A study of the mechanisms of erosion in silicon single crystals using Hertzian fracture tests

WEAR Wear186187(1995)

105-116

A study of the mechanisms of erosion in silicon single crystals using Hertzian fracture tests Jyh-Woei Lu a, Gordon A Sargent b, Hans Conrad ’ aSupertech Consultants Co., Chmdler, AZ 85224, USA b University of Dayton, Dayton, OH 45469-1620, USA ’North Carolina State University, Raleigh, NC 27695, USA

Abstract The fundamental mechanisms of erosion in brittle solids were investigated using Hertzian fracture tests. Tests were carried out at room temperature on the ( 111) surface of silicon single crystals under both static (Instron) and dynamic (free-fall) conditions, using spherical indenters of radii from 1.19 to 4.76 mm. Considered were the effects of specimen surface roughness, indenter size, indenter material and loading rate on the Hertzian fracture load and ring crack radius. The surface energy and fracture stress, associated with the formation of the Hertzian ring crack, were calculated using existing fracture mechanics theories for Hertzian fracture. Keywords: Erosion; Silicon; Hertzian fracture tests; Brittle solids; Fracture

1. Introduction

Erosion damage to solid surfaces by impacting particles is recognized as a serious industrial problem. However, because of the complexity of the interaction between the particles and the surface, a fundamental understanding of the phenomenon is still not well developed. Among the most significant results from previous studies [ l-31 was the empirical finding that two different modes of erosion occur for two different classes

of target material. The ductile mode, characteristic of most metal targets, where the maximum erosion is found to occur at a particle angle of impingement of about 20-30”. This has been interpreted to mean that the ductile erosion mechanism involves cutting or ploughing, with the sharp edges of individual particles acting as microcutting tools. The brittle mode, which is characteristic of ceramics and glasses, were the erosion is a maximum at a normal (90”) angle of impingement. This has been interpreted as being consistent with a mechanism of failure involving brittle fracture. In the present study, an attempt is made to develop an understanding of the damage produced by single particle impacts on a brittle surface. The results may then be correlated with more complex multi-particle erosion behavior. Previous work has shown that material removal during the erosion of brittle solids occurs by the formation [ 4,5] and intersection [6,7] of cracks. If a rigid and isotropic particle strikes a nominally brittle solid at a normal (90”) angle of impact the 0043-1648/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDIOO43-1648(95)07128-S

target may crack in one of two ways depending upon the

particle radius and the load developed at the particle-target interface. As the load is increased, fracture will be completely elastic in nature if the particle radius is larger than some critical value. Whereas, if the particle size is less than the critical radius, initial fracture will occur in an elastic-plastic manner. The advantage of studying the completely elastic interaction is that the stress field below the impacting particle, although complex, is well defined up to the point of fracture [ 81. Therefore, the system lends itself to an analysis in terms of the Griffith theory of fracture for elastic-brittle solids. Hence, the Hertzian fracture test, in which a hard sphere (indenter) is loaded onto a brittle flat surface, was chosen for these single particle erosion studies. Under Hertzian fracture conditions a crack is first produced on the surface of the flat specimen as a ring, which encircles the area of contact where the stress is a maximum, Fig. 1, and then propagates below the surface in the form of a truncated cone in accordance with the requirements of an energy balance condition. The radius of the circular elastic contact, a, is given by [8,9] ;

a3=--

4kPR 3E

where R is the indenter radius, P is the normal load and a constant given by

(1) k

is

Jyh-Woei Lu et al. /Wear 186187

106

(a! Soherical

(1995) 105-116

studies of Hertzian fracture have established that, within a certain particle size range, the critical load to produce a ring crack is approximately proportional to the radius R [9,12,13,15]:

P

I

P,=AR

W

Compressive

Stress

Shear Stress

Fig. 1. (a) Hertzian indentation geometry. (b) Location of maximum tensile, compressive and sheer stresses in elastic half space.

1

(l-u2)f(l-d2)E E’

(2)

where E and E’ are the Young’s moduli and v and v’ are the Poisson’s ratios of the specimen and indenter, respectively. Important features of the stress field that influence fracture behavior are illustrated in Fig. 1(b) . These are: (i) the region below the contact circle and extending into the body of the specimen in which all normal stresses are compressive; (ii) a region close to the surface and outside the contact circle where cr, is tensile and (iii) the maximum shear stress located at a distance of approximately 0.6a below the center of the contact circle. The maximum tensile stress, (TV,in the radial direction on the surface occurs at the contact circle and is given by (3) The tensile stress urr, which decreases away from the periphery of the contact circle, is [ 9,101; cr=--*

l-2lJ 2

P

a2

Ira2 0 r

(4)

where r is the distance on the surface taken radially from the point of contact. When a spherical particle of radius R impacts a brittle target surface with a velocity V the maximum load P,, developed at the contacting interface can be described by [61 pm= ($p)-K-2/5V6/5R2

