Displacement and Stress Fields in a Functionally Graded Fiber- Reinforced Rotating Disk With Nonuniform Thickness and Variable Angular Velocity

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1 Y. Zheng Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 025 H. Bahaloo Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 025 D. Mousanezhad Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 025 A. Vaziri Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 025 H. Nayeb-Hashemi Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA Displacement and Stress Fields in a Functionally Graded Fiber- Reinforced Rotating Disk With Nonuniform Thickness and Variable Angular Velocity Displacement and stress fields in a functionally graded (FG) fiber-reinforced rotating disk of nonuniform thickness subjected to angular deceleration are obtained. The disk has a central hole, which is assumed to be mounted on a rotating shaft. Unidirectional fibers are considered to be circumferentially distributed within the disk with a variable volume fraction along the radius. The governing equations for displacement and stress fields are derived and solved using finite difference method. The results show that for disks with fiber rich at the outer radius, the displacement field is lower in radial direction but higher in circumferential direction compared to the disk with the fiber rich at the inner radius. The circumferential stress value at the outer radius is substantially higher for disk with fiber rich at the outer radius compared to the disk with the fiber rich at the inner radius. It is also observed a considerable amount of compressive stress developed in the radial direction in a region close to the outer radius. These compressive stresses may prevent any crack growth in the circumferential direction of such disks. For disks with fiber rich at the inner radius, the presence of fibers results in minimal changes in the displacement and stress fields when compared to a homogenous disk made from the matrix material. In addition, we concluded that disk deceleration has no effect on the radial and hoop stresses. However, deceleration will affect the shear stress. Tsai Wu failure criterion is evaluated for decelerating disks. For disks with fiber rich at the inner radius, the failure is initiated between inner and outer radii. However, for disks with fiber rich at the outer radius, the failure location depends on the fiber distribution. [DOI: 0.5/ ] Keywords: functionally graded material, rotating disk, variable angular velocity, nonuniform thickness, finite difference method, fiber-reinforced composite material Introduction Functionally graded materials are a new class of composite materials used as an alternative to traditional materials in engineering devices. These materials provide more resistance to crack initiation and propagation, and have a higher strength-to-weight ratio compared to their homogenous counterparts [ 4]. Unidirectional fiber-reinforced functionally graded (FG) materials are now being considered in the design of many rotating disks. These disks could be manufactured by filament winding techniques and using tapes with different fiber volume fractions. Disks with a high fiber volume fraction at their outer radius may result in a reduced radial displacement and thus any seizure against their stationary hub. There are a number of investigations toward understanding inplane and out-of-plane behaviors of FG rotating disks made of an isotropic material [5 9]. C allioglu et al. [6] obtained closed-form solutions for stress field in a FG rotating disk with constant angular velocity. It was found that lower radial displacement is obtained by increasing elastic modulus from inner radius to the outer radius. In contrast, both radial and hoop stresses increase by increasing elastic modulus from inner radius to the outer radius. Dai and Dai [7] studied stress field in a decelerating FG disk with a uniform thickness using the Runge Kutta method. They Corresponding author. Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received October 27, 206; final manuscript received February 20, 207; published online April 9, 207. Assoc. Editor: Vikas Tomar. observed the existence of shear stress and showed that stresses decrease when the gradient index increases. Bayat et al. [8,9] obtained solutions for stress field in a FG rotating disk with nonuniform thickness for free free and fixed-free boundary conditions. They found that a FG rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. Horgan and Chan [9] found that for an inhomogeneous disk, the location of maximum radial and hoop stresses shift compared to the homogeneous disk problem. Similar investigations have been carried out by Damircheli and Azadi [0], Hassani et al. [], Afsar et al. [2,3], and Go et al. [3], to obtain stress field in FG rotating disks under thermal loads. Also, vibrations of FG disks have been studied by numerous researchers [4 8]. Asghari and Ghafoori [20] obtained semi-analytical three-dimensional solutions for thick FG rotating disks by modifying the two-dimensional planestress solutions which had been previously proposed for thin FG rotating disks. Kadkhodayan and Golmakani [2] conducted a nonlinear bending analysis on an FG rotating disk and presented a parametric study on the effects of material gradient index, angular velocity, and disk geometry on the deformation pattern. Fiber-reinforced composites are a class of FG materials that utilize the combined stiffness of fiber and flexibility of the matrix to achieve higher performance compared to homogenous materials. The mechanical behavior of anisotropic rotating disks has been the subject of a number of studies. Tang [22] obtained closedform solutions for elastic stress field in rotating disks with a uniform thickness and different boundary conditions and gave Journal of Engineering Materials and Technology JULY 207, Vol. 39 / Copyright VC 207 by ASME

2 numerical examples. Peng and Li [23] presented an alternative method, which transforms the arbitrarily variable-gradient problem into a Fredholm integral equation, to investigate the effect of orthotropic material properties on stress field in FG rotating disks and verified the method with numerical results. Murthy and Sherbourne [24] developed an analytical solution for annular disks with a nonuniform thickness for different boundary conditions and gave numerical examples of the stress behavior. Reddy and Srinath [25] presented closed-form solutions for displacement and stress fields in a FG rotating disk with a nonuniform thickness and density. It is found that the stresses and displacement are lower when the mass density increases radially. Tahani et al. [26] developed a semi-analytical method to analyze displacement and stress fields in circumferentially fiber-reinforced composite rotating disks. Alexandrova and Vila Real [27] presented a mathematical model for displacement and stress fields in elastic perfectly plastic anisotropic annular disks. Sayer et al. [28] performed thermoelastic stress analysis on a thermoplastic curvilinearly fiberreinforced composite disk. They found that disks under linearly changing temperature produce lower stresses than under uniform temperature. To the best of our knowledge, there is no previous research on displacement and stress fields in FG fiber-reinforced transversely isotropic decelerating tapered disks in the literature. This paper presents displacement and stress fields of such disks for two cases of fiber volume fraction distribution: fiber-rich at inner radius and fiber-rich at the outer radius. Furthermore, Tsai Wu failure criterion for these disks is evaluated. 2 Material Properties Figure shows a schematic diagram of the disk with inner radius, a, and outer radius b, rotating around the z-axis with an angular velocity, x, and an angular deceleration, _x. The disk is assumed to have a stress-reducing thickness profile in the form of a=r þ b, where a and b are constant parameters, and r is the radial coordinate [29] and is made of a unidirectional fiber-matrix composite material with fibers oriented in the circumferential direction, Fig. 2. Based on these assumptions, material properties at any location in circumferential (i.e., direction ) and radial (i.e., direction 2) directions are obtained by using the rule of mixture, as follows: E ¼ E f V f þ E m V m () E f E m ¼ (2) E f V m þ E m V f t 2 ¼ t f V f þ t m V m (3) G 2 ¼ G f G m G f V m þ G m V f (5) q ¼ q f V f þ q m V m (6) where E,, 2, and 2 are directional elastic moduli and Poisson s ratios of the composite, and G 2 and q are shear modulus and mass density. Moreover, E f, G f, q f, and f are elastic modulus, shear modulus, mass density, and Poisson s ratio of the fiber, while E m, G m, q m, and m are the corresponding values for the matrix. V f and V m are volume fractions for fiber and matrix, respectively, and are related by V f þ V m ¼ (7) Two different types of fiber-reinforced composites were investigated in this work. The first one is fiber-rich composites at the outer radius, while the second one is fiber-rich composites at the inner radius. Here again, the disks are assumed to be manufactured by filament winding techniques. Therefore, all fibers are assumed to be circumferentially located within the disks, Fig. 2. For each case, the fiber volume fraction at any location is assumed to be a power-law function of the radial coordinate, r, as follows: V f ¼ r a n ; n 0 (8a) b a V f 2 ¼ b r n ; n 0 (8b) b a where subscripts and 2 represent cases and 2, and n is the gradient index representing the level of material gradation. For n ¼ 0, V f becomes equal to unity, which means that the disk is made of fiber material (i.e., V m ¼ 0); while for other values of n, the disk is made of a fiber-reinforced composite material (i.e., FG material). 