On the burst strength and necking behaviour of rotating disks

Int. Z mech. Sci., Vol. 20, pp. 109-120. Pergamon Press 1978. Printed in Great Britain.

ON

THE BURST BEHAVIOUR

STRENGTH AND NECKING OF ROTATING DISKS

VIGGO TVERGAARD D e p a r t m e n t of Solid Mechanics, The Technical University of Denmark, Lyngby, D e n m a r k (Received 16 September 1977; received .[or publication 2 November 1977)

S u m m a r y - - T h e behaviour of an elastic-plastic rotating disk is analysed both in the context of three dimensional theory and within the f r a m e w o r k of the plane stress approximation. For an a x i s y m m e t r i c disk the possibility of bifurcation into a non-axisymmetric mode is investigated. C o m p u t a t i o n s are also made for the behaviour of a disk with initial imperfections either in the form of a thickness variation or in the form of material inhomogeneities. For a ductile, bored disk of uniform thickness it is found that bursting occurs after the critical bifurcation point, which m a y occur before or after the point of m a x i m u m angular velocity in the axisymmetric solution. T h u s , necking has started to develop in the disk, when ductile bursting occurs.

1. I N T R O D U C T I O N

The behaviour of rotating disks at burst has been the subject of a large number of investigations, both by experimental methods and by theoretical ones. Quite early Robinson' made an extensive series of experiments for steel disks of various shapes and suggested a useful criterion for the bursting speed of a rotating disk. According to this semi-empirical criterion the nominal average hoop stress at the bursting speed equals approximately the tensile strength of the material. The nominal average stress criterion was further confirmed by Skidmore 2 in a series of experiments for disks of steel and aluminium, and by several later investigations, so that now this criterion is generally accepted as a useful engineering estimate of the bursting speed. Some more recent experiments on the bursting of rotating disks reported by Waidren, Percy and Mellor 3 are of particular interest because the use of vacuum melted steel has reduced the scatter considerably compared with results obtained using air method stock. These experiments include disks of uniform thickness, disks of uniform strength and a model turbine rotor with a wide hub to reinforce the bore. For a sufficiently ductile material a maximum angular velocity is reached before fracture occurs, whereas the experiments for a less ductile material show that fracture occurs before this maximum is reached. Particularly for high hardening materials the strains predicted at the maximum are so large that the ductility required to reach this point is often not present in practice. Further experiments based on vacuum melted steel have appeared in a recent paper of Percy, Ball and Mellor, a considering uniform thickness disks with a central bore. A fine point in these experiments is that some disks have been stopped just before bursting and that necking at the bore has been observed in these disks. The theoretical results obtained in this paper and in several earlier papers on rotating disks 3'5-7 are based on axisymmetric computations. However, a recent investigation by Andersen 8 includes an analysis of bifurcation into an eccentric shape. In the present paper the mechanism of disk bursting is analysed in more detail, assuming fully ductile material behaviour in the whole range considered. The interest is focussed on the possibility of bifurcation away from the axisymmetric state and on the sensitivity to imperfections to see if these phenomena are important for the tensile instabilities in consideration, as has been found earlier for a spherical shell under internal pressure 9:° and for a plate under equal biaxial stretching. ~ A large strain elastic-plastic theory is used to describe the material behaviour, and the present investigation is confined to uniform thickness disks. Numerical computations are made by a linear incremental method both for the axisymmetric case and for non-axisymmetric, imperfect disks. Computations based on 109

1 l0

VIGG{) TVERGAARD

the three dimensional approximation the

general

theory

H i l l . L''~3 S o m e is i n c l u d e d

theory

are included further

of

as well as computations in t h e i n v e s t i g a t i o n .

uniqueness

discussion

in a v e r y r e c e n t

and

bifurcation

of applying

paper

The

in t h e c o n t e x t bifurcation in

this theory

of the plane stress

analysis

elastic-plastic with centrifugal

is b a s e d

solids force

due

on to

loading

o f S t o r ~ . k e r s . ~4

2. P R O B L E M F O R M U L A T I O N The analysis of the rotating circular disks is performed in the rotating coordinate system, in which each material point is at rest or, at least, has negligible accelerations, so that the centrifugal force loading can be considered as quasi-static. A Lagrangian formulation of the field equations is adopted, with a material point identified by the coordinates x ~ in the reference configuration. Here, cylindrical coordinates are employed, with radius x ~ - r, circumferential angle x z = 0, and axial coordinate x ~= z, In the reference configuration the volume and surface of the body are denoted by V and S, respectively, and the covariant c o m p o n e n t s of the metric tensor are denoted by g#. with determinant g" while in the current configuration these quantities are denoted by G.~ and G. The Lagrangian strain increments are given by ]

}

"k

/~ -

7j,i = ~ (ti~.j +tii., ) + ~ ( u , u~., + u ~u~.i )

(2.1)

where u~ are the covariant c o m p o n e n t s of the displacement vector on the reference base vectors, I t., denotes covariant differentiation in the reference coordinate s y s t e m , and ( ) denotes differentiation with respect to some monotonically increasing parameter that characterizes the loading history. Here, and subsequently Latin indices range from 1 to 3, and Greek indices range from 1 to 2. The contravariant c o m p o n e n t s ~-'J of the Kirchboff stress tensor on the e m b e d d e d deformed coordinates are related to the contravariant c o m p o n e n t s of the C a u c h y stress tensor o-~ by (2.21

r"-~ k/(G/gjo"L

The incremental principle of virtual work takes the f o r m "

f lk"6il,,+'r"u~.~6tJ,~}dV=fsi"'Za,dS+f p/'6a,dV

12.3)

where the variations 3~ii of the strain increments are related to the displacement increment variations by (2.1)..p is the m a s s density in the reference configuration,/~ are the increments of body force per unit mass. and T' are the nominal traction increments given by Ti = (÷,i + ÷k~u,~ + ,r~u,}ni

