Poole Ch.P., Jr. Handbook of Superconductivity

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566

Chapter

12:

Magnetic Properties

Ac Susceptibility

An ac field Bac(t ) = B 0 cos cot applied to a superconductor causes the magnetiza-

tion

M(t)

to trace out a magnetic hysterisis loop during every cycle of the applied

field. During the cycle

M(t)

does not follow Bac(t ) directly, but tends to lag, to

distort in shape, and to shift in phase relative to the applied Bac(t ), so it acquires

an out-of-phase component that varies as sin cot. To account for this we define in-

phase dispersion g' and out-of-phase (quadrature) absorption Z" susceptibilities

(Matsumoto

et at,

1991):

7/-- ~o J M(t)

cos

cot dcot

(72)

;~" - la~ l M(t)

sin

cot dcot.

(73)

Figure 12.13 presents the temperature dependences of Z' and g" determined for

several values of the ac field amplitude applied at the frequency co/2rc - 73 Hz

Fig. 12.13.

, 0.4

0.2

'~,~_~ 0.0

-0.2

~, -0.4

-0.6

..~

-0.8

~ -I.0

~ -1.2

-1.4

XI"

.u, oHdc-O 11 " ?'JHz

xl'

,.~.

#0Hac - 212

0424

7e 're eo e2 e4 ee ee 90

Temperature

(K)

Real (Z') and imaginary (Z") components of the susceptibility of YBa2Cu3OT_6 measured in

the applied ac magnetic fields #0Hac = 0.0424, 0.424, and 2.12mT as a function of the

temperature below T c for the frequency 73 Hz. For this experiment no dc field was present, and

the data were not corrected for the demagnetization factor. [Ishida and Goldfarb (1990).]

References

$67

with no dc field present. A simultaneously applied dc field affects the temperature

dependence of Z' and Z"-

References

J. R. Chem,

Phys. Rev. B

43, 7837 (1991).

T. Ishida and R. B. Goldfarb,

Phys. Rev. B

41, 8937 (1990).

W. E. Lawrence and S. Doniach, in

Proc. 12th Int. Conf. Low Temp. Phys. Kyoto, 1970

(E. Kanda, Ed.),

p. 361. Keigaku, Tokyo, 1971.

Y. Matsumoto, M. Katada, and T. Nishida,

Physica C

185, 1229 (1991).

C. P. Poole, Jr., H. A. Farach, and R. J. Creswick,

Superconductivity.

Academic Press, New York, 1995;

vide Chapters 9-12.

S. Senoussi, M. Oussena, and S. Hadjoudi,

J. AppL Phys.

63, 4176 (1988).

A. R. von Hippel,

Dielectrics and Waves,

p. 255. MIT Press, Cambridge, MA, 1954.

N.-C. Yeh,

Phys. Rev. B.

40, 4566 (1989); see also

ibid.

39, 9708 (1989).

This Page Intentionally Left Blank

Chapter 13

Mechanical Properties

Ronald G. Munro

Ceramics Division, National Institute of Standards and Technology,

Gaithersburg, Maryland

A. Introduction 570

B. Elastic Properties 570

a. Terms and Basic Relations 572

b. Measurement Methods 574

c. Corrections for Porosity 575

d. Property Data 576

C. Strength 576

a. Terms and Basic Relations 596

b. Measurement Methods 597

c. Property Data 601

D. Hardness 601

a. Terms and Basic Relations 601

b. Measurement Methods 606

c. Property Data 606

E. Toughness 610

a. Terms and Basic Relations 611

b. Measurement Methods 611

c. Property Data 614

E Conclusion 614

Acknowledgements 619

References 619

ISBN: 0-12-561460-8

$30.00

HANDBOOK OF SUPERCONDUCTIVITY

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569

570 Chapter 13: Mechanical Properties

Introduction

Mechanical properties are important to the study and application of super-

conductors in three respects: (1) scientifically through the electron-phonon

interaction; (2) technologically through processing conditions and product

design parameters; and (3) economically through the durability and reliability

of the manufactured products. In all cases, superconductors require cryogenic

environments within which the material may be required to survive long-term

exposure to significant electromagnetic and thermomechanical stresses. Poor

mechanical properties or low structural reliability can result in premature failure

of these materials during normal service conditions. If the material is fragile,

stresses that arise from handling during routine production operations can

introduce deleterious cracks in the material. Knowledge of the mechanical

properties of the material can be particularly helpful in considering safeguards

against the occurrence of such situations. More beneficially, it may be possible to

exploit that same knowledge to produce an improved material, even, perhaps,

with tailored properties. For example, the elastic-plastic deformation properties

can be exploited during the material processing stages to produce more highly

textured (grain aligned) materials that tend to achieve higher critical current

densities.

In the following text, the elastic properties are discussed first, followed by

discussions of flexural and tensile strengths, hardness, and fracture toughness. In

each case, basic terminology, critical issues, measurement methods, and tables of

property data are presented. To facilitate comparisons of results, all of the data

tables have a similar structure, although some variation occurs in the column

headers to emphasize the variables and measurement features that are important

to the determination of specific property values. The first column in each table

identifies the material using the abbreviations defined by Table 13.1. The

structure of each table is otherwise self-explanatory.

Elastic Properties

The elastic properties of superconducting materials are important to the theore-

tical understanding of the phenomenon of superconductivity, as well as being

important to its practical application. In the BCS theory (Bardeen

et aL,

1957) of

superconductivity, the electron-lattice interaction is crucial to the formation of

superconducting states. Since the long-wavelength phonon spectrum is intimately

related to the elastic properties of the material, it follows that the superconducting

characteristics also must be related to the elastic properties of the material. In the

B. Elastic Properties 571

Table 13.1.

Material abbreviations.

Phase Nominal formula

Bi : 2212 Bi 2 Sr2CaCu208+ x

Bi(Pb) :2223

Bi2_yPbySr2Ca2Cu3Olo+x

Dy: 123 DyBazCu307_ x

Eu : 123 EuBa 2 Cu 3 O7_ x

Gd : 123 GdBazCu 3 O7_ x

Ho: 123 HoBazCu307_ x

La(Sr) : 21 Laz_ySryCuO 4

Nd(Ce) : 21

Nd2_yCeyCuO4_ x

Sm: 123 SmBa2Cu3 O7_ x

T1 : 2212 T12Ba2CaCu208+ x

Tm : 123 TmBa 2Cu 3 O7_ x

Y: 123 YBa2 Cu3 O7_ x

BCS theory, the critical temperature, T c, is

related to the

Debye temperature, |

of the lattice and the electron-phonon interaction parameter (Poole

et al.,

1988),

Tc ~ 1.134| D exp (- ~). (1)

Anderson (1965) showed that the Debye temperature can be expressed as

(h)(3qNAP'] 1/3

OD-

Vm kBB ~,

4~tM ] ' (2)

where h is Planck's constant, k B is Boltzmann's constant, q is the number of atoms

per molecule, ]71 is Avogadro's number, p is the density, M is the molar mass,

and Vm is a mean sound velocity computed as

Vm -- + ' (3)

where V s

and

V c are the shear

and longitudinal sound velocities, respectively. For

high-T c mateirals, V L ~ 2Vs; hence, V m ~ V s. Given that V s -

(G/p) 1/2,

where

G is the shear modulus, it follows approximately that

(h)( 3qN A

)1/3 (G)1/2

OD ~ kBB ~,47Zf~mol,] '

(4)

where f~mol is the molar volume;

hence, the

critical temperature should vary as

the

square root of the shear modulus,

Tc oc G 1/2.

(5)

572

Chapter

13:

Hechanical Properties

a. Terms and Basic Relations

When a material is subjected to a stress, a (force per unit area), the material

undergoes a deformation. For a given direction in a solid material, the engineer-

ing tensile strain, e, of the material is defined (Timoshenko, 1940) as the relative

change of a linear dimension in that direction:

L,~ - L o

e - L0 (6)

where Lo is the length of a linear dimension of a material segment after the

application of the stress, and L 0 is the length of that material segment before

deformation.

The elastic deformation of a material refers to a limited domain of stress

and strain in which the material behaves reversibly. In this domain, Fig. 13.1, the

elastic response of the material is described by Hooke's law, that is, the tensile

stress, a, is proportional to the strain:

o- = Ee. (7)

The proportionality constant, E, is known as the elastic modulus (also known as

Young's modulus, modulus of elasticity, coefficient of elasticity, modulus of

Fig. 13.1.

,~ ~ Fracture

Schematic illustration of the relation between stress a and strain e for a brittle material. Such

materials are said to be linearly elastic up to the point of failure (yield point) where the material

fractures.

B. Elastic Properties 573

extensibility, stretch modulus, tensile modulus, and numerous other less common

terms).

In general, when a tensile or normal stress is applied, the deformation of the

material involves a strain that is transverse to the direction of the applied stress

(called variously the transverse or lateral strain), as well as the strain in the

direction that is parallel to the applied stress (called variously the tensile, axial, or

longitudinal strain). For example, the diameter of a cylindrical rod tends to

decrease as the rod is stretched along its length, as illustrated in Fig. 13.2. The

negative of the ratio of the lateral strain to the axial strain is called Poisson's ratio,

V -- -- elatera-------~l. (8)

~axial

Taking Poisson's ratio into account, the strain,

e x,

in the x direction depends on

the stress components in the x, y, and z directions. Hence, the generalized Hooke's

law for ex becomes

~x -- 2 [O'x -- 1;(O'y + O'z)], (9)

and similarly for

and

with appropriate permutations of the indices.

For a homogeneous, isotropic material, E and v completely describe its

elastic properties. However, it is sometimes convenient to consider another elastic

property called the bulk modulus, B = -V(dP/dV), where V is the volume and

P is an applied hydrostatic pressure. For this case, the volumetric strain is

dV/V =e x

-+-g,y-'~-g'z

to first order in the strain, and the stresses are

ax = ay = az = -P.

Hence,

B -- . (10)

3(1 - 2v)

Fig.

13.2.

L, Lo ;

', ,

' ' ,L

L ,,,I

An illustration of tensile strain with lateral contraction.

574

Chapter

13:

Mechanical Properties

Comparable quantities can be defined for conditions of shear stress, z, which

results in a shear deformation, 7. Hooke's law for shear conditions is

"v -- GT, (11)

where G is called the shear modulus (also called the rigidity modulus). For a

homogeneous, isotropic material, G is related to E and v according to

G - 2(1 + v)" (12)

Within the realm of linear elasticity, E > 0 and G > 0. The definition of the bulk

modulus also requires B > 0. Hence, from Eqs. (10) and (12), the bounds of

Poisson's ratio are readily seen to be -1 _< v < 1/2. In practice, 0 < v < 1/2 for

most materials.

In some cases, it is more convenient to express E and v in terms of B and G:

9BG

E - ~ (13)

3B+G

3B-2G

v -- ~. (14)

6B + 2G

b. Measurement Methods

Several different types of measurement methods are used to determine the elastic

properties of solid materials: direct stress-strain measurements in tensile tests

(ASTM D3379), static pressure-volume measurements using crystallographic

techniques, ultrasonic techniques (Krautkr~imer and Krautkr~imer, 1990), and

resonance measurements (ASTM C 1198). The tensile tests use strain gauges,

extensometers, and calibrated loads to determine the elastic properties via direct

applications of Hooke's law. The crystallographic methods use X-ray or neutron

diffraction to determine the change in the lattice parameters as a function of an

applied stress and then calculate the bulk modulus from the pressure and volume

relation, often using a model isothermal equation of state (Munro

et aL,

1984).

The most widely used methods involve ultrasonic techniques that can provide a

complete determination of the elastic properties in nondestructive measurements.

Ultrasonic techniques commonly measure the longitudinal and shear sound

velocities, V L and V s, from which the elastic properties may be calculated if the

mass density, p, is also known. The quantities most simply related to the sound

velocities are the shear modulus and the bulk modulus:

(15)

B-p[v 4

- gV~]. (16)

Given G and B, Eqs. (13) and (14) can be used to evaluate E and v.

Elastic Properties

575

c. Corrections for Porosity

Porosity in the range of 15 to 40% is quite common for ceramic superconductors.

A considerable effort has been made in the literature to normalize the measured

results to the limiting condition of zero porosity. Both empirical models and

computations based on elasticity theory have been used. The empirical methods

generally are extrapolations based on trend analyses and least squares fitting

techniques. Early investigations suggested that the elastic and shear moduli of

ceramics decreased exponentially with increasing porosity (Spriggs and Brissette,

1962),

Y = Xo e-axp ,

(17)

where X is either E or G; X 0 is the modulus at zero porosity; p = 1

P/Pmax

the volume fraction of porosity for a material whose density, p, is less than the

theoretical maximum density, Pmax; and

is an empirical fitting parameter.

When comparable trends are found for the elastic and shear moduli, it follows

from Eqs. (12) and (17) that Poisson's ratio should vary approximately linearly

with the porosity,

v = v 0 - r/p; (18)

where r/~ (1 + Vo)(a E -aG).

Other studies have reported linear (Hasselman and Fulrath, 1964),

X = X o - bxp,

(19)

and power-law results (Ishani and Cohen, 1967; Wagh

et al.,

1993),

X -- Yo(1 _p2/3)

(20)

X = X0(1

-p)q,

(21)

where

and q are fitting parameters.

Applications of elasticity theory have generally assumed spherical voids or

inclusions dispersed in an otherwise homogeneous, isotropic medium. Differ-

ences among the various models have resulted from the specific approximations

that are invoked to obtain useful solutions. Mackenzie (1950) applied elasticity

theory to a spherical void surrounded by a spherical shell of matrix material that

was, in turn, surrounded by a self-consistent bulk medium representing the

macroscopic ensemble of matrix material and voids; estimates for the bulk

modulus and the shear modulus were found in the forms

1 1

= B0(1 _ p) + 4G0( 1 _ p) (22)

G o - G (3B 0 + 4G0)

Go = 5p (9Bo + 8Go), (23)

in which higher order terms in p were neglected.