Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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Experimental Mechanics in Nano-engineering 299

Fig. 18. Intensity distribution received by the CCD camera sensor observing in the direction

normal to the plane of the grating.

The corresponding depth of penetration of the evanescent field computed with

Eq. (2) is 89.6 nm. Consequently, the size of the region of illumination caused by

the evanescent field is 750 nm.

Figure 18 shows in the insert the distribution of light intensity in the image as a

function of the coordinates of the field of view. It can be seen that the bright

illumination corresponds to a very small region of the saw tooth surface. The

figure shows the intensity distribution in the brightest area. It is indicated the

region where the direct influence of the evanescent field manifests itself. In this

region, there is saturation of the sensor. At about 3.5 times the theoretical distance

computed with Eq. (2) the intensity begins to decay reaching a minimum of about

20% of the maximum intensity in the central valley. Figure 18 indicates that the

main source of light energy in the Fabry-Perot cavity is the evanescent field.

order

0 order

Fig. 19. FT pattern of the 5 μm pitch grating imaged by the CCD. The first harmonic

(corresponding to the pitch of 5 μm) and the second harmonic (corresponding to the pitch

of 2.5 μm) are visible in the FT spectrum.

300 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti

Figure 19 shows the diffraction orders for the 5 μm pitch grating. The third

order (1.25 μm pitch) is not present in the FT pattern because the numerical

aperture of the utilized microscope objective limited the angular spectrum to the

first two orders. However, by utilizing double illumination it is possible to carry

out measurements with sensitivity corresponding to the 1.25 μm pitch.

3.3.1 Determination of the Size of the Field of View

In order to perform the measurement of the geometric parameters defining the

tooth shape one needs to know precisely the size of the field of view. The

following procedure has been applied. The grating superimposed on the top of the

standard is focused and the corresponding order n

of the grating is determined

from the FT pattern. Then, it can be written:

pnL

(28)

where p is the pitch of the grating. The field of view is hence known with an

accuracy determined by the pitch of the grating. In order to increase the accuracy

it is possible to utilize several orders of one grating and proceed to apply the

method that will be explained later in the paper.

3.3.2 Determination of the Saw Tooth Geometry

The determination of geometric dimensions of the tooth includes two different

steps. First, the pitch of the saw tooth is determined from the FT pattern of the

image: the order n

corresponding to the number of saw teeth included in the field

of view is extracted. Then, m harmonics of the order n

are considered.

The pitch of the tooth L

can be obtained by dividing the size of the field of

view L

by the order corresponding to the number of teeth included in the image.

Hence, one must interpolate the fraction of pixels that will correspond to the actual

tooth pitch. For the first harmonic, the pitch of the tooth is given by the equation:

pnL

(29)

The number of teeth N

included in the recorded image is equal to the ratio:

twt

L/LN =

(30)

By replacing in Eq. (30) the expressions (28) and (29) written for L

and L

, it

follows:

N =

⋅

(31)

Experimental Mechanics in Nano-engineering 301

In order to determine precisely the ratio n

, the higher harmonics of the order n

available in the FT pattern must be considered. In Fig. 20, the multiple values of

the tooth pitch are plotted as a function of the corresponding order. By utilizing

the minimum least square regression of the computed values, a line going through

the origin of coordinates is obtained. The slope of this line is the most accurate

value of the saw tooth pitch. Thus the quantization in pixels of the coordinates of

the space can be compensated.

Fig. 20. Determination of the pitch of the saw tooth

The height h

of the saw tooth (Figure 21a) is determined by considering that

the equivalent grating formed in the process of illumination and projected onto the

standard is modulated by the slope of the surface.



Fig. 21. a) Main geometric dimensions of the saw tooth; b) Relationship between total

phase, carrier phase and modulation function.

The intensity distribution of the modulated carrier is (see [30,31] and the

references cited in the former paper):

302 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti

The phase of the moiré fringes thus formed can be determined as [30,31]:

where the different quantities entering in the above relationship are represented in

Fig. 21b.

The phase of the carrier is:

=φ

(34)

The modulation function can be expressed as:

)cx(

sin)x( ⋅

⋅θ=Ψ

(35)

where θ is the illumination angle of the projection moiré equivalent scheme (see

Fig. 16). In the present case, c corresponds to the constant slope of the tooth

surface.

The total phase of the modulated carrier for one saw tooth hence is:

tttooth1,mc

)h2(

sin)x( ⋅

+⋅

⋅θ=φ

(36)

The term 2h

must be introduced in Eq. (36) in order to account for the fact that

each tooth is comprised of two sides with positive and negative slopes

respectively.

