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The effect of ﬁlament winding mosaic pattern on the stress state of ﬁlament wound composite ﬂywheel disk Md. Sayem Uddin, Evgeny V. Morozov ⇑, Krishnakumar Shankar School of Engineering and Information Technology, The University of New South Wales, Canberra ACT, Australia

a r t i c l e

i n f o

Article history: Available online 7 August 2013 Keywords: Composite ﬂywheel disk Filament winding pattern Finite element analysis Mosaic unit

a b s t r a c t A ﬁlament wound spinning composite disk is characterised by the mosaic pattern conﬁguration produced during the ﬁlament winding process. In this structure, each helically wound layer consists of curved triangular shaped units alternating in the radial and circumferential directions. Filament wound composite shells are usually analysed based on the implementation of mechanics of composite laminates. Usually, the mosaic pattern conﬁguration is not considered in the general stress analysis procedures based on the conventional modelling of laminated composite structures, including those available in ﬁnite element analysis packages. However, the ﬁlament winding mosaic pattern of the composite layer could signiﬁcantly affect the stress ﬁeld developed due to rotational loading. This paper presents results of the ﬁnite element modelling and stress analysis of a ﬁlament wound spinning composite disk with different types of ﬁlament winding mosaic pattern. The disk has also been modelled and analysed using conventional method and differences in the predicted stress values from both techniques are demonstrated through stress distributions. As illustrated, the level of stresses in the thin ﬁlament wound composite ﬂywheel disk could be underestimated by the structural analysis based on conventional mechanics of laminated structures. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Composite materials currently have gained momentum as high performance structural materials because of their high strengthto-density and stiffness-to-density ratios. The blending of composite materials’ directional properties with the design optimisation provides greater opportunities to improve the performance of composite structures compared to their metallic counterparts. Moreover, the simpliﬁed fabrication of composite structures with greater accuracy using latest computer-controlled automatic ﬁlament winding machines has broadened the range of applications of composite materials in different products. A ﬁlament wound composite ﬂywheel disk with the optimal ﬁbre trajectories resulting in uniform strength [1,2] is an example of progressively increasing number of optimal composite structures. A ﬂywheel (spinning disk) is a device designed to store energy in the form of kinetic energy and then release when required. Emerging ﬂywheel technology has encompassed a variety of applications in different ﬁelds. Flywheels provide the substantial improvement of performance and service life in different applications such as, spacecraft, aircraft power systems, uninterruptible power supplies and planetary outposts and rovers [3,4]. Flywheels are used in alternating engines, compressors, press and strike ⇑ Corresponding author. Tel.: +61 2 6268 9542; fax: +61 2 6268 8276. E-mail address: [email protected] (E.V. Morozov). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.07.004

machinery. Flywheels can be made of either metals or composite materials. However, a composite ﬂywheel can have comparatively higher energy storage capacity per unit mass [5]. Optimal composite ﬂywheels can attain much higher rotational speed than metallic ones and thus increased rotational speed compensates for lower mass of the composite ﬂywheel rotor without compromising the kinetic energy storage capacity of the rotor [6]. In general, a structure’s mass can be reduced by 20–30% through the replacement of metal alloys by composite materials [2]. Composite ﬂywheels have increased energy density storage capability that is better than batteries which have slow loading and unloading cycle due to chemical process as well as short life [7]. One of the prominent applications of composite ﬂywheels is energy storage for satellites. A ﬂywheel stores the energy generated by photovoltaic cells during passing half of the orbit with greater reliability and lower weight [8]. A ﬂywheel also facilitates the recovery of the kinetic energy of a vehicle lost as heat during braking. It can store and release this kinetic energy in automotive vehicles in fractions of a second [9]. Flywheels can play an important role in preventing blackouts and also in regulating the functioning of not well-meshed networks in electricity distribution [10]. Although composite ﬂywheels hold great promises, a number of challenges need to be overcome for the better service and longterm performance which require the implementation of advanced design and analysis approaches. Different studies have been performed to date dealing with composite ﬂywheels to achieve the

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maximum performance under certain conditions. Most of the research on ﬂywheels aims to maximize the energy storage capability while keeping the mass of ﬂywheels at minimum possible level. The concept of uniform stress ﬁlament wound spinning composite ﬂywheel disk composed of structural ﬁlaments of uniform cross-sections was presented by Kyser [1]. The ﬁbres arranged as of a ﬁne-mesh circular net forming curved load-carrying paths that spiral outward from the centre of the disk and maintain constant tension through the cancellation of decreasing ﬁbre tension towards the centre due to radially directed loading and increasing ﬁbre tension towards the centre because of inertia forces due to rotation. Bokov et al. [11] examined the optimum design of a ﬂywheel in the form of a membrane shell of revolution formed by winding or placing of orthotropic reinforced strips and explored reinforcing paths for ensuring maximum bearing capacity while assessing the energy storage capacity of these shells. Seleznev and Portnov [12] demonstrated the chord winding process of composite-tape disk and showed that the variation of elastic properties along the radius must be taken into account in the stress analysis. An approximate method of the calculation of both mass and volume energy capacity of a chord ﬂywheel was presented by Portnov and Kustova [13] and an analysis was also carried out to reveal the most effective combination of materials. Valiullin et al. [14] demonstrated the use of ﬁnite element method to calculate the stress–strain state of a composite ﬂywheel having varying thickness and consisting of a ring with ﬁbres orientated in the circumferential direction and a peripheral shell made by winding along the lines of constant deviation from geodesic lines. Vasiliev and Morozov [2] discussed the ﬁlament wound composite spinning disk of uniform strength based on the monotropic model of unidirectional ﬁlaments as an example of optimal composite structure. Structural assessment of two circumferentially wound composite ﬂywheel rotor assemblies was performed by Abdul-Aziz et al. [3] using ﬁnite element method (FEM). Dems and Turant [15] analysed optimally designed composite ﬂywheels to maximize the accumulated kinetic energy of the ﬂywheel. However, these studies are limited to certain considerations that cannot fully encompass the actual structure of composite materials. Filament winding mosaic pattern (MP) of the ﬁlament wound composite structures is one special feature that is generally overlooked in modelling and stress analysis of such structures as laminated shells using analytical or/and ﬁnite element solutions with the exploitation of mechanics of composite laminates that takes into consideration the number of plies, stacking sequence and ﬁbre orientation [2,16,17]. The ﬁlament wound composite ﬂywheel disk under consideration also has this special in-built mosaic pattern produced by the ﬁlament winding process. As can be

Fig. 1. A carbon-epoxy ﬂywheel [2].

