International Journal on Mechanical Engineering and Robotics (IJMER)
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Behavioural Study of a Functionally Graded Material Disk Subjected to Pressure Conditions
1Ishrat Meera Mirzana, 2G. Krishna Mohana Rao, 3S. Irfan Sadaq, 4Kaushal Dasari, 5N.B.V. Lakshmi Kumari
1,3,4,5MED, MJCET, INDIA
2JNTUH, INDIA Abstract - Functionally Graded Materials are
microscopically inhomogeneous and the material properties vary smoothly or continuously from one surface to another. Typically, these materials are made from a mixture of ceramic and metal or a combination of different materials. Many components are subjected to mechanical, thermal or chemical loads that are unevenly distributed across their section. Gradient materials offer the possibility to combine two materials properties avoiding most of the disadvantages of a bi material. Consider a functionally graded material disc with an inner material 𝐚, and an outer radius 𝐛, subjected to internal and external pressure 𝐏𝐢 and 𝐏𝐨 , respectively. It is assumed that the Poisson’s ratio υ, takes a constant value and modulus of elasticity E. In this work mathematical modeling and simulation has been carried out on hollow disk made of Aluminium and Zirconium. Stresses are obtained as a function of radial direction by using the theory of elasticity. The pressure, inner radius and outer radius are considered constant.
Keywords – Functionally Graded materials, gradient materials, theory of elasticity
I. INTRODUCTION
Functionally Graded Materials are microscopically inhomogeneous and the material properties vary smoothly or continuously from one surface to another.
Typically, these materials are made from a mixture of ceramics and metal or a combination of different materials. Many components are subjected to mechanical, thermal or chemical loads that are unevenly distributed across their section. Gradient materials offer the possibility to combine two materials properties avoiding most of the disadvantages of a bi material. In contrast, traditional composites are homogeneous mixtures, and they therefore involve a compromise between the desirable properties of the component materials. Since significant proportions of an FGM contain the pure form of each component, the need for compromise is eliminated.
The properties of both components can be fully utilized.
For example, the toughness of a metal can be mated with their refractoriness of a ceramic, without any compromise in the toughness of the metal side or there refractoriness of the ceramic side. Consider for example a turbine blade which must with stand high non-
stationary heat fluxes and centrifugal accelerations. An ideal structure for this application would consist of a tough metal core and a heat and corrosion resistant ceramic at the hot surface of the blade. If the ceramic is directly bonded to the metal, spilling may occur during thermal cycling as very high thermal stresses occur at the interface. A gradient material that has a smooth transition from the ceramic surface to the metal core can avoid the thermo mechanical stress concentration at the interface.
The main feature of a gradient material is that its properties changes gradually with position used as coatings and interfacial zones, they help to reduce mechanically and thermally induced stresses caused by the material property mismatch and to improve the bonding strength.
II. LITERATURE SURVEY
Mehdi Bayat, et.al., [1] have carried out their studies on a rotating functionally graded (FG) disk with variable thickness under a steady temperature field. Thermo elastic solutions and the weight of the disk are related to the material grading index and the geometry of the disk.
Fernando Viegas Stump, et.al., [2] have explained a topology optimization framework to design the material distribution of a functionally graded rotating disk considering mechanical stress constraints. In the case of a rotating disk, under body forces, the problem of interest consists of maximizing the rotating inertia subjected to a global stress constraint. Mehdi Bayat, et.al., [3] have studied a semi-analytical investigation intended to determine the axisymmetric elastic response of functionally graded (FG) disks. The material properties of the disk are assumed to be graded continuously along the radial direction. Mandira Bhattacharyya, et.al., [4] have studied two-multilayered functionally graded materials (FGMs), namely aluminium–silicon carbide (Al/SiC) and nickel–alumina (Ni/Al2O3) systems which are designed, synthesized and characterized considering 10, 20, 30 and 40 vol.%
ceramic concentrations. S B Singh, et.al ., [5] have investigated Steady-state creep response in a particle- reinforced isotropic functionally graded material (FGM) disc with linear variation of particle distribution along
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the radial distance and comparing with that of a disc containing the same amount of particle distributed uniformly. In view of the application of rotating discs in friction drives, turbines, and a number of other machine components, which are often exposed to elevated temperatures, weight saving without impairing the creep response may be a desirable goal.
