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Shrink fit with solid inclusion and functionally graded hub
Composite Structures 121 (2015) 217–224
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Shrink fit with solid inclusion and functionally graded hub Eray Arslan a,⇑, Werner Mack b a b
Department of Mechanical Engineering, Inonu University, 44280 Malatya, Turkey Institute of Mechanics and Mechatronics, Vienna University of Technology, Getreidemarkt 9, 1060 Vienna, Austria
a r t i c l e
i n f o
Article history: Available online 7 November 2014 Keywords: Interference fit Solid inclusion Functionally graded material Rotation Thermo-elastic analysis
a b s t r a c t Since in a shrink fit the transferable moment essentially depends on the interface pressure between inclusion and hub, the interface pressure should be as large as possible. This may be facilitated by a partially plastic design, which however also has some drawbacks like a possible permanent redistribution of the stresses after operating at high angular speeds and temperatures. In contrast to that, in the present study the use of a functionally graded hub in an elastically designed interference fit with solid inclusion is proposed. It is shown that for an appropriate grading not only the weight of the hub can be reduced noticeably as compared to a homogeneous hub, but also that particularly a much better performance at rotation can be achieved, which predominates over marginal disadvantages at high temperatures. The generally valid analytical results provide a comprehensive means for the practicing engineer for the design of this kind of shrink fits. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction As shrink fits are a simple and cost-effective means of transfer of moment, they frequently are found in mechanical engineering: examples are shrunk-on rings, armature bandages in rotating machines, or tires of railway wheels [1], to mention but a few. Hence, due to their importance, many studies of their behavior were performed by (semi-) analytical as well as numerical methods. Since under certain circumstances a partially plastic design for better utilization of the material is admissible, often not only elastic but also elastic–plastic states were taken into consideration. For both cases, an application-oriented basic survey of the design of shrink fits can be found in the monograph by Kollmann [2]. Later on, special attention was paid to the widely-used thermal assembly process, see, e.g., the studies by Cordts [3], Mack [4,5], Bengeri and Mack [6], Mack and Bengeri [7], Sen and Aksakal [8], Dolezˇel et al. [1], and Lorenzo et al. [9]. Moreover, several investigations on (transient) heating during operation were performed (e.g. [10–14]), and also effects of various material laws and/or geometrical properties were studied in a number of papers (e.g. [15–20]); in some of these investigations rotation of the shrink fit was taken into account, too. Furthermore, special design procedures were proposed [21]. Since for given geometry of the shrink fit and friction coefficient at the interface between inclusion and hub the transferable ⇑ Corresponding author. Tel.: +90 422 377 4819. E-mail address: [email protected] (E. Arslan). http://dx.doi.org/10.1016/j.compstruct.2014.10.034 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
moment depends solely on the interface pressure, the latter should be as large as possible. However, maximizing the interface pressure under all operating conditions is limited by several constraints. As was mentioned above, a possibility to achieve a high interface pressure is to admit partially plastic behavior; if however purely elastic behavior is required a priori – e.g., if the shrink fit shall be easily dismountable without permanent deformation –, an elastic–plastic design cannot be chosen, of course. Nevertheless, the most important issue is a reduction of the interface pressure with increasing angular speed and/or heating during operation. This reduction may be a transient one (which is particularly pronounced for a high rate of the outer surface temperature, see [13]), or in case of an elastic–plastic design also a permanent one, accompanied by a permanent redistribution of the stresses in the entire device. And a further point, which becomes increasingly important in engineering design, is to minimize the weight of the device while maintaining a good or even excellent performance of the shrink fit. Hence, an interesting alternative (or at least supplement) to admitting partial plasticization might be the use of a functionally graded material (FGM), particularly for the hub. As is well known, in a machine part of FGM the material properties like modulus of elasticity, density, coefficient of thermal expansion, and yield stress vary continuously and can - to a certain extent - be tailored in an appropriate way (for survey articles on this topic see, e.g., [22–25]). Thus, the aim of the present study is to analytically investigate the essential features of a purely elastic shrink fit with
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solid inclusion and functionally graded hub, taking both rotation and an elevated temperature into account. The material properties are presupposed to vary according to a power law in the radial direction; in particular, the case of radially decreasing density of the hub is considered. The latter property may be achieved, e.g., by using a steel/aluminum FGM, which can be produced by a powder metallurgical process, i.e. by appropriate mixing of the respective pure powders, cold pressing, and subsequent sintering (for details see [26]). Then, for a sufficiently large ratio of outer surface radius to interface radius a considerably better performance at rotation may be achieved, accompanied by a substantial saving of weight as compared to a homogeneous hub. These two significant advantages must be weighted against the fact that a marginally worse evolution of the interface pressure with increasing temperature may occur. It is the intention of this paper to thoroughly discuss these effects and to support the practicing engineer in deciding whether a shrink fit with FGM-hub might be advantageous for the specific application under consideration. The paper is organized as follows: in Section 2, the statement of the problem is given, and the governing equations are derived. In Section 3, the stress distribution is discussed, and particularly the interface pressure under various operating conditions is studied. Finally, some concluding remarks are made in Section 4. 2. Statement of the problem and governing equations 2.1. Statement of the problem The subject of the investigation is a cylindrically symmetric shrink fit with homogeneous solid inclusion (0 6 r 6 a) and an FGM-hub (a 6 r 6 b) with free cylindrical outer surface,
r¼b:
rr;h ¼ 0;
ð1Þ
it is presupposed that the axial length c of the device under consideration is much smaller than the diameter of the inclusion (see Fig. 1), and therefore a treatment as a plane stress problem is feasible [2,21], so that
rz;i ¼ 0; rz;h ¼ 0;
ð2Þ
where the indices i and h are here and in the following assigned to the inclusion and the hub, respectively. As a matter of course, the radial displacement u has to comply with the condition
r¼0:
ui ¼ 0;
ð3Þ
and the relations
2.2. Governing equations First, the basic equations shall be summarized. Taking a variable density q ¼ qðrÞ into account, the equation of motion in radial direction reads
dðr rr Þ rh ¼ qðrÞx2 r 2 : dr
ð6Þ
For small deformations, the geometric relations are
r ¼
du ; dr
h ¼
u : r
ð7Þ
Furthermore, considering a variable modulus of elasticity E ¼ EðrÞ and variable coefficient of thermal expansion a ¼ aðrÞ, but constant Poisson’s ratio m (as discussed below), the generalized Hooke‘s law reads
1 ðrr mrh Þ þ aðrÞT; EðrÞ 1 ¼ ðrh mrr Þ þ aðrÞT; EðrÞ
r ¼
ð8Þ
h
ð9Þ
where T means the difference of absolute and reference temperature. Next, for the FGM-hub the dependence of q; E, and a on the radial coordinate must be specified: as was already mentioned in the Introduction, a power law is presumed (compare e.g. related studies for disks by Horgan and Chan [27] or Tutuncu and Temel [28]). The basic grading law is however not postulated for the volume fractions of the constituents (e.g. [29]), but for the modulus of elasticity, and the dependence of the other physical quantities on r then is derived by the rule of mixture. The reason for this is that in this case a closed-form solution of the differential equations can be found, and therefore a purely analytical discussion of the problem is possible. The general linear rule of mixture reads [29]
Preff ðrÞ ¼ Pr 1 V 1 ðrÞ þ Pr2 V 2 ðrÞ;
there, Pr eff means an effective material property, and V j is the volume fraction of material j with property Pr j ; j ¼ 1; 2. In the following, it is further postulated that the material at the inner surface of the hub (denoted by the index s) is pure constituent s of the FGM (i.e., V s ðaÞ ¼ 1) and the same material as used for the homogeneous inclusion. If the index l denotes the second constituent of the FGMhub, it is obviously
V l ðrÞ ¼ 1 V s ðrÞ:
r¼a:
rr;i ¼ rr;h ;
ð4Þ
r¼a:
uh ui ¼ d
ð5Þ
with the interference d hold. A purely elastic design of the shrink fit under all operating conditions is required, and slowly varying angular speed is presupposed.