For the entire field of view containing N

saw teeth, the total phase is:

ttttooth1,mctTOT

N)h2N(

sin)x(N)x(

⋅+⋅⋅

⋅θ=φ⋅=φ

(37)

In order to compute precisely the phase modulation term

⎥

⎦

⎤

⎢

⎣

⎡

⋅⋅

⋅θ )h2N(

sin

for the entire field of view, one can use a process

similar to that described previously for the tooth pitch.

Figure 22 shows the FT of the image containing the carrier frequency f

and the

modulated carrier frequency f

for the first three harmonics. The difference

between the phase of the modulated carrier and the phase of the carrier can be

determined for each harmonic in the whole field of view.

(32)

(33)

Experimental Mechanics in Nano-engineering 303

Fig. 22. Schematic representation of the different harmonics extracted from the FT

spectrum. The carrier frequency f

is a known quantity. The frequency of the modulated

carrier f

is close to the carrier frequency.

In Fig. 23, the multiple values of the modulated phase are plotted as a function

of the corresponding order. By fitting data of Fig. 23 in the least square sense, a

line going through the origin of coordinates is obtained. The slope of this line

represents the most accurate value of the average change in phase per order <Ψ>

that can be computed for the recorded image. From this value, it is possible to

determine the height of the saw tooth pitch as follows:

sin

h ⋅

⎟

⎠

⎞

⎜

⎝

⎛

⋅

⎟

⎠

⎞

⎜

⎝

⎛

>Ψ<

(38)

Fig. 23. Determination of the height of the saw tooth

The value of R

evaluated, accordingly to ANSI B46.1, from the experimental

data gathered with the present advanced digital moiré contouring technique falls in

the 3.0175÷3.0784 μm range certified by NIST. The average measured pitch is

101.24 μm with a standard deviation of ±0.322 μm. The average measured depth

is 6.078 μm: the average value of Ra is hence 3.039 μm, well within the range of

304 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti

NIST’s measurements. The difference between the value of roughness measured

optically and the average value of roughness indicated by NIST is only 0.231%

(i.e. 3.039 μm vs. 3.032 μm).

Fig. 24. a) Detail of one tooth; b) 3D MATLAB representation of the reconstructed surface

of half of one tooth.

Finally, a square area has been extracted from the image and resized to

2048×2048 pixels in order to precisely reconstruct the profile of one saw tooth.

Figure 24 shows the tooth profile and the reconstructed 3D shape of the tooth. The

local height of the tooth profile measured in this region is 6.0645 μm, practically

the same as the nominal height of 6 μm. The local length of the tooth is 101.3 μm,

very close to nominal length of 100 μm. The average R

is 3.0505 μm and

oscillates between 3.0175 μm and 3.0785 μm. By extracting different profiles it is

possible to make an estimate of the average surface finish of the standard. The

average depth thus determined is 6.035±0.1367 μm. Therefore, the finish of

surface standard can be estimated as 0.1367 μm / 0.635 μm, that is about λ/5.

4 Determination of Contact Strains

The analyzed metallic surface is made of copper and presents a system of furrows

that contain some frequencies that can be utilized as carrier fringes. In preceding

papers, this procedure was referred to as holographic moiré. Although there is not

a physical carrier (i.e. a grating), the frequency of the reference grating can be

extracted from the FT spectrum of the image. The in-plane strains can be directly

derived from the basic equation of holographic moiré [32-38]:

)90sin(2

θ−°

(39)

This is the fundamental equation of holographic moiré to get in-plane

displacements. It corresponds to the double illumination setup that makes the

sensitivity vector

→

S equal to:

Experimental Mechanics in Nano-engineering 305

→→→→

•−= d)kk(S

(40)

where

→

k and

→

k are the two symmetrical illumination vectors with respect to

the normal to the observed surface and

→

d is the displacement vector.

The strains in the contact surface can be computed through direct

differentiation in the Fourier space of the unwrapped phase of the displacement

field. This procedure was successfully applied to measurements in the micron

range in a number of cases [38,39].

4.1 Determination of the Contact Strains of a Small Cylinder

In this work, the contact strains of a small cylinder of 10 mm in diameter and 8

mm in thickness were measured using an SPR based optical setup. The specimen

was made of copper and the contact surface was finished with grooves created by

scratching the surface with sandpaper. The same procedures of calibration of the

pixel values and of the field of view applied in the contouring experiments

described in Section 3.3 were utilized also in the strain determination experiments.

The modulus of elasticity of the copper specimen is 117 GPa while the Poisson’s

ratio is 0.3. The yield limit is 48 MPa.

Fig. 25. Loading device with the cantilever load measuring system used in the strain

determination experiments. The specimen, a small cylinder, is compressed between the

aluminum plate shown in the picture and the glass surface. The specimen is not visible.