261

seen from Fig. 1, each layer of the ﬁlament wound composite disk is composed of two plies with +/ and / orientation of the ﬁbres which are interlaced in the process of continuous helical ﬁlament winding. As a result, the structure of the layer has the distinctive mosaic pattern. The effect of ﬁlament winding mosaic pattern on the strength of thin-walled composite cylindrical shells has been investigated by Morozov et al. [16,17]. The inﬂuence of ﬁlament winding pattern on the compressive stability of ﬁlament wound composite cylinders has been investigated by Jensen and Pai [18]. The effect of mosaic pattern on the ﬁlament wound composite pressure vessels has been demonstrated by Mian and Rahman [19]. As has been shown, the assessment of mechanical response of the ﬁlament wound composite structures was considerably inﬂuenced by the incorporation of the mosaic pattern units in the analyses. However, thickness variation of ﬁlament wound structures has not been considered in these studies to explore the effect on stress state. In the current paper, the results of the stress analysis of the ﬁlament wound composite ﬂywheel disk with a variety of ﬁlament winding pattern around the circumference and with actual varying thickness are presented. The analysis is aimed to demonstrate the effect of mosaic pattern created during a ﬁlament winding process on the stress distributions and effectively on the strength characteristics of the ﬁlament wound spinning composite ﬂywheel disk with varying stiffness properties. 2. Filament winding pattern architecture of ﬁlament wound disk The ﬁlament wound composite ﬂywheel disk (see Fig. 1) composed of antisymmetric ±/ angle-plies is manufactured following the inverse fabrication process as the shape and reinforcement of the ﬁnal product are given [20]. Therefore, for manufacturing ﬁlament wound composite ﬂywheel disk, it is required to determine the shape of the preform made by geodesic winding provided the ﬁnal shape and the law of reinforcement of the structure are given. Mitkevich et al. [20] determined the shape of the preform based on the ﬁnal form of ﬁlament wound composite ﬂywheel disk for various dimensions of both inner and outer radii. Consequently, the shell of revolution is fabricated by ﬁlament winding onto an inﬂated elastic mandrel at the ﬁrst stage. After the shell with proper winding pattern is wound, the shell is compressed in the axial direction between two plates while the pressure in the mandrel is continuously reduced. As a result, the shell is transformed into a ﬂat disk. The resin in the composite material is cured after this transformation is complete. So, the ﬁnished ﬁlament wound ﬂywheel disk has an annular band of smooth continuously turning ﬁbre paths tangent both to the outer and inner periphery [1,2] as shown in Fig. 2. The ﬁlament wound disk shown in Fig. 1 consists of an even number of angle-ply antisymmetric layers [2,17]. Each layer is composed of two plies with +/(r) and /(r) angles of ﬁbre orientation to the radius of the disk as shown in Fig. 2. These plies are interlaced in the process of helical ﬁlament winding. Thus, the angle-ply layer has a particular type of repeating ﬁlament winding pattern as can be seen in Fig. 1. This pattern consists of the ‘‘curved triangular’’ shaped twoply units with alternating ±/ and / ﬁbre orientation repeating in a chess-board fashion [16,17]. The plies are not interlaced within a unit. They are combined into the angle-ply antisymmetric laminates. The units are arranged in regular geometric pattern around the circumference and along the radius of the disk. So, the ﬁlament wound composite ﬂywheel disk comprises multiple numbers of mosaic units of varying size depending on the ﬁlament winding parameters. Arrangement of mosaic units of a ﬁlament wound ﬂywheel disk having four units around the circumference and corresponding ﬁbre orientations are shown in Fig. 3. As can be seen in the ﬁgure, curved triangular area ABD having / ﬁbre orientation is

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+φ −φ R r0

angle-ply layer is treated as orthotropic and homogeneous. Such homogeneous orthotropic angle-ply layer is described by the following set of constitutive equations [2]:

rx ¼ A11 ex þ A12 ey ; ry ¼ A21 ex þ A22 ey sxy ¼ A44 cxy ; sxz ¼ A55 cxz ; syz ¼ A66 cyz

ð1Þ

The stiffness coefﬁcients Amn are speciﬁed as follows:

A11 ¼ E1 c4 þ E2 s4 þ 2E12 c2 s2 A12 ¼ A21 ¼ E1 m12 þ ðE1 þ E2 2E12 Þc2 s2 A22 ¼ E1 s4 þ E2 c4 þ 2E12 c2 s2

ð2Þ

2

A44 ¼ ðE1 þ E2 2E1 m12 Þc2 s2 þ G12 ðc2 s2 Þ A55 ¼ G13 c2 þ G23 s2 A66 ¼ G13 s2 þ G23 c2 where

E1;2 ; 1 m12 m21

E1;2 ¼ Fig. 2. Schematic representation of ﬁbre trajectories in a ﬁlament wound disk.

surrounded by other areas such as, ABC, DEB that have ±/ ﬁbre orientation. The texture of the angle-ply layer depends on the process parameters of the ﬁlament winding for the same ﬁbre orientation /(r) [17]. 3. Analysis of ﬁlament wound structures 3.1. Conventional approaches for modelling ±/ angle-ply layers Filament wound ±/ angle-ply layers are usually modelled as laminated shells based on the utilisation of mechanics of composites that takes into consideration the number of plies, stacking sequence and ﬁbre orientation. Two main approaches to the modelling of the ﬁlament wound angle-ply layers are generally used for the mechanical analysis of ﬁlament wound composite shells [16]. Both techniques are widely employed for the stress analysis of composite plates and shells using analytical or/and ﬁnite element solutions. 3.1.1. Homogeneous orthotropic ±/ angle-ply layer The ﬁrst technique assumes that the ﬁlament wound shells are usually composed of a large number of ±/ angle-ply layers. Each ±/

