Poole Ch.P., Jr. Handbook of Superconductivity

606 Chapter 13: Mechanical Properties

b. Measurement Methods

The empirical definition of indentation hardness has a strong intuitive appeal

making hardness a useful comparative characteristic. Its application, however,

must be approached with some caution. More direct empirical studies of the size

of an indentation as a function of the applied load have found that the load and

indentation size are often related by a power law, such that

F = ~d '7, (40)

where ~ and r/ are parameters for a given material (Mayer, 1908). Using this

relation in the expressions for H v and HK, one can see that the indentation

hardness will be a well-defined constant for a material only in the exceptional

case that the value of r/ is exactly 2. This observation is very important to the

measurement of hardness. In general, a reported value of hardness can be

considered meaningful only when the load used to determine the hardness is

also reported. It is fairly common for measurements of hardness to be conducted

over a range of loads to provide an explicit accounting for the load dependence.

At sufficiently high loads, significant cracks, subsurface damage zones, and

fragmentation around the indentation site change the general nature of the

deformation. At these higher loads, the hardness value, to the extent that it

remains measurable, may become constant, i.e., independent of the load, but the

interpretation of this constant value, even as a comparative statistic, is an open

issue.

In both tests, the chosen load is applied along an axis that is normal to the

surface of the specimen. Typically, the indenter is pressed into the specimen at a

rate of 15 to 70 #m/s, and the maximum static load is sustained for 10 to 15 s.

The lengths of the diagonals of the resulting impression commonly are measured

using an optical microscope with a graduated ocular. Usually, multiple indenta-

tions are made on one specimen, and it is important that there be sufficient space

between individual indentations so there is no overlap of any of the associated

deformation features.

c. Property Data

Data from hardness measurements are given in Table 13.8. For most of the studies

referenced in this table, indentation impressions began to form at very low loads

on the order of 0.05N. The largest value of hardness, 12.7 GPa, occurred for a

single-crystal specimen of Y: 123. In contrast, the largest value for a Bi : 2212 or

a Bi(Pb) : 2223 material was less than 1 GPa.

Table 13.8.

Hardness (H).

Phase

Relative Crystal

Density (%) Face

Applied T Hardness

Load (N) [K] (GPa)

Measurement

method Notes Ref.

,,q

Bi:2212

Bi(Pb) : 2223

Dy: 123

Ho: 123

La(Sr) : 21

100

100 (001)

9O

91

83

100

(001)

50 or 100 296 0.76

0.66

0.6 77 0.62

296 0.33

296 0.35

0.02 to 0.75 296 0.95

296 0.85

2 296 0.6

10 to 50 296 0.35

0.8 to 1.3 296 0.38

0.5 to 50 296 0.4

1.5 296 0.42

0.15 300 5.7

0.15 300 5.9

0.02 to 0.8 296 6

0.15 83 11.1

84 11.7

109 11

110 10.6

Vickers indentation Hot isostatically pressed

Sinter forged

Vickers indetation

Vickers indentation

Knoop indentation

Chu

et al.

(1992)

Single crystal; load dependent below Muralidhar

et al.

(1991)

0.4 N load

Single crystal; load dependent below

0.4 N load

Polycrystal; load dependent below

0.4 N load

Single crystal

Bil.7Pb0.3

Vickers indentation Average for grain aligned specimens

Vickers indentation Bil.7Pb0.3, p = 4.989 g/cm 3

Vickers indentation p = 5.44 g/cm 3

Vickers indentation

Vickers indentation Single crystal

Vickers indentation Single Crystal

Vickers indentation Single crystal

Orlova

et aL

(1990)

Joo

et aL

(1994)

Low

et aL

(1995)

Low

et aL

(1995)

Muralidhar

et al.

(1992)

Lo

et al.

(1995)

Luo

et aL

(1995)

Bobrov (1993)

Orlova

et al.

(1990)

Bobrov (1993)

(continued)

Table 13.8.

(continued)

Phase

Relative Crystal

Density (%) Face

Applied T Hardness

Load (N) [K] (GPa)

Measurement

method Notes Ref.

