A STUDY ON THE ELASTO-PLASTIC BEHAVIOUR OF A ROTATING SOLID DISK HAVING VARIABLE THICKNESS S. Bhowmick1, D. Das2 and K. N. Saha3 Department of Mechanical Engineering, Jadavpur University, Kolkata – 700032, India 2 3 1
[email protected],
[email protected],
[email protected] Abstract: A computational model based on Von-Mises yield criterion and linear strain hardening rule is developed to study the elasto-plastic behaviour of rotating solid disks having uniform and variable thickness and a numerical solution of the stress and deformation states of the disk is reported. The problem is formulated through a variational method, where the radial displacement field is taken as unknown variable. Assuming a series solution and using Galerkin’s principle, the solution of the governing partial differential equation is obtained. The rotational speed is high enough to cause yielding of the disk and the stress distribution is estimated for various rotational speeds. Keywords: Elasto-plastic, Variational method, Von-Mises criterion, Plastic front, Limit angular speed 1. Introduction Due to widespread applications, the analysis of rotating disk behavior has been of greatest interest to many researchers. The study has become much more feasible in the past few decades due to the intensified application of numerical methods as well as advent of computational machines.The plane stress state in a rotating disk of constant thickness of elastic plastic material with linear strain hardening behaviour has been studied in [1-3]. The analyses were carried out using Tresca’s yield condition and its associated flow rule and initiation of a plastic core at the axis of the disk was reported. The plastic core was shown to consist of two plastic adjacent regions governed by different forms of yield condition. The work of [1-3] was extended to rotating annular disks with variable thickness in [4-7]. In [4] the effect of radial density gradient on elastic-plastic stresses and radial displacement of a rotating disk with variable thickness under the assumption of Tresca’s yield condition and its associated flow rule have been studied. In [5] the study of linearly hardening rotating solid disk with variable thickness for fully plastic state has been carried out. The thickness was considered to vary hyperbolically along the radius. Exponential variation of thickness was first discussed in [6] although it could not present a solution satisfying all boundary and continuity conditions. In [7] analysis of rotating disks with power function thickness profile has been carried out. Elasto-plastic deformation of rotating solid and annular disks of uniform thickness made of elasticperfectly plastic material and comparison of the solutions obtained from the two different failure criterion have been reported in [8]. The study of elasto-plastic deformations of disks with different parameter values of parabolic thickness functions, representing a wide range of non uniform crosssectional profiles has been carried out in [9] where closed form solutions in terms of hypergeometric functions, by performing displacement based formulation was reported. In [10] an analytical solution of a solid convex disk with exponentially varying thickness has been obtained. A similar work was carried out for power function thickness variation in [11]. However, in [12] elasto-plastic deformation behaviour of variable thickness solid disks having concave profiles and its difference from that of uniform thickness disk was reported. Due to non linearity involved with the application of Von Mises criterion, the analysis demands for a numerical solution. But the advantage in using Von Mises criterion is that unlike Tresca’s Criterion, here a single formulation takes care of the whole plastic region. In [13] inelastic stress state of linear hardening solid disks with exponential thickness variation using both Tresca’s and Von Mises criterion has been studied. In a recent paper [14] analytical solution for rotating disks with elliptical thickness variation and made of linearly hardening material using Tresca’s criterion and its associated flow rule has been presented. In [15] numerical schemes based on perturbation method and power series solution method to investigate elastic-plastic deformations of rotating disks with uniform thickness have been presented. This work was further extended in [16] by using Runge-Kutta numerical procedure, to compute elastic-plastic stresses in
1
rotating annular disks of variable thickness and variable density. Recently, the application of variational method, proposed in [17] has yielded a generalized approach to study the behaviour of rotating disks of variable thickness in the elastic regime.
Fig. 1. An annular disk having varying thickness
Fig. 2. Linear strain hardening material
The present work employs variational principle to study the elasto-plastic behaviour of rotating disks of variable thickness using Von Mises yield criterion and linear strain hardening material behaviour. The validations of the present model with the existing works have been presented. Also the result showing the advancement of the plastic front with angular speed has been reported. 2. Mathematical Formulation A disk of exponentially varying thickness subjected to centrifugal loading is shown in Fig. 1. The mathematical model is based on the assumptions that the material of the disk is homogeneous, isotropic and follows a linear strain hardening flow rule (Fig. 2). Whenever the Von Mises stress at a particular radial location of a rotating disk reaches the uni-axial yield stress value, the plastic front initiates at that location and the corresponding rotational speed is termed as elastic limit angular speed (Ω1). The determination of Ω1 has been carried out in [17] by the variational principle. The present work is carried out to obtain the solution of a rotating disk above elastic limit angular speed. Variational principle states that δ (U + V ) = 0
(1)
Where, U = U e + U p i.e., total strain energy, U consists of an elastic ( U e ) and a plastic (U p ) part and V is the potential of the external forces. The interface between the outer elastic and the inner post elastic region is demarcated by the radius r = xc .