Hasan Callioglu, et.al ., [6] have studied stress analysis of functionally graded rotating annular disc subjected to internal pressure and various temperature distributions such as uniform reference temperature (To), linearly outer surface temperature(Tb) and decreasing inner surface temperature(Tb) in radial direction. Infinitesimal deformation theory of elasticity and for graded parameter power law functions are used in the solution procedure. M. Bayat, et.al. [7]have studied semi- analytical investigation intended to determine the axisymmetric elastic response of functionally graded (FG) disks. The material properties of the disk are assumed to be graded continuously along the radial direction. An exponential function and the Mori-Tanaka scheme are used for estimating the effective material properties. Two kinds of functionally graded materials (FGMs) namely metal-ceramic and ceramic-metal are considered. Hollow disks are considered and the solutions for the displacements and stresses are given under appropriate boundary conditions. M. Rattan, et.al., [8] have investigated creep response for isotropic axisymmetric rotating disc made of a particle-reinforced FGM. The result obtained for non linear variation of particle distribution along the radial distance of the disc are compared with that of discs containing the same amount of particle distributed uniformly or linearly along the radial distance. Hasan Callioglu, et.al.,[9]have studied stress analysis of annular rotating discs made of functionally graded materials (FGMs). Elasticity modulus and density of the discs are assumed to vary radially according to a power law function, but the material is of constant Poisson’s ratio. Xu-long PENG, et.al.,[10]have investigated a thermoelastic problem of a circular annulus made of functionally graded materials with an arbitrary gradient. The analysis neither requires a special form of the gradient of material properties nor demands partitioning the entire structure into a multilayered homogeneous structure.
F. A.Sohag, et.al.,[11] have investigated the thermoelastic characteristics of a thin circular disc having a concentric hole and a functionally graded material (FGM) coating at the outer surface. The disc is subjected to a temperature gradient field and an inertial force due to rotation of the disc. The coating region is assumed to have an exponential variation of all the material properties except the Poisson’s ratio which is assumed to be constant throughout the disc. A. M. Afsar, et.al.,[12] have focused on the analysis of thermo elastic characteristics of a thin circular disc having a concentric hole and a functionally graded material (FGM) coating at the outer surface. The disc is subjected to a temperature gradient filed and an inertial force due to
rotation of the disc. The coating region is assumed to have an exponential variation of all the material properties except the Poisson’s ratio which is assumed to be constant throughout the disc. G. ANLAS, et.al.,[13] have studied finite element method for its use in cracked and uncracked plates made of functionally graded materials. The material property variation is discretized by assigning different homogeneous elastic properties to each element. Finite Element results are compared to existing analytical results and the effect of mesh size is discussed. Nilanjan Coomar, et.al.,[14]
have carried out Internal cooling passages and thermal barrier coatings (TBCs) are presently used to control metal temperatures in gas turbine blades. Functionally graded materials (FGMs), which are typically mixtures of ceramic and metal, have been proposed for use in turbine blades because they possess smooth property gradients thereby rendering them more durable under thermal loads. S. Akbarpour, et.al.,[15] have studied finite element method and micromechanical modeling of FG thermal barrier coatings, stresses under thermal and mechanical loadings of the same and different phases have been investigated. Also, the effect of some parameters such as refinement and offsetting of particles on stresses are studied. As for the loading, thermal cycle and in-phase and out-of-phase thermo-mechanical cyclic loadings are considered.
S.B. Singh, et.al.,[16]have studied the analysis of steady state creep in a rotating disc made of Al-SiCp composite having variable thickness using Sherby’s constitutive model. The creep parameters have been evaluated using the available experimental results in the literature using regression analysis. Three variations in the thickness (constant, linearly and hyperbolic varying thickness) of the disc have been considered while keeping other material parameters same. Mohammadi F, et.al.,[17]
have analyzed a composite rotating disk under internal and external pressure subjected to temperature distribution. The modified Tsai-Wu FC is used in this study. It has been shown that FC is strongly depends on temperature distribution. Polynomial function is selected for thermal distribution. For a limited range of the temperature the coefficient of the polynomial has a great effect on failure criterion. Glaucio H. Paulino, et.al.,[19]
have studied functionally graded materials with an additional length scale associated to the spatial variation of the material property field which competes with the usual geometrical length scale of the boundary value problem. By considering the length scale of non homogeneity, it presents the weak patch test (rather than the standard one) of the graded element for non homogeneous materials to assess convergence of the finite element method (FEM). Both consistency (as the size of elements approach zero, the FEM approximation represents the exact solution) and stability (spurious mechanisms are avoided) conditions are addressed.
