NA SA / TM--2001-210578

Deformation and Life Analysis of Composite

Flywheel Disk and Multi-Disk Systems

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

A.F. Saleeb and N.R. A1-Zoubi

University of Akron, Akron, Ohio

January 2001
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NASA / TMm2001-210578

Deformation and Life Analysis of Composite

Flywheel Disk and Multi-Disk Systems

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

A.F. Saleeb and N.R. A1-Zoubi

University of Akron, Akron, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

January 2001
Acknowledgments

The authors would like to thank Jeff Trudell for his assistance in supplying the FEA validation results and

for his many helpful suggestions during the review of this manuscript.

NASA Center for Aerospace Information

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Available electronically at http:Z Zgltrs.g-rc.na_sa.gov/GLTRS
DEFORMATION AND LIFE ANALYSIS OF COMPOSITE FLYWHEEL

DISK AND MULTI-DISK SYSTEMS

S. M. Arnold

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

A. F. Saleeb and N. R. Al-Zoubi

University of Akron

Akron, Ohio 44325

SUMMARY

In this study an attempt is made to put into perspective the problem of a rotating disk, be it a single disk or a

number of concentric disks forming a unit. An analytical model capable of performing an elastic stress analysis

for single/multiple, annular/solid, anisotropic/isotropic disk systems, subjected to both pressure surface tractions,

body forces (in the form of temperature-changes and rotation fields) and interfacial misfits is derived and discussed.

Results of an extensive parametric study are presented to clearly define the key design variables and their associated

influence. In general the important parameters were identified as misfit, mean radius, thickness, material property

and/or load gradation, and speed; all of which must be simultaneously optimized to achieve the "best" and most

reliable design. Also, the important issue of defining proper performance/merit indices (based on the specific stored

energy), in the presence of multiaxiality and material anisotropy is addressed. These merit indices are then utilized

to discuss the difference between flywheels made from PMC and TMC materials with either an annular or solid

geometry.

Finally two major aspects of failure analysis, that is the static and cyclic limit (burst) speeds are addressed. In

the case of static limit loads, upper, lower, and out-of-plane bounds for disks with constant thickness are presented

for both the case of internal pressure loading (as one would see in a hydroburst test) and pure rotation (as in the case

of a free spinning disk). The results (interaction diagrams) are displayed graphically in designer friendly format. For

the case of fatigue, a representative fatigue/life master curve is illustrated in which the normalized limit speed versus

number of applied cycles is given for a cladded TMC disk application.

SYMBOLS

Material Parameters

EL

E T

K

cxI_

%

f_

v L

P

(YUL,(YUT

longitudinal Young's modulus

transverse Young's modulus

transverse yield stress under shear loading

longitudinal thermal expansion

transverse thermal expansion

ratio of anisotropy, i.e., (E_-LT )

longitudinal Poisson's ratio, i.e., (e r/e 0, load is in 0 direction)

ratio of longitudinal to transverse ultimate stresses, i.e., _,Cur )

material density

ultimate tensile strength (UTS) in the longitudinal and transverse directions respectively.

longitudinal yield stress

NASA/TM--2001-210578 1
StressesandStrains

O/J

F( )

Sij

crij

second order direction tensor

unit vector denoting local fiber direction

limit function (static fracture, endurance limit and normalization surfaces)

deviatoric stress tensor

Cauchy stress tensor

radial stress and strain components respectively

tangential stress and strain components respectively

Subscripts and Superscripts

( )L'( )T

(.)

longitudinal and transverse properties (static fracture, endurance limit and normalization surfaces)

respectively

denotes differential with respect to time

Geometry

a

b

h

Rm

Ri

F

It

6

inner radius

outer radius

height

mean radius

radial location of the ith interface

radius

radial deflection

misfit between concentric disks

width of a given disk

Forces

Pin

Pout

k

AT

0)

internal applied pressure

external applied pressure

Generalized Pressure Loading:

(a) When internal pressure loading applied; equal to internal pressure Pin

(10)When rotational loading is applied; equal to the equivalent centripetal pressure load, i.e.,

/3= p02 (b 2 _ a2)12

temperature at the inner diameter, r = a

temperature at the outer diameter, r = b

radial temperature distribution

speed of rotation

1.0 INTRODUCTION

1.1 General

Over the years, there has been an increasing demand for effective energy storing systems (ref. 1). To meet such

needs, many alternative systems have been proposed, as listed in table I, with composite flywheel energy storage

systems rising to the top of the list of candidates, because of there high power and energy densities with no fall-off

NASA/TM--2001-210578 2
incapacityunderepeatedcharge-discharge cycles.Inrecentyears,ahost(e.g.,refs.1to3)ofdifferentflywheel

systemdesigns(composedof various fiber-reinforced material systems, i.e., metallic composites, polymeric composites, etc.) have been identified with the goal of improving the performance of these systems. These flywheel systems can be categorized into principally three design topologies: (1) preloaded, (2) growth matching, and (3) mass

loaded, with each approach having its own set of advantages and disadvantages.

Energy storage in all these flywheel designs is basically sustained by a spinning mass, called a rotor. This

important "rotor" component has therefore been the subject of research investigations over many years, generating

extensive literature on its many analysis and design related aspects. At different stages of developments, a number of

comprehensive state-of-art reviews (refs. 1 and 2) have been completed. For instance, representatives of these in the

area of material-system selections for flywheel devices are given in Genta (ref. 1) and DeTeresa (ref. 4) whereas

optimization for integrated system design are given by Bitterly (ref. 3).

In the following section, a brief summary is given for the state-of-art that is presently available on the analysisrelated aspects of the rotor-type components (also disks, multiple-disks, etc.). However, we emphasize at the outset

that the present review is not intended to be exhaustive; rather we are primarily concerned with works and studies on

the important factors that should be considered in any valid assessment of the performance (both at working/service

and at limiting operational conditions) of flywheel systems. To this end, we will group the pertinent list of references in tables II to IV according to three major considerations; i.e., (1) factors related to additional loading conditions beyond those of rotation (e.g., internal pressure, temperature, and interference fits, etc.), (2) factors related to

material behavior (e.g., isotropic versus anisotropic, elastic versus inelastic, etc.) and (3) factors related to geometry

(e.g., uniform versus variable thickness, solid versus annular). This background then puts into perspective the subsequent parametric studies and related conclusions in the later part of the paper.

1.2 Background Literature

Starting as early as 1906 the rotating isotropic disk has been studied by Grubler (ref. 6) followed by Donatch

(ref. 7) in 1912. One of the first "modern" dissertations in analyzing the flywheel rotor was the seminal work done

by Stodola (ref. 8), whose first translation to English was made in 1917. In 1947 Manson (ref. 9) published a finite

difference method to calculate elliptic stresses in gas-turbine disks that could account for the point-to-point variations of disk thickness, temperature, and material properties. Millenson and Manson (ref. 10) extended the method to

include plastic flow and creep in 1948. Both methods have been widely used and extended by industry (ref. 11).

More recently, Genta (ref. 11) modified Manson's method to include orthotropic materials. The 1960s and 1970s

lead to numerous other serious efforts in analyzing the rotor, and introducing different designs for the flywheel, with

the onset of composite material development giving added impetus. A detailed review of the rotating disk problem

up through the late 1960's is given by Seireg (ref. 12), whereas, Habercom (refs. 13 to 15) provides a collection of

abstracts and reference of government-sponsored flywheel related research (late 1960's until the late 1970's). The

past 40 years showed a tremendous amount of work done concerning the flywheel, e.g., see Genta's (ref. 11) extensive review, including a historical perspective, on research work conducted up to the early 1980's related specifically to flywheel systems.

For convenience in presentation, we have grouped in tables H and III, respectively, the different representative

works with regard to the two major material behavior classifications treated; i.e., elastic-isotropic and elastic anisotropic. These classifications in fact have had a long-standing history in the available literature. The more complex

nonlinear counterpart (i.e., plasticity, creep, fracture, etc.) is of more recent origin, and consequently comparatively

smaller in size, and primarily focused on isotropic material behavior. A sampling of work on these latter nonlinear

areas is given in table IV.

With reference to tables II and Ill we note that most of the listed investigations have focused on the disk of uniform thickness under rotation. Alternatively, some investigators such as Kaflanoglu (ref. 5), Abir (ref. 41), Huntington (ref. 70), Nimmer (ref. 90), Daniel (ref. 94) and Portnov (ref. 111), considered the effect of misfit in addition to

disk rotation as they studied the multirim disk case. Others like Leopold (ref. 18), Potemkina (ref. 31), Prasek

(ref. 39), Shanbhag [46], Chakrabarti [61], Gurushankar [66], Dick [74], Baer [82], Misra [100], Ferrero [102] and

Portnov (ref. 108), considered the temperature effect as well as the disk rotation case.

Considering the list in table IV, we refer to Shevchenko (ref. i 19), Weber (ref. 120), Pisarenko (ref. 121), Reid

(ref. 122), Gururaja (ref. 127) and Gueven (ref. 136), for a representative nonlinear elastoplastic analysis of rotating

disks, and to Lenard (ref. 147), Gupta (ref. 150), and Mukhopadhyaya (ref. 152), for a number of time-dependent

(viscoelastic) solutions.

NASA/TM--2001-210578 3
1.3ObjectivesandOutline

Insummary, tablesII toIVwiththeirbroadclassificationfpreviouswork,conveythecomplexityoftheproblem,inthataverylargenumberofconditionsandfactorsmustbeconsidered. Inparticular, eachofthelistedinvestigationshavenecessarilyfocusedonsomespecifiedconditions, particularmaterial behaviorandrelated

conclusions. Anurgentneedthereforeexistsforacompilation, inasconciseaformaspossible, oftheextensive

results,majortrendsobservedandthemanyconclusions reachedinthesepreviousstudies. Totheauthorsknowledgeastudyofthistypeispresentlyacking;inwhichtheprobleminits"totality"hasbeenconsidered. Asafirst

steptowardthisend,anannotativerviewstudyisconductedherein,withthemajorrestrictionbeingthatofdisks

withconstant thickness. Inparticular, emphasisisoncombinedloading(rotation, pressure, misfit,andtemperature)

andtheeffectofmaterialanisotropy(withitsgreatimpactonstressdistributionsandanyensuingsingularitiesin

stress, strengthandfailuremodes, etc.).Tofacilitatethepresentation ofthewell-established elasticanisotropic

solutionprocedure(singleandmultipledisks)specificreferencetothenumerouspreviousworkscitedintablesII

andIII willnotbeindividuallymentioned. However, weemphasisattheoutsetthatalltheconclusionsmadeherein

areinagreement withthesepreviousworks,exceptofcoursethenotationpresentlyemployed. Inadditiontothis

review,oursecondprimaryobjectiveistopresent anumberofnewanalyticalndnumerical resultswithregardto

limitingconditionsofburst(pressureaswellasrotation),fatigue,etc.Thesewill thenenableustoreachanumberof

important conclusionsinparallelunderbothserviceconditions(stressanalysis), aswellasattheoverloadconditions

for"firing"studiesoftheproblem.

Inanoutlineform,theremainder ofthepaperisdescribedasfollows.InsectionII, wegiveadetailedderivationoftheanalytical modelusedintheelasticstressanalysisforsingle/multiple, annular/solid, anisotropic/isotropic

disksystems, subjectedtobothpressuresurfacetractions, bodyforces(intheformoftemperature-changes androtationfields)andinterracial misfits.SectionIII containstheresultsofextensiveparametricstudiesandimportant

observations obtainedtherefrom. InSectionIV,weaddresstheimportant issueofdefiningproperperformance/

meritindices(basedonthespecificstoredenergy), inthepresenceofanisotropy. SectionV dealswithtwomajor

aspectsoffailureanalysis, i.e.,staticandcyclic,providingthecorresponding limit(burst)loadandrepresentative

fatigue/lifelimitspeedcurve,respectively.

2.0ELASTICSTRESSANALYSISVIAANALYTICALMODEL

2.1SingleDiskSolution

2.1.1 Annular Disk.--Here, as classically done the problem of a quasi-static rotating disk subjected to both thermal and mechanical boundary conditions is summarized by stating the governing equations required to define a

well-posed boundary value problem. We begin with the equation of equilibrium:

dry r

tYr - tYO + pco2r = 0 (1)

dr r

or

(YO = d"ff-(rtYr ) + P (02r2

(if"

(2)

in which the centrifugal "internal force" (po)2r) is included as a body force term. Next the pertinent compatibility

equation,

e,. = d (reo) (3)

is obtained from the well known strain----displacement conditions

NASA/TM--2001-210578 4
and

du

G- dr

II

_0 =--

1"

so that only the specific constitutive equations to be utilized, need be defined. Given that the problem is an

axisymmetric, transversely isotropic disk (with the fiber directed along the circumferential direction of the disk

fig. 1), the plane stress constitutive relations are as follows:

G r VLGO

G = + OtTAT

ET EL

(4)

(5)

(6)

e 0 = _ vLcr,_____.__ + ao + aLAT

EL EL

(7)

where AT is the assumed linear temperature gradient across the disk and is defined as

AT = Ta - To + Tb - Ta (r - a)

b-a

(8)

where a and b define the inner and outer radius of the disk, To and T b are the imposed temperature at the inner and

outer radius respectively, and TOis the reference temperature of the disk.