Figure 25 shows the loading device built for the strain determination

experiments. A rubber bag is inflated to compress the specimen between an

aluminum plate and a glass plate. The cantilever beam shown in the figure is part

of a sensor that has a strain gage (Wheastone bridge) to measure the strain level.

306 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti

The sensor was calibrated utilizing dead weights resulting in a linear function load

in N vs. the readings in με of the bridge. Three cycles of loading were applied.

The “unloaded” images for the specimens contact stress were taken with an initial

small load to eliminate rigid body motions.

Equation (39) allowed in-plane displacements of the loaded copper cylinder to

be computed. Images from the central portion of the specimen (35x35 μm) were

recorded for the unloaded state and then for the successive loads. The changes in

the surface shape caused by the presence of contact loads were determined by

subtracting the phase of the “loaded” image from the phase of the “unloaded”

image. Then, strains were obtained by the direct differentiation of the unwrapped

phase pattern. Figure 26a shows the recorded pattern corresponding to the

“unloaded” state while Fig. 26b shows the recorded pattern corresponding to the

contact load of 286 N. The Fourier transforms of those images are shown in Fig.

27a and Fig. 27b, respectively.

Fig. 26. Portion of contact region (35x35 μm) in the central part of the cylindrical

specimen (diameter 10 mm and thickness 8 mm): a) pattern corresponding to zero loading;

b) pattern corresponding to the load of 286 N.

Fig. 27. a) Fourier transform of Fig. 26a; b) Fourier transform of Fig. 26b.

Experimental Mechanics in Nano-engineering 307

After the specimen was loaded, the changes in the reflectivity experienced by

the specimen surface made it possible to observe the contact area between the

glass and the copper cylinder. The contact area is an ellipse of major axis 5 mm

and minor axis 3.5 mm. As a first approximation, one can assume that the

maximum Hertzian pressure is 1.5 times the average contact pressure of 20.8 MPa

(i.e. 286N /π(2.5mmx1.75mm). Therefore, the expected maximum value of stress

at the center of the contact area should be σ

max

=31.2 MPa. From the stress-strain

curve of the copper plotted in Fig. 28, the corresponding deformation of the

surface in the point of maximum contact pressure would be approximately 500 με.

Such a value is well below than the 0.2% (i.e. 2000 με) off-set yield strain.

Fig. 28. Typical constitutive behavior of copper under tension: a) Engineering and true σ-ε

curves; b) Enlarged view of the initial region where yield strain is defined.

However, it can be seen from Fig. 29 that the strain values in the central region

of the specimen are much higher than 200 με. This fact indicates the presence of a

local strain field produced by the contact between the asperities of the copper

surface and the glass surface. At the edges of the circular region shown in Fig. 29

Fig. 29. a) Map of the strain ε

obtained for the contact load of 286 N; b) 3-D view of the

same map.

308 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti

strains are mostly tensile and very high. The order of magnitude reached by the

strain in this region corresponds to the part of the stress-strain curve limited by the

red parenthesis in Fig. 28a. The region where high tensile stresses take place is

outside of the contact area while the central part of contact area is under

compressive stresses.

The strain distribution along the horizontal cross-section corresponding to the

X-axis is plotted in Fig. 30. Since strain peaks are located at the path edges, micro-

cracking may occur at the corresponding points. In order to verify this hypothesis,

the topography of the contact region was analyzed in detail.

Fig. 30. Distribution of strain ε

along the horizontal diameter of Fig. 29 (white line control

path).

Figure 31a shows the surface profile before loading: depth values ranged from

1 to 3 μm. Figure 31b shows the final configuration taken by contact surface when

the 286 N load was applied. It can be seen that depth values changed significantly

as they go from 0.2 μm to a maximum of 1.4 μm. The differences of depth

between the initial and the final surface configurations are shown in Fig. 31c:

these values were obtained by simply subtracting the data of Fig. 31b from the

data of Fig. 31a. The contact area indeed experienced large plastic deformations. It

was flattened in agreement with the results of the strain analysis of the surface.

The results of the present experiments are another verification of the model

introduced by Johnson in Contact Mechanics to explain the wear mechanism of

rolling surfaces [40]. Modeling this mechanism was proven to be a very difficult

challenge. The resulting wear particles are platelets that lie parallel, or nearly so, to

the rolling surface, on the planes of maximum compressive stress. The large plastic

strains measured in this experiment confirm the hypothesis that the resulting cracks

are ductile fractures, driven by plastic strain rather than the elastic stress intensity

instability and that these severe plastic strains are the consequence of the contact

between asperities. In the present experiment one of the contacting surfaces, the

glass plate, is very smooth. Therefore, the operating mechanism is one of flattening

the copper asperities against the supporting glass plate. The results presented in this