(5)

In the Hertzian fracture test, when the load reaches a critical value PC, a characteristic ring crack appears at the surface of the specimen. Observation by Auerbach, and confirmed by others [ 1l-151, is that the ring crack usually forms outside of the circle of contact between the sphere and the target frequently as much as 1.6 times the contact radius. Other

(6)

where the constant A is usually referred to as Auerbach’s constant, which is a characteristic of the target material and can be related to the surface energy of the brittle solid. Neither of the above experimental observations can be predicted from the Hertzian fracture equations. Wilshaw [lo], Frank and Lawn[16],Lawn[17],Johnsonetal.[18],Warren[19], Mouginot and Maugis [ 201, and most recently Zeng et al. [ 21,221, have attempted to explain these apparent anomalies of Hertzian fracture using fracture mechanics. However, each of the above theoretical studies used different assumptions and equations to calculate the stress field. For example Lawn [ lo] used Huber’s equation [ 231 and assumed a Poisson’s ratio of u= 0.33. Mouginot and Maugis [ 201 used Sneddon’s equation [ 241 and a Poisson’s ratio of u= 0.22. This produced some difference in the results, particularly in the principle tensile stress value u,. Zeng et al. [ 211 concluded that Lawn’s analysis [ 171 was not in a useful form because it required numerical integration and no solution was presented. They also argue that Sneddon’s equation is not suitable for use in the case of a spherical indenter on a flat surface since it was derived for a flat punch. Zeng et al. [ 2 1 ] are able to provide the stress field solution for a spherical indenter on a flat surface in the three dimensional case both along the loading axis and on the surface. The above theories each predict the shape of the stress field caused by the interaction of the sphere with the flat surface. It is the interaction of this stress field with pre-existing surface flaws which results in the formation of the ring crack in the surface, which subsequently propagates as a cone crack below the surface. Thus the distribution of surface flaws plays an important role in determining the diameter of the surface ring crack. Johnson et al. [ 181, showed that the difference in elastic constants between the indenter and the target lead to frictional traction at the surface, which modifies the Hertzian stress distribution. Therefore, the maximum tensile stress occurs at some distance outside of the circle of contact, and the observed ring crack radius to contact radius ratio values ( r* / a) will always be greater than 1.0. Wilshaw [lo] concluded that Auerbach’s law is not a general phenomenon. He proposed that the load P is proportional to the ring crack radius r* to the power n, where n may have a value between 1 and 2, depending on the distribution of the surface micro-cracks and the size of the indenter. Greenwood and Trip [25] have shown that the effective pressure distribution can be modified by surface roughness. The quantitative effect being similar to that of surface roughness. Conrad et al. [ 151 also observed a variation in, r*, for indenter of different elastic moduli, and roughness of the indenter and target surfaces from which they concluded that

Jyh-Woei Lu et al. /Wear 186-187 (199.5) 105-116

friction and surface roughness modified the Hertzian stress field such that the maximum tensile stress occurred outside the contact circle. This lead them to propose that a more fundamental relationship than is expressed by Auerbach’s law may be given by u~=BR,“~

R,=Ca

(8)

where C depends upon the target surface condition. After the ring crack is completely formed, at some critical load, an additional load is necessary to cause the ring crack to develop into a cone crack. Roesler [26] has shown that the radius R, of the ring crack may be related to that of the radius R* of the base of the cone crack by P

R”

0 f’

“EC’2

Table 1 Composition and properties of the silicon target (A) Composition B P 02

(ppm) 0.001 0.001 lo6 atoms cm-’

(7)

where R, is the ring crack radius and gf is the fracture stress calculated at R = R, using Eq. (4). In this case the constant B is independent of surface condition or the type of particle material. The ring crack radius R, may also be related to the radius a of the area of contact at the instant of fracture by the following relationship

R

107

213

(9)

More recently, Lu et al. [ 271 have studied the fundamental mechanisms of erosion of Pyrex glass under both static and dynamic loading conditions using Hertzian fracture tests. They found that the ratio of the ring crack radius to contact radius decreased with increased load, indenter size and loading rate. Also, the ratio and fracture load increased with increase in elastic modulus mismatch between the indenter and specimen and with abrasion of the indenter and/ or specimen. They concluded that the effects of abrasion and indenter material on Hertzian fracture behavior were in quantitative accord with predictions of friction and roughness effects proposed by Johnson et al. [ 181. Kinetic effects, associated with the presence of water vapor at the crack tip, were concluded to cause the increase in the critical load to fracture. This was based on the increase in critical load observed in the presence of a grease coating and on dynamic tests when compared with static tests. The present study extends the work of Lu et al. [ 271, where the target material was Pyrex glass, to silicon as the target material. As opposed to Pyrex glass, which is an amorphous, elastically isotropic, and inherently brittle solid, silicon is an elastically anisotropic crystalline solid, which allows for an investigation of the effect of crystallographic orientation on Hertzian fracture behavior.