ð10Þ
ð11Þ
Now, presuming that in the hub the modulus of elasticity varies according to
EðrÞ ¼ Es
r m a
ð12Þ
;
there follows from (10)–(12) that the volume fraction of constituent s should vary with r according to
V s ðrÞ ¼
Es
r m
El a : Es El
ð13Þ
Then, applying the rule of mixture (10) to the density qðrÞ, the coefficient of thermal expansion aðrÞ, and also to the uniaxial yield limit ry ðrÞ, respectively, one finds
Preff ðrÞ ¼ Apr
r m a
þ Bpr
ð14Þ
with
Apr ¼ Fig. 1. Sketch of the shrink fit (prior to the assembly); r inclusion, s hub.
Es ðPrs Pr l Þ ; Es El
Bpr ¼
Es Prl El Prs ; Es El
ð15Þ
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and Pr ¼ q; a, and ry , respectively. While these properties depend on the radial coordinate, Poisson’s ratio m is however assumed to be constant [27,30]. Moreover, the influence of an elevated temperature on the material properties is disregarded, since this – at least for common operating conditions – also would have a minor effect on the results only (Woo and Meguid [31]; compare also Blanke [32]). Now, one is in a position to derive the governing equations for the functionally graded hub; for m ¼ 0 they hold also for the homogeneous inclusion. Using (7)–(9) and (12), and (14), (15) for the coefficient of thermal expansion, the stresses can be expressed in terms of u and its derivative with respect to r (denoted by a prime),
Es r 1þm rm mu þ ru0 ð1 þ mÞ Aa m þ Ba Tr ; m 2 ð1 m Þa a Es r 1þm rm ¼ u þ r mu0 ð1 þ mÞ Aa m þ Ba Tr : m 2 ð1 m Þa a
rr ¼
ð16Þ
rh
ð17Þ
For (slow) heating of all free surfaces of the shrink fit it is reasonable – for a problem with small ratio of axial length to diameter as sketched in Fig. 1 – to approximate the temperature distribution as a homogeneous one throughout the device, and in this case the integrals in the above equations can be written in closed form. Thus, one ends up with the constitutive equations for the hub,
uh ¼ C 1h rðmMÞ=2 þ C 2h r ðmþMÞ=2 1 m2 x2 r 3m Aq r m Bq am þ 8 þ mð3 þ mÞ 8 mð3 mÞ Es 2Aa r m Ba Tr; þ ð1 þ mÞ þ ð3 þ 2m þ mÞam 1 þ m
Es C 1h ðm þ M 2mÞr ð2þmMÞ=2 2ð1 m2 Þam þ C 2h ðm M 2mÞr ð2þmþMÞ=2
Aq ð3 þ mÞr m Bq ð3 m þ mÞ þ x2 r 2 ½8 þ mð3 þ mÞam 8 mð3 mÞ
rr;h ¼
Substituting (16) and (17) into the equation of motion (6) yields a differential equation for the displacement,
r
1þm 00
m 0
1þm
u þ ð1 þ mÞr u ð1 mmÞr u
2 2 1 m m m Aq r þ Bq a x r ¼ ð1 þ mÞ Es rm rm r m m 2Aa m þ Ba T þ Aa m þ Ba T 0 r a a
rh;h ¼ ð18Þ
with the solution
Aq r m Bq am u ¼ C 1 rðmMÞ=2 þ C 2 rðmþMÞ=2 þ 8 þ mð3 þ mÞ 8 mð3 mÞ 1 m2 x2 r 3m ð1 þ mÞrðmMÞ=2 þ Es 2Mam Z ð3mþMÞ=2 ð2 m þ M Þ þ Ba am rðmþMÞ=2 Tdr Aa r Z ð3mMÞ=2 ð19Þ Aa r þ Ba am r ðmMÞ=2 Tdr ð2 m M Þr M
M¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ mðm 4mÞ:
ð20Þ
Therefrom, one obtains for the stresses the expressions
Es C 1 ðm þ M 2mÞr ð2þmMÞ=2 2ð1 m2 Þam þ C 2 ðm M 2mÞr ð2þmþMÞ=2
Aq ð3 þ mÞr m Bq ð3 m þ mÞ E rð2þmMÞ=2 þ x2 r 2 s ½8 þ mð3 þ mÞam 8 mð3 mÞ 2Ma2m Z ð2 þ m þ M Þ ðAa r m þ Ba am ÞTrðmþMÞ=2 dr ð2 þ m M ÞrM Z ð21Þ ðAa rm þ Ba am ÞTrðmMÞ=2 dr ;
rr ¼
rh ¼
Es C 1 ½2 mðm þ M Þr ð2þmMÞ=2 2ð1 m2 Þam þ C 2 ½2 mðm M Þrð2þmþMÞ=2
Aq ð1 þ 3mÞr m Bq ½1 þ mð3 mÞ E r ð2þmMÞ=2 þ x2 r2 s ½8 þ mð3 þ mÞam 8 mð3 mÞ 2Ma2m
Z ½2 mð1 þ 2mÞ M rM ðAa rm þ Ba am ÞTrðmMÞ=2 dr Z ½2 mð1 þ 2mÞ þ M ðAa r m þ Ba am ÞTrðmþMÞ=2 dr
Es r m ðAa r m þ Ba am ÞT: a2m
ð22Þ
Aa Es Tr2m ; ð3 þ 2m þ mÞa2m
ð24Þ
Es C 1h ½2 mðm þ M Þr ð2þmMÞ=2 2ð1 m2 Þam þ C 2h ½2 mðm MÞrð2þmþMÞ=2
Aq ð1 þ 3mÞr m Bq ½1 þ mð3 mÞ þ x2 r 2 ½8 þ mð3 þ mÞam 8 mð3 mÞ
Aa Es ð1 þ 2mÞTr2m ; ð3 þ 2m þ mÞa2m
ð25Þ
and by setting m ¼ 0 with those for the inclusion,
ui ¼
C 1i r ð1 þ mÞ ð1 mÞqs x2 r 2 þ C 2i r as T ; 2 r 4Es
rr;i ¼
where the C j are constants of integration, and
ð23Þ
rh;i ¼
C 1i Es C 2i Es 1 ð3 þ mÞqs x2 r2 þ 4Es as T ; þ r 2 ð1 þ mÞ 1 m 8
C 1i Es C 2i Es 1 ð1 þ 3mÞqs x2 r 2 þ 4Es as T : þ r2 ð1 þ mÞ 1 m 8
ð26Þ
ð27Þ
ð28Þ
As mentioned in the Introduction already, an essential feature may be the weight of the hub (per length in axial direction),
W h ¼ 2p
Z
b
qðrÞrdr:
ð29Þ
a
As yield criterion the one by von Mises is applied [33], predicting the onset of plastic deformation if at some radius
rM ¼ ry ðrÞ with rM ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2r rr rh þ r2h :
ð30Þ
3. Stress distribution and interface pressure Based on the above equations, the stress distribution in the shrink fit now can be calculated. To this end, the four constants of integration must be determined: from (26) and condition (3) one obtains immediately C 1i ¼ 0, and C 2i ; C 1h , and C 2h follow from conditions (1), (4), and (5). In principle, these constants of integration could also be given explicitly, but the respective expressions would be quite lengthy so that a numerical solution of this system of equations is preferable. For the numerical calculations it is convenient to introduce the following non-dimensional quantities:
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(a)
(b) 1
volume fractions
0.8
0.6
0.4
0.2 steel aluminum 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
non-dimensional modulus of elasticity and density
1
0.8
0.6
0.4
0.2
0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 non-dimensional radial coordinate
non-dimensional radial coordinate
(d) 1
1.4
0.8
non-dimensional uniaxial yield limit
non-dimensional thermal expansion coeff.