3 Governing Equations for In-Plane Deformation and Stress Fields Figure 3 shows an infinitesimal element of the disk represented in a polar coordinate system, where r and h are radial (i.e., direction 2) and circumferential (i.e., direction ) coordinates. The equilibrium equations for the element with a variable thickness, hðrþ, can be presented r rrhðþr r hr ðþ r hhhðþ¼ q r ðþx r ½ ðþ t Š 2 r 2 hr ðþ (9) t 2 ¼ðt f V f þ t m V m Þ =E s rhhðþr r Šþ s rh hr ðþ¼ qðþr r 2 hr (0) where r rr and r hh are normal stresses in radial and circumferential directions, while s rh is the in-plane shear stress, and time is denoted by t. Since the thickness/diameter ratio of the disk is far less than one, it is assumed that the disk is under plane-stress condition, thus the stress strain relations can be presented as r rr ¼ e rr þ t 2E e hh () t 2 t 2 t 2 t 2 r hh ¼ t 2 e rr þ e hh (2) t 2 t 2 t 2 t 2 E Fig. Schematic diagram of tapered disk with nonuniform thickness, where a and b are the inner and outer radii, respectively, and the disk is rotating around the z-axis by a variable angular velocity s rh ¼ G 2 c rh (3) where e rr and e hh are normal strains in radial and circumferential directions, while c rh is the in-plane shear strain. The strain fields / Vol. 39, JULY 207 Transactions of the ASME

3 can be obtained from the deformation field by using the linearized von Karman theory, as e rr ðr; tþ rðr; e hh ðr; tþ ¼ u rðr; tþ r h ðr; tþ (4) (5) c rh r ðr; tþ hðr; tþ u hðr; tþ r where u r and u h are the element s displacement in radial and circumferential directions, respectively. For an axisymmetric disk geometry and material properties, displacement fields are function of r only. Equations (4) (6) are then simplified to e rr ðr; tþ rðr; e hh r; t ð Þ ¼ u rðr; tþ r c rh hðr; tþ u hðr; r Substituting Eqs. (7) (9) into Eqs. () (3) will result in r rr r t 2 t þ t 2E u r t 2 t 2 r r hh ¼ t r t 2 t þ E u r t 2 t 2 h s rh ¼ G u h r (7) (8) (9) (20) (2) (22) If the disk is fixed at the inner edge and free at the outer radius, the boundary conditions can be presented as u r ðaþ ¼0; r rr ðbþ ¼0 (23) u h ðaþ ¼0; s rh ðbþ ¼0 (24) Substituting Eqs. (20) (22) into the governing (i.e., Eqs. (9) and (0)) will result 2 u 2 r dr þ 2E r þ drh ð Þ rhdr 2 r 2 de þ u r r dr 2E r 2 þ 2E drh ð Þ hr 2 dr E r 2 ¼ qx2 rð 2 2 Þ 2 u 2 drh ð Þ þ u h drh ð Þ hrdr hr 2 dr ¼ qr dx G 2 dt (26) The angular velocity of the disk is assumed to be, xðtþ ¼x 0 e kt, where x 0 and k are two constant parameters [7]. Employing the method of separation of variables, the displacement field is represented as u r ðr; tþ ¼R r ðrþk r ðtþ (27) u h ðr; tþ ¼R h ðrþk h ðtþ (28) where K r ðtþ ¼e 2kt, K h ðtþ ¼e kt, and R r ðrþ and R h ðrþ are two unknown functions of the radial coordinate, r, to be determined. Substituting Eqs. (27) and (28) into Eqs. (25) and (26) will result in the following set of ordinary differential equations for R r ðrþ and R h ðrþ: where R 00 r ðrþþf ðrþr 0 r ðrþþf 2ðrÞR r ðrþ ¼f 3 ðrþ (29) R 00 h ðrþþf 4ðrÞR 0 h ðrþþf 5ðrÞR h ðrþ ¼f 6 ðrþ (30) f ¼ d dr þ t 2E r þ drh ð Þ rhdr t 2 r (3a) f 2 ¼ t 2dE r dr t 2E r 2 þ t 2E drh ð Þ hr 2 dr E r 2 f 3 ¼ qx 0 2 rð t 2 t 2 Þ (3b) (3c) Fig. 2 Schematic diagram of unidirectional fiber-matrix composite, where the fibers are aligned with direction (circumferential direction), and direction 2 (radial direction) is perpendicular to direction Fig. 3. An element of the disk with all in-plane tractions presented in a polar coordinate system located at the center of the disk, where r and h are the radial and angular coordinates Journal of Engineering Materials and Technology JULY 207, Vol. 39 /