(2.4~

on a surface with normal n i in the reference state. For the disk rotating with angular velocity ~o the increments of the body force v e c t o r / ' per unit mass, resulting " ~m the centrifugal forces, are of the form ]' = (w-~ oh~+ oJ-~&~

12.5)

where ,b' are linear functions of uL In particular for the cylindrical coordinate system with rotation about the xLaxis ~' and ~ ' are given by dp I

x l + u I,

&e-u

~,

~b'=0

12.6)

The disks to be considered here are axisymmetric with a bore at the center. In the undeformed state, to be used as reference, the inner radius is R~, the outer radius is R,. and the thickness is t. Apart from a special computation accounting for the flexibility of the shaft, which will be discussed later, all surfaces are taken to be traction free. T h u s . the boundary conditions for the disk are I"~='/'2=T'~=0,

at r - R ~ , R , ,

andat z=+-t[2.

12.8)

For an elastic-plastic solid with a smooth yield surface the stress increments are related to the strain increments by a constitutive equation of the form .i-'~ = L"k~'ih~

(2.9~

where the instantaneous moduli have two branches, one corresponding to plastic loading, the other to elastic unloading. Here, a finite strain generalization of J,-flow theory shall be used, which has been proposed by Prof. B. Budiansky (unpublished work) and has been reviewed in detail by Hutchinson. '~ The elastic strain-rate is a s s u m e d to be given by • v_

1

r/~j - ~ {( I + v)GikGi~ - vG,iGk~} *k~

(2.10)

On the burst strength and necking behaviour of rotating disks

111

where E is Y o u n g ' s modulus, v is P o i s s o n ' s ratio, and ~.~i are the contravariant c o m p o n e n t s of the s y m m e t r i c J a u m a n n rate-of-change of the Kirchhoff stress, which are related to the convected rate by ~ii = ~ii +

Gitris.iltl + GJJ,r,.iha.

(2.11)

The plastic strain-rate is taken to be given by

e "0. ~i

3 = a ~ (liE, - IlE)so#,.Itr,

(2.12)

where s is =

13

I

ks\ Is"

#

o',=t~Gi, Giis s ) a=

1, 0,

for for

-

(2.13)

r ~s- ~ G'SGi,sr u

6",=

,

tr,=(o'Dm~x cr,<(~r,)m,,

Oi,Oiss'SSr 's

(2.14)

6-,>0 d,<0.

(2.15)

and or

The parameter E, is determined from a uniaxial s t r e s s - s t r a i n curve, and for the special case X / ( G I g ) = I E, is exactly the slope of the true stress-natural strain curve at stress level cry. As discussed by various authors t6 the approximation X / ( G / g ) ~ I and thus ¢~ = M j involves little error, as the relative volume change V ' ( G / g ) - 1 is entirely due to elastic strains, which remain small. T h e total strain increment is the s u m of "q"Eli and "q"pii , which can be inverted to e x p r e s s ~-~Jin terms of ~ks. Using (2.11), the tensor of moduli for the finite strain J,-flow theory is obtained as

L#k~ =

E

{~ (GikGi ~+ G . G i k)+ ~

-- 1_ ~ i t ,

2to

G#Gk s _

a

3

E/E~- I

s'Js 'w}

2 EII~OT2v)I3

it + Gi~.,t.it + G,t,riJ, + Git.ro,}.

(2.16)

~"

This tensor of moduli has the s y m m e t r i e s L #kt = L i~kt = L k~#. A similar large strain generalization can be made for the simplest deformation theory of plasticity. J:-deformation theory. H Here. the tensor of moduli L °ks is given by an expression of the same form as (2.16), with E and v replaced by E, and v,, respectively, where E~ is the ratio of stress and strain for the uniaxial true s t r e s s - n a t u r a l strain curve at stress level tr,, and E

,,=

I

,2,7,

The uniaxial s t r e s s - s t r a i n behaviour is represented as a piecewise power-law with continuously-varying tangent m o d u l u s tr E'

for

cry
for

tr>o-,.

=

(2.18)

E

~-y

-

+1 ,

where • is the natural strain, o- is the true stress, o-,. is the uniaxial yeild stress, and n is the strain hardening exponent. Part of the present investigation will be performed within the f r a m e w o r k of a plane stress approximation. In this approximation the stresses, strains and displacements are replaced by their thickness averages• As only the inplane stresses are taken to be non-zero, the constitutive relation can be written as

where the in-plane moduli are given by /~-~

= L-#~8

/" aO33ry833 - ~- ~ .

(2.20)

For the rotating disks to be considered here the plane stress approximation of the incremental principle of virtual work (2.3) takes the form

• fRR"foZ~{4""oSil,o+'c~ot~k.~SCt~n}trdOdr=fRi"fo""p/"StJJrdOdr.

(2.21)

The very useful approximate burst criterion for rotating disks established by Robinson ~ states that a disk will burst when the average tangential stress equals the tensile strength of the material. This semi-empirical criterion is based on the m a x i m u m nominal traction Tma~ obtained in a uniaxial tensile test and on the centrifugal forces acting on the u n d e f o r m e d disk. In particular for a circular disk with inner and outer radii Ri and Ro, respectively, and with uniform thickness, this nominal average stress criterion gives the

IJMS Vol. 20 No. 2 - - D

112

VIGGO TVERGAARD

following estimate of the m a x i m u m angular velocity •

3~l'm,, R,, p

R,

(2.22~

R,, -

3. B I F U R C A T I O N A N A I , Y S I S For an initially axisymmetric disk rotating at a certain angular velocity an axisymmetric equilibrium solution is available. This axisymmetric solution shall be determined here and furthermore the possiblility of bifurcation into a non-axisymmetric mode shall be investigated.