π E b u 2 Elastic part of strain energy, U e = +2 µ 1 − µ 2 x∫c r
du du u + r dr dr
2
hdr
and post elastic part of strain energy, U p = ∫ ( area under radial 'σ − ε ' curve + area under tangential 'σ − ε ' curve )dv . = ∫ ( dU r + dU t )dv
dU r and dU t are given by, (considering ε r p =
(2)
(3)
du u − ε r 0 and ε t p = − ε t 0 ) dr r
du u E − E1 0 1 E1 du 0 du + + + ν ε νε r t dr 2 (1 −ν 2 ) dr dr r (1 −ν 2 ) 2
dU r =
1 E1 − E 0 2 u ε t 0 − ε r 0 ν and + (ε r ) + νε r 0ε t 0 + 12 1 −Eν1 2 du 2 (1 − ν 2 ) dr r ( )
(4)
2
du u E − E1 0 1 E1 u 0u + ν + ε + νε t r r 2 (1 −ν 2 ) r dr r (1 −ν 2 ) 1 E1 − E 0 2 1 E1 u 0 du 0 + ε t ) + νε r 0ε t 0 + ε r − ε t ν ( 2 2 (1 −ν ) dr 2 (1 −ν 2 ) r 2
dU t =
So, substituting Eq. (4) and Eq. (5) in Eq. (3), we have 2 xc 2 du 1 E1 u du r Up = + + 2ν u 2π hdr 2 ∫ dr 2 (1 − ν ) a r dr xc E − E1 0 du + ε + νε t 0 ) r + ( ε t 0 + νε r 0 ) u 2π hdr 2 ∫ ( r dr (1 −ν ) a x c 2 2 1 E − E1 + ε 0 r + ( ε t 0 ) r + 2νε t 0ε r 0 r 2π hdr 2 ∫ ( r ) 2 (1 − ν ) a
{
(5)
(6)
}
b
Again, V = − ∫ u (ω 2 r )dm = −2 π ρ ω 2 ∫ (r 2uh)dr Vol
(7)
a
The normalization of the radial coordinate (r) is carried out with three parameters ( ∆, ∆1 and ∆ 2 ) and three normalized coordinates ( ξ , ξ1 and ξ 2 ), where
∆ = b − a and ξ = ( r − a ) ∆ , ∆1 = xc − a and ξ1 = ( r − a ) ∆ 1 , ∆ 2 = b − xc and ξ 2 = ( r − xc ) ∆ 2 (8)
Substituting the expressions of U e , U p and V from Eq. (2), Eq. (6) and Eq. (7) respectively in Eq. (1) and using the normalization given by Eq. (8), the governing equilibrium equation becomes, 1 µ du du ( ∆2ξ2 + xc ) du du E uδ u + δ u + δ uδ + h∆ dξ 2 dξ2 dξ2 2 2 1 − µ 2 ∫0 ( ∆2ξ2 + xc ) ∆2 dξ2 dξ2 ∆ ( ) 2 1 du E1 uδ u ν du ( ∆1ξ1 + a ) du du + δ δ u + uδ + h∆ dξ + 2 dξ1 1 1 1 −ν 2 ∫0 ( ∆1 ) dξ1 dξ1 ( ∆1ξ1 + a ) ∆1 dξ1 1 ∆1ξ1 + a ) du ∆1ξ1 + a ) du ( ( E − E1 0 0 + δ δ ε νδ u + + ε t δ u + ν h∆1dξ1 ∆1 ∆1 (1 −ν 2 ) ∫0 r dξ1 dξ1
(
)
1
{
(9)
}
− ρ ω 2 ∫ ( ∆ξ + a ) hδ (u ) ∆dξ = 0 0
2
The global displacement function u (ξ ) in Eq. (9) is approximated by u (ξ ) ≅ ∑ ciφi , i=1, 2,…, n, where φi is the set of orthogonal functions developed through Gram-Schmidt scheme. The necessary starting function to generate the higher order orthogonal functions is selected by satisfying the relevant boundary conditions ( σ r ( a ) = 0 and σ r (b ) = 0 ), of a rotating disk in elastic regime. For the purpose of computation, displacement functions in the elastic and post-elastic regions are expressed as p e u (ξ1 ) ≅ ∑ ciφi and u (ξ 2 ) ≅ ∑ ciφi respectively. Substituting these assumed displacement functions and replacing operator δ by ∂ ∂c j , Eq. (9) is obtained in matrix form as follows: 1 φi eφ j e (∆ ξ + x ) E n n µ c + (φie'φ ej + φieφ ej ' ) + 2 2 2 c φie'φ ej ' h∆2 dξ2 2 ∑∑ i ∫ 1 − µ j =1 i =1 0 (∆2ξ2 + xc ) ∆2 (∆ 2 ) p p 1 n n φ φ (∆1ξ1 + a) p' p' i j ν p' p E1 p p' c (10) + + + + φ φ φ φ φ φ h∆1dξ1 { } ∑∑ i i j i j i j (1 −ν 2 ) j=1 i=1 ∫0 (∆1 )2 ( ∆1ξ1 + a ) ∆1 1 n ( ∆1ξ1 + a ) p' E − E1 1 0 p ( ∆1ξ1 + a ) p' 2 2 0 p h d ε νφ φ ε φ ν φ ξ = ∑ ρω ∫ {(∆ξ + a) φ j } h∆dξ − + + + ∆ r j j t j j 1 1 2 ∫ ∆ ∆ 1 ν − j =1 ( ) 1 1 0 0 In Eq. (10), ( )’ indicates differentiation with respect to normalised coordinates.
3
