPa)

R = 25 nm, c = 0 nm

Compressive

Tensile

Bending stress contours

Single asperity

B-B

Throughout the nanobeam, p =50nm

p = 50 nm p = 100 nm

B-B

–2

GPa

B-B

–2

–8

–16

GPa

–3

–10

–26

GPa

–3

GPa

– 8 GPa

42 GPa

– 10 GPa

42 GPa

A-A

B-B

A-A

18 GPa

Fig. 14.27. Eﬀect of trans-

verse semicircular asper-

ities located at diﬀerent

pitch values on the maxi-

mum bending stresses af-

ter loading trapezoidal Si

nanobeams (w

= 200 nm,

= 370 nm, t = 255 nm,

 = 6 µm, E = 169 GPa,

ν = 0.28, Load = 70 µN).

Bending stress contours ob-

tained in the beam with

semicircular single asperity

and semicircular asperities

throughout the nanobeam

surface at p = 50 nm [72]

value of maximum tensile stress is 42GPa which is much larger than the maximum

tensile stress value with no asperity of 16 GPa or when the semicircular asperity is

present in the longitudinal direction. It is also observed that the maximum tensile

stress does not vary with the number of asperities or the pitch while the maximum

780 Bharat Bhushan

compressive stress does increase dramatically for the asperities present throughout

the beam surface from its value when a single asperity is present. Maximum ten-

sile stress occurs at the ends and an increase in p does not add any asperities at the

ends whereas asperities are added in the central region where compressive stresses

are maximum. The semicircular asperities present at the center cause the local per-

turbation in the stress distribution at the center of the asperity where load is being

applied leading to a high value of maximum compressive stress [101]. Figure 14.27

also shows the stress contours obtained at a section of the beam from front and from

the side for both cases when there is a single semicircular asperity and when asper-

ities are present throughout the beam surface at a pitch equal to 50nm. Trends are

similar to that observed earlier for a smooth nanobeam (Fig. 14.25).

In the study pertaining to scratches, the number of scratches are varied along

with the variation in the pitch as well. Furthermore the load is applied at the center

of the beam and at the center of the scratch near the end as well for all the cases. In

all of these c value was kept equal to 50nm and L value was equal to 100nm with

h value being 20nm. Figure 14.28 shows that the value of maximum tensile stress

remains almost the same with the number of scratches for both types of loading that

is when load is applied at the center of the beam and at the center of the scratch near

the end. This is because that the maximum tensile stress occurs at the beam ends

no matter where the load gets applied. But the presence of scratch does increase the

maximum tensile stress as compared to its value for a smooth nanobeam although

the number of scratches no longer matter as the maximum tensile stress occurring at

the nanobeam end is unaﬀected by the presence of more scratches beyond the ﬁrst

scratch in the direction towards the center. The value of the tensile stress is much

lower when the load is applied at the center of scratch and it can be explainedas fol-

lows. The negative bending moment at the end near the load applied decreases with

load oﬀsetafter two-thirdsof the lengthof the beam[102].Since thisnegativebend-

ing moment is responsible for tensile stresses, their behavior with the load oﬀset is

the same as the negative bending moment. Also the value of maximum compressive

stress when load is applied at the center of the nanobeam remains almost the same

as the center geometry is unchanged due to the number of scratches and hence the

maximum compressive stress occurring below the load at the center is same. On the

other hand when the load is applied at the center of the scratch we observe that the

maximum compressive stress increases dramatically because the local perturbation

in the stress distribution at the center of the scratchwhere load is being applied leads

to a high value of maximum compressive stress [101]. It increases further with the

number of scratches and then levels oﬀ. This can be attributed to the fact that when

there is another scratch present close to the scratch near the end, the stress concen-

tration is more as the eﬀect of local perturbation in the stress distribution is more

signiﬁcant. However, this eﬀect is insigniﬁcant when more than two scratches are

present.

