Composites: Part B 33 (2002) 433–459
www.elsevier.com/locate/compositesb
Deformation and life analysis of composite flywheel disk systems
S.M. Arnolda,*, A.F. Saleebb, N.R. Al-Zoubib
a
NASA Glenn Research Center, 21000 Brookpark, Road, Mail Stop 49-7, Cleveland, OH 44135, USA
b
University of Akron, Akron, OH 44325, USA
Received 13 August 2001; revised 11 March 2002; accepted 15 May 2002
Abstract
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 pressure surface tractions, body forces (in the form of temperature-changes and rotation fields) and interfacial
misfits is summarized. 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, a
lower (first fracture) bound for disks with constant thickness is presented. 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. q 2002 Elsevier Science Ltd. All rights reserved.
Keywords: A. Anisotropy; A. Failure; B. Fatigue; B. Elasticity; Flywheels; Multiple disks
1. Introduction
1.1. General
Over the years, there has been an increasing demand for
effective energy storing systems [1]. To meet such needs,
many alternative systems have been proposed, as listed in
Table 1, with composite flywheel energy storage systems
rising to the top of the list of candidates, because of their
high power and energy densities with no fall-off in capacity
under repeated charge – discharge cycles. In recent years, a
host [1 – 5] of different flywheel system designs (composed
of 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.
* Corresponding author.
E-mail address: [email protected] (S.M. Arnold).
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 [1,2,6] have been completed.
For instance, representatives of these in the area of materialsystem selections for flywheel devices are given in Genta
[1] and DeTeresa [7] whereas optimization for integrated
system design are given by Bitterly [5].
In Section 1.2, a brief summary is given for the state-ofart that is presently available on the analysis-related 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 group the pertinent list of references in Tables
2 –4, according to three major considerations; i.e. (1) factors
1359-8368/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 8 3 6 8 ( 0 2 ) 0 0 0 3 2 - X
434
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Nomenclature
Material parameters
suL ; suT ultimate tensile strength (UTS) in the longitudinal and transverse directions, respectively
sYL
longitudinal yield stress
j
ratio of longitudinal to transverse ultimate
stresses, i.e. suL =suT
EL
longitudinal Young’s modulus
ET
transverse Young’s modulus
b
ratio of anisotropy, i.e. ET =EL
nL
longitudinal Poisson’s ratio
aL
longitudinal thermal expansion
aT
transverse thermal expansion
r
material density
Stresses and strains
sr,1r
radial stress and strain components, respectively
su,1u tangential stress and strain components, respectively
F( )
limit function (static fracture, endurance limit
and normalization surfaces)
Sij
deviatoric stress tensor
Dij
second order direction tensor
di
unit vector denoting local fiber direction
sij
Cauchy stress tensor
related to additional loading conditions beyond those of
rotation (e.g. 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).
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 [9] followed by Donatch [10] in
1912. One of the first ‘modern’ dissertations in analyzing
the flywheel rotor was the seminal work done by Stodola
[11], whose first translation to English was made in 1917. In
1947 Manson [12] published a finite difference method to
calculate elliptic stresses in gasturbine disks that could
account for the point-to-point variations of disk thickness,
temperature, and material properties. Millenson and Manson [13] extended the method to include plastic flow and
creep in 1948. Both methods have been widely used and
extended by industry [14]. More recently, Genta [1]
modified Manson’s method to include orthotropic materials.
The 1960s and 1970s led 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 1960s is given
Subscripts and superscripts
ð ÞL ; ð ÞT longitudinal and transverse properties (static
fracture, endurance limit and normalization
surfaces), respectively
(˙)
denotes differential with respect to time
Geometry
R
radius
A
inner radius
B
outer radius
T
height
l
thickness ‘in radial direction’
Rm
mean radius
Ri
radial location of the ith interface
Ro
reference mean radius
U
radial deflection
d
misfit between concentric disks
Forces
v
speed of rotation
DT
radial temperature distribution
Ta
temperature at the inner wall
Tb
temperature at the outer wall
To
reference temperature
Pin
internal applied pressure
Pout
external applied pressure
by Seireg [15], whereas, Habercom [16 – 18] provides a
collection of abstracts and reference of governmentsponsored flywheel related research (late 1960s until the
late 1970s). The past 40 years showed a tremendous amount
of work done concerning the flywheel, e.g. see Genta’s [1]
extensive review, including a historical perspective, on
research work conducted up to the early 1980s related
specifically to flywheel systems.
For convenience in presentation, we have grouped in
Tables 2 and 3, respectively, the different representative
works with regard to the two major material behavior
Table 1
Energy storage types [8]
Storage type
Specific energy (W h/kg)
Magnetic fields
Superconducting coil
1– 2
Elastic deformations
Steel spring
Natural rubber band
0.09
8.8
Electrochemical reaction
Lead–acid battery
Nickel–cadmium battery
17.9
30.6
Kinetic energy
Managing steel flywheel
4340 steel flywheel
Composite flywheel
55.5
33.3
213.8a
a
Using longitudinal strength of Kevlar without any shape factor.
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
435
Table 2
Rotating disks composed of isotropic materials
Conditions/type of study
Analytical
Numerical
Experimental
Variable thickness
Uniform thickness
Misfit
Temp.
[8–10,19,20,23,25,26,29,32,35,47,58]
[8–10,19–26,28,29,35,36,39,47,48,54,58]
[11,15,27,33,45,52,53]
[11,15,27,30,33,37,38,40–42,44–46,49,52–54,56,57]
[44]
[42,49]
[31,34,43]
[31,34,43]
[21,51]
classifications treated; i.e. elastic isotropic and elastic
anisotropic. These classifications in fact have had a longstanding 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 4.
With reference to Tables 2 and 3 we note that most of the
listed investigations have focused on the disk of uniform
thickness under rotation. Alternatively, some investigators
such as Kaftanoglu [8], Abir [44], Huntington [73], Nimmer
[93], Daniel [97] and Portnov [115], considered the effect of
misfit in addition to disk rotation as they studied the multirim disk case. Others like Leopold [21], Potemkina [34],
Prasek [42], Shanbhag [49], Chakrabarti [64], Gurushankar
[70], Dick [77], Baer [85], Misra [103] and Portnov [112],
considered the temperature effect as well as the disk rotation
case.
Considering the list in Table 4, we refer to Shevchenko
[123], Weber [124], Pisarenko [125], Reid [126], Gururaja
[131] and Gueven [139], for a representative nonlinear
elastoplastic analysis of rotating disks, and to Lenard [151],
Gupta [154], Mukhopadhyaya [156] and Emerson [164,
165], for a number of time-dependent (viscoelastic)
solutions.