2. Experimental methods and procedures 2.1. Target material The Hertzian fracture tests were all carried out on silicon ( 111) single crystals obtained from Materials Research Cor-

( B ) Mechanical properties [ 28,291

Young’s modulus E Poisson’s ratio Y Hardness (Mob’s scale)

1.88x105Nmn-2 0.3 7

poration. The surfaces of the as-received specimens had been polished by mechanical-chemical means to yield, 2.5 cm diameter by 0.5 cm thick, optical grade discs. The chemical composition and mechanical properties of the silicon are given in Table 1. X-ray Laue patterns were taken to check the surface orientation to within 1O of each plane. Surfaces in the as-received condition and abraded with 400 grit Sic were studied. 2.2. Indenters The majority of Hertzian fracture tests were carried out using hardened chrome alloy steel balls ranging in size from 1.19 to 3.18 mm radius. More limited studies were performed with alumina and tungsten carbide. The composition and physical properties of the indenters are given in Table 2. All test were carried out in air and at room temperature.

2.3. Test procedure The Hertzian fracture tests were carried out under both quasi-static and dynamic conditions, using an Instron machine and free-fall to load the indenter on to the surface of the silicon wafer. In the quasi-static tests the silicon wafer was clamped to the cross-head of a table top Instron testing machine which was driven at a constant speed of 8.5 X 10m6m s-l. The indenter was held in a jig attached to the compression load cell of the In&on. The indenter was pressed against the specimen surface until a ring crack formed. The critical load P, at which the ring crack first formed was detected by means of an acoustic transducer mounted on the underside of the specimen. The critical load was found to exhibit appreciable scatter, in keeping with the fracture of brittle materials, so it was necessary to repeat the test 20-40 times for each ball size and test condition and the results were subjected to a statistical analysis. The dimensions of the ring crack, r*, formed at the critical load was also recorded for each test condition. Dynamic Hertzian fracture tests were carried out using free-fall of the indenter on to the specimen surface In this test the ball indenter was allowed to fall freely onto the specimen, which was mounted horizontally on a thick steel block. Ten to twenty tests were carried out at each height, for each ball size and test condition, beginning at a height sufficiently low

108

Jyh-Woei Lu et al. /Wear

186-187(1995)

Table 2 Composition

105-116

3. Experimental and properties

(A) Composition

(wt. 8)

(a) Steel balls C Mn P s Cf Si Ni CU MO

0.95-1.10 0.254.45 0.025 0.025 1.3-1.6 0.2-0.35 0.35 0.25 0.08

(b) Aluminum

oxide balls 99.5

Al103

(c) Tungsten carbide balls WC co

(B ) Mechanical

results

of the indenters

93.5-94.5 5.5-6.5

properties

Ball material

Young’s modulus E (N mm-‘)

Poisson’s ratio Y

Hardness

Steel Al@, WC

2.07 X lo5 3.59 x lo5 6.77 X IO5

0.30 0.23 0.26

63-66 (C-scale) 81 (N scale45 kg) 91-92 (A-scale)

that no ring cracks were observed and ending with a height at which all impacts produced full ring cracks. The critical load to form the ring crack was calculated by converting the potential energy of the free-falling ball indenter to elastic energy transferred to the specimen surface.

3.1. Geometry of Hertzian cracks in Si (1 I I) single crystals

Previous studies on diamond [ 301, germanium [ 3 I], silicon [ 321 and this current work all show that crystallographic anisotropy play a significant role in the Hertzian fracture of crystalline solids. In these materials the Hertzian cracks are polygonal in form, unlike the circular cracks found in isotropic materials such as Pyrex glass. The cracks tend to lie along the trace of the { Ill} cleavage planes, producing a hexagonal outline on the ( 111) surface as shown in Fig. 2. Removal of the surface layer by mechanical polishing to a depth of 0.0129 cm below the original surface, Fig. 3, shows that the hexagonal outline of the surface cracks was replaced by a triangular configuration of cracks defined approximately by three (110) directions. An SEM photomicrograph of a Hertzian crack produced on the surface of as-received Si( 111) is shown in Fig. 4. Here the surface was etched for 1 h to reveal the contours of the surface and subsurfacecracks. In profile the cracks were found to lie along a plane which made a smaller angle to the surface than the { 11 1} planes. That is, the crack profile did not lie along cleavage planes, but on surfaces determined primarily by the stress field produced by the indenter as it was loaded onto the surface. Previous studies on germanium [ 31,331 and silicon [ 321 also showed that fracture occurs on non-crystallographic surfaces. In sections taken perpendicular to a (1 lo), it can be seen after etching that the cracks have a zig-zag shape, as shown in Fig. 5. The angle of the plane connecting the steps in the zig-zag crack was measured to be close to 70”, which suggests that the cracks individual, small cracks forming the steps in the zig-zag lie on cleavage planes. To determine whether or not plastic deformation occurred around the tip of the internal cracks due to the stress concen-