4 f 4 ¼ drh ð Þ hrdr f 5 ¼ drh ð Þ hr 2 dr (3d) (3e) f 6 ¼ kx 0qr G 2 (3f ) Furthermore, using Eqs. (27) and (28), the boundary conditions, Eqs. (23) and (24), can be presented as R r ðaþ ¼0 R 0 r t 2 t ðbþþ t 2E R r ðþ b 2 t 2 t 2 b G 2 R h ðaþ ¼0 R 0 h ðbþ b R hðbþ ¼ 0 ¼ 0 (32a) (32b) (33a) (33b) 4 Finite Difference Method and Boundary Conditions Finite difference method was utilized to solve the governing equations with the boundary conditions given in Eqs. (29) (33). The radial distance, b a, was divided into p segments, pdr ¼ b a. The optimum mesh size, Dr, for each simulation depended on the fiber distribution pattern. A mesh size corresponding to p > 3000 was satisfactory for all simulations. For this reason, we selected p ¼ 4000 in order to be conservative. A mesh sensitivity analysis was performed to ensure that the results were independent of the selected mesh size. No exhaustive effort was made to find optimum mesh size for each case. The finite difference form of Eq. (29) can be presented as! R m ðdrþ 2 þ R m 2 ðdrþ 2 f! Dr þ f 2 þ R mþ ðdrþ 2 þ f! ¼ f 3 Dr (34) where R m ¼ R r ðr m Þ, r m ¼ a þðm ÞDr, and m is an integer varying between [R ¼ R r ðaþ] and p þ [R pþ ¼ R r ðbþ]. The finite difference form of the boundary conditions (Eq. (32)) can be also presented as R pþ R p t 2 t 2 Dr R ¼ 0 (35a) þ t 2E t 2 t 2 R pþ b ¼ 0 (35b) Using the above formulations, Eqs. (34) and (35) can be put into a matrix form as ½AŠ ðpþþðpþþ ½R r Š ðpþþ ¼½BŠ ðpþþ (36) where [A] and [B] are known matrices. Equation (36) was solved to evaluate [R r ], using MATLAB VR. R h can be evaluated by following the same process of Eqs. (34) (36). The only difference is to use Eqs. (30) and (33). The results were then postprocessed to evaluate displacement and stress fields for disks with identical thickness profiles and angular decelerations but different fiber volume fractions. All parameters used in numerical calculations are presented in Table (unless otherwise stated). 5 Results and Discussion In this study, the disk thickness profile is taken as, a=r þ b, where a ¼ 0:00075 m 2, b ¼ 0:0025 m, and a r b. This thickness profile results in a disk having the same volume as of a disk with uniform thickness of 0.0 m. The results are presented for the time at which the disk is at the beginning of deceleration (i.e., at t ¼ 0), and therefore, subjected to the maximum stress field. We first validated our numerical results by comparing them with the analytical closed-form solutions presented in the literature for the case of uniform thickness, constant fiber volume fraction, and constant angular velocity. Our results were in perfect agreement with the closed-form solutions presented by Tang [22], as shown in Fig. 4. Here, the radial displacement, radial stress, and hoop stress are normalized as u r ¼ u r E =r 0, r r ¼ r r =r 0, and r h ¼ r h =r 0, where E ¼ 0:3E f þ 0:7E m ¼ 24:345 GPa, and r 0 ¼ qx 2 0 b 2 ¼ð0:3q f þ 0:7q m Þx 2 0 b 2 ¼ 234 kpa. Furthermore, we compared some of our results with the finite element results by using ABAQUS VR finite element codes. Finite element analyses were performed using axisymmetric elements for FG disks with variable thickness under constant angular velocity. Figure 5 shows the radial displacement, radial stress, and circumferential stress for FG tapered disk with fiber-rich at the outer surface and gradient index n ¼ 5. The results show that our numerical solution is in close agreement with the finite element results. However, since our material was an FG material, finite element simulations were much more exhaustive and required further mesh refinements to converge to the correct stress field. Since finite element formulation is based on the assumed displacement field, it may not satisfy equilibrium condition and may not converge to the correct stress field, especially if there is a sharp gradient in the structure. Our numerical solution is based on the satisfying equilibrium condition and thus is much more efficient algorithm for finding the correct stress field. Our validated numerical method can now be extended to further study displacement and stress fields in FG fiber-reinforced rotating disks with a nonuniform thickness and variable angular velocity. Figure 6 shows the variations of fiber volume fraction along the radial direction (i.e., Eqs. (8a) and (8b), for different values of gradient index n, confirming the concentration of fiber density at the outer and inner radii for cases and 2 of FG fiber-reinforced disks, where the fiber concentration becomes more pronounced for greater values of n. Figures 7 show the distribution of radial and circumferential displacements, u r and u h, and radial, circumferential, and shear stresses, r rr, r hh, and s rh, as functions of the normalized radial coordinate, (r a)/(b a) or(b r)/(b a), for different values of the gradient index, n, for disks with