Stress and Deformation Analysis of Variable Thickness Clamped Rotating Disk of Functionally Graded Orthotropic Material

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Science Direct Materials Today: Proceedings 18 (2019) 4431–4440

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ICMPC_2019

Stress and Deformation Analysis of Variable Thickness Clamped Rotating Disk of Functionally Graded Orthotropic Material Lakshman Sondhia, Amit Kumar Thawaitb,Subhashis Sanyalc, and Shubhankar Bhowmick d* a,c,d

Department of Mechanical Engineering,National Institute of Technology Raipur (C.G)-492010, India b Department of Mechanical Engineering , IIT Mumbai 400076, India

Abstract The present study deals with the linear elastic stress analysis of rotating disks made of functionally graded polar orthotropic material whose mechanical properties vary in radial direction. The analysis is carried out using element based gradation of material properties over the discretized domain. Stress and deformation distribution for four types of thickness profiles namely uniform, linear profile, parabolic concave and parabolic convex are investigated and effects of the grading parameter and other parameters, governing the orthotropy of the material are studied for clamped-free boundary condition. The results obtained are in good agreement with established results and it can be seen that there is a significant reduction of stresses in variable thickness disks as compared to uniform disks. © 2019 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.

Keywords: Elastic Analysis, Functionally Graded Material, Polar orthotropic material, Variable Thickness Clamped Rotating Disk, Element Based Material Gradation. 1. Introduction Functionally graded materials (FGMs) are special inhomogeneous materials that have continuous variations of physical and mechanical properties. Functionally graded structures, in recent years, are widely used in space vehicles, aircrafts, nuclear power plants and many other engineering applications. Many engineering components are

* Corresponding author. Tel.: +91 9575 955040. E-mail address:[email protected]

2214-7853© 2019 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.

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modeled as rotating circular disks such as turbines, gears, flywheels, centrifugal compressors, circular saws, etc. The total stresses and deformation due to centrifugal load have important effects on their strength and safety. Functionally gradation of the material properties and variable thickness profile optimize the component strength by controlling the changes of the local material properties. A few researchers have reported work on analysis of rotating FGM disks and plates by analytical as well as finite element method. Tang [1] obtained a closed-form solution for stresses in uniform thickness rotating anisotropic disks for three different conditions: a solid disk; a disk mounted on a circular rigid shaft; and a disk with a circular hole at the center. Furthermore in [2], using the closed-form solution, optimization for stresses in rotating disks with various boundary conditions is presented. Liang et al. [3] investigated natural frequencies and stability of a spinning polar orthotropic disk subjected to a stationary concentrated transverse load. Zenkour [4] derived an analytical solution for functionally graded annular rotating disks under the plane stress assumption with exponentially variable material properties. Callioglu et al. [5] studied the elastic– plastic stress analysis of a curvilinear orthotropic rotating annular disk by analytical methods for strain-hardening material. Vibration of rotating polar orthotropic disks were studied by Koo [6], which shows that critical rotation speed, can be improved by circumferentially reinforcement. Callioglu [7] reported work on the thermal stress analysis in curvilinear orthotropic rotating disks. In present research work clamped rotating disks made up of orthotropic functionally graded material having different thickness profiles and constant mass are analyzed using element based gradation. Geometric, material and finite element formulation is presented and governing equations are derived using principle of stationary total potential. Three types of variable profile along with uniform thickness are considered and effects of thickness variation on stress and deformation distribution are investigated. Further the effects of grading parameter and other governing parameters representing degree of orthotropy are also investigated and presented as shown in below Fig 1. 2. Mathematical formulation For an annular disk considered in present study the thickness variation follows (Eq. 1)

Fig.1. Disks of varying thickness; sectional isometric view



h  r  = h0 1- q  r - a  b-a  

m  

(1)

Material properties of the disk are graded in radial direction according to the following [8]:

 r  - a   r  - a  Y  r  = Yi  1-    +Yo        b -a  b -a     

(2) 

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Where a and b are inner and outer radius, h(r) and h0 are half of the thickness at radius r and at the root of the disk, respectively. Symbols m and q are the geometric parameters that control the thickness profiles of the disk. For uniform thickness disk q is taken as zero and for variable thickness disk, q >0.Where Y(r) denotes material property to be determined and Yi and Yo denotes material property at inner and outer radius, a and b denotes inner and outer radius and grading parameter β.In polar orthotropic material Young’s modulus in radial and circumferential directions are different and are related by following [8]:

νθr  r 

ν r  = rθ Eθ  r  E r  r 

(3) 

 

Where ν and E denotes Poisson’s ratio and Young’s modulus respectively. The displacement field is presented by (Eq. 4) (4) 

u = Nδ  

Where u is element displacement vector, N is matrix of quadratic shape functions and δ is nodal displacement vector.