E12 ¼ E1 m12 þ 2G12 ;

c ¼ cos /;

s ¼ sin /

E1 and E2 are the longitudinal and transverse moduli, m12 and m21 are Poisson’s ratios (E1m12 = E2m21), and G12, G13 and G23 are the shear moduli of the unidirectional ply. The thickness of the homogeneous orthotropic layer, h is double that of the unidirectional ply. The constitutive equations that link stress resultants and couples with the corresponding strains of the reference middle surface of the homogeneous orthotropic laminate are as follows [2,16]:

Nx ¼ B11 e0x þ B12 e0y ;

Ny ¼ B21 e0x þ B22 e0y

B44 0xy ;

c M x ¼ D11 jx þ D12 jy M y ¼ D21 jx þ D22 jy ; M xy ¼ D44 jxy V x ¼ S55 cx ; V y ¼ S66 cy Nxy ¼

ð3Þ

where the stress resultants and couples are speciﬁed as

Nx ¼

Z

h=2

rx dz; Ny ¼

Z

h=2

Mx ¼ Vx ¼

Z Z

h=2

ry dz; Nxy ¼

h=2

Z

h=2

rx z dz; My ¼ h=2 h=2

sxz dz; V y ¼

h=2

Z

Z

h=2

sxy dz

h=2

h=2

ry z dz; Mxy ¼

h=2

Z

h=2

sxy z dz

h=2

h=2

syz dz

h=2

and the stiffness coefﬁcients are calculated as follows: 3

Bmn ¼ Amn h;

Dmn ¼ Amn

h ; 12

Smn ¼ Amn h

The strains of the reference surface of the homogeneous ortho e0x ; e0y , in-

tropic laminate are in-plane tension or compression

plane shear (c0xy ), bending in xz- and yz-plane (jx, jy), twisting (jxy), and transverse shear (cx, cy). 3.1.2. Antisymmetric balanced ±/ angle-ply layer The second approach deals with the angle-ply layer as an antisymmetric balanced laminate consisting of +/ and / unidirectional plies. The angle-ply layer in this case consists of two plies with the same thickness and ﬁbre orientation angles +/ and /. The constitutive equations for this layer are (considering midplane as reference surface) [16]

Nx ¼ B11 eox þ B12 eoy þ C 14 jxy Ny ¼ B21 eox þ B22 eoy þ C 24 jxy Fig. 3. Curved triangular mosaic pattern with alternating ﬁbre orientations.

Nxy ¼ B44 c0xy þ C 41 jx þ C 42 jy

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M x ¼ C 14 c0xy þ D11 jx þ D12 jy

30.00

c þ D21 jx þ D22 jy M xy ¼ C 41 e0x þ C 42 e0y þ D44 jxy V x ¼ S55 cx þ S56 cy ; V y ¼ S65 cx þ S66 cy My ¼

n=4 n=8 n = 14

C 24 0xy

25.00

ð4Þ

where the stiffness coefﬁcients are calculated as follows:

Bmn ¼ Amn h;

C mn ¼

h Amn ; 4

20.00

3

Dmn ¼

h Amn ðh ¼ 2dÞ 12

The coefﬁcients characterising the transverse shear, S55, S66, and S56 = S65 are calculated in terms of A55, A66 and A56 = A65 = (G13 G23)cs [2]. The set of stiffness coefﬁcients in Eq. (2) should additionally include the following coefﬁcients

A14 ¼ A41 ¼ E1 c2 E2 s2 E12 ðc2 s2 Þ cs 2 A24 ¼ A41 ¼ E1 s E2 c2 E12 ðc2 s2 Þ cs

h (mm)

2

15.00

10.00

ð5Þ

5.00

In this approach, the layer is treated as anisotropic because +/ and / plies are located in different planes with regard to the reference surface.

0.00

3.1.3. Actual structure of the ±/ angle-ply layer Both techniques mentioned above overlook some effects of the ﬁlament winding manufacturing process on the formation of laminated composite structures. Filament winding pattern conﬁguration described previously is one of such inherited characteristics speciﬁc to the composite structure made by ﬁlament winding [2,16,17,19]. The ﬁrst technique (see Section 3.1.1) neglects both the ﬁlament winding mosaic pattern and the anisotropy of the layer as angle-ply layer is treated as homogeneous orthotropic one. Although, the anisotropy of the angle-ply layer is reﬂected in the second approach (see Section 3.1.2) but it does not take into account the real texture of the ﬁlament wound layer. Namely, interlacing of ﬁlaments in the winding process is neglected in this approach. Furthermore, the angle-ply layer is considered to be composed of two plies where +/ ply is placed on top of / ply or vice versa. Although, this model works well for the laminated composite plates, however it overlooks the actual construction of alternating triangular-shaped areas as a result of interlacing of unidirectional plies [16]. As discussed before, the real angle-ply layers of ﬁlament wound composite ﬂywheel disk are composed of triangular-shaped repeating mosaic pattern units placed around the circumference and along the radius of the disk. Each unit consists of two plies with either [+// /] or [//+/] structure and these plies are not interlaced within the unit area. The difference between alternating units of two different layer structures are in the stiffness coefﬁcients A14, A24 and A56

Table 1 Thickness values for the disks composed of 4, 8 and 14 plies. Number of plies, n 4 8 14

k

hro (m)

484 968 1694

hR (m) 2

1.0 10 2.0 102 2.6 102

0.4 102 0.8 102 1.5 102

Table 2 Dimensions and material properties for a ﬁlament wound disk.