1.3. Objectives and outline
In summary, Tables 2 –4 with their broad classification
of previous work, convey the complexity of the problem, in
that a very large number of conditions and factors must be
considered. In particular, each investigation has necessarily
focused on some specified conditions, particular material
behavior and related conclusions. An urgent need therefore
exists for a compilation, in as concise a form as possible, of
the extensive results, major trends observed and the many
[34]
conclusions reached in these previous studies. To the
authors knowledge a study of this type is presently lacking;
in which the problem in its ‘totality’ has been considered.
As a first step toward this end, an annotative review study is
conducted herein, with the major restriction being that of
disks with constant thickness. In particular, emphasis is on
combined loading (rotation, pressure, misfit, and temperature) and the effect of material anisotropy (with its great
impact on stress distributions and any ensuing singularities
in stress, strength and failure modes, etc.). To facilitate the
presentation of the well-established elastic anisotropic
solution procedure (single and multiple disks) specific
reference to the numerous previous works cited in Tables 2
and 3 are not individually mentioned. However, we
emphasis at the outset that all the conclusions made herein
are in agreement with these previous works, except of
course the notation presently employed. In addition to this
review, our second primary objective is to present a number
of new analytical and numerical results with regard to
limiting conditions of burst (pressure as well as rotation),
fatigue, etc. These will then enable us to reach a number of
important conclusions in parallel under both service
conditions (stress analysis), as well as at the overload
conditions for ‘lifing’ studies of the problem.
In an outline form, the remainder of the paper is
described as follows. In Section 2, we summarize the
analytical model used in the 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. Section 3 contains the results of extensive
parametric studies and important observations obtained
there from. In Section 4, we address the important issue of
defining proper performance/merit indices (based on the
specific stored energy), in the presence of anisotropy.
Section 5 deals with two major aspects of failure analysis,
i.e. static and cyclic, providing the corresponding limit
Table 3
Rotating disks composed of anisotropic materials
Conditions/type of study
Analytical
Numerical
Experimental
Variable thickness
Uniform thickness
[62,64,67,69,74,91]
[6,8,50,55,59 –67,69– 74,76,78,79,
91,95,98,104,105,107,117,119]
[6,8,73,104]
[6,64,70]
[12,13,111,113,120]
[68,75,77,80,85,87,97,101,
103,108–114,118,120]
[97,115]
[77,85,103,112]
[63,77,78,81–84,86,88–90,92– 94,
96,99,100,102,106,114,116,118]
[93]
[77,106]
Misfit
Temp.
436
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
with
Table 4
Nonlinear analysis classification for rotating disks
m1 ¼ 21 þ S;
Conditions/type of study
Isotropic
Anisotropic
Elasto-plastic
Creep/viscoelastic
Fracture
Fatigue
[121–130, 132–145]
[145–153, 155–159]
[166,168–175, 177–179]
[6,131]
[154, 160 –165]
[6, 167, 176]
[6, 180]
(burst) load and representative fatigue/life limit speed
curve, respectively.
2. Elastic stress analysis via analytical model
2.1. Single disk solution
2.1.1. Annular disk
Here, as classically done the problem of a quasi-static,
transversely isotropic, rotating disk (Fig. 1) subjected to
both thermal (linear temperature gradient) and mechanical
(internal and external pressure) boundary conditions1 is
summarized.
The general solution of the resulting Euler – Cauchy
governing equation is:
sr ¼ C1 r m1 þ C2 rm2 þ Q1 ðrÞ 2 Q2 ðrÞ
ð1Þ
b¼
1
QðrÞ ¼
rEL ðTa 2 Tb Þð2aL 2 aT Þ þ a{EL ðTb 2 To Þ
b2a
ðaL 2 aT Þ þ r 2 rv2 ð3 þ nL Þ} 2 b{EL ðTa 2 To Þ
ð5Þ
ðaL 2 aT Þ þ r 2 rv2 ð3 þ nL Þ}
and C1 and C2 are constants to be determined from the
applied mechanical boundary conditions. Implied in this
solution is the assumption that the material parameters (i.e.
EL, ET, nL and GL, aL and aT) 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 b are those less than or equal to one, as the fiber
stiffness is always greater than the matrix (thereby resulting
in EL $ ET).
The constants C1 and C2 are obtained by imposing the
following boundary conditions along the inner and outer
radius of the disk
ðsr Þr¼b ¼ 2Pout
where Pin and Pout are the internal and external pressures,
respectively. The resulting expressions are:
pffiffiffiffi
pffiffiffiffi
a1þ 1=b ðPin þ Q2 ðaÞ 2 Q1 ðaÞÞ 2 b1þ 1=b ðPout þ Q2 ðbÞ 2 Q1 ðbÞÞ
pffiffiffiffi
pffiffiffiffi
C1 ¼
ð6Þ
a2 1=b 2 b2 1=b
2 ðm2 þ 1ÞQ2 ðrÞ þ rv2 r 2
ð2Þ
and
pffiffiffiffi pffiffiffiffi h
pffiffiffiffi
pffiffiffiffi
i
a 1=b b 1=b 2ab 1=b ðPin þ Q2 ðaÞ 2 Q1 ðaÞÞ þ ba 1=b ðPout þ Q2 ðbÞ 2 Q1 ðbÞÞ
pffiffiffiffi
pffiffiffiffi
C2 ¼
a2 1=b 2 b2 1=b
ur ¼ r
su
n ·s
T 2 Ta
ðr 2 aÞ
2 L r þ aL Ta 2 To þ b
EL
EL
b2a
ð3Þ
where
Q1 ðrÞ ¼
ð
r m1
r 2ðm1 þ1Þ QðrÞ dr
m1 2 m2
Q2 ðrÞ ¼
ð
r m2
r 2ðm2 þ1Þ QðrÞ dr
m1 2 m2
1
ET
EL
and
ðsr Þr¼a ¼ 2Pin ;
su ¼ C1 ðm1 þ 1Þrm1 þ C2 ðm2 þ 1Þrm2 þ ðm1 þ 1ÞQ1 ðrÞ
m2 ¼ 21 2 S;
sffiffiffiffi
1
;
S¼
b
The problem considered is that of plane stress and axisymmetry.
ð4Þ
Fig. 1. Schematic of n concentric disks.