3. Solution algorithm The governing equation can be expressed in matrix form as, [K] {c} = {R} - {R’} and the required solution of unknown coefficients {c} is obtained numerically by using an iterative scheme. To obtain the values of ε r 0 and ε t 0 , to generate the right hand side of Eq. (10) for a particular load step above elastic limit angular speed, the ratio of σ t and σ r in each radial coordinate of the complete field is assumed to be same to that of previous load step. As the plastic front originates from r=a (for the disk profiles taken for analysis) at the elastic limit angular speed, with each subsequent increase in the speed above the elastic limit angular speed, plastic front proceeds towards the outer radius. For each load step, the location of plastic front is given a small increment starting from its exact location solved for the previous load step and attainment of Von Mises stress at the plastic front location equal to the value of unidirectional yield stress gives the required solution for that load step. 4. Numerical Results The validation of this work is carried out with [15] which gives a solution methodology for the elastoplastic analysis of rotating disk for a non-linearly strain hardening material using Von Mises yield criterion. An equivalent linear strain hardening stress-strain relation and the solution using that relation by finite element method are also given in [15]. The normalized (by yield stress) radial and tangential stresses for a solid disk with uniform thickness obtained by the present scheme using the equivalent linear strain hardening stress-strain relation mentioned in [15] is compared with corresponding finite element solution and the graphical comparisons for ω=1620 rad/sec are given in Fig. 1 and Fig. 2. The comparison is carried out by using E=207 GPa, E1=33.63 GPa, ρ=7850 kg/m3, σy=232.97 MPa, ν =0.3 and b=0.2 m.
Fig. 1. Comparison of radial stress
Fig. 2. Comparison of tangential stress
The fully plastic speed in non-dimensional form, Ω2 (speed at which the whole disk attains the plastic stress state) has been calculated for a solid disk having uniform thickness and following linear strain hardening material behavior using Von Mises yield criterion in [13], taking hardening parameter H=0.5 and ν =1/3. Using these parameters, the comparison of the fully plastic speed obtained by the present method and by [13] is given in Table 1. The speed Ω2 is represented in non-dimensional form by using the relation Ω=ωb √(ρ/σy). It may be pointed out that the ratio Ω2/Ω1 gives the limit design factor for rotating disks. This factor is very important to the designers because it gives an indication of the reserve in hand, once the yielding has started.
4
Table 1: Comparison of fully plastic speed
Ω2
Present work 2.16997
By [13] 2.11747
The variation of the advancement of the plastic front (initiated at the root) with angular speed has been carried out and shown in Fig. 3 for a solid disk with uniform thickness and for solid disks with exponential variation of thickness with n=0.5, k=2.0 and n=0.5, k=2.5. The graphical variations of the disk profiles have been shown in Fig. 4 in which the exponential variation of thickness is given by h(ξ ) = h0 exp[− n(ξ ) k ] . The analysis is carried out for E=210 GPa, E1=70, GPa, ρ=7850 kg/m3,
σy=350 MPa, ν=0.3, and b=1.0 m.