Now we address the eﬀect of pitch on the maximum compressive stress when

the load is applied at the center of the scratch near the end. When the pitch is up to

a value of 200nm the maximum compressive stress increases with the number of

14 Mechanical Properties of Nanostructures 781

p = 100 nm

p = 200 nm

p = 300 nm

Scratches

Traverse direction

L = 100 nm, h = 20 nm, c = 50 nm

Number of scratches

10 234

Maximum bending stress (GPa)

Compressive

Load at the center

of nanobeam

Tensile

Load at the center of

scratch near the end

Number of scratches

10 234

Maximum bending stress (GPa)

Compressive

Load at the center

of nanobeam

Tensile

Load at the center of

scratch near the end

Number of scratches

10 234

Maximum bending stress (GPa)

Compressive

Load at the center

of nanobeam

Tensile

Load at the center of

scratch near the end

Load at the center of

scratch near the end

Load at the center of

scratch near the end

Load at the center of

scratch near the end

Fig. 14.28. Eﬀect of number

of scratches along with the

variation in the pitch on the

maximum bending stresses

after loading trapezoidal Si

nanobeams (w

= 200 nm,

= 370 νμ, τ = 255 νμ,  =

6 µm, E = 169 GPa, ν = 0.28,

Load = 70 µN.) Also shown

is the eﬀect of load when

applied at the center of the

beam and at the center of the

scratch near the end [72]

scratches as discussed earlier. On the other hand, when the pitch value goes beyond

225nm this eﬀect is reversed. This is because the presence of another scratch no

longer aﬀects the local perturbation in the stress distribution at the scratch near the

end. Instead more scratches at a fair distance distribute the maximum compressive

stress at the scratch near the end and the stress starts going down. Such observations

782 Bharat Bhushan

of maximum bending stresses can help in identifying the number of asperities and

scratches allowed separated by an optimum distance from each other.

14.4.4 Eﬀect on Stresses and Displacements for Materials Which are Elastic,

Elastic-Plastic or Elastic-Perfectly Plastic

This section deals with the beam modeled as elastic, elastic–plastic and elastic–

perfectly plastic to observe the variation in the stresses and displacements from an

elasticmodel used so far [72].Figure 14.29 showsthe typical stress-strain curvesfor

the three types of deformation regimes and their corresponding load–displacement

curves obtained from the model of an Si nanobeam which are found to exhibit same

trends.

Table 14.4 shows the comparison of maximum von Mises stress and maximum

displacements for both smooth nanobeam and nanobeam with a deﬁned roughness

which is single semicircular longitudinal asperity of R value equal to 25nm for the

three diﬀerent models. It is observedthat the maximum value of stress is obtained at

a given load for elastic material whereas the displacement is maximum for elastic–

perfectly plastic material. Also the pattern that the maximum bending stress value

increases for a rough nanobeam still holds true in the other models as well.

Stress-strain curve

Strain

Stress

Elastic

Elastic-perfectly plastic

Elastic-plastic

Displacement (μm)

40812

Load-displacement curve

Elastic

Elastic-perfectly plastic

Elastic-plastic

Load (μN)

Fig. 14.29. Schematic rep-

resentation of stress–strain

curves and load-displacement

curves for material when it

is elastic, elastic-plastic, or

elastic-perfectly plastic for

a Si nanobeam (w

= 200 nm,

= 370 nm, t = 255 nm,

 = 6 µm, E = 169 GPa,

tangent modulus in plastic

range = 0.5E, ν = 0.28) [72]

14 Mechanical Properties of Nanostructures 783

Table 14.4. Stresses and displacements for materials that are elastic, elastic–plastic orelastic–

perfectly plastic (load = 70 µN, w

= 200 nm, w

= 370 nm, t = 255 nm,  = 6 µm, R = 25 nm,

E = 169 GPa, tangential modulus in plastic range = 0.5E, ν = 0.28)

Elastic Elastic–plastic

Smooth

nanobeam

Single

semicircular

longitudinal

asperity

Smooth

nanobeam

Single

semicircular

longitudinal

asperity

Max. von Mises stress

(GPa)

18.219.313.515.2

Max. displacement

(µm)

1.34 1.40 3.35 3.65

Elastic–perfectly plastic

Smooth

nanobeam

Single

semicircular

longitudinal

asperity

Max. von Mises stress

(GPa)

7.89.1

Max. displacement

(µm)

11.512.3

14.5 Closure

Mechanicalpropertiesof nanostructuresare necessary in designingrealistic MEMS/

NEMS and BioMEMS/BioNEMSdevices.Most mechanical propertiesare scale de-

pendent. Micro/Nanomechanical properties, hardness, elastic modulus, and scratch

resistance of bulk materials of undoped single-crystal silicon (Si) and thin ﬁlms of

undopedpolysilicon, SiO

,SiC,Ni

−

P, and Au are presented.It is found that the SiC

ﬁlm exhibits higherhardness, elastic modulus and scratch resistance as compared to

other materials.