Substituting the constitutive relations equations (6) and (7) into the compatibility expression, equation (3), and

utilizing equilibrium, equation (2); the following governing equation for the radial stress component is obtained:

r2 d2o'r _ do"r 1

--dr 2 +3r'-'_r +(1 - -_)0" r = QO')

(9)

where

rEL (Ta - Tb)(2Ct L - OCr) |

]

[J= E.---E-TEL and Q(r)= b_al+a{EL(rb-To)(O_L-_XT,+ r2ptO2(3+ VL )}..[ (10)

Implied in equation (9) is the assumption that the material parameters (i.e., EL, E T, vL, v T, and G L, cxL and c_/-) are

not strongly temperature dependent and therefore can be taken to be independent of radial location. Inclusion of this

temperature dependence would only contribute higher order effects in the solution and distract attention from the

primary emphasis of examining the effects of rotation. Also, the only practical values for 13are those less than or

equal to one, as the fiber stiffness is always greater than the matrix (thereby resulting in EL > ET).

The general solution of this Euler-Cauchy governing equation is:

where

a r = clrmt + C2 rm2 + Ql(r)- Q2(r) (11)

NASA/TM--2001-210578 5
''____2___' f

r-O"l +t)Q(r),_ r

QI (1")= ml _ m2

m..._r 2 J"1.-(,12+l)Q(r)dr Q2(r) = ml - m2

=-1+5_, m 2=-1-52 and E= /1 rill

(12)

and C l and C 2 are constants to be determined from the applied mechanical boundary conditions.

The general solution for the tangential stress is obtained by merely substituting equation (11) back into (2):

O"0 = C 1(m I + l)r nh + C2(m 2 + l)r m2 + (m I + I)Q1 (r) - (n72 + I)Q2 (r) + pa)2r 2 (13)

Whereupon, the radial deflection is derived by substituting equations (l 1) and (13) into (7) and the resulting expression into equation (5):

LEL EL

To obtain the constants C 1and C 2 appearing in equations (11) and (13), we impose the following boundary

conditions along the inner and outer radius of the disk, i.e.,

(o-_)_=a = -pi_

((Yr )r=b = -Pout

where Pin and Pout are the internal and external pressures, respectively. Solving the resulting 2x2 matrix equation

results in the following expressions:

a (Pin +Q2(a)-Ql(a)) -b "Vfl(Pout +Q2(b)-Ql(b)) (15) C 1 =

a -b

and

C2 =

a b -ab (pin + Q2 (a) - Q1 (a)) + ba (Pout + Q2 (b) - Q1 (b))

(16)

where we may separate the thermal and rotational terms such that

QI(r)=QT(r)+QIR(r) and Q2(r)=Qf (r)+Q_(r)

with the generalized thermal forces being defined as,

(17)

NASA/TM--2001-210578 6
QIT(r) = 9_-_[EL__ (_L --aT ) a(Tb[_lb)-To)+ + _-fl b(-Ta )(a - + TO)

rEL (Ta - Tb)(2aL - a T ) ]

9-_[ -_T" a(Tb----T°)+b(-Ta+'T°) +rEL(Ta-Tb)(2aL-aT)]

Qr(r) =---EL(OCL )[-l-f-_](a-b) (-2-_-_)(a-b)J

and those due to rotation as,

QR(r) = r2p092( 3 + v L)

(18)

and QR(r)= r2po)2(3+VL) (19)

2.1.2 Solid Disk.--For the special case of a solid disk, the radial stress must remain finite at the center of the

disk, therefore given either an isotropic o1"anisotropic material, it can be immediately seen that C 2 in equation (11)

must be equal to zero (thus both displacement and stress fields at the center will be finite), such that,

for the isotropic case, and

0",. = C1 + QI (r) - Q2 (r)

cr,. = Clrm' + Ql(r) - Q2(r)

(20)

(21)

for the anisotropic case. Now assuming that an applied external pressure Pout exist, the constant C 1 is determined to

be:

Cl =Pout +Q2(b)-QI(b) (22)

for the isotropic case, or

1

Cl = 7 [Pout + Q2 (b) - QI (b)] (23)

for the anisotropic case. Where in the definitions for QT QR and Q_', Q_2 given above in equations (18) and (19), a

(the inner radius) is taken to be zero. Similarly, the tangential stress is

°'0 = CI + QI (r) + Q2 (r) + pg.02r 2 (24)

for the isotropic case, and

dr0 = (m 1+ 1)CIrml + (m I + 1)Ql(r) - (m2 + 1)Q2(r) + po92r 2 (25)

NASA/TM--2001-210578 7
fortheanisotropiccase.Note,fortheisotropic case, Q1 and Q2, can be greatly simplified by taking [3= 1. with the

resulting expressions being given in equation (29), see section 2.4.1.

2.2 Multiple Number of Concentric Disks

Extension of the formulation to "n" number of annular concentric disks, with possible misfit (see fig. 2), is simply accomplished through application of appropriate boundary conditions at the interface of each disks, so as to

ensure continuity of radial stresses and displacements. Continuity of radial stress demands the following:

cr,_ (r= q) = Pl

a,, (r = i_) = o', 2 (r = 1"2)=/'2

o',.2 (r = 13) = O'r3(r = r3) = P3

cr,,,__(r = 1s,) = or,;,(1+= 1;_) = e,,

Or,s,(r= iS++j) = P,,+l

(26)

where PI and Pn+l are the applied internal and external pressures, respectively; whereas the remaining continuity

conditions in (26) will then lead to the determination of the remaining interfacial pressures P2" P3"'" Pn (which are

redundant unknowns, for the common case of imposed (known) interference fit). Similarly, kinematic constraints are

introduce at the interfaces corresponding to the imposed misfit (5) at each interface. These are:

Ur2 (r = r2) - Ur, (r = r2) = 81

Ur3 (r = r3)- Ur: (r = r3) = _52

Ur, 1 (r = r4) - Ur3 (r = r 4) = _3

Urn (r = r n) - Ur,.l (r = rn) = 8n. 1

(27)

Where _51, 8 2 .... _n-I are the imposed misfits between disks that will in turn generate interface pressures. Combining

equations (26) and (27) with equations (11) and (14) specifies a system of 2n equations with 2n number of unknown

constants; i.e.:

[K]{C}={R} (287

where

{C} is the vector of unknown constants; {C} = C 1,C2,C l ,C 2 ,.

[K] is the geometry and material matrix

[R] is the vector of applied forces and/or misfit parameters

2.3 Verification of the Solution

As indicated in the introduction, the analysis of isotropic and anisotropic disks subjected to rotation, internal

and external pressure, temperature variation and misfit are not new (refs. l to 4). Consequently, the present unified

solution can be, and has been, validated through comparison with a variety of other solutions, see table V for a brief

summary. Clearly all aspects of the solution have been verified with at least one other independent numerical or

analytical solution. In all cases the present solution agrees identically with all other analytical solutions and for all

practical purposes identically with all FEA results as well. Figures 3 to 5 illustrate three of the many comparisons

made between the present formulation results and other previous analytical results. In figure 3 the thermal and internal/external pressure analysis is compared against a previous analytical solution derived by Arnold (ref. 175), with

the agreement being excellent. In figure 4 the analysis of an anisotropic disk subjected to pure rotation is compared

NASA/TM--2001-210578 8
withLekhnitskii'ssolution(ref.58),againtheagreement isexcellent. Anotherexampledemonstrates theabilityto

include misfit within the formulation, see figure 5. Here comparison with Ugural and Fenster's (ref. 176) solution of

two isotropic disks subjected to a misfit of 0.1 mm is made. The calculated contact pressure is, 12.305 MPa, while

the tangential stresses in the outer and inner cylinder are:

(_0(r = 200 mm) = 56.055 MPa

c0(r= 250 mm) = 43.750 MPa

and

(_0(r = 150 mm) = -56.250 MPa

(Y0(r = 200 mm) = -43.945 MPa

respectively. These values agree exactly with those calculated by Ugural and Fenster (ref. 176)

2.4 Observations Regarding The Analytical Solution

Before conducting a parametric study to identify the important parameters in the design of flywheel systems, let

us make a number of observations regarding the current analytical approach for conducting stress analysis by examining a number of special cases.

2.4,1 Material Isotropy.--For the special case of material isotropy (i.e., [3= 1), the character of the governing

equation (see eq. (9)) will change as the zeroth order term will disappear and consequently, the particular solution,

i.e., equation (12), will be appropriately modified. The resulting expressions for Q Land Q2 coming from

equation (12) and including both thermal and rotational loading are:

Ql(r) r(E(Ta-Tb)a rpc°2(3+v))

= + (29)

2(a - b) -4

Q2(r)=r E(Ta-Tb)a _

-6(a - b)

2.4,1 Thermal Isotropy.--Here let us consider the case when only thermal loading is applied to the disk. Under

this condition it is apparent from equations (10) to (19) that, in general, the stress distribution (tangential and radial)

is dependent upon geometry, material properties (i.e., strength of anisotropy), mismatch in thermal expansion coefficients, and the magnitude of the thermal load itself. In the case of thermal isotropy (i.e., when the tangential (orL)

and radial (ctT) thermal coefficients are equal) the stress distribution is primarily dependent only on [3 (the ratio of

moduli) and EL, as the constants C l and C2 become a function of Q2- QI, that is:

Q2 - QI = rEL (Ta - T°)°: (30) 1

(a - b)(4 - _)

Consequently, under a uniform temperature distribution no stresses are developed as one might expect. Also in general, from equations (10) and (14), it is apparent that the longitudinal thermal coefficient is the dominant parameter

with the longitudinal modulus dictating the magnitude of thermal stresses being developed.

2.4.2 Numerical Singularities.--It is well know that nonphysical numerical singularities can exist within an

analytical solution, and these vary depending upon the approach taken. In the above derivation the final governing

equation to be solved is stress based, therefore we will examine the resulting radial stress under various special conditions so as to identify all potential singularities. As the solution is limited to elastic behavior the concept of superposition applies; consequently individual-loading conditions can be examined separately. This will greatly simplify

the current task. The following four possible loading cases (rotation, pressure, thermal and interfacial misfit) were

taken under consideration.

NASAfrM_2001-210578 9
Rotation Only: Given a single, annular, disk subject to only rotation, the radial stress equation (11) becomes:

(3 + v L )pw2 r(-1+__) r- -- + t.2

a -b a 2 -b 2

(31)

Clearly, a singularity would exist in the solution whenever 13approaches 1/9 (i.e., whenever E L = 9ET). This type of

singularity has been found before in other solutions, such as Lekhnitskii (ref. 58) and more recently for the case of

solid disks, as discussed in reference 177, for values of 13> I. Once again confirming the accuracy of the present

formulation.

Pressure Only: Solving for the single, annular, disk case subjected to both inner and outer pressure (Pin and

Pout, respectively), the following equation for the radial stress can be derived:

I" a b -ab Pi,, + ba Pout + Pin + b Pout

_,. = (32) /

For any finite disk, a can not equal b in equation (32), and consequently there is no possible singularity for this type

of loading.

Thermal Loading Only: Again considering a single, annular disk, subjected to a pure thermal loading condition,

the derived expression for the radial stress becomes somewhat lengthy; consequently, for convenience, only the

denominator (which would be the source of any numerical problem) of the troublesome term will be shown here.

Numerator

t_,. = (33)

l bta2 _t _(a- - 13- 1)(4_- 1)

Clearly the solution has two singularities, namely whenever 13= 1/4 (i.e., EL = 4ET) or 13= 1 O.e., E L = Er, which is

the isotropic case).

Misfit only: The simplest problem containing a misfit is that of two disks with a misfit, 5, imposed between the

inner and outer disks, see figure 6, where a and b are the inner and outer radii, respectively and c designates the

radial location of the misfit. Solving for the radial stress in both the inner and outer disks, respectively, we get the

following expressions:

NASAITM--2001-210578 10
Innerdisk

c_

22-')

-tl3 _1. _r1318

J

Outer disk

(34)

It is obvious from examining equations (34) and (35) that for the case of misfit only, no singularities exist over the

domain.

Given the results of the above four loading cases it is apparent that a pattern has emerged, thus prompting a

generalization regarding the potential for the existence of numerical singularities. One can exoect sinmalarities to be

present whenever body fgrces are imposed upon a given solution; however, when the excitation is arising due to the

application of imposed boundary conditions (i.e., applied tractions or displacements) no singularities should be

expected to occur.

3.0 INFLUENCE OF KEY DESIGN PARAMETERS

Having developed and verified a general analytical solution for multiple, anisotropic, concentric disks, with

misfit, subjected to rotation, pressure and thermal loading, we are now in a position to identify and examine the

influence of important design parameters such as: mean radius, thickness, the amount of misfit, rotational speed, and

material gradation. Two classes of anisotropic (composite) materials are primarily employed throughout this study.