fiber-rich at the outer radius [part (a)], and fiber-rich at the inner radius [part (b)]. From Figs. 7 8, it can be seen that for both cases of fiber rich at outer and inner radii, a smaller gradient index n leads to a reduced displacement field due to having over all more fibers in the disk. The composite with fiber rich at outer radius (Figs. 7(a) and 8(a)) Table Disk material properties, geometrical characteristics, gradient index, and angular velocity and deceleration used for numerical calculations [30,3] Elastic modulus of fiber E f 73. GPa Elastic modulus of matrix E m 3.45 GPa Mass density of fiber q f 2550 kg=m 3 Mass density of matrix q m 2250 kg=m 3 Poisson s ratio of fiber f 0.22 Poisson s ratio of fiber m 0.35 Shear modulus of fiber G f GPa Shear modulus of matrix G m GPa Inner radius a 0.02 m Outer radius b 0. m Deceleration factor k 0.5 Angular velocity before deceleration x 0 00 rad/s / Vol. 39, JULY 207 Transactions of the ASME

5 Fig. 4. Comparison between our FDM results and analytical solutions by Tang [22], presented for radial displacement, and radial and hoop stresses versus normalized radial coordinate exhibits lower displacement field in the radial direction but a higher one in the circumferential direction, compared to the disk with fiber rich at inner radius. These results could be justified considering that the fiber rich zone at the outer radius prevents radial displacement of the disk. However, fiber rich zone is bonded to the soft matrix material and disk deceleration results in a tangential force that will simply move this fiber rich layer in the circumferential direction as a rigid body. This fiber rich layer may not provide any resistance to the tangential displacement. While in the case of disk with fiber rich at inner radius, the circumferential displacement is significantly lower with the same values of n, because the inner radius is fixed on the shaft and the fiber rich zone is substantially stiffer than the matrix, thus resisting circumferential displacement, Fig. 8(b). With a smaller value of n, the width of the fiber rich zone increases, enhancing its resistance to tangential displacement and therefore leads to a lower circumferential displacement field. Our results in Fig. 9(a) show that in contrast to a homogenous disk with pure tensile radial stress at any point, compressive radial stresses can be developed near the outer radius of the disks with fiber-rich at the outer radius. This compressive radial stress can prevent the propagation of circumferential cracks from initiating at the outer radius of the disk. In contrast to radial stress, the hoop stress for this type of FG disks is higher compared to that of a homogenous disk, Fig. 0(a). However, FG disks in this region that has more fibers are much stronger than homogenous disks. For disks with fiber-rich at the inner radius, the radial stress is lower in the region close to the inner radius compared to the homogenous disk, Fig. 9(b). The Fig. 6 (a) and (b) Fiber volume fraction distribution along the disk radius for disks with a fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume radial stress distribution then approaches to that of homogeneous disk for all gradient indices. Furthermore, for disks with fiber-rich at the inner radius and gradient index less than n ¼ 5, the hoop stress is lower at the outer radius compared to the homogenous disk. This will provide a mechanism to mitigate crack initiation and growth in the radial direction, Fig. 0(b). Fig. 5 (a) (c) Comparison between our FDM results and finite element analysis results, presented for radial displacement, radial stress, and hoop stresses versus normalized radial coordinate Journal of Engineering Materials and Technology JULY 207, Vol. 39 /

6 For the disk deceleration and geometry used in this study, the shear stress is fairly low in magnitude and tends to coincide with the shear stress in a homogeneous disk, Fig.. Moreover, the material gradient index has little effect on the shear stress distribution. In order to understand the effect of fiber distribution on the disk failure, Tsai Wu failure criterion was adopted in this study [32]. For a disk under plane-stress condition, Tsai Wu failure criterion is expressed as follows: F 2 ¼ Y t þ Y c (40) F 22 ¼ Y t Y c (4) F 66 ¼ S 2 (42) F r rr þ F 2 r hh þ F r rr 2 þ F 22 r hh 2 þ F 66 s rh 2 þ 2F 2 r rr r hh ¼ q (37) F 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi F F 22 2 (43) where F i and F ij are strength tensors and are defined as [32]: Composite is assumed failed when Tsai Wu index q F ¼ X t þ X c (38) F ¼ X t X c (39) where X t ; X c ; Y t ; Y c are tensile and compressive strength of unidirectional composite in h and r directions, and S is the shear strength. These strengths strongly depend on the fiber volume fraction. The tensile strength, X t, in the fiber direction (i.e., circumferential direction) can be defined as X t ¼ V f þ E m E f V m r f (44) Fig. 7 (a) and (b) Radial displacement along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume Fig. 8 (a) and (b) Circumferential displacement along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume / Vol. 39, JULY 207 Transactions of the ASME