Mehdi Bayat, et.al.,[20] have studied a rotating functionally graded (FG) disk with variable thickness under a steady temperature field. Thermo elastic solutions and the weight of the disk are related to the
material grading index and the geometry of the disk. It is found that a disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared to a uniform thickness disk.
Maximum radial stress due to centrifugal load in the solid disk with parabolic thickness profile may not be at the center unlike uniform thickness disk.
III. METHODOLOGY
A. Mathematical Modeling:
Figure 1: Configuration of a FGM disk with radially varying properties
Consider a functionally graded material disc with an inner material a, and an outer radius b, subjected to internal and external pressure Pi and Po , respectively as shown in the Fig.1. It is assumed that the poisson’s ratio υ, takes a constant value and modulus of elasticity E, is assumed to vary radially according to parabolic form as follows
E R = Ei 1 + n 1−R1−Kηη ---1 Where R = ra, n = EEo
i , K = ba
Ei and Eo are modulus of elasticity in inner and outer surfaces respectively. Here, n and η are material parameters.
Radial and circumferential strains εr , εθ , in the polar coordinates are as follows,
εr = dudr --- 2 εθ = u
r --- 3 Where u is radial displacement
The stress-strain relations for non- homogenous and isotropic materials are
σr
σθ = E(R)a A BB A
du dRu R
--- ---4
Where σr and σθ are radial and circumferential stresses.
A and B are related to Poisson’s ratio υ as
A = 1−υ12
B = 1−υυ2 ---5
The equilibrium equation in the absence of body forces is expressed as
dσr dR + σr−σθ
R = 0 ---6 Here, primes denotes differentiation with respect to R.
Substituting Eqs. (4) and (6), the equilibrium equation is expressed as
R2 ddR2u2+ R 1 +REE′ dudR− 1 −υ∗ REE′ u = 0
υ∗= BA ----7 The general solution of Eq. (7) is as follows
u(R) = C1G(R) + C2H(R) --- ---8
where C1 and C2 are arbitrary integration constants.
Here G and H are homogenous solutions.
Substituting Eq. (8) into Eqs. (4), yields σr
σθ = E R a A B
B A C1G′+ C2H′
C1GR+ C2HR ---9 The form of G and H will be determined next
Substituting Eq. (1) into Eq. (7) the governing differential equation is as follows
R2 1 −n+KnRηη−1 ddR2u2+ R 1 − 1+η nRn+Kη−1η dudr- 1 −
1−υ∗η nRη
n+Kη−1 u = 0 ---10
Equation 10 is a homogenous hyper geometric differential equation.
Using a new variable x = mRη = n/ n + Kη− 1 Rη and applying the transformation u R = Ry x , the result Eq. (10) is
x 1 − x dRd2y2+ 1 +2η − 2 1 +1η x dRdy-1+υ∗
η y = 0 -- 11
The solution of Eq. (11) is given as
y x = C1FH α, β, δ; x + C2x−2 ηFH(α − δ + 1, β − δ + 1,2 − δ; x) --- 12
With FH α, β, δ; x being the hypergeometric function defined by Abramowitz and Stegun,
FH α, β, δ; x = 1 + (α)(δ)k(β)k
k
∞k=1 xk
k! -- 13 Where
α k = α α + 1 α + 2 … … α + k − 1 ---14 Thus
FH α, β, δ; x = 1+αβδ 1!x + α α+1 β(β+1) δ(δ+1) x2
2!+α α+1 α+2 β β+1 (β+2) δ δ+1 (δ+2)
x3
3!+... ---15
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In Eq. (15), the argument α, β and δ are determined as α = 2 1+υ
∗ 2+η + 2+η 2−4η 1+υ∗
β = 2+η − 2+η 2 1+υ2∗−4η 1+υ∗
δ = 1 +η2
--- 16
From u R = Ry mRη , the homogenous solutions G and H are found in the form
GR = R Fh (α, β, δ; mRηɳ) --- 17 HR = 1
R Fh (α + β + 1, β − δ + 1,2 − δ; mRη) ---18 The Eqs. (8) and (9) may be rewritten with non- dimensional parameters as
U R = C3G R + C4H R --- 19 σr
σθ
= 1 + n 1−R1−Kηη A B
B A C3G′+ C4H′ C3G
R+ C4H R
---- 20 Where
U =uEaPi
i , σ = Pσ
i, CC2
1= CC4
1= aPEi
i ---21 Integration constants C3and C4 are determined by using the following boundary conditions
σ R = 1 = −1 , σr R = K = −r PPo
i = −P ---- 22 Thus
AG′ 1 + BG 1 C3+ AH′ 1 + BH 1 C4] = −1
AG′ K + BG K K C3+ AH′ K + BH K K C4= −1+nP − ---23