ð7Þ
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
where we may separate the thermal and rotational terms
such that
Q1 ðrÞ ¼
QT1 ðrÞ
QR1 ðrÞ
þ
and Q2 ðrÞ ¼
QT2 ðrÞ
þ
QR2 ðrÞ
ð8Þ
with the generalized thermal forces being defined as
0
B
B
1 B
aðT 2 To Þ þ bð2Ta þ To Þ
B
T
Q1 ðrÞ ¼ sffiffiffiffi BEL ðaL 2 aT Þ 0b
sffiffiffiffi1
B
1 B
@ 2 1 þ 1 Aða 2 bÞ
@
2
b
b
1
for the isotropic case, and
sr ¼ C1 r m1 þ Q1 ðrÞ 2 Q2 ðrÞ
ð12Þ
for the anisotropic case. Now assuming that an applied
external pressure Pout exists, the constant C1 is determined
to be
C1 ¼ Pout þ Q2 ðbÞ 2 Q1 ðbÞ
ð13Þ
for the isotropic case, or
C1 ¼
1
r m1
½Pout þ Q2 ðbÞ 2 Q1 ðbÞ
ð14Þ
for the anisotropic case. In the definitions for QT1 ; QR1 and
QT2 ; QR2 given above in Eqs. (9) and (10), a (the inner radius)
is taken to be zero. Similarly, the tangential stress is
C
C
rEL ðTa 2 Tb Þð2aL 2 aT Þ C
C
þ 0
C
sffiffiffiffi1
C
1
Aða 2 bÞ C
@22þ
A
b
su ¼ C1 þ Q1 ðrÞ þ Q2 ðrÞ þ rv2 r2
ð15Þ
for the isotropic case, and
0
ð9Þ
B
B
1 B
aðT 2 To Þ þ bð2Ta þ To Þ
B
T
Q2 ðrÞ ¼ sffiffiffiffi BEL ðaL 2 aT Þ 0b
sffiffiffiffi1
B
1 B
@ 2 1 2 1 Aða 2 bÞ
@
2
b
b
1
C
C
rEL ðTa 2 Tb Þð2aL 2 aT Þ C
C
1
0
þ
C
sffiffiffiffi
C
1
Aða 2 bÞ C
@222
A
b
su ¼ ðm1 þ 1ÞC1 rm1 þ ðm1 þ 1ÞQ1 ðrÞ 2 ðm2 þ 1ÞQ2 ðrÞ
þ rv2 r 2
ð16Þ
for the anisotropic case. Note, for the isotropic case, Q1 and
Q2, can be greatly simplified by taking b ¼ 1; with the
resulting expressions being given in Eq. (20), see Section
2.4.1.
2.2. Multiple number of concentric disks
and those due to rotation as
r 2 rv2 ð3 þ nL Þ
sffiffiffiffi#
QR1 ðrÞ ¼ sffiffiffiffi"
1
1
23 þ
2
b
b
437
ð10Þ
Extension of the formulation to ‘n’ number of annular
concentric disks, with possible misfit (Fig. 1), 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 following2:
sr1 ðr ¼ r1 Þ ¼ P1
sr1 ðr ¼ r2 Þ ¼ sr2 ðr ¼ r2 Þ ¼ P2
and
sr2 ðr ¼ r3 Þ ¼ sr3 ðr ¼ r3 Þ ¼ P3
2
2
r rv ð3 þ nL Þ
sffiffiffiffi#
QR2 ðrÞ ¼ sffiffiffiffi"
1
1
23 2
2
b
b
ð17Þ
..
.
srn21 ðr ¼ rn Þ ¼ srn ðr ¼ rn Þ ¼ Pn
srn ðr ¼ rnþ1 Þ ¼ Pnþ1
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 or anisotropic material, it can be immediately
seen that C2 in Eq. (1) must be equal to zero (thus both
displacement and stress fields at the center will be finite),
such that
where P1 and Pnþ1 are the applied internal and external
pressures, respectively; whereas the remaining continuity
conditions in Eq. (17) 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 introduced at the interfaces corresponding to the
2
sr ¼ C1 þ Q1 ðrÞ 2 Q2 ðrÞ
ð11Þ
In the case of solid hub, the first expression in Eq. (17) is not needed,
however the condition finite radial displacement must also be imposed.
438
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
imposed misfit (d ) at each interface. These are:
ur2 ðr ¼ r2 Þ 2 ur1 ðr ¼ r2 Þ ¼ d1
ur3 ðr ¼ r3 Þ 2 ur2 ðr ¼ r3 Þ ¼ d2
ur4 ðr ¼ r4 Þ 2 ur3 ðr ¼ r4 Þ ¼ d3
ð18Þ
..
.
urn ðr ¼ rn Þ 2 urn21 ðr ¼ rn Þ ¼ dn21
where d1 ; d2 ; …; dn21 are the imposed misfits between disks
that will in turn generate interface pressures. Combining
Eqs. (17) and (18) with Eqs. (1) and (3) specifies a system of
2n equations with 2n number of unknown constants; i.e.:
½K{C} ¼ {R}
ð19Þ
where {C} is the vector of unknown constants; {C} ¼
{C11 ; C21 ; C12 ; C22 ; …; C1n ; C2n }T ; ½K is the geometry and
material matrix, ½R is the vector of applied forces and/or
misfit parameters.
2.3. Verification of the solution
As indicated in Section 1, the analysis of isotropic and
anisotropic disks subjected to rotation, internal and external
pressure, temperature variation and misfit are not new
[1 –7]. Consequently, the present unified solution can be,
and has been validated through comparison with a variety of
other solutions, see Ref. [6].
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. b ¼ 1), the
character of the governing equation will change (as the
zeroth order term will disappear, see Ref. [6]) and
consequently, the particular solution, i.e. Eq. (4), will be
appropriately modified. The resulting expressions for Q1
and Q2 coming from Eq. (4) and including both thermal and
rotational loading are:
!
EðTa 2 Tb Þa
r rv2 ð3 þ vÞ
ð20Þ
þ
Q1 ðrÞ ¼ r
24
2ða 2 bÞ
!
EðTa 2 Tb Þa
r rv2 ð3 þ vÞ
Q2 ðrÞ ¼ r
þ
28
2 6ða 2 bÞ
2.4.2. Thermal isotropy
Here let us consider the case when only thermal loading
is applied to the disk. Under this condition it is apparent
from Eqs. (1) – (10) 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 (aL) and radial (aT) thermal coefficients are
equal) the stress distribution is primarily dependent only on
b (the ratio of moduli) and EL, as the constants C1 and C2
become a function of Q2 2 Q1 ; that is:
Q2 2 Q1 ¼
rEL ðTa 2 Tb Þa
1
ða 2 bÞ 4 2
b
ð21Þ
Consequently, under a uniform temperature distribution no
stresses are developed as one might expect. Also in general,
from Eqs. (5) and (3), it is apparent that the longitudinal
thermal coefficient is the dominant parameter with the
longitudinal modulus dictating the magnitude of thermal
stresses being developed.