7 where r f is the tensile strength of the fiber. The compressive strength in the fiber direction, X c, assuming the transverse fiber buckling mode is given as [33] X c ¼ 2 V f þ ð V f Þ E s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m V f E m E f E f 3 ð V f Þ (45) ðr m Þ yield Y t ¼ 0 t f þ t m E f E m þ V B m A E m E f V f (47) whereas in the case of fiber buckling in the shear mode, it is given as [33] X c ¼ 0:63G m V f (46) where G m is the shear modulus of the matrix material. In this paper, we assumed fiber buckling in the shear mode is the dominate mechanism when composite is subjected to compression in the direction of fibers and used Eq. (46) to evaluate X c : It has been shown fiber buckling in the shear mode is the dominant mechanism when fiber volume fraction is greater than 20% [33]. The composite tensile strength normal to the fiber direction (i.e., radial direction), Y t, assuming that interfacial strength is stronger than the matrix strength (based on matrix yielding) is presented as [34] where ðr m Þ yield is the matrix yield strength. Moreover, it can be shown that the compressive strength normal to the fiber direction, Y c, can be obtained from Eq. (47). However, we used compressive strength of matrix from Table 2 to evaluate Y c. Finally, the shear strength of the composite assuming matrix yielding can be presented as [33] S ¼ 0:5ðr m Þ yield V m þ G f V f (48) G m where G f is shear modulus of the matrix. The Tsai Wu failure criterion is evaluated for disks with various fiber volume fraction distributions (i.e., fiber-rich at the outer and inner radii). Table 2 summarizes the yield strength values used to evaluate the Tsai Wu failure in this study. For the time at which the disk is at the beginning of deceleration (i.e., at t ¼ 0), Fig. 2 shows that Tsai Wu failure criterion is Fig. 9 (a) and (b) Radial Stress along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume Fig. 0 (a) and (b) Circumferential stress along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume Journal of Engineering Materials and Technology JULY 207, Vol. 39 /

8 Table 2 Yield strengths of fiber and matrix used to evaluate the Tsai Wu failure criterion [35,36] Tensile strength of fiber (E-glass) r ft 3445 MPa Compressive strength of fiber (E-glass) r fc 080 MPa Tensile strength of matrix (epoxy resin) r mt 85 MPa Compressive strength of matrix (epoxy resin) r mc 90 MPa mostly dominated by radial and hoop stresses, while the shear stress has little effect on the disk failure. Figure 3 shows the Tsai Wu failure criterion for disks with various angular decelerations (i.e., various k values). The results show that Tsai Wu failure index is weakly dependent on the disk deceleration. These results can be justified by the fact that both radial and hoop stresses merely depend on the disk angular velocity. In contrast, the shear stress strongly depends on the disk angular deceleration, Fig. 4. For all cases, the maximum shear stress is located at the inner radius, and its value increases with an increase in disk deceleration parameter, k. For disks with fiber rich at the inner radius, failure always initiated between inner and outer radii for any value of gradient index. In contrast, for disks with the fiber rich at the outer radius, failure location depends on the gradient index. To be specific, for disks Fig. 2 (a) and (b) Tsai Wu failure criterion, along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n Fig. (a) and (b) Shear stress along the disk radius for disks with fiber-rich at the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with the same volume Fig. 3 The effect of disk deceleration on the Tsai Wu failure index for disks with fiber-rich at the outer radius and gradient index, n / Vol. 39, JULY 207 Transactions of the ASME