3.1 Method of analysis In the axisymmetric solution all field quantities are independent of 0 and u2 = 0. This solution is computed numerically by a linear incremental method based on the equilibrium equation (2.3), or on equation (2.21) in the case of the plane stress approximation. At each stage of the loading history equation (2.3t or (2.21) are solved approximately by the finite element method. The integrals in (2.3) or t2.21) are evaluated numerically within each element. The active branch of the tensor of moduli (2.16) for Y,-flow theory is determined in each increment as follows. If the stress state at an integration point is on its current yield surface, the plastic branch is taken to be active. If 6,, for that integration point turns out to be negative, the elastic branch is taken to be active in the next loading increment. This procedure is sufficiently accurate if small increments are used and if the transition from loading to unloading (or vice versa) occurs only few times during the loading history. The prescribed quantity in each increment is taken to be (co21 " initially. However, around and after the point of m a x i m u m angular velocity this would lead to numerical difficulties. The difficulty is avoided, without loosing the s y m m e t r y of the stiffness matrix, by using a special additional Rayleigh-Ritz procedure to prescribe a node displacement increment instead of (w'-) (see Tvergaard"~). The bifurcation analysis is based on the general theory of uniqueness and bifurcation in elastic-plastic solids due to H i l l . ' " Searching for a non-unique solution, we let (-) denote the difference between two solution increments that both correspond to the same increment of the prescribed quantity. Then, a bifurcation point has been reached if at a certain deformation level a solution can be found for the eigenvalue problem given by the variational equation

61 = 0

f3.11

t f{/.'~'-6,,~,+r"&,,~,}dV f t'a, dS-f d'a, dV'.

(Y2)

Here. the tensor of moduli is that of a linear comparison solid defined by using the plastic branch where the stress is currently on the yield surface and the elastic branch elsewhere. The nnn-axisymmetric bifurcation modes are of the form

a, tSJ,{r.:)cos(mOil

fi~ = U~(r. z ) s i n ( m 0 ) ~" u~= Udr, z)cos(mO)J

(3.3)

where (./, are independent of O, and m is an integer. With these modes the governing equations for different values of m are completely decoupled. The components of the corresponding strain tensor ~,~ take the form

G~ ~= /~'~ cos(.lO).

"6..: = ~ ' - c o s ( , . O ) .

5,, = l::',,cos {mO) I

~ : = E,.sin(mO).

il:,: [2~sin(mO).

fi,,= E . cos(mO)l

I~"4)

where E,; are independent of 0. Substitution of (3.3) and (3.41 into {3.2) and integration in the 0-direction, using the boundary conditions (2.8) and the e x p r e s s i o n s (2.51-(2.7) for the body force increment, leads to the following reduced form of the bifurcation functional

I

7rf.,.j I{l~'i~E,f-~k~ + r"(l~ ,(/!,}

~o2p(J"(/,, lr dr dz.

3 5)

Here. A is the cross-sectional area in the r - z - p l a n e in the reference configuration. If w is the prescribed quantity, the functional (3.2) takes the form (3.5) due to the constraint o3 :: 0. If the angular m o m e n t u m Jw is prescribed, where J is the current polar moment of inertia, the term - o5w] should be added to (3.5). with O5 given by the constraint o5 =-~o.fIJ. However, the non-axisymmetric modes (3.31 give J = 0 and thus o5 0. so that the functional is still given by (3.'~). Axisymmetric non-uniquenes~ for these constraints shall be discussed later in Section 4. In the plane stress approximation the displacements tL and fi~ are also of the form (3.3). with O'~ and l), only dependent on r, and ~ is of the form (3.41. Thus, in the plane stress approximation the bifurcation functional for the disks in consideration is i3.6 With a finite element approximation of the displacement fields fi,, the variational equation (3. It results in a s y s t e m of linear, h o m o g e n e o u s , algebraic equations. A bifurcation point has been reached, when the determinant of the stiffness matrix equals zero, and the corresponding eigenvector gives the shape of the bifurcation mode.

On the burst strength and necking behaviour of rotating disks

113

In the a x i s y m m e t r i c finite element analysis based on three dimensional theory the nonvanishing displacement increments are taken to be linear functions of r and z within each triangular, axis ymmetric element, and the same approximation is made for the amplitude functions U~, U21r and U~ in the bifurcation functional (3.5). The integrals in (2.3) and (3.5) are evaluated approximately by one central integration point within each element. The grid employed consists of quadrilaterals, each built up of four triangular elements, and the c o m p u t a t i o n s here are made with 20 quadrilaterals in the radial direction and 2 through the half thickness, a s s u m i n g s y m m e t r y about the middle plane. For the similar problem of a biaxially stretched annular plate L7 results based on this grid were found to agree very well with results based on double the n u m b e r of quadrilaterals in both directions. The grid size is varied continuously in the radial direction, so that the elements are smallest at the bore. Note that although only disks of initially uniform thickness are considered here. this p r o g r a m m e based on the three dimensional theory can be used to analyse disks of any a x i s y m m e t r i c shape. In the plane stress analysis the a x i s y m m e t r i c displacement u~ and the mode amplitudes /-)E, Uflr are taken to be linear functions of r within each element. The computations are made with 20 constant thickness elements along the radius, using also here a continuously varied element length. 3.2 Bifurcation results The first rotating disk to be analysed is chosen in order to compare with the experimental results obtained by Percy et alfl for disks made of v a c u u m remelted alloy steel. Their disks had an outer radius R,, = 0-0653 m, and t/R,, = 0.2. Various bore radii were used, and here we shall choose RJR,, = 0.1. The uniaxial s t r e s s - s t r a i n curve m e a s u r e d in these experiments. 4 using a torsion test, is reasonably well approximated by the piecewise power-law (2.18), with E = 2.2. 10 ~ N / m 2, o,J E = 0.004491 and n = 35. Furthermore, u = 0.3 and p = 7800 kg/m ~ are used. With this geometry and these material properties the m a x i m u m rotational speed 98,500 rpm obtained in the present computation deviates only 1.5% from the value that Percy et al. 4 obtained by an axisymmetric rigid-plastic analysis. For this disk the angular velocity squared versus the radial expansion U of the inner edge are plotted in Fig. I. Here. U is normalised by R,, while to-" is normalised by a parameter o~,,: defined as 3o~ R,, - Ri to,," = p R,,~ R, ~.