2.4.3. Numerical singularities
It is well know that nonphysical numerical singularities
can exist within an analytical solution, and these vary
depending upon the approach taken, based on a detailed
examination [6].
Of four independent loading (rotation, pressure, thermal
and misfit) one can expect numerical singularities to be
present whenever body forces (e.g. rotation or thermal) 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. 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% volume fraction graphite/epoxy
system. The second class, titanium matrix composite
(TMCs) systems, is represented by a 35% volume 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 are
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
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
439
Table 5
Room temperature PMC material property data
Property
Fiber [184]
Resin [185]
vf ¼ 40%
vf ¼ 60%
vf ¼ 80%
EL (Msi)
ET (Msi)
nL
nT
GL (Msi)
aL (in./in./8F)
aT (in./in./8F)
r (kip s2/in.4)
UTS (ksi)
40.0
2.0
0.25
0.25
2.9
20.22 £ 1026
6.28 £ 1026
1.6511 £ 1027
747–790
0.677
0.677
0.33
0.33
0.2544
42.0 £ 1026
42.0 £ 1026
1.2163 £ 1027
17.5
16.4
1.06
0.295
0.42
0.425
0.845 £ 1026
33.7 £ 1026
1.3033 £ 1027
110a
23.1
1.26
0.282
0.373
0.62
0.269 £ 1026
25.4 £ 1026
1.4777 £ 1027
302(L)a, 10(T)b
32.1
1.58
0.267
0.303
1.5
20.032 £ 1026
17.0 £ 1026
1.5657 £ 1027
398a
a
b
Determined via rule of mixture with a 0.3 knockdown for in situ strength.
Suggested from experiment.
Rotor-Safe-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 5 and 6,
where the composite properties were obtained using the
gðb; Rm ; l; nL Þ ¼
to pure rotation, the influence of the three key material and
geometric parameters (i.e. the degree of material anisotropy
(b ), mean radius (Rm) and thickness (l )) on the radial and
tangential stress distribution will be investigated parametrically. Analytically, these parameters can be clearly seen
by normalizing both the radial and tangential stress with
respect to the density, rotational speed, and outer radius, b,
that is:
sr
rv2 b2
ffi!
ffi !
8
9
3þpffiffiffi
pffiffiffi
1=b
>
>
a
a 1=b
a 3
>
>
>
p
ffiffiffiffi
p
ffiffiffiffi
p
ffiffiffi
ffi
12
2
>
>
212 1=b
1=b 1þ 1=b 2 >
<
b
b
b
ð3 þ nL Þ
r
a
b
r =
pffiffiffiffi !
¼
2
2
ð22Þ
!
2pffiffiffiffi
1
> a 2 1=b
1=b
b
b
r
b >
>
a
>
>
29 >
>
>
21
21
:
;
b
b
b
su
ð3 þ nL Þ
¼
2 2
1
rv b
29
b
ffi!
8
9
!
3þpffiffiffi
pffiffiffiffi
1=b
1=b a 3
0 1
1
>
>
a
a
>
>
p
ffiffiffi
ffi
>
ffi sffiffiffiffi
sffiffiffiffi
2
>
>
21þpffiffiffi
pffiffiffiffi
1þ 1=b
2 >
þ
3
n
L
< 1 12 b
1=
b
1=
b
b
b
B b
C r =
r
1
a
b
B
C
ffi !
ffi !
þ@
£
þ
ð23Þ
2pffiffiffi
2pffiffiffi
>
1=b
1=b
b
b
r
b
b
3 þ nL A b >
>
>
a
a
>
>
>
>
21
21
:
;
b
b
f ðb; Rm ; l; nL Þ ¼
micromechanics analysis code based on the generalized
method of cell, MAC/GMC [182]. All properties given are at
room temperature and assumed to be independent of
temperature. The flywheels to be considered in the RSL
program are expected to have a mean radius of approximately 4 in. and a thickness of approximately 3 in.;
therefore, these geometric properties constitute our baseline
case.
3.1. Single disk case
Considering the case of a single annular disk, subjected
where the inner and outer radii are defined as
a ¼ Rm 2
l
;
2
b ¼ Rm þ
l
;
2
l
a
2Rm
¼
l
b
1þ
2Rm
12
so that gðb; Rm ; l; nL Þ and f ðb; Rm ; l; nL Þ will be denoted as
the radial and tangential distribution factors, respectively,
for an anisotropic annular disk.
Figs. 2 and 3 depict these factors as a function of
normalized radial location, r/Rm, for two limiting geometric
representations, i.e. a thick disk ðl=Rm < 2:0Þ and a thin
440
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Table 6
Room temperature TMC material property data
Property
Fiber SCS-6 [193]
Matrix Ti-15-3 [193]
SCS6/Ti-15-3, vf ¼ 15%
SCS6/Ti-15-3, vf ¼ 35%
SCS6/Ti-15-3 ¼ 55%
EL (Msi)
ET (Msi)
nL
nT
GL (Msi)
aL (in./in./8C)
aT (in./in./8C)
r (kip s2/in.4)
UTS (ksi)
58
58
0.19
0.19
24.36
2.2 £ 1026
2.2 £ 1026
3.085 £ 1027
550–650
13.0
13.0
0.33
0.33
4.924
8.1 £ 1026
8.1 £ 1026
4.21 £ 1027
126.2
19.8
15.7
0.306
0.366
5.65
5.54 £ 1026
7.66 £ 1026
4.041 £ 1027
185a
28.8
20.2
0.272
0.37
7.0
3.99 £ 1026
6.38 £ 1026
3.816 £ 1027
257a
37.8
27.0
0.249
0.319
9.62
3.18 £ 1026
5.24 £ 1026
3.591 £ 1027
347a
a
26.1
17.4
201.6(L), 60.9(T)
Determined via rule of mixture: bolded values are experimentally measured [193].
disk ðl=Rm < 0:125Þ; given a slightly anisotropic ðb ¼ 0:7Þ
TMC, and strongly anisotropic ðb ¼ 0:05Þ PMC material
description, respectively. Further, in Table 7 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.