(b)

(d)

Fig. 2. Photomicrographs of irregular cracks produced on the surface of as-received 221 N, (b) 240 N. (c) 255 N, (d) 264 N.

Si ( 111) using a stee1 ball ( r = 1.19 mm) with lnstron at various loads (a)

Jyh-Woei Lu et al. /Wear 186-187 (1995) 105-116

109

tration, the top surface layers of the Si specimen were progressively removed by polishing and were then treated with Dash enchant to reveal dislocation etch pits. It was found that many dislocation etch pits were present along the (111) directions, see Fig. 6. These etch pits reveal the presence of dislocations which presumably climb up to the free surface from the tip of the internal crack, to relieve the internal stress. It was also found that more dislocations form when the load is increased. 3.2. The critical load toform Hertzian ring cracks

,0.2mm,

Experiments were first carried out to determine if the observed contact radius was in agreement with that predicted theoretically by the Hertzian equations. In order to verify this, some free-fall impact tests were made on as-received silicon ( 111) surfaces which were coated with a thin layer of grease. Balls of various sizes and of different materials were impacted onto the surfaces from various heights. The impression in the grease due to contact between the ball and the specimen surface was measured and compared with that calculated using the Hertzian equations. Since the stresses are in all directions in the surface, the Young’s modulus and Poisson’s ratio values that were used for calculating the contact radius a were Eay, (=1.64X lo5 NmmV2) and v,,, (=0.225). A plot of the calculated vs. the measure contact radius is shown in Fig. 7, from which it was concluded that the calculated

Fig. 3. Subsurface views of Hertzian cracks (0.0129 cm below the surface) produced on Si( 111).

:a)

(5)

0.02mm

IFig. 4. SEM micrograph of the Hertzian cracks produced on the surface of as-received Si( 111). Surface etched for 1 h with Dash etchant.

Fig. 5. Section perpendicular to a ( 110) diction showing the profile of the internal crack produced by lnstron with steel ball.

Jyh-Woei Luetal./Wear

110

l&S-187(1995)

(b)

105-116

(c)

Fig. 6. Photomicrographs of the etch pits around the Hertzian crack on the surface of as-received Si( 111). The cracks were produced by a WC ball with Instron at increasing loads. Cracks surface annealed at 650°C for 45 min and then etched for 15 min: (a) 196 N; (b) 343 N; (c) 382 N. I

1

I

I

I

t

I

I

07

0.6

09

I

-

Indenter 0 R=3 175 mm A R=3 175 mm ??R= 3.175 mm 0 R= I 588mm fhR=2 36 mm OR=2 38 mm

c i G _= v

05-

Steel Boll A1203 Ball w c BatI WC Boll A1203 Boll WC Ball

0

0

01

0.2

03

0.4

0.5

0 (mm)Contoct

06

Rodfus

I0

(Measured)

Fig. 7. Calculated contact vs. measured contact radius with various indenter sizes and materials on the as-received Si ( 111) surface.

I

R=l

19mm

1.59

I

3 I8

I 96

(r=

1.59 mm)

percent probability, fF, for a ring crack to form. This was derived by arranging the N total values of P, in increasing order and assigning a number IZto each value& is then given by n/N+ 1. For the free-fall test, the percent probability fF was taken as the fraction of the tests at a given load which resulted in ring cracks. Typical results are shown in Figs. 8 and 9, where fF is plotted as a function of the critical load to fracture, PC, as measured under quasi-static conditions using the Instron, for as-received and abraded (400 grit Sic) surfaces, respectively. More scatter in the data was observed for a larger indenter and for the as-received surface compared with the abraded surface. Other results from dynamic tests using free-fall of the indenter onto an abraded silicon ( 111) surface are shown in Fig. 10. To use these tests to study the influence of other variables, such as indenter size, composition and surface roughness on Hertzian fracture, the value of the load for fF= 0.5 was taken as a representative parameter of the fracture behavior. This value of load was designated as P, 0.5. A log-log plot of load (P, O,s)vs. ball radius R is shown in Fig. 11, where it can be seen that for the quasi-static Ins&on tests on both the asreceived and the abraded surfaces Auerbach’s law appears to be obeyed. This was not the case for the dynamic free-fall tests. However, a plot of PC&R vs. R (Fig. 12) indicates that Auerbach’s law may not be strictly valid for any of the experimental conditions.