0.1016

0.1512

0.2008

0.2504

0.3

r (m) Fig. 4. Variation of thickness of the ﬁlament wound ﬂywheel disks (n = 4, 8 and 14).

that are responsible for the anisotropic behaviour of the angle-ply layers [16]. Consequently, two alternating mosaic pattern units exhibit antisymmetric stretching-twisting and bending-shear coupling effects. For antisymmetric angle-ply laminates, the stretching stiffnesses (B11, B12, B22, B44) and the bending stiffnesses (D11, D12, D22, D44) (see Eq. (4)) are independent of the number of layers. There is neither coupling between tension and shear (B14 = B24 = 0) nor coupling between bending and twisting (D14 = D24 = 0). However, the antisymmetric angle-ply laminates show coupling between tension and twisting (C14, C24) and coupling between shear and twisting. These couplings decrease progressively as the number of layers increases (C14 and C24 are inversely proportional to the number of layers) [21]. Coupling between stretching and bending does not exist in symmetric laminates. However, nonsymmetric laminates need to be employed for certain applications. For example, the stretching-bending coupling may be utilised in the design of turbine blades that have a warped proﬁle and twist under centrifugal forces. Also, in some other cases like robotic parts undergoing complicated deformation under simple loading and airplane wings twisting under bending, it is necessary to have layers possessing different orientations [2,21]. An antisymmetric angle-ply laminate is formed of an even number of layers of which the distribution of thicknesses is symmetric, and that of the orientations of the reinforcement is antisymmetric with respect to the middle plane. Existence of a stretching-twisting coupling in antisymmetric angle-ply laminate results from the coefﬁcients C14 and C24. The present work investigates the inﬂuence of mosaic pattern on the stress state developed in the rotating ﬁlament wound composite ﬂywheel disk. The ﬁlament wound spinning composite disk has been modelled using different mosaic pattern conﬁgurations and then analysed exploiting ﬁnite element analysis technique to investigate the mechanical response of the ﬁlament wound ±/ angle-ply structures. Results of the stress analyses have been compared with those obtained from conventional ﬁnite element modelling of laminated shells.

Dimensions

Material

Properties

Outer radius (R) = 0.30 m Inner radius (ro) = 0.1016 m

IM6 – Epoxy (Carbon–Epoxy)

E1 = 203.00 GPa E2 = 11.20 GPa G12 = 8.40 GPa

4. Finite element modelling of ﬁlament wound composite ﬂywheel disk

wo = 4.00 103 m d = 0.70 103 m

Density = 1600 kg/m3

m12 = 0.32

In order to investigate the effect of ﬁlament winding mosaic pattern, the stress analysis of a ﬁlament wound composite

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Fig. 5. Three different designs chosen to model the ﬁlament wound ﬂywheel disk.

(i) Conventional model

(iii) 6–unit disk

(ii) 4–unit disk

(iv) 8–unit disk

Fig. 6. Conventional, (i); 4-unit, (ii); 6-unit, (iii); and 8-unit, (iv) disk models.

ﬂywheel disk with different number of mosaic pattern units around the circumference has been performed. Fibre orientation

angle in the ﬁlament wound disk of a radius R and with a central opening of a radius ro has been deﬁned using the monotropic

Md. Sayem Uddin et al. / Composite Structures 107 (2014) 260–275

265

Fig. 7. Contour plots of rr in various models of the 4-ply disk.

material model and assuming that transverse (r2) and shear (s12) stresses in the unidirectional ply are zero (r2 = s12 = 0). The equation for the ﬁbre orientation angle is speciﬁed as [2]

sin / ¼

R r

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k r4 1 4 1 2 R

ð6Þ

where

k¼

1þ

2 r 2

ð7Þ

0

R

The disk thickness is speciﬁed by

hðrÞ ¼

kwo d 2pr cos /

ð8Þ

where k is the number of ﬁbrous tapes passing through the circumference r = constant, and wo and d are the tape width and thickness, respectively. The disk thickness varies depending on the values of radial location r and ﬁbre orientation angle / as can be observed from Eq. (8). Three different designs composed of 4, 8 and 14 plies have been considered to demonstrate the change in level of stresses in different layers and the extent of the inﬂuence of ply interlacing. The overall thickness of the disk is determined by Eq. (8). However, total thickness values both at the inner (hro ) and outer (hR) radii

cannot be determined using this equation as ﬁbre orientation angle at these locations of the ﬁlament wound ﬂywheel disk is obtained to be 90° from Eq. (6). Hence, for the three disks chosen for the ﬁnite element modelling, constant values have been assumed up to the distance wo from the inner and outer edges (i.e., hðr ¼ r o Þ ¼ hðr ¼ r o þ wo Þ ¼ hro and h(r = R) = h(r = R wo) = hR). The approximate variations of thicknesses according to Eq. (8) along the radial cross-section of the disk for the chosen values of wo, d and k from Tables 1 and 2 are presented in Fig. 4. Total thickness values both at the inner (hro ) and outer (hR) radii and corresponding number of plies are given in Table 1. The corresponding ﬁnite element models of the disks are shown in Fig. 5. Both the development of ﬁnite element models and the analyses of ﬁlament wound composite spinning disk have been performed using ANSYS [22]. The composite spinning disk has been modelled using SHELL281 element. Modelling of composite laminate using this element is governed by the ﬁrst order shear deformation theory. First, the ﬁlament wound ﬂywheel disk has been modelled using conventional approach (CA) according to which the disk is modelled as a laminated circular plate composed of different number of plies (see Fig. 6(i)). The 4-ply disk (Fig. 5 (i)) comprises four plies [+/(r)//(r)]s. The 8-ply (Fig. 5(ii)) and 14-ply (Fig. 5(iii)) disks are composed of eight plies [+/(r)//(r)/+/(r)/ /(r)]s and fourteen plies [+/(r)//(r)/+/(r)//(r)/+/(r)//(r)/ +/(r)]s, respectively. The middle plane is taken as the reference

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(i)

(iii)

(conventional model)

(6-unit disk)

(ii)

(4-unit disk)

(iv)

(8-unit disk)