The first being that both the radial and tangential stresses
increase as Rm and l are increased (Table 7). The second is
that the degree of material anisotropy (b ) impacts the
magnitude of increase in the radial stress component 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. l=Rm , 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 Fig. 2, 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
Fig. 2. Normalized radial g and tangential f stresses (aka distribution factors) versus normalized radius given a single rotating disk made from PMC. (a) and (b)
correspond to a 3 in. thick disk, while (c) and (d) to a 7 in. thick disk.
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
441
Fig. 3. Normalized radial g and tangential f stresses (aka distribution factors) versus normalized radius given a single rotating disk made from TMC. (a) and (b)
correspond to a 3 in. thick disk, while (c) and (d) to a 7 in. thick disk.
TMC material system, see Fig. 3, the maximum tangential
stress is always at the inner radius. Similarly, the location of
the maximum radial stress shifts from approximately
0.75Rm to Rm as the mean radius increases for a TMC
(Fig. 3). Whereas for a PMC material (Fig. 2) the maximum
location moves from 1.25Rm, 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 [183] 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 Figs. 2 and
3. This is discussed in Section 4. Finally, it is obvious from
Eqs. (22) and (23) that the density (r ) of the material
impacts the actual magnitude of the stress field (both radial
and tangential components) linearly, while the rotational
speed (v ) impacts both components quadratically.
The radial and tangential stress distributions as a function
Table 7
Maximum normalized radial and tangential stress fields
Rm
sr =rv2 ¼ gðb; Rm ; lÞb2
l¼2
4
8
16
32
PMC
1.08
1.46
1.6
1.64
su =rv2 ¼ f ðb; Rm ; lÞb2
l¼4
TMC
1.6
1.64
1.65
1.65
PMC
2.1
4.3
5.8
6.4
l¼8
TMC
6.3
6.5
6.6
6.5
PMC
3.8
8.5
17.4
23.4
l¼2
TMC
19.5
25.1
26.1
26.3
PMC
17.8
72.8
276
1066
l¼4
TMC
22.0
75.3
278
1067
PMC
22
71
291
1106
l¼8
TMC
29
88
301
1111
PMC
39
88
286
1165
TMC
25
116
353
1205
442
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 5. Influence of applying pressure along the inner surface of a single
disk. l is the thickness of disk.
Fig. 4. Influence of temperature gradient on a single disk.
of normalized radial location for pure thermal loading3 and
internal pressure loading are shown in Figs. 4 and 5;
assuming the baseline material parameters corresponding to
the 60% volume fraction PMC system in Table 5 for a
relatively thick disk (Rm ¼ 4 in. and l ¼ 3 in.). From Fig. 4,
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
3
The reference temperature T0 is assumed to be higher in magnitude than
the applied temperature field.
curvature of the hoop stress distribution and the magnitude
of the maximum tensile hoop stress.
In Fig. 5 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
Fig. 5 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 Ref. [181].
Now the superposition of these three classes of loads
(thermal, internal pressure, and rotation) is illustrated in
Figs. 6 and 7; where both radial and hoop distributions are
shown for a relatively thick disk (l ¼ 3; Rm ¼ 4) rotating at
15 000 rpm (Fig. 6) and 60 000 rpm (Fig. 7) subjected to
both uniform and gradient temperature profiles and an
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 6. Influence of thermal, internal pressure, and rotation on a single disk.
Speed of rotation is 15 000 rpm.
internal pressure of 1 ksi. Note how for the lower speed case
(Fig. 6), 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. Fig. 7 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.
3.2. Three concentric disks
In the RSL program the flywheels of interest are being
443
Fig. 7. Influence of thermal, internal pressure, and rotation on a single disk.
Speed of rotation is 60 000 rpm.
manufactured with a pre-load design in mind, that is the
flywheel is composed of a number of concentric rings pressfit together with a specified misfit between each ring.
Therefore, the resulting laminated flywheel has a compressive radial stress at the interface of each concentric ring.
Consequently, the desire here is to investigate the influence
of a key design parameter (i.e. misfit, d between disks) on
the corresponding radial and tangential stress distribution
within the laminated system. This parameter is in addition to
the two previous geometric parameters: Rm, mean radius
and l, ring thickness in the radial direction. When
examining the influence of d, 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)
444
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
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 material properties
To simplify the study, only three (each with the 60%
volume fraction PMC system given in Table 5) concentric
disks are investigated with baseline geometry similar to the
previous single disk case, that is, a total thickness (i.e.
l ¼ l1 þ l2 þ l3 ) 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, d1,
and that between disk 2 and disk 3, d2, see Fig. 8. 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. di ¼ 1h Ri ) such
that a constant hoop strain (1h) 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 (v ) of
30 000 rpm, analysis results for the disk system are shown
in Figs. 9 and 10, when the overall mean radius and misfit
hoop strain are varied as follows: Rm ¼ 2; 4, and 6 in. and
1h ¼ 0; 0.1 and 0.2%. In Figs. 9 and 10, the radial stress
Fig. 9. Normalized radial stress for disk subjected to misfit and rotation.
Case (a) has no misfit, i.e. 1h ¼ 0; (b) has 1h ¼ 0:1% and (c) has 1h ¼ 0:2%
applied at all interfaces. R0 ¼ 4:0 in:
Fig. 8. Schematic representation of three concentric disks with misfit.
versus radial location and tangential stress versus radial
location are shown, respectively, for each of the above
cases. Examining Fig. 9 it is apparent that in the case of no
misfit (1h ¼ 0; see Fig. 9a) all radial stresses are tensile.
Recalling from Figs. 6 and 7, it is clear that even if internal
pressure was applied to offset these rotationally induced
stresses only the inner most surface would experience
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 10. Normalized tangential stress for disk subjected to misfit and
rotation. Case (a) has no misfit, i.e. 1h ¼ 0; (b) has e h ¼ 0:1% and (c) has
1h ¼ 0:2% applied at all interfaces. R0 ¼ 4:0 in:
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 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.
445
Alternatively, if one imposes a pre-stress through
application of misfit at the interfaces between disks, a
compressive radial stress state can be induced throughout.
These interfacial compressive radial stresses (denoted in
Fig. 9b and c by an i1 and i2) 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, i1, is more
compressive as it benefits from the compressive state at i2 as
well. Furthermore, Fig. 9 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.
Fig. 10 illustrates the tangential stress distribution for
various mean radii and magnitudes of hoop strain mismatch
(misfit) between the disks. Examining Fig. 10, 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.
l=Rm . 0:3: Nevertheless, it is apparent that a complex
relationship between the various geometric parameters, d, l,
and Rm exists.