3.3. Effects of sur$ace condition and load on the ratio of the ring crack radius to contact radius (r-“/a) I 0

200

400

I 600

I

I 600

I

I 1000

I

P, (NJ

Fig. 8. The cumulative frequency& vs. P, for as-receivedSi( 111) using the Instron.

values of the contact radius are sufficiently accurate to be used in evaluating other relationships. The critical load PC to form a Hertzian ring crack showed appreciable scatter; so the results were subjected to a statistical analysis. The procedure followed for the quasi-static Instron tests was to determine the cumulative frequency or

The Hertzian ring crack which forms on the Si( 111) surface, at the critical load PC, is geometric in shape, unlike the circular ring cracks observed in glass. Therefore it is necessary to define two radial distances r; and r; to characterize its shape as shown in Fig. 13. The variation of these two parameters with load is shown in Fig. 14, where it is seen that the two crack dimensions increase with load at about the same rate. The mean value of the ratio of r,’ to r-J is about 1.06, and is less than the ratio of the same dimensions of a regular hexagon, which is 1.15.

Jyh-Woei Lu et al. /Wear 186187

I

’R = I’.19 (mm)

1.0 -

(1995) 105-116

111

’

‘1.59 ‘1.98 i.38 ‘2.78’ ‘3.18 ’

’I

0.6 -

0.4 -

0.2Method.

100

0

200

300 P,

Instron(V=

400

8.5 x lO‘sm/s)

500

(NJ

Fig. 9. The cumulative frequency& vs. PCfor abraded Si( 111) using the Instron machine.

The effect of surface condition and load on the mean values of r; /a for the Si( 111) surface impacted with a 3.18 mm steel ball in a free-fall test is shown in Fig. 15. The mean values of r; la were found to be larger for the abraded surfaces

I

I

SI (ill)

Abraded

With

400

Grit

SIC

’

I

’

1’1

: Si ( III)

Specimen Indenter:

Steel

Environment

Speumen

’

Method:Instron(V=8.5~ 10‘6m/~) Free Fall (V= 1.3-4.8 m/s)

A : lnstron (Polished) 0 : lnstron (Abraded, Sic 400 Grit) 0. Free Fall (Abraded)

100 O80

Ball

: Ambient

O-

60 O-

40 O-

0.6 t 0.4 0.2 O.$ 1

20 OL H

(cm) Height

c

1.0

/

0.8

IO

0.6

Ol

4

2

6

8

IO

Rlmm)

0.4

Fig. 11. Plot of log PCo,5 vs. log R as a function of loading method for Hertzian ring cracks in as-received and abraded Si ( 111)

0.2 0.0

0 Pc (N)

Fig. 10. The fraction of tests at a constant load which produced Hertzian ring cracks in abraded Si( 111) vs. the load PC using free fall in abraded Si(ll1).

than for the as-received surfaces; however, the mean values of r; la are almost independent of load for either surface condition.

Jyh-Woei Lu et al. /Wear 186-187 (1995) 105-116

112

I

I

I

I

I

I

I

I

I

I

I

Specimen: Si(lll)

As-Received (PolIshedI Steel Ball (R=l 19mm) Instron(V=8.5 x 10-6m/s)

Indenter Method:

Methods’

lnstron

and

Free

Fall

240 Free

t

2nnL I

s

“V

A

Fall(Pollshed

O.lOh

Surface)

J I

,/

LT \

0 22 -+I

0061 I50

0 0

a”

Sic

Surface

400

I 250

I

200

Grit

I

I

I

300

350

400

P, (NJ

40

Fig. 14. Values of rfl and rsz vs. the fracture load PCfor as-received 0

I

I

I

I

I

l

2

3

4

5

I 6

I

I

7

8

a(mm)

R(mm)

Fig. 12. PC&R vs. R as a function of loading cracks in as-received and abraded Si ( 111)

Si( 111)

I 0

I

method for Hertzian ring

’

0.160

0.202

I

I

Speclmen.S~

(Ill)0

0.231

1 Abraded

0 254

I

I

0 274

I

Surface(SlC

400Grit)

1.81.6 -

I

I

1

I

600

800

I

400

/

1000

PC (N)

Fig. 15. The ratio of ring crack radius vs. the fracture load PCfor the various surface conditions of Si( 111) Fig. 13. Hertzian crack on the surface of Si( 111)

I

I

I

I

‘i

I

Specimen : Si (Ill) As-Received Method: lnstron ond Free Fall

3.4. Effects of indenter size and loading rate on the ratio of

the ring crack radius to contact radius (rJ/a)

1.8

I

A Free Follon

-

0 lnstron

I

and Abraded

I Surface

Abraded

on Polished

Surface Surface

The mean values of r; /a as a function of indenter radius R for steel balls loaded onto both as-received and abraded Si ( 111) surfaces using quasi-static (In&on) and dynamic (free-fall) tests are shown in Fig. 16. For both surface conditions and test methods the slopes of the curves are negative, i.e. the larger the ball size the smaller the r-;/a ratio. In addition, Fig. 17 shows a plot of the ratio of the crack area A, to contact area A,, where AI = 3(r,‘?*) and A,=&. Here the slope is even more negative than in the previous plot. Extrapolation of this line shows that the crack area will equal the contact area at a ball radius of about 10 mm.