(3.7)

It is directly seen from (2.22) and (3.7) that the m a x i m u m value of to2la~,,: would precisely equal Tm,Jo, if the nominal average stress criterion gave a precise estimate of the m a x i m u m angular velocity. Fig. I s h o w s that after the disk has yielded completely w 2 c h a n g e s little, according to the a x i s y m m e t r i c solution+ while U grows considerably. The values of U at which bifurcation occurs for various circumferential wave n u m b e r s m are also s h o w n in Fig. I+ as obtained by a n u m b e r of different approaches. The plane stress approximation s h o w s bifurcation points before the m a x i m u m angular velocity is reached+ whereas the J,-flow theory computation based on three dimensional theory predicts bifurcation considerably after the m a x i m u m . This discrepancy is due to the effect of out-of-plane shear deformations+ which increase for increasing wave number+ and which are completely neglected in the plane-stress approximation. T h u s . the three dimensional computation made for a ten times thinner disk+ t/R,, = 0.02, s h o w s good agreement with the plane stress approximation for the lowest mode numbers. Finally. J~-deformation theory predictions are also included in Fig. 1+ showing bifurcation into the m = 2 mode shortly after the m a x i m u m . This result is mainly of interest i

~

42

i

i

- Fully ~lastic

08

3

r

Initial yield

3 04

i 004

i 008

i 0 !2

0116

0'20

0 ~24

0128

0132

i

i

;

036

U/R, i

i ~11

f

J z-flow theory (I t/Ro=0. 02 ' ~ o J2- flow theory tlR,)=O 2

(I J2- flow theory • plane stress •

dz - deformation

t/Ro=o2

theory

\o ©

L

i

I

J

FIG. 1. Angular velocity squared vs radial bore expansion for disk with R;/Ro = 0.1, o-JE = 0.004491, n = 35 and v = 0.3. Lower diagram s h o w s bifurcation mode n u m b e r vs critical bore expansion.

114

VIGGO TVERGAARD

because analyses of necking in biaxially stretched sheets H'~ have shown that J,-deformation theory, used as a simple model of a solid with a vertex on its yield surface, does considerably improve the agreement with experimental results. For m = 1 the bifurcation point depends on the flexibility of the shaft. The results shown in Fig. I land in Figs. 5 and 6) for m = I correspond to a completely rigid shaft, obtained by prescribing O~(R~, z ) = O. For simplicity a flexible shaft shall be represented here as a linear spring with stiffness k, which is accounted for in the bifurcation analyses by adding the term kI/--Jj(R, ll:

13.8)

to the bifurcation functional (3.5) or (3.6). Results of a plane stress computation are shown in Fig. 2, giving a plot of the shaft stiffness k (expressed by the critical speed ~o], for a rigid disk) vs the bore expansion U 20

I

16 t

L bz~

2

k

i ~o

~ 8i

/,0

0.04

008

O 12

016

020

0 24

U/R, FIG. 2. Shaft stiffness vs radial bore expansion for disk with Ri/R,, = 0.1, o - J E = 0-004491, n =35 and v = 0 - 3 . at bifurcation. Clearly, the critical U increases for increasing k, but for k growing towards infinity the bore expansion approaches asymptotically the value U/R~ = 0.20 also given in Fig. 1 for a rigid shaft. For m >! 2 the shaft flexibility is unimportant, as the average displacement of the bore edge in any given direction vanishes. Thus, for m ~> 2 the term (3.8) does not appear, and the boundary conditions (2.8) are also satisfied at the bore edge. Now, in Fig. 1 the m = 2 mode is the critical one according to any of the computations. However, due to shaft flexibility the m = I mode may be critical at any value of U / R , smaller than that given in Fig. 1. In fact, in the experiments of Percy et al. 4 the first critical speed was encountered in the elastic range, at oJ-'/oJ,,2 = 0-22. In such cases it is well known from classical studies of rotor dynamics that the critical speed can be passed and that the rotor motion is stable at higher speeds. For a shaft flexibility chosen so that the n~ = 1 mode is critical in the completely plastic range, before any other mode, this bifurcation point may have more serious consequencies than those of the critical speed in the elastic range. However, as this depends on the shaft flexibility, and as the first critical speed is in practice usually passed in the elastic range, the interest shall here be concentrated on the modes m >/2. Among the bifurcation modes for m ~ 2 the m - 2 mode is first critical, both in the plane stress approximation and even more so when three dimensional effects are accounted for, as the stiffening effect of the out-of-plane shear deformations increases for increasing wave number (Fig. I). This also agrees with the experimental observations of Percy et al. ~ that just before bursting the disks showed pronounced necking at two diametrically opposite points of the bore. The critical bifurcation mode in terms of Lagrangian strain increments is shown in Fig. 3, as obtained by the plane stress approximation. It is clearly seen that the thinning due to bifurcation ('6::) is most pronounced near the bore. Fig. 4 shows stresses and strains in the axisymmetric state at the point of maximum angular velocity in Fig. 1, obtained by the plane stress approximation. Note the rather small variations of the hoop stress o'os in this fully plastic state, which is part of the reason for the reasonable results obtained by the nominal average stress criterion (2.22). For the strains it is of interest to note the axisymmetric thinning of the disk in the region of smaller radii. The strains agree neatly with the values obtained experimentally 4 at the maximum. The effect of a more strain hardening material, n = 8, is shown in Fig. 5 for an otherwise unchanged disk. In this case the maximum value of ~o2/oJ,,-" is higher and is reached for a considerably larger circumferential strain at the bore. The bifurcation results, based on the plane stress approximation, are rather similar to those obtained in Fig. 1, with the mode m = 2 being critical and occurring a little before the maximum. The results in Fig. 6 are obtained for a disk identical with that of Fig. I except for a larger radius of the bore, Ri[R., = 0.4. Again the bifurcation mode for m = 2 is the critical one, but here the bifurcation points for larger m occur relatively later.