Selecting a proper disk thickness configuration is another
important aspect that was investigated. Fig. 11 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 with 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. Nonuniform material properties 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. The fact that not
only the magnitude of misfit is important in the determination of the radial stress at the interface but also the
material properties 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 8 were
446
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 11. Influence of varying internal disks thickness on the radial and
tangential stresses. Case (a): li ¼ {2; 0:5; 0:5}; case (b): li ¼ {0:5; 0:5; 2};
case (c): li ¼ {0:5; 2:0:5}: The misfit here is 1h ¼ 0:1%: R0 ¼ 4:0 in:
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
variation (induced by changing the fiber volume fraction)
will be small and in practicality higher in order when
compared to the associated variation in stiffness and
strength. Even so, the induced body forces due to rotation,
and thus influence of graduation 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 (v ) of 30 000 rpm, variation of mean radius (Rm ¼ 2;
4, and 6 in.) and imposition of constant hoop strain (1h) at
each interface of 0, 0.1 and 0.2% (i.e. variable misfit, d, per
interface such that di ¼ 1h Ri ). The results of these analyses
are presented in Figs. 12 –15, 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 maximum value
and location of this stress are also greatly impacted by
gradation. For example, note the difference between case
two and three (compare, Fig. 13b 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. 15b) would initiate burst failure in the
outer ring and case 6 (Fig. 15c) 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 Figs. 16 –18. In Figs. 16 and 17 it is apparent
that material gradation naturally leads to higher compressive (or at the least, smaller tensile) interfacial 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 i2 is now greater
than i1. This result is in contrast to those given in Fig. 9.
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 (Fig. 4) regarding temperature
gradient results, one can envision that imposing an
Table 8
Description of material gradation by case
Case
Disk 1
One
Two
Three
Four
Five
Six
vf
vf
vf
vf
vf
vf
¼ 60%; PMC
¼ 20%; PMC
¼ 80%; PMC
¼ 35%; SCS-6/Ti-51-3
¼ 15%; SCS-6/Ti-51-3
¼ 55%; SCS-6/Ti-51-3
Disk 2
vf
vf
vf
vf
vf
vf
¼ 60%; PMC
¼ 60%; PMC
¼ 60%; PMC
¼ 35%; SCS-6/Ti-51-3
¼ 35%; SCS-6/Ti-51-3
¼ 35%; SCS-6/Ti-51-3
Disk 3
vf
vf
vf
vf
vf
vf
¼ 60%; PMC
¼ 80%; PMC
¼ 20%; PMC
¼ 35%; SCS-6/Ti-51-3
¼ 55%; SCS-6/Ti-51-3
¼ 15%; SCS-6/Ti-51-3
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
447
Fig. 13. 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 (Table 9).
Fig. 12. Radial 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 (Table 9).
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.
Finally, from Fig. 18, it can be clearly seen that
increasing the rotational speed (v ), decreases the compressive stress at an interface (possibly eliminating it altogether
depending upon the magnitude of mean radius, for a given
misfit) as expected. One must keep in mind, that increasing the
rotational speed significantly increases (proportional to v 2)
the hoop (tangential) stress thus leading to potential burst
failure of the system. Once again, from a design point of view
all of these important parameters (misfit, mean radius, speed,
material properties, and gradients of properties and/or loads)
must be simultaneously optimized to achieve the ‘best’ design.
4. Material and configuration selection
In the Section 3 we understand the key design parameters
that influence a single (or system of) rotating disks; such as
448
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 14. 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 (Table 9).
mean radius, thickness, misfit, speed, material properties,
and gradients of properties and/or loads. The interrelationship among these various parameters can be quite
complicated and therefore typically requires a rigorous
analysis be performed for each configuration (material and/
or geometry) investigated. In this study we have focused
primarily on two classes of composite materials, PMC and
TMC, given a range (from relatively thick to thin) sizes of
annular disks. In this section we 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
Fig. 15. 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 (Table 9).
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 result of several factors (e.g. overspeed (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 7) due to centrifugal forces reaches the tensile
strength (under burst conditions) or the fatigue strength
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Fig. 16. Radial stress for material case 2 for different applied misfits
subjected to rotational speed of 30 000 rpm. (a) 1h ¼ 0; (b) 1h ¼ 0:1% and
(c) 1h ¼ 0:2%:
(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 that
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 concern ourselves with the
449
Fig. 17. Radial stress for material case 5 for different applied misfits
subjected to rotational speed of 30 000 rpm. (a) e h ¼ 0; (b) 1h ¼ 0:1% and
(c) 1h ¼ 0:2%:
kinetic energy of the spinning disk, only. Consequently,
given that this energy is defined as, U ¼ J v2 =2; with J
being defined as the polar moment of inertia of the annular
disk JAnnular ¼ prtðb4 2 a4 Þ=2; or solid (with a ¼ 0; i.e.
Jsolid ¼ prtb4 =2) and the mass of the annular disk being
mAnnular ¼ prtðb2 2 a2 Þ; or solid (with a ¼ 0; such that
msolid ¼ prtb2 ), one can obtain easily the following ratio,
which needs to be optimized, that is:
U
m
2
¼v
solid
b2
4
!
ð24Þ
450
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
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 strength is approximately 30:1, whereas in the case
of TMCs this ratio is 3:1. This together with the fact that the
induced stress field is biaxial makes the identification of an
appropriate multiaxial failure criterion very important.
To illustrate this fact we examine three separate failure
conditions to determine the maximum speed of rotation for
both a solid and annular flywheel. The first two conditions,
limit the speed based solely on a single (uniaxial) stress
component (i.e. tangential ‘longitudinal direction’ and
radial ‘transverse direction’ stress, respectively). Alternatively, the third is based on an equivalent transversely
isotropic effective (J2) stress measure (multiaxial) described
more fully in Section 5, see Eq. (35), as well as Refs. [187,
189]. Returning to Eqs. (22) and (23) it can be shown that
the maximum rotation speed is either
su L
1
Hoop stress only
v2max #
ð26Þ
r b2 f ða; b; b; r; nL Þ
or
Radial stress only
v2max #
suT
r
1
b gða; b; b; r; nL Þ
ð27Þ
2
or
Multiaxial condition
su L
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2max #
2
2
2
r b {j g þ f 2 2 g·f }
Fig. 18. Radial stress for material case 2 for different speeds of rotation. (a)
v ¼ 15,000 rpm; (b) v ¼ 30,000 rpm; and (c) v ¼ 60,000 rpm.
or
U
m
¼v
Annular
2
b2 þ a2
4
!
ð25Þ
depending upon the desired geometry.