The first class, polymeric matrix composite (PMCs) systems, is represented by a 60 vol % fraction graphite/epoxy

system. The second class, titanium matrix composite (TMCs) systems, is represented by a 35 vol % fraction SiC/

Ti-15-3. Subsequent to examining a single disk composed of a uniform volume fraction of material, examination of

material gradation and misfit effects will be undertaken by considering three concentric disks with varying fiber

volume fractions.

Flywheels composed of PMC material systems are currently being investigated under the NASA funded RotorSafe-Life Program (RSL); a collaborative government/industry program designed to establish a certification procedure for composite flywheels. Consequently, this material class will dominate the parametric study. The associated

material properties for the two classes of composite systems (and their individual fiber and matrix constituents) used

in this study are listed in tables VI and VII, where the composite properties were obtained using the micromechanics

analysis code based on the generalized method of cell, MAC/GMC (ref. 178). All properties given are at room temperature and assumed to be independent of temperature. The flywheels to be considered in theRSL program are

expected to have a mean radius of approximately 4 in. and a thickness of approximately 3 in.; therefore these geometric properties will constitute our baseline case.

NASA/TM--2001-210578 11
3.1SingleDiskCase

Consideringthecaseofasingleannulardisk,subjectedtopurerotation,theinfluenceof thethreekeymaterial

andgeometricparameters (i.e.,thedegreeofmaterialanisotropy([3),meanradius(Rm) and thickness ([3.'. L)) on the

radial and tangential stress distribution will be investigated parametrically. Analytically, these parameters can be

clearly seen by normalizing both the radial (see eq. (31)) and tangential stress (see eq. (2)) with respect to the density, rotational speed, and outer radius, b, that is:

,_aC J/ ) -(_)_ a/

(36)

a _/_ a 3

(37)

where the inner and outer radii are defined as

a= Rm --- 1---

2 and a =--2R m_

_, b l+--

b = Rm + _ 2R m

so that g(13, R m, _., VL) andfl[3, Rrn, _,, v/) will be denoted as the radial and tangential distribution factors, respectively, for an anisotropic annular disk.

Figures 7 and 8 depict these factors as a function of normalized radial location, r/R m, for two limiting geometric

representations, i.e., a thick disk (_URm -_ 2.0) and a thin disk (L/R m = 0.125), given a slightly anisotropic ([3 = 0.7)

TMC, and strongly anisotropic (13= 0.05) PMC material description, respectively. Further, in table VIII the maximum values of the modified normalized radial and tangential stress are given for a range of thickness and mean

radii. From these results one can draw a number of conclusions.

NASAfrM--2001-210578 12
Thefirst being that both the radial and tangential stresses increase as Rm and _, are increased (see table VIII).

The second is that the degree of material anisotropy (13)impacts the magnitude of increase in the radial stress cornponent more so than the tangential stress component, and this influence increases as the thickness is increased;

whereas, it diminishes as the mean radius is increased. In fact, once the disk is sufficiently "thin," i.e., k/R m < 0.125,

the degree of material anisotropy becomes irrelevant. Thirdly, the location of the maximum stress is also greatly

impacted by the degree of anisotropy. For example in figure 7, we see that for the strongly anisotropic PMC material

system the maximum tangential stress location varies greatly depending upon mean radius; whereas for the slightly

anisotropic TMC material system, see figure 8, the maximum tangential stress is always at the inner radius. Similarly, the location of the maximum radial stress shifts from approximately 0.75 Rm to R m as the mean radius

increases for a TMC (see fig. 8). Whereas for a PMC (see fig. 7) material the maximum location moves from

1.25 R m, for small mean radii, towards the center (i.e., Rm) as the mean radius, Rm, is increased. These observations

were substantiated by the work of Gabrys and Bakis (ref. 179) wherein they designed and tested a flywheel that

employed a much softer urethane matrix, which precluded the development of a large radial stress. This then

allowed the use of thick flywheels, wherein the maximum hoop stress was shifted so that it occur near the OD;

thereby theoretically leading to a less catastrophic failure mode and fail-safe flywheel design. Fourthly, irrespective

of the degree of material anisotropy the most efficient use of material is made when one considers the case of relatively thin disks rather than thick ones, see figures 7 and 8. This will be discussed further in the subsequent material

and configuration selection section. Finally, it is obvious from equations (36) and (37) that the density (p) of the

material impacts the actual magnitude of the stress field (both radial and tangential components) linearly, while the

rotational speed (0_) impacts both components quadratically.

The radial and tangential stress distributions as a function of normalized radial location for pure thermal loading

and internal pressure loading are shown in figures 9 and 10; assuming the baseline material parameters corresponding to the 60 vol % fraction PMC system in table VI for a relatively thick disk (R m = 4 in. and _,= 3 in.). From

figure 9, we see that temperature (whether uniform or gradient) induces compressive hoop stress in the outer portion

of disk and tensile hoop stress in the inner portion. However, in the presence of a linearly varying temperature field,

the slope (positive or negative) of the gradient of temperature will shift the maximum location of the tensile radial

stress from one side of the disk to the other, as well as change the curvature of the hoop stress distribution and the

magnitude of the maximum tensile hoop stress.

In figure 10 the influence of applying pressure along the inner surface of the disk is illustrated. This type of

loading distribution would be consistent with the presence of a solid hub (shaft), albeit in practice it is clear that the

magnitude of this internal pressure would be a function of the density, size and speed of rotation of this solid hub.

Consequently, in figure 10 both the radial and tangential stress profiles are normalized with respect to the applied

internal pressure. The basic influence of internal pressure is to induce a compressive radial stress state and tensile

hoop stress throughout the annular disk, with a maximum in both stress components occurring at the inner radius.

Further, discussions regarding the influence of internal/external pressure and temperature profiles for single cylindrical or disk configurations can be found in reference 175.

Now the superposition of these three classes of loads (thermal, internal pressure, and rotation) is illustrated in

figures 11 and 12; where both radial and hoop distributions are shown for a relatively thick disk (_, = 3, R m = 4)

rotating at 15 000 rpm (fig. 11) and 60 000 rpm (fig. 12) subjected to both uniform and gradient temperature profiles

and an internal pressure of 1 ksi. Note how for the lower speed case (fig. 11), the overall conclusions for the combined loading situation are similar to those of the pure thermal cases; with the exception of tensile hoop stress being

present even in the outer region of the disk for both the uniform and inner gradient case. Tensile hoop stresses are

present in these two cases because the tensile hoop stress generated, due to rotation, are sufficiently high to overcome the thermally induced compressive hoop stresses in the outer region. In addition we see immediately that at the

inner radius the radial stress becomes compressive due to the applied internal pressure, while it is zero at the outer

surface, as it must be. Figure 12 clearly shows that as the rotational speed is increased to 60 000 rpm the stresses due

to rotation dominate those induced by either the internally applied pressure or the imposed thermal profiles. Note,

however, that the hoop stress distribution becomes almost uniform across the disk, thus making more efficient use of

the material.

NASA/TM--2001-210578 13
3.2ThreeConcentricDisks

IntheRSLprogramtheflywheelsofinterestarebeingmanufactured withapreloadesigninmind,thatisthe

flywheeliscomposedofanumberofconcentricringspress-fittogetherwithaspecifiedmisfitbetweeneachring.

Therefore, theresultinglaminatedflywheelhasacompressiveradialstressattheinterfaceofeachconcentricring.

Consequently, thedesirehereistoinvestigatetheinfluenceofakeydesignparameter (i.e.,misfit,5betweendisks)

onthecorresponding radialandtangential stressdistributionwithinthelaminatedsystem. Thisparameter isinadditiontothetwopreviousgeometricparameters: R m - mean radius and _. - ring thickness in the radial direction. When

examining the influence of _i, tracking of the radial stress distribution is important, since a positive value at a given

interface will constitute (by definition) disk separation. Similarly, if the radial stress exceeds some critical value,

delamination within a given disk may also take place. Alternatively, the tangential (or hoop) stress is significant in

that it will dictate the burst failure of a given ring (disk) within the system, which may or may not (depending upon

the location) lead to an immediate catastrophic failure of the entire system. The limit speed of a given,

circumferentially reinforced, disk is dictated by the circumferential (or hoop) stress, thus excessive hoop stress will

result in fiber breakage and a subsequent total loss in load carry ability. Consequently, this failure criterion is

defined as the burst criterion.

3.2.1 Uniform Matefi_! Properties.--To simplify the study, only three (each with the 60 vol % fraction PMC

system given in table VI) concentric disks will be investigated with a baseline geometry similar to the previous

single disk case, that is, an Rm = 4 and a total thickness (i.e., _'t + _2 + _'3) equal to 3 in. A three disk minimum is

required so that the influence of two misfits can be examined, that is the misfit between disk 1 and disk 2, _j, and

that between disk 2 and disk 3, 52, see figure 13. When considering applying a misfit at an interface one could

specify a similar misfit directly for each interface or alternatively specify a given mismatch in hoop strain for all

interfaces. The first option would effectively result in applying different amounts of hoop strain at each interface,

whereas, the second option would result in different misfits being specified for each interface. In an attempt to isolate the interaction of mean radius and misfit, the constant applied hoop strain approach to selecting the appropriate

misfit at each interface is taken. Consequently, each displacement misfit applied is proportional to the radial location

of the corresponding interface (i.e., _5i = ehR i) such that a constant hoop strain (eh) at each interface is imposed. In

this way the impact of the misfit will be consistently felt even when the mean radius is increased.

Assuming only rotational loading with a speed (co) of 30 000 rpm, analysis results for the disk system are shown

in figures 14 and 15, when the overall mean radius and misfit hoop strain are varied as follows: Rm = 2, 4, and 6 in.

and eh = 0., 0.1 and 0.2 percent. In figures 14 and 15, the radial stress versus radial location and tangential stress

versus radial location are shown respectively, for each of the above cases. Examining figure 14 it is apparent that in

the case of no misfit (eh = 0, see fig. 14(a)) all radial stresses are tensile. Recalling from figures 11 and 12, it is clear

that even if internal pressure was applied to offset these rotationally induced stresses only the inner most surface

would experience compressive radial stress, while the remainder of the disk would remain in a state of tension. The

extent of the compressive region is dependent upon (as shown in fig. 10) the magnitude of the imposed inner pressure. However, increasing this inner pressure (e.g., in the case of a mass loaded design) is of only limited value as

this compressive region rapidly becomes tensile as one proceeds toward the outer diameter of the disk system.

Alternatively, if one imposes a prestress through application of a misfit at the interfaces between disks, a compressive radial stress state can be induced throughout. These interfacial compressive radial stresses (denoted in figures 14(b) and (c) by an I! and/2) then increase as the misfit hoop strain is increased for a given mean radius. Note

that even though the imposed hoop strain mismatch is the same at all interfaces, within the system, the first interface, 11, is more compressive as it benefits from the compressive state at 12 as well. Furthermore, figure 14 clearly

demonstrates that as the overall mean radius of the system is increased, the effectiveness of the imposed hoop strain

is reduced; even to the extent of not inducing any compressive stresses.

Figure 15 illustrates the tangential stress distribution for various mean radii and magnitudes of hoop strain mismatch (misfit) between the disks. Examining figure 15, it is apparent that, (1) the location of the maximum tensile

hoop stress (and thus the initiation of fiber breakage and failure) changes and (2) the magnitude of the hoop stress

increases as the mean radius is increased, given a specified amount of hoop strain mismatch. In addition, in the presence of misfit, the tangential (hoop) stress profile is discontinuous and forms a stair step pattern, with each subsequent disk within a given system, typically having a higher stress state than the previous. Consequently, given an

interfacial misfit and the given material anisotropy, one can be assured that failure would initiate at the inner radius

of the outer most disk, provided the overall system is not relatively thin, i.e., _3Rm > 0.3. Nevertheless, it is apparent

NASA/TM--2001-210578 14
thatacomplexrelationshipbetweenthevariousgeometricparameters, 5,)_, and R,, 7exists (as suggested by eqs. (34)

and (35) for the case of two disks).

Selecting a proper disk thickness configuration is another important aspect that was investigated. Figure 16

shows three arrangements with different individual disk thickness but all adding up to the same overall total thickness. Examining the figure it can be concluded that the most beneficial arrangement is that of two thinner inner rings

surrounded by a thicker outer ring. As this arrangement provides a maximum compressive radial stress state while

giving a lower maximum hoop stress within the outer ring. Clearly, the thicker outer ring could fail (as could all

other configurations) by delamination due to excessive tensile radial stress. However, for this particular case delamination and burst would be confined to the outer ring, which should be a more fail-safe design than either of the other

two configurations examined.

3.2.2 Nononiform Material Pro_tyerties and/or Loads.--The goal of any design is to maximize the allowable rotational speed before either a disk separation (in a preload design) or burst criterion is exceeded. Although, in equations (34) and (35), both disks were assumed to be comprised of the same transversely isotropic material, it is

apparent that not only is the magnitude of misfit important in the determination of the radial stress at the interface

but so too are the material properties. Consequently, this led us to investigate the influence of gradation of the material properties within the laminated system. As a result, the six cases identified in table IX were examined. The first

three cases being PMC (strongly anisotropic) laminated systems and the last three TMC (slightly anisotropic) systems. An additional motivation (besides the degree of anisotropy) behind investigating an idealized TMC system as

well as a PMC system, is the ability to examine the trade-offs between density and stiffness/strength. Since in a

PMC system, as the volume fraction of reinforcement is increased the density (weight) of the material as well as

stiffness and strength also increase. Alternatively in a TMC as the fiber volume fraction is increased the density of

the material is reduced, while the stiffness and strength are increased. Albeit, this later density consideration will be

small and in practicality higher order when compared to the variation in stiffness and strength. Even so, the induced

body forces due to rotation, and thus influence of gradation effects is different between the two classes of materials.