On the burst strength and necking behaviour of rotating disks

0.5

115

--

-0.5

I 02

I 04

I 06

Q8

Do

•/Ro

Fro. 3. Critical bifurcation mode (m = 2) according to plane stress approximation for disk with Ri[R,, = 0.l, o ' J E = 0-004491, n = 35 and u = 0.3. 15

l

l

I

i

I 0.4

I

I

02

0.6

0.8

I

I

I0

O"

D

05

1.0

rlRo I

r

0.I0

0.05

~rr

Z

,

~ -0.05

t

~zt I

I

I

FIG. 4, Axisymmetric stresses and strains at maximum according to plane stress approximation for disk with R J R o = O. 1, o.y/E = 0.004491, n = 35 and v = 0.3. With a still larger radius of the bore the limit of a thin rotating ring, R J R o ~ 1, is approached (see Appendix). For such a thin ring with current radius R the circumferential stress is o'00 = pR2o~ 2

(3.9)

assuming incompressibility. The maximum angular velocity is reached when o'00 equals half the current tangent modulus o'eo = Et/2, at mm,x (3.10) and at bifurcation into the m'th mode the current circumferential stress is m2-3 o'oo=Elm2_2,

for m~>2.

(3.11)

Thus, the critical bifurcation mode is m = 2, and this bifurcation point coincides with the point of maximum angular velocity (3.10). Furthermore, bifurcation has occurred for all modes when o'00 = E,.

1 16

VIGGO TVERGAARD

20[

T

i6

•

T

7

~/~-~ F u l I y plastic

/

2

No \3

3

F

08

-Im~al yield 04

I

0

__

~

m _ ~ _

01

02

03

~

T

7-

i

04

. . . . .

05

I

06

07

J

08

09

U/R, I0

.....

T-

~

--

J z - flow theory

8

•

~

•

plane stress

7

T. . . .

.H

•

E 6

-I

•

5

i

•

4

7

•

I

]

.

FIG. 5. A n g u l a r v e l o c i t y s q u a r e d vs r a d i a l b o r e e x p a n s i o n f o r d i s k w i t h R , / R , = 0. I, c r , / E : 0 - 0 0 4 4 9 1 . n = 8 a n d t, = 0.3. L o w e r d i a g r a m s h o w s b i f u r c a t i o n m o d e n u m b e r vs critical b o n expansion.

I

I

I

i

12

o,:,ic 3

04

I

initial yield

Jo2

o~oa

I

006

I

008

oio

U/R, 6 5 4

I

I

1

ID

I

t

J 2 - flow theory plane stress

3 2 I

t

•

FIG. 6. A n g u l a r v e l o c i t y s q u a r e d vs r a d i a l b o r e e x p a n s i o n f o r d i s k w i t h R , / R , , = 0.4. o ' J E = 0 . 0 0 4 4 9 1 , n = 35 a n d v = 0-3. L o w e r d i a g r a m s h o w s b i f u r c a t i o n m o d e n u m b e r vs c r i t i c a l bore expansion.