To maximize the energy per unit mass, the task now
becomes one of determining the maximum rotation speed
(v ) 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
ð28Þ
where j ¼ suL =suT depending upon the desired failure
condition utilized. The above expressions are quite general
in that they can be used for both an annular ða – 0Þ or solid
ða ¼ 0Þ disk as well as a transversely isotropic ðb , 1Þ or
isotropic material ðb ¼ 1Þ: Ashby [186] has derived a
similar expression to that of Eq. (26) for the special case of a
solid, isotropic disk, In Ref. [186] Ashby denoted the term
suL =r as a performance index and employed it as a
discriminator for the selection of the optimum material
from which to construct a flywheel. The optimum material
selected was a graphite (CFRP) or glass (CFRP) reinforced
epoxy system, which can store between 150 – 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 [186] 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
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
451
Table 9
Maximum specific stored energy (kJ/kg) for a single disk made with PMC
Criterion
Annular disk case ðl ¼ 3Þ
a
Solid disk case ða ¼ 0Þ
Rm ¼ 4
Rm ¼ 8
Rm ¼ 16
Rm ¼ 32
b ¼ 5:5
b ¼ 9:5
b ¼ 17:5
b ¼ 33:5
Hoop only
vf ¼ 40%
vf ¼ 60%
vf ¼ 80%
327.86
792.16
983.93
295.29
713.46
886.17
299.96
724.75
900.19
313.04
756.37
939.46
275.62
665.94
827.14
275.67
666.06
827.29
275.74
666.23
827.51
275.79
666.36
827.67
Radial only
vf ¼ 40%
vf ¼ 60%
vf ¼ 80%
113.74
267.56
338.8
250.12
588.40
745.05
87.285
205.34
260.01
87.357
205.51
260.22
87.514
205.88
260.69
87.64
206.17
261.06
Multiaxial
vf ¼ 40%
vf ¼ 60%
vf ¼ 80%
115.36
278.72
346.19
211.14
510.15
633.65
89.848
217.09
269.64
90.048
217.57
270.24
90.109
217.72
270.42
90.01
217.48
270.12
a
825.3
1942.1
2458.1
3127.1
7357.1
9316.1
299.96
724.75
900.19
313.04
756.37
939.46
Rm, b and l have the unit of inches.
monolithic (isotropic) and composite (anisotropic)
materials.
In the past others (e.g. Refs. [1,2]) have expressed the
performance index coming from the specific energy for
isotropic systems as
su L
U
¼ K
ð29Þ
m
r
where K was defined to be geometric shape factor.
Furthermore, it was only noted in passing in the literature
that this shape factor would become significantly more
complex in the presence of anisotropic material behavior.
Now by combining Eqs. (25) and (28) one can define this
complex ‘shape’ factor, which accounts for both material
anisotropy and stress state multiaxiality to be:
!
1
b 2 þ a2
K ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð30Þ
4b2
{j2 g2 þ f 2 2 g·f }
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
b; j; a=b; nL Þ such that now both K and
attributes, K ¼ Kð
suL =r become integral to the performance index.
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 5 and 6, are given in Tables
9 and 10, respectively. Examining the results a number
of conclusions become obvious. First, as expected form
Eqs. (24), (26) – (28), 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
Table 10
Specific stored energy (kJ/kg) for a single disk made with TMC
Criterion
Annular disk case ðl ¼ 3Þ
a
Rm ¼ 4
Rm ¼ 8
Hoop only
vf ¼ 15%
vf ¼ 35%
vf ¼ 55%
105.53
155.24
222.74
119.76
176.18
252.79
Radial only
vf ¼ 15%
vf ¼ 35%
vf ¼ 55%
225.36
332.41
452.87
802.10
1183.10
1612.10
Multiaxial
vf ¼ 15%
vf ¼ 35%
vf ¼ 55%
105.53
155.24
222.74
119.76
176.18
252.79
a
Rm, b and l have the unit of inches.
Solid disk case ða ¼ 0Þ
Rm ¼ 16
131.69
193.74
277.97
3119.1
4610.0
6267.1
131.69
193.74
277.97
Rm ¼ 32
139.10
204.63
293.60
12 361.0
18 231.0
24 831.0
139.1
204.63
293.6
b ¼ 5:5
b ¼ 9:5
b ¼ 17:5
b ¼ 33:5
190.68
280.51
402.48
190.75
280.61
402.62
190.68
280.50
402.46
190.69
280.53
402.50
72.94
107.58
146.56
72.97
107.64
146.64
72.94
107.58
146.57
72.97
107.63
146.63
71.79
105.61
151.53
71.8
105.62
151.54
71.79
105.62
151.54
71.84
105.69
151.64
452
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Table 11
Ratio of annular disk to solid disk maximum specified stored energy
U U =
m A m S
Criterion
Hoop only
Radial only
Multiaxial
a
PMC ðl ¼ 3Þ
TMC ðl ¼ 3Þ
Rm ¼ 4a
Rm ¼ 8
Rm ¼ 16
Rm ¼ 32
Rm ¼ 4
Rm ¼ 8
Rm ¼ 16
Rm ¼ 32
1.19
1.303
1.284
1.071
2.863
2.345
1.088
9.43
3.329
1.135
35.686
3.478
0.553
3.09
1.47
0.628
10.991
1.668
0.691
42.758
1.834
0.729
169.359
1.936
Rm and l have the unit of inches.
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. 2 and 3), 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 9 and 10 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 11. This is in sharp contrast to the fact that for
isotropic materials, the specific energy of solid disk is
somewhere between 20% (for a thin annular disk) to
100% (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) solid disk is more efficient than an
annular disk. Once again demonstrating the importance
of using a multiaxial criterion in the presence of
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 12). 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 12 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 5 and 6). 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 –0.5 as well as TMC flywheels, see Table
12. 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. 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 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
Table 12
Ratio PMC to that of TMC flywheels specific energy
U U
=
m PMC m TMC
Annular disk cases ðl ¼ 3Þ
Rm ¼ 4a
Rm ¼ 8
Rm ¼ 16
Rm ¼ 32
Hoop only
40/15%
60/35%
80/55%
3.107
5.103
4.417
2.466
4.05
3.506
2.278
3.741
3.238
2.25
3.696
3.2
Radial only
40/15%
60/35%
80/55%
0.505
0.805
0.748
0.312
0.497
0.462
0.265
1.422
0.392
0.253
0.404
0.375
Multiaxial
40/15%
60/35%
80/55%
1.093
1.795
1.554
1.763
2.896
2.507
2.278
3.741
3.238
2.25
3.696
3.2
a
Rm and l have the unit of inches.