Fig. 8. Contour plots of rh in different models of the 4-ply disk.

one. In the conventional model, each ply has ﬁbre orientation angle of either +/(r) or /(r) all over the disk. However, interlacing of plies due to ﬁlament winding process is not considered. Clearly, the conventional modelling approach does not reﬂect the real structure of ﬁlament wound composite ﬂywheel disk since the ﬁlament winding mosaic pattern architecture realised due to the interlacing of the plies and, subsequently, the anisotropy of the antisymmetric angle-ply structure of the units are neglected. In order to model the mosaic pattern structure, the disk has been partitioned into mosaic units. Correspondingly, the ﬁnite elements have been combined into the respective alternating groups. The composite material for the ﬁnite elements from each of these groups has been deﬁned as either [+/(r)//(r)/+/(r)//(r)] or [/(r)/+/(r)//(r)/+/(r)] laminate for the 4-ply disk (Fig. 5(i)). Similarly, for 8-ply disk (Fig. 5(ii)) the composite material of alternating mosaic pattern groups has been deﬁned as either [(+/(r)// (r))4] or [(/(r)/+/(r))4] and for 14-ply disk (Fig. 5(iii)) as either [(+/(r)//(r))7] or [(/(r)/+/(r))7]. Four, six and eight mosaic units around circumference of the disks (4, 8 and 14-ply) have been considered to model the mosaic pattern conﬁguration. The corresponding ﬁnite element models of the ﬁlament wound disk are shown in Fig. 6(ii–iv). As mentioned earlier, the number of mosaic units in the ﬁlament wound ply is determined by the appropriate selection of manufacturing parameters for ﬁlament winding process.

For each unit, the layer conﬁguration through the thickness of the disk is deﬁned as layer-by-layer assembly starting from the bottom of the reference plane to the top. So, the bottom layer is counted as layer 1 and additional layers are stacked and numbered from the bottom to the top in the positive direction of the coordinate normal to the reference plane. The layers of the ﬁlament wound composite ﬂywheel disk under consideration have been reinforced along the radially varying ﬁbre trajectories having orientation angle /(r) deﬁned by Eqs. (6) and (7). 5. Stress analysis of ﬁlament wound composite ﬂywheel disk Finite element stress analyses considering plane-stress condition have been performed in order to investigate the effect of ﬁlament winding mosaic pattern for the ﬁlament wound 4, 6 and 8-unit composite disks (see Fig. 6) composed of 4, 8 and 14 plies. The disk has been rotated about its central axis at a constant angular velocity. Dimensions of the ﬁlament wound composite disks under consideration and material properties used in the ﬁnite element analyses are listed in Table 2 in which E1, E2 are Young’s moduli, G12 is shear moduli and m12 is Poisson’s ratio. Displacement and rotation boundary conditions have been applied at the nodes on inner and outer radii along both x and y axes (see Fig. 6(i)) of the reference plane to prevent in-plane rigid body motion in both radial and circumferential directions and out-of-plane rigid body motion in axial direction.

Md. Sayem Uddin et al. / Composite Structures 107 (2014) 260–275

(i)

(conventional model)

(iii)

(6-unit disk)

(ii)

(iv)

267

(4-unit disk)

(8-unit disk)

Fig. 9. Contour plots of srh in different models of the 4-ply disk.

Radial (rr), tangential (rh) and shear (srh) stress distributions have been determined for the top layer (layer 4, 8 and 14 for 4, 8 and 14-ply disks, respectively) of both conventional and mosaic patterned models of the ﬁlament wound spinning disk. The variation of stresses observed at different locations of the disk and its dependency on the arrangement of mosaic units are demonstrated using results obtained from ﬁnite element analyses of the ﬁlament wound composite ﬂywheel disk. Radial stress distributions as calculated using ANSYS for conventional and mosaic patterned models of the spinning 4-ply disk are illustrated by contour plots shown in Fig. 7. As can be seen, the stress distributions in the disk modelled using mosaic pattern differ substantially from those calculated on the basis of conventional modelling of laminated shells. It follows from contour plots of rr that in the real structure of the ﬁlament wound disk, radial stress is varied considerably from one mosaic unit to the adjacent ones both in the radial and circumferential directions. The levels of rr experienced by the ﬂywheel disk due to rotation and also the location of maximum rr are signiﬁcantly affected by the consideration of the actual structural characteristics. The number of mosaic units around circumference of the disk also has a substantial inﬂuence over the distribution and level of magnitude of rr. Maximum radial stress in 4-ply disk obtained from conventional model is found to be uniform over a wider range of radial distance as can be seen in Fig. 7(i). On the other hand, locations of maximum rr in models with different number of mosaic units can be traced at different positions.

Similarly, the predicted contour plots of hoop stress, rh for different models of the 4-ply disk as can be seen from Fig. 8 also reﬂect the signiﬁcant effect of mosaic pattern on the distributions of rh. Distribution of rh obtained from conventional model of the disk differs signiﬁcantly from that determined from models with various number of mosaic units. Also, the magnitude of maximum rh calculated using conventional model of 4-ply disk varies considerably from that obtained from mosaic patterned models. Maximum value of rh from conventional model reaches the level of 238 MPa, whereas this value from 4-unit disk model is found to be 352 MPa. The maximum value of rh changes with the increasing number of mosaic units. It is predicted to be 350 MPa for 6-unit model and 336 MPa in case of 8-unit model of the 4-ply disk. Therefore, the predicted values of rh reﬂect the effect of mosaic pattern on the stress distribution. Furthermore, the distribution of shear stress, srh is also inﬂuenced by the incorporation of mosaic pattern in the ﬁnite element models of the ﬁlament wound disk. As can be seen from Fig. 9, shear stress distribution for the top layer of conventional 4-ply disk does not reﬂect the distinctive variation of magnitude that is predicted using 4-ply disk model with different number of mosaic units. For mosaic patterned models, the predicted srh values at different mosaic units ﬂuctuates over a wide range of magnitude, whereas conventional model does not capture this effect. For instance, the maximum and minimum values of srh for 8-unit model of 4-ply disk (see Fig. 9(iv)) are 70.9 MPa and 70.7 MPa,