On the burst strength and necking behaviour of rotating disks

117

Now, comparing equation (3.1 I) with the plane stress results in Figs. 1, 5 and 6 the following pattern appears. For the thin circular ring the critical bifurcation, m = 2, occurs at the point of m a x i m u m angular velocity. For decreasing values of R~/R,, the m = 2 mode remains the critical one and occurs before the m a x i m u m . At the same time the critical hoop stress tr0, at the bore, for m = 2, increases above the value I/2E, given by (3.11) and has exceeded two times the current tangent m o d u l u s in Figs. I and 5. for RJR, = 0-1. For comparison it is interesting to note that a cylindrical bar under uni-axial tension reaches the m a x i m u m load when o- = E, and that, in the limit of a long thin bar. bifurcation occurs immediately after this point (see H u t c h i n s o n and Miles~). Bifurcation into the modes with higher wave n u m b e r s (m >I 3) is strongly delayed by three dimensional effects (Fig. I). However, for the smaller bore radii considered, R,/R,, = 0-I, the plane stress approximation predicts bifurcation into these modes (m ~ 3) shortly after the critical bifurcation point (m = 2). This probably is connected with the fact that here the ratio troo/E, at the bore has exceeded the critical value for bifurcation in the related problem of a biaxially stretched annular plate, t7 A computation made for the limit of a solid disk (R, = 0) s h o w s that here bifurcation occurs after the m a x i m u m with a critical wave n u m b e r m varying from about 7 in the plane stress approximation to about 3 when accounting for three dimensional effects. 4. B E H A V 1 O U R O F I M P E R F E C T DISKS Initial deviations from the perfect a x i s y m m e t r y of a rotating disk cannot be avoided in practice. The imperfections m a y appear as initial thickness variations in circumferential direction or as various types of material inhomogeneities. The bebaviour of such imperfect disks shall be analysed here within the context of the plane stress approximation. The linear incremental solution is based on the equilibrium equation (2.21). following the numerical procedure described in Section 3.1. The disk is divided into elements, each bounded by two radial lines and by two concentric circles, and within each element the displacement increments ti~ and u,/r are expanded in t e r m s of quadratic "'serendipity" functions. 2~ T h e integral in (2.21i is evaluated by 4 point ( 2 . 2 ) G a u s s i a n quadrature within each element, and it has been checked that this gives good agreement with results based on 9 point ( 3 . 3 ) Gaussian quadrature. The computations to be discussed here are performed with 5 elements over the circumferential interval considered and with 8 elements of continuously varied length in radial direction. The main interest is in imperfections that will cause a significant growth of the critical bifurcation mode for m = 2. Therefore. we shall consider a disk with an initial thickness inhomogeneity of the form t = t , - A t cos (20)

(4.1)

where At is the amplitude of the thickness variation and t~ is the initial thickness of the perfect disk. This c h a n g e s both the m a s s distribution and the distribution of strength s u c h as to cause particularly large radial displacements of the heavier parts around 0 = *r/2 and 31r/2. In the numerical computation with (4.1) only one quarter of the disk, between the radii 0 = 0 and 0 = 7r/2. need be considered, and s y m m e t r y boundary conditions are used along these radii. Fig. 7 s h o w s results for an imperfect disk corresponding to that of Fig. 1 with R,/R,, = 0. I, t./R,, = 0.2 and n = 35. The parameter U* used here as a m e a s u r e of disk deformation is the average radial expansion of the bore edge. It is seen that the large imperfection. At~to = 0. I, gives a considerable reduction of the m a x i m u m angular velocity and of the corresponding average expansion U * of the bore. At the m a x i m u m a great deal of the disk is elastic, while plastic flow has localized in a region around the diameter 0 = 0. The strains at this diameter are considerably larger than those s h o w n in Fig. 4, even though the average e x p a n s i o n U* is s m a l l e r . . F o r the imperfection At~to = 0-01 there is a little reduction of the m a x i m u m angular velocity, but the m a x i m u m still occurs at a relatively small value of U*. For the very small imperfection At/to = 10 ~ the axisymmetric solution is followed closely untill the critical bifurcation point. Here, elastic unloading starts at two opposite points of the bore edge and subsequently spreads into the disk. while the angular velocity reaches a m a x i m u m and starts to decay m u c h more rapidly than for the a x i s y m m e t r i c solution. This initial elastic unloading at the bifurcation point with s u b s e q u e n t growth of unloading regions for growing bifurcation mode amplitude is typical of bifurcation and post-bifurcation behaviour in the plastic range (see Hutchinson"~).

i

./ ~xlsym

<+~08/ / '

"A,/,o=O~ 1

04 I 004

0108

J 012

UPR,

0~16

20

d24 "120

FIG. 7. Angular velocity squared vs average bore expansion for disks with a thickness inhomogeneity and with R;IR,, = 0.1, crJE = 0"00491. n = 35 and v = 0.3. x, M a x i m u m ; El, (J~)' =0.

VIGGO TVERGAARD

118

In addition to the m a x i m u m angular velocity it is also of interest to find the point at which the disk continues to expand without any more energy supply, i.e. without a twisting m o m e n t acting on the shaft. This point occurs when the angular m o m e n t u m of the disk is stationary. For a disk rotating about the x~-axis in the cylindrical coordinate system employed here the current angular m o m e n t u m is given by the expression Jw = tof~ 4 ~ t ~ p d V

(4.2}

where J is the current polar m o m e n t of inertia of the disk and ~b" is defined by (2.6). For the imperfect disks considered in Fig. 7 the point of stationary angular m o m e n t u m is reached a little after the point of m a x i m u m angular velocity. However, for the axisymmetric solution no point of stationary angular m o m e n t u m is found in the range considered (up to U/R~ = 0.5), as the relative decay of the angular velocity to is too small to balance the simultaneous relative increase of the polar m o m e n t of inertia J. The same is found for the axisymmetric solutions in Figs. 5 and 6, and for the thin rotating ring it has been proved that the angular m o m e n t u m increases monotonically for increasing radius (see Appendix). Now, the behaviour of a perfect disk does not everywhere follow the axisymmetric solution, but is well illustrated by the computation for the very small imperfection At/to = 10 5 in Fig. 7. T h u s , the solution is axisymmetric up to the critical bifurcation point, but then for growing bifurcation mode displacements the angular velocity starts to decay more rapidly and a point of stationary angular m o m e n t u m is reached on the post-bifurcation path s o m e w h a t after the m a x i m u m . If three-dimensional effects are accounted for, the critical bifurcation point may occur considerably after the m a x i m u m as shown in Fig. l, but still the point of stationary angular m o m e n t u m has not been reached before bifurcation and it is expected that this point will be reached on the post-bifurcation path, as suggested by the plane stress approximation in Fig. 7. The fact that bursting occurs after the m a x i m u m at a slightly reduced angular velocity was also observed in the e x p e r i m e n t s of Percy et a l : , but was explained as a feature of the low-powered spinning rig. In these e x p e r i m e n t s the reduction in angular velocity before bursting was up to I% of the m a x i m u m , while the plane stress approximation for At~to = l0 - ' in Fig. 7 s h o w s a reduction about 0.2%. Although a thickness inhomogeneity may reasonably represent some actual deviation from axisymmetry, one can hardly expect any noteworthy imperfection of the form (4.1) in a disk machined on a lathe. A material inhomogeneity may be more likely to appear in such a disk. Here, a special type of material imperfection shall be investigated, given by a circumferential variation of the uniaxial yield stress m and the strain hardening e x p o n e n t n of the form o'~ = o',o- Ao', cos (20)