N =  N 1 , N 2 ... .. .. . .. .. .. .N 8 

(5) 

 

The strain components are related to elemental displacement components as

ε = εr  u   r

εθ

T



 u =  r

T

u  u  = B1  r  r

T

u  r  

u z

(6) 

T

u  r  

(7) 

T

T

 u  u u   u  u u    =B2    r  z r   ξ  η r  T

(8) 

 

T u u u    = B3 u1 ,u2 ...........u8 ξ η r   





(9) 

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Where εr and εθ, are radial and tangential strain respectively. By transforming the global co-ordinates into natural coordinates (ξ η),The above elemental strain-displacement relationships can be written as:

ε = Bδe  

(10) 

Where B is strain-displacement relationship matrix, which contains derivatives of shape functions. From the constitutive relations, hooks law, components of stresses in radial and circumferential direction are related to components of total strain as (Peng and Li. 2012)

ν (r) εr = σr - θr σθ Er (r) Eθ (r) σ ν (r) εθ = θ - rθ σ r Eθ (r) Er (r)

 (11) 

 

(12) 

   

By solving above equations, one can find stress strain relationship as:

σr =

σθ =

Er  r   ε +ν ε  1- νθr νrθ   r θr θ  Eθ  r 

(13) 

 

 εθ + νrθ εr 

1 - νθr νrθ  

(14) 

 

In standard finite element matrix notation above stress strain relations can be written as: (15) 

σ = D(r)ε   Where

σ = σr D r  =

σθ

T



 Er  r   1 - νrθ νθr   νrθ Eθ  r  1

(16) 

  νθr E r  r     E θ  r  

(17) 

Upon rotation, the disk experiences a body force which under constrained boundary results in deformation and stores internal strain energy U.

U=

1 T ε σdv 2 v  

(18) 

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The work potential due to body force resulting from centrifugal action is given by

V = -  δ T qv dv v

(19) 

 

U e =  Πrh δeT BT D(r)Bδ edr r v

(20)  (21) 

V e =-2 πrh δeT NT q dr r

v

v

For a disk rotating at ω rad/sec, the body force vector for each element is given by  ρ  r  ω2 r      0  

(22) 



(23) 

qv  = 

   T T 1  e    e   e  e e    e  πp =  δ K δ    -  δ   f            2     



Using the MPEP, the total potential is set to be stationary with respect to small variation in the nodal degree of freedom that is

 πp  δT

(24) 

=0  

By transforming the global coordinates into natural co-ordinates, the stiffness matrix and load vector are obtained as 1

 -1



 K e = 2π   B  D  r  B r dξ   e

1

T

T

 f  = 2π   N  qv  rdξ  

(25) 

(26) 

-1

The element matrices are then assembled to yield the global stiffness matrix and global load vector respectively. (Eq. 24) is solved for clamped-free boundary conditions, ur (a)= 0, σ r (b)= 0 . In FEM, the functional grading is popularly carried out by assigning the average material properties over a given geometry followed by adhering the geometries thus resulting into layered functional grading of material properties. The downside of this approach is that it yields singular field variable values at the boundaries of the glued geometries. To get better results, it is an established practice to divide the total geometry into very fine geometries. However, a better approach is to assign the average material properties to the elements of mesh of the single geometry. This is, in other words, better described as assigning material properties to the finite elements instead of geometry. In (Eq. 17) the [D(r)] matrix, being a function of r, is calculated numerically at each node and these results into continuous material property variation throughout the geometry. The element matrices are then assembled to yield the global stiffness matrix and global load vector respectively. The element based grading of material property yields an appropriate approach of

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functional grading as the shape functions in the elemental formulations being co-ordinate functions make it easier to implement the same. N e  K  =   K  = Global Stiffness matrix   (27)  n =1

N

e  F  =   f  = Global load vector  

(28) 

n =1

N is number of elements.

8

 e = i Ni

(29)