9 with more reinforcement, n, the incipient of the failure is located at the inner region of the disk. It shifts to the outer radius for n >. Comparing results for disks with fiber-rich at the inner radius with those with fiber-rich at the outer radius, it can be concluded that the latter case with n is more preferable in design. 6 Conclusions Displacement and stress fields in a decelerating FG fiberreinforced disk with a specific tapered thickness profile are obtained numerically. For a disk with fiber-rich at the outer radius, radial displacement, circumferential displacement, and radial stress are significantly lower compared to the homogenous disk with the same volume. In contrast to a homogenous disk, the distribution of radial stress is not purely tensile since it becomes compressive at the outer radius at higher gradient indices. However, the hoop stress in such disks is higher compared to that of the homogenous counterpart. A significant increase in hoop stress at the outer radius of the disk is observed at higher gradient index. However, overall composite in this region is much stronger than the matrix. On the other hand, for a disk with fiber-rich at the inner radius, radial and circumferential displacements increase with an increase in the gradient index, and they approach to those of homogenous disk. The hoop stress again is higher compared to the homogenous disk. For disks with any fiber distribution, both the maximum hoop stress and radial stresses are independent of the disk angular deceleration. In contrast, shear stress strongly depends on the disk angular deceleration. Furthermore, for the disk geometry and properties investigated, the shear stress was little affected by the disk gradient index. Finally, a Tsai Wu based failure criterion was developed to understand the effect of fiber distribution on the disk failure. The results show that for fiber-rich at the outer radius, by increasing the gradient index, failure location shifts from inner radius of the disk toward the outer radius. However, for a disk with fiber-rich at the inner radius, failure location is always between the inner and outer radii. Furthermore, for the range of angular deceleration studied, the Tsai Wu failure index is little affected by the disk angular deceleration. Comparing results for disks with fiber-rich at inner and fiber-rich at the outer radius, one may conclude it is preferable to manufacture composite disks with fiber-rich at the outer radius and gradient index of less than unity. Such a disk may prevent any interaction of the disk with stationary hub and thus any seizure. Acknowledgment This research was made possible by a NPRP award (NPRP ) from the Qatar National Research Fund (a member of the Qatar Foundation). The statements herein are solely the responsibility of the authors. Fig. 4 (a) (c) Maximum radial, hoop, and shear stress distributions for disks under various angular deceleration References [] Suresh, S., and Mortensen, A., 998, Fundamentals of Functionally Graded Materials (Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites), IOM Communications Ltd., London. [2] Miyamoto, Y., Kaysser, W. A., Rabin, B. H., Kawasaki, A. and Ford, R. G., 999, Functionally Graded Materials: Design, Processing and Applications, Springer, New York. [3] Mousanezhad, D., Ghosh, R., Ajdari, A., Hamouda, A. M. S., Nayeb-Hashemi, H. and Vaziri, A., 204, Impact Resistance and Energy Absorption of Regular and Functionally Graded Hexagonal Honeycombs With Cell Wall Material Strain Hardening, Int. J. Mech. Sci., 89, pp [4] Ajdari, A., Nayeb-Hashemi, H., and Vaziri, A., 20, Dynamic Crushing and Energy Absorption of Regular, Irregular and Functionally Graded Cellular Structures, Int. J. Solids Struct., 48(3 4), pp [5] Durodola, J. F., and Adlington, J. E., 997, Functionally Graded Material Properties for Disks and Rotors, Key Eng. Mater., 27 3, pp [6] C allioglu, H., Bektaş, N. B., and Sayer, M., 20, Stress Analysis of Functionally Graded Rotating Discs: Analytical and Numerical Solutions, Acta Mech. Sin., 27(6), pp [7] Dai, T., and Dai, H.-L., 205, Investigation of Mechanical Behavior for a Rotating FGM Circular Disk With a Variable Angular Speed, J. Mech. Sci. Technol., 29(9), pp [8] Bayat, M., Saleem, M., Sahari, B. B., Hamouda, A. M. S. and Mahdi, E., 2008, Analysis of Functionally Graded Rotating Disks With Variable Thickness, Mech. Res. Commun., 35(5), pp [9] Bayat, M., Sahari, B. B., Saleem, M., Ali, A. and Wong, S. V., 2009, Thermoelastic Solution of a Functionally Graded Variable Thickness Rotating Disk With Bending Based on the First-Order Shear Deformation Theory, Thin-Walled Struct., 47(5), pp Journal of Engineering Materials and Technology JULY 207, Vol. 39 /

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