(c) 1.5
1.3
1.2
1.1
1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.6
0.4
0.2
0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
non-dimensional radial coordinate
non-dimensional radial coordinate
Fig. 2. (a) Volume fractions, (b) modulus of elasticity and density, (c) thermal expansion coefficient, (d) uniaxial yield limit vs. radius.
1.0 1.10 pi = 0.4 b = 5.0
0.9
stand-still
1.05
T=0 d = 1.011
b = 2.5
0.7
b = 5.0 0.6
0.5
m
non-dimensional interference
non-dimensional weight of hub
0.8 1.00
mext = -0.652 0.95
0.90
= -1.146
W = 0.525
m
= -0.652
W = 0.474
0.85 d = 0.833
0.4
0.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.80 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
grading index m
grading index m
Fig. 3. Weight of the hub vs. grading index.
Fig. 4. Necessary interference for an interface pressure of pi ¼ 0:4 at stand-still and T ¼ 0.
E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
(a)
(b)
(c)
(d)
221
Fig. 5. Stresses and radial displacement in a shrink fit with FGM-hub: (a) at stand-still, (b) at rotation and elevated temperature, and with homogeneous hub: (c) at stand-still, (d) at rotation and elevated temperature.
r ¼ r=a;
u ¼ uEs = ary;s ;
T ¼ as Es T=ry;s ;
a ¼ a=as ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ¼ xa qs =ry;s ;
E ¼ E=Es ;
ri ¼ ri =ry;s ;
q ¼ q=qs ;
mext ¼ ð31Þ
hence, the volume fraction of constituent s can (with El ¼ El =Es ) be written as
V s ðr Þ ¼
r m El 1 El
ð32Þ
:
The weight of the hub is non-dimensionalized by the weight of a homogeneous hub of material s, that is,
Wh ¼
Wh
pqs b2 a2
:
ð33Þ
An important note is the following one: since the minimum possible volume fraction of constituent s at the outer cylindrical surface of the hub is equal to zero, it follows from (32) that the extremum grading index mext is
ln El ; ln b
ð34Þ
in contrast to that, for a homogeneous hub m is equal to zero, and therefore, depending on whether El P 1 or El 6 1, the relations 0 6 m 6 mext or mext 6 m 6 0 must hold, respectively. For a numerical evaluation of the equations, specific values are assigned to the radii ratio and the material constants. In this paper, the focus is on an FGM-hub with the high-strength – and therefore usually heavier – constituent at the inner edge of the hub. The reason for this is that on the one hand in general the homogeneous inclusion will be of a high-strength material, and that on the other hand for an FGM-hub with radially increasing density a detrimental effect on the interface pressure at rotation is to be expected. Hence, the ratios of the parameter values chosen for the example discussed below,
El ¼ El =Es ¼ 0:35;
al =as ¼ 1:40; ql =qs ¼ 0:35; ry;l =ry;s ¼ 0:15; ð35Þ
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E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
(b)
(a)
Fig. 6. Interface pressure: (a) for T ¼ 0 vs. angular speed X, (b) for X ¼ 0 vs. temperature T (in the considered parameter range plasticization does not occur).
b = 5.0 0.9
m=0
0.3
0.2 m =-0.652 0.1
0 0 0.2 0.4 0.6 0.8 non-dimensional temperature
1
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
non-dimensional angular speed
Fig. 7. Interface pressure vs. angular speed and temperature for a homogeneous hub (m ¼ 0, the surface is cylindrically curved) and an FGM-hub with m ¼ mext ¼ 0:652.
correspond to a hub with grading from steel (s) at the inner cylindrical surface to a steel/aluminum-composite or to mere aluminum (l) at the outer cylindrical surface; moreover, ml ¼ ms ¼ 0:3 is presupposed. Since the effects to be discussed are especially pronounced and clearly visible particularly for larger radii ratios, b ¼ b=a ¼ 5:0 is chosen. First, Fig. 2(a) shows the volume fractions of the constituents for various grading indices, with m < 0 for the FGM-hub under consideration; the extremum grading index leading to mere aluminum at r ¼ b is mext ¼ 0:652. Then, Figs. 2(b)–(d) present the dependencies of E, q; a, and ry on the radius, respectively. In Fig. 3 the weight reduction of the hub that can be achieved by an FGM can be seen (for comparison also the corresponding values for b ¼ 2:5 are depicted), and one observes that in this case about halving the weight as compared to a homogeneous hub is possible. Of course, as a general trend the achievable gain will be larger for higher b=a-ratios and smaller for lower (absolute) values of the grading index. At the design of a shrink fit, in most cases the moment to be transferred and hence the necessary interface pressure is the essential input parameter. Therefore, in the following a nondimensional interface pressure of pi ¼ rr ð1Þ ¼ 0:4 is chosen, a rather high load but still not leading to plasticization at stand-still and T ¼ 0 for the entire admissible range of m (and the thermal mounting process can also be performed without plastic deformation, see Appendix A). For this interface pressure, Fig. 4 shows the
non-dimensional interface pressure / non-dimensional weight of hub
non-dimensional interface pressure
1 0.4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
grading index m
Fig. 8. Ratio of non-dimensional interface pressure to non-dimensional weight of the hub vs. grading index.
necessary interference d vs. the grading index m. One can see that it quasi-linearly increases with increasing (absolute) values of m, which may be explained by the increasing fraction of the softer FGM constituent. As the maximum weight reduction occurs for the extremum grading index, indeed, in Fig. 5 a comparison of the stresses and the displacement in a shrink fit with FGM-hub with mext ¼ 0:652 and with homogeneous hub (m ¼ 0) is shown, both at stand-still and at rotation at elevated temperature. First, it can be observed that in all cases the ratio rM =ry < 1, hence no plasticization occurs (one should be aware of the fact that rM =ry is not just a non-dimensional stress as rr or rh since ry depends on r, see Fig. 2(d)). Second, the circumferential stress is somewhat larger for the FGM-hub, and this also holds true for the radial displacement; generally, a striking feature is the increase of the
E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
displacement with radius in the hub, which is caused by rotation and thermal expansion. Third, and this is the most important observation, rotation causes due to the centrifugal forces in both cases a reduction of the interface pressure, of course, but this reduction is noticeably less pronounced for the FGM-hub (compare Figs. 5(b) and (d)). Hence, the transferable moment during operation is higher, which obviously is an advantage of the FGM-hub. While Fig. 5 shows a general case of operating conditions with both rotation and thermal load, it is nevertheless worthwhile to study the effects of rotation and elevated temperature on the interface pressure separately, too. Fig. 6(a) presents the evolution of the interface pressure with increasing angular speed and T ¼ 0, and Fig. 6(b) depicts that for increasing temperature and X ¼ 0 for different values of the grading index. One observes from Fig. 6(a) that for X ¼ 0:2 a homogeneous hub loses almost the contact with the inclusion and hence its ability to transfer a moment, whereas the interface pressure at the same angular speed for an FGM-hub with extremum grading still is almost one half of its initial value. On the other hand, Fig. 6(b) shows that for a homogeneous hub the interface pressure is insensitive to homogeneous heating, whereas for the steel-aluminum-FGM under consideration it slightly decreases with temperature for higher (absolute) values of m; this disadvantageous effect may be attributed to the more pronounced thermal expansion of aluminum, but is comparatively small. To get a still more comprehensive overview of the benefits and drawbacks of using an FGM-hub with respect to the interface pressure, in Fig. 7 a homogeneous hub is compared with one with extremum grading index. The surfaces for the two types of hub intersect each other at a curve, and one can – according to the operating conditions for the shrink fit – decide whether applying a functionally graded material might be an advantage. Generally speaking, this will be the case at high angular speeds and moderate operating temperatures. Finally, in case that also the weight of the device plays an important role, it might be interesting to look at the ratio of the nondimensional interface pressure to the non-dimensional weight of the hub, too. This quantity is plotted in Fig. 8 for different operating conditions vs. the grading index, and it is clearly visible that using an FGM-hub is advantageous even for an only thermally loaded shrink fit if a reduction in weight has highest priority at its design.