Fig. 3. Advancement of plastic front with angular speed
Fig. 4. Disk profiles corresponding to the cases of Fig. 3.
5. Conclusion The present work gives an approximate solution in the elasto-plastic region of a solid rotating disk of varying thickness assuming linear strain hardening material behavior following Von Mises yield criterion. The results obtained by the present methodology have been validated and it showed a close conformity with the existing results of similar problem. Also a plot showing the advancement of the plastic front with increase in rotational speed has been presented. The present work has all the potential to determine the limit design factors, and essential tool for the rotating disk design, in loaded and unloaded condition. Nomenclature a, b Inner and outer radii of the disk respectively ci The vector of unknown coefficients h0 ,h Thickness at the root and thickness at any radius r of the disk n, k Parameters controlling the thickness variation of disk r, t Subscripts, correspond to radial and tangential directions u Displacement field of the disk xc Radial location of yield front U Strain energy of the disk V Potential energy of the disk due to rotation ρ, ν , E, E1 Density, Poisson’s ratio, elasticity modulus and tangent modulus of the disk material
5
H ω,Ω
Ω1 Ω2 εr ,εt εr0,εt0 σr , σt σy
φi ξ ξ1 ,ξ2 e, p
Hardening parameter as mentioned in [13] Angular speed and dimensionless angular speed of the disk Elastic limit angular speed Plastic limit angular speed Strains in radial and tangential direction Strains in radial and tangential direction in yield condition Radial and tangential stresses Yield stress of the disk material The set of orthogonal polynomials used as coordinate functions Normalized radial co-ordinate Normalized radial co-ordinate in plastic and elastic region Superscripts, correspond to elastic and plastic state of stress
References: 1. U. Gamer, 1984, Elastic–plastic deformation of the rotating solid disk, Ing.-Arch., 54, 345– 354. 2. U. Gamer, 1984, The rotating solid disk in the fully plastic state, Forsch. Ing. Wes., 50, 137– 140. 3. U. Gamer, 1985, Stress distribution in the rotating elastic–plastic disk, ZAMM, 65, T136– T137. 4. U. Güven, 1992, Elastic-plastic stresses in rotating annular disk of variable thickness and variable density, Int. J. Mech. Sci., 34(2), 133-138. 5. U. Güven, 1994, The fully plastic rotating disk of variable thickness, ZAMM, 74, 61–65. 6. U. Güven, 1995, On the applicability of Tresca’s yield condition to the linear hardening rotating solid disk of variable thickness, ZAMM, 75, 397–398. 7. U. Güven, 1995, Tresca’s yield condition and the linear hardening rotating solid disk of variable thickness, ZAMM, 75, 805–806. 8. D. A. Rees, 1999, Elastic–plastic stresses in rotating disks by Von Mises and Tresca, ZAMM, 79, 281–288. 9. A. N. Eraslan, 2003, Elastic-plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions, Int. J. Mech. Sci., 45, 643-667. 10. A. N. Eraslan and Y. Orcan, 2002, Elastic–plastic deformation of a rotating solid disk of exponentially varying thickness, Mech. Mat., 34, 423–432. 11. Y. Orcan and A. N. Eraslan, 2002, Elastic–plastic stresses in linearly hardening rotating solid disks of variable thickness, Mech. Res. Commun., 29, 269–281. 12. A. N. Eraslan and Y. Orcan, 2002, On the rotating elastic–plastic solid disks of variable thickness having concave profiles, Int. J. Mech. Sci., 44, 1445–1466. 13. A. N. Eraslan, 2002, Inelastic deformations of rotating variable thickness solid disks by Tresca and Von Mises criteria, Int. J. Comp. Engrg. Sci., 3, 89–101. 14. A. N. Eraslan, 2005, Stress distributions in elastic-plastic rotating disks with elliptical thickness profiles using Tresca and Von Mises criteria, ZAMM, 85, 252-266. 15. L. H. You and J. J. Zhang, 1999, Elastic–plastic stresses in a rotating solid disk, Int. J. Mech. Sci., 41, 269–282. 16. L. H. You, Y. Y. Tang, J. J. Zhang and C. Y. Zheng, 2000, Numerical analysis of elastic– plastic rotating disks with arbitrary variable thickness and density, Int. J. Solids Struct., 37, 7809–7820. 17. S. Bhowmick, G. Pohit, D. Misra and K. N. Saha, 2004, Design of High Speed Impellers, Proc. Int. Con-HERP, IIT, Roorkee, 229-241.
6