i =1   Where ϕe is element material property, ϕi is material property at node i and Ni is the shape function. 3. Results and discussions The validation of orthotropic rotating disk under clamped free boundary condition is presented with Peng and Li.2012 in Fig.2 Radial stresses in dimensionless form for clamped-free boundary condition is validated and good agreement is reported. In this section rotating annular orthotropic FG disks of uniform and variable thickness profiles having clamped-free boundary condition are analyzed and distribution of displacement, radial, tangential and von-Mises stress are presented for each type of profile. In this investigation also the material model and properties are so taken that inner surface of the disk (r = a) is 100% injection molded Nylon 6 composite containing 40 wt% of short glass fiber and the outer surface of the disk (r = b) is 100% glass-fiber/ epoxy prepreg as presented with the material properties in Table1.Symbol E, ρ and ν represents the usual meanings of Young’s modulus, density and Poisson’s ratio respectively and subscript rand θ denotes radial and tangential direction. The normalized parameters are as follows,Displacement:urE0/ρ0ω2b3,Stresses: σ/ρ0ω2b2, Where ur is displacement, σ is stress and ω is angular velocity. Fig.(7-9) shows the comparison of displacement, radial, tangential and von-Mises stress respectively in dimensionless form for four types of thickness profiles namely uniform, linear varying, concave and convex thickness profile for grading parameter, β = 10. From figures, it is seen that variable thickness profiles yield significant reduction in displacement and stresses as compared to uniform disk. Radial stress is maximum and displacement is zero at the inner radius, which confirms clamped-free boundary condition. Further it is observed that radial stress is more critical as compared to tangential and von- Mises stress in case of clamped free boundary conditions and among the four profiles, concave profile reports least radial stress at the inner radius. Fig. (10-12) the distribution of displacement and radial stress in dimensionless form for the different values of grading parameter β in concave disk is reported. It can be seen that β has limited influence on the displacement but has significant effect on radial stress. Displacement is maximum at the outer radius and zero at inner radius for all values of β. It slightly increases with increasing value of β. Radial stress is maximum at inner radius and is zero at outer radius for all values of β . It decreases with increasing value of β but it is also evident that lower range (β= 1, 4) of n influences radial stress more as compared to higher range (β= 7, 10). Fig. (13-14) reveals the effect of parameter μ which is the ratio of Y0 and Yi (of (Eq. 2)) on the distributions of displacement and stress for β = 1 and λ = 20/12 which is the ratio of Eθ and Er of injection molded nylon 6 composite containing 40 wt% of short glass fiber, λ represents the degree of orthotropic of the disk material. From Fig. (13-14) it can be seen that parameter μ significantly effects both displacement and radial stress and both increases with increasing μ. Fig. (15-16) shows the distribution of dimensionless displacement and stresses for the different values of parameter λ at β = 1 and μ = 2. From Fig. (15-16) it is observed that both displacement and stress decrease with increasing value of λ as shown in below Fig (3-6).

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Table 1 Material Properties Material Injection molded Nylon 6 composite containing 40 wt% short glass fiber Glass-fiber/ epoxy prepreg

Er (GPa)

Eθ (GPa)

ρ (kg/m3)

νθr

12

20

1600

0.35

21.8

26.95

2030

0.15

Fig.2. Comparison of the results of current work with Reference (Peng and Li. 2012)

Fig.3. Distribution of Er for different values of β

Fig.5. Distribution of ρ for different values of β

Fig.4. Distribution of Eθ for different values of β

Fig.6. Distribution of νθr for different values of β

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Fig.7. Distribution of displacement for β = 10

Fig.9.Distribution of tangential stress for β = 10

Fig.8. Distribution of radial stress for β = 10

Fig.10. Distribution of von-Mises stress for β = 10

Fig.11. Distribution of displacement for different values of β

Fig.12. Distribution of radial stress for different

for concave disk

values of β for concave disk

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Fig.13. Effect of μ on displacement for β = 1 and λ = 20/12 for concave disk

Fig.15. Effect of λ on displacement for β = 1 and μ = 2 for

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Fig.14. Effect of μ on radial stress for β = 1 and λ = 20/12 for concave disk

Fig.16. Effect of λ on radial stress for β = 1 and μ = 2

concave disk

for concave disk

Conclusion In present research paper stress and deformation analysis of orthotropic FGM rotating disks of variable thickness is studied. Geometric modeling, material modeling and finite element modeling is done. Functionally gradation of the material properties in radial direction is achieved by element based gradation. The governing equations are derived using principle of stationary total potential. Four types of thickness profile disks subjected to clamped-free boundary condition is considered and stress and deformation distribution are presented. Further the effects of grading parameter and degree of orthotropy on stresses and displacement distribution is investigated. It is observed that there is a significant reduction in stresses and deformation behavior of the variable thickness FGM disks compared to uniform disk. Moreover it is observed that concave thickness profile has least stresses and therefore is most suitable among the considered thickness profiles for clamped-free boundary condition. References [1] Tang S. Elastic stresses in rotating anisotropic disks. Int J Mech Sci 1969; 11:509–517. [2]Ari-J Gur, Stavsky Y. On rotating polar-orthotropic circular disks. Int J Solids Strut 1981;17:57–67.

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[3] Liang DS, Wang HJ, Chen LW.Vibration and stability of rotating polar orthotropic annular disks subjected to a stationary concentrated transverse load. J Sound Vib 2002; 250:795–811. [4] Zenkour, A.M.: Analytical solutions for rotating exponentially-graded annular disks with various boundary conditions. Int. J. Struct. Stab. Dyn. 5, 557–577 (2005) [5]Callioglu H, Topcu M, Tarakcilar AR. Elastic–plastic stress analysis of an orthotropic rotating disk. Int J Mech Sci 2006;48:985–990. [6] Koo KN.Vibration analysis and critical speeds of polar orthotropic annular disks inrotation. Compos Struct 2006;76:67–72. [7] Callioglu H. Thermal stress analysis of curvilinearly orthotropic rotating disks. J Thermoplastic Compos Mater 2007; 20:357–369. [8] X.-L. Peng, X.-F.Li . Elastic analysis of rotating functionally graded polar orthotropic disks. International Journal of Mechanical Sciences. 2012; 61(1): 84-91p. [9] Seshu P., 2003, A text book of finite element analysis, PHI Learning Pvt. Ltd., First Edition.