14.3)

n = no + An cos (20).

t4.4)

In general the parameters o'y0, Act, no and An should be determined so that equation (2.18) gives the best possible fit of the two extreme uniaxial s t r e s s - s t r a i n curves. However, here we shall show the effect of varying each of the parameters o-y and n separately. In Fig. 8 o'y is varied circumferentially according to (4.3). For the low hardening material (n = 35) considered here this does give two nearly parallel uniaxial stress-strain curves, and in fact the imperfection Ao'v/o'y0 = 0.012 represents rather closely a variation between the two extreme s t r e s s - s t r a i n curves obtained experimentally by Percy et al. 4 In addition to this rather small imperfection, a computation for Ao'Jo'r.0 = 0. I is also s h o w n in Fig. 8. Obviously, the sensitivity to this type of material imperfection is quite strong, and the behaviour observed in Fig. 8 is rather similar to that induced by a thickness inhomogeneity (Fig. 7). In Fig. 9 the strain-hardening e x p o n e n t n is varied circumferentially according to (4.4), while or, is kept fixed. The effect of this type of material inhomogeneity is found to be rather limited, even for an imperfection as large as An = 10, with no = 35. This is partly due to the fact that with the power hardening law (2.18) the weakening effect of adding An to no around 0 = 0 is less pronounced than the strengthening effect of subtracting An from no around 0 = ¢r/2. "[he results obtained here for various imperfections may be regarded together with the predictions of the nominal average stress criterion (2.22), by comparing the m a x i m u m values obtained for ~o-'/co,;" with the value of Tm,~/~y, using (3.7). The tensile strength reported by Percy et al. 4 for the material in consideration I

I

--

i

I

I

j Axisym

- ~

[

1.2 - Acrr/~ro: 0012

t

i - A c r y Aryo = 0. I ~ " 0.8 3

/

0.4

l /

O0

t

0.04

OtOO

1

I

t

~

1

0.t2 0 16 020 0.24 0.28 U 7,q', FIG. 8. Angular velocity squared vs average bore expansion for disks with a material inhomogeneity and with R J R . = 0-1, O,yo/E = 0.004491, n = 35 and v = 0-3. × , m a x i m u m ; D, (Jw)' = 0.

On the burst strength and necking behaviour of rotating disks

119

/ Axisym.

1.2 ~n

- 25

~

n =10

,~

08

3

04l 0

0

I 0.04

0'08

0112

I

0.16

0120

I

O.24

0.28

U'/RI FIG. 9. Angular velocity squared vs average bore expansion for disks with a material inhomogeneity and with Ri/Ro =0.1, tr,/E = 0.004491, no= 35 and ~, =0-3. x, Maximum; 0, (J~o)' = 0. is Tmax/~, = 1"08-1.23, which corresponds to maximum angular velocities ranging from a little below that of the axisymmetric solution to a little above that obtained for the largest imperfection in Fig. 8. Thus. the maximum angular velocities predicted by the semi-empirical criterion (2.22) seem to give a useful rough estimate of the actual maxima, even for imperfect disks with the imperfection levels to be expected in practice. It may be added that the experimental support of (2.22) in a number of investigations ~-3has been obtained for disks of several different shapes with and without a central bore. In most of these experimental investigations the actual maximum was within 5-10% of (2.22), but in a number of cases the experimentally obtained maximum was more than 20% below (2.22).

5. CONCLUSIONS T h e i n v e s t i g a t i o n of b i f u r c a t i o n in r o t a t i n g disks m a d e of ductile material has s h o w n that a p o i n t of b i f u r c a t i o n a w a y f r o m the a x i s y m m e t r i c state m a y be r e a c h e d b e f o r e the p o i n t of m a x i m u m a n g u l a r velocity. F o r thin disks b i f u r c a t i o n o c c u r s r e l a t i v e l y early as p r e d i c t e d by the p l a n e stress a p p r o x i m a t i o n , w h e r e a s for t h i c k e r disks b i f u r c a t i o n is s o m e w h a t d e l a y e d by three d i m e n s i o n a l effects. A p o i n t of i n t e r e s t to the m e c h a n i s m of ductile b u r s t i n g is that b u r s t i n g does not o c c u r in the a x i s y m m e t r i c m o d e , as no p o i n t of s t a t i o n a r y a n g u l a r m o m e n t u m is r e a c h e d o n the a x i s y m m e t r i c s o l u t i o n s in the r a n g e s c o n s i d e r e d . T h u s , b i f u r c a t i o n is e s s e n t i a l for b u r s t i n g , a l t h o u g h r a t h e r little g r o w t h of the b i f u r c a t i o n m o d e a m p l i t u d e is r e q u i r e d , b e f o r e the p o i n t of m a x i m u m a n g u l a r m o m e n t u m is r e a c h e d . Both a m a t e r i a l i n h o m o g e n e i t y a n d a t h i c k n e s s i n h o m o g e n e i t y m a y c o n s i d e r a b l y r e d u c e the m a x i m u m a n g u l a r velocity. H o w e v e r , for the i m p e r f e c t i o n levels to be e x p e c t e d in p r a c t i c e the m a x i m u m is n o t m u c h b e l o w that c o r r e s p o n d i n g to the a x i s y m m e t r i c solution.