0 8-

I

I

I

I

I

I

1

0

I

2

3

4

5

6

7

R(mm)

Fig. 16. Mean value of r* ,/a vs. ball radius for free-fall and Instron test of Si(ll1).

113

Jyh-Woei Lu et al. /Wear 186-187 (1995) 105-116


Speamen:

Si (Ill)

As-Received

Method:

lnstron

Test

loading method. To avoid geometric scale effects in the Hertzian testing, the indenters of steel, alumina, and tungsten carbide of ball radius 1.19 mm, 1.59 mm and 1.59 mm, respectively, were chosen so as to produce the same contact area on the Si( 111) surface at the same load, taking into account the differences in Young’s modulus and Poisson’s ratio between the ball and the silicon surface. It is seen from Fig. 18 that for the steel indenter, the mean values or r* /a are slightly less than those for the alumina or tungsten carbide indenters, and that all of the slopes are slightly negative.

(Polished)

(V=8.5x

10-6ms-‘)

1.6-

4. Discussion 4. I. The fracture load R(mm)

Fig. 17. Mean values of crack area A, to contact as-received Si ( 111)

Environment:

SI (III)

Indenter

0

Steel

A

Al,03Ball,

0

WC

lnstron

I

2.0

A:,vs. ball radius R for

Ambient

Specimen,

Method

area

(IS-

Received

Ball,

R -

(V=

(Pollshad)

I. I9

R = I

Ball,R= 8.5

It was found that for the same indenter size, the Pco.5 values for Si( 111) as-received, polished surfaces are greater than those obtained for the abraded (400 grit Sic) surfaces. This is contrary to the behavior observed in Pyrex glass [ 271, and may be interpreted to mean that for the abraded surface in Si, surface flaws dominate fracture rather than surface roughness.

59

mm

1

mm

l59mm Y lO+m/s)

I

I

I Steel

Symbol Spec.Cond. lndeter

I .6

I

Mean = I.18

2

Abraded Steel Free Fall O (grit 400 Si C)

08

A As-Received 0 As-Received 0 As-Received

b 2.0 Al203

n

< ?? L-

Method

Steel AI,O, WC

lnstron I nstron lnstron

I .6 t A

I2

t

Mean =I.21

O.E+

T-

2.0WC -

I .6-

P,

(NJ

Fig. 18. Values of r* ,/a vs. PC for the indentation of as-received Si( 111) with various indenters.

3.5. Effects of indenter material on the ring crack radius to contact radius ratio (r-;/a) To quantitatively evaluate the ring crack radius radius, which is affected by the elastic constant between the indenter material and the Si( 111) series of test were undertaken using the quasi-static

to contact mismatch surface, a (Instron)

,I

I

I

0.2

0.4

I

0.6

I

0.8

II

1.0

rf:(mm) Fig. 19. Plot of log P,,, vs. log r’, for the indentation of as-received polished and abraded Si( 111) with various methods and indenter materials.

114

Jyh-Woei Lu et ul. /Wear

denter’ Steel ethod lnstr

Ball

Radius

with direction because silicon is elastically anisotropic. It was experimentally determined that the direction of the tensile stress which initiates the crack at the critical load is [ 2111, so the Hertzian fracture stress was calculated using Y’instead of an average value Y,,, where u’ is the Poisson’s ratio when the stress is in the [ 2111 direction and on the ( 111) plane. Plots of the cumulative frequency fF vs. fracture stress err are shown in Figs. 20 and 21 for quasi-static Instron tests on the as-received polished specimen surfaces and for the freefall test on the abraded specimen surfaces, respectively. The distribution of the data in Fig. 20 is similar for the different ball sizes, but in Fig. 21 the values of oF increased in scatter with increased ball size. It was shown by Johnson et al. [ 181 and Lawn and Wilshaw [ 91 that interfacial friction changes the position of the maximum fracture stress away from the contact circle to a larger radius, hence the r* /a ratio becomes greater than 1.O. The equation used to calculate the fracture stress at the location of the ring crack is Eq. 4., which may also be written as