Case one is identical to the baseline material examined thus far; consequently, case two through six will be subjected to the same conditions. These are a rotational speed (o)) of 30 000 rpm, variation of mean radius (Rm = 2, 4,

and 6 in.) and imposition of constant hoop strain (Eh) at each interface of 0, 0.1 and 0.2 percent (i.e., variable misfit,

5, per interface such that _5i = eh Ri)' Again the representative model TMC material properties are given in table VII.

The results of these analyses are presented in figures 17 to 20, where the radial stress versus radial location and tangential stress versus radial location are shown, respectively, for all three cases.

Examining these figures it is clear that material and density gradation play a major role in the resulting radial

and tangential stress distributions; the geometric and strength of anisotropy effects being similar in nature to those

discussed at length previously. In particular, note how the radial compressive stress at the disk interfaces can be

increased or completely eliminated depending upon the imposed material gradient. Similarly, the magnitude of the

stair steps in the tangential stress as well as the maximu.m value and location of this stress are also greatly impacted

by gradation. For example, note the difference between case two and three (compare, figs. 18(b) and (c)) in that for

case two burst failure would initiate in the outer ring, whereas in case three it would most likely initiate in the

middle ring, all else being equal. A similar trend is observed for the TMC cases, wherein case 5 (fig. 20(b)) would

initiate burst failure in the outer ring and case 6 (fig. 20(c)) clearly would initiate failure in the inner ring given sufficient rotational speed.

Since cases 2 and 5 had the most favorable material gradient with respect to minimizing the potential for disk

separation (i.e., maximizing the speed of rotation prior to disk separation) the influence of misfit and rotational

speed will be examined given these configurations. The results are presented in figures 21 to 23. In figures 21 and

22 it is apparent that material gradation naturally leads to higher compressive (or at the least, smaller tensile) interfaciaI stress as the mean radius is increased until the ratio of thickness to mean radius becomes small and the influence

saturates. Also, increasing the misfit (imposed hoop strain) leads as before to an increase in radial compressive

stress. However, now since the outer disk is significantly Stiffer than the inner disks, the compressive stress at interface 12 is now greater than I 1. This result is in contrast to those given in figure 14. Further, for a fixed misfit strain,

the radial stress generally increases with increasing mean radius; although this is dependent upon the magnitude of

the mean radius. Given the previous discussion (see fig. 9) regarding temperature gradient results, one can envision

that imposing an appropriate temperature gradient would lead to similar results as that of material gradation, since in

essence the material properties as well as imposed internal thermal stresses will be nonuniform and naturally lead to

increases in radial compressive stresses.

NASA/TM--2001-210578 15
Finally,fromfigure23,it canbeclearlyseenthatincreasingtherotational speed(co),decreases thecompressive

stressataninterface(possiblyeliminatingit altogether dependinguponthemagnitudeofmeanradius,foragiven

misfit)asexpected. Onemustkeepinmind,thatincreasingtherotational speedsignificantlyincreases(proportional

to_2)thehoop(tangential) stressthusleadingtopotential burstfailureofthesystem.Onceagain,fromadesign

pointofviewalloftheseimportant parameters (misfit,meanradius,speed,material properties, andgradientsof

propertiesand/orloads)mustbesimultaneously optimizedtoachievethe"best"design.

4.0MATERIALANDCONFIGURATIONSELECTION

Intheprevioussectionwespentasignificant amountoftimeunderstanding thekeydesignparameters that

influenceasingle(orsystemof)rotatingdisks;suchasmeanradius,thickness, misfit,speed,materialproperties,

andgradientsofpropertiesand/orloads.Theinterrelationship amongthesevariousparameters canbequitecomplicatedandthereforetypicallyrequiresarigorousanalysisbeperformedforeachconfiguration(material and/or

geometry) investigated. Inthisstudywehavefocusedprimarilyontwoclassesofcompositematerials, PMCand

TMC,givenarange (from relatively thick to thin) sizes of annular disks. In this section we will confine ourselves to

single (solid or annular) disk configurations with constant height and attempt to determine the best configuration for

maximizing energy storage. Clearly, an efficient flywheel stores as much energy per unit weight as possible, without

"failing". Failure can be defined in a number of different ways (e.g., uniaxial, multiaxial, burst criterion, or disk

separation), induced as a result of several factors (e.g., over-speed (burst), decay of material properties (due to environmental factors), fatigue and/or creep conditions) and is dependent upon the specific application. Typically failure

is dictated when the largest stress (i.e., hoop stress, see table VIII) due to centrifugal forces reaches the tensile

strength (under burst conditions) or the fatigue strength (under cyclic conditions) of the material. The utilization,

however, of anisotropic materials has made defining failure even more challenging as now the material strengths and

properties are different in the various material directions and new modes of failure may be introduced.

The "best" flywheel system is one that maximizes the kinetic energy per unit mass without failure. In practice, it

should be remembered that the mass considered should be that of the whole system and the energy stored should be

assessed only as the energy that can be supplied in normal service. Here, however, we will concern ourselves with

the kinetic energy of the spinning disk, only. Consequently, given that this energy is defined as, U = Jco2/2, with J

being defined as the polar moment of inertia of the annular disk JAnnular = r_Pt(b4 - a4)/2, or solid (with a = 0, i.e.,

Jsolid = rcPtb4/2) and the mass of the annular disk being mAnnula r = rtpt(b 2 - a2), or solid (with a = 0, such that

msoli d = _ptb2), one can obtain easily the following ratio, which needs to be optimized, that is:

(38)

or

I lann , r O2162 I

(39)

depending upon the desired geometry.

To maximize the energy per unit mass, the task now becomes one of determining the maximum rotation speed

(co) that a given disk configuration (be it material and geometric) can withstand before the induced stresses (or

strains) exceed some allowable limit. In the case of determining the limit (burst) speed of the disk one would use the

ultimate tensile strength of the material; whereas in the case of fatigue (where the disk is spun up and down

repeatedly) one would most likely use the endurance limit as the design allowable stress. Note, that in the case of

anisotropic materials (e.g., PMCs and TMCs) these allowable limits will be significantly different depending upon

orientation and fiber volume fraction. For example in the case of PMCs the ratio of longitudinal to transverse ultimate tensile s_engtfiis-approximately 30 to 1, whereas in the case 0fTMfs t-hi_ ratio is 3 to 1. This toge(her with

the fact that the induced stress field is biaxial, makes the identification of an appropriate multiaxial failure criteria

very important.

NASA/TM--2001-210578 16
Toillustratethisfactwewillexaminethreeseparatefailureconditionstodeterminethemaximumspeedof

rotationforbothasolidandannularflywheel.Thefirsttwoconditions, limitthespeedbasedsolelyonasingle

(uniaxial)stresscomponent (i.e.,tangential (longitudinal direction)andradial(transversedirection)stress,respectively).Alternatively, thethirdisbasedonanequivalent transversely isotropiceffective(J2)stressmeasure(multiaxial)describedmorefullyinthenextsection, seeequation(59),aswellasinreferences185and188.Returningto

equations(36)and(37)it canbeshownthathemaximumrotationspeediseither

HoopStressOnly

or

COmax < b2f(a,b,_,r,vL)

'

Radial Stress Only

(40)

or

Multiaxial Condition

2 <

O)max - b-g(a, b, 13,r, v L)

O)max <

1

(42)

depending upon the desired failure condition utilized. The above expressions are quite general in that they can be

used for both an annular (a * 0) or solid (a = 0) disk as well as a transversely isotropic (13< I) or isotropic material

(13=1). Ashby (ref. 182) has derived a similar expression to that of equation (40) for the special case of a solid, isotropic disk. In (ref. 182) Ashby denoted the term c_u/p as a performance index and employed it as a discriminator

for the selection of the optimum material from whicLh to construct a flywheel. The optimum material selected was a

graphite (CFRP) or glass (GFRP) reinforced epoxy system, which can store between 150 to 350 KJ/kg of energy. It

is interesting to note that the current work provides similar overall trends, however, the specific form now has an

additional anisotropic factor that will modify the performance index suggested by Ashby (ref. 182) thereby making

it more accurate. Inclusion of this additional factor requires that the appropriate anisotropic material properties be

included in any database used to select between both monolithic (isotropic) and composite (anisotropic) materials.

In the past others (e.g., see refs. (1) and (2)) have expressed the performance index coming from the specific

energy for isotropic systems as

U ,_( _.& ) (43)

m

where K was defined to be geometric shape factor. Furthermore, it was only noted in passing in the literature that

this shape factor will become significantly more complex in the presence of anisotropic material behavior. Now by

combining equations (39) and (42) one can define this complex "shape" factor, which accounts for both material

anisotropy and stress state multiaxiality to be:

-- 1 (b2+a2)

4_7_)

Clearly this shape factor (only valid as shown here for constant height disks) is no longer just a geometric parameter

but in actuality has both material and geometric attributes, K = K (13,4, a/b, vL) such that now both K and t_UL/p

become integral to the performance index.

NASA/TM--2001-210578 17
Invoking the corresponding maximum speed resulting from the three criteria; the calculated stored kinetic

energy per unit mass (specific energy), given an annular and solid disk made of the three PMC and TMC materials

defined in tables VI and VII, are given in tables X and XI, respectively. Examining the results a number of conclusions become obvious. First, as expected from equations (38) and (40) to (42), the specific energy within a solid disk

is independent of its geometry and only influenced by the degree of material anisotropy (stiffness and strength

(UTS)). Secondly, for given annular disk, the specific energy increases as the mean radius is increased. This conclusion is particularly apparent when using the multiaxial criterion. Alternatively, however, the hoop only criterion

does not show such a clear monotonic increase with increasing mean radius (as the location of the maximum hoop

stress moves, this is particularly true for the case of strong anisotropy and relatively thick disks, see figs. 7 and 8),

thus demonstrating the advantage of using a multiaxial criterion. Thirdly, it is apparent that the multiaxial criterion

naturally accounts for the geometry of the disk; in that if a relatively thick anisotropic (with a weak bond between

fiber and matrix) disk is analyzed the controlling stress is the radial component, whereas in the case of a relatively

thin disk the hoop stress dominates. From tables X and XI it is apparent that the analyst would need to appreciate

and account for this fact if they restricted themselves to using a simple uniaxial criterion. Fourthly, annular anisotropic disks are more efficient than solid anisotropic disks, see table XII. This is in sharp contrast to the fact that for

isotropic materials, the specific energy of a solid disk is somewhere between 20 percent (for a thin annular disk) to

100 percent (for a thick annular disk) more efficient than its annular counterpart. Note also, that if one where to only

utilizes the hoop stress criterion, one would incorrectly conclude that in the case of only slightly anisotropic materials _TMCs) a solid disk is more efficient than an annular disk. Once again demonstrating the importance 0f using a

multiaxial critcri0rl in the presence of a multiaxial stress field.

Finally, if we examine the ratio of the specific energy of PMC flywheels to that of TMC flywheels, it is apparent that flywheels made from PMCs are at least comparable to those made with TMCs, and more often they are

approximately three times better (particularly for large sized flywheels (see table 13)). This result is not unexpected

as PMCs are typically 2.5 times lighter than TMCs and as suggested by the above performance index, density plays

a significant role. Also, as noted in table XIII the manufacturable fiber volume fraction of PMCs are typically significantly higher than TMCs and consequently the ultimate longitudinal strength of PMCs are typically significantly

greater (1.5) than that of TMCs (compare tables XI and XII). However, the transverse strength of PMCs are

approximately 0.2 that of TMCs; thus explaining why if one were to employ only a radial failure criterion, PMC

flywheels would only perform approximately 0.3 to 0.5 as well as TMC flywheels, see table XIII. Remember however, that thus far only a static failure condition (ultimate tensile strength) at room temperature has been used to

determine the maximum allowable rotational speed; which is key to calculating the maximum stored energy. Under

fatigue conditions and/or at elevated temperatures TMC flywheels may out perform flywheels made of PMCs. This

will be a topic of future study.

5.0 FAILURE ANALYSIS

A key factor in any design is the ultimate load capacity of a given structure. Flywheels are no exception. The

desire here is to maximize the energy storage per unit mass without experiencing catastrophic failure of the system.

Again, for simplicity we will limit our discussion to that of a single rotating annular disk of constant height and

examine the allowable limit speed under both monotonic and cyclic load histories. Multiple disk configurations with

imposed misfit will be addressed in the future.