Using Eqs .(23), the constants C3 and C4 are determined as follows
C3 = D2D5+ D4
D2D3− D1D4 C4 = D1D5+ D3
D1D4− D2D4 ---24 Where
D1= AG′+ BG(1) D2= AH′+ BH(1)
D3= AG′+ BG(K)K D4= AG′(K) + B G(K)K
D5= −1+nP
---25
IV. RESULTS AND DISCUSSION
In this present study, stresses in rotating disc made of functionally graded material consisting of Aluminum metal and Zirconia ceramic are obtained as a function of radial direction by for obtaining elastic solutions. The internal radius of (a =) 30 mm and outer radius of (b =) 150 mm is considered. The applied internal pressure is 10 MPa. In addition, ranges from 0.5 to 3.5. The properties of aluminum and Zirconia considered for analysis are as follows
Specific Properties of Aluminium:
• Chemical Symbol: Al
• Atomic Number: 13
• Atomic Weight: 26.98
• Density: 2700 kg/m3
• Melting Point: 933.47 K
• Thermal Conductivity: 237 W·m−1·K−1
• Electrical Resistivity: (20 °C) 28.2 nΩ·m
• Crystal Structure: Face Centered Cubic Specific Properties of Zirconia:
• Chemical Symbol: Zr
• Atomic Number: 40
• Atomic Weight: 91.224
• Density: 6.49 g.cm-3 at 20°C
• Melting Point: 2128 K
• Thermal Conductivity: 22.6 W·m−1·K−1
• Electrical Resistivity: (20 °C) 421 nΩ·m
• Crystal Structure: hexagonal close-packed
Figure 2: Variation of vonmises stress w.r.t radial distance
Figure 3: Variation of Deformation w.r.t radial distance
Figure 4: Radial Stress (𝛔𝐫)
Figure 5: Tangential Stress (𝛔𝛉)
Figure 6: Vonmises stresses (𝛔𝟎)
Figure 7: Deformation (𝐔𝐫)
Figure 8: Variation of vonmises stress w.r.t radial distance
Figure 9: Variation of Deformation w.r.t radial distance
Figure 10: Vonmises comparison of Analytical and ANSYS results
Figure 11: Deformation comparison of Analytical and ANSYS results
V. CONCLUSIONS
An FGM disk consists of Zirconia and Aluminium has been considered.
It has been observed that for varying values of from 0.5 to 3.5 i.e., material constant the behaviour of FGM disk varies.
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When the varies from 0.5 to 1.5, it has been observed that there is a steady increase in the deformation in internal and external disk material but in the interfacial region the variation is very little due the application of internal and external pressure applied of same magnitude.
A drastic variation has been observed when =2, i.e., there is drop in deformation in the last two layers of the disk. There is little variation in the deformation in zirconia material for =2.5 to 3.5 whereas unsteady variation has occurred in the interfacial region of the disk due to higher values of material constant and then there is a steady increase in the aluminium region. Thus, it is advisable to use an FGM disk with lower values of material constant to have better results, performance and application.
For the vonmises stresses, when the varies from 0.5 to 1.5, it has been observed that there is a uniform increase in the stress values in the disk.
when varies from 2 to 2.5 a steep increase has been observed in the vonmises stress of ceramic rich material zirconia and then a variation has been observed in interfacial region which is fluctuating and then a steady increase has been observed in metal rich material aluminium then for =3 the behaviour of vonmises stresses varies from the other values.
From the present study it is observed that for an FGM disk made of ceramic rich and metal rich materials, better performance and application can be obtained for material constant less than 2.
Further work can be carried out by applying the same principles to FGM cylinder and various materials inputs can also be evaluated to determine their deformation and vonmises stresses.
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