Ball

R (mm)=

0 crF x 102~N/mm2)

Fig. 20. The cumulative polished Si( 11 I ).

frequency

vs. gF using the lnstron for as-received

A log-log plot of P, 0.5vs. r; for various specimen surfaces and indenter materials is presented in Fig. 19. It is found that the critical load P, ,,5 is proportional to (r; )3’2 and is relatively independent of the indenter material. i.e. PC05

= C’(

rl * y*

(10)

where the values of C’ are 4000 and 2370 N mme3’* for the Instron test on the as-received polished surfaces and for the free-fall test on the abraded surfaces respectively 4.2. The fracture stress In determining the true fracture stress from the fracture load it is necessary to consider that the elastic constants vary

Specimen:

Ball

3.5

43

Si (III)

l-2V G=-‘“(r*,a)2 2

1

(11)

where PO= PITa* is the mean pressure. A plot of P, vs. contact radius is shown in Fig. 22 for the quasi-static test on an as-received silicon surface using, steel, alumina, and tungsten carbide balls as indenters. Since the ideal Hertzian equation does not include surface friction effects caused by surface roughness or elastic constant mismatch, it is necessary to consider the applicability of this equation to the observed phenomenon of Hertzian fracture. To check its practical applicability it is necessary to compare the value of u,lP,, for the case of a steel indenter, since for a steel indenter on a silicon specimen surface we

Abraded

(grit

400

Sic)

R(mm)=

3.5

4

I .59

1.98

2.38

3.18

I

3

Radius

186-187 (1995) 105-116

4

4.5

5

3.5

4

4.5

5

(To x IO* (N/mm*) Fig. 21. The fraction of tests at a constant load which produced Hertzian cracks in as-received

polished Si( 111) vs. cF using free fall

115

Jyh-WoeiLuetal./Wearl%-187(1995)105-116

51 Specimen

6ooo-

??

Si (III)As-Received.Pollshed

A

; E

4000

Indenter

\

5 CL”

3000 5ooo!

CR= 1.5 9 mm)

0 Steel A AlEO 0 WC

\

2000-

based on this calculation yields, yl, 1= 1.14 J cm- *, which is in good accord with the value obtained by Gilman [ 341 of yll 1= 1.24 J m-* using the cleavage method. Lawn [ 321 obtained a value of ylll = 0.14 J m-* by applying the Hertzian test to measure the fracture surface energy. In the present study, Eq. (13) derived by Warren [ 191, was used to obtain the value of the surface energy for fracture on { 111) planes: p,c

kyR

a

’ 4(b)

db

27 ,,i[(c*-b*)“*]

IOOO-

I 0. I

I

I

I

I

I

0.2

0 3

0.4

0.5

0.6

-* (13)

Values of y1 ,, = 2.1-6.8 J m-* were obtained. The larger values obtained here may be due to plastic deformation associated with the fracture.

a(mm)

Mean pressure POvs. contact radius a using Instron test method in as-received polished Si( 111) with different indenter materials.

Fig. 22.

I

I

I

I

I

I

I

Indenter

5. Conclusions

I

CR = 1.5 9 mm)

0 Steel A AI,O,

a=0

15mm

0 I-

0

I I.0

I 1.2

I

I rl*/

(r;/a)’

2

I

I.4

I

I-2V’

=‘F -=-_ PO

I 1.6

I

I 1.8

a

Fig. 23. uF/P,, vs. r* ,/a using Instron test on the as-received polished Si( 111) surface with different intender materials.

have an elastic mismatch parameter, K, value of about 0.16, which is very close to the ideal Hertzian fracture condition. Fig. 23, shows a plot of gF/ a vs. a from which it can be seen there is reasonably good agreement between the experimental data and the ideal Hertzian equation represented by the solid curve which supports the applicability of this approach to the study of Si. 4.3. Sueace

energy calculations

In silicon, the cleavage planes were found to play a significant role in Hertzian fracture in that the cracks tend to lie along cleavage planes resulting in a hexagonal outline on the ( 111) surface. It was observed that the cracks initiate at the surface, not at the position of maximum shear stress below the surface. Furthermore, at room temperature dislocations were observed around the tips of the internal cracks. It was found that Auerbach’s law is not strictly valid for silicon. The ring crack radius to contact radius ratio increased with increased surface roughness. A similar effect was observed with increasing the friction constant, K, due to elastic constant mismatch caused by the different indenter materials on the silicon surface. The value of the critical load was higher for the polished silicon surface than for the abraded surface. From this it was concluded that the critical load for the abraded surface is dominated more by surface flaw size than by surface roughness. The observed experimental results are in accord with the recent theoretical predictions of Zeng et al. [ 211. For silicon, it was concluded the ideal Hertzian fracture stress equation applies when the proper anisotropic moduli are used. None of the existing entirely elastic energy balance Hertzian fracture theories yielded surface energy values in accord with those determined theoretically or by other experimental techniques.