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
453
can be defined, e.g. disk separation, delamination within a
effective stress) are
disk, or a burst criterion. Here our concern will be with
suL
1
calculating the burst pressure or burst speed of a single
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð31Þ
ðPin Þlimit ¼
2
2
F:S:
{j g þ f 2 2 g ·f}
annular disk. Although, within the current RSL program
disk separation is the primary design consideration, burst
and
conditions must still be established. Currently, hydroburst
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
tests are being considered as potential certification tests for
su L
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
¼
ð32Þ
limit
establishing the quality of the flywheel configurations being
ðF:S:Þr b2 {j2 g2 þ f 2 2 g·f }
manufactured. However, one should always remember that
for a general flywheel configuration the character of the
respectively. Where F.S. is the factor of safety imposed, g
problem could change significantly when going between
and f were define previously, see Eqs. (22) and (23) for the
case of pure rotation, and now
internal pressure (hydroburst test) and rotation (spin test).
0
8
9
ffi1
2pffiffiffi
1=b
>
>
b
>
>
>
@
A
>
>
pffiffiffiffi
pffiffiffiffi >
>
>
a
<
=
21þ
1=
b
212
1=
b
1
r
r
g ðb; Rm ; lÞ ¼ 0 pffiffiffiffi 1
2 0 pffiffiffiffi 1
ð33Þ
>
>
a
a
2 1=b
2 1=b
>
>
b
b
>
>
@
@
>
>
21A
21A
>
>
:
;
a
a
and
0
8
9
ffi1
2pffiffiffi
1=b
>
>
b
>
>
>
A
>
>
pffiffiffiffi sffiffiffiffi @
pffiffiffiffi >
sffiffiffiffi
>
>
a
< 1
=
21þ
1=
b
212
1=
b
1
r
1
r
fðb; Rm ; lÞ ¼
0
1
0
1
þ
p
ffiffiffi
ffi
p
ffiffiffi
ffi
>
>
b
a
b b 2 1=b
b b 2 1=b
>
>
>
>
@
A
@
A
>
>
21
21
>
>
:
;
a
a
That is, under internal pressure the ordering of the stresses is
such that sr is always minimum (compressive), sz is
intermediate and su is the maximum (tensile) value;
whereas, under rotation all stresses are tensile (given a
single free spinning disk), with sr being either an
intermediate values, 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 test 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 of 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 Section 4, 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
ð34Þ
are taken directly from Ref. [181] 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 [6,188]. It is important to realize that the above first
fracture analysis (using the effective J2 criterion4), provides
a minimum lower bound as compared to that coming from a
ductile plastic limit analysis, see Ref. [6] and Table 13. 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, z ) 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 during failure, which
is determined primarily 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
4
Note that in the case of internal pressure a significant difference
between the first fracture limit using maximum hoop stress only versus the
effective J2 (multiaxial stress measure) criterion is observed (again since
radial stress is always compressive the effective J2 is always larger than the
hoop stress, maximum stress, alone).
454
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
Table 13
Comparison of brittle and ductile limits
b=a ¼ 1:35; z ¼ 0:93
Limit pressure (ksi)
Limit speed (rpm)
Ductile
Upper bound
Lower bound
61.038
54.237
55 320
54 760
Brittle
First fracture ðJ2 Þ
First fracture ðsmax Þ
39.1
58.3
51 180
51 180
brittle and ductile failures become important, therefore this
will be an area of future study.
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 utilize a previously developed transversely
isotropic fatigue damage model [189,190]. 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 (I1)—matrix cracking, (ii) the
longitudinal shear stress ðI2 Þ—dictates interfacial degradation, and (iii) the maximum normal stress ðI3 Þ 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:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(
)ffi
u
2
u 1
ð4
j
2
1Þ
9
ðÞ
ð35Þ
Fð Þ ¼ t
ð4j2ð Þ 2 1ÞI1 þ
I2 þ I3
ð ÞL
4
h2ð Þ
where
I1 ¼ J2 2 I^ þ
1
I ;
4 3
I2 ¼ I^ 2 I3 ;
I3 ¼ I 2
in which
J2 ¼
1
S S ;
2 ij ij
Dij ¼ di dj ;
I ¼ Dij Sij ;
Sij ¼ sij 2
I^ ¼ Dij Sjk Ski ;
1
s d
3 kk ij
and di (i ¼ 1,2,3) are the components of a unit vector
denoting the local fiber direction. Note this effective J2
invariant (Eq. (35)) has been the multiaxial criterion used
throughout this study, but specialized to the pertinent case of
circumferential reinforcement with h ¼ 1 (equal longitudinal and transverse shear response). This special case is
simply written in terms of the radial and hoop stress as
follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
$
1 # 2
Fð Þ ¼
j ð Þs2r þ s2u 2 sr su
ð36Þ
ð ÞL
where the open bracket subscript notation in Eq. (35) and
the above is used to distinguish between the static fracture,
endurance limit and normalization surfaces, see Ref. [190].
This fatigue model has already been successfully applied
to the problem of a circumferentially reinforced disk
subjected to internal pressure, see Refs. [191,192]. Consequently, here we 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. [189] 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 confine our investigation to a previously
examined SiC/Ti system; the associated damage model
parameters being completely defined in Ref. [192].
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
Figs. 19 –21. In Fig. 19 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 Eq. (35), that is
1 2 FU ¼ 0; see Refs.[189,190]. This analysis is defined as
an uncoupled approach, see Ref. [191]. Fig. 19a clearly
shows that the limit speed greatly decreases with increasing
mean radius; while Fig. 19b illustrates that as the relative
thickness (l/Rm) 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
Fig. 20, 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 were 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 Fig. 21. A similar master curve
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
455
Fig. 21. Normalized speed versus cycle master curve denoting fatigue
initiation at the inner diameter. vL is the static limit speed and v is the
maximum applied speed within a cycle.
ing 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. Conclusions
Fig. 19. 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.
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. [191,192]) a single master design life curve represent-
Fig. 20. Generalized speed versus cycles curve indicating the onset of
fatigue initiation cracks at the inner radius of the composite core.
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 pressure surface tractions, body forces
(in the form of temperature-changes and rotation fields) and
interfacial misfits has been 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, a lower bound for disks with
456
S.M. Arnold et al. / Composites: Part B 33 (2002) 433–459
constant thickness was 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).
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.
Acknowledgments
The second and third authors are grateful to NASA Glenn
Research Center for supplying the required funding, see
NCC3-788, to conduct this research.
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