5.1 Burst (Rupture) Limit

As mentioned previously, numerous failure criterions can be defined, e.g., disk separation, delamination within

a disk, or a burst criterion. Here our concern will be with calculating the burst pressure or burst speed of a single

annular disk. Although, within the current RSL program disk separation is the primary design consideration, burst

conditions must still be established. Currently, hydroburst tests are being considered as potential certification tests

for establishing the quality of the flywheel configurations being manufactured. However, one should always remember that for a general flywheel configuration the character of the problem changes significantly when going between

internal pressure (hydroburst test) and rotation (spin test). That is, under internal pressure the ordering of the stresses

is such that 6 r is always minimum (compressive), 6_ the intermediate and 60 the maximum (tensile) value; whereas,

NASA/TM--2001-210578 18
underrotation all stresses are tensile, with 6,. being either an intermediate value, in the case of plane stress, or a

minimum in the case of plane strain. This stress state may be altered depending upon the design selected (e.g., a

preloaded multidisk design will have compressive radial stress component throughout provided the preload is sufficient). Therefore, although spin tests are considerably more expensive than their hydroburst counterparts, they still

remain the most prototypical and representative tests and thus the ultimate certification tests. Consequently, we are

also interested in calculating the limit (burst) speed for a given rotating flywheel configuration.

5.1.1 First Fracture.--An approach to calculating the burst limit for a circumferentially reinforced disk, for

either internal pressure or rotation, is to define burst failure to be at the instant that the maximum hoop stress

exceeds the material's ultimate tensile strength in the fiber direction (longitudinal). However, as we showed in the

previous section, a more consistent criterion (than just the hoop stress alone) is to utilize an equivalent transversely

isotropic effective stress (multiaxial) criterion. Consequently, the limit pressure and limit speed based on first fracture (at the location of maximum effective stress) are:

and

=(_tq ] 1

(Pi,,)limit _F.S.) ff{_2_2 + j_2 ff.i }

(45)

= ( _,,L ] 1

°)limit I,,(F.S.)P)b23j{_2g2+f2_g.f }

(46)

respectively. Where F.S. is the factor of safety imposed, g andfwere define previously, see equations (36) and (37)

for the case of pure rotation, and now

_([3, Rm ,_) =

-1+

(r]

_1]

(47)

and

./2(_, Rm ' _.) =,

_l]

(48)

are taken directly from Arnold (ref. 175) for the case of internal pressure. The above first fracture based limits can

be considered lower bounds on the corresponding limit loads, particularly for brittle like materials (e.g., PMC).

However, in the case of more ductile materials (e.g., TMCs) one would prefer to employ plastic limit analysis to

obtain a set of bounds.

NASA/TM--2001-210578 19
5.1.2 Fully-Plastic Limit.---Considering the case of the fully plastic, limit (burst) load (e.g., pressure or speed),

extensive studies have been made previously for the case of isotropic tubes and disks under pressure and rotation;

see Hill (ref. 183) and Nadai (ref. 184). Following these approaches, applications for the extended case of material

transverse anisotropy under pressure for (plane-strain) cylinders and thin disks, were reported in Robinson and

Pastor (ref. 185). A generalization of this anisotropic case under rotation will be outlined here. For convenience, we

first summarize the goveming equations of the present limit load problem, and outline the solution procedure. This

will then be followed by a summary of the various proposals for the different limit loads. More detailed studies and

derivations can be found in references 184 to 187.

The required three conditions for the limit load solution are the satisfaction of (i) equilibrium (see eq. (1)) and

any associated stress boundary conditions, (ii) compatibility for the "plastic" strain rates (see eq. (3)), and iii) the

yield function (see eq. (59)). in particular when equilibrium and yield are combined we get for the generalized case

of plane strain stress the following general equilibrium expression:

+ - + pco2r = 0 (49)

"_ ' 2_r

where

_2 2

= -2 + -4 (3¢- 2) 2

-5 2 2 4

= :1-'2 + --3--4 (3¢- 2) -526

2

?= z12 4 z2 (3¢- 2) zg

2 2 4

and

. _ (2-_,1) _ -(1+_,2)

-1 - 3 z2 3

(2_Ll - 1) (2_, 2 - l)

-'3 = -"4 =

3 3

--(1 + _'1)

-5 = 3 --6 =

with _'1 and 3"2defined for the case of plane stress and plane strain as follows:

(50)

Plane Stress: _1 = _2 = 0

2+¢

Plane Strain: _l = 4----'_

Note the strength of anisotropy (¢) is defined as, _= 4(_ 2- 1)/(4_ 2 - 1), where _ = 0 u/6 u is the ratio of longitudinal to transverse yield (or ultimate) stress. Similarly, the required kinematic constrain(is t_e satisfaction of strain

compatibility, that is:

d ,

_r = _(re0) (51)

Finally, the yield (or ultimate) transverse shear stress, K, (first used in eq. (49)) is related to the longitudinal tensile

yield stress by the following expression,

K 2 (O'_)2 - (O_)2 (52)

= _- (4-;) 1

(1-¢)

NASA/TM--2001-210578 20
Internal Pressure: As an example, for the case of internal pressure only, under plane stress, the final expression

(including stress boundary conditions) for the lower bound of failure obtained from equation (49) will be:

0

f 2drl = In(b); -Plimit < 1 (53)

-elimit (l-aq 2) -1]

2K

where generally this integration requires a numerical procedure. However, for the special case of plane strain under

internal pressure only, analytical integration (between 1"= a, c,.(a) = -Plimit' and i" = b, 6,.(b) = 0) is possible leading

to the following lower bound solution.

Plimit _ 1 _ In b

2K 23_ 1-_ a

(54)

It is also interesting to note the fact that the above solution also satisfies compatibifity thus making it the "exact"

limit pressure for the plane strain case and consequently a valid upper bound for the case of plane stress, as discussed by Robinson and Pastor (ref. 185).

Rotation: In the case of rotation (i.e., nonzero body-force term) the resulting differential equations for the lower

bound estimates for both the cases of plane-strain and plane-stress are not separable and must be solved for numerically. A straightforward iterative solution technique can be employed in conjunction with Euler integration until the

boundary conditions, err(a) = 0 and _r(b) = 0, are satisfied explicitly. Unfortunately, in the case of rotation the

resulting stress state arising from the solution of equation (49) for the case of plane strain cannot be shown to be

associated with a kinematically admissible strain field, as in the case of internal pressure loading. Consequently,

both the plane stress and plane strain conditions merely yield lower bound (limit) solutions.

An alternative analytical proposal for the upper, lower, as well as out-of-plane bounds have recently been

developed and are described more fully in reference 187 with the final expressions taking on the following forms

respectively:

Lower Bound:

Upper Bound:

where

Out of Plim¢ Bound:

(55)

t a "_1+1 [[_

1-[b) "/ _l-(b)2]

/ a _3+ 1

'-t J '

(56)

(57)

1

_=--

8-1

= 41 + T + (g" y)2

(58)

NASA/TM--2001-210578 21
Representative lmit(upperandlowerbound)curvesforagivengeometry(i.e.,b/a = 1.8) are shown in

^

figure 24; that is normalized effective pressure, P/2K, versus strength ofanisotropy, _. Note in the case of rotation

this effective pressure is defined to be the "equivalent" centripetal load, P= po_2(b 2 - a2)/2, and in the case of internal pressure this is merely the applied internal pressure, P = Pin"

From figure 24 it is apparent that in the case of rotation the difference between upper and lower bounds is

greater than in the case of internal pressure, for slightly anisotropic materials; with the averaged effective centripetal

limit load (for an isotropic material, _ = 0) being just slightly lower ( P/2K ---0.52) than the corresponding averaged

limit pressure load (/3/2K = 0.58). Similarly, as the strength of anisotropy is increased (_> 0.8) it is interesting to

note (and yet coincidental) that the plane strain solution for the case of internal pressure follows closely the average

limit speed obtained for the rotation case.

Also, from figure 24 we see that the critical value of reinforcement (referred to by Robinson and Pastor

(ref. 185) as the condition of over-reinforcement) for the case of internal pressure occurs at a strength of anisotropy

equal to, _ = 0.713 for a b/a = 1.8. This point is denoted in figure 24 by the on-set of the plateau. Whereas in the

case of rotation, a potential shear induced out-of-plane mode (see eq. (56)) is identified which is purely dependent

upon the ratio of outer radius b, to height h, (or length) of the disk (or cylinder). An insightful design oriented presentation of this out-of-plane mode is shown in figure 25, where it is immediately apparent that the critical height (or

length) to avoid (or cause) axial splitting before in-plane failure is dependent upon: (i) the strength of anisotropy, (ii)

in-plane geometry of the disk (a/b) and (iii) the ratio of in-plane to out of plane shear flow stress (K/K*).

These curves are simply obtained by equating a given in-plane bound to that of the out-of-plane bound. Though

in actuality the modes will most likely interact in a complex way, following the notion of nonlinear interaction

curves. Figure 25(a) corresponds to setting the lower in-plane bound equal to the out-of-plane bound while that of

figure 25(b) corresponds to equating the average of the upper and lower in-plane bounds to that of the out-of-plane

bound. Examining figure 25 it is clear that in the case of isotropy (4 = 0) the relative size of the disk (a/b) has a significant impact on the allowable height (h). Whereas, in the case of strongly anisotropic yielding (or failure) behavior, the relative size of the disk is not as important as is the ratio of the in-plane to out-of-plane shear flow stress (K/

K*). Furthermore, one can utilize the results displayed in figure 25 to either design a given disk to fail in-plane before axial splitting takes place (i.e., adjusting h/b(K/K*) to be below the line) or to make sure that what is believed

to be the more benign failure mode of axial splitting takes place prior to the more catastrophic in-plane failure (i.e.,

selecting an h/b(K/K*) such that it is above the associated line).

Returning to the case of internal pressure, we see that the condition of over-reinforcement corresponds to a

point in which the mode of failure switches from that of overall in-plane to that of localized out-of-plane flow within

the vicinity of the inner radius (r = a). This alternate upper bound limit is important as it suggests either the maximum amount of reinforcement (volume fraction of fibers) one should utilize or alternatively the maximum relative

size of the disk for a given amount of reinforcement. If one were to exceed either of these parameters, see figure 26,

the resulting design would be inefficient and the resulting limit pressure would be limited by the value of the allowable transverse shear flow stress of the material, as /3 = 2K. For example, in the case of PMC' s with relatively high

volume fractions of fibers (_ > 0.95), the relative disk size, )_/Rm < O. 15, should be small to ensure in-plane failure.

From a practical point of view, it is interesting to note that from such a configuration one has the potential to obtain

an accurate measure of the transverse shear flow stress (K), since at burst and for this geometry, Pin = 2K.

Recall, in section 4 we demonstrated that the stored energy per unit mass increases as the relative size ()dR m) of

the disk is decreased. Consequently, one might conclude that the most efficient flywheel design is that of a relatively

"thin" disk. If thicker disks are to be utilized, than one would naturally desire as high a transverse shear stress (K) as

possible; thus suggesting the utilization of some type of 3-D weave so as to increase the transverse flow (ultimate)

stress and consequently the controlling limit load of the given disk.

Finally, it is important to realize that the first fracture analysis (using the effective "/2criteri°nl), initially discussed, provides a minimum lower bound as compared to that coming from the ductile plastic limit analysis, see

table XIV. This is consistent with the fact that in the first fracture case yielding (or fracture) is satisfied only at a

single point, whereas in the plastic limit analysis yielding (or fracture, depending on the values used in the anisotropic strength measurement, 4) is satisfied at all points throughout the disk. Consequently, one must be cognizant

(when estimating the actual limit load) of whether the material tends to exhibit more ductile or brittle behavior dur1Notethat in the case of internal pressure a significant difference between the firstfracture limit using maximum hoop stress only verses the

effective j2 criterion is observed (again since radial stress is always compressive the effective j2 (multiaxial stress measure) is always larger

than the hoop stress, maximum stress, alone).

NASA/TM--2001-210578 22
ingfailure,whichisdeterminedprimarily by material selection (e.g. resign/sizing/flow) and the chosen manufacturing process (e.g., cure temperature, environment, etc.). In materials that exhibit both characteristics, interaction effects between brittle and ductile failures become important, therefore this will be an area of future stud),.

5.2 Fatigue Life

Employing flywheels as energy storage systems inherently demand that the flywheel be repeatedly spun up and

down, thus requiring the designer to be concerned about the fatigue capacity of any given flywheel system. To

address this concern we will utilize a previously developed transversely isotropic fatigue damage model (refs. 188,

189). This model has a single scalar damage state variable (damage is the same in all directions) that represents both

mesoscale-initiation and -propagation of damage but whose rate of accumulation of damage is anisotropic (dependent upon the orientation of the preferred material direction relative to the loading direction). Furthermore, it has

both a static fracture limit and endurance limit, accounts for mean stress effects and captures the typically observed

nonlinear-cumulative effects in multiple block-program loading tests. The anisotropic fatigue (endurance) limit and

static fracture surfaces are comprised of physically meaningful invariants that represent stress states that are likely

to strongly influence the various damage modes within composites. For example, (i) the transverse shear stress

(ll)-matrix cracking, (ii) the longitudinal shear stress (/E)-dictates interfacial degradation, and (iii) the maximum

normal stress (13) in the fiber direction which dictates fiber breakage. All three of these invariants are combined into

a function representing the effective transversely isotropic J2 invariant:

£_ ) = - (4_()2 _ 1)i1 + rl( )2 12 4

(59)

where

1

I1 = J2 - _ + _ 13

12 = I - 13

13 = 12

in which

J2 = 2SijS(i

I =Ofiij

i = DoSjkSki

D,j = dfl j

Sit = _ij - 3_kk_ij

,i

and di (i = 1,2,3) are the components of a unit vector denoting the local fiber direction. Note this effective J2 invariant (eq. (59)) has been the multiaxial criterion used throughout this study, but specialized to the pertinent case of

circumferential reinforcement with rI = 1 (equal longitudinal and transverse shear response). This special case is

simply written in terms of the radial and hoop stress as follows:

1 2

]_ )=I_L{ _ ()(yr 2+_2 _Cyr(Y0}

(60)

where the open bracket subscript notation in equation (59) and the above is used to distinguish between the static

fracture, endurance limit and normalization surfaces, see reference 189.