T1:2212

Y: 123

100

86

88

76

100

62

100

(100)

(001)

0.15

10 to 100

2

0.25 to 2

2

0.2 to 0.4

136 11.1

140 10.9

161 10.6

167 10.3

188 10.3

192 10

200 10.1

207 10

217 9.3

230 9.9

233 9.6

265 9.3

292 9

300 2.8

296 2

296 4.3

4.4

296 7

296 2.2

296 10.1

8.4

Vickers indentation Single crystal; T1-Ba-Ca-Cu-O Bobrov (1993)

Vickers indentation Grain size -- 15 #m Cook

et aL (1987b)

Knoop indentation Grain size = 16 #m Ihm

et aL (1990)

Grain size- 11/~m

Knoop indentation Single crystal; varies with orientation Fang

et al.

(1993)

Knoop indentation Low

et al.

(1994)

Vickers indentation Single crystal Anderson (1989)

Y: 123

Y" 123

100

90

100

1

O0

1 O0

100

(001)

(OOl)

(lOO/OlO)

(OOl)

(OLO)

5 to 20

0.05 to 0.5

0.02 to 0.5

0.1 to2

0.25 to 0.5

0.1 to2

0.15

0.02

296

87

110

150

187

226

259

293

296

5.3

11

9.7

5

8.7

9.6

6.7

3.8

10.3

13

12.5

12.7

11.7

11

10.3

10

10.8

Vickers indentation Varies with load

Indentation Single grains, 50 nm to 200 nm

penetration

Vickers indentation Single crystal, T c -- 65 K

Vickers indentation Single crystal; T c = 90 K

Vickers indentation Single crystal

Vickers indentation Single crystal; twinned and

untwinned

Vickers indentation Single crystal

Vickers indentation

Vickers indentation Single crystal

Indentation Single crystal

Ni

et al.

(1993)

Marinez

et al.

(1992)

Demirskii

et al.

(1989)

Orlova

et al.

(1990)

Cook

et al.

(1987a)

Raynes

et al.

(1991)

Goyal

et al.

(1992)

Lucas

et al.

(1991)

Bobrov (1993)

Goyal

et al.

(1991)

610 Chapter 13: Mechanical Properties

Toughness

Fracture toughness refers to the resistance of a material to the extension of a

crack. When a stress is applied to a brittle material, there is a maximum stress that

can be sustained within the material before an existing crack begins to propagate.

That stress, however, is not the same as the applied stress at fracture (which is the

fracture strength discussed in Section C). Rather, propagation of a crack depends

on the local stress that occurs at the tip of the crack. If one considers, for example,

an idealized crack in the form of an elliptical hole in a semi-infinite plate, as

shown in Fig. 13.7, with major axis 2a and minor axis 2b, in a medium beating an

applied uniform tensile stress, O-, it can be shown (Inglis, 1913) that the maximum

local tensile stress is

O-max -- (1 nt- b) O-, (41)

which occurs at the tip of the ellipse along the major axis. The presence of the

crack is said to intensify the stress. When amax -- O-a, the specimen fractures. It is

evident from Eq. (41) that O'ma x depends on the geometric shape of the crack and,

in particular, on the radius of curvature at the crack tip. As the ratio

a/b

increases,

O-max increases, indicating that sharper cracks produce a greater intensification of

the stress. Thus, the extent to which a specimen can resist crack extension

depends on the particular flaw distribution that is present in the material.

Fig. 13.7.

G G G

...)z

T T T

z'l"

Maximmn

Tensile

Stress

G (Y I~

Schematic of an elliptical hole in a stressed medium.

..=

Z

E. Toughness 611

a. Terms and Basic Relations

The development of linear elastic fracture mechanics has allowed the extension of

a crack in a material body to be characterized using the methods of linear stress

analysis. In this analysis, a parameter K, called the stress intensity factor, is

introduced to represent the enhanced state of stress in the vicinity of the crack9

The symbol K I is used for tensile stress, which is formally classified as Mode I

stress. The critical value of the stress intensity factor, Kic, commonly called the

fracture toughness, is defined as the value for the onset of fast crack extension.