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Fig. 10. Contour plots of rr and rh in the 8-unit disk with different number of plies.

respectively, whereas conventional ﬁnite element analysis predicts the corresponding values to be 10.2 MPa and 75.5 MPa as shown in Fig. 9(i). It clearly signals the susceptibility of the ﬁlament wound disk to the inﬂuence of mosaic pattern conﬁguration. Finite element analysis of mosaic patterned models of 8-ply and 14-ply disks also illustrates the signiﬁcant variation of stress values in various mosaic units. Contour plots of rr, rh and srh determined for the 8-unit model of both 8 and 14-ply disks are shown in Fig. 10. These stresses obtained from mosaic patterned models are

noticeably distinguishable from those estimated with the help of conventional modelling technique of the disk. Considerable ﬂuctuation of predicted levels of stresses calculated using mosaic patterned models not only clearly indicates the effect of mosaic pattern on stress distributions, but also emphasizes the necessity of incorporating the manufacturing characteristics in the modelling and analysis of ﬁlament wound structures using ﬁnite element technique. Stress distributions around circumference (0° 6 h 6 180°) have been plotted with values calculated at a radial distance of

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r

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(i)

(iii) 5.49x10

7

4.86x10

7

4.24x10

7

3.61x10

7

2.98x10

7

2.36x10

7

1.73x10

7

1.11x10

7

4.80x10

6

(ii)

(4-unit)

r -Zone A(8MP)

(6-unit)

(iv)

r -Zone B(8MP)

(4-unit)

(6-unit)

r - (CA)

r at TOP of Layer 4 (r = 0.23885)

0

30

60

(v)

90

120

150

180

(8-unit)

(vi)

(8-unit)

Fig. 11. Circumferential distributions of rr and rh for 4-ply disk.

0.23885 m for all the models of the disk. At this radial location, two mosaic units of different sizes are arranged circumferentially in alternating fashion. These units are denoted as ‘‘Zone A’’ and ‘‘Zone B’’ in the stress plots. As can be seen from Fig. 3, if the mosaic unit EBD and other units of the same size are denoted as ‘‘Zone A’’ then the unit DBA and other similar units will be denoted as ‘‘Zone B’’.

Distributions of both, radial (rr) and tangential (rh) stresses around circumference for conventional and three mosaic patterned models (4, 6 and 8-unit models) of the ﬁlament wound 4-ply disk (see Fig. 11) show that the stress values are substantially different from those obtained from conventional ﬁnite element analysis. The circumferential distributions of stresses predicted by ﬁnite

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(i)

(4-unit)

(ii)

2

(4-unit)

(iv)

2

(6-unit)

8

1.60x10

8

1.40x10

8

1.20x10

8

1.00x10

7

8.00x10

at TOP of Layer 8 (r = 0.23885) 7

6.00x10

0

30

60

90

1

150

180

(6-unit)

1

(Pa)

(iii)

120

(v)

1

(8-unit)

(vi)

2

(8-unit)

Fig. 12. Circumferential distributions of r1 and r2 for 8-ply disk.

element analysis of the 4-ply disk considering mosaic pattern conﬁguration maintain noticeably different levels of stress values at various locations within a mosaic unit compared to that calculated by conventional modelling technique. The pattern of the variation of rr and rh and also the levels of their values are

greatly inﬂuenced by the number of mosaic units around the circumference and the locations of mosaic units. As can be observed from Fig. 11, both radial and tangential stresses from mosaic patterned models follow various repeating patterns and magnitudes of stresses ﬂuctuate within a range of

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(i)

1

(ii)

2

Fig. 13. Effect of increasing number of mosaic units on circumferential stress distributions for 4-ply disk.

minimum and maximum values depending on the number of mosaic units. On the other hand, conventional technique predicts almost constant values of both rr and rh. The maximum levels of deviation of stress values obtained from 4-unit model (see Fig. 11(i and ii)) reach 65% for rr and 67% for rh with respect to those from conventional model. 8-unit model also yields this deviation of stress values that reach maximum 54% for rr and 69% for rh as shown in Fig. 11(v and vi), respectively. Circumferential distributions of stresses along (r1) and across (r2) ﬁbres obtained from ﬁnite element analysis of the ﬁlament wound disk based on the mosaic pattern conﬁguration also reﬂect the effect of structural characteristics on the stress distribution in different layers. As can be observed from Fig. 12, conventional ﬁnite element analysis technique predicts constant values of both r1 and r2 at the top layer of the 8-ply disk, whereas distributions of these stresses obtained from models with different number of mosaic units vary distinctively within mosaic units located at various positions. Depending on the number of mosaic units incorporated in the disk model, the level of maximum values of stresses changes slightly. For instance, the maximum values of r1 and r2 are obtained to be 155 MPa and 13 MPa for 4-unit model, 154 MPa and 15 MPa for 6-unit model and 151 MPa and 12 MPa for 8-unit model. However, conventional technique calculates the values of r1 and r2 to be 159 MPa and 10 MPa, respectively that are constant all over the selected circumferential location. With respect to results from conventional analysis, values of both r1 and r2 predicted by mosaic patterned models of the 8-ply disk offer different levels of variation. The maximum level of difference is found to be 52% for r1 and 42% for r2 which are estimated for 6-unit model of the disk as shown in Fig. 12(iii and iv). The effect of increasing mosaic units on the magnitude of stresses could be observed from the comparison of stress distributions obtained from conventional model as well as mosaic patterned (4, 6 and 8-unit) models. As can be seen from Fig. 13, the level of magnitudes of r1 and r2 from 4-ply disk model is decreasing with the increasing number of mosaic units around the circumference of the disk. However, the difference of values obtained from mosaic patterned models still exists compared to those determined using conventional technique. Variation of stresses along radial cross-section is shown for one orientated at 45° counter clockwise from +x axis of ﬁnite element models (see Fig. 6) of the ﬁlament wound disk. The ﬁrst unit of the mosaic pattern located in the selected radial cross-section