NASA/TM--2001-210578 23
Thisfatiguemodelhasalreadybeensuccessfully appliedtotheproblemof a circumferentially reinforced disk

subjected to internal pressure, see (refs. 190 and 191). Consequently, here we will focus our attention only on the

fatigue life of a circumferentially reinforced, rotating disk and in particular on the influence of the key geometric

parameters, i.e., mean radius and thickness. The fatigue life model of (ref. 188) requires at a minimum the availability of both longitudinal and transverse S-N curves to describe the evolution of damage for a given material system.

As this data is presently not available for the PMC system of interest we will confine our investigation to a previously examined SiC/Ti system; the associated damage model parameters being completely defined in reference

(ref. 191).

Results for the case of a single, matrix cladded (inner and outer monolithic disk), disk subjected to multiple

cycles of rotational loading (triangular waveform) is shown in figures 27 to 29. In figure 27 the limit burst speed

versus the geometry of the disk is presented, wherein this limit speed corresponds to a given induced stress state at

the various radial locations (i.e., inner radius (ID) and outer radius (OD)) that fulfills the static fracture condition of

equation (51), that is F L - 1 = 0. This analysis is defined as an uncoupled approach, see reference 190. Figure 27(a)

clearly shows that the limit speed greatly decreases with increasing mean radius; while figure 27(b) illustrates that as

the relative thickness (XlRrn) of the disk compared to the mean radius increases the difference between the inner and

outer limit speed increases greatly as well, as one would expect. In figure 28, the resulting speed versus cycles to

failure curve defining fatigue initiation at the inner radius (ID) of the composite core for various mean radii are displayed. Again, we observe the expected trend associated with increasing the mean radius of the disk; that is, a

marked decrease in either the limit speed (for similar life times) or number of cycles to end of life (for the same

speed). If one where to normalize the calculated fatigue limit speed with the corresponding static limit (burst) speed

at the same radial location for the given geometry one can obtain a master fatigue initiation life curve that is essentially independent of mean radius variation, see figure 29. A similar master curve can be obtained corresponding to

fatigue initiation at other radial locations within the disk as well. Also, if one were to conduct a coupled deformation

and fatigue damage analysis, as done previously for the case of internal pressure (see refs. 190 and 191) a single

master design life curve representing the global failure of the disk due to fatigue could be produced. Such a curve

will be produced in the near future for the PMC reinforced disks of interest to the RSL program when the necessary

experimental data for characterizing the current life model become available.

6.0 CONCLUSIONS

In this study we have attempted to put into perspective the problem of a rotating disk, be it a single disk or a

number of concentric disks forming a unit. An analytical model capable of performing an elastic stress analysis for

single/multiple, annular/solid, anisotropic/isotropic disk systems, subjected to both pressure surface tractions, body

forces (in the form of temperature-changes and rotation fields) and interfacial misfits has been derived and discussed. Results of an extensive parametric study were presented to clearly define the key design variables and their

associated influence. In general the important parameters are misfit, mean radius, thickness, material property and/or

load gradation, and speed; all of which must be simultaneously optimized to achieve the "best" design. All of the

observations made herein regarding the deformation analysis are in total agreement with previous conclusions and

observations made in the past by other investigators.

Also, the important issue of defining proper performance/merit indices (based on the specific stored energy), in

the presence of multiaxiality and material anisotropy have been addressed and utilized to discuss the difference

between flywheels made from PMC and TMC materials with either an annular or solid geometry. An interesting

observation made is that an annular anisotropic disk is more efficient than its solid counterpart; this is in sharp contrast to the fact that an isotropic solid disk is more efficient than an annular disk.

Finally two major aspects of failure analysis, that is the static and cyclic limit (burst) speeds were addressed. In

the case of static limit loads, upper, lower, and out-of-plane bounds for disks with constant thickness were presented

for both the case of internal pressure loading (as one would see in a hydroburst test) and pure rotation (as in the case

of a free spinning disk). The results (interaction diagrams) were displayed in a designer friendly format. For the case

of fatigue, a representative fatigue/life master curve was illustrated in which the normalized limit speed versus number of applied cycles was described for a cladded TMC disk application.

NASA/TM--2001-210578 24
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NASA/TM--2001-210578 31
TABLE I.--ENERGY STORAGE TYPES (ref. 5)

Storage type

Mal_netic fields

Superconducting coi'f

Elastic deformations:

Steel sprin$

Natural rubber band

Electrochemical reaction

Lead-acid batter';,,

Nickel-cadmium batteQ'

Kinetic energy .

Mara$in_ steel flywheel

4340 steel flTwheel

Composite fl_cwheei

_Using longitudinal strength of Kevlar

without any shape factor.

Specific energy.

Wh/k_:

1-2

0.09

8.8

17.9

30.6

55.5

33.3

_2t3.8

TABLE 17. RI )TATING DISKS COMPSED OF ISOTROPIC MA

Conditirms/type of stud}, Analytical l" N'umerical

Variable thickness [5]. [6]. [7]. [16]. [171, [20]. ] [8]. 1t2] [24]. [301.

[22]. [23]. [26], [321. [441. [42], [491, [50],

, [55],

Uniform thickness. [5], [6], [7], [16], [17], [18], [8], [12 L [241, [27],

[19], [20]. [21], [22], [23]. [30], [34], [35], [37],

. [25], [26], [32], [33], [36], [38], [39], [41]; [42],

!

[44t, [45]. [51]. [551. [43]. [467.N7]_ [491,

• [50t.[5i1.[53],[541,

Misfii

Temp

"ERIALS

ExperimentaJ

[281, [311. [401,

[281. [311. [40].

[411. "

[t81. [481, [39]. [.1.61, [311,

TABLE III.--ROTATING DISKS COMPOSED OF ANISOTROPIC MATERIALS

C0nditi0,g[type of Shady: Anal yt ic al

Variable thickneSS: [5% [61], [64], [66]. [71], [881,

Uniform thickness i [-_, [56], [57], [58], [59], [60],

[6i]; [62],[63]. [64], [66]. [67].

[68]; [69], [70], [71]. [73]. [75].

........... .... [7@ [88], [92]; [95], [101],

_ii! ! !!!

............ ....... ................... [103li [1131,

[51, [70],

[611. [67].

[ [% [lO1_[to71i[1o% [t161;

[65]. [72], [74], [771_ [82],

[_]_ [94]:[981;[1001;[!041;

[105i;[i06t [i07] [i081;

[109]. [1101; [I 14]; [I16], :

[74],[821;[lOO1;[io8]; .......

Experimental

[601, [74], [75i, [78]; [79],

[80], [81], [83], [85], [86],

[87], [891, [901, [911, [93],

[96], [97], [99], [102].

[1101, [1121, [1141,

[9o1........

[741, [99],

NASA/TM--2001-210578 32
TABLE IV.--NON-LINEAR ANALYSIS CLASSIFICATION FOR ROTATING DISKS

Condltions/type of study Isotropic

Elasto-Plastic :f_

:: : ii!i :__l, [ 1211, [122]. [ 1231. [1241,

.... ] [128]. [I291. [130I. [131] [1321 [133],

Uniform thickness J [t371, [138]. [139], [141]

remp _'_ _J_}ld1261.1134,1d J351, [140].

Creep/Viscbeia_ti¢ ..... = -

,--,e-, . ..............................

Temp _ : [I67],

Fatigue ..... _:!_J--

Variable thickness .....

Uniform thickness

Temp

TABLE V.--SUMMARY OF PROBLEMS INVESTIGATED TO VALIDATE

PRESENT ANALYTICAL SOLUTION.

T_,pe/Altemate Method

Solid Anisotropic Disk

FEA

Single Annular Anisotropic Disk

Lekhnitskii [58] & FEA

Arnold [ 175]

Hub/Anisotropic Disk

FEA

Two Annular, isotropic Disks

Ugural and Fenster [176]

Rotation

X

X

Pressure

X

X

The rmal Mi silt

X

X

X

TABLE VI.--ROOM TEMPERATURE PMC MATERIAL PROPERTY DATA

Property

El,,, Msi

E r, Msi

Fiber,

(ref. 180)

40.0

Resin,

(ref. 181)

vf = 40 percent vf = 60 percent vf = 80 percent

0.677 16.4 23.1 32.1

2.0 0.677 1.06 1.26 1.58

_302 L

hi0 T

v L 0.25 0.33 0.295 0.282 0.267

v r 0.25 0.33 0.42 0.373 0.303

G t, Msi 2.9 0.2544 0.425 0.62 1.5

_c, in.tim/°F '-0.22x104z 42-0x10-°6 0"845x 10_z 0"269 x104_6 -0"032x 104z

ar, in./in.l°F 6.28x 10-°6 42.0x10-_ 33"7x 10-°6 25"4x10-_ 17"0x10°6

p._p-sec_/in 4 1.6511x10 "°7 1.2163x10 _ 1.3033x10 -°7 '£.477xi0 -°7 1.5657x10 -°7

UTS, Ksi 747 to 790 17.5 "110 "398

_Determined via rule of mixture with a 0.3 knockdown for in-situ strength.

bSuggested from experiment.

NASAfrM--2001-210578 33
E L, Msi

E r, Msi

V L

V T

G/., Msi

cq_, in/in/° C

{xr, in/in/°C

p, kip-sec-/in

UTS, Ksi

TABLE VII.--ROOM TEMPERATURE TMC MATERIAL PROPERTY DATA

Property Fiber

SCS-6.

¢ref. 192)

58

58

0.19

0.19

24.36

2.2x10 _"

212x10_ , "

3.085× 10I_7

550 to 650

Matrix

Ti-15-3.

(ref. 192)

13.0

13.0

0.33

0.33

4.924

8.1xlO -_

8.1x10 "-_

4.21x10 mr

126.2

SCS6/Ti-15-3

•,,f =15 percent

19.8

15.7

0.306

0.366

5.65

5.54×10 "_

7.66x10 _s

4.04x 10 -°_

_185

SCS6/Ti- 15-3

v_ =35 percent,

(ref. 192)

28.8 26.1

20.2

0.272

0.37

7.0

3.99x10 _

6.38x10 _

3.81x10 _7

17.4

SCS6/Ti- 15-3

vf =55%

_257 201.6 L

60.9 T

37.8

27.0

0.249

0.319

9.62

3.18 xl0 "m

5.24 xl0 _

3.591x10 _

' 347

aDetermined via rule of mixture: Botded values are experimentally measured, (ref. 192).

TABLE VIII.--MAXIMUM NORMALIZED RADIAL AND TANGENTIAL STRESS FIELDS

Rm (_r/pO}2= g( _, R,,,, _.) b z

k=2 _.=4 _=8

PMC TMC PMC TMC P/vIC TMC

4 1.08 1.6 2.1 6.3 3.8 19.5

8 1.46 1.64 4.3 6.5 8.5 25.1

16 1.6 1.65 5.8 6.6 17.4 26.1

32 1.64 1.65 6.4 6.5 23.4 26.3

e_o/pO)"= f( _, R,,, _.) b:

k=2 _=4 k=8

PMC TMC PMC TMC PMC TMC

17.8 22.0 22. 29. 39. 25

72.8 75.3 71 88 88 116

276 278 291 301 286 353

1066 1067 1106 1111 1165 1205

TABLE IX.--DESCRIPTION OF MATERIAL GRADATION BY CASE

Case Disk 1 Disk 2 Disk 3

One v_= 60%, PMC vf= 60%, PMC .......... V[= 60%, PMC

Two vf= 20%, PMC vt= 60%, PMC vf= 80%, PMC

Three vf= 80%, PMC vf= 60%, PMC v_= 20%, PMC

Four vf= 35%, SCS-6/Ti-51-3 vt= 35%, SCS-6,rFi-51-3 _'t= 35%, SCS-6/Ti-51-3

Five vr= 15%, SCS-6fri-51-3 vf= 35%, SCS-6/Ti-51-3 vt= 55%, scs-6fri-51-3

Six vt= 55%, scs-6fri-51-3 vf= 35%, scs-6fri-51-3 v_= 15%, SCS-6/Ti-51-3

Criterion

vt = 40%

vf = 60%

v t = 80%

vr = 40%

v t = 60%

vr = 80%

vf =

vf =

vf=

"R,,, b

TABLE X.--MAXIMUM SPECIFIC STORED ENERGY (KI/KG) FOR A SINGLE

DISK MADE WITH PMC

Annular Disk Case (k = 3) I Solid Disk Case (a = 0)

Rm=4_[ R,,=8 I Rm=16 I Rm=32 I b=5"5], b=9"51 b=17"5 I b=33"5

327.86 295.29

792.16 713.46

983.93 886.17

113.74 250.12

267.56 588.40

338.8 745.05

, ,,,,

40% 115.36 211.14

60% 278.72 510.15

80% 346.19 633.65

and _ hasthe units ofinches.