Linear stress analysis leads to the relation

KZc = FE'= 27E' (42)

for tensile cracks. Recalling the Griffith criterion, Eq. (29), it is easy to see that a

relation between the fracture toughness of the material and the measured fracture

strength, af, of the material should be expressible in the form

KI c

.7

1/2

(43)

l~TfCo ,

where Y is a dimensionless numeric factor that depends on the test configuration

and the shape of the initial crack. Numerous test methods for measuring Kic

exploit this relation using an initial crack that has a measurable shape and size.

b. Measurement Methods

Designs for fracture toughness test methods may be grouped rather broadly into

two basic types (Anderson; 1989). In one type, an initial flaw is created in the

surface of the specimen, and then the specimen is fractured, usually in a bend test.

In the other type, an indenter is used in the manner of a hardness test, but with

sufficient load to create an indentation flaw with radial cracks extending from the

comers of the impression. Both approaches have advantages and disadvantages

that add or detract from their use, and numerous variations of these basic ideas

have been pursued to enhance or minimize specific features. For high-T c

materials, the predominant methods of determining Kic have been the single-

edge notched-beam (SENB) method and the Vickers indentation toughness

method.

In the basic SENB method (Rief and Kromp, 1988; Srinivasan and

Seshadri, 1981; Quinn

et al.,

1992), Fig. 13.8a an initial crack is cut into a

bend test specimen using a saw blade or a wire. The specimen is then fractured in

a three-point or a four-point bend test to determine o-f, and the initial depth of the

cut is taken as the initial crack size. The fracture toughness is evaluated as

3F-A s 1/2

9 c 0 , (44)

KIc- YSENB "

2wh2

612 Chapter 13: Hechanicai Properties

Fig. 13.8.

(a)

W

) ..s -''~

,r

(b)

T

h

1

L W d

P 1

22T

i l

:"

co ',1

' _L I

I._ .......... ,,,1

Schematic of (a) the SENB (single-edge notched-beam) test in the three-point bend config-

uration, and (b) a chevron notch.

where A s is s in the three-point test and s o - s i in the four-point test. In the three-

point test when

s/h

= 4, YSENB is

(--~) (-'~) (h) 3 (h) 4

2

YSENB,3 = 1.93- 3.07 + 13.66 23.98 + 25.22

(45)

and in the four-point test

(h) (h) 2 (h)3 (h)4

YSENB,4 = 1.99 -- 2.47 + 12.97 23.17 + 24.8

(46)

where the geometric parameters are defined in Figs. 13.5 and 13.8a. Although the

simplicity of the SENB method is highly desirable, it is sometimes argued that the

notch produced in this manner is relatively blunt. Since the result may vary with

the radius of the notch tip, the measured values can be somewhat unreliable and

typically too high. To correct that situation, a sharp crack at the notch tip may be

induced by a mechanical procedure (Nose and Fujii, 1988). Use of such

E. Toughness 613

precracked specimens, however, leads to other concerns, such as a greater

uncertainty in the value of the initial crack size.

A modification of the SENB method that retains the simplicity of the test

while providing a sharp tip within the notch is the chevron notch method (Munz

et al.,

1980), Fig. 13.8b. In this method, the straight notch is replaced with a notch

in the shape of a chevron. The point of the chevron provides a deliberate stress

concentration and serves as the primary origin for fracture. In the four-point test

configuration, Kic is evaluated as

F. (s o - Si)

KIc- YEN,4 "

wh3/2

(47)

YCN,4-

3.08 + 5.00(-~)+ 8.33(h) 2 9 [1 + 0.007\ h2 ,/]. Co,

where the geometric parameters are defined in Figs. 13.5 and 13.8b.