and connected to the inner radius has been denoted as ‘‘Zone 1’’ and other consecutive units have been numbered sequentially. Results obtained from conventional models of 4, 8 and 14-ply disks illustrate dissimilar distributions of rh compared to that from mosaic patterned models (see Fig. 14). Distribution of r2 along the selected radial cross-section of the conventional model follows a trend that is similar to that for mosaic patterned models. With regard to the results produced from 4, 8 and 14-ply conventional models, various levels of variation of stress values are determined using models with mosaic pattern conﬁguration. As can be observed from Fig. 14(i and ii), the maximum levels of variation in the magnitudes of rh and r2 from 4-ply disk reach 58% and 55%, respectively compared to the results from conventional solution. However, the levels of variation in values of rh and r2 gradually diminish with the increase of total number of plies, i.e., the thickness of the ﬁlament wound disk. The radial distribution of rh for 8ply disk approaches the values calculated using conventional technique and gets closer in case of 14-ply disk as shown in Fig. 14(iii and v). Similar trend is visualized also for the radial distribution of r2 shown in Fig. 14(iv and vi). It implies that the effect of mosaic pattern conﬁguration on the radial stress distributions in the ﬁlament wound ﬂywheel disk diminishes gradually with the increasing number of plies. Maximum levels of different stresses are considerably inﬂuenced by the incorporation of mosaic pattern conﬁguration. For instance, maximum values of stress along ﬁbres, r1 and circumferential stress, rh obtained from mosaic patterned models are compared to those from conventional models as shown in Fig. 15. As follows from the presented data, ﬁnite element analysis of conventional and mosaic patterned models provides diverse levels of maximum values of stresses depending on various number of plies and mosaic units around circumference. For 4, 8 and 14-ply models, each having 4,6 and 8 mosaic units around circumference, maximum values of both r1 and rh are substantially different from those obtained from conventional ﬁnite element analysis. For instance, the maximum value of r1 at top of layer 3 in 4-ply disk is determined to be 239 MPa from conventional model, whereas for 8-unit mosaic patterned model this value reaches 357 MPa as shown in Fig. 15(i). Similarly, 8 and 14-ply disk models with 8 mosaic units produce 27 and 17% higher values of r1 at the top of layer 7 and 13, respectively compared to the results obtained using conventional technique. Furthermore, maximum values of tangential stress, rh obtained from mosaic patterned models are observed

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(i)

(4-ply)

(ii)

(4-ply)

(iii)

(8-ply)

(iv)

(8-ply)

(14-ply)

(vi)

(14-ply)

(v)

Fig. 14. Radial distributions of rh and r2 for 4-unit disk with different number of plies.

to be higher compared to the values calculated using conventional models as shown in Fig. 15(ii) (e.g., for 4-ply disk, rh = 328 MPa (4unit), 326 MPa (6-unit) and 325 MPa (8-unit) at top of layer 3; for conventional model, rh = 236 MPa). In addition, the maximum values of rr and srh from mosaic patterned models are higher than

those determined by conventional ﬁnite element analysis (e.g., for 8-ply disk, rr = 103 MPa (4-unit), 102 MPa (6 and 8-unit) at top of layer 7; for conventional model, rr = 71 MPa). Both of conventional and mosaic patterned models provide close values of r2 and s12 in different plies (e.g., for 14-ply disk, r2 = 12 MPa (4,

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4-ply disk

200.0

8-ply disk

334

325 302

321

326 298

284 236 236

239 240

300.0

328 293 310

335

348

357 303

322 299

350.0

σθ (MPa)

σ1 (MPa)

300.0

304 320

288

350.0

250.0

358

400.0

250.0

4-ply disk

200.0

8-ply disk 14-ply disk

14-ply disk

150.0

150.0

100.0

100.0

50.0

50.0 0.0

0.0 Conventional

4-unit

6-unit

8-unit

Conventional

4-unit

6-unit

8-unit

(ii)

(i)

Fig. 15. Comparison of maximum values of r1 and rh at top of layer 3(4-ply), layer 7(8-ply) and layer 13(14-ply) of different models of the disk.

and

(Pa)

(i) Radial distributions of

(ii) Circumferential distributions of

and

Fig. 16. Radial and circumferential distributions of rh and r2 for 14-ply disk (8-unit).

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6 and 8-unit) at top of layer 13; for conventional model, r2 = 11 MPa). Distributions of stresses and the comparison of maximum values provide an indication that the level of stresses in the ﬁlament wound composite spinning disk could be underestimated to a certain extent in some cases by the general stress analysis based on the conventional mechanics of laminated structures that does not take into account the ﬁlament winding mosaic pattern of the ±/ angle-ply layers. The effect of ﬁlament winding mosaic pattern of the ±/ angle-ply layers is most pronounced for ﬁlament wound structures having comparatively lower thickness [16]. Multilayered laminated shells with a large number of angle-ply layers experience the effect of mosaic pattern to a lesser degree. The stretching-twisting and bending-shear coupling coefﬁcients for multilayered laminates are calculated as follows [2]:

1 C mn ¼ Amn hd 2

ð9Þ

where h is the laminate thickness, d the ply thickness, and Amn are materials stiffness coefﬁcients. If the thickness of the single ply, d is small compared to the overall thickness of a laminate composed of large number of ±/ angle-ply layers, then the effect of the ﬁlament winding mosaic pattern is not very pronounced [16]. Radial distributions of stresses from 14-ply disk model of the ﬁlament wound disk illustrate the diminishing effect of the ﬁlament winding mosaic pattern on the stress state in different plies of the disk. As can be seen from Fig. 16(i), both conventional and mosaic patterned models produce very close values of both rh and r2. It is also true for radial distributions of other stresses. Circumferential distributions of stresses obtained from mosaic patterned models of 14-ply disk maintain various levels of difference with respect to the magnitudes of stresses calculated using conventional ﬁnite element analysis technique. As can be observed from Fig. 16(ii), distributions of both rh and r2 follow dissimilar trend and values at different points of the selected circumferential location maintain 4–15% difference compared to the values determined from conventional model. Thus, the effect of the ﬁlament winding pattern conﬁguration would be diminished gradually with the increasing thickness of the ﬁlament wound shell. The effect of ﬁlament winding mosaic pattern is also dependent on the positioning of the angle-ply layers in the ﬁlament wound laminate. If the curved triangular units with the same structure are placed one on top of the other at the time of winding, then the effect of mosaic pattern could propagate through the overall thickness of the ﬁlament wound laminate and transmit to the whole ﬁlament wound shell. In the other way, stacking of the triangular mosaic units with the opposite structure ([±/] and [/]) results in a symmetric laminate. The stretching-twisting and bending-shear coupling effects would balance each other inducing, however, additional interlaminar stresses [16]. Hence, ﬁlament winding process requires special adjustments to maintain the regular structure of laminated composite material across the thickness of the ﬁlament wound shells. Otherwise, the positions of consecutive ﬁlament wound ±/ angle-ply layers would be shifted arbitrarily and produce some uncertainty for the modelling and estimation of stress state in ﬁlament wound structures where the effect of mosaic pattern could not be overlooked. 6. Conclusions The ﬁnite element modelling and analysis of the ﬁlament wound composite ﬂywheel disk have been presented in this paper to demonstrate the effect of mosaic pattern created during a ﬁlament winding process on the stress distributions and effectively on the strength of the structure. The inﬂuence of mosaic pattern

architecture of ﬁlament wound angle-ply laminates has been presented using the ﬁnite element modelling, more accurately reﬂecting the real structure of the ﬁlament wound composite shell composed of the interlaced plies. It has been illustrated that the variation of calculated stresses in different layers of the spinning composite disk obtained from ﬁnite element analysis of the mosaic patterned models with different levels of thickness differs substantially from the stresses determined by conventional analysis of the laminated shells. The levels of variation in predicted stress values from both conventional and mosaic patterned models ﬂuctuate substantially among different mosaic units located at varied positions all over the ﬁlament wound ﬂywheel disk. It implies that the mechanical behaviour of the disk is sensitive to the mosaic pattern conﬁguration and stress distributions are affected by the number of mosaic units around circumference and along radius together with the level of thickness of the disk. With the increment of the level of thickness, the effect of mosaic pattern on the stress state gradually diminishes. However, it can be stated that the level of maximum stresses in the ﬁlament wound disks having low thickness could be underestimated by conventional ﬁnite element analysis technique. The obtained results clearly show the sensitivity of the mechanical behaviour of ﬁlament wound spinning composite ﬂywheel disk and the effect of mosaic pattern on the stress distributions in different plies. It also demonstrates the necessity of incorporating the actual thickness variation induced in the ﬁlament wound spinning disk for ﬁnite element modelling and analysis. Actual stress distributions could remain undetermined if the mosaic pattern as well as actual level of thickness is disregarded. The methodology proposed in this work would be helpful in designing ﬁlament wound rotating disks with radially varying ﬁbre orientation angles. References [1] Kyser AC. The uniform-stress spinning ﬁlamentary disk. National Aeronautics and Space Administration (NASA); 1964. [2] Vasiliev VV, Morozov EV. Advanced mechanics of composite materials. 2nd ed. Elsevier; 2007. [3] Abdul-Aziz A, Baaklini G, Trudell J. Structural analysis of composite ﬂywheels: an integrated NDE and FEM approach. National Aeronautics and Space Administration (NASA); 2001. [4] Ashley S. Flywheels put a new spin on electric vehicles. Mech Eng 1993;115:44–51. [5] Arnold SM, Saleeb AF, Al-Zoubi NR. Deformation and life analysis of composite ﬂywheel disk systems. Composites: Part B 2002;33:433–59. [6] Arslan MA. Flywheel geometry design for improved energy storage using ﬁnite element analysis. Mater Des 2008;29:514–8. [7] Pérez-Aparicio JL, Ripoll L. Exact, integrated and complete solutions for composite ﬂywheels. Compos Struct 2011;93:1404–15. [8] Christopher DA, Beach R. Flywheel technology development program for aerospace applications. IEEE AES Syst Mag 1998;25:9–14. [9] Jefferson CM, Ackerman M. Flywheel variator energy storage. Energy Convers Manage 1996;37:1481–91. [10] Suzuki Y, Koyanagi A, Kobayashi M, Shimada R. Novel applications of the ﬂywheel energy storage system. Energy 2005;30:2128–43. [11] Bokov YV, Vasil’ev VV, Portnov GG. Optimum shapes and paths of reinforcement in rotating composite shells. Mech Compos Mater 1982;17:572–9. [12] Seleznev LN, Portnov GG. Chord winding of composite-tape disks. Mech Compos Mater 1977;13:832–5. [13] Portnov GG, Kustova IA. Energy capacity of composite ﬂywheels with continuous chord winding. Mech Compos Mater 1988;24:688–94. [14] Valiullin AK, Mukhambetzhanov SG, Antsilevich YG, Golovanov AI. Finite element calculation of elements of a composite ﬂywheel. Strength Mater 1987;19:738–44. [15] Dems K, Turant J. Two approaches to the optimal design of composite ﬂywheels. Eng Optimiz 2009;41:351–63. [16] Morozov EV. The effect of ﬁlament-winding mosaic patterns on the strength of thin-walled composite shells. Compos Struct 2006;76:123–9. [17] E.V. Morozov, M. Hoarau, K.E. Morozov, Filament-winding patterns and stress analysis of thin-walled composite cylinders, in: Proceedings of ﬁber society 2003 symposium, Loughborough University, Loughborough, UK, 2003. [18] D. Jensen, S. Pai, Theoretical sensitivity of composite cylinders in compression to ﬁlament-winding pattern, in: The 9th international conference on composite materials (ICCM-9), Madrid, Spain, 1993, pp. 447–454.

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