Hoop Only"

299.96 313.04 275.62 275.67 275.74 275.79

724.75 756.37 665.94 666.06 666.23 666.36

900.19 939.46 827.14 827.29 82_/.51 827.67

Radial Only

825.3 3127.1 87.285 i 87.357 87.514 87.64

1942.1 7357.1 205.34 205.51 205.88 206.17

2458.1 9316.1 260.01 260.22 260.69 261.06

Multiaxial

299.96 313.04 89.848 90.048 90.109 90.01

724.75 756.37 217.09 217.57 217.72 217.48

900.19 939.46 269.64 270.24 270.42 270.12

NASA/TM--2001-210578 34
TABLE XI.--MAXIMUM SPECIFIC STORED ENERGY (KJ/KG) FOR A SINGLE

Critcrion

v t = 15%

vf= 35%

vf= 55%

v_ = 15%

vf= 35%

vf= 55%

DISK MADE V_qTH TMC

I Annular Disk Case (X=3) 1 Solid Disk Case (a = 0) R_=4_ I Rm=8 I Rm=16 IRm=32 b=5.5 I b=9.5 Ib=17.5l b=33.5

105.53 119.76

155.24 176.18

222.74 252.79

225.36 802.10

332.41 1183.10

452.87 1612.10

Hoop Only

131.69 139.10 190.68 190.75 190.68 190.69

193.74 204.63 280.51 280.61 280.50 280.53

277.97 293.60 402.48 402.62 402.46 402.50

Radial Only,

3119.1 12361.0 72.94 72.97 72,94 72,97

4610.0 18231.0 107.58 107.64 107.58 107.63

6267.1 24831.0 146.56 t46.64 146.57 146.63

Multiaxial

vf = 15% 105.53 119.76 131.69 139.1 71.79 71.8 71.79 71.84

vf= 35% 155.24 176.18 193.74 204.63 105.61 105.62 105.62 105.69

vf= 55% 222.74 252.79 277.97 293.6 151.53 151.54 151.54 151.64

aRm, band X has the units of inches.

TABLE XII. RATIO OF ANNULAR DISK TO SOLID DISK MAXIMUM SPECIFIED STORED

ENERGY

r //I l I

Criterion

Hoop Onl_¢ 1.19 1.071 1.088 1.135 0.553 0.628 0.691 0.729

Radial Only' 1.303 2.863 9.43 35.686 3.09 10.991 42.758 169.359

Multiaxial 1.284 2.345 3.329 3.478 1.47 1.668 1.834 1.936

"R m and _.has the units of inches.

TABLE XIII.--RATIO PMC TO THAT OF TMC

FLYWHEELS SPECIFIC ENERGY

(U) /(U) Annutar Disk Cases (L=3)

eMC TMC Rm = 4_ I Rm = 8 I Rm = 16 R m = 32

I

I

Hoop Onl_¢

40%/15% 3.107 2.466 2.278 2.25

60%/35% 5.103 4.05 3.741 3.696

80%/55% 4.417 3.506 3.238 3.2

Radial Onlv

40%/15% 0.505 0.312 0.265 0.253

60%/35% 0.805 0.497 0.422 0.404

80%/55% 0.748 0.462 0.392 0.375

Multiaxial

40%/15% 1.093 1.763 2.278 2.25

60%/35% 1.795 2.896 3.741 3.696

80%/55% 1.554 2.507 3.238 3,2

aR m and _.has the units of inches.

TABLE XIV.--COMPARISON OF BRrI'TLE AND

DUC TILE LIMITS

b/a = 1.35, _=0.93 Limit pressure,

Ksi

Limit speed,

_m

Ductile

Upper Bound 61.038 55 320

Lower Bound 54.237 54 760

Brittle

39.1 51 180

58.3

First Fracture (Jg)

First Fracture (om,_) 51 180

NASA/TM--2001-210578 35
Fibers --, 11, L, 0

22, T, r

Figure 1 .------Single disk, reinforced in the circumferential

direction.

Figure 2._Schematic of "n" concentric disks.

NASA/TM--2001-210578 36
.08 -- _ Current solution

• Lekhnitskii [58]

0.0 _[175]_ Current solution _.°J .06 --

•-o.5 _ _ .o4

_IQ:-_-1.o .o2

-1.5 i I ] e__

00 11.8

-2.0 1 l J_ I " 0.2 0.6 1.0 1.4

r

0.0 0.5 1.0 1.5 2.0

r

R m

3 m

•--.--- Current solution

1 -- • Arnold [175]

¢1_-_ _--I-

-5 f I

0.0 0.5 1.0 1.5 2.0

r

Rm

Figure 3._Comparison between the current study and

Arnold [175].

p

Rm

.70 -

¢___¢..55- _ _#"_ Cu_nt _os

¢ -- .40 -

lution

.25 - ,_/ • Lekhnitskii [58]

.10 I I I I

0.2 0.6 1.0 1.4 1.8

r

Rm

Figure 4._Comparison between the current study and

Lekhnitskii [58].

NASA/TM--2001-210578 37
0.0

_.1

_ 1[4.4

_.7

-1.o I

0.7 0.8 0.9 1.0 1.1 1.2 1.3

r

Rm

1.5

1.0

0.5

_ E_- 0.0

-0.5

-1.0

--'---- Current solution

O Ugural and Fenster [176]

-1.5 I I 1 I

0.70 0.85 1.00 1.15 1.30

r

Rm

Figure 5.mNorTnatized radial and tangential stresses.

Comparison with maximum given by Ugural and

Fenster [176],

Misfit -_

\

\

Figure 6.---Schematic representation of two concentric

disks with misfit.

NASA/TM--2001-210578 38
PMC, X = 3

.06 --

--= ,

.05 --

.04 --

g .03 --

.02 f

.01

.00 1 I

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

(a) R m

1°0 --

0.9--

0.8--

f 0.7 --

0.6--

0.5--

0.4

(b)

PMC, _ = 3

%0.094

o

"%¢

I I I I I I t I I

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

.06

.05

.04

g

.03

.02

.01

.00

(c)

.07 -- PMC, X = 7

_ ,-,_'--'_ __ =1.75 I/\ .... k"m

/

I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2,0

r

Rm

PMC, ,_ = 7

1.0 I

0.9 "-,,0.21 9

0.8 "'-

0.7 0,438_'-,.,.,_

0.6 -- .,,,-" "" _,-""-.,_

f 0.5 -- s° ./"

0.4 -- _,,,,.,,, s

0.875_

0.2--0.3-- /_m=1.75

0.1 --.._t

o.o I t I I 1 I I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

r

(d) Rm

Figure 7.mNormalized radial "g" and tangential "f" stresses (aka distribution factors) versus normalized radius

given a single rotating disk made from PMC. Figure (a) and (b) correspond to a 3 in. thick disk, while (c) and (d)

to a 7 in. thick disk.

NASA/TM--2001-210578 39
.14 --

TMC, X = 3

(a)

.12

.10

.08

.06

.04

.02

.00 I t

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

1,0 --

0.9 --

0.8 --

0.7 --

f

0.6 --

0.5 --

0.4--

0.3

TMC, h =3

• "_-_"",. 0.094

%% "_ "t

• ,, "_._0.188

%%

%***0.375

l t I I I 1 I I I

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

(b) Rm

.20

g .15

.10

.05

.00

(c)

.30 --

.25 --

TMC, X = 7

.75

0:4,3.8

, I I !I ¢.._-"t....._1 T t I\ I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

r

Rm

1.0 _rL TM , X = 7

0.g _ "_

0.8 t "_ "-,0.219

,,, ,,

f 0.6 -- _ _'•0.438

0.5 _ "

0.4 _, 0.875

0.3 -- Rm

0.2 I I I I I I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

r

(d) Rm

Figure 8._Normalized radial "g" and tangential "f" stresses (aka distribution factors) versus normalized radius

given a single rotating disk made from TMC. Figure (a) and (b) correspond to a 3 in. thick disk, while (c) and (d)

to a 7 in. thick disk.

NASA/TM--2001-210578 40
Applied T-T o

temperature profile

./..ii_; profile

L_Temperature a b a b

Constant temperature

--, -- • Linearly increasing from

inside temperature

3.0 ........ Linearly decreasing from

inside temperature

2.5

2.0

_"_" 1.5 ...................................

1.0 ,," .....

_.@ ",.. '_%

,.'" ........ "'-,

o.o I I

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

Constant temperature

20 -- -= -- • Linearly increasing from

inside temperature ........ Lineady decreasing from

10 -",,._ inside temperature

u_ -"""_--'=,--

.... .....

-10 --

-2O I ] I I ] I I I

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

Figure 9.--Influence of temperature gradient on a

single disk.

0.00

-0.5o

- ,........................ !:;:;._. ...................

E j," ,W A,

-0.25- -_ in.

-

.- =' _ 0.5

: i /71 "-- 1.0

; ! a

--': ! a --,, 3.0

: I ,,o,- 6.0

-0.75 - i

_ J li I ] J ] ]

-1.00 i

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

r

Rm

in.

8.8- _ _ 0.5

\

B _ 1.0

.... 3.0

• .... 6.0

t

t

=. i t%

2.2 - _"

-- _

.I,.,,. I I [ I

0.o , ......... ........ _L-.:_,..._.,,.........

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

r

Rm

Figure 10.--Influence of applying pressure along the

inner surface of a single disk. X - thickness of disk.

NASA/TM--2001-210578 41
Uniform temperature

=.- ,,,, Linearly increasing from

0.4 Inside temperature

0.2-- %*_

_" " o.1

0.0

-0.1 -_

-0.2 I 1 I , I I I I l

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

4

3

0

-1

_- Uniform temperature

-- ,-, Linearly increasing from

.14 inside temperature

--

H E .08

_ .06

.04

.02

.00

-0.2 i

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

F _ Uniform temperature

i =. Linearly increasing from 1.5

inside temperature l_

0.5

-%

I I I I I t I'_1

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Figure 11.mlnfluence of thermal, internal pressure,

and rotation on a single disk. Speed of rotation is

15 000 rpm.

2.0 --

Uniform temperature

--- ,-, Linearly increasing from

_ inside temperature

0.0 I I I I I I I I

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

Figure 12.mlnfluence of thermal, internal pressure,

and rotation on a single disk. Speed of rotation is

60 000 rpm.

NASA/TM--2001-210578 42
(a)

.20

.15

.10

.05

.00

0.0

R m ,

in.

-- 2

"= " 4

==1111 6

__ le'"= %%

I

i

I

I

I

V/2 ,

0.5 1.0

r

Rm

_t

: I

: l

- t

t

.-: t I

1.5

=

J

2.0

Figure 13.---Schematic representation of three concentric disks with misfit.

.05

.00

*" -.o5

-.10

-.15 I

0.0 2.0

0o)

-- _w,,

......_/\......._

. '_

\ /',, '_

-

n == 4

II Illll J 6

I I I

0.5 1.0 1.5

r

R m

.00

-.05

H E -.10

_. -.15

-.20

-.25

0.0

t "_I ,F

t "-/ :

t ,¢.

I =- .,,,,..:

t J "

- ! !1 ;;

,l l, I'_ --2

-- LI --.. 4

...... 6

'l I [ I

0.5 1.0 1.5 2.0

r

(C) Rm

Figure 14.--Normalized radial stress for disk subjected to misfit and rotation. Case (a) has no misfit,

i. e., Eh = 0, (13)has Eh = 0.1% and (c) has ch = 0.2%

applied at all interfaces.

NASA/TM--2001-210578 43
3.2

2.4

Rm7

in.

------- 2

lW 4

==lWgl 6

"%"', I,,,, ,, ,,,, ,a, ,=.mo#,.,

-__] I

o.o I I

0.0 0.5 1.0 1.5 2.0

r

(a) R m

4.0

3.0

T_ 2.0

mr

1.0

0.0

Rm,

in. ,,

2 "t"*".

-- - 4 """,,,,,

%%

-1 .o I ..... I I I

0.0 0.5 1.0 1.5 2.0

r

CO) Rm

4 F

Rm' ".

in. ""'...

3 - __. 2

I I 4 _,, "'°....

E 2 ....... 6 "", ""

_. 1 s .....

-- *apts.

0

-1 I I

0.0 0.5 1.0 1.5 2.0

r

(c) Rm

Figure 15.--Normalized tangential stress for disk subjected to misfit and rotation. Case (a) has no misfit,

i. e., Eh : O, CO)has _h : 0.1% and (c) has Eh = 0.2%

applied at all interfaces.