REFERENCES 1. E. L. ROBINSON,Trans. A S M E 66, 373 (1944). 2. W. E. SKIDMORE,E S A 8, 27 (1951). 3. N. E. WALDREN,M. J. PERCYand P. B. MELLOR,Proc. Inst. Mech. Engrs. 180, 111 (1965). 4. M. J. PERCY, K. BALLand P. B. MELLOR,Int. J. Mech. Sci. 16, 809 (1974). 5. A. NADAIand H. DONNELL, Trans. A S M E Sl, APM-51-16, 173 (1929). 6. H. J. WEISSand W. PRAGER,J. Aero. Sci. 21, 196 (1954). 7. M. J. PERCYand P. B. MELLOR,Int. J. Mech. Sci. 6, 421 (1964). 8. L. V, ANDERSEN,M.Sc. project, Dept. Solid Mech., Techn. University of Denmark (1973). 9. m. NEEDLEMAN,J. Mech. Phys. Solids 23, 357 (1975). 10. V. TVEROAARD,J. Mech. Phys. Solids 24, 291 (1976). ! 1. A. NEEDLEMANand V. TVERGAARD,J. Mech. Phys. Solids 25, 159 (1977). 12. R. HILL, J. Mech. Phys. Solids 6, 236 (1958). 13. R. HILL, Problems of Continuum Mechanics, p. 155. S.I.A.M. Philadelphia (1961). 14. B. STORAKERS,J. Mech. Phys. Solids 2S, 269 (1977). 15. B. BUDIANSKY,Problems of Hydrodynamics and Continuum Mechanics, p. 77. S.I.A.M. Philadelphia (1969). 16. J. W. HUTCHINSON,Numerical Solution o[ Nonlinear Structural Problems (Edited by R. F. HARTUNG), p. 17. ASME, New York (1973).

IJMS Vol. 20 No. 2--E

120

VIGGO TVERGAARD

17. V. TVERGAARD, On the numerical analysis of necking instabilities and of structural buckling in the plastic range, Proc. Int. Conf. on Finite Elements in Non-linear Solid and Structural Mechanics, Norway, August 1977, Tapir Publishers, Trondheim, N o r w a y (to appear). 18. S. STOREN and J. R. RICE, J. Mech. Phys. Solids 23, 421 (1975). 19. J. W. HUTCHINSON and J. P. MILES, J. Mech. Phys. Solids 22, 61 0974). 20. O. C. ZIENKIEWICZ, The Finite Element Method in Engineering Science, 2nd Edn. McGraw-Hill, London (1971). 21. J. W. HUTCHINSON, J. Mech. Phys. Solids 21, 163 (1973).

APPENDIX T H I N R O T A T I N G RING A thin rotating ring can be considered as the limit RJRo ~ 1 of a rotating disk with a central bore. This model is given special consideration here, because it is simple e n o u g h to allow an analytical solution. T h e bifurcation analysis given here is quite similar to that earlier given ~7 for a thin circular ring subject to prescribed uniform radial expansion. W h e n the ring is very thin relative to its current radius R all other stresses than the circumferential can be neglected. In the following incompressibility is a s s u m e d , and the stress, strain and displacements are replaced by the cross-sectional averages. Equilibrium gives the circumferential stress or = pRZto 2

(AI)

and according to this expression the m a x i m u m angular velocity is reached w h e n or = Et/2,

at to ....

(A2)

Using the current configuration as reference, the bifurcation functional (3.5) for non-axisymmetric m o d e s reduces to the one-dimensional form I =

{ E , ~ + or(~b - ~ 1 - to2p(tir6 ~ + 6otie)}AR d0 -~ ) / R

\ ~0

(A3) (A4)

in terms of the physical displacements u~ and u0 in radial and circumferential direction, respectively. Inserting an a s s u m p t i o n of the form (3.3) for tL and tie and using ( A l L equations (3.1) and (A3) lead to the following condition for bifurcation into the m ' t h mode m2-3 or = E, m--~_2,

for m

92.

(A5)

Thus, for m = 2 bifurcation occurs at the point (A2), while for increasing m the critical stress increases to reach the value E, for m ~ . For m = 1 the condition tit = 0 used in Figs. 1, 5 and 6 leads to bifurcation at or = 17.,, for m = I.

(A6)

However, as discussed in relation to Fig. 2, the critical stress m a y be decreased arbitrarily by a shaft flexibility. The angular m o m e n t u m in the axisymmetric solution is given by the expression Jto = 2¢rR3 pAto.

(A7)

Differentiation of (A7), using incompressibility, gives the following condition for reaching the point of zero rate-of-change of the angular m o m e n t u m go/to = - 2 R / R .

(A8)

However, according to l A D the increment of to for given increment of R is tblto ~ (~ E d o r - l ) R,R.

(A9)

Thus, with Edor >~0 (no strain softening), the condition (A8) is never satisfied, and any increase of radius requires external work, even after that o~ has become negative. The very recent work of Stor~.kers ~4 also includes a discussion of stability for a thin rotating ring. Here, oJ is taken as prescribed, and loss of u n i q u e n e s s is f o u n d to be associated with the a x i s y m m e t r i c mode, corresponding to (A2). In practice, however, rather than the angular velocity, for e x a m p l e the rate of energy input will be prescribed, and as this remains positive (see A8 and A9) loss of u n i q u e n e s s will be associated with the m = 2 bifurcation mode, corresponding to (A5).