Acknowledgements

The surface energy of silicon for the ( 111) plane can be calculated based on the work needed to separate atoms across the plane to an infinite distance. Thus;

This paper is based on work supported in part by the National Science Foundation under Grant DMR-75- 10347.

ET, Y=G

References

(12)

where x0, the interatomic distance, equals 0.24 X lop9 m, Elll = 1.88 X lo5 N mm-*. The value of the surface energy

[l] I. Finnie, Wear, 3 (1960) 87. [2] J.G.A. Bitter, Wear, 6 (1963) 5

Jyh-Woei Lu Edal. /Wear 186-187 (1995) 105-116

116

131 G. L Sheldon, Truns ASME, J. Basic Eng., (Sep. 1970) 619.

[17] B.R. Lawn, JAppl. Phys., 39 (1968) 4828.

[41A.G. Evans, M.E. Gulden, G.E. Eggum, and M. Rosenblatt, Tech. Rep.

[I81 K.L. Johnson,

Contract #NOOO14-75-C-0669, Project #471 (Office of Naval Research, 800 N. Quincy St., Arlington, VA 22217, USA). [5] A.G. Evans and T.R. Wilshaw, Acra Metall., 24 (1976) 939. [6] W.F. Adler, Tech. Rep. AFML-TR-74-210, Sept. 1974 (Air Force Materials Laboratory, Air Force Systems Command, Wright Patterson Air Force Base, OH, USA). [7] P.K. Mehrotra, G.A. Sargent and H. Conrad, Corrosion-erosion behavior of materials, in K. Natesan (ed), Proc. Conj h4erullurgical Sot. AIME, St. Louis, MO, October 17-18, 1979, p. 127. [8] H. Hertz, J. Reine Angew. Math., 92 ( 1881) 156; reprinted inEnglish, Miscelluneous Pupers, Macmillan, London, Chapter 5, 1896. [9] B. Lawn and R. Wilshaw, J. Muter. Sci., 10 (1975) 1049. [IO] T.R. Wilshaw, J. Phys. Appl. Phys., 4 (1971) 1567. [ 111 E. Auerbach, Ann. Phys. Chew 43 ( 1891) 61. [ 121 J.P.A. Tillet, Proc. Phys. Sot., 698 ( 1956) 47. [ 131 D.N. Turner, P.D. Smith and W.B. Rotsey, J. Amer. Ceram. Sot., 50 (II) (1967) 594. [ 141 A.S. Argon, Y. Hori and E. Orawan, J. Amer. Ceram. Sot., 43 (II) (1960)

86.

[ 151H. Conrad, M.K. KeshavanandG.A.

Sargent,J.

Muter. Sci., 14 (1979)

1473. [ 161 F.C. Frank and B.R. Lawn, Proc. R. Sot. London, Ser. A, 229 (1967) 291.

J.J. O’Connor

and AC.

Woodward,

Proc R. Sot.

London, Ser. A, 334 (1973) 95.

[ 191 R. Warren, Acta Metall., 26 (1978) 1759. [ZO] R Mouginot and D. Maugis, J. Muter. Sci., 20 (1985) 4354. [21] K. Zeng, K. Breder, and D.J. Rowcliffe,Acta

Me&l. Mater., 40 (1992)

2595. [22] K. Zeng, K. Breder, and D.J. Rowcliffe,AcraMetallMarer.,40

(1992)

2601. [23] M.T. Huber, Ann. Physik., 14 (1904)

1.53.

[24] I.N. Sneddon, Proc. Camb. Philos. Sot., 42 ( 1946) 27. [25] J.A. Greenwood 89 (1967)

and J.H. Tripp, J. Appl. Mech. Trans. ASME Ser. E,

153

[26] F.C. Roesler, Proc. R. Sot., London, Ser. A, 69 ( 1956) 55. [27] J-W. Lu, G.A. Sargent and H. Conrad, Wear, 162-164

(1993)

856

[28] ASM Metals Handbook (Desk Edition), ASM, Metals Park, OH, 1985, pp. 1117. 1291 A. Kelly, Strong Solids, Clarendon

Press, Oxford,

( 1973) p 267.

[30] B.R. Lawn and H. Komasu, Philos. Msg., 14 (1966) 689 [31] E.N. Pugh and L.E. Samuels, Philos. Mug., 8 (1963) 301. [32] B.R. Lawn, J. Appl. Phys., IO (1968) 39. [33] E.N. Pugh and L.E. Samuel% J. Electrochem. Sot., 108 (1961) [34] J.J. Gilman, J. Appl. Phys., 31 (1960) 22.

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