.10

.05

HE .00

_. -.05

-.10

-.15

0.6

i ,l_l,,gl,,I,iR_. _. ,._o%

_" / I k 1

l m " "- A2 -- t

\ ss_,-t ....... A3

IVl I ] 1 I I I

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

r

Rm

2 1

1 --

0 IQ'l=q'ql=l

-1 ]

0.6 0.7

i A1

1 1 A2

....... A3

• m#a'

q%%%__h..._ I r_ _at

t**'#r'"r ft =nl w=l =,,11 ,i r, _'

ot _I I w mt I Ol,f_=_r,

%%1=

I I I I I I I

0.8 0.9 1.0 1.1 1.2 1.3 1.4

_Lr

Rm

Figure 16.--Infiuence of varying internal disks

thickness on the radial and tangential stresses.

k 1 {2, 0.5, 0.5}, A2 {0.5, 0.5, 2}, A3 {0.5, 2, 0.5}. The

misfit here is Eh = 0.1%.

NASA/TM--2001-210578 44
(a)

1

0

-1

-2

-3

0

.,,% ....

.,,,,,..... ,.,_ "...

_--2 2

__ 4

_ .... 6

I I I I I I I I

1 2 3 4 5 6 7 8

r, in

Rms

JR.

1 2

4

6

0 ___ f_---_'_

E _

c -1 • s% I +-. :

-3 I I I I I I

0 1 2 3 4 5 6 7 8

(b) r, in

(c)

Rm s ,,_'* in / \

\

•----- 2 ."

,

.- - 4 /" _'_/

...... 6

- i

.: t

=

\

1 -- ,,=J

-1

-2

-3 I

0 1 2 3 4

r, in

I 1 I I

5 6 7 8

Figure 17.--Radial stress for different material

configurations subjected to rotational speed

of 30 000 rpm. (a) material case 1, (b) materlal case 2 and (c) material case 3. (see

Table 9)

90

70

50

30

10

0

-10

0

(a)

R m ,

JR, "+,.,,°%,

----- 2

am 4 "*e+'*s+*+,

1 2 3

I 1 I I I

4 5 6 7 8

r, irt

110

9O

70

50

b

50

(b)

Rm,

-- irl.

,--,,- 2

-" - 4

I ! II III II I 6

-- %%

lO0 ""

-1o I I

0 1 2 3 4 5

r, in

.r°f,,,

I 1 I

6 7 8

70

R m, ,,,,.,,,,,

irl. %, .......

50 ----,_, 2

-- - 4

llllll 6 "--"

30 -- '.........

_) _ =1, illl Q

-

0

I_ I I I 1 I ]

-10

0 1 2 3 4 5 6 7

(c) r, in

Figure 18.--Tangential stress for different material configurations subjected to rotational

speed of 30 000 rpm. (a) material case 1,

(b) material case 2 and (c) material case 3.

(see Table 9).

NASA/TM--2001-210578 45
Ii

b

(a)

9

7

1

0

-1

-3

-5

0

R m ,

- in..,/"\

----_- 2 .."

-- - 4 t"% _ \

_ # =l/ • -

• "" 6 # "._

I

_ , _',

I I I I I I rl I

1 2 3 4 5 6 7

r, in

_-2

b

-4

(b)

-6

0

,'\ .• ...,_

I i= # • :, .% ...,, z ?'''.'"

_ 4

6

IVl I I I I I I

1 2 3 4 5 6 7 8

r, in

200--

150 --

& 100 --

50--

0

0

(a)

&

b

(b)

R m ,

in.

------ 2

mm 4

Illlll 6

i i J i

1 2 3 4 5 6 7

r, in

200--

100 --

0 --

-10o l

0 1

m

m mB

uxluel

Rml

in. "'%,,,

4

6 _'_'*

I 1 I I I I 1

2 3 4 5 6 7 8

r, in

20--

Rm,

in. ...

15--._... 2 - _.

-- - 4 -"

_10-- # -

' V " b I

# .:t

5-- I .-'i :

0 I I_.1 :--I

0 1 2 3 4 5 6 7 8

(c) r, in

Figure 19.--Radial stress for different material

configurations subjected to rotational speed

of 30 000 rpm. (a) material case 4, (b) material

case 5 and (c) material case 6. (see Table 9).

.t

b

240 --

(c)

R m ,

in. ",,,,.

-- 2 %'-.,,

eu

180 --

120

o --_-I I i I I

0 1 2 3 4 5 6

r, in

Figure 20.--Tangential stress for different material

configurations subjected to rotational speed of

30 000 rpm. (a) material case 4, (b) material

case 5 and (c) material case 6. (see Tab!e 9).

NASA/TM--200 i-210578 46
C

(a)

1.5

1.0

0.5

0.0

-0.5

-1.0

0

r t, _

t _ .."%

I t "':" '"

I ',/'_

I =wsww;

I [ I ] ]

1 2 3 5 6 7

I

4

r, in

C

(b)

1

-3 I I

0 1 2

r '_/\l _ .-

t I tl " .e'; "

V _ -" _---2_

_ ====wj

I I I I I _J

3 4 5 6 7 8

r, in

.M

0

-1

-2

-3

-4

-5

-6 I I

0 1 2

R m ,

in.

2

4

6

I

8

(c)

Figure 21 .--Radial stress for material case 2 for

different applied misfits subjected to rotational

speed of 30 000 rpm. (a) Eh = 0, (b) Eh : 0.1%

and (c) £h = 0-2°'/°-

R m ,

in.

2

4

6

( V ----

t-- .= i &

II , [ Rm, t ! "

,I.,!/.. _ in.

¥ _] ,,,"-- 2

" - 4

xwwwll 6

I I t I I 1

3 4 5 6 7 8

r, in

(a)

12--

9 m

6--

3--

0

0

Rip

in.

2

I _ t l ......

\" ,, .... °

t .."%

I\ _....s .,...

:\ l, \

I I:t_t/_',f i\1

1 2 3 4 5 6 7 8

r, in

-6

0 4

(b) r, in

2 --

l/

1_1 I I I I I I

1 2 3 5 6 7 8

o _ ./ -...._/_

-6 -- ,-_j

R m ,

"_ In.

_. -_2

_ _ _/ ------ 2

-" - 4

-18 -- "'* 6

-24 I I I I 1 I I I

0 1 2 3 4 5 6 7 8

(c) r, in

Figure 22.mRadial stress for matedal case 5 for

different applied misfits subjected to rotational

speed of 30 000 rpm. (a) _h = O, (b) Eh = 0.1%

and (c) Eh = 0.2%.

NASA/TM--2001-210578 47
C

t_

(a)

0

-1

-2

-3

-4

-5

-6

-7

0

:\ \/ /

-- "l \ J ;

,-V\ ;., :

1- '\J ',! "__

I- V _ .7_"

/ I I t i I I I

1 2 3 4 5 6 7

r, In

Rmy

in.

2

4

6

I

8

i!

--5 --

_6 I

0 1

-2

-4

(b)

•

I -1

E1

1

F.

t IE

Z IE

}' I '

I I I I 1

2 3 4 5 6

r, in

i

mu

Illll_

I I

7 8

Rml

in.

2

4

6

2 -- __ /% ,,'%

,. ,_.. ; \ /_

o--\ ,,,, u; ;;h

C

-4t- -_ --

".: IL_

-6 _ ......

-8 t 1 I I I I 1 I

0 1 2 3 4 5 6 7 8

(c) r, in

Figure 23.--Radial stress for material case 2 for

different speeds of rotation. (a) _o= 15 000 rpm,

(b) (o = 30 000 rpm, and (c) to = 60 000 rpm.

Rm_

in.

2

4

6

3.0

2.5

2.0

"1_ 1.5

1.0

0.5

Out-of-plane Upper bound --_\_ _ J

- ..

Ave_ °

| /

Lower bound --_

o.o t 1 I I I

0.0 0.2 0.4 0.6 0.8 1.0

Strength of anisotropy,

Figure 24.--Illustrates the upper and lower bounds

for both rotation (solid lines) and internal pressure

(symbols) limit loads by plotting the normalized

effective pressure versus strength of anisotropy.

v

J_I.Q

v

VIo = 0.9 "thin annular"

,-- a/b = 0.5

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.0

I I I

0.2 0.4 0.6 0.8 1.0

(a)

4.0_ 3.5 ,-- a/b = 0.9 "thin"

.0 _.,,_ PMCs

2.5

2.0 an_

1.5

1.0

0.5

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 25.--Interaction curves for the out-of-plane

failure bound: (a) represents the limit as compared to the in-plane lower bound, and (b)

represents the limit as compared to the average

in-plane bound. The h is the height of disk, b outer

radius, _ the strength of anisotropy.

NASA/rM--2001-210578 48
3.0

2.5

_1¢ 2.0

1.5

1.0

0.0

_ Out-of-plane flow

0.2 0.4 0.6 0.8 1.0

Figure 26.--illustrates the interaction curve differentiating in-plane failure (below the line) and

out-of-plane flow (on or above the line). _-is the

ratio of outer to inner radius and _ is a measure of

the strength of anisotropy.

E

.3

3

120000

100 000

80000

60 000

40 000

20 000

0

0

i

-- __jj-- ID

5 10 15 20 25

R m, in

(a) Mean radius affect

E

.3

$

120 000 -- ID --_

100000 -- OD--_ \\J _

60 000

40 000

20 000

0 I I l

0.0 0.1 0.2 0.4 0,5 0.6

h

Rm

(b) Thickness affect

Figure 27.---Static limit or burst speed as a function

of geometry, (a) maximum speed versus mean

radius, (b) maximum speed versus the ratio of disk

thickness to that of mean radius.

NASA/TM--2001-210578 49
E

Fiber

reinforcement -_7_k I Rm,

120 000 -- in.

Life at ID

-- 2.00

100 000 -- /- Increasing --- 2.96

_ radius -- 6.00

80000 -- -- 12.0

- - 24.0

60ooo- .._

20 000

0 ...... f ...... 1...... 1...... I ...... I...... 1

100 102 104 106 108 1010 1012

Fatigue life, Nf

Figure 28.--Generalized speed versus cycles curve indicating the onset

of fatigue Initiation cracks at the inner radius of the composite core.

1_O

L.,,uat,u _ /-- Increasing

0.8 -- m_ _ m_an radius

Rm,

0.6 - in': / _-=-_--__

-- 2.00 /

0.4_-- -- 2.96

0.2F -- 2410

!

o.o/ t I [ I I I

100 102 104 106 108 1010 1012

Fatigue life, Nf

Figure 29.--Normalized speed versus cycle master curve denoting

fatigue initiation at the inner diameter, o_L is the static limit speed

and to is the maximum applied speed within a cycle.

NASA/TM--2001-210578 50

REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, Including the time for reviewing instructions, searching existing data sources,

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

January 2001 Technical Memorandum

4. TITLE AND"SUBTITLE ' ' 5. FUNDING NUMBERS

Deformation and Life Analysis of Composite Flywheel Disk and

Multi-Disk Systems

6. AUTHOR(S)

S.M. Arnold, A.F. Saleeb, and N.R. Al-Zoubi

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

g. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

11. SUPPLEMENTARY NOTES

WU-755-A4-4)9--00

NCC3-788

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-12547

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--2001-210578

S.M. Arnold, NASA Glenn Research Center; A.E Saleeb and N.R. AI-Zoubi, University of Akron, Akron, Ohio 44325,

(work funded by NASA Cooperative Agreement NCC3-788). Responsible person, S.M. Arnold, organization code 5920,

216--433-3334.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited

Subject Categories: 24 and 39 Distribution: Nonstandard

Available electronically at ht_p_://gltrs._c.nasa._ov/GLTRS

This publication is available from the NASA Center for AeroSpace Information, 301-621-0390.

13. ABSTRACT (Maximum 200 words)

In this study an attempt is made to put into perspective the problem of a rotating disk, be it a single disk or a number of concentric

disks forming a unit. An analytical model capable of performing an elastic stress analysis for single/multiple, annular/solid,

anisotropic/isotropic disk systems, subjected to both pressure surface tractions, body forces (in the form of temperature-changes and

rotation fields) and interfacial misfits is derived and discussed. Results of an extensive parametric study are presented to clearly define

the key design variables and their associated influence. In general the important parameters were identified as misfit, mean radius,

thickness, material property and/or load gradation, and speed; all of which must be simultaneously optimized to achieve the "best"

and most reliable design. Also, the important issue of defining proper performance/merit indices (based on the specific stored energy),

in the presence of multiaxiality and material anisotropy is addressed. These merit indices are then utilized to discuss the difference

between flywheels made from PMC and TMC materials with either an annular or solid geometry. Finally two major aspects of failure

analysis, that is the static and cyclic limit (burst) speeds are addressed. In the case of static limit loads, upper, lower, and out-of-plane

bounds for disks with constant thickness are presented for both the case of internal pressure loading (as one would see in a hydroburst

test) and pure rotation (as in the case of a free spinning disk). The results (interaction diagrams) are displayed graphically in designer

friendly format. For the case of fatigue, a representative fatigue/life master curve is illustrated in which the normalized limit speed

versus number of applied cycles is given for a cladded TMC disk application.

14. SUBJECT TERMS

Deformation analysis; Elasticity; Misfit; Thermal; Mechanical;

Rotating disk; Fatigue limit loads

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

56

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A04

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18 29B-102