An alternative to the notch methods is the surface crack in flexure method

(Tracy and Quinn, 1994). This method uses a Vickers hardness indenter with

sufficient load to create a crack under the indentation impression. The impression

and the stress damage zone immediately below the impression, however, are

removed by mechanical lapping so that only the crack remains. Then, the

specimen is fractured in a four-point bend test. Subsequently, scanning electron

microscopy is used to examine the fracture surface and to determine the initial

crack depth. The fracture tougness is evaluted directly as

KI c .r 1/2

(49)

I SCFO-fC 0 ,

where YSCF is an empirical factor discussed extensively by Newman and Raju

(1981). Although this method is not yet widely used for high-T c materials, it has

become recognized as one of the more reliable methods, and substantial progress

has been made toward standardizing the test procedure (Quinn

et al.,

1994).

Fracture toughness tests using bend test configurations require relatively

large specimens. A useful alternative method that requires only a small amount of

material is provided by the indentation method. (Evans and Charles, 1976). In this

approach Fig. 13.9 a Vickers indenter is pressed into the surface of the specimen

until cracks develop radially from the comers of the impression. The analysis of

the method by Anstis

et al.

(1981) found that Klc could be evaluated as

gic-~

(50)

where E is the elastic modulus, H is the indentation hardness, and ( is a numeric

factor determined empirically to be ( = 0.016 4- 0.004 based on the average and

standard deviation of values for a wide range of ceramics. The analysis assumes

that a well-defined crack morphology, known as median cracks, develops around

the impression. At sufficiently low loads, only shallow cracks, known as

614 Chapter

13:

Mechanical Properties

Fig. 13.9.

_

Schematic of the Vickers indenter impression in the indentation toughness test.

Plamqvist cracks, develop, and a different analysis is required for determining

Kic. Various studies of the indentation method have found that average values

computed across laboratories are consistent with average results from other

methods, but the variance of the values among laboratories is greater for the

indentation method than for other methods. Often, the advantages of the

simplicity of the method and the need for only a small specimen outweigh the

concerns for interlaboratory consistency, with the result that the method is widely

used for high-T c materials (Ponton and Rawlings, 1989a,b).

c. Property Data

Fracture toughness data are given in Table 13.9. Three-point bend SENB tests and

Vickers indentation toughness tests account for all of the data in this table. The

largest value, 3.9 MPa. m 1/2 for Bi:2212, is comparable to the fracture tough-

ness values found for structural ceramics such as sintered alumina or silicon

carbide. However, most of the reported values were less than 2 MPa. m 1/2 and

often less than 1 MPa-m 1/2.

Conclusion

The mechanical properties of high-T c superconductors are important to the

development of commercial applications of these material, particularly for

Table

13.9.

Fracture toughness (Kic).

Phase

Relative

Density (%)

T

[K]

Kic

(MPa m 1/2)

Measurement

method

Notes

Ref.

o,,

t.~

Bi:2212

Bi(Pb) : 2223

Y: 123

90

77

90

99

86

296

77

293

296

3.2

3.9

2.7

3.2

1.9

1.2

1.8

1.4

1.8

2.9

2.8

2.4

1.3

0.3

3

2.6

1.3

SENB a

SENB b

SENB c

SENB d

Vickers indentation

Hot isostatically pressed; notch

perpendicular to pressing direction

Hot isostatically pressed; notch

parallel to pressing direction

Sinter forged; notch perpendicular to

pressing direction

Sinter forged; notch parallel to

pressing direction

2mm x 3 mm x 22mm

Bil.2Pb0.8; 2 mm • 5 mm • 30 mm

Bil.7Pb0.3; 3 mm x 12 mm x 60 mm

B i 1.8 Pb0.3 Sr 2 Ca 1.9 Cu3010+x;

2 mm x 6.4 mm x 31.75 mm

Assumes

E/H -- 40; 2 N to 100 N;

10s

Chu

et al. (1992)

Joo

et aL (1994)

Martin

et aL (1993)

Alford

et al. (1990)

Low

et aL (1995)

Yuan

et al. (1996)

Cook

et aL (1987b)

(continued)

Poole Ch.P., Jr. Handbook of Superconductivity

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