Applied Mathematics, 2012, 3, 451-456 http://dx.doi.org/10.4236/am.2012.35068 Published Online May 2012 (http://www.SciRP.org/journal/am)

Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane

Circular Disk: Analytical Approach

Dmytro М. Lila1, А. А. Martynyuk2 1Cherkasy National Bohdan Khmelnytsky University, Cherkasy, Ukraine

2Stability of Processes Department, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

Email: dim_l@ukr.net

Received March 5, 2012; revised March 26, 2012; accepted April 3, 2012

ABSTRACT

For a rotating inhomogeneous circular disk a way of calculating dynamics of boundary shape perturbation and failure of bearing capacity is proