FILE COPY

NBSIR 76-837 DO NOT REMOVE

CHARACTERIZATION OF

A SUPERCONDUCTING COIL COMPOSITE AND ITS COMPONENTS

W.F. Weston

V.D. Arp

J.G. Hust

RJ. Trapani

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1976

INTERIM REPORT

Prepared for:

Annapolis Laboratory

Naval Ship Research and Development Center

AnnapoJis, Maryland 21402

RECEIVE!

A.F. Clark

NBSIR 76-837

CHARACTERIZATION OF

A SUPERCONDUCTING COIL COMPOSITE AND ITS COMPONENTS

A.F. Clark

W.F. Weston

V.D. Arp

J.G. Hust

RJ. Trapani

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1976

INTERIM REPORT

Prepared for:

Annapolis Laboratory

Naval Ship Research and Development Center

Annapolis, M aryland 21402

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Act.ng Director

TABLE OF CONTENTS

Page

List of Tables v

List of Figures vi

Abstract , 1

1. Introduction 1

2. Mechanical Properties 2

2.1 Young's Moduli and Poisson's Ratio 6

2.2 Estimate of Ultimate Strengths 6

2.3 Shear Moduli 14

2.4 Compressibility of Glass Cloth 24

References 28

3. Thermal Properties . . . .

' 29

3.1 Thermal Conductivity of the Coil Composite 29

3.2 Thermal Expansion of the Superconducting Wire 37

3.3 Specific Heat of the Coil Composite and Fiberglass Epoxy 44

References 52

4. Elastic Properties by W. F. Weston 53

4.1 Introduction 53

4.2 Experimental Procedure 55

4.3 Results , 60

4.4 Discussion 74

4.5 Conclusions 79

References 80

5. Stress Analysis by V. D. Arp 82

5.1 Introduction 82

5.2 Consultation 82

5.3 Transversely Isotropic Material Parameters in a Cylindrical Winding .... 83

5.4 Solution for Stress Distributions in the Absence of Internal Stress .... 85

5.5 Internal Stress Due to a Uniform Temperature Change .... 86

5.6 Stresses Due to Winding Tension 88

5.7 Epoxy Impregnation of the Coil and Bobbin Removal ..... 99

5.8 Magnetic Forces 100

iii
Page

5.9 Summary of Numerical Results 105

5.10 Conclusions and Recommendations 118

5.11 Nomenclature 123

5.12 Computer Program 124

References 129

6. Discussion 130

6.1 Comparison with Predicted Values , 130

6.2 Preliminary Coil Values 131

6.3 The Effects of Anisotropy and Temperature 134

References 134
LIST OF TABLES

1-1. Characterization of superconducting coil composite.

1-2. Considerations of superconducting coil stress analysis for NSRDC.

2-1. Young's moduli and Poisson's ratios for superconducting coil materials.

2- 2. Shear moduli for superconducting coil composite.

3- 1. Thermal conductivity of longitudinal and transverse sections of superconducting coil

composite.

3-2. Thermal expansion of copper-niobium titanium superconducting wire.

3-3. Specific heat of NbTi/Cu-f iberglass eopxy coil composite in temperature range 4.0 to

27.2 (run No. 1).

3-4- Specific heat of NbTi/Cu-f iberglass epoxy coil composite in temperature range 4.5

to 22.2 K (run No. 2)

3-5. Specific heat of f iberglass-epoxy composite.

3-6. Low-temperature specific heat coefficients for f iberglass-epoxy composite.

3-7. Low-temperature specific heat of fiberglass-composite at specific temperatures.

4- 1. Dimensions and orientations of specimens.

4-2. Experiments performed on the superconducting coil composite.

4-3. Parameters in equation 4-7.

4-4. Elastic compliances of the superconducting coil composite at selected temperatures

(10

1U m

Z /N).

4-5. Elastic stiffnesses of the superconducting coil composite at selected temperatures

(10

10 N/m2 ).

4-6. Comparison of static and acoustic measurements of elastic constants.

6-1. Comparison of calculated and measured values of coil composite properties.

\

V
LIST OF FIGURES

1-1. Superconducting coil composite characterization program schematic.

2-1. Schematic of solenoid coil showing construction and coordinate axes.

2-2. Cross-section of the superconducting coil tested.

2-3. Young's modulus versus temperature for superconducting coil materials.

2-4. Poisson's ratio versus temperature for superconducting coil materials.

2-5. Failure of ultimate strength test specimens.

2-6. Schematic of shear modulus apparatus.

2-7. Schematic of shear displacement measurement system (optical lever arm).

2-8. Shear modulus specimens.

2-9. Shear modulus apparatus calibration data.

2-10. Static shear modulus of coil composite.

2-11. Glass cloth compressibility test apparatus.

2-12. Compressibility of the fiberglass cloth as a function of applied stress.

3-1. Coil composite thermal conductivity specimens.

3-2. Thermal conductivity fixed temperature point apparatus.

3-3. Thermal conductivity of coil composite materials. (Interpolated curves are based on

the temperature dependence of similar materials)

.

3-4. Dilatometer schematic.

3-5. Superconducting wire thermal expansion specimen.

3-6. Comparison of measured and predicted dilatometer calibration curves.

3-7. The thermal contraction of Cu-NbTi wire compared to niobium, titanium, copper, and

the longitudinal coil composite.

3-8. Specific heat of the superconducting coil composite and the fiberglass epoxy at

low temperatures.

3-9. Specific heat of the coil composite at the critical temperature.

4- 1. Resonant piezoelectric oscillator apparatus.

4-2. Young's modulus of the superconducting coil composite, 6 = 90°.

4-3. Young's modulus of the superconducting coil composite, 0=0°.

4-4. Shear modulus of the superconducting coil composite, 0 = 0°.

4-5. Shear modulus of the superconducting coil composite, 0 = 90°.

4-6. Young's modulus of the superconducting coil composite, 0 = 60°.

4-7. Shear modulus of the superconducting coil composite, 0 = 60°.

4-8. Shear modulus of the superconducting coil composite.

vi
4-9. Young's modulus of epoxy-f iberglass , 6=0°.

4-10. Shear modulus of epoxy-f iberglass , 6=0°.

5-1. Radial and circumferential stresses developed by cooldown of a representative magnet

coil from 338 K to 4 K, calculated from equations 5-25 and 5-26

5-2. Circumferential stress developed at the inner radius due to a uniform temperature

change, normalized by the parameter LAaAT (eq. 5-26), as a function of coil thickness, for several values of the mechanical anisotropy parameter K (eq. 5-17).

5-3. Circumferential stress developed at the outer radius due to a uniform temperature

change, normalized by the parameter LAaAT (eq. 5-26), as a function of coil thickness, for several values of the mechanical anisotropy parameter K (eq. 5-17)

5-4. Maximum radial stress due to a uniform temperature change, normalized by the parameter LAaAT (eq. 5-26), as a function of the coil thickness, for several values of

the mechanical anisotropy parameter K (eq. 5-17).

5-5. Radial and circumferential stress calculated from equations 5-42, 43 for a particular coil wound at constant strain of 0.001, equivalent in this example to a tensile

2 stress of 7.6 kN/cm , shown as the dashed line. Because of the very low value of

E^, strain relaxation of up to 55% occurs within the winding.

5-6. Raidal and circumferential stresses for a particular coil as wound, and after epoxy

impregnation and removal of the bobbin on which the coil was wound.

5-7. Radial and circumferential stresses for a particular coil when it generates a field

of 3 Tesla at the coil center. As discussed in the text, these results are quite

insensitive to reasonable variations in coil material parameters.

5-8. Circumferential stresses in a particular coil after epoxy impregnation and bobbin

removal, for four different programmed winding tensions, equivalent to the four

dashed curves

5-9. Radial stresses after epoxy impregnation and bobbin removal, for the four different

programmed winding tensions illustrated in Figure 5-8. Note the change from radial

tension in curve A to radial compression in the other three curves.

5-10. Circumferential stress in a particular coil after epoxy impregnation and bobbin

removal, for four different values of the radial modulus T (eq. 5-8) prior to epoxy

impregnation.

vii
5-11. The circumferential stress for a particular coil and programmed winding strain calculated for four different radial moduli before epoxy impregnation, E^, using the

linear theory in this report (solid lines), and using iterative techniques and a

non-linear modulus E^ (dashed line, data calculated by W. Young).

5-12. The radial stress for a particular coil and programmed winding strain calculated for

four different radial moduli before epoxy impregnation E^, using the linear theory

in this report (solid lines) , and using iterative techniques and non-linear modulus

E (dashed line, data calculated by W. Young).

5-13. The change in circumferential stress for a particular coil as the bobbin is removed,

calculated for four different radial moduli before epoxy impregnation, E , using the

T linear theory in this report (solid lines) and using iterative techniques and a

non-linear modulus E^ (dashed line, data calculated by W. Young).

5-14. The circumferential stress for a particular coil after epoxy impregnation and bobbin

removal, calculated for three different values of a (= i.d. of the bobbin t o.d. of

the bobbin)

5-15. The circumferential stress after epoxy impregnation and bobbin removal, for a given

set of material properties, but for four different coil thickness specified by the

ratio (outside coil radius t inside coil radius).

5-16. Cumulative circumferential stresses for a particular coil, calculated for a winding

tension which decreases linearly from inside to outside radius (solid lines) and for

a winding tension which increases parabolically from the minimum at the inner radius

to a maximum in mid-winding and then decreases slightly (dashed lines). The maximum

and minimum winding strains are the same in both cases. The dashed line (4) exhibits

a significant reduction in maximum operating stress compared to the solid line (4)

5-17. Cumulative radial stresses for a particular coil, calculated for a winding tension

which decreases linearly from inside to outside radius (solid lines) and for a winding tension which increases parabolically from the minimum at the inner radius to a

maximum in mid-winding and then decreases slightly (dashed lines). The maximum and

minimum winding strains are the same in both cases. The dashed line (4) exhibits a

significant reduction in maximum operating stress compared to the solid line (4)

viii
CHARACTERIZATION OF A SUPERCONDUCTING COIL COMPOSITE

AND ITS COMPONENTS

A. F. Clark, W. F. Weston, V. D. Arp, J. G. Hust, and R. J. Trapani

ABSTRACT

The physical properties of a superconducting coil composite and

its components are studied in order to accurately predict the coil behavior under a variety of operating conditions. This second interim

report includes data on the Young's moduli, the Poisson's ratios, the

shear moduli, and the thermal conductivity of the coil composite as

well as the compressive modulus of the fiberglass cloth and the thermal

expansion of the superconducting wire. In addition, the development

of an acoustic technique for the determination of the elastic moduli

of the coil composite and the stress analysis of transversely isotropic

magnet coils are reported.

Key Words: Fiberglass cloth, mechanical properties, superconducting

coil composite, superconducting wire, stress analysis, thermal properties.

1 . INTRODUCTION

The heart and essential uniqueness of any superconducting motor or generator is the

superconducting coil used as a magnetic field source. This coil is subject to a wide range

of thermal, mechanical, and magnetic environments and proper understanding of the coil behavior in these environments is necessary to assure not only desired operating behavior

but long-term reliability as well. The objective of this research effort has been to

determine the physical properties of a superconducting coil composite and its components

in order to accurately predict the coil behavior under a variety of conditions.* This second

interim report supplements the first"'" and reports the Young's moduli, the Poisson's ratios,

the shear moduli, the thermal conductivity and specific heat of the coil composite as well as the

compressive modulus of the fiberglass cloth and the thermal expansion of the superconducting

* The particular coil composite studied is that developed by the Naval Ship Research and

Development Center (NSRDC), Annapolis, MD for use in one of their laboratory machines.
wire. In addition, the development of an acoustic technique by one of the authors (W.F.W.)

for the determination of the elastic moduli of the coil composite is reported in Section 4,

and the stress analysis by another (V.D.A.) of transversely isotropic magnet coils is reported in Section 5.

A pictorial representation of the general program is shown in Fig. 1-1. The measured properties of the individual components can be related by micromechanical analysis

to the measured properties of the coil composite as a whole. This not only provides the

composite properties for design application and a check on the micromechanical analysis,

but also permits selected modification of the composite by an educated choice of its

components. With macromechanics the anisotropic coil properties can be used to develop a

stress analysis to predict coil stresses due to the variety of initial and operating conditions such as winding, potting, and cooldown of the coil and magnetic and thermal excursions of the system. Instrumentation of an actual coil will provide a check on the applicability of the stress analysis method and permit selection of coil properties necessary for

desirable coil behavior.

The properties measurement program is outlined in Table 1-1 which shows what has been

measured and reported in the first report^, what is reported here, and what is still to be

measured. No symbol indicated no intention of measurement either because it's inapplicable

or not needed at this time. The Young's moduli and Poisson's ratios in the first report

were not complete so they are reported again in this report. Shown in Table 1-2 are the

conditions considered in the development of the stress analysis.

To briefly summarize the first report, first the macroscopic description of an anisotropic composite material was derived and then related to individual constituents by a

variety of micromechanical predictive schemes. After describing the mechanical properties

and thermal expansion measurements, the results are compared to the values predicted and

preliminary coil values are given. For this report the new experimental results are presented and are also compared to predicted values. The effects of temperature and the

anisotropy are discussed.

2. MECHANICAL PROPERTIES

Several more mechanical properties of the coil composite and its components are reported here. They include the shear moduli of the composite, the compressive moduli of the

fiberglass cloth and the Young's modulus along the axial direction E^» and tne Poisson's

ratio associated with this axial measurement , v.^. A convention of labeling was established

2
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Table 1-2 Considerations of Superconducting Coil Stress Analysis for NSRDC

I. Macromechanical Analys is mandrel rigidity

winding tension

anisotropic contraction

anisotropic moduli:

w/ linear glass-cloth

w/ highly non-linear cloth

(requires analytical technique

applicable to future windings)

magnetic field 2 D calculations

3 D calculations

external constraints

II. Micromechanical Analysis

yielding in epoxy around each wire

thermal cycling or fatigue

III. Transient Thermal Stresses

cool down

magnet quench

5
in the first report with (1) being the longitudinal axis along the wires of the solenoid,

(2) perpendicular to the wires in the radial direction of the solenoid, and (3) perpendicular

to the wires in the axial direction of the solenoid. This is shown schematically in

Figure 2-1. That the (2) and (3) directions are not identical can be seen in the photographic cross section of Figure 2-2 which shows the gap created between the layers of the

nearly rectangular wires created by the wrap of fiberglass cloth between each layer.

Thus in the photograph the (3) direction is along this layer of cloth, (2) perpendicular

to it, and (1) perpendicular to the page. The measured volume fraction of the wire taken

from enlargements of similar micrographs is 0.73.''"

2.1 Young's Moduli and Poisson's Ratios

The specimen preparation and experimental arrangement are described in detail in the

preceding report.^ Briefly, the specimens were machined from actual coil material, strain

gauges applied, and measurements performed at the three temperatures, 293, 76 and 4 K, in

a normal low temperature tensile arrangement

The only difference for the tests reported here was that the longitudinal and transverse gauges were both monitored at the same time using two bridge networks and an X-Y-Y

recorder. This reduced the number of loadings required for each specimen.

Because of the relatively weak fracture properties of the epoxy bond between wires,

2 the load for the axial specimen was limited to ^ 1.0 MN/mm O 1000 psi) . The resultant

strains were quite small (especially for the Poisson gauge output) and the noise limited

resolution yields an estimated uncertainty of about 5% for the Young's modulus and 12-15%

for the Poisson's ratio. The data for all of the Young's moduli and Poisson's ratios are

tabulated in Table 2-1 and shown in Figures 2-3 and 2-4.

The behavior of the axial Young's modulus, is, as expected, very similar to that

of the radial specimen, ^22' It: is P r°bably due almost entirely to the epoxy. The

Poisson's ratio, ^3' ^as a value between the longitudinal v,^, which would be expected

to behave more like the Cu-NbTi wire, and the radial V^^, which behaves similar to the

fiberglass epoxy. This is as might be expected. The slight increase with decreasing temperature is similar to the Young's moduli.

2.2 Estimate of Ultimate Strengths

Specimens for all three principal directions were pulled to failure at 4 K in an

attempt to estimate the ultimate strengths. The radial specimen failed at a very low load

' 6
2 a RADIAL

7

Table 2-1 Young's Moduli and Pois son's Ratios for

Superconducting Coil Materials

4 2 6 YOUNG'S MODULI 10 MN/m (10 psi)

Temperature, Kelvin

Specimen 293 JJ ±

X\.d. Q ld.1 #1 2. 82 (4. 10) 3. 82 (5.53) 4. 15 (6.01)

Radial #2 (E22) 2. 68 (3. 89) 3. 75 (5.43) 4. 08 (5.9D

Long

41 (En) 7. 72 (11. 21) 8. 26 (11. 97) 8. 74 (12. 67)

Long

#3 (En) 7. 39 (10. 72) 7. 89 (11.44) 8. 31 (12. 05)

F. E. #1 ( E ll) 2. 82 (4. 10) 3. 33 (4. 83) 3. 56 (5. 16)

Transv. #1 (E33) 2. 28 (3. 3) 3. 10 (4. 5) 3. 10 (4.5)

POISSON'S RATIOS

Temperature, Kelvin

Specimen 293 77 4

Radial (v 21 ) 0. 146 0. 202 0. 196

Radial #2 (v 21 ) 0. 199 0. 248 0. 256

Long

#3 (V13) 0. 333 0. 352 0. 352

F. E. #1 (v 13 ) 0. 149 0 209 0. 221

T rans v. #1 N32) 0. 278 0 300 0. 321

9
15x10'

c

- 10

CO

0

0

Longitudinal No. I ,E

Longitudinal No. 3, E

Radial No. I, E.

Radial No.2,E,

Fiberglass Epoxy No. I, E

J I I L

Axial , E

J L

33

10x10

CM

CO

5 E

O

CO

Id

IDO>-

100 200 300

TEMPERATURE,

Figure 2-3. Young's modulus versus temperature for superconducting coil materials.

10
' 0 100 200 300

TEMPERATURE,

Figure 2-4. Poisson's ratio versus temperature for superconducting coil materials.

11
at one of the epoxy joints of its fabrication^ and is therefore meaningless. Both the

longitudinal and axial specimens failed in the grips, but with measurements of crosssectional areas and estimates of stress concentration factors rough approximations of

ultimate strengths can be calculated. The failure modes are shown in Figure 2-5, from

which it can be seen that the axial specimen failed in tension and the longitudinal In

shear.

One method of estimating ultimate strength in the axial specimen would be to consider

the net area at the failure line, using

\ = A

G

" A

c (2 -1}

where

= net area at failure line

A„ = gross cross-sectional area at failure line

A^ = area of circular hole at failure line

and by using

a = P/^ (2-2)

where

a = apparent stress

P = measured output of load cell

and also

ay = ka (2-3)

where

= estimated ultimate strength

k = stress concentration factor.

2 A value of 3 for the stress concentration factor is used. The approximate ultimate

strength for the axial specimen at 4 K, as estimated without a stress concentration factor,

2 is 57 MN/m (8300 psi) . Calculated with a stress concentration factor of 3, an estimate fo

ultimate strength is 171 MN/m2 (24,800 psi).

The failure in the longitudinal specimen was due to shear between wires of the composite perpendicular to the glass cloth. An estimate of ultimate shear strength can be

made using the equation:

a_ = P/A (2-4)

>• b

12
Figure 2-5. Failure of ultimate strength test specimens.

13
where

a = ultimate shear strength

P = ultimate load

A = area of shear failure.

2 Using this equation, estimated ultimate shear strength is about 107 MN/m (15,500 psi) at

4 K. No stress concentration factor has been used. These estimates should be considered

just that and nothing more.

2.3 Shear Moduli

The shear moduli were measured using a version of the apparatus developed by Mikesell

3 and McClintock. The apparatus consists of two concentric stainless steel cylinders in

which the specimen is secured and is schematically shown in Figure 2-6. The inner cylinder

moves in a teflon bearing attached to a brass plate, while the outer cylinder is securely

attached to the plate. The plate rests on a glass dewar with the cylinders suspended in

the cryogen. The torque is applied to the specimen by slotted weights on a pan which, via

a pulley and butterfly system, is transferred to the inner cylinder and specimen. The

torsion of the specimen is measured by observing the displacement of an optical image of

a laser beam formed on a distant translucent scale from a mirror attached to the top of the

inner cylinder.

Modifications to the original apparatus included a strip-silvered outer glass dewar

which allowed visual monitoring of liquid levels and also provided a liquid nitrogen shield.

The original optical lever arm was modified to allow the use of a laser beam image. A

low power helium-neon laser was used. Alignment of the laser optical arm is accomplished

by adjusting the laser to shine on the mirror and then adjusting the mirror so the image

falls on the translucent scale (see Figure 2-7) . Due to the various sizes of the specimens

tested, in some cases a heavier set of weights was used. These heavier weights caused a

slight torsion in the inner cylinder itself, however the effect was calculated to be

negligible.

Two specimen geometries were tested. The first type was made by winding layers of

wire and cloth around a flat plate. One linear section was then used. This specimen

configuration is referred to as "longitudinal". The second type of specimen was manufactured by winding the superconducting wire around a cylindrical mandrel, with a layer

of fiberglass cloth applied between each winding of wire. This type of specimen is basically

14
15
Translucent

Scale

Less"

note:

Beam, Front Surface of Mirror, and

Center of Rotation should be Coincident,

Figure 2-7. Schematic of shear displacement measurement system (optical lever arm)

16
a very small radius solenoid and is referred to as "cylindrical". After the specimens

were wound, they were placed in a potting mold and impregnated with the epoxy. The entire

systems were then cured for eight hours at 150°F.

Materials used for calibration specimens included aluminum 1100 and 17-4PH stainless

steel. These specimens were in the cylindrical rod specimen configuration as used in

reference 3. All the specimens are pictured in Figure 2-8.

The longitudinal specimen and also the calibration specimens were mounted using the

method in 3. The cylindrical specimen required special adaptors as shown in Figure 2-8.

The ends of the specimen were notched to fit over a matching pin when inserted into the

adaptor. Epoxy was then poured into the dished portion and the specimen was cured. This

provided a positive mechanical bond which withstood many low-temperature cycles.

The convention for the notation of the shear moduli is that G.. is for shear with the

force in the direction in the j*" plane (the plane perpendicular to j ) . This convention

is also shown in Figure 2-8.

Room temperature runs began with mounting the specimen and then assembling the apparatus.

After checking the distance from the mirror to the scale, the optical lever arm was aligned

to the translucent scale. Pairs of weights are then successively placed on the weight pans,

and the angular displacements of the specimen are noted. Five pairs of weights were

applied. To minimize the effects of hysteresis of the specimens and apparatus, the load

is increased to a maximum and then back to a minimum (load-unload cycle) , with angular

displacements measured for both.

The procedure for low-temperature runs is similar to the room temperature procedure

except the specimen and apparatus are first cooled with liquid nitrogen. For 4 K measurements the inner dewar is then purged of liquid nitrogen and filled with liquid helium.

After the transfer is completed, the mirror and teflon bearing are ventilated with room

temperature air to prevent frost and subsequent binding between the inner cylinder and

teflon bearing. The specimen is then put through several load-unload cycles and the

angular displacements are measured.

The apparatus was calibrated by measuring the shear moduli of two calibration materials

(aluminum 1100 and stainless steel 17-4PH) and comparing the measured values to known values

determined previously^ and by an acoustic technique.^ Figure 2-9 sho-ws the known and measured values plotted versus temperature. It is noted that there is a 3% and 4.5% discrepancy

17
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19
at low temperatures. This inconsistency is still unresolved and the data are presented for

the coil material uncorrected.

Experimental data were reduced by first plotting angular displacement vs load and

then fitting a line to the data by using a linear least squares program. The first and

last data points of each run were not included in the fit so as to minimize the effect of

hysteresis. After the slope of the line was determined, the shear modulus was computed

from:

G _ AS(2_5)

J L where

w = the weight on specimen

R = the radius arm

I = the effective length of specimen

L = the optical lever arm length

J = the polar moment of inertia of the specimen

AS = the deflection of the optical image.

This equation is derived from the basic equation for torsion in a shaft:

e = || (2-6)

where, 6 = the amount of twist and T is the applied torque.

In equation (2-6), it can be noted that the specimen length and also the diameter of

the specimen will change with the test-temperature. The values were corrected by using a

thermal expansion coefficient for the material, which was found by Fowlkes et al."*" This

correction varied for the different specimen geometries. This correction was significant

because the polar moment of inertia for the longitudinal specimen is:

J = ^

(2-7)

where,

d = the specimen diameter

and for the cylindrical specimen:

TT(d4 - d

4

) J = 1_ — (2-8)

20
where,

d = the outer diameter of the specimen

od^ = the inner diameter of the specimen.

Since the specimen diameter for both cases enters into the calculation to the fourth

power, the thermal expansion correction must be carefully applied.

As reported in 3, the most significant errors arise from d and AS. Due to irregulari-

_3 ties on the surface of the specimens, the diameter was measureable to 2 x 10 inches,

4 which gives an uncertainty in d of about 1%. Using a 3 meter optical lever arm, the

maximum standard deviation for a set of five measurements (at each temperature) on the coil

material was about 7%.

Average values and the data found are presented in Figure 2-10 and Table 2-2. A size

effect was noted on further testing of the cylindrical specimen. A layer of superconducting

wire was machined off and the specimen was tested at room-temperature and in liquid nitrogen.

An increase in shear modulus of about 20% and 15% were noted at 293 K and 77 K, respectively.

This increase can be explained by the fact that a greater volume fraction of wire was

present in the machined-down specimen, thus causing an increase in the shear modulus.

Shear modulus of the coil material can be predicted by the solution of the mixture

laws of composite materials, which are derived from simplified microstructural models.

In the previous report Fowlkes et al.^" predicted shear modulus values for the cylindrical

specimen at room temperature ranging from .346 x 10^ MN/m^ (.502 x 10^ psi) to .916 x 10^

2 6 MN/m (1.33 x 10 psi), depending on the microstructural model chosen. It is noted that

the predicted values bracket the experimental value for this specimen configuration.

The acoustic measurements of the elastic constants were made at about the same time

as these static tests, and initial results indicated that the two techniques were yielding

nearly identical results (See Table 4-6) . In addition, it was determined that for a

transversely isotropic material should be approximately equal to G^^ and The

difference between the two static tests (^ 20%) is therefore an indication of the effect

of curvature and/or the averaging effect in the longitudinal specimen of and G^^.

Because of these results the static tests were considered a sufficient check on the acoustic

technique and were abandoned in favor of the more rapid and accurate acoustic measurements

reported in Section 4. The comparisons are more accurately described in Section 4.4.

21
22
Table 2-2 Shear Moduli for Superconducting Coil Composite

10

4 MN/m2 (10

6 psi)

Temperature, Kelvin

293 77 4

Longitudinal (Ave. ) 0. 82 (1. 19) 1. 27 (1 84) 1. 33 (1. 93)

G

21,31

(Data) 0. 82 (1. 19) 1. 28 (1 86) 1. 26 (1 83)

0. 78 (1. 13) 1. 21 (1 76) 1. 38 (2. 00)

0. 83 (1. 21) 1. 30 (1 88) 1 34 (1. 95)

0. 84 (1. 22) 1. 26 (1 83)

0. 82 (1. 19) 1. 28 (1 86)

1. 27 (1 84)

Cylindrical (Ave. ) 0. 67 (0. 975) 1. 05 (1 52) 1. 11 (1. 61)

G

13

(Data) 0. 75 (1- 09) 1. 05 (1 52) 1. 11 (1. 61)

0. 68 (0. 988) 1. 06 (1 54) 1 07 (1. 55)

0. 68 (0. 979) 1. 00 (1 45) 1. 15 (1. 67)

0. 58 (0. 844) 1. 12 (1. 62)

1. 01 (1 46)

23
2.4 Compressibility of Glass Cloth

The compressibility of the glass cloth which separates succeeding magnet layers is an

important factor determining internal magnet stresses. A brief literature survey disclosed

2 no information on this topic for stress levels up to tens of Mega-Newtons per m , as are

relevant for typical coils. Consequently, we assembled and performed the following compressive modulus tests on the glass cloth used by the sponsors and supplied by them.

The coil winding is done at room temperature so only room temperature measurements

were made. A test jig shown in Figure 2-11 was assembled for use in our tensile testing

machine. Nine layers of glass cloth, seperated one from another by flat brass washers,

cumulatively provided enough compression that its magnitude could be measured with reasonable accuracy using clip-on strain gauges which are standard in our laboratory. The brass

washers were carefully machined so that the maximum variation in thickness around any given

washer was no more than about 0.0010 cm, so as to insure a reasonably uniform loading of the

0.010 cm thick (nominal) glass cloth. We wereaided in this respect by the extremely low

compressive modulus of the cloth at low pressures, such that an initial loading of no more

5 2 than about 10 N/m is sufficient to cause a compression of 0.001 cm per layer of cloth.

Layers of gasket rubber, totalling about 0.5 cm thick, are placed on either end of the

test jig (outside of the gauge length) to take up any lack of parallelism which may exist

in the elements which apply the compressive loads.

2 Two sets of washers were used; a large washer, of area 23.67 cm , for obtaining good

resolution at lower stress levels with minimized effects due to any residual force mis2 alignment, and a small washer, of area 4.434 cm , for obtaining higher compressive

pressures within the constraints imposed by the tensile testing equipment. Where the

respective data with these two washer sizes overlapped, the derived values of da/de(o)

agreed within a factor of two, which is fair considering the large scatter and hysteresis

which ware observed.

Results are plotted in Figure 2-12. Because of the very large hysteresis and nonlinearity in the curves of e(c0, the logarithmic plots are necessary in order to bring the

results into a useable form. Considerable scatter is also seen, especially when the

testing machine is reversed under high load, so that a begins to decrease steadily with

time. As can be seen, results can be expressed approximately by

24
25
Figure 2-12. Compressibility of the fiberglass cloth as a function of applied stress.

26
— = 22a , for increasing loads (2-9)

—da

= 50a , for decreasing loads. (2-10)

These expressions cannot be valid as a -* 0, since an infinite strain would be predicted.

To compute the cumulative strain, assume that, for increasing stress

4^ = 22a , a < a de o o

Ha (2 -n> ^ = 22a , a > a de o

where a is a verv small stress below which elastic behavior does occur. Presumably it is

o

'

near or below the lowest stress plotted in Figure 2-12. Then, for z > c , integration

gives

l + log(a/o ) E = 22 ~ • (2 " 12)

2 The following table gives the total strain at an applied stress of 10 MN/m , for several

assumed values of a . o

a (10

4 N/m2

) E(0 = 10 MN/m2) o

1 0.359

3 0.310

10 0.255

20 0.223

2 Thus, it is reasonable to estimate that at an applied load of 10 MN/m (typical of that

which may be experienced during magnet winding) the nominal 0.004" thick cloth would

compress by 0.001" - 0.0015".

2 Inspection of the washers after a test in which the compressive load reached 50 MN/m

revealed a clear imprint of the cloth in the form of a square array of small dots. Evidently

the localized stress on the brass where the fibers crossed each other was high enough to

o cause slight yielding of the brass, about 100-200 MN/m . On the other hand, microscopic inspection of the glass cloth revealed no evident damage to it.

27
These tests have yielded useful data for the stress analysis of the sponsor's model

coil done in Section 5 of this report. More tests should be run, on different cloth materials, in order to find whether this non-linear behavior is typical of cloth materials in general

REFERENCES

1. Fowlkes, C. W. , Angerhofer, P. E. , Newton, R. N. , and Clark, A. F. , "Characterization

of a Superconducting Coil Composite," NBSIR 73-349, National Bureau of Standards,

Boulder, CO (December 1973).

2. Griffel, W. , Handbook of Formulas for Stress and Strain , Frederick Ungar Publishing

Co., New York (1966), p. 248.

3. Mikesell, R. P. and McClintock, R. M. , "A Method of Measuring Shear Modulus From

-424° to 70° F

, " Advances in Cryogenic Engineering , Vol. 7, K-ll, p. 509-513,

Plenum Press, NY (1961).

4. Cryogenic Materials Data Handbook, Air Force Materials Laboratory, ML-TDR-64-280

,

August 1964.

5. Weston, W. F. , Naimon, E. R. , and Ledbetter, H. M. , Unpublished Results, NBS,

Cryogenics Division, Boulder, CO.

28
3. THERMAL PROPERTIES

The thermal conductivity of the composite coil material and the thermal expansion of

one of its constituents, the Cu-NbTi superconducting wire, are reported in this section.

The thermal conductivity is highly anisotropic and significantly lower, particularly in the

1* transverse direction, than predicted in the first report. This anisotropy contributes

significantly to coil stresses as will be seen in Section 5. More accurate predictions

here are still somewhat high. The thermal expansion of the wire behaves more or less as

expected. Several thermal properties have yet to be measured as indicated in Table 1-1.

The most immediately useful, specific heat, is partially reported here.

3.1 Thermal Conductivity of the Coil Composite

Thermal conductivity measurements were performed on longitudinal and transverse

sections of the superconducting coil composite. The superconducting coil is composed of

a copper-stabilized niobium-titanium wire coil impregnated with epoxy. The cross-sectional

area fractions perpendicular to the wires are: OFHC Cu/NbTi = 1.8 and wire/epoxy = 3.

The longitudinal specimen was measured in direction 1, as illustrated in Figure 2-1 and

the transverse specimen was measured in direction 2 (radial) . No measurements were performed in the transverse-axial direction but estimates of the thermal conductivity in this

direction are calculated. The thermal conductivity specimens are shown in Figure 3-1.

The measurements were performed in a fixed-point apparatus designed to measure

technically important materials to engineering accuracy. This apparatus requires less

measurement time and smaller specimens, but is inherently less accurate, than the variabletemperature apparatus used for nearly a decade to measure thermal and electrical properties

of solids in the Cryogenics Division. The design accuracy of the new apparatus for midrange

conductors is + 10%. Preliminary tests with standard reference materials indicate that

the design accuracy has been achieved . This apparatus is used to measure the thermal conductivities of solids near the temperatures of fixed-point baths. Measurement time and

cost is about one-fourth that of the variable-temperature apparatus.

The fixed-point apparatus, shown schematically in Figure 3-2, is based on linear

heat-flow and is a modification of the Forbes cut-bar apparatus. The specimen heater

and the differential-temperature thermocouples are permanently mounted on the copper end

pieces, to which the specimen is attached. This eliminates the time-consuming task of

* The references in each chapter are treated separately.

29
2.00'^

(5.08 cm)

LONGITUDINAL

i

0.24" Did. (0.61cm)

7

0.25

(0.64 cm)

.50"° (1. 27 cm)

i

0.24" Did. (0.61cm)

(3.81 cm)

TRANSVERSE

-0.25"

(0.64 cm)

Figure 3-1. Coil composite thermal conductivity specimens.

30
MPERATURE

LINKS

Figure 3-2. Thermal conductivity fixed temperature point apparatus.

31
connecting these elements to each specimen, but also leads directly to the larger uncertainty of this apparatus. Specimen assembly involves only the relatively simple task

of soldering (woods alloy or indium is used) the specimen and vacuum can in place. The

differential-temperature observed includes not only the specimen temperature gradient but

also the temperature difference caused by heat flow through the specimen end sections, the

soldered thermal contact, and the copper end pieces. These latter temperature differences

are generally less than 15% of the total and can be estimated sufficiently accurately to

reduce total uncertainty to below + 10%.

The principle of operation is as follows. For each fixed-point bath, a series of

empty chamber gradients are established to determine = f (AT) of the specimen chamber.

This chamber, shown as a cylinder in Figure 3-2, is actually a bellows to reduce both strain

on the specimen and the values of Q . Plots of Q versus AT are obtained for each of the

o o baths used. Q

q is the electrical power input required to establish the temperature differential, AT. Since the empty chamber configuration remains permanently fixed, the above

procedure is performed only once.

A specimen is then installed and the power input to establish a temperature-differential,

AT, is determined. This AT is generally small compared to T, but if the specimen thermal

conductivity varies weakly or linearly with temperature, this restriction can be relaxed.

The heat-flow through the specimen, Q - Q = Q - f (AT) , is determined and thermal conductio vity of the specimen is calculated from

, _ [Q - f(AT)]& , . AAT ' (3 1}

where £ is the length of the specimen between the copper end pieces and A is the crosssectional area of the specimen. Corrections are applied to the measured AT to account for

the temperature gradients in the specimen ends and clamps. These estimates are based in

part on qualitative arguments of the heat-flow streamlines at the ends of the specimen.

Detailed heat flow calculations based on the solution of the general heat flow equation

are not performed, because the magnitude of the correction does not warrant such complexities.

In addition, these corrections are based on check measurements of NBS standard reference

materials of thermal conductivity. The correction for a conventional cylindrical, isotropic,

heterogeneous specimen is estimated as + 10%.

32
Principal sources of error are the end effects discussed above, thermocouple calibration errors, bath temperature drift, and heat loss by radiation and gaseous conduction.

For specimens with diameters which are small compared to the specimen chamber diameter

(3.2 cm), the effect of the radiation heat transfer is nearly equal in the Q and Q

q determinations and, therefore, is diminished in the calculation of A. Temperature drift

and radiation errors will be greatest for large diameter specimens. The optimum range of

conductivities measureable with this apparatus is about the same as with the variabletemperature apparatus, 0.001 to 50 W/cm K. Conductivities outside of this range can be

measured with reduced accuracy. Calibration of this apparatus has been performed by

measuring the thermal conductivity of NBS Standard Reference Materials, SRM's, 734 and

735. These SRM's were certified for thermal conductivity, to approximately 2% accuracy,

using the variable-temperature apparatus.

The results of the measurements on the longitudinal coil composite specimen are given

in Table 3-1 and Figure 3-3 . Corrections have been applied to the measured data to compensate for end effects in the apparatus. These corrections are based on measurements

on standard reference materials and decrease the measured values by 7% for this specimen.

The uncertainty in the corrected data is + 15%. The interpolated curve given for the

longitudinal specimen in Figure 3-3 is based on the typical temperature dependence of

OFHC copper.

It is interesting to compare the measured data to values ' calculated from the properties of the constituents of the coil. The longitudinal specimen can be separated

into three parallel components, Cu, NbTi, and Epoxy. The corresponding relative crosssectional areas of these are 0.47, 0.26, and 0.27, respectively. The effective conductivity of parallel specimens is obtained by summing the area-weighted conductivities of each

of the parts. The estimated values of the conductivities of Cu, NbTi, and epoxy, listed in

2 Table 3-1, were obtained from Childs, et al. The uncertainties of these values are high

because of the lack of adequate characterization data. OFHC copper is the wire matrix

material, but the fabrication procedure may have considerably altered its composition.

The low-temperature conductivity of pure metals is significantly dependent on this composition and may also be altered. The calculated values given in Table 3-1 differ from the

observed values by as much as 23%. These differences are considered reasonable in view

of the large uncertainties in defining the actual nature of the constituents.

33
Table 3-1 Thermal Conductivity of Longitudinal and Transverse

Sections of Superconducting Coil Composite

Specimen Temp. obs

ycalc

\A cu

X epoxy IsfbTi

(K) (W/mK) (W/mK) (W/mK) (W/mK) (W/mK)

Longitudinal 4. 90 169 136 290 0. 05 0. 1 78. 8 201 257 550 0. 92 4

196 165 187 400 1. 62 6 276 169 187 400 1. 82 10

T rans verse 7. 50 0. 24 0. 20 10* 0. 05

(radial) 78. 7 0. 81 0. 92 10* 0. 09

197 1. 13 1. 33 10* 0. 16

276 1.28 1. 33 10* 0. 18

Transverse 7. 50 0. 91

(axial

79. 0 3. 3 197 4. 3 277 4. 3

These values are for the transverse direction of Copper - NbTi composite

(see text).

34
1000

Longitudinal, I

Figure 3-3. Thermal conductivity of coil composite materials. (Interpolated curves

are based on the temperature dependence of similar materials.)

35
The observed thermal conductivities for the transverse (radial) specimen are also

given in Table 3-1 and Figure 3-3. These values required no end effect corrections and

the uncertainty is + 10%. Again the solid curve is based on the temperature dependence

of similar materials.

The transverse specimens can be viewed as two parallel segments. The first is composet

of the epoxy strip directed along the specimen axis. The second is composed of epoxy and

wire segments arranged in series with the heat flow direction. The dimensions of the two

parallel conductivity paths are different for the radial and axial transverse sections.

The epoxy strip for the radial transverse specimen has a relative cross-sectional area of

0.04. The epoxy-wire segmented strip is 0.96 of the total area. As before the conductivity of the two parallel paths is obtained by summing the component conductivities weighted

by their respective relative areas. However, first one must obtain the effective conductivity of the epoxy-wire segmented portion. The effective conductivity of segments arranged

in series can be obtained by summing the thermal resistances of the various segments.

Since the cross-sectional areas of all the segments are equal, it can be shown that

1 ll l2 x= a;

+ r2 + ••' (3 - 2)

where 2,. is the relative length of the i*" segment in the direction of heat flow. According to reference 1 the relative length of epoxy in this radial direction is 0.2 and that

of the wire is 0.8. It would be desireable to further divide the wire into the Cu and NbTi

components, but because of insufficient characterization data, this is not done in the

transverse case. Instead we just make a crude estimate of the effective conductivity of

the wire as 10 W/mK at all temperatures. The calculated values for the transverse (radial)

specimen are tabulated in Table 3-1.

Because of the reasonable agreement obtained it may be useful to estimate values for

the transverse-axial case by the same technique. The results are 0.91, 3.3, 4.3, and

4.3 W/mK at 7.5, 79, 197, and 277 K, respectively. These values are also shown in Figure

3-3.

The thermal conductivity is the most highly anisotropic of all the properties measured

on the coil composite. The longitudinal value being a factor of about 1000 times greater

than the poorest conducting transverse value in the radial direction. Even the axial and

36
radial thermal conductivities differ by a factor of 10. These differences are not unexpected

because of the high thermal conductivity of copper compared to epoxy and the obvious

anisotropy in conducting paths.

These differences will be of no consequence in steady state operation but will have a

significant effect on any transient behavior. The thermal contraction (and its anisotropies

as well) creates significant thermal stresses during cooldown and these can be compounded

by the anisotropy of heat removal. The final residual stress distribution due to contraction

from cooling to 4 K will not be affected but significant differential stresses may occur in

the coil depending upon the rate of heat removal. Also, propagation of a normal conducting

zone in a wire will tend to be along the wire instead of the desired transverse path to the

cooling surface. The buildup of local stresses due to this local heating and anisotropic

heat removal is difficult to assess but may be significant, particularly if repetitive.

These effects could be mitigated with an appropriate filler in the epoxy to increase the

conductivities transverse to the wires and reduce the anisotropy. Measurements of thermal

diffusivity could quantify the magnitude of these effects and permit an assessment of

their significance.

3 . 2 Thermal Expansion of the Superconducting Wire

The superconducting wire properties are essential components of the composite analysis

and thermal contraction of the wire during cooldown is one of the major sources of stress.

Toward this end the low-temperature thermal expansion of the Cu-NbTi wire was measured

using a quartz tube dilatometer from room temperature to 4 K utilizing the apparatus of

3 Clark. The dilatometer is schematically shown in Fig. 3-4. The experimental procedure

4 for the wires was the same as the procedure used by McConnell, et al for measuring fibers.

The specimen tested was the superconducting wire which is one of the constituent

materials of the NSRDC coil composite. The wire specimen was attached to a flanged copper

plate and had 3 individual wires soldered on a 1 cm diameter, spaced at 120° intervals. The

lower attachment for the wires is a solid brass disc, upon which the quartz rider sits.

The specimen length was 179.0 + 2.5 mm (7.04 + .10 inches). The specimen is shown in

Figure 3-5. The wires themselves are 180 filament, .056 x .072 cm (.022 x .028 in) drawn

and annealed wire with Cu/S.C. ratio of 1.8/1.0 and a twist of about 1/cm (2.4/in).

Since this type of thermal expansion measurement is relative, it must be compared to

a specimen of known material. The calibration was accomplished by measuring several times

37
Bell jar

Dial gauge

Brass cap

0" ring seal He gas (I psig)

Quartz rider Jacket Solder

Copper top cap

Samples

Thermocouples

Copper heat shield

Stainless steel tube Brass disk

Figure 3-4. Dilatometer schematic.

3d

B- 83581
Quartz Rider

Copper Support

and Heat Shield

Superconducting

Wire (3)

Figure 3-5. Superconducting wire thermal expansion specimen.

39
a specimen machined from OFHC copper. The difference between the average of the calibr

tion data and the standard reference data"* was due to the differential expansion between

the quartz tubes and the copper heat shield. This difference was applied to the specimen

measurement as a correction factor. The correction was less than 10% of the copper values

throughout the entire temperature range.

The observed strain in the sample is

AL0BS

/L

o

= (ALCu " AL

s + AL

c )/Lo

(3"3)

where AL /L is the sample contraction, AL„ /L the reference data contraction of the OFHC so Cu o copper heat shield, and AL^/L^ is the correction factor determined by measuring the sample

of OFHC copper. For this calibration measurement of copper

AL /L = AL /L (3-4) so Cu o

and

AL,/L = AL /L . (3-5)

OBS o co

This correction arises from the difference between the contraction of the quartz support

tube and the copper heat shield over the 2 cm segment between the end of the quartz rider

and the bottom of the support tube. From the known expansion of these materials, the

predicted correction curve and the measured correction curve can be compared as shown in

Fig. 3-6. The measured curve is an average of four calibration measurements of a high

purity copper sample.

This correction applied to the two measurements of the wire sample results in the

averaged data shown in Fig. 3-7 obtained from equation (3-3) rewritten as

AL /L = (AL - AL _ + AL )/L . (3-6) so Cu OBS c o

This averaged data was fit with a polynomial limited to T^ and the values of the expansion

and its derivative, the expansion coefficient, are shown in Table 3-2. at standard temperature intervals. Because of the accumulation of errors arising from the summation in

equation (3-6), the estimated uncertainty from experimental deviations is about + 10 x 10

from the expansion and + 1 x 10 ^ K or the expansion coefficient. Variations due to

inhomogeneities in the wire would not be expected to be this large.

40
O to

41
0

CO

02

CM

CO

05

CM

100 -

O

200 -

300

f/

Ti-33%Nb

Niobium

Titanium

Cu-NbTi Wire7

7

7

7

Coil Composite

(parallel to wires)

7

Wire

Predicted

Copper

1 00 20t) 300

TEMPERATURE, K

Figure 3-7. The thermal contraction of Cu-NbTi wire compared to niobium, titanium,

copper, and the longitudinal coil composite.

42
Table 3-2. Thermal Expansion of Copper-Niobium Titanium Superconducting Wire

-.5

L293L

T 10

6 dL -1

Temperature 10 —- —, K

(K) ^293 293

0 265a 0.0 a

10 265a 0.16a

20 265 0.60a

30 264 1.3

40 262 2.1

50 259 3.1

60 256 4.2

70 251 5.2

80 245 6.3

90 238 7.4

100 231 . 8.4

120 212 10.1

140 190 11.2

160 167 11.8

180 142 12.2

200 117 12.4

220 91 12.5

240 66 12.6

260 4.1 12.7

273 25 12.8

280 16 12.8

293 0 12.9a

300 - 9a 12.9a

Extrapolated

43
We can now compare the results with previous measurements of the coil composite and

with measurements of the constituents of the wire. There is no data available on the

thermal expansion of the Nb - 48% Ti which is used in the superconducting wire. Data are

6 6 7 5 available, however, for niobium , titanium , Ti - 33% Nb , and copper . The thermal

expansion curve of the coil composite in the direction parallel to the wires is shown also

in Fig. 3-7 and is just slightly larger than that of the wires. This would be expected

since the coil expansion should be dominated by the wires but slightly enlarged by the

weaker epoxy whose thermal expansion coefficient is significantly larger.

The thermal expansion curve of Ti - 33% Nb is also shown in Fig. 3-7 as well as the

total length change to 0 K for Ti and Nb. As a first approximation to the composite wire

value whose copper-to-superconductor ratio is 1.8:1, the total length change to 0 K would

be (1.8 AL„ + 1 AL^. ... )/2.8 where the AL„,. is an average of the data available. This

Cu li—Nb ii—Nb

predicted value of the total length change to 0 K is also shown in Fig. 3-7 (the rest of

the curve would be similar to the others) and the 2% difference from the measured value is

well within the experimental error. Since the thermal expansion of alloys appears to be

3 well behaved at low temperatures unless phase transformations are involved this is

probably a safe procedure to predict the thermal expansion for other wires from the copperto-superconductor ratio with an accuracy of the order of 10%.

3.3 Specific Heat of the Coil Composite and Fiberglass Epoxy

The specific heat at low temperatures was measured for both the superconducting coil

composite and the f iberglass-epoxy composite. Both specimens were portions of the same material used for the thermal expansion measurements."^ The coil composite is 27% fiberglass

epoxy by volume and the f iberglass-epoxy specimen was constituted to have the same proportions of fiberglass and epoxy as in the coil composite. The measurements were taken only to

about 25 K and either further measurements or extrapolations will be done to extend the

data to room temperature. The measurements reported here were performed by E. W. Collings

of Battelle Columbus Laboratories, Columbus, OH.

The specific heat at low temperature measured here was by adiabatic calorimetry (the

Nernst method). In this method, small increments of heat are supplied to a thermally insulated sample, and the accompanying temperature increase is measured. The capacity is calculated using the relationship:

44
= d£ „

AQ^ =

I 2 RAt

p dT AT AT

(3-7)

where current, I, is supplied for a time- interval , At, to a heater winding of resistance, R,

in thermal contact with the sample; and AT is its resultant rise in temperature. The experiment is of course carried out at constant pressure, hence the subscript p. (However, in

the temperature range of interest the difference between and is negligible)

In general, a calorimeter can is immersed in liquid helium during the course of the

experiment, and thermal insulation of the sample is achieved with the aid of a vacuum.

Inside the calorimeter can the sample is secured to a copper block which carries a manganin

heater (1900 ohms) and a calibrated germanium resistance thermometer. In performing a

measurement, a pulse of current is passed through the heater coil. The duration of the heat

pulse (typically 6 to 60 seconds) is measured by a clock, also actuated by the current. A

correction for heat leakage is applied to the data.

In a test of performance, a specimen of high-purity copper was measured yielding a fit

2 of y = 7.07 mJ/mole-deg and = 338 K to the data. These values agree reasonably well

with those reported in the literature. The accuracy of the heat-capacity results are

estimated to be about f 15%.

The results of these measurements are shown in Fig. 3-8 and Tables 3-3 , 4 and 5 . The

tabular data is included to aid in future fitting and extrapolation. One can readily see

the similar behavior except for the small discontinuity in the coil composite due to the

superconducting- to-normal transition of the superconductor. This transition has been plotted

on an expanded scale in Fig. 3-9 which reveals the broad nature of the critical temperature

(T ) transition centered about 9.3 K. That the transition is several tenths of a degree

c wide implies that there are regions of different 1^ due to cold working or diffusion of

the superconductor itself.

The fiberglass-epoxy data is smooth and can be fit to a polynomial. The four term

polynomial

(3-8)

45
(M/B)j/r) 1V3H 3ld!03dS

0)M

3

00 H

46
CO CM -H (>l/BVr) 1V3H 3NI33dS

47
Table 3-3. Specific Heat of NbTi/Cu-Fi berglass Epoxy Coil Composite

in Temperature Range 4.0 to 27.2 (Run No. 1)

Temperature Specific Heat Temperature Specific Heat

( K) (J/kg-K) ( K) (J/kg-K)

4,,0 0,.338 9,,2 3,.060

4. 2 0,,381 9,.6 2..868

4., 5 0.,464 10,.0 3,.165

4. 6 0.,520 10,.5 3,,935

4..9 0.,592 10.,9 3,,775

5, 1 0.,680 11

,6 4.,415

5,,4 0.,786 12..2 4.,875

5. 7 0.,914 12.,7 5.,263

5.,9 1,029 13.,1 5.,709

6.,1 1068 13..6 6.,194

6.,1 1

.112 14..1 6.,715

6. 3 1

,242 14.,7 6,. 764 ?

6,.6 1 , ,397 15, 3 8.,108

6..9 1,553 16..3 9..348

7 ,2 1,769 17.,0 10,.317

7,,5 1 , ,940 18..3 12,.120

7 .8 2,,162 19,.2 13,.515

8 , l 2.,362 20. i

, i 15,.013

8..4 2.,606 21

,0 16,.563

8 .8 2. 937 22..0 18.,139

23,.2 29,,577

24.,3 23,,058

25.,6 25,.425

27,,2 27. 389

48
Table 3-4. Specific Heat of NbTi/Cu-Fiberglass Epoxy Coil Composite

in Temperature Range 4.5 to 22.2 K (Run No. 2)

Temperature Specific Heat Temperature Specific Heat

( K) (J/kg-K) ( K) (J/kg-K)

4.5 0.468 9.04 3.099

4.6 0.521 9.15 3.104

4.8 0.576 9.27 2.890

4.9 0.612 9.39 2.783

5.1 0.669 9.52 2.733

5.3 0.744 9.66 2.876

5.5 0.828 9.80 2.904

5.8 0.958 9.94 3.081

6.0 1 .077 10.10 3.124

6.3 1.202 10.27 3.244

6.5 1.335 10.4 3.369

6.7 1 .473 10.6 3.533

6.9 1.547 10.8 3.650

7.1 1 .652 11.0 3.848

7.2 1.744 11.2 3.956

7.3 1 .821 11 .4 4.219

7.50 1 .950 11.9 4.594

7.56 1.952 12.2 4.851

7.61 2.017 12.5 5.074

7.71 2.097 12.9 5.417

7.85 2.207 13.3 5.814

7.99 2.281 13.7 6.188

8.15 2.426 14.1 6.814

8.30 2.526 14.6 7.057

8.42 2.622 15.3 8.055

8.50 2.649 17.1 10.227

8.62 2.801 18.1 11 .633

8.72 2.835 19.3 13.372

8.81 2.921 20.1 14.907

8.93 2.973 21.1 15.988

22.2 19.123

49
Table 3-5. Specific Heat of Fiberglass-Epoxy Composite

Temperature Specific Heat Temperature Specific Heat

(K) (J/kg-K) (K) (J/kg-K)

4. 26 1 . 106 8. 18 5. 470

4. 35 1

144 8. 39 5.,898

4, 47 1.239 8.,73 6.,359

4. 62 1

,340 9.,21 7.,412

4, 78 1

.449 9.,78 8. 277

4. 94 1

598 10.,57 9,,482

5, 09 1,709 10,.90 10..528

b

9 C

1 , I I

97 , CI I I

5. 43 2.,065 11

.68 12,.014

5,,68 2,,288 12,.24 13.,357

5..95 2,,586 12,.85 14,,643

6..19 2,,852 13,.44 15.,724

6..47 3,.105 14,.08 17 ,497

6,,71 3.,402 14,.68 19,.302

6 ,90 3,.673 15 .29 20 .758

7,.09 3..971 16 .39 24,.001

7 .29 4..175 17,.05 25,.626

7 .52 4,.467 18 .01 28,.134

7 .77 4,.860 19..42 31.987

3 .03 5,.293

*

50
Table 3-6. Low-Temperature Specific Heat Coefficients for

Fi bergl ass-Epoxy Composite

Value (J/kg,K

n

) Coefficient below 10K above 10K

a: -1 .81 * 0.14 -1 .81 ± 0.05

a2 9.48 - 0.46 9.48 ± 0.15 a 3 1.99 - 0.45 1 .99 - 0.15

a* -1 .05 4 0.13 -1 .05 - 0.04

Table 3-7. Low-Temperature Specific Heat of Fiberglass-Composite

at Specific Temperatures

Temperature (K) Specific Heat (J/kg-deg)

' 4.0 0.89 - 0.15

5.0 1 .65 - 0.25

10.0 8.6 - 1.2

15.0 20.0 - 2.2

20.0 33.4 - 5.1

51
was determined to give the best fit to the experimental data. The values of the polynomial

coefficients are listed in Table 3-6 . Because of the higher precision of the data above

10 K, two sets of coefficients (and estimated uncertainties) are given.

As indicated in Table 3-6, the quadratic term is dominant. Usually this term is

absent in the expression for specific heat of a solid, as dictated by the physics of the

three-dimensional model. The presence of a dominant quadratic term indicates the presence

of two-dimensional, rather than three-dimensional, arrays of atomic oscillators. Whether

the two dimensionality is due to the long, chain-like polymer molecules or to the reinforcement by the fiberglass is uncertain and could be the subject of further study.

The specific heat values of the f iberglass-epoxy at specified temperatures is given "in

Table 3-7 along with the estimated uncertainties.

REFERENCES

1. Fowlkes, C. W. , Angerhofer, P. E. , Newton, R. N. , and Clark, A. F. , "Characterization of

a Superconducting Coil Composite," NBSIR 73-349, National Bureau of Standards, Boulder,

CO (1973).

2. Childs, G. E. , Ericks, L . J. and Powell, R. L. , "Thermal Conductivity of Solids at Room

Temperature and Below," NBS Monograph 131, National Bureau of Standards, Boulder, CO

(September 1973)

.

3. Clark, A. F. , "Low Temperature Thermal Expansion of Some Metallic Alloys," Cryogenics £5,

282 (1968).

4. McConnell, P. M. , Daney , D. E. , and Kirgis, J. B. , "Thermoelastic Expansion and Creep

of PET and PPMI Films and PET Fibers from 20 to 295 K," J. Appl. Phys. 41, 5066-70 (Dec. 1970)

5. Hahn, T. A., "Thermal Expansion of Copper from 20 to 800 K - Standard Reference Material

736," J. Appl. Phys. 41, 5096-5101 (Dec. 1970).

6. Corrucini, R. J. and Gniewek, J. J,, "The Thermal Expansion of Technical Solids at Low

Temperatures," National Bureau of Standards Monograph 29, U.S. Government Printing

Office, Washington, D.C. (May 1961).

7. Jelinek, F. J. and Collings, E. W. , "Low Temperature Thermal Expansion and Specific

Heat Properties of Structural Materials," in: Materials Research for Superconducting

Machinery - II , Clark, A. F . , Reed, R. P., and van Reuth, E. C, Editors, ADA004586

(October 1974)

.

52
4. ELASTIC PROPERTIES

by W. F. Weston

4. 1 Introduction

The elastic properties of composite materials are currently of considerable interest.

Many composites, because they have high mechanical strength and a high modulus-to-density

ratio, are useful as strong, lightweight structural materials. Some composites have been

designed for use at low temperatures, for example as superconducting coils in generators

or in motors. However, few elastic data are available for composite materials, particularly

at low temperatures. Such data are useful as design parameters, and elastic constants

are among the most accurately known fundamental physical properties of solids. Because

composite materials are usually highly anisotropic, the determination of their elastic

constants is much more difficult than for the more usual quasi-isotropic engineering

materials.

Conventional methods of measuring the elastic properties of solids have been applied

to composites with only limited success. These methods usually require relatively large

specimens and considerable time and effort, especially if a complete set of elastic constants

1 2 3 is required. * These methods, as applied to composites, include vibrating ' or resonantbeam^'"''^ tests, tensile or compressive tests,"''' ' '** torsion tests, "'"'^ bending tests, ^>^® and

11 12 13

pressure tests ' ' . Generally, the elastic constants measured by these methods are

less accurate than those acquired by ultrasonic tests.

Ultrasonic pulse techniques are limited by the necessity of using wavelengths that

are large compared with fiber diameters and at the same time sufficiently small compared

with the dimensions of the specimen so that true plane-wave conditions can exist. Since

ultrasonic pulse methods are typically performed at megahertz frequencies, this condition

cannot be met for many composites. If the wavelength is not larger than the fiber diameter,

the wave is attenuated and scattered. Ultrasonic pulse-echo methods have been applied to 14 15 16 some metal-matrix composites with extremely small fibers. ' ' Even in these cases the

pulse-echo trains are poor because of high attenuation and dispersion^. In fact,

Achenbach and Herrman^ predicted large dispersive effects for shear waves propagated in

the fiber direction, even when the wavelength is much larger than fiber diameters. Zimmer

18 and Cost verified this prediction by measuring sound velocities in a unidirectional

* The references in each chapter are treated separately.

53
glass-reinforced epoxy-fiber composite. For wavelength-to-fiber diameter ratios of roughly

50, dispersion increased some phase velocities by a factor of 2. Elastic constants of some

carbon-fiber-reinforced plastics have also been measured by ultrasonic pulse-echo

. . 19,20,21 techniques.

22

The ultrasonic immersion technique, as described by Markham, is fast and accurate

and offers the possibility of measuring all the elastic properties on one specimen.

23 24

However, this technique is also limited by fiber size and by sample dimensions. '

The choice of ultrasonic modes that can be propagated in a given direction in a specimen

is also limited, and the technique is inapplicable to low-temperature measurements.

25

Zecca and Hay avoided the problems encountered with megahertz frequencies by using

electrostatic transducers to generate and to detect resonant kilohertz frequencies of a

metal-matrix composite. Electrostatic transducers, however, cannot be used with nonmetallic materials.

The most widely used ultrasonic technique, using kilohertz frequencies, is the

resonant piezoelectric oscillator. This method is due principally to Quimby and associates.

Quimby thoroughly analyzed the vibrations of solid rods driven by a piezoelectric crystal,

27 and Balamuth first employed drivers and specimens of matching fundamental frequencies.

28

Rose extended this method to torsional oscillators. Basically, the piezoelectric properties of quartz are used to generate and to detect resonant frequencies of an oscillator

consisting of one or two quartz crystals, a specimen, and perhaps a dummy rod, all cemented

29 30 together. This method has been used to measure elastic moduli at both high ' and

. 27,28,30,31 . , , -j mm , - . , .« 32,33 , . . 27-31 low temperatures, and has worked well with both metals and non-metals.

Thus, this technique is well understood both theoretically and practically, and it would

seem to be well suited for testing composites. The long wavelengths used (typically on

the order of centimeters in the kilohertz region) are much larger than fiber diameters,

and, since the technique employs standing waves rather than traveling waves, specimen

sizes can still be kept small.

In this section it is described how this resonance technique was used to measure the

complete set of elastic compliances of a superconducting coil composite, which was essentially composed of an epoxy matrix and unidirectional copper-niobium-titanium fibers. The

compliances were measured semi-continuously from 300 to 4 K. Results are given also for

54
an epoxy-fiber glass layered composite. The results generally agreed closely with the

few existing static measurements on the same composite reported in Section 2.3.

4.2 Experimental Procedure

4.2.1 Resonant Piezoelectric Oscillator

The three-component resonant piezoelectric oscillator technique was originally

34 35 described by Marx , and was discussed in detail by Fine . The technique consists of

bonding quartz-driver and quartz-gauge piezoelectric crystals to a specimen to produce and

to detect a standing longitudinal (or torsional) resonant wave. Each component's length

is adjusted so that its resonant frequency is closely matched to that of the other components. The system is then driven by the driver transducer at its resonant frequency,

which is monitored by the gauge crystal.

Longitudinal waves are excited by a-quartz bars of square cross section, excited into

longitudinal vibration by an ac signal applied to full-length, adherent electrodes (see

Fig. 4-1). The driver quartz was fully gold plated on two parallel sides (z-faces) with

a shallow notch cut in the center of each electrode. Fine, gold-plated tungsten wires,

seated in these notches, served to suspend the resonator assembly at displacement nodes

and provided the necessary electrical contact. The gauge quartz was gold-plated only over

the center third of the two parallel faces and electrical contact was made through fine

wires adhered into the notches with silver conducting paint. The electric axes of the

driver and the gauge crystals were perpendicular to reduce electrical pickup.

The torsional quartz crystals were circular cylinders with their length in the

36 x-direction and with four gold-plated electrodes running the length of the crystal.

Opposite pairs of electrodes were electrically connected. The driver quartz was suspended

vertically by means of fine nylon threads attached with varnish to opposite sides of the

quartz near a displacement node of vibration.

The quartz crystals were bonded together with a semi-permanent adhesive, Eastman 910*.

The specimens were right-circular cylinders and were also bonded to the gauge quartz with

the same adhesive. For some torsional experiments, this cement occasionally failed at low

temperatures; such experiments were repeated until successful. Other materials such as

37 vacuum grease and epoxy resin have been used as low-temperature bonding agents.

* The use in this paper of trade names of specific products is essential to a proper

understanding of the work presented. Their use in no way implies any approval,

endorsement, or recommendation by NBS

55
CS9

irn

56
4.2.2 Electronics

Most workers who have used the piezoelectric oscillator technique have employed

34 electronics similar to those described by Marx . This system requires manual adjustment

of the input frequency to the driver crystal to keep the oscillator at resonance. A

less time-consuming and more accurate system is the regenerative system designed by

38 39 Gerk and slightly modified by Johnson

This system consists primarily of a closed loop containing an amplifier and the

driver-gauge combination of piezoelectric crystals, which act as a resonant element, with

a feedback circuit to control the gain of the amplifier. This system is based on the

principles of an oscillator; that is, the output of an amplifier is fed into the input

through a frequency-selective network (the quartz crystals) . The phase shifts are corrected

with a variable phase shifter. A more complete description of the system was given by

38

Gerk. This system locks onto the resonant frequency and automatically stays at the

resonant frequency if this frequency changes, due to a change of the specimen's environment,

for example.

4.2.3 Apparatus

The longitudinal oscillator apparatus is illustrated in Figure 4-1. The driver-quartz

crystal was suspended by 0.008 cm gold-plated tungsten wires. The driver mounts, which

were aluminum, were removed during the torsional measurements, since the torsional driver

crystal was suspended between the bakelite posts by nylon threads. The specimens were

cylindrical for all measurements. The chromel-constantan thermocouple was placed near,

but not touching, the specimen. The entire apparatus was sealed in a thin-walled stainlesssteel can. The can was partially evacuated, placed in the ullage of a helium dewar and

lowered stepwise to achieve cooling.

The quartz crystals used had resonant frequencies of 60 and 100 kHz. Most of the

measurements were done with the 60 kHz quartz crystals (3/16 in (0.48 cm) square for

longitudinal, and 3/16 in (0.48 cm) diameter for torsional), but some were done with the 100 kHz

crystals (1/8 in (0.32 cm) diameter). The mass of each driver-gauge combination was noted and the

resonant frequencies were monitored from 300 to 4 K. The length of each specimen was

determined such that its resonant frequency for the entire temperature range was within

approximately 5% of the frequency of the oscillator assembly although in some cases the

specimen frequency differed by as much as 10%. The mass, length, and diameter of each

57
specimen were noted and its mass density was determined by Archimedes' method using distilled

water as a standard. The specimen was then cemented to the quartz crystals and the resonant

frequency of the oscillator was monitored from 300 to 4 K. No thermal contraction corrections

were made; for the coil composite described below this introduces a maximum error (over a 40

300 K range) of about 0.5/o. Maximum uncertaintites in the frequency measurements are

estimated to be about one part in 10"*.

4.2.4 Specimens

The superconducting coil composite is fully described in reference 40. Basically,

the ceil is composed of a copper-stabilized niobium-titanium wire (0.56 mm by 0.72 mm

cross section) coil impregnated with epoxy. The wire layers are separated by layers of dry

fiberglass cloth (0.1 mm thick). The copper-to-superconductor volume ratio of the wire is

1.8 to 1. The wire-to-epoxy cross-sectional area fraction is about 3 to 1. The coil dimensions were approximately 19 cm I.D., 24 cm O.D., and 10 cm long.

Coordinate axes were chosen to coincide with the specimen axes. The longitudinal axis

(1-axis) coincides with the axis of the wire, which was assumed to have zero helix angle.

The 2- and 3-axes lie along the radius and coil axes, respectively, and form an orthogonal

set with the 1-axis. The symmetry of the coil (orthorhombic) requires nine elastic constants to characterize the material. By assuming the material to be transversely isotropic,

and by neglecting the effects of curvature, the number of independent elastic constants is

reduced to five. An orthotropic body with transverse isotropy has the same symmetry as

close-packed hexagonal crystals such as magnesium and zinc. Justification for equating

the 2- and 3-axes is given in reference 40, and further experimental data show that the

41 effect of curvature on the moduli is minor.

The masses, lengths, and angles between the specimen axis and the 1-axis are given

in Table 4-1. The density of the coil composite was found to be 6.000 gm/cm.

Besides the superconducting coil, some specimens composed of fiberglass cloth and

epoxy were made available. The f iberglass-to-epoxy volume ratio in these samples was

approximately the same as in the coil composite. Measurements were made on two specimens

of this material. Dimensions and orientations of these specimens are also reported in

Table 4-1, with the angle 8 corresponding to the angle between the specimen axis and the

direction direction parallel to the layers of fiberglass cloth. The density of this

3 jiaterial was found to be 1.931 gm/cm . This material can also be considered transversely

58
Table 4-1. Dimensions and orientation of specimens

jpcL i in en Numb er Diameter

:~ Mass Length

(cm) (degrees)

Superconducting Coil Composite Specimens

1 0.3175 0.540 1.1128 90

2 0.3175 0.925 1.9439 0

3 0.4732 1.081 1.0818 0

4 0.4801 1.248 1.1481 90

5 0.4763 1.874 1.7600 60

6 0.4702 1.327 1.2555 60

Epoxy-Fiberglass Composite Specimens

1 0.4763 0.885 2.5387 0

2 0.4770 0.520 1.4862 0

59
isotropic. Thus, five constants are again needed to completely characterize the material

elastically. Cylindrical specimens of only one orientation were available; thus, only two

elastic constants of the f iberglass-epoxy matrix material were determined.

4.3 Results

The directly measured quantity in these measurements is f , the frequency of the

three-component piezoelectric resonator. For longitudinal resonance the frequency of the

specimen (f

g

) can be found from f Q, the frequency of the driver-gauge quartz assembly (f ),

42

the mass of the specimen (m

g

) > and the mass of the quartz (m^) :

f

2 = f 2 + (f2 - f2 so oq )m /m . (4-1) qs

This formula is more exact than the approximation

f = f + (f - f )mJm , (4-2)

s o o q q s

. 34 which is usually quoted and used for resonant oscillators.

The formula for the torsional oscillator is somewhat more complicated since the

36

moments of inertia (rather than the masses) of the components are involved:

2 2 2 2 2 2 f = f

Z + (f - fZ)m r /(m r ). (4-3)

s o o qqqss

Here, r is the radius of the quartz and r is the radius of the specimen,

q s The Young's modulus, E, or shear modulus, G, for the particular orientation of the

specimen is given by:

E or G = 4 L

2

f

2 p. (4-4)

s

E is found from the longitudinal-mode fundamental frequency and G is found from the

torsional-mode fundamental frequency. In Eq. 4-4, L is the specimen length and p is the

mass density.

Thus, the measured quantities are the Young's and shear moduli, which are directly

related to the elastic compliances, . Elastic compliances rather than elastic stiffnesses, C_ , are measured because of the fundamental difference between the behavior of

high-frequency and low-frequency elastic waves. In the case of a high-frequency wave

(uiegahertz and higher), there is insufficient time in a cycle for the material to relax

60
and to relieve the stresses due to the wave. Thus, for high-frequency measurements, the

strains are relatively simple but the stresses are complicated and the elastic stiffnesses

are the constants used in solving the elastic-wave equation. For low-frequency measurements, the reverse is true, and the elastic compliances are the constants of interest.

However, for low frequencies, if the wavelength is not much larger than the sample dimensions, a correction for the Poisson contraction (or lateral motion) must be applied to

35 Young's modulus. When the length-to-diameter ratio is 10 to 1, this correction amounts

to about one part in a thousand.

The value of Young's modulus for an arbitrary direction in a specimen with transverse

isotropy, with the 1-axis the unique axis, is given in terms of the elastic compliances

by

|

= S 33

sin4

9 + S

1:L cos

4

e + (2 S 12 + S 55

)sin2e cos

2

9 , (4-5)

where 8 is the angle between the specimen axis and the unique axis. Similarly, the shear

modulus is

i = S 55 + 1/2(S

3 - S 23

- 1/2S

55

) sin2

6 + 2(S

33 + S u

- 2S

12

- S 55

)sin 2 6 cos

2 8. (4-6)

4.3.1 Superconducting Coil Composite

Measurements made on the coil composite are listed in Table 4-2. Again, 8 is the angle

between the specimen axis and the 1-axis (longitudinal axis) . The mode specifies either

longitudinal or torsional oscillations. The subscripts on E and G designate the direction

of the force and the plane on which the force is applied, respectively. The compliances

S 33' S ll'anc* S 55 are ^oun^ directly from experiments 1, 2, and 3, respectively. From

experiment 4, S 23

is found using the relation 2(S

33

- S 23 ) = an^ t ^xe value of S^.,.

from experiment 3. Either specimen 5 or 6 can be used to determine S^. This compliance

must be determined from a measurement on a specimen with its axis at an angle to the

longitudinal direction. As can be seen from Table 4-2, however, the Young's or shear

modulus measured on specimens with 0° < 8 < 90°, is only partly dependent on S^

2

« Thus

S^

2 will be the least accurately determined compliance. The measurement of this compliance

corresponds to the measurement of the elastic stiffness C^

3 by megahertz ultrasonic techniques,

61
Table 4-2. Experiments performed on superconducting coil composite

Exp. No. Spec. No. Mode 0 Measured Quantity

1 1 L 90 E 99 = F

2 2 L 0 En =

11 is> n

3 3 T 0 = -|

4 4 T 90 G'

S 55 +S 44

60

60

16

) = 60° - 9S

33 + S n

+ 6S

12 + 3S

55

=60 3S Q

„ + 3S,, - 6S 10 + 2S_ C + 3S..

33 11 12 55 44

62
Zimmer and Cost"^ encountered difficulty in measuring f°r a composite material and

their estimated uncertainty is this elastic constant (100%) was ascribed to dispersion

as well as to the relatively high inaccuracy.

The experimental data are presented in Figures 4-2,7. Temperature dependences of

44 these moduli were fitted to a theoretical relationship suggested by Varnshni

C = C° - -f— (4-7)

e -1 where C°, s, and t are adjustable parameters and T is temperature. The value of C at

T = 0 K is C°, and -s/t is the high-temperature limit of the temperature derivative dC/dT.

By invoking an Einstein oscillator model of solids, it can be shown (in the absence of

electronic effects) that t is the Einstein characteristic temperature. This function has

44 44 been used successfully to represent the temperature dependences of metals, ionic solids,

45 o and several alloys. The parameters C , s, and t are given in Table 4-3. The overall

fit of this function to these data is not nearly as good as for other materials. In

general, the Varshni function is flatter at low temperatures than these data, giving lower

curve values at 0 K. It also has smaller temperature derivatives, giving larger curve

values near room temperature. The goodness of the fit is also listed in Table 4-3 as

the average percentage difference between the curve values and the experimental values.

45 These percentage differences are larger than was found for alloy data. One set of data,

E,^, could not be fitted to the Varshni function because of its concave-upwards curvature.

A smooth curve was drawn through the data points and the actual point values were used in

calculations.

The shear modulus corresponding to shearing the axial planes in the radial direction

^23' ° r v^- ce-versa i- s shown in Figure 4-8. This modulus was calculated from the curve

values of G, _ and G', since G.„ = 1/S. ,

13 23 44

Because of the difficulty in machining the specimens to small radii, the ratios of

lengths to diameters were in some cases on the order of four or five to one. The estimated

correction to the Young's moduli in these cases is still only about one percent.

The elastic compliances as functions of temperature are given in Table 4-4. The

compliances S^j s -q> and S,.,. are just reciprocals of e-q> an<i G^3> as mentioned

previously. The error in the directly measured moduli is estimated to be about 1%, thus

these compliances should be accurate to 1%, also. The compliance is found from

63
,^IW

0I 0I)

ZZ

3

64
oICD

09o—Eo

COc o 3 c

CM

Co

LlJ

y

0£

M

=> 01 1— a. «C 3 £ CO LU o a

in

i—i UJ 4J hCD33aos

COto3O

>-

enI

egu3 ( z m OI

oi;

it.

65
UJ

en

o o o o o o o o o o o CNJ CNJ CM o o CT> co CO i

WD

1—

.—

o o o o o o o

\ z

/ft QI

UL EL

66
oC

I

O >X) CM CO «3" O <X> CNJCO^J-O co cm r— i—i <o «—i un 0"=J"0^ ^i- n co cm cm >—1 —< O oo^cocc ^_;^h'^h'^^ ^ o d d

(^/N

0I 0I) ,t)

67
a:

o co CM O CO CM to o r-- CM T—

ID

CM o co IT) co CO CO co CM CM CM CM CM

o

: zm OT oi) o 09=e

3

68
onZD

{ zm/H

oi

OT) 0 °9=9lj

69
Table 4-3. Parameters in equation 7

Mode r>0 s t Avg. % Dirf.

(10 10 N/m2

) (1010 N/m2) (K)

E 22 4.137 1.373 264.5 0.63

G13

1.099 1.352 469.0 0.89

G' 1.298 0.930 372.8 0.67

E e=6o°

3.419 4.431 533.5 0.74

G0=6O°

1.550 0.724 320.7 0.94

70
Xo-szuco3cooKV

&.3

oa

013

•aoEH

cdv

X.XI -T

a>u3

(f/n 0T

ot:

71
Table 4-4. Elastic compliances of the

i -10 2 superconducting coil composite at selected temperatures (10 m /N)

T(K) S n

S 12S 23S 33

S 44S 55

0 0.112 -0.061 -0.074 0.242 0.631 0.910

50 0.113 -0.061 -0.074 0.242 0.632 0.910

100 0.115 -0.066 -0.076 0.248 0.648 0.920

150 0.118 -0.073 -0.083 0.260 0.684 0.964

200 0.119 -0.079 -0.089 0.275 0.728 1.046

250 0.121 -0.082 -0.094 0.294 0.776 1.170

300 0.123 -0.084 -0.096 0.316 0.823 1.350

72
Table 4.5. Elastic stiffnesses of the superconducting

coil composite at selected temperatures (10^^N/m2)

T(K) C llC.12C,23 C 33C 44

C 55

0 10.59 2 .39 1 .94 5.10 1.58 1.10

50 10.46 2 .39 1 .93 5.10 1.58 1.10

100 10.27 2 .34 1 .90 4.99 1.54 1.09

150 10.05 2 .27 1 .87 4.80 1.46 1.04

200 9.86 2 .15 1 .79 4.54 1.37 0.96

250 9.64 1 .99 1 .63 4.21 1.29 0.85

300 9.35 1 .77 1 .39 3.82 1.22 0.74

73
combining the results of three measurements. Thus, S^^ maY De less accurately known. It

was found that in solving for S^

2 from either Eq. 5 or 6, a 1% error in the measured

modulus led to a 10-20% error in S^2> Thus, the error associated with S^

2 is considerable,

perhaps as much as 50%. The values reported for S^

2 are the average of the values found

from experiments 5 and 6. As expected, the material was found to be highly anisotropic

with S 33 » 2S U

.

4.3.2 Epoxy-Fiberglass Composite

Measurements were also made on the epoxy-f iberglass specimens listed in Table 4-1.

These specimens were oriented with the layers of fiberglass cloth running the length of

the specimen; these were the only orientations available. The Young's modulus and shear

modulus are shown in Figures 4-9 and 4-10. The shear modulus was fitted to the Varshni

function with C° = 0.885 x 10

10 N/m2

, S = 0.227 x 10

10 N/m2

, and T = 193.8 K. The average

percentage difference between the curve value and the measured value was 0.28%. A smooth

curve was drawn through the Young's modulus data, which could not be fitted to the Varshni

function because of the low-temperature maximum.

4 . 4 Discussion

Some elastic data, taken by conventional static methods, are available in Section 2.3

for the superconducting coil composite and the epoxy-f iberglass composite. These data

generally compare very favorably with the resonance data. See Table 4-6. For example,

resonance values of at room temperature and 4 K are 12% and 1% larger than the static

data taken at these temperatures. The inaccuracy of the static values is given as about

10%. Also, resonance values of compare to within 7-5% of the static data, and resonance values of agree within 3-9% of the static values.

Other elastic constants of interest include the Poisson ratios:

v

23

= S 23

/S

33 (4-8)

and

v

21

= -S

12

/S n

(4-9)

where V„ represents the negative ratio of strain in the j -direction to strain in the

i-direction.

40 The value of v

23 at room temperature is 0.304. The reported static value is 0.335,

with the inaccuracy again believed to be about 10%. Thus, this result gives added confidence to the resonance value of S^. However, the Poisson ratio ^

2 ^, involves the

74
75
oLO

o CO CSJ 10 o co CSJ LO o o ro to CSJ CO LO i—

CO

cr. CO co r-~- lo LO LO LO O O O o o o o o o o o

( 2 w/n

oi ol) smnaow uvbhs ssviaaaaij-Axoda

7ft
Table 4-6. Comparison of static and acoustic measurements

of elastic constants (units of 10"'^ N/m2

).

Elastic Constant Temperature (K)

193 77_ 4

E

ll acoustic 8 .14 8.77 8.96

static 7 .58 8.08 8.53

E22 acoustic 3 .14 4.07 4.17

static 2 .76 3.79 4.12

3 13 acoustic 0 .76 1 .09 1.12

static 0 .74 1.16 1 .22

Ef iberglass- acoustic 3 .14 3.47 3.42

epoxy

static 2 .80 3.33 3.56

77
compliance S^

2

, which has already been noted to be much more inaccurate than the other

four compliances. In fact, if the room-temperature values of S^

2 and are used,

40

V21

= 0*^83, whereas the static value is 0.333. Thus, the value of i s highly

suspect, especially since a value near 1/3 seems much more realistic for Poisson's ratio.

It is also of interest to consider the compressibilities. For a material with

transverse isotropy, the linear compressibilities are

k

3 = S 33 + S 23 + S 12 (4-10)

and

k

l 2S

12 + S ll

where the subscripts designate the axis. Also, the volume compressibility is

k = 2k

3 + Y. (4-11)

If the room-temperature values of the compliances, as given in Table 4-4, are used, then

is seen to be negative. Since this is unlikely, a minimum value for S^

2 would seem to -10 2 be S = -S

i;L /2 = -0.0615 x 10 N/m . This, of course, leads to a value of 1/2 for the

Poisson's ratio. If the Poisson ratio is taken as 1/3, then = -0.041 at room tem-

-10 2 -10 2 perature, and k3 = 0.179 x 10 N/m , k^ = 0.041 x 10 N/m , and the volume compressi-

-10 2 bility k = 0.399 x 10 N/m . These are probably reliable values for the compressibilities,

This leads to a value for the bulk modulus (reciprocal compressibility) of 2.506 x 10"^

N/m2

. By comparison, the bulk modulus of copper is about 1.4 x 10"^ N/m2

. 4 ^

The elastic stiffnesses (C.) can be directly calculated by inverting the S.„ matrix

47 if all the compliances are known. The explicit relationships are : 2

S 33S 11 " S 12

C 33

C23

(s 33

- s 23 )§

2 S 12 "

S 23S 11

(s

33

- s 23 )§

S 12

C 12

=

~ ~ » (4 ~12)

C ll

S 33 + S 23

C 55

1 55

78
and

C = 1—

44 S 44

'

x 2 where S = s -j_j_^33 + ^23^~^12 " ^"P11 f° r an^ ^44" these relationships depend

strongly on S^- In calculating the elastic stiffnesses, a value of 1/3 for was used,

and this Poisson's ratio was taken as constant with temperature

4 ^ It is believed that

the elastic stiffnesses obtained this way are more accurate than those obtained using the

resonance value of V

2_2' T^ie e^- ast ^ c stiffnesses are given in Table 4-5.

As expected, the elastic properties of this composite are highly anisotropic. The

anisotropy is best described by considering the percent elastic anisotropy as discussed

, 48

by Chung and Buessem. For transversely isotropic symmetry, this scheme involves using

the conventional anisotropy ratios (the linear compressibility ratio A = k-/k_, and the

c 13

* * shear anisotropv A = C_./C.,) to calculate anisotropv parameters A and A with the

s 55 44 fjf c s * following properties: A is zero for materials that are elastically isotropic, i.e.,

* A = 1; A is always positive and a single-valued measure of the elastic anisotropy of a

material regardless of whether A > 1 or A < 1; and A gives the relative magnitude of the

elastic anisotropy present in the material. For this composite A - 0.23, A = 8.4%,

* * * 48 A

g

= 0.61, and A

g

- 12.3%. For comparison, A^ - 9.9% and A

g = 13.7% for zinc, which

is one of the most anisotropic hexagonal metals.

40 The static value of Young's modulus of the f iberglass-epoxy is also available.

This Young's modulus was measured on a specimen oriented like the one used for the

resonance measurements. The resonance and static values differ by 4 - 10%. There was no

evidence of a maximum in the static data, but it is believed that the maximum is real.

4.5 Conclusions

The resonant piezoelectric oscillator method used in this work is generally suitable

for measuring the elastic properties of composite materials. This method is accurate,

fast, and requires only small specimens. It is suitable for measurements at all tempera29 30 tures from 0 K to greater than 1000°C. ' The general techniques and theory of this

method are well understood. This method does not appear to have the limitations of static

and ultrasonic techniques.

The elastic compliance data presented in this work agree well with existing data

taken by conventional static methods. These data can be used to calculate other elastic

properties, such as the elastic stiffness coefficients and compressibilities.

79
The superconducting coil composite is highly anisotropic, with elastic anisotropies

similar to those of zinc. This high anisotropy, characteristic of many composite materials,

necessitates careful measurements of elastic properties of these materials.

The measured elastic moduli of the composite are not as well represented by the

Varshni function as they are for most metals and alloys; i.e., the elastic moduli of the

superconducting coil composite are less regularly behaved with respect to temperature.

REFERENCES

1. P. R. Goggin, J. Mater. Sci. 8, 233 (1973).

2. T. J. Dudek, J. Compos. Mater. 4, 232 (1970).

3. T. Hirai and D. E. Kline, J. Compos. Mater. 1_, 160 (1973).

4. A. B. Schultz and Stephen W. Tsai, J. Compos. Mater. 2, 368 (1968).

5. R. C. Rossi, J. R. Cost, and K. R. Janowski, J. Amer. Ceramic Soc . 55, 235 (1972).

6. G. C. Wright, J. Sound Vibr. 21, 205 (1972).

7. R. E. Allred and W. R. Hoover, J. Compos. Mater. 8_, 15 (1974).

8. L. J. Cohen and 0. Ishai, J. Compos. Mater. 1, 1030 (1967).

9. N L. Hancox, J. Mater. Sci. ]_, 1030 (1972).

10. T. R. Tauchert, J. Compos. Mater. 5, 456 (1971).

11. A. Smith, J. Phys. E. Sci. Instrum. 5, 274 (1972).

12. A. Smith, W. N. Reynolds, and N. L. Hancox, J. Compos. Mater. 7, 138 (1973)

13. A. Smith, S. J. Wilkinson, and W. N. Reynolds, J. Mater. Sci. 9^, 547 (1974) .

14. J. V. Grabel and J. R. Cost, Metall. Trans. 3, 1973 (1972).

15. A. E. Lord and D. Robert Hay, J. Compos. Mater. 6, 278 (1972).

16. W. Sachse, J. Compos. Mater. 8^, 378 (1974).

1 -7 17

J0. Achenbach and G. Hermann, AIAA J. 6, 1832 (1968).

18. J. E. Zimmer and J. R. Cost, J. Acoust. Soc. Amer. 4_7, 795 (1970)

19. w. N. Reynolds and S. J. Wilkinson, Ultrasonics 12, 109 (1974).

20. s. J. Wilkinson and W. N. Reynolds, J. Phys. D: Appl. Phys. 7_, 50 (1974) .

21. T. R. Tauchert and A. N. Guzelsu, J. Appl. Mech. , Trans. ASME 39, 98 (1972) .

22. M. F. Markham, Composites 1, 145 (1970)

23. G. D. Dean and P. Turner, Composites, July 174 (1973).

24. R. S. Smith, J. Appl. Phys. 43, 2555 (1972).

25. A. R. Zecca and D. R. Hay, J. Compos. Mater. 4_, 556 (1970).

80
26. S

L. Quimby, Phys. Rev. 25, 558 (1925).

27. L. Balamuth, Phys. Rev. 45, 715 (1934).

28. F. C. Rose, Phys. Rev. 49, 50 (1936).

29. L. Hunter and S. Siegel, Phys. Rev. 61, 84 (1942).

30. J. W. Marx and J. M. Silvertsen, J. Appl. Phys. 24, 81 (1953).

31. M. E. Fine, Rev. Sci. Instrum. 25, 1188 (1954).

32. S. Siegel and S. L. Quimby, Phys. Rev. 49, 663 (1936).

33. J. Zacarias, Phys. Rev. 44, 116 (1933).

34. J. Marx, Rev. Sci. Instrum. 22, 503 (1951).

35. M. E. Fine, ASTM Special Tech. Publ. No. 129, 48 (1952).

36. W. H. Robinson, S. H. Carpenter and J. L. Tallon, J. Appl. Phys. 45, 1975 (1974).

37. A. J. Matheson, J. Phys. E. (Sci. Instru.) 4, 796 (1971).

38. A. P. Gerk, Rev. Sci. Instrum. 43, 1786 (1972).

39. D. L. Johnson, University of Illinois, private communication.

40. C. W. Fowlkes, P. E. Angerhofer, R. N. Newton, and A. F. Clark, NBSIR 73-349 (1973).

41. A. F. Clark, National Bureau of Standards, private communication.

42. W. N. Robinson and A. Edgar, IEEE Trans. Sonics and Ultrasonics 21, 98 (1974).

43. W. Voigt, Lehrbuch der Kristallphysik, (Teubner, Berlin, 1928), p. 746-7.

44. Y. P. Varshni, Phys. Rev. B2, 3952 (1970).

45. E. R. Naimon, W. F. Weston, and H. M. Ledbetter, Cryogenics 14, 721 (1973).

46. H. M. Ledbetter and E. R. Naimon, J. Phys. Chem. Ref . Data 3, (1974)

47. R. F. S. Hearmon, Rev. Mod. Phys. 18, 409 (1946).

48. D. H. Chung and W. R. Buessem, Anisotropy in Single-Crystal Compounds (Edited by

F. W. Vahldiek and S. A. Mersol, Plenum, N.Y., Vol. 2, p. 217, 1968)

81
5. STRESS ANALYSIS OF TRANSVERSELY ISOTROPIC MAGNET COILS

by V. Arp

5 . 1 Introduction

The NBS Cryogenics Division has been assisting the sponsor in several ways: (a)

evaluation of the elastic constants of various coil and cryostat materials, for use in

stress calculations, (b) evaluation of maximum allowable stresses in various coil materials,

stresses which could cause either mechanical failure, or degradation of the nominal superconducting properties of the wire, and (c) representative calculation of operating stresses

for a particular test coil of the sponsor using data from (a) and constraints from (b)

This part of the report presents the calculations and results obtained under this last item.

Stresses in superconducting magnet windings originate from four separable causes: (1)

internal stresses due to the fabrication process, (2) thermal stresses due to differential

contraction between various parts as the magnet is cooled to helium temperature, (3) stresses

due to the magnetic field produced by the coil, and (4) time-dependent stresses caused by

localized temperature inhomogeneities or eddy-currents from changing magnetic, 'fields. Reliable coil design involves, among other considerations, prediction of these stresses as a

function of various material, geometrical, and operational factors, and consequent design

selection such that allowable stresses are not exceeded.

The geometry of interest to the sponsor is a simple cylindrical coil wound with superconducting wire, with glass cloth separating wire layers, which is then epoxy-impregnated to

form a monolithic structure. Stress analysis of such a structure may be separated into

sequential steps: (1) macroscopic analysis, which assumes a uniform material characterized

by anisotropic elastic and thermal moduli, and (2) microscopic analysis, in which the detailed stress distribution from one wire through the epoxy to the next wire is calculated

from knowledge of the macroscopic average stress found in (1) . here we consider the first

step, the macroscopic analysis.

5. 2 Consultation

Early in this study on magnet stresses, telephone discussions on the subject were held

with a number of superconducting magnet experts in government laboratories throughout the

country. The purpose was to find what stress calculations, if any, they had been doing on

stresses in anisotropic magnet structures. Several laboratories have the capability of

82
doing this kind of calculation using sophisticated finite-element computer programs, with

running times reported for some magnet studies of up to an hour on a CDC 6600 computer.

Such elaborate calculations may be necessary in the final stages of a very large system

design, but are hardly appropriate for exploring the general effects of properties variations and optimization of fabrication techniques.

A group at Rutherford laboratory in England has reported studies of composite magnet

1*

materials but they have not developed the macroscopic stress analysis as is done in

this report.

A group at the University of Wisconsin has been studying the problems of superconducting magnetic energy storage. As part of this effort, Professor Warren Young has been developing a less general but very practical program for stress calculations in superconducting

coils. After discussing the problem with Professor Young in some detail, he felt that one

of their two dimensional programs could be modified to accept a specified magnet winding

tension at the input. An intermediate output would be prediction of stresses up to but not

including the magnetic field stresses; these would then serve as input to a much longer

3-dimensional finite element program for evaluation of axial as well as radial and circumferential stresses due to the magnetic field. This latter calculation is certainly a necessary

part of the program, in that there is no hope of obtaining an analytical solution in place

of the long finite-element 3-D calculation. Accordingly a small contract was given to

Professor Young to perform these calculations. As of writing this report. Professor Young has

incorporated the winding stresses into his 2-dimensional calculations and reported results

which are compared with our analytical calculations in section 5.8, but he has not completed

the 3-dimensional magnetic field stress calculation.

Also as a result of our telephone discussions, it was decided to proceed with setting

up a general analytical framework for study and evaluation of 2-dimensional stresses in

anisotropic magnet coils. While the intermediate output from Frofessor Young's calculations will duplicate this analysis in part, the analytical study is expected to fall in the

more general description of coil-winding and thermal stresses as they relate to coil fabrication

techniques and material properties. The majority of this report describes this analytical

work.

: - Transversely Isotropic Material Parameters in a Cylindrical Winding

The basic material-symmetry approximation used in these calculations is that the elastic

noduli and thermal expansivities are isotropic in the plane perpendicular to the wire axis,

*The references in each chapter are treated separately.

83
but different than the moduli parallel to the wire. This is sometimes referred to as orthotropic symmetry. It is a reasonable approximation for tightly-wound, filled, magnet coils.

For our case of a cylindrically symmetric, transversely isotropic coil, strain as a

2 3 4 function of stress is given by ' ' (nomenclature is in section 5.11).

E e

= E^ *

( I} (°r

+

°z )

- a v V

8 r = r - ( E > °e " F °z (5-2 ) r t a v

£

z = f ~ ( I

} °e ' f°r (5-3)

z t

where

E = E = E

t r z

v = v = v

t rz zr

and / V ^9r _ ^r9

( E )= IT = E~ • 6 r" W

This last identity is a consequence of the fact that the strain energy is a scalar quantity.

In this work we use only plane-stress (a^ = 0) or plane-strain (e^ = 0) approximations, so

that the three additional equations relating torsional strain to shear stress are not needed

or presented. Based on these expressions, simple algebra shows that both plane stress and

plane strain relationships can be formulated as

and

where

e e-r " Vo

r

(5 " 5)

e r

= ^ -Vo

e

(5-6)

L = e for plane stress

9

E0

t = — , for plane strain

v 2 d-Ee

E

t

(¥

) )

1 = E , for plane stress

E, X = —— for plane strain

(5-7)

(5-8)

84
V = (—)

, for plane stress

£

V = (tt)(1 + v ), for plane strain.

Ji t

(5-9)

It is convenient also to express stress as a function of strain, by inverting equations

(5-4) and (5-5),

»

r " £+ «„> i (5-10)

and e °e

= ( T

+ VV 5 (5-ii)

where

= ^ " V • (5-12)

Force balance in a cylindrical element of radius r under either plane-stress or plane-

. 1

strain requires that

8a

r + J - at = 0. (5-13)

dr r 9

5 . 4 Solution for Stress Distributions in the Absence of Internal Stress

The term "internal stress" as used here and elsewhere within this report refers to

those stresses which exist within the body in the absence of all external forces applied to

it. Within the elastic limit any such internal stresses are simply additive to the stresses

which are produced by external forces. It is convenient here to write down the formal

solution for plane-strain or plane-stress distributions in our cylindrically symmetric,

transversely isotropic body in the absence of such internal strain.

Following standard texts, tne relationships between strain, £, and an infinitesimal

radial displacement, u, are:

e

9 = - (5-14)

and

e = £ . (5-15)

r or

These expressions, rather than any of the equations (5-1) - (5-13), must be modified

when internal strains are present.

Substitution of equations (5-14) and (5-15) into equations (5-10) and (5-11) yields

expressions for the stresses in terms of the displacement u. When these latter are substituted

85
into the equation of force balance, eq. (5-13), we find, after multiplying through by L:

ru

f

' + u' - k

2 - = 0 (5-16)

r

where the primes denote differentiation with respect to r, and the parameter

k = ( | ) 1/2

(5-17)

is a measure of the mechanical anisotropy of the structure. Realistic values of k for

epoxy- impregnated magnet windings may be in the range 1 to 3. The solution to this

differential equation is

u = ark + br"

k

. (5-18)

where a and b are coefficients determined by boundary conditions. Simple substitution leads

to the stresses

°r

= t

( L + V)rk_1 " I( L" V > r "k-1

< 5 -19 >

°e

= T ( l + v)rk_1 + 7T ( l

" v)r

~k~1

- (5 " 20)

For k=l, these equations reduce to the form

-2 o = const + const2 r ,

which is found in standard texts for stresses in isotropic cylinders.

5.5 Internal Stress Due to a Uniform Temperature Change

The thermal expansion coefficients of the composite magnet winding will in general

form a transversely isotropic set, just as do the elastic moduli. For a uniform temperature change AT, assume that the fractional length changes of thin samples are given by

(M/l) n = aQ AT in the longitudinal (theta) direction

W H and

(A£/&) = a AT in the transverse (radial) direction,

r r

Because our experimental work shows that filler materials, such as epoxies, generally have

higher expansion coefficients than do metals, we generally expect that a > OLq.

For our cylindrical winding, under either plane-strain or plane stress conditions,

the relationships between strain, expansion coefficient, and an infinitesimal radial expansion u(r) are

86
e

0 = 7

- a

eAT (5 " 21)

e = — - a AT. (5-22)

r 9r r

For an isotropic structure, with otg = a^, a uniform temperature change produces a radial

expansion given by u = raAT, but internal stresses and strains remain at zero.

As in section 5.4, substitution of equations (5-21) and (5-22) into (5-10) and (5-11),

and from there into (5-13), yields the differential equation

ru" + u' - k

2 - = [a - k2a„ + LV(aQ - a )]AT. (5-23)

r r 6 8 r

Neglecting the temperature variation of the elastic moduli, the solution to this differential

equation is

u = ar^ + br ^ + r (5-24)

1-k

where A is the expression in brackets on the right-hand side of equation (5-23). Further

substitution into (5-21) and 5-22), and then into (5-10) and (5-11) yields

k-i -k-i +

L(a

r - a

e )AT

a = cr - dr + r (5-25)

r 1-k

and

k_i _k_i

L < a

r

- VAT

aQ = kcr 1 + kdr

K 1 + — ^— (5 -26)

9 1-k2

and c and d are a different pair of arbitrary constants. The thermal stresses are those

which are present in the absence of external forces, i.e., at the inner (r=l) and outer

(r=g) boundaries, when

0.(1) = 0 = a

r (g).

Evaluation of c and d for this condition is straightforward, and both are found to be

proportional to the quantity

L(a - aQ )AT = LAaAT.

r 9

Thus it is convenient to normalize the internal stresses and Oq by dividing through by

this quantity. Within the elastic limit, the thermal stresses are simply added to any

externally-produced stresses to obtain the total stresses on the winding under operating

conditions

87 U
At first sight the equations appear to blow up as K 4 1, i.e., for an isotropic

material — a most unphysical result. However, expansion of the equations in terms of the

small quantity (k-1) shows that in this limit the equations reduce to the form

0 „ -2 , LAaAT

= const

i

* nr const2 * r + const^,

where the constants are independent of k. This is the form given by standard texts, e.g.,

Timoshenko and Goodier^, for internal stresses in isotropic cylindrical geometries.

Numerical evaluation of these equations for a particular coil geometry, set of material

properties, and cooldown from 338 to 4 K is shown in figure 5-1. Three features of these

curves should be noted: (a) Maximum circumferential compressive stress occurs at the outside radius, (b) Maximum circumferential tensile stress occurs at the inside radius, and

(c) Maximum radial tensile stress occurs within the body of the coil, but is generally

smaller in magnitude than the maximum circumferential stresses.

Generalized representation of these three maximum stresses as a function of coil

shape ( r 0

/ r -^) and mechanical modulus anisotropy (k) is shown in figures 5-2, 5-3, and 5-4,

all normalized to the thermal expansion anisotropy (LAaAT) . For practical epoxy- impregnated

windings, cooled f^om a curing temperature above room temperature down to 4 K, (LAaAT) may

3 2 3 well be as large as -35 x 10 N/cm , or -50 x 10 psi. One can see from the figures that

the maximum internal stresses due to cooldown of practical superconducting magnet coils

may typically be in the range 0.05 to 0.5 of this number, and thus may be quite significant.

On a macroscopic scale, these thermal stresses in principle could be reduced to zero

by adding a filler to the epoxy such that its expansivity matches that of the superconducting wire, or a^_ = a^. There may even be an advantage in carrying this further, to obtain

a

r

< a

Q (though in practice this would be very difficult to achieve). Under this condition,

dool-down of the coil would cause compressive circumferential stress at the inner radius,

and tensile stress at the outer radius. If such a stress distribution were added to the

stresses subsequently generated by the magnetic field, it would tend to produce a more

uniform circumferential stress distribution throughout the coil, rather than having it

peaked near the inner radius where the magnetic field is highest.

5. 6 Stresses due to Winding Tension

We consider the layer-by-layer winding of a simple solenoid in which relatively soft

insulating material, such as fiberglass, is wrapped between each layer of superconducting

88
CM CM E E o u

z. VO VO «3- o o VO

»—

CM X X o 00 VO

ir> r-~ o CM 1

ii n II

i- I

<

oVOCM

cooCM

VO

o

CM

If) o

oo

o • O—

oCO

I o

o

oCM

,lud/n 'SS3H1S

89
Figure 5-2. Circumferential stress developed at the inner radius due to a uniform temperature

change, normalized by the parameter LAaAT (Eq. 5-26), as a function of coil thickness, for several values of the mechanical anisotropy parameter K (Eq. 5-17).

175X 4453 "I

90
iP ( a D--TD)

W 3/ I O I CO CJ u-1 U

0) 14-1 • d. o a a w

a;

C

w

U

Oo u C V 3 4J

(1) E«H

re co re

CO O. ou - >, —

cu \D o 3 csi p

u-l o

05 CO 3 • -H •h cr c

•O

W CO

CO "-^

l-i .-I H U <3

0) S

4J <3 iJ O M

cfl U

•Hc

COo

01 0) 0) J3 4-1 6

4-i a)E n>

4-i co j;

CO

U 4J

CO o

T3

0)C 0) i—I 4-1 0)> .h

Ci) CO T3 >

T3

II) HI

H

CO

N CO <U -H U

HI 4J CO >

CO E 0)

CO : uo

4J <*-l a o

0) 60 « p c CD

01 CO Sac 3 J«i CJ 0) o M P i-l •H 3 J3 U U *J

01IM3

oo

91
1 1—I—I—I I I I

I

1—-

1—I—I—I I I I

Figure 5-4. Maximum radial stress due to a uniform temperature change, normalized by the

parameter LAaAT (Eq. 5-26), as a function of the coil thickness, for several

values of the mechanical anisotropy parameter K (Eq. 5-17).

92
wire. During this process the wire is maintained under tension, which we allow to vary as

a function of the dimensionless radius, r. For tractable analysis, we make the following

reasonable assumptions and approximations:

(1) After each layer is wound, the composite windirg at that moment may be characterized

on a macroscopic basis by a set of orthotropic material parameters (which can be estimated

from the mechanical properties of the wire and the cloth) . This further implies (a) that

all strains are elastic, and (b) that friction between the wire and the cloth is high enough

to prevent sliding of the wire along its length.

(2) As each layer is completed, there is no variation in either stress or elastic

moduli in the axial direction. This allows either plane-stress or plane-strain approximations to be used. Considering that each turn falls into place under relatively low axial

stress, compared to the winding tension, the plane-stress approximation is probably the

better one, though numerical results are little influenced by this choice.

(3) The coil is wound on a cylindrical bobbin which is characterized by isotropic

elastic moduli. Mathematically we define

inside diameter , , , . , . ol, = — r-. of the bobbin

B outside diameter

V = Poisson's ratio of the bobbin

B

E = Young's modulus of the bobbin.

B

Following standard texts, the relationship between radial deflection u of the outside

B diameter and radial stress o „ at the o.d. is rB

1

' a

2 ?- fc <-:- ° - V I < 5-">

rB B 1 - a_

a Thus, B is an effective modulus of the bobbin.

As each layer is wound onto the coil it adds a compressive stress to the layers underneath it, causing them to move radially inward by a small amount. The inward movement causes

a proportionate reduction in the circumferential stress which was originally established by

the winding tension. The radial compressive stress is infinitesimal at the outer layer,

accumulating to substantial values at the innermost layer. However, radial deflection of

this innermost layer will be very small if the bobbin is quite rigid, while radial deflection

93
of the middle layers will be determined much more by the radial compressibility of the

composite winding itself (which may be very high because the glass cloth interlayer is soft

compared to metals).

Analysis of internal strains due to winding tension can follow either of two methods:

(1) We can use an integration technique, adding winding thickness dr at each increment until

the final diameter is reached^ or (2) we can deduce the form of the equations for the stress

(and strain) distributions and evaluate arbitrary constants from boundary conditions. In

this report we use method (2). However, this method proves to be an approximation: see

note added in proof, p. 121.

We assume that the programmed winding tension produces a strain ey( r ) given by a power

series

n

ew(r) = I s.r

1 o

when s^ are constants which can be evaluated by fitting a polynomial to the programmed

winding tension. The corresponding circumferential strain within the body of the coil is

n . e Q = - + E s.r (5-28)

0 r i o

where, u is the radial deflection of any given layer due to the compressive load of subsequently-wound overlayers and by definition, u = 0 at r = g, the outer surface of the coil.

The corresponding expression for radial strain is

e = + I 3.r\ (5-29)

r 3r i o

where 3^ are unknown constants which in fact turn out to be dependent on boundary conditions.

Following the same mathematical steps outlined in section 5.3, utilizing equations (5-4)

(5-12), we obtain the differential equation

u" r + u' - k

2 u = - I [(1 + i - LV)3. + (d + i)LV - k2 )s

i ]r

±

. (5-30)

o

The solution to this is found to be

u = arK + br"K

- I (1 + i - LV^r1"*" 1 - E s.r

1+1

(5-31)

o o

where the unknown c^'s are related to the unknown 3^'s in eq. (5-29) by

94
The corresponding stresses are found to be

°r

= t

( L + V)rk-1 " J

(I

" V ) r "k_1

" L ^ c

i r± ( 5 " 33 > o

and

°0 = F" ( L + V)rk_1 + T ( L

" V ) r

~k_1

- L " d+Dc.r1

. (5-34)

o

The boundary conditions which must be met are:

(1) There is no radial stress at the outer surface, or

a

r

(g) = 0 (5-35)

(2) The deflection, u, is so defined by equation (5-23) that it must be zero at the

outer surface, or

u(g) = 0 (5-36)

(3) The ratio u/a^ at the inner surface must match that of the bobbin, given by

equation (5-27) , or

JiOL. = I . (5-37) a (1) B

13 in

Now, however, we have a procedural problem, since we have 3 boundary conditions and

n + 3 unknown constants where (n+1) is the number of terms in the polynomial expression of

the programmed winding tension. However, the final stress distribution i_s_ uniquely

determined by the given conditions, and the apparent excess of arbitrary constants in

method (2) demands additional manipulation but not additional boundary conditions.

The way to continue with method (2) is to recognize that we are working within the

domain of linear stress-strain theory, and that stresses from separate external forces are

additive. This suggests that we rewrite equations (5-31, 33, and 34) in the form

95
n n , . -k u = 2 u = 2 [a r + b.r - ((1 + i - LV)c. - s.)r ] (5-38)

o o

n n a

i k k-1 bi k -k-1 i a

r = Z °ri

= S r 2T (I + V)rk1 -^ f V)r k 1 - Lc/] (5-39)

o o

n n

k k-1 bi k -k-1 i o Q

= 2 o ei= 2 [-^ q + V)r

k + ^ (i - V)r k X-L(l+i)c

i

r

1

]

.

' ( 5-40)

o o We then require that each of the n components of strain u^ and stress a obey the boundary

conditions (5-30-32), — a procedure which will be justified at a later stage. When this

til is done for the i component, we obtain equations of the form

ga + g-k

b4

'-+ (LV-l-i)c. = s jg

2 (5-41)

i l i i°

(I

+ V) Va

i - <I- V)

± - Lg

1 c. = 0

<£ + v) |-v

(1 - -ba— )«

± + <1 + > \ +

(J +LV-i-i)c

i

= s r

The constants a^, b^, and c^ are readily determined from these three equations. An immediate

result, from the form of equations (5-41), is that for the general case

a.ill = b. = c. = 0 if s. = 0. l

For example, if the coix is wound at constant tension,

s ^ 0, s. = 0 for i > 0

o 1

we obtain equations equivalent to (5-25) and (5-26) in their dependence on r:

_ k-1 _ -k-1 + D (5-42)

o = Cr - Dr o

r

aQ = kCr

k_1 + kDr"^"1 + D . (5-43)

8 o

On the other hand, if a higher order polynomial, of degree n, must be used to describe the

programmed winding tension, then in eqs. (5-42) and (5-43)

C = I

( L + V) "E a

i' (5_44)

o

1 k n_1

D = f (£ - V) Z b., (5-45)

o

96
and for each non-zero s. an additional term D.r = Lc.r must be added to the right hand

1 11

side of equation (5-42), and (l+i)D_^r

1 must be added to the right hand side of equation

(5-43).

The justification for the unusual procedure of applying boundary conditions to each

th

of the i components u_^ and in equations (5-38) to (5-40) , rather than just to the

sums u = Zu. and a = la., is that in doing so we have obtained a final solution which fits

l i °

all physical conditions to the problem, namely plane-stress orthotropic material behavior

(eqs (5-5,6)), force balance (eq (5-13)), programmed winding tension (eq (5-28)) and

boundary conditions at the inner and outer radius, (eqs (5-35, 36, 37)). Since on a

physical basis these conditions to the problem are sufficient to determine a unique stress

and strain distribution within the coil, this mathematical solution must likewise be

unique.

Figure 5-5 shows circumferential and radial stresses calculated from these equations

for a particular coil geometry, assuming a constant strain of 0.001 due to winding tension,

i.e. ,

e

0 = u/r + 0.001. (5-46)

Multiplying this by the longitudinal modulus, L, yields the corresponding winding stress,

which is plotted in figure 5-5 as the dotted line. This applied winding stress must equal

the calculated circumferential stress at the outer radius, where there is no overwinding

to cause stress relaxation. In the example of figure 5-5, circumferential stress relaxation of up to 55% has occurred within the body of the coil during the winding process.

A significant test of the numerical results is to approximate the constant winding

tension of the above numerical example by the expression

e a = u/r + .002676r - .002379r 2 + .0007025r 3 (5-47)

which differs in magnitude from equation (5-46) by less than 0.04% when r is in the range

from 1 to 1.26, appropriate to the sponsors test coil. Thus, physically it describes

essentially the same programmed winding tension as was used for the example of figure 5-1,

but mathematically its solution invokes equations (5-38-40), with n = 3, rather than the

simpler equations (5-31-34) appropriate to a constant winding tension. We find that the

calculated stresses obtained with equation (5-47) agree within 0.1% with those used in

97
Bobbin: a = 0.8 E = 19.3 x 10

6 N/cm

2

Coil : E r

= 0.08 x 10

6 N/cm2

E0

= 7.6 x 10

6 N/an

2

5-5. Radial and circumferential stress calculated from Eqs. 5-42, 43 for a particular coil wound at constant strain of 0.001, equivalent in this example to

a tensile stress of 7.6 kN/cm2, shown as the dashed line. Because of the

very low value of E r , strain relaxation of up to 55% occurs within the winding.

98
plotting figure 5-5. This further confirms the validity of the analytical assumptions

which were made in extending equations (5-31-34) to the more general form of equations

(5-38-40).

Another test of the analysis is made by assigning an artifically high modulus to the

bobbin, such that it becomes almost infinitely rigid. Under this condition the winding

strain at r=l cannot change as subsequent layers are overwound, or in other terms, u(l) = 0,

regardless of the shape and magnitude of the programmed winding strain curve away from the

point r=l. Calculated results, not shown, are in agreement with this prediction.

5. 7 Epoxy Impregnation of the Coil and Bobbin Removal

After being wound, the coil is impregnated with epoxy to form a rigid, monolithic

structure.- This takes place at an elevated temperature, the curing temperature of the

epoxy. The change in stress as the unimpregnated coil is warmed from room temperature

to the curing temperature, and recooled to room temperature after impregnation can be

calculated separately using the results of section 5.5.

We assume that the coil stresses do not change during the curing operation. Actually

the epoxy may shrink during the curing operation, resulting in some change in the stresses,

but in practice the change in stress distribution is probably not large. This point needs

further study, which we will address in the future.

Assume that the stresses in the coil at the beginning of the impregnation process are

known. Since we are working within the framework of elastic behavior, these stresses can

be uniquely divided into (a) "internal" stresses satisfying the boundary conditions

o^(l) = 0 and O (y) = 0, and (b) "external" stresses, due to the compressive pressure P

of the bobbin on the inside diameter of the coil. When the bobbin is removed, the "external"

stresses disappear, leaving just the "internal" stress.

The stresses due to a compressive pressure P at the inside diameter are easily determined from equation (5-19) by applying the boundary conditions

to determine the two unknown coefficients in the equation. The same coefficients occur

in equation (5-20), so that the circumferential stress distribution Og is likewise determined.

The material property, k, which appears in these two equations is that of the composite coil

a

r (l) = -P »

99
after epoxy impregnation. When the bobbin is removed, the two stress distributions

o and a„, so calculated, are simply subtracted from the total stresses which existed

r 6 as the beginning of the impregnation process. The corresponsing dimensional change of

the bobbin is

Au = rAe„ = r( —— - VAa ).

a L r

This process of epoxy impregnation and bobbin removal results in a general reduction of o fl (r) from that established by the winding process, such that

a' _(r) = 0. 1

6

Since in general, for the coils we consider, 0^ is much less than the Og due to winding

tension, the curve CqCf) does not change shape greatly during this process. This change

is illustrated in figure 5-6, which shows the initial and Og, taken from figure 5-5,

and the final a and Og after epoxy impregnation and bobbin removal, calculated from the

above equations for a particular model coil.

5.8 Magnetic Forces

The magnetic field generated by a cylindrical coil will be a function of both r and z,

resulting in a stress distribution which may be significantly different from either the

(2-dimensional) plane stress or plane strain solutions which we have used up to this point.

The only practical way to obtain complete answers to this problem is to use a 3-dimensional,

finite-element analysis. However, for at least some geometries, the axial stresses will

be somewhat smaller than the circumferential tensile stresses, and our simple 2-dimensional

analysis will provide a rough approximation to the stress distribution at the midplane

where (Jq is largest. Furthermore, changes in the operating stresses following changes

in fabrication techniques and/or material properties can perhaps be estimated to reasonable accuracy even though the true stress distribution may be more uncertain. We give

here the 2-dimensional solution for magnetically-induced stresses as the midplane of our

orthotropic magnet winding.

The magnetic field causes a body force dF^ on every element of the coil, proportional

to the vector cross-product of field and current. Considering just the axial component of

magnetic field, which produces a radial force, eq (5-13) becomes"''^

100
1.000 1.052 1.104 1.156 1.208 1.260

r/r

i

Radial and circumferential stresses for a particular coil as wound, and aft

epoxy impregnation and removal of the bobbin on which the coil was wound.

101
9a 3F

r dT +

°r " °6

=

"r W- ' (5 " 48)

For a rectangular coil cross-section carrying a uniform current density,

dF B

2 r B B

r -~ = h A

,° ri 1 ^ = H ^ r (5-49) dr 4ttc> (g-1) J B

q

B

q

where B

q is the magnetic field at the center of the coil, and

2 2 1/2

* - -S- £n ( g + (g + BV/Z

, 9 e-1 V 2 1/2 ; g

1 + (1 + 3 )

is the Fabry factor. In this expression 3 is the coil length divided by inside diameter.

The parameter H, defined by eq 5-49, is proportional to the square of the magnetic field.

Consistent with the accuracy of the 2-dimensional calculation itself, we may take

r

= h U • (5 " 50)

i.e., assume that B

z falls off lineraly from the value hBQ

(h ~ 1 to 2, depending on 3 and

g) at the inner surface of the windings to zero at the outer surface. The parameter h,

defined by eq 5-50, is the field at the inner radius divided by the field at the center

of the coil; for the sponsors test coil, h = 1.53.

Now, following the procedure of section 5.4, the components of stress due to the

magnetic field can be obtained by solving the differential equation

, , 2 u LHhg , LHh ,

+ u - k — = r +

8-1 • ' 8-1

4

• (5 "51)

The general solution to this equation is

u = ark + br

k + cr

2 + dr

3 (5-52)

where c = LHhgA

(g-l)(k

2 4)

LHhA and d = (g-l)(k2 -9)

The corresponding stresses are

O

r = Ark 1 - Br

k 1 + (|- + V) | r + (| + V) f

r

2 (5-53)

102
aQ = kAr

k-1 + kBr -k-1 + (| + 2V) | r + (| + 3V) | r

2 (5-54)

where, as before

and

B-!<r-v,.

These two integration constants must be evaluated from the two boundary conditions

a

r

(l) = 0 = a

r (g).

2 Evaluation is straightforward. All coefficients are proportional to B , as is anticipated.

Numerical evaluation of these magnetic stresses for a particular coil is shown in

figure 5-7. Further work discloses that the calculated stresses are remarkably independent

of material anisotropy parameter, k, and the Poisson's ratio. Varying k between 1 and 3,

and V between zero and 0.4/L changes the calculated circumferential stresses by only

about 1%, which is far smaller than the probable accuracy of the 2-dimensional calculation

itself. The radial stress exhibits greater variation but at the same time it is generally

one or more orders of magnitude smaller than the circumferential stress, and hence probably

inaccurate in the face of axial compressive stresses which have been neglected in the

2-dimensional calculation. These numerical results agree with those presented by Hord [15]

for an isotropic coil winding in the same two-dimensional approximation.

Greater sensitivity to k and V would probably be found if the coil were overwound with

a load-bearing wrap carrying no current. In this case there would be a finite O at r=g

which would reduce z r within the coil by an amount dependent on both coil and wrap parameters.

We have not performed this calculation, but plan to do so within the next reporting period.

We conclude that, within the two-dimensional approximation, and for cylindrical coils

unsupported by external wraps, previously existing magnetic stress analyses based upon

isotropic magnet winding structure are satisfactorily accurate for this stage of our calculations. Improvement in the accuracy of the calculated stresses due to the magnetic field

requires primarily a three-dimensional calculation (using finite element techniques)

Results from these calculations (Professor Young's work) will be reported later, when

they become available.

103
10,000

CO

CO _ UJ 5,000

or

y—CO E r = ZSMN/cm*

Ej= 7.6MN/cm2 B„=4Tesla

o

-Rodiol

1.000 1.026 1.052 1.078 1.104 1.130

r

r

i

.156 1.182 1.208 1.234 1.260

Figure 5-7. Radial and circumferential stresses for a particular coil when it generates

a field of 3 Tesla at the coil center. As discussed in the text, these results

are quite insensitive to reasonable variations in coil material parameters.

104
5 . 9 Summary of Numerical Results

Within the framework of elastic stress-strain behavior used in this study, the

cumulative stresses and strains under operating conditions will be just the sum of the

stresses and strains developed at each stage of the fabrication and operation procedure.

Thus, it is meaningful to analyze the effects of materials and fabrication variables

just in terms of the stresses and strains which are immediately affected, without totalling

all the effects. In particular, for simple solenoidal magnetic fields generated by an

unsupported coil, the calculated stresses and strains generated by the magnetic field are

nearly independent of the material anisotropy, as discussed in section 5.8. We consider

here only the stresses and strains related to fabrication and cooldown of the coil.

As the various stages of analysis were developed, the equations were programmed into

our timeshare computer. The program as it exists on July 1, 1976 is listed in section

5.12. As will all such programs, developed by stages over a period of time, it is far from

optimized in operation and format, but it does give useful results. The numerical results

presented here were obtained with this program.

5.9.1 Winding Tension

First we consider the variation in internal stresses with different programs of

winding tension. On an early model coil, the sponsor used a winding tension which decreased

in four equally-spaced steps from the inside radius to the outside radius, as described

in the following table (the coil fabrication data were presented in more detail in the

previous report to the sponsor NBSIR-73-349)

layer wire stress strain

? 6 2 (N/cm ) (assuming E=10.4xl0 N/cm ) 1-10 20500 0.001973

11-20 15400 0.001480

21-30 10300 0.000986

31-40 7400 0.000493

This stepwise programming is not easily handled by our analytic model, but it can be

approximated by assuming a linear variation between the inner and the outer radius. In

figures 5-8 and 5-9, the resulting circumferential stress and radial stress (curves A) after

epoxy impregnation and bobbin removal are compared with that obtained by three other

possible winding tension programs: curve (B) a constant winding tension equal to the

105
o

1—1

( p d ill z oi) 'ssms "iviiNaaBdwnDHio

106
n 1 1 1 r

A

1T000 1.052 1.104 1.156 1.208 1.260

r/r.

1 Figure 5-9. Radial stresses after epoxy impregnation and bobbin removal, for the four

different programmed winding tensions illustrated in Figure 5-8. Note the

change from radial tension in curve A to radial compression in the other

three curves

107
maximum tension of (A); curve (C) , linearly increasing tension, with limits just reversed

from that of curve (A) ; curve (D) , a quadratically variable tension which rises rapidly

from the lowest value used in (A) to the maximum, about 2/3 of the way towards the outer

radius, and then decreases. The dashed lines in the figures give a stress which is

proportional to the programmed winding tension (actually it is LZS^r 1

, which equals the

winding tensile stress times the area ratio, 0.73), and the solid lines give the resulting

internal stresses after epoxy impregnation and bobbin removal, ignoring any thermal effects.

The material and geometry parameters used in this calculation approximate those of the

sponsor's model coil. Several important conclusions can be discussed with reference to

these figures.

A decreasing winding tension, as in curve A, results in the lowest average circumferential

stress, but also results in circumferential tension at the inner radius and radial tension

throughout the coil. Radial tensile stress will tend to cause crack propagation at any

bonding defect between the epoxy and the wire, and should thus be avoided. Furthermore,

subsequent magnetic-field-induced stresses are higher at the inner radius than at the outer

radius, so that a lower maximum stress under operating conditions will be obtained by

creating a compressive circumferential stress at the inner radius during the fabrication

and impregnation procedure.

Curves B, C, and D of figures 5-8 and 5-9 relate to different possible programmed

winding tensions which were investigated in this study. All three have the advantage of

providing both circumferential compression on the inner layers and radial compressive

stress throughout the coil. Of these, the quadratic curve D provides the largest and most

linear variation in circumferential stress from inside its outside radius, and thus will

provide the lowest and most uniform stresses in the coil when the magnetic field is turned

on.

It is important to note that the magnitude of the internal stresses (solid curves of

figure 5-6) will scale linearly with the magnitude of the winding tension (dashed curves

of figure 5-6). Hence, the higher the maximum tensile stress during fabrication, the

higher will be the built-in internal strains before the coil is energized, and, assuming

a programmed tension like C or D, of figure 5-8, the lower will be the maximum stresses

when the coil is energized. The limit to this procedure for minimizing the operating

108
stresses will be reached when the winding tension causes either yielding or a degradation

in the superconducting properties of the wire. We conclude that this last topic needs

careful attention.

5.9.2 Effects of Properties Variations

Numerical predictions, and the Conclusions drawn from them, do depend on the assumed

values of material properties, and on the coil configuration. It is important to investigate

the sensitivity of the conclusions to changes in coil parameters.

Properties of the impregnated coil have been measured in our laboratory, but, for

calculation of winding stresses, we must estimate the moduli of the un-impregnated coil.

The circumferential modulus, Eg, is that parallel to the wire, and can be estimated

with reasonable accuracy knowing Young's modulus for the wire E and the packing fraction, w

A, of the wire in the winding. Since the glass cloth does not provide much strength

parallel to the wire,

For the sponsor's coil

EQ = 0.73 x 10.4 = 7.6 MN/cm2. 0

This is about the same as we have measured on the impregnated coil, which indicates that

the epoxy does not add much strength parallel to the wire. The transverse modulus, E , is

best estimated by considering the average modulus of a thickness d^ of glass cloth of

modulus E in series with a thickness d of wire of modulus E , with the result

p WWE E (d + d ) E = P w p w

J r E d + E d

p w w p

2 3 Since E ~ (10 to 10 ) x E , this reduces to w p

d + d E z E -E- = . r p dP

Our measurements on samples of the cloth show that its stress-strain behavior is quite

hysteretic, represented approximately by —da _ „ _ . . „ ^ 22a for increasing 0

d£ 6

109
4~ ~ 50a for decreasing a

2 for a in the range 5 to 5000 N/cm . To apply this result to our formalism based on linear

or elastic material behavior, we take do/de as a local modulus, valid for small variations

around an average compressive stress within the coil. This is obviously not a good

approximation, but to improve upon it within the framework of this general analysis would

require very cumbersome non-linear stress-strain theory, and would not advance the project

measurably.

The calculations discussed in 5.9.1 above, suggest that the magnitude of for

realistic values of the other parameters will be in the range from near-zero to about

2 2 2000 N/cm ; 1000 N/cm is a reasonable average, though it is difficult to know what relative weights should be given to the lower and higher values. Thus, with d^ - .004" and

d - .021", we estimate w

(E ) ^ 22 x 1000 x a .15 MN/cm2. r avg .004

After epoxy impregnation, our measured value is

E =5 2.8 MN/cm2

. r

Poisson's ratio in the transversely isotropic body is given by eqs (5-4) and (5-9).

In this case with E n > E , we take

8 r

( ^ ) ~ ~" s 0.04 cm

2 /MN.

Figure 5-10 shows the effect of varying the radial compressive modulus over a

realistic range encompassing the value which is estimated above. It is seen that the

results are sensitive to the assumed value of E before epoxy impregnation, but not so

much so that the conclusions drawn in 5.7 above are significantly affected. Figures

5-11-13 compare calculated stresses with those obtained by Professor Young at Wisconsin,

who was able to obtain a more accurate non-linear solution by iteration, using twodimensional finite-element program. (Professor Young's work was discussed earlier, in

section 5.2). In these latter comparisons, a parabolic distribution of winding tension

was assumed, approximately like curve D of figure 5-8. Figure 5-11 gives the circumferential stresses after winding but before epoxy impregnation. Figure 5-12 gives the

110
o

CSJ

00o

CVJ

ID

CMo

(Bd^OI) 'SS3U1S 1VUN3a3JWn3cJI3

111
r •

CO u

00 I CD C 00 13

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rH

T3 co

at

X CU TJ 0 c CU a •H J=

cO cu rH CO M CO OC cu T3 TD 0 •r-l o iH0 M

cu CO

W

piQ ti CO

03 •H 4J 3

r-l

M rH rH 3 O 3

i-l 13 a. -a 0 O cu 0 U 3 n s M i—

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co cO •H CO i—

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co

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p.. cu

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112
H 01

cfl > • U ti 13 C !>% H w n .O

CO OJ C u 13 o •H 01

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—I cd 3 -h o -a

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113
corresponding radial stresses. Figure 5-13 gives the change in circumferential stress

following epoxy impregnation and bobbin removal. It is seen that the non-linear solution

2 agrees reasonably well with the linear solution for Eg ~ 0.08 MN/cm . This latter value

is within the range which we would expect for an appropriate average value of do/de over

the cross-section, though it is lower than estimated above.

The variation in circumferential stress with bobbin characteristics is shown in

figure 5-14. It is seen that the softer bobbin, i.e., as a + 1, gives a more uniform

variation in stress with radius, and a larger total variation from inside to outside radius.

Both of these are desirable. However, the variation is not very large and probably comparable with the uncertainty in the calculations due to the non-linear glass cloth

characteristics.

The circumferential stresses after epoxy impregnation and bobbin removal are plotted

in figure 5-15 for coils of various thicknesses. It is seen that the thicker coils have

a quite non-linear internal stress distribution, and considerable restraint must be used

in extrapolating the results prescribed in this report for the sponsors test coil (g=1.26)

to other geometries.

The sensitivity of the calculated stresses to changes in Poisson's ratio, eq 5-4,

has also been tested. The results vary only about one percent as (v/E) changes from 0 to

0.4/Eq for these two-dimensional calculations. However, we expect that the threedimensional stresses, from the magnetic field, will be reasonably influenced by the

Poisson's ratio.

5.9.3 Thermal Stresses

Thermal stresses have been discussed in some generality in section 5.5. For the

sponsors coil, our mechanical and thermal properties measurements on impregnated coil

samples give

LAaAT - -20900 N/cm

for contraction from 65°C to 4 K and

k = 1.59.

With g-1 = 0.26, figures 5-2-4 give the results, that the circumferential stress at the

inner radius will increase by

114
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115
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X>

M 13 0 O 0) J3

<4-4 4-1 CO cu

CO rH j=

co 3 4J

ai o U r-l <H

4-1 CO

O

CO o H « -a

CO i-l •H CO o

4J > C O

CU e u cu e

CU

u

H IH 43 6 C ^ 3 -H C O 43 43 M 43

•H

O cu

O 43 43

44

CU T) 4= C IH H CO o ( B dz oi) * sshhis iviiN3a3Jwnaun

116
o

c

^1 00

QJ r-l ^£ > (J o •H a ricuu CO

,—

3

•r-l •H o

CJ cfl n

o 4J

r-|

CU • H U o

11 o

U-J M-l 0) •H XI 0 X) •rH •H CO 4J (_l n) 3 H

0

bO 14-1 •I* 0)

t_i CO D. o g *H •H T3 U CO%4 o t—

pi •H

0) co o

cu

(-1 H

CD cu 4J T3

C1J •H

cfl D. CO O 4-1 CO 3

(0 O

o> v—

H 4J tfl o

10 H •H

rJ 4->

i—

Q) nj CD j-i U

•H td

4-) E CD a sz

<a 14-1 jj u o

(U >^

4-1 <D

CO T) U 01 u C •r-l •H 4-1 U > •H H CJ

o> W) cu JS cx H cfl CO

( e d

z 0l) ssaais iviiN3y3dwnoyiD

117
Aa (r.) ~ 2700 N/cm2 c 1

the circumferential stress at the outer radius will increase by

Aa (r ) 2 -2000 N/cm2 c o

and the maximum radial stress will increase by

Aa ~ + 1300 N/cm2. r

These are not large stresses, but unfortunately they all go the wrong direction, i.e.,

they tend to increase the maximum circumferential stress (at the inner radius) when the

magnet is turned on, and to increase the opportunity for delamination due to radial

tension. Considering the last point, it is important that the radial compression due to

winding tension always be larger than the radial tension developed during cooldown, so

that the cuculative radial stresses are always compressive, and delamination or crack

propagation will be inhibited. This is another reason for using high winding tension

during the fabrication process.

5.9.4 Cumulative Stresses

The cumulative circumferential and radial stresses during fabrication, cooldown,

and magnetic field operation are illustrated in figures 5-16 and 5-17. The solid curves

in the figures refer to a model coil previously built by the sponsor. The dashed curves

give predicted stresses for a coil of the same geometry, but fabricated differently:

(a) wound on a thin-shell aluminum bobbin rather than a solid stainless steel one, (b)

wound with a variable tension (curve "D") which increases from inside to outside radius

about like curve D of figure 5-8, rather than decreasing from inside to outside radius,

like curve A of figure 5-8. The dashed curves show a more uniform operating stress than

do the solid curves, with a reduction of about 26% in the maximum circumferential stress

and 44% in the maximum radial stress. These stress reductions are significant, and the

potential of achieving them should be considered in future designs.

5 . 10 Conclusions and Recommendations

The analysis developed here provides a reasonable means for relating anisotropic

materials and fabrication parameters to stresses and strains in transversely isotropic

coil structures. Results are quite sensitive to both elastic modulus and thermal expansion anisotropy, especially as coil thickness is increased. When applied to the design

118
Figure 5-16. Cumulative circumferential stresses for a particular coil, calculated for

a winding tension which decreases linearly from inside to outside radius

(solid lines) and for a winding tension which increases parabolically from

the minimum at the inner radius to a maximum in mid-winding and then

decreases slightly (dashed lines). The maximum and minimum winding strains

are the same in both cases. The dashed line (4) exhibits a significant

reduction in maximum operating stress compared to the solid line (4)

119
Figure 5-17. Cumulative radial stresses for a particular coil, calculated for a winding

tension which decreases linearly from inside to outside radius (solid lines)

and for a winding tension which increases parabolically from the minimum at the inner radius to a maximum in mid-winding and then decreases slightly

(dashed lines) . The maximum and minimum winding strains are the same in

both cases. The dashed line (4) exhibits a significant reduction in maxi- mum operating stress compared to the solid line (4).

120
of a test coil for the sponsor, we can develop the following recommendations:

(1) The tension on the wire during the fabrication process should be programmed

approximately as curve (D) in figure 5-8. On a practical basis it would probably have

to be stepped from layer to layer.

(2) The maximum winding tension, e.g., the peak in curve D, should be as high as

possible without causing subsequent degradation of superconducting wire performance. What

this limit is should be determined.

(3) Peak operating stresses will be reduced if the composite radial modulus during

the winding process were higher. This could be achieved by using a thinner cloth, or

perhaps preimpregnating the wire and cloth and partially curing as winding progresses.

(4) Shrinkage of the epoxy during the curing process needs to be determined

(5) Peak operating stress will be lowered as the bobbin is made "softer". This

could be accomplished by using a thin aluminum rather than a solid stainless steel mandrel.

An equivalent effect could be obtained by wrapping the mandrel with several layers of

glass paper before starting the winding.

Future work which should be pursued includes the following:

(a) The analysis should be enlarged to accomodate the effect of an overwrap or

other mechanical constraint at the outer radius.

(b) Shrinkage stress during curing needs to be evaluated.

(c) The maximum allowable tensile stress during the winding process should be

evaluated

(d) Microscopic analysis needs to be done to evaluate r.he stress distribution from

one wire through the epoxy to the next wire as a function of the average stress and

component properties. This should lead to prediction of inter-laminar shear and, perhaps,

to additional recommendations on materials and fabrication techniques which are beyond the

scope of this macroscopic analysis.

Note added in proof : Recent work discloses that the method developed here for calculation

of winding stresses, Eqs. (5-28) to (5-41), is an approximation to the (exact) integration

method mentioned on page 94. The results as outlined in this report agree within a percent or so with the exact calculations when the bobbin is quite stiff, i.e. when the bobbin

modulus, B, defined by Eq. (5-37), is comparable with the winding modulus, L, Eq. (5-7).

For softer bobbins (lower values of B) , the exact calculation predicts a load relaxation

at the inner winding layer which is up to 25% greater than predicted here (load relaxation

121
is illustrated in Figure 5-5, p. 98). The errors from using this approximate calculation

rather than the exact integration technique are not very serious for design purposes, and

our conclusions above remain valid.

122
5. 11 Nomenclature

a,b,c,d,C,D

B

E

g

h

H

k or K

L

s

r

T

u

V

a

a

Subscripts

r,6,z

B

i

i and o

t

I

adjustable coefficients

defined by eq. 5-27; an elastic modulus of the bobbin

elastic modulus

outside coil radius v inside coil radius

magnetic field at inner layer f magnetic field at center (defined by equation 5-54)

a quantity proportional to the square of the magnetic

field (defined by equation 5-53)

mechanical anisotropy parameter (defined by equation 5-17)

longitudinal elastic modulus (defined by equation 5-7)

a strain coefficient, defined by equation 5-28

radius t inside coil radius

transverse elastic modulus (defined by equation 5-8)

= 6r, a small radial displacement v inside coil radius

Poisson's ratio t elastic modulus (defined by equation 5-9)

thermal expansivity or inside radius t outside radius

of bobbin (eq. 5-27)

inverse square of a modulus (defined by equation 5-12)

strain

stress

cylindrical coordinates

refers to bobbin

integers 1,2,3 ....

inside and outside diameter (or radius)

transverse to wire; the r or z directions

123
5 . 12 computer Program

1 REM GO TO 9900 FOR INDEX TO VARIOUS SUBSECTIONS

5 12 = 0 ! NONZERO VALUE SURPRESSES PRINTOUT OF WINDING

20 DIM RC 1

,2) ! RADIAL STRESS

2 1 MAT R= ZER

22 DIM Q( 1 1 ,2) ! THETA STRESS

23 MAT Q = ZER

24 DIM U(11 ,2) ! RADIAL DISPLACEMENT

25 MAT U = ZER

26 DIM xcn

)

! RADIAL POSITIONS

30 DIM A(4,4)

32 DIM B(4,4)

34 DIM C(4) ! GENERAL CONSTANTS

36 DIM F(4) ! ii ii 38 DIM W(4) ! WIRE STRAIN POLONOMIAL COEFFICIENTS

40 DIM S(4) ! INPUT WINDING STRAINS

42 DIM D(6) ! ALPHAS IN THE U EQUATION

44 DIM E(5,2) ! MODULI

46 DIM Y(5,2) ! ALPHA DELTA T*S

50 DATA .07,5.0 , 2*9,7.8, 3.8,8.4, 4.15,8,8, 7.1,7.1

51 MAT READ E 52 MAT E=(1.0E+06)»E

6 0 DATA 1,1,1,1 ,-.00470, -.00275, 1,1, -.00255,- .00255

6 1 MAT READ Y 102 R1= 1

104 R2=12.6

106 R3=R2+( .250*2.54)

108 R0=8

110 H8=1.32 ! LENGTH/I. D. 112 H0=5*0.897 ! CENTRAL FIELD, TESLA

114 H9=1/0.897 ! PEAK FIELD/CENTRAL FIELD

116 H7=-0.2 ! OUTSIDE FIELD/CENTRAL FIELD

1 18 M7=(H7-H9)/(R2/R1-1

)

124 GOSUB 7000

130 FOR J=1 TO 11 140 X(J)=R1+(R2-R1)«(J-1)/10

150 NEXT J 170 TYPE "COIL R1, R2" , R1,R2

200 PAUSE

215 DATA 0.3,0.9,1.1 ,0.9

220 MAT READ S 222 MAT S=(.001)«S

230 TYPE "EFFECTIVE BOBBIN MODULUS", BO

295 TYPE "WINDING STRAIN", S ( 1 )

, S ( 2 ), S ( 3 ), S ( 4

)

300 REM EVALUATE W*S

310 MAT A= IDN

320 C( 1 )=R1

322 C(2) = R1 + 0.3*(R2-R'I ) 324 C(3) = R1 + 0.7*(R2-R1 ) 326 C(4)=R2

330 FOR 1=1 TO 4 340 FOR J=1 TO 4 34 2 A(I,J)=C(ir (J-1 ) 344 NEXT J 346 NEXT I 350 MAT B=INV(A)

352 MAT W=B*S

360 N=1

365 GOSUB 8000

375 TYPE "COIL, ER, EL, K =", E(N,1), E(N,2), K 400 REM SET UP STRESS EQUATIONS

405 D1=1+B0*K6

124
410 D2=1-B0»K7

420 X1=R2

430 GOSUB 700

440 17=19 ! VALUE OF INTEGRAND AT THE LAST STEP

446 16=0 ! CUMULATIVE VALUE OF THE INTEGRAL

450 R ( 1 1 ,1 )=0

455 Q( 1 1 ,1 ) = T1

460 U( 1 1, 1 )=0

470 X2=(R2-R1 )/10

500 FOR J=2 TO 11 510 Z=X(12-J)

520 X1=Z+X2/2

530 GOSUB 700

540 15=19

550 X1=Z

555 GOSUB 700

560 I6=I6+(X2/R1/6)*(I9+4«I5+I7)

600 C9=(-D1 )«X9*I6

605 D9= (-D2 ) *I6/X9

610 R(12-J,1 )=(C9-D9)/Z

615 Q(12-J,1)=T1+K*(C9+D9)/Z

620 U(12-J,1)=K7*C9+K6*D9

630 17=19

650 NEXT J 660 FOR J=1 TO 11 665 GOSUB 7500

670 NEXT J 680 P1=R( 1

, 1

)

690 GO TO 900

700 REM X1 IS THE LOCAL VALUE OF R 710 GO TO 800

720 X9= (X1/R1 ) *K

730 I9=R1*T1/(D1*X9-D2/X9)

750 RETURN

800 REM LOCAL VALUE OF WINDING STRESS

805 T1=0

810 FOR J1=1 TO 4 820 T1=T1+W( J1 )*X1~ (J1-1 ) 830 NEXT J1

840 T1=T1«L

850 GO TO 720

900 PAUSE

1000 TYPE "EPOXY IMPREGNATION AND BOBBIN REMOVAL"

1020 MAT A=IDN

1040 MAT C=ZER

1050 MAT F=ZER

1060 N=2

1065 GOSUB 8100

1 070 PRINT "ER,EL,K" ,E(N , 1 ) ,E(N ,2) ,K

1150 R6=R1"K

1152 R7=R2*K

1155 A(1 ,1 )=R6/R1

1160 A( 1 ,2)=-1/R6/R1

1165 A(2,1)=R7/R2

1 170 A(2,2)=-1/R7/R2

1175 C( 1 )=P1

1200 MAT B=INV(A)

1220 MAT F=B«C

1300 FOR J=1 TO 11 1320Z=X( J)

1325 Z6=Z~K

1330 R(J,1)=-(F(1)*Z6/Z-F(2)/Z6/Z)

1335 Q(J,1 )=-(K«F(1 )*Z6/Z+K*F(2)/Z6/Z)

125
1 340 U(J,1)=Z»(Q(J,1)/L-R(J,1)*V)

1360 GOSUB 7500

1400 NEXT J 1420 12=0

1600 PAUSE

2800 TYPE "FOR COOLDOWN WITH OVERWRAP GO TO 30C0"

2850 PAUSE

2900 GO TO 3500

3000 PRINT "COOLDOWN OF COIL AND OVERWRAP"

3034 TYPE "WRAPPER, EL, R3" , E ( 5 , 1 )

, E ( 5 , 2 )

, R3

3036 TYPE "WRAP, A R , AL", Y(5,1), Y(5,2)

3038 K9=2

3040 GO TO 3520

3060 A(4 , 1 )=K7*R2*K

3065 A(4 ,2)=K6/R2*K

3070 L1=L

3075 V1=V

3080 K1=K

3090 N=5

3095 GOSUB 8100

3100 A(3,3)=R3*(K-1)

3105 A(3,4)=-R3~ (-K-1

)

3110 A(2,3)=-R2"(K-1)

31 15 A(2,4)=R2* (-K-1

)

3140 A(4 f 3)=-(K/L-V)»R2~K

3150 A(4 ,4 ) =-(K/L+V)/R2'> K 3160 MAT B=INV(A)

3240 Y5 = Y(5 , 1 )-Y(5 ,2)

3244 IF ABS (K-1)<0.01 GO TO 3280

3250 S4=L*Y5/ ( 1-K»K)

3255 S5=S4

3258 S6=S4

3260 GO TO 3300

3280 S4=L»Y5»L0G(R2)/2

3282 S5=L*Y5«L0G(R3)/2

3285 S6=S4+L»Y5/2

3300 F(2)=F(2)+S4

3305 F(3)=-S5

3310 F(4)=-R2»(S3/L1-S2*V1+Y(3,2))+R2*(S6/L-S4*V+Y(5,2))

3350 N=3

3360 GOSUB 8100

3400 GO TO 3710

3500 TYPE "COOLDOWN OF COIL WITHOUT OVERWRAP"

3507 K9=0

3520 N=3

3525 GOSUB 8100

3530 TYPE"ER, EL, K", E ( N , 1 ) , E ( N , 2 ) , K 3540 T Y P E " A R , AL ", Y ( N , 1 )

, Y ( N , 2

)

3600 MAT A = I DN

3605 MAT C=ZER

3610 MAT F=ZER

3615 A( 1 , 1 ) = R1

" (K-1

)

3620 A(1 ,2)=-R1~(-K-1)

3640 A(2,1)=R2~(K-1)

3645 A(2,2)=-R2"(-K-1

)

3650 Y3=Y(3,1 )-Y(3,2)

3655 IF ABSCK-1 ) <«0Q1 GO TO 3680

3660 S1=L«Y3/(1-K«K)

3665 S2=S1

3667 S3=S1

3670 GO TO 3700

' 3680 S1=L*Y3*L0G(R1 )/2

3685 S2=L*Y3*L0G(R2)/2

126
3690 S3=S2+L*Y3/2

3700 F( 1

) =-S1

3705 F(2)=-S2

3708 IF K9 > 1 GO TO 3060

3710 MAT B=INV(A)

3715 MAT C=B«F

3720 FOR J=1 TO 11 3721 Z=X(J)

3722 Z6=Z~K

3723 IF ABSU-1 )>0.001 GO TO 3730

3724 S1=L*Y3*LOG(Z)/2

3726 S2=S1+L*Y3/2

3730 R(J,1)=C(1)«Z6/Z-C(2)/Z6/Z+S1

3740 Q(J,1)=K»C(1 )*Z6/Z+K*C(2)/Z6/Z+S2

3 750 U( J,1)=Z»(Y(3,2)+Q( J, 1 )/L-R( J,1 )*V)

3780 GOSUB 7500

3800 NEXT J 3810 PAUSE

4800 TYPE "FOR MAGNETIC FIELD WITH OVERWRAP GO TO 5000"

4880 M7=(H7-H9)/(R2/R1-1)

4900 PAUSE

4905 M5=0

4910 GO TO 5100

5000 N=5

5030 GOSUB 8100

5040 G2=(R3/R2)*K

5 06 0 M5=( 1/G2-G2)/(K7/G2+K6»G2)/G

5080 TYPE "OVERWIND, E R , EL , DR" t E ( N , 1 )

, E ( N , 2 ) , ( R 3-R2

)

5100 MAT A= I DN

5110 MAT F = ZER

5120 N=4

5140 GOSUB 8100

5150 G=R2/R1

5160 G6=G*K

5180 TYPE "COIL,ER r EL,K f M" f E(N f 1 ) ,E(N,2) ,K,M5

5185 TYPE "LENGTH/I. D. =", H8

5187 TYPE "BZ(R1)/B0", H9

5188 TYPE "BZ(R2)/B0", H7

5190 TYPE "CENTRAL FIELD, TESLA =", HO

5200 A(1 ,1 )=G6* (M5-1/G/K7)

5205 A(1 ,2) = (M5 + 1/G/K6)/G6

5210 A(2,1 )=1/K7

5220 A(2,2)=-1/K6

5 300 H2 = H0»H0/4/3* 14159/ (H8*LOG( (G + SQR(G*G+H8«H8) )/( 1+SQR( 1 + H8»H8) ) ) ) 5305 H2=-1000»H2 ! FIELD IN TESLA STRESS IN N/CM2

5320 C2=H2* (H9-M7 ) *L»E9/ (4-K*K)

5330 C3=H2«M7»L«E9/ (9-K»K)

5340 L2= (2/L+V)«C2/E9

5345 L3=(3/L+V)*C3/E9

5347 T2= ( 1/T+2»V) *C2/E9

5348 T3=(1/T+3*V)»C3/E9

5 35 0 F( 1 )=L2»G-M5*C2*G«G+L3*G*G-M5«C3*G~3

5360 F(2)=-L2-L3

5400 MAT B=INV(A)

5410 MAT C=B*F

5450 FOR J=1 TO 11 5455 Z=X(J)/R1

5460 Z6=Z"K

5 470 R( J,1 )=C( 1 )»Z6/Z/K7-C(2)/Z6/Z/K6+L2*Z+L3*Z»Z

5 4 80 Q ( J , 1

) =K«C( 1 )«Z6/Z/K7+K«C(2)/Z6/Z/K6+T2»Z+T3*Z«Z

5 49 0 U( J,1 )=Z»(Q( J,1 )/L-V«R( J,1 ) ) 5495 U(J,1)=U(J,1)»R1

127
5540 GOSUB 7500

5600 NEXT J 5700 PAUSE

7000 REM CALCULATION OF BOBBIN STIFFNESS

7002 N=5

7004 GOSUB 3100 ! ASSUMES PLANE STRAIN FOR BOBBIN

7006 X7=(R0/R1)"2

7010 B0=1/((1+X7)/(1-X7)/L-V)

7020 RETURN

7500 U( J,2)=U( J,2) + U( J,1 ) 7520 R( J,2)=R( J,2)+R( J,1

)

7540 Q(J,2)=Q(J,2)+Q(J,1)

7599 RETURN

7999 GO TO 9999

8000 REM PLAIN STRESS

8010 L=E(N,2)

8020 T = E ( N , 1 ) 8030 V=0,33/E(N,2)

8040 GO TO 8200

8100 REM PLANE STRAIN, ASSUME ISOTRCPY IN RZ PLANE

8110 V = 0 * 3 3* 1 «33/E(N,2)

8120 L=E(N f 2)/(1-0.33*(0,33*E(N,1)/E(N,2)))

8130 T=E(N f 1 )/( 1-0*33*0*33)

8200 E9=1/L/T-V*V

8205 K=SQR(L/T)

8210 K6=K/L+V

8220 K7=K/L-V

8250 RETURN

9000 FOR J=1 TO 11 9020 TYPE R( J , 1 ) ,Q( J , 1 ) , U( J, 1

)

9040 NEXT J 9060 PAUSE

9100 FOR J=1 TO 11 9120 TYPE R( J, 2) ,Q( J, 2) ,U( J, 2)

9140 NEXT J 9900 REM INSTRUCTIONS

9910 TYPE "WINDING STRESS 205"

9920 TYPE "BOBBIN REMOVAL 1000"

9940 TYPE "CCOLDOWN 3500 OR 3000"

9960 TYPE "MAGNETIC FIELD 4905 OR 5000"

9980 PAUSE

9999 END

128
REFERENCES

1. A. J. Middleton, P. D. Hey, and B. Colyer, "The Impregnation of Superconducting Windings

with Epoxy Resins,'' Part I. Theoretical Considerations" (B. Colyer, only), and "Part II.

Resin Data," Rutherford Laboratory Reports RHEL/R264 and RHEL/R265 (1972).

2. R. E. Sherrer, "Filament-Wound Cylinders with Axi-symmetric Loads," Journal of Composite

Materials 1, 345 (1967).

3. W. L. Fourney, "Residual Strain in Filament-Wound Rings," Journal of Composite Materials

2, 408 (1968).

4. B. R. Dewey and C. E. Knight, "Residual Strain in Layered Rings," Journal of Composite

Materials 3, 583 (1969).

5. S. Timoshenko and J. N. Goodier, "Theory of Elasticity," Second Edition, McGraw-Hill

Book Co. (1951).

6. J. Hord, "Stress Analysis of Tape-Wound Magnet Coils," Journal of Res. of NBS 69C ,

287 (1965).

7. J. Case, The Strength of Materials , Arnold and Company, London (1938).

129
6. DISCUSSION

6 . 1 Comparison with Predicted Values

With a more complete test matrix we can now make further comparisons with the predicted values for the coil properties from the first report.

1 Outlined in Table 6-1 are

the calculated property values and the corresponding measured values at room temperature

and 4 K for the NbTi/Cu-f iberglass epoxy superconducting coil composite. Generally, the

calculated values are quite close to those measured and in several cases within a few

percent. Calculation of property values at 4 K are not included because many of the

constituent properties at 4 K are not available. Discussed below are some of the newly

reported measurements.

The Young's modulii for the 1 and 2 directions were previously discussed and the

calculated E^^ is seen to be about 35% lower than the measured value. For the calculation

the equation

E

f Em

E

c =E £ V +mE Vc (6_1)

f m m f

was used, where the subscripts c, f , and m denote the composite, the wires, and the fiberglass epoxy matrix, respectively. is the volume fraction of the wires which in the 3

direction is 0.803."^ As with ^22' t^is discrepancy is probably due to the very complex

stress distribution in the epoxy around the wires and the assumption that the modulus of

the matrix is that of the epoxy without fiberglass. Both of these would tend to raise the

1 2 calculated values. A more sophisticated calculation ' gives a value of 3.16 which is now

too high by 25%. The calculations do, however, bracket the experimental value.

The measured value of the Poisson's ratio, is about 17% lower than that calcu2 lated using a complex equation assuming near isotropy in the 2 and 3 directions. This

discrepancy probably reflects the small anisotropy in these two directions.

None of the shear modulii were previously reported and the agreement of with the

calculated value is quite good (less than 6% difference). The dynamic value (measured

acoustically) may be more realistic since the static measurement was on a tightly wound

coil (1.25 cm O.D). That the experimentally averaged value of 31 is different from G^

probably reflects again the slight anisotropy in the transverse directions, although it is

quite close to the dynamic measurement.

130
The shear modulus in the plane of isotropy, G^^ is calculated from

G

23

=

: E33 /2(1 + v

23 } (6_2)

and depending upon the choice of the calculated E.^ from above either is much too low or

agrees quite well. We are now compounding calculations and agreement must be considered

fortuitous.

The thermal conductivity calculations are discussed in detail in Section 3.1 and the

agreezer.t is very satisfactory considering the approximations necessary fcr values for

the fiberglass epoxy. The thermal expansion was discussed earlier and no calculations

were done for the specific heat.

6.2 Preliminary Coil Values

Most of the experimental values have been obtained and we can generate a better set

of constitutive relationships for the superconducting coil composite than those reported

earlier.''" These should still be considered preliminary but would probably create much

less error than any model used for their application. The experimental values used are

those in Table 6-1 and the relationships are the same as in the first report. The few

remaining calculated values are noted by an asterisk. Values at 4 K are in parentheses

just below the room temperature values.

The compliance matrix (for a transversely isotropic, orthotropic material) is in

, in-10 -1 2, units of 10 N m ;

0.132 -0.065 -0.065 0 0 0 (0.117) (-0.055) (-0.055)

0.364 -0.101 0 0 0 (0.234) (-0.0781)

0.364 0 0 0 (0.243)

S..(295K)

(S *4K) =

0.929 0 0 (6_3) ^ (0.643)

SYMMETRIC 1.31 0 (0.901)

1.31

(0.901)

131
Table 6-1. Comparison of Calculated and Measured Values of Coil Composite Properties

293 K 4 K

Calculated Measured Measured

ao„, 2, Young ' s Modulus

Ell ( 10 N /m ) 6.66 7.55 8.52

10 2 E

22 (10 N/m ) 1.96 2.75 4.11

i n ? E (10 N/m ) 1.45 2.28 3.10 _ (3.16)

Poisson's Ratio

V 0.345 ' 0.333 0.352

v 0.11 0.172 0.226

v 0.335 0.278 0.321

Shear Modulus

10 2 G -(10 N/m ) 0.63 0.67(STATIC) 1.11

0.76 (DYNAMIC)

G

21 3 1

( lol°N /m2 ) °- 63 °- 82 1 ' 33

'23

10 2 Go ^(10 N/m ) 0.54 1.21 1.58

(1.18)

Thermal Conductivity

A u

(W/mk) 208 169 169

A 22 (W/mk) 1.33 1.28 0.24

X 33

(W/mk) 4.3

Thermal Expansion

a

11 (10

_6

K -1

) 14.2 14.0 0.0

a99 (10~ 6K"

1

) 21.2 22.9 0.0

c P

Specific Heat

(Jkg"^

-1

) — ~ 0.338

132
where the S.. are used in the relation

e. . = S..O.. (6-4)

ij ij ij

between stress, a.., and strain, e...

The thermal conductivity matrix simplifies to:

169 0 0 (169)

A..(295K)(Wm-1K1

) 0 1.28 0

(X

±j

X4K) (Wm

1K ) (0.24)

{b ~ 5)

0 0 4.3*

(0.91*)

The thermal expansion matrix is as before:

14.0 0 0 (6.8)

a^SKMK"1

) Q 22 9 Q

(a

±y(77K)(K ) (12.0) ^ b-b;

0 0 22.9

(12.0)

and the expansion at 4 K is, of course, zero so the values at 77 K have been given in

parentheses

The thermal diffusivity can also be calculated now from:

(6-7)

with the 4 K value for the specific heat and a measured density, p, of 6.0 gm/cm . Not to be

confused with the thermal expansion we have used D as the symbol for thermal diffusivity.

Its matrix is:

(D

ij )(4K)(cm

2

s 1

) =

(83.3)

0

0

133

0

(1.18)

0

0

0

(4.49*)

(6-8)
6. 3 The Effects of Anisotropy and Temperature

The superconducting coil which is made up of wires is obviously anisotropic. The

elastic properties, generally, have an anisotropy of about 2.5:1, the thermal expansion

about 1.5:1, and the thermal conductivity (and diffusivity) of about 700:1 at 4 K. As a

function of temperature from room temperature to 4 K, the elastic modulii increase 10-40%,

the thermal expansion goes to zero, and the thermal conductivity (and diffusivity) decreases

by a factor of 1 to 5, depending upon the direction. In all cases the anisotropy increases

with decreasing temperature, being largest at 4 K.

Basically, there are no unexpected trends in the data so far generated. At a fixed

temperature or under fixed conditions, all of the behavior can be accomodated and the

response of the coil to small or slow changes in its environment are predictable. The aniso

tropy in thermal expansion and very high anisotropy in the thermal diffusivity, however,

could create significant stresses during cooldown or any thermal transients. The propagation of the normal zone in the superconductor will also follow the heat conduction down

the wire. With further refinement of these measurements and comparison to measured coil

behavior, even these effects should be calculable. These steps are part of the overall

program and will be addressed in later reports.

REFERENCES

1. Fowlkes, C. W. , Angerhofer, P. E. , Newton, R. N. , and Clark, A. F. , "Characterization of

a Superconducting Coil Composite," NBSIR 73-349, National Bureau of Standards, Boulder,

CO (1973).

2. Adams, D. F. and Doner, D. R. , "Transverse Normal Loading of a Unidirectional Composite," J. Composite Materials 1, 152 (1967).

3. Advanced Composites Design Guide, Volume 11, Analysis, Air Force Materials Laboratory

(AFML/LC), Wright-Patterson Air Force Base, Ohio 45433 (January 1973).

134
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLIC A I ION OR REPORT NO. NBSIR 76-837

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Characterization of a Superconducting Coil Composite and Its

Components

5. Publication Date July 1976

6. Performing Organization Code

275.03

7. author(S) A. F. Clark, W. F. Weston, V. D. Arp, J. G. Hust,

R. J. Trapani

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2750538

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Annapolis Laboratory

Naval Ship Research & Development Center

Annapolis, MD 21402

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The physical properties of a superconducting coil composite and its components

are studied in order to accurately predict the coil behavior under a variety of

operating conditions. This second interim report includes data on the Young's

moduli, the Poisson's ratios, the shear moduli, and the thermal conductivity of

the coil composite as well as the compressive modulus of the fiberglass cloth and

the thermal expansion of the superconducting wire. In addition, the development

of an acoustic technique for the determination of the elastic moduli of the coil

composite and the stress analysis of transversely isotropic magnet coils are

reported.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Fiberglass cloth; mechanical properties; superconducting coil composite; superconducting wire; stress analysis; thermal properties.

18. AVAILABILITY [Xj Unlimited

1 For Official Distribution. Do Not Release to NTIS

1

1 Order From Sup. of Doc, U.S. Government Printing Office Washington. D.C. 20402. SD Cat. No. C13

K Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

142

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

$6.00

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