Bending tests have been performed on the Si and SiO

nanobeams and Ni

−

Au, PPMA and PMMA microbeams using an AFM and a depth-sensing nanoin-

denter, respectively. The bending tests were used to evaluate elastic modulus, bend-

ing strength (fracture stress), fracture toughness (K

), and fatigue strength of the

beam materials. The Si and SiO

nanobeams exhibited elastic linear response with

sudden brittle fracture. The notched Ni

−

P beam showed linear deformation behav-

ior followed by abrupt failure. The Au beam showed elastic–plastic deformation

behavior. Elastic modulus values of 182±11GPa for Si(110) and 85 ±3GPafor

SiO

were obtained, which are comparable to bulk values. Bending strength val-

ues of 18±3GPaforSiand7.6±2GPa forSiO

were obtained, which are twice

as large as values reported on larger scale specimens. This indicates that bend-

784 Bharat Bhushan

ing strength shows a specimen size dependence. Fracture toughness value esti-

mates obtained were 1.67± 0.4MPa

√

m for Si and 0.6. ± 0.2MPa

√

m for SiO

which are also comparable to values obtained on larger specimens. At stress am-

plitudes less than 15% of their bending strength and at mean stresses of less than

30% of the bending strength, Si and SiO

displayed an apparent endurance life

of greater than 30,000 cycles. SEM observations of the fracture surfaces revealed

a cleavage type of fracture for both materials when subjected to bending as well as

fatigue.

The hardness, elastic modulus and creep behavior of PPMA and PMMA poly-

mericbeams arealso presented.Thehardness(0.41GPa),elasticmodulus(5.0GPa),

and creep resistance of PMMA beams are higher than the hardness (0.19GPa, elas-

tic modulus (3.7GPa), and creep resistance of PPMA beams. The scratch behavior

of PPMA and PMMA ﬁlms is diﬀerent from metallic and ceramic ﬁlms in that the

in situ displacement oscillated and coeﬃcient of friction are greater than one during

the scratch test. This might be because the polymer is so soft and easily plowed. Af-

ter 36hours soaking in DI water, the hardness and elastic modulus of PPMA beam

were decreased. The soaking seems to have had no eﬀect on elastic modulus of

PMMA beam. At 37.5

◦

C, the hardness and elastic modulus of PPMA beam were

decreased as comparedto room temperature, while this temperature appears to have

no eﬀect on elastic modulus of PMMA beam.

The AFM and nanoindenters used in this study can be satisfactorily used

to evaluate the mechanical properties of micro/nanoscale structures for use in

MEMS/NEMS.

FEM simulations are used to predict the stress and deformation in nanostruc-

tures. The FEM has been used to analyze the eﬀect of type of surface roughness and

scratches on stresses and deformation of nanostructures. We ﬁnd that roughness af-

fects the maximum bendingstresses. The maximum bendingstresses increase as the

asperity number increases for both semicircular and grooved asperities in longitu-

dinal direction. When the semicircular asperity is present in the transverse direction

the maximum tensile stress is much larger than the maximum tensile stress value

with no asperity or when the semicircular asperity is present in the longitudinal

direction. This observation suggests that the asperity in the transverse direction is

more detrimental. The presence of scratches increases the maximum tensile stress.

The maximum tensile stress remains almost the same with the number of scratches

for two types of loading, that is, when load is applied at the center of the beam

or at the center of the scratch near the end, although the value of the tensile stress

is much lower when the load is applied at the center of the scratch. This means

that the load applied at the ends is less damaging. This analysis shows that FEM

simulations can be useful to designers to develop the most suitable geometry for

nanostructures.

14 Mechanical Properties of Nanostructures 785

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