223
is much less pronounced than the gain in performance at rotation. (iv) If both the interface pressure under operating conditions and the weight of the device are assessment quantities for the shrink fit, an FGM-hub generally may be considered superior if weight reduction has priority at the design. Although the above results correspond to a negative grading index and in the numerical example a steel/aluminum FGM was presupposed, it must be emphasized that the equations derived above are universally valid, and that any other FGM can be treated analogously, provided that the shrink fit behaves purely elastically. Indeed, it was presupposed that no elastic–plastic design was intended, and using an FGM-hub shows possibilities to improve the performance of a shrink fit without admitting plasticization. Nevertheless, possibly a combination of using a functionally graded material and a partially plastic design could have the potential for further improvements, which however remains a topic of future investigations. Finally, it should be pointed out once more that the above results could be obtained by analytical means and hence also may serve as a countercheck for purely numerical studies e.g. by FE-methods. Acknowledgments The authors are indebted to Prof. Dr. Udo Gamer for many helpful discussions. Furthermore, Eray Arslan gratefully acknowledges the financial support by Vienna University of Technology during the performance of parts of this investigation, there. Appendix A For a purely elastic design of the shrink fit plasticization must not occur also during the mounting process, of course. If the hub is thermally assembled to the inclusion, it must be heated until (at least) uh ð1Þ ¼ d. The necessary temperature can be found from (23), and the stresses then follow from (24) and (25); in this case,
4. Concluding remarks In the above, an elastic shrink fit with solid homogeneous inclusion and a hub of functionally graded material obeying a power law has been investigated, and the performance of the shrink fit both at rotation and at elevated temperature has been studied. In particular, for the numerical example a negative grading index corresponding to, e.g., a steel/aluminum FGM has been considered, and the most relevant features for this case can be summarized as follows: (i) Generally, the effect of grading will become quantitatively significant only if the ratio of outer surface radius to interface radius is sufficiently large compared to 1. Nevertheless, in any case a reduction of weight is possible, of course, and in the above example more than half of the weight could be saved. (ii) An FGM-hub with large (absolute) value of the grading index – which however is limited by zero volume fraction of the ‘‘inner’’ FGM-constituent at the outer surface – shows significant advantages for the interface pressure and hence the transferable moment for a rotating shrink fit as compared to a homogeneous hub. (iii) On the contrary, a (homogeneous) thermal load causes a somewhat worse performance in case of an FGM-hub, which nevertheless for temperatures in a realistic operating range
Fig. A.1. Stresses and radial displacement in the heated hub prior to the assembly (uð1Þ ¼ d ¼ 1:011, compare Fig. 4).
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the two constants of integration come from the conditions that the radial stress vanishes at the inner and outer cylindrical surface of the hub, respectively. While a homogeneous hub remains stressfree for (homogeneous) heating, this is – due to the radially varying coefficient of thermal expansion – not the case for an FGM-hub. Nevertheless, Fig. A.1 reveals that for the parameter values of the above example the stresses in the hub remain far beyond the yield limit, and the same holds true for the entire range of m considered in this paper. References [1] Dolezˇel I, Kotlan V, Ulrych B. Design of joint between disk and shaft based on induction shrink fit. J Comput Appl Math 2014;270:52–62. [2] Kollmann FG. Welle-Nabe-Verbindungen. Konstruktionsbücher, Bd. 32. Berlin: Springer; 1984. [3] Cordts D. Numerische Simulation des Fügens von Querpreßverbindungen. Konstruktion 1990;42:278–84. [4] Mack W. Spannungen im thermisch gefügten elastisch-plastischen Querpreßverband mit elastischer Entlastung. Ing Arch 1986;56:301–13. [5] Mack W. Thermal assembly of an elastic–plastic hub and a solid shaft. Arch Appl Mech 1993;63:42–50. [6] Bengeri M, Mack W. The influence of the temperature dependence of the yield stress on the stress distribution in a thermally assembled elastic–plastic shrink fit. Acta Mech 1994;103:243–57. [7] Mack W, Bengeri M. Thermal assembly of an elastic–plastic shrink fit with solid inclusion. Int J Mech Sci 1994;36:699–705. [8] Sen S, Aksakal B. Stress analysis of interference fitted shaft-hub system under transient heat transfer conditions. Mater Des 2004;25:407–17. [9] Lorenzo M, Pérez-Cerdán JC, Blanco C. Influence of the thermal assembly process on the stress distributions in shrink fit joints. Key Eng Mater 2014;572:205–8. [10] Lippmann H. The effect of a temperature cycle on the stress distribution in a shrink fit. Int J Plast 1992;8:567–82. [11] Kovács Á. Thermal stresses in a shrink fit due to an inhomogeneous temperature distribution. Acta Mech 1994;105:173–87. [12] Kovács Á. Residual stresses in thermally loaded shrink fits. Period Polytechn Ser Mech Eng 1996;40:103–12. [13] Mack W, Plöchl M. Transient heating of a rotating elastic–plastic shrink fit. Int J Eng Sci 2000;38:921–38.
[14] Mack W, Plöchl M, Gamer U. Effects of a temperature cycle on an elastic– plastic shrink fit with solid inclusion. Chin J Mech 2000;16:23–30. [15] Güven U. The shrink fit with elastic–plastic hub exhibiting variable thickness. Acta Mech 1991;89:65–72. [16] Gamer U. A concise treatment of the shrink fit with elastic–plastic hub. Int J Solids Struct 1992;29:2463–9. [17] Megahed MM, Abdel-Kader MS. Elastoplastic analysis of rotating shrink-fitted discs with nonlinear hardening characteristics. Int J Solids Struct 1993;30:751–65. [18] Gao X-L, Atluri SN. An elasto-plastic analytical solution for the shrink-fit problem with a thin strain-hardening hub and an elastic solid shaft. Math Mech Solids 1997;2:335–49. [19] Özel A, Temiz Sß, Aydin MD, Sß en S. Stress analysis of shrink-fitted joints for various fit forms via finite element method. Mater Des 2005;26:281–9. [20] Baykara C, Güven U. The shrink fit with plastic orthotropic hollow inclusion. J Reinf Plast Comput 2006;25:1117–23. [21] Baldanzini N. A general formulation for designing interference-fit joints with elastic–plastic components. J Mech Des 2004;126:737–43. [22] Kieback B, Neubrand A, Riedel H. Processing techniques for functionally graded materials. Mater Sci Eng 2003;A362:81–105. [23] Shin KH, Natu H, Dutta D, Mazumder J. A method for the design and fabrication of heterogeneous objects. Mater Des 2003;24:339–53. [24] Birman V, Byrd LW. Modeling and analysis of functionally graded materials and structures. Appl Mech Rev 2007;60:195–216. [25] Sobczak JJ, Drenchev L. Metallic functionally graded materials: a specific class of advanced composites. J Mater Sci Technol 2013;29:297–316. [26] Nemat-Alla MM, Ata MH, Bayoumi MR, Khair-Eldeen W. Powder metallurgical fabrication and microstructural investigations of aluminum/steel functionally graded material. Mater Sci Appl 2011;2:1708–18. [27] Horgan CO, Chan AM. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J Elast 1999;55: 43–59. [28] Tutuncu N, Temel B. A novel approach to stress analysis of pressurized FGM cylinders, disks and spheres. Comput Struct 2009;91:385–90. [29] Tung HV. Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties. Acta Mech 2014;225:1795–808. [30] Zenkour AM. Elastic deformation of the rotating functionally graded annular disk with rigid casing. J Mater Sci 2007;42:9717–24. [31] Woo J, Meguid SA. Nonlinear analysis of functionally graded plates and shallow shells. Int J Solids Struct 2001;38:7409–21. [32] Blanke W, editor. Thermophysikalische Stoffgrößen. Berlin: Springer; 1989. [33] Chen WF, Han DJ. Plasticity for structural engineers. New York: Springer; 1988.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Shrink fit with solid inclusion and functionally graded hub Eray Arslan a,⇑, Werner Mack b a b
Department of Mechanical Engineering, Inonu University, 44280 Malatya, Turkey Institute of Mechanics and Mechatronics, Vienna University of Technology, Getreidemarkt 9, 1060 Vienna, Austria
a r t i c l e
i n f o
Article history: Available online 7 November 2014 Keywords: Interference fit Solid inclusion Functionally graded material Rotation Thermo-elastic analysis
a b s t r a c t Since in a shrink fit the transferable moment essentially depends on the interface pressure between inclusion and hub, the interface pressure should be as large as possible. This may be facilitated by a partially plastic design, which however also has some drawbacks like a possible permanent redistribution of the stresses after operating at high angular speeds and temperatures. In contrast to that, in the present study the use of a functionally graded hub in an elastically designed interference fit with solid inclusion is proposed. It is shown that for an appropriate grading not only the weight of the hub can be reduced noticeably as compared to a homogeneous hub, but also that particularly a much better performance at rotation can be achieved, which predominates over marginal disadvantages at high temperatures. The generally valid analytical results provide a comprehensive means for the practicing engineer for the design of this kind of shrink fits. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction As shrink fits are a simple and cost-effective means of transfer of moment, they frequently are found in mechanical engineering: examples are shrunk-on rings, armature bandages in rotating machines, or tires of railway wheels [1], to mention but a few. Hence, due to their importance, many studies of their behavior were performed by (semi-) analytical as well as numerical methods. Since under certain circumstances a partially plastic design for better utilization of the material is admissible, often not only elastic but also elastic–plastic states were taken into consideration. For both cases, an application-oriented basic survey of the design of shrink fits can be found in the monograph by Kollmann [2]. Later on, special attention was paid to the widely-used thermal assembly process, see, e.g., the studies by Cordts [3], Mack [4,5], Bengeri and Mack [6], Mack and Bengeri [7], Sen and Aksakal [8], Dolezˇel et al. [1], and Lorenzo et al. [9]. Moreover, several investigations on (transient) heating during operation were performed (e.g. [10–14]), and also effects of various material laws and/or geometrical properties were studied in a number of papers (e.g. [15–20]); in some of these investigations rotation of the shrink fit was taken into account, too. Furthermore, special design procedures were proposed [21]. Since for given geometry of the shrink fit and friction coefficient at the interface between inclusion and hub the transferable ⇑ Corresponding author. Tel.: +90 422 377 4819. E-mail address: [email protected] (E. Arslan). http://dx.doi.org/10.1016/j.compstruct.2014.10.034 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
moment depends solely on the interface pressure, the latter should be as large as possible. However, maximizing the interface pressure under all operating conditions is limited by several constraints. As was mentioned above, a possibility to achieve a high interface pressure is to admit partially plastic behavior; if however purely elastic behavior is required a priori – e.g., if the shrink fit shall be easily dismountable without permanent deformation –, an elastic–plastic design cannot be chosen, of course. Nevertheless, the most important issue is a reduction of the interface pressure with increasing angular speed and/or heating during operation. This reduction may be a transient one (which is particularly pronounced for a high rate of the outer surface temperature, see [13]), or in case of an elastic–plastic design also a permanent one, accompanied by a permanent redistribution of the stresses in the entire device. And a further point, which becomes increasingly important in engineering design, is to minimize the weight of the device while maintaining a good or even excellent performance of the shrink fit. Hence, an interesting alternative (or at least supplement) to admitting partial plasticization might be the use of a functionally graded material (FGM), particularly for the hub. As is well known, in a machine part of FGM the material properties like modulus of elasticity, density, coefficient of thermal expansion, and yield stress vary continuously and can - to a certain extent - be tailored in an appropriate way (for survey articles on this topic see, e.g., [22–25]). Thus, the aim of the present study is to analytically investigate the essential features of a purely elastic shrink fit with
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solid inclusion and functionally graded hub, taking both rotation and an elevated temperature into account. The material properties are presupposed to vary according to a power law in the radial direction; in particular, the case of radially decreasing density of the hub is considered. The latter property may be achieved, e.g., by using a steel/aluminum FGM, which can be produced by a powder metallurgical process, i.e. by appropriate mixing of the respective pure powders, cold pressing, and subsequent sintering (for details see [26]). Then, for a sufficiently large ratio of outer surface radius to interface radius a considerably better performance at rotation may be achieved, accompanied by a substantial saving of weight as compared to a homogeneous hub. These two significant advantages must be weighted against the fact that a marginally worse evolution of the interface pressure with increasing temperature may occur. It is the intention of this paper to thoroughly discuss these effects and to support the practicing engineer in deciding whether a shrink fit with FGM-hub might be advantageous for the specific application under consideration. The paper is organized as follows: in Section 2, the statement of the problem is given, and the governing equations are derived. In Section 3, the stress distribution is discussed, and particularly the interface pressure under various operating conditions is studied. Finally, some concluding remarks are made in Section 4. 2. Statement of the problem and governing equations 2.1. Statement of the problem The subject of the investigation is a cylindrically symmetric shrink fit with homogeneous solid inclusion (0 6 r 6 a) and an FGM-hub (a 6 r 6 b) with free cylindrical outer surface,
r¼b:
rr;h ¼ 0;
ð1Þ
it is presupposed that the axial length c of the device under consideration is much smaller than the diameter of the inclusion (see Fig. 1), and therefore a treatment as a plane stress problem is feasible [2,21], so that
rz;i ¼ 0; rz;h ¼ 0;
ð2Þ
where the indices i and h are here and in the following assigned to the inclusion and the hub, respectively. As a matter of course, the radial displacement u has to comply with the condition
r¼0:
ui ¼ 0;
ð3Þ
and the relations
2.2. Governing equations First, the basic equations shall be summarized. Taking a variable density q ¼ qðrÞ into account, the equation of motion in radial direction reads
dðr rr Þ rh ¼ qðrÞx2 r 2 : dr
ð6Þ
For small deformations, the geometric relations are
r ¼
du ; dr
h ¼
u : r
ð7Þ
Furthermore, considering a variable modulus of elasticity E ¼ EðrÞ and variable coefficient of thermal expansion a ¼ aðrÞ, but constant Poisson’s ratio m (as discussed below), the generalized Hooke‘s law reads
1 ðrr mrh Þ þ aðrÞT; EðrÞ 1 ¼ ðrh mrr Þ þ aðrÞT; EðrÞ
r ¼
ð8Þ
h
ð9Þ
where T means the difference of absolute and reference temperature. Next, for the FGM-hub the dependence of q; E, and a on the radial coordinate must be specified: as was already mentioned in the Introduction, a power law is presumed (compare e.g. related studies for disks by Horgan and Chan [27] or Tutuncu and Temel [28]). The basic grading law is however not postulated for the volume fractions of the constituents (e.g. [29]), but for the modulus of elasticity, and the dependence of the other physical quantities on r then is derived by the rule of mixture. The reason for this is that in this case a closed-form solution of the differential equations can be found, and therefore a purely analytical discussion of the problem is possible. The general linear rule of mixture reads [29]
Preff ðrÞ ¼ Pr 1 V 1 ðrÞ þ Pr2 V 2 ðrÞ;
there, Pr eff means an effective material property, and V j is the volume fraction of material j with property Pr j ; j ¼ 1; 2. In the following, it is further postulated that the material at the inner surface of the hub (denoted by the index s) is pure constituent s of the FGM (i.e., V s ðaÞ ¼ 1) and the same material as used for the homogeneous inclusion. If the index l denotes the second constituent of the FGMhub, it is obviously
V l ðrÞ ¼ 1 V s ðrÞ:
r¼a:
rr;i ¼ rr;h ;
ð4Þ
r¼a:
uh ui ¼ d
ð5Þ
with the interference d hold. A purely elastic design of the shrink fit under all operating conditions is required, and slowly varying angular speed is presupposed.
ð10Þ
ð11Þ
Now, presuming that in the hub the modulus of elasticity varies according to
EðrÞ ¼ Es
r m a
ð12Þ
;
there follows from (10)–(12) that the volume fraction of constituent s should vary with r according to
V s ðrÞ ¼
Es
r m
El a : Es El
ð13Þ
Then, applying the rule of mixture (10) to the density qðrÞ, the coefficient of thermal expansion aðrÞ, and also to the uniaxial yield limit ry ðrÞ, respectively, one finds
Preff ðrÞ ¼ Apr
r m a
þ Bpr
ð14Þ
with
Apr ¼ Fig. 1. Sketch of the shrink fit (prior to the assembly); r inclusion, s hub.
Es ðPrs Pr l Þ ; Es El
Bpr ¼
Es Prl El Prs ; Es El
ð15Þ
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and Pr ¼ q; a, and ry , respectively. While these properties depend on the radial coordinate, Poisson’s ratio m is however assumed to be constant [27,30]. Moreover, the influence of an elevated temperature on the material properties is disregarded, since this – at least for common operating conditions – also would have a minor effect on the results only (Woo and Meguid [31]; compare also Blanke [32]). Now, one is in a position to derive the governing equations for the functionally graded hub; for m ¼ 0 they hold also for the homogeneous inclusion. Using (7)–(9) and (12), and (14), (15) for the coefficient of thermal expansion, the stresses can be expressed in terms of u and its derivative with respect to r (denoted by a prime),
Es r 1þm rm mu þ ru0 ð1 þ mÞ Aa m þ Ba Tr ; m 2 ð1 m Þa a Es r 1þm rm ¼ u þ r mu0 ð1 þ mÞ Aa m þ Ba Tr : m 2 ð1 m Þa a
rr ¼
ð16Þ
rh
ð17Þ
For (slow) heating of all free surfaces of the shrink fit it is reasonable – for a problem with small ratio of axial length to diameter as sketched in Fig. 1 – to approximate the temperature distribution as a homogeneous one throughout the device, and in this case the integrals in the above equations can be written in closed form. Thus, one ends up with the constitutive equations for the hub,
uh ¼ C 1h rðmMÞ=2 þ C 2h r ðmþMÞ=2 1 m2 x2 r 3m Aq r m Bq am þ 8 þ mð3 þ mÞ 8 mð3 mÞ Es 2Aa r m Ba Tr; þ ð1 þ mÞ þ ð3 þ 2m þ mÞam 1 þ m
Es C 1h ðm þ M 2mÞr ð2þmMÞ=2 2ð1 m2 Þam þ C 2h ðm M 2mÞr ð2þmþMÞ=2
Aq ð3 þ mÞr m Bq ð3 m þ mÞ þ x2 r 2 ½8 þ mð3 þ mÞam 8 mð3 mÞ
rr;h ¼
Substituting (16) and (17) into the equation of motion (6) yields a differential equation for the displacement,
r
1þm 00
m 0
1þm
u þ ð1 þ mÞr u ð1 mmÞr u
2 2 1 m m m Aq r þ Bq a x r ¼ ð1 þ mÞ Es rm rm r m m 2Aa m þ Ba T þ Aa m þ Ba T 0 r a a
rh;h ¼ ð18Þ
with the solution
Aq r m Bq am u ¼ C 1 rðmMÞ=2 þ C 2 rðmþMÞ=2 þ 8 þ mð3 þ mÞ 8 mð3 mÞ 1 m2 x2 r 3m ð1 þ mÞrðmMÞ=2 þ Es 2Mam Z ð3mþMÞ=2 ð2 m þ M Þ þ Ba am rðmþMÞ=2 Tdr Aa r Z ð3mMÞ=2 ð19Þ Aa r þ Ba am r ðmMÞ=2 Tdr ð2 m M Þr M
M¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ mðm 4mÞ:
ð20Þ
Therefrom, one obtains for the stresses the expressions
Es C 1 ðm þ M 2mÞr ð2þmMÞ=2 2ð1 m2 Þam þ C 2 ðm M 2mÞr ð2þmþMÞ=2
Aq ð3 þ mÞr m Bq ð3 m þ mÞ E rð2þmMÞ=2 þ x2 r 2 s ½8 þ mð3 þ mÞam 8 mð3 mÞ 2Ma2m Z ð2 þ m þ M Þ ðAa r m þ Ba am ÞTrðmþMÞ=2 dr ð2 þ m M ÞrM Z ð21Þ ðAa rm þ Ba am ÞTrðmMÞ=2 dr ;
rr ¼
rh ¼
Es C 1 ½2 mðm þ M Þr ð2þmMÞ=2 2ð1 m2 Þam þ C 2 ½2 mðm M Þrð2þmþMÞ=2
Aq ð1 þ 3mÞr m Bq ½1 þ mð3 mÞ E r ð2þmMÞ=2 þ x2 r2 s ½8 þ mð3 þ mÞam 8 mð3 mÞ 2Ma2m
Z ½2 mð1 þ 2mÞ M rM ðAa rm þ Ba am ÞTrðmMÞ=2 dr Z ½2 mð1 þ 2mÞ þ M ðAa r m þ Ba am ÞTrðmþMÞ=2 dr
Es r m ðAa r m þ Ba am ÞT: a2m
ð22Þ
Aa Es Tr2m ; ð3 þ 2m þ mÞa2m
ð24Þ
Es C 1h ½2 mðm þ M Þr ð2þmMÞ=2 2ð1 m2 Þam þ C 2h ½2 mðm MÞrð2þmþMÞ=2
Aq ð1 þ 3mÞr m Bq ½1 þ mð3 mÞ þ x2 r 2 ½8 þ mð3 þ mÞam 8 mð3 mÞ
Aa Es ð1 þ 2mÞTr2m ; ð3 þ 2m þ mÞa2m
ð25Þ
and by setting m ¼ 0 with those for the inclusion,
ui ¼
C 1i r ð1 þ mÞ ð1 mÞqs x2 r 2 þ C 2i r as T ; 2 r 4Es
rr;i ¼
where the C j are constants of integration, and
ð23Þ
rh;i ¼
C 1i Es C 2i Es 1 ð3 þ mÞqs x2 r2 þ 4Es as T ; þ r 2 ð1 þ mÞ 1 m 8
C 1i Es C 2i Es 1 ð1 þ 3mÞqs x2 r 2 þ 4Es as T : þ r2 ð1 þ mÞ 1 m 8
ð26Þ
ð27Þ
ð28Þ
As mentioned in the Introduction already, an essential feature may be the weight of the hub (per length in axial direction),
W h ¼ 2p
Z
b
qðrÞrdr:
ð29Þ
a
As yield criterion the one by von Mises is applied [33], predicting the onset of plastic deformation if at some radius
rM ¼ ry ðrÞ with rM ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2r rr rh þ r2h :
ð30Þ
3. Stress distribution and interface pressure Based on the above equations, the stress distribution in the shrink fit now can be calculated. To this end, the four constants of integration must be determined: from (26) and condition (3) one obtains immediately C 1i ¼ 0, and C 2i ; C 1h , and C 2h follow from conditions (1), (4), and (5). In principle, these constants of integration could also be given explicitly, but the respective expressions would be quite lengthy so that a numerical solution of this system of equations is preferable. For the numerical calculations it is convenient to introduce the following non-dimensional quantities:
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E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
(a)
(b) 1
volume fractions
0.8
0.6
0.4
0.2 steel aluminum 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
non-dimensional modulus of elasticity and density
1
0.8
0.6
0.4
0.2
0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 non-dimensional radial coordinate
non-dimensional radial coordinate
(d) 1
1.4
0.8
non-dimensional uniaxial yield limit
non-dimensional thermal expansion coeff.
(c) 1.5
1.3
1.2
1.1
1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.6
0.4
0.2
0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
non-dimensional radial coordinate
non-dimensional radial coordinate
Fig. 2. (a) Volume fractions, (b) modulus of elasticity and density, (c) thermal expansion coefficient, (d) uniaxial yield limit vs. radius.
1.0 1.10 pi = 0.4 b = 5.0
0.9
stand-still
1.05
T=0 d = 1.011
b = 2.5
0.7
b = 5.0 0.6
0.5
m
non-dimensional interference
non-dimensional weight of hub
0.8 1.00
mext = -0.652 0.95
0.90
= -1.146
W = 0.525
m
= -0.652
W = 0.474
0.85 d = 0.833
0.4
0.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.80 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
grading index m
grading index m
Fig. 3. Weight of the hub vs. grading index.
Fig. 4. Necessary interference for an interface pressure of pi ¼ 0:4 at stand-still and T ¼ 0.
E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
(a)
(b)
(c)
(d)
221
Fig. 5. Stresses and radial displacement in a shrink fit with FGM-hub: (a) at stand-still, (b) at rotation and elevated temperature, and with homogeneous hub: (c) at stand-still, (d) at rotation and elevated temperature.
r ¼ r=a;
u ¼ uEs = ary;s ;
T ¼ as Es T=ry;s ;
a ¼ a=as ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ¼ xa qs =ry;s ;
E ¼ E=Es ;
ri ¼ ri =ry;s ;
q ¼ q=qs ;
mext ¼ ð31Þ
hence, the volume fraction of constituent s can (with El ¼ El =Es ) be written as
V s ðr Þ ¼
r m El 1 El
ð32Þ
:
The weight of the hub is non-dimensionalized by the weight of a homogeneous hub of material s, that is,
Wh ¼
Wh
pqs b2 a2
:
ð33Þ
An important note is the following one: since the minimum possible volume fraction of constituent s at the outer cylindrical surface of the hub is equal to zero, it follows from (32) that the extremum grading index mext is
ln El ; ln b
ð34Þ
in contrast to that, for a homogeneous hub m is equal to zero, and therefore, depending on whether El P 1 or El 6 1, the relations 0 6 m 6 mext or mext 6 m 6 0 must hold, respectively. For a numerical evaluation of the equations, specific values are assigned to the radii ratio and the material constants. In this paper, the focus is on an FGM-hub with the high-strength – and therefore usually heavier – constituent at the inner edge of the hub. The reason for this is that on the one hand in general the homogeneous inclusion will be of a high-strength material, and that on the other hand for an FGM-hub with radially increasing density a detrimental effect on the interface pressure at rotation is to be expected. Hence, the ratios of the parameter values chosen for the example discussed below,
El ¼ El =Es ¼ 0:35;
al =as ¼ 1:40; ql =qs ¼ 0:35; ry;l =ry;s ¼ 0:15; ð35Þ
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E. Arslan, W. Mack / Composite Structures 121 (2015) 217–224
(b)
(a)
Fig. 6. Interface pressure: (a) for T ¼ 0 vs. angular speed X, (b) for X ¼ 0 vs. temperature T (in the considered parameter range plasticization does not occur).
b = 5.0 0.9
m=0
0.3
0.2 m =-0.652 0.1
0 0 0.2 0.4 0.6 0.8 non-dimensional temperature
1
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
non-dimensional angular speed
Fig. 7. Interface pressure vs. angular speed and temperature for a homogeneous hub (m ¼ 0, the surface is cylindrically curved) and an FGM-hub with m ¼ mext ¼ 0:652.
correspond to a hub with grading from steel (s) at the inner cylindrical surface to a steel/aluminum-composite or to mere aluminum (l) at the outer cylindrical surface; moreover, ml ¼ ms ¼ 0:3 is presupposed. Since the effects to be discussed are especially pronounced and clearly visible particularly for larger radii ratios, b ¼ b=a ¼ 5:0 is chosen. First, Fig. 2(a) shows the volume fractions of the constituents for various grading indices, with m < 0 for the FGM-hub under consideration; the extremum grading index leading to mere aluminum at r ¼ b is mext ¼ 0:652. Then, Figs. 2(b)–(d) present the dependencies of E, q; a, and ry on the radius, respectively. In Fig. 3 the weight reduction of the hub that can be achieved by an FGM can be seen (for comparison also the corresponding values for b ¼ 2:5 are depicted), and one observes that in this case about halving the weight as compared to a homogeneous hub is possible. Of course, as a general trend the achievable gain will be larger for higher b=a-ratios and smaller for lower (absolute) values of the grading index. At the design of a shrink fit, in most cases the moment to be transferred and hence the necessary interface pressure is the essential input parameter. Therefore, in the following a nondimensional interface pressure of pi ¼ rr ð1Þ ¼ 0:4 is chosen, a rather high load but still not leading to plasticization at stand-still and T ¼ 0 for the entire admissible range of m (and the thermal mounting process can also be performed without plastic deformation, see Appendix A). For this interface pressure, Fig. 4 shows the
non-dimensional interface pressure / non-dimensional weight of hub
non-dimensional interface pressure
1 0.4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
grading index m
Fig. 8. Ratio of non-dimensional interface pressure to non-dimensional weight of the hub vs. grading index.
necessary interference d vs. the grading index m. One can see that it quasi-linearly increases with increasing (absolute) values of m, which may be explained by the increasing fraction of the softer FGM constituent. As the maximum weight reduction occurs for the extremum grading index, indeed, in Fig. 5 a comparison of the stresses and the displacement in a shrink fit with FGM-hub with mext ¼ 0:652 and with homogeneous hub (m ¼ 0) is shown, both at stand-still and at rotation at elevated temperature. First, it can be observed that in all cases the ratio rM =ry < 1, hence no plasticization occurs (one should be aware of the fact that rM =ry is not just a non-dimensional stress as rr or rh since ry depends on r, see Fig. 2(d)). Second, the circumferential stress is somewhat larger for the FGM-hub, and this also holds true for the radial displacement; generally, a striking feature is the increase of the
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displacement with radius in the hub, which is caused by rotation and thermal expansion. Third, and this is the most important observation, rotation causes due to the centrifugal forces in both cases a reduction of the interface pressure, of course, but this reduction is noticeably less pronounced for the FGM-hub (compare Figs. 5(b) and (d)). Hence, the transferable moment during operation is higher, which obviously is an advantage of the FGM-hub. While Fig. 5 shows a general case of operating conditions with both rotation and thermal load, it is nevertheless worthwhile to study the effects of rotation and elevated temperature on the interface pressure separately, too. Fig. 6(a) presents the evolution of the interface pressure with increasing angular speed and T ¼ 0, and Fig. 6(b) depicts that for increasing temperature and X ¼ 0 for different values of the grading index. One observes from Fig. 6(a) that for X ¼ 0:2 a homogeneous hub loses almost the contact with the inclusion and hence its ability to transfer a moment, whereas the interface pressure at the same angular speed for an FGM-hub with extremum grading still is almost one half of its initial value. On the other hand, Fig. 6(b) shows that for a homogeneous hub the interface pressure is insensitive to homogeneous heating, whereas for the steel-aluminum-FGM under consideration it slightly decreases with temperature for higher (absolute) values of m; this disadvantageous effect may be attributed to the more pronounced thermal expansion of aluminum, but is comparatively small. To get a still more comprehensive overview of the benefits and drawbacks of using an FGM-hub with respect to the interface pressure, in Fig. 7 a homogeneous hub is compared with one with extremum grading index. The surfaces for the two types of hub intersect each other at a curve, and one can – according to the operating conditions for the shrink fit – decide whether applying a functionally graded material might be an advantage. Generally speaking, this will be the case at high angular speeds and moderate operating temperatures. Finally, in case that also the weight of the device plays an important role, it might be interesting to look at the ratio of the nondimensional interface pressure to the non-dimensional weight of the hub, too. This quantity is plotted in Fig. 8 for different operating conditions vs. the grading index, and it is clearly visible that using an FGM-hub is advantageous even for an only thermally loaded shrink fit if a reduction in weight has highest priority at its design.
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is much less pronounced than the gain in performance at rotation. (iv) If both the interface pressure under operating conditions and the weight of the device are assessment quantities for the shrink fit, an FGM-hub generally may be considered superior if weight reduction has priority at the design. Although the above results correspond to a negative grading index and in the numerical example a steel/aluminum FGM was presupposed, it must be emphasized that the equations derived above are universally valid, and that any other FGM can be treated analogously, provided that the shrink fit behaves purely elastically. Indeed, it was presupposed that no elastic–plastic design was intended, and using an FGM-hub shows possibilities to improve the performance of a shrink fit without admitting plasticization. Nevertheless, possibly a combination of using a functionally graded material and a partially plastic design could have the potential for further improvements, which however remains a topic of future investigations. Finally, it should be pointed out once more that the above results could be obtained by analytical means and hence also may serve as a countercheck for purely numerical studies e.g. by FE-methods. Acknowledgments The authors are indebted to Prof. Dr. Udo Gamer for many helpful discussions. Furthermore, Eray Arslan gratefully acknowledges the financial support by Vienna University of Technology during the performance of parts of this investigation, there. Appendix A For a purely elastic design of the shrink fit plasticization must not occur also during the mounting process, of course. If the hub is thermally assembled to the inclusion, it must be heated until (at least) uh ð1Þ ¼ d. The necessary temperature can be found from (23), and the stresses then follow from (24) and (25); in this case,
4. Concluding remarks In the above, an elastic shrink fit with solid homogeneous inclusion and a hub of functionally graded material obeying a power law has been investigated, and the performance of the shrink fit both at rotation and at elevated temperature has been studied. In particular, for the numerical example a negative grading index corresponding to, e.g., a steel/aluminum FGM has been considered, and the most relevant features for this case can be summarized as follows: (i) Generally, the effect of grading will become quantitatively significant only if the ratio of outer surface radius to interface radius is sufficiently large compared to 1. Nevertheless, in any case a reduction of weight is possible, of course, and in the above example more than half of the weight could be saved. (ii) An FGM-hub with large (absolute) value of the grading index – which however is limited by zero volume fraction of the ‘‘inner’’ FGM-constituent at the outer surface – shows significant advantages for the interface pressure and hence the transferable moment for a rotating shrink fit as compared to a homogeneous hub. (iii) On the contrary, a (homogeneous) thermal load causes a somewhat worse performance in case of an FGM-hub, which nevertheless for temperatures in a realistic operating range
Fig. A.1. Stresses and radial displacement in the heated hub prior to the assembly (uð1Þ ¼ d ¼ 1:011, compare Fig. 4).
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the two constants of integration come from the conditions that the radial stress vanishes at the inner and outer cylindrical surface of the hub, respectively. While a homogeneous hub remains stressfree for (homogeneous) heating, this is – due to the radially varying coefficient of thermal expansion – not the case for an FGM-hub. Nevertheless, Fig. A.1 reveals that for the parameter values of the above example the stresses in the hub remain far beyond the yield limit, and the same holds true for the entire range of m considered in this paper. References [1] Dolezˇel I, Kotlan V, Ulrych B. Design of joint between disk and shaft based on induction shrink fit. J Comput Appl Math 2014;270:52–62. [2] Kollmann FG. Welle-Nabe-Verbindungen. Konstruktionsbücher, Bd. 32. Berlin: Springer; 1984. [3] Cordts D. Numerische Simulation des Fügens von Querpreßverbindungen. Konstruktion 1990;42:278–84. [4] Mack W. Spannungen im thermisch gefügten elastisch-plastischen Querpreßverband mit elastischer Entlastung. Ing Arch 1986;56:301–13. [5] Mack W. Thermal assembly of an elastic–plastic hub and a solid shaft. Arch Appl Mech 1993;63:42–50. [6] Bengeri M, Mack W. The influence of the temperature dependence of the yield stress on the stress distribution in a thermally assembled elastic–plastic shrink fit. Acta Mech 1994;103:243–57. [7] Mack W, Bengeri M. Thermal assembly of an elastic–plastic shrink fit with solid inclusion. Int J Mech Sci 1994;36:699–705. [8] Sen S, Aksakal B. Stress analysis of interference fitted shaft-hub system under transient heat transfer conditions. Mater Des 2004;25:407–17. [9] Lorenzo M, Pérez-Cerdán JC, Blanco C. Influence of the thermal assembly process on the stress distributions in shrink fit joints. Key Eng Mater 2014;572:205–8. [10] Lippmann H. The effect of a temperature cycle on the stress distribution in a shrink fit. Int J Plast 1992;8:567–82. [11] Kovács Á. Thermal stresses in a shrink fit due to an inhomogeneous temperature distribution. Acta Mech 1994;105:173–87. [12] Kovács Á. Residual stresses in thermally loaded shrink fits. Period Polytechn Ser Mech Eng 1996;40:103–12. [13] Mack W, Plöchl M. Transient heating of a rotating elastic–plastic shrink fit. Int J Eng Sci 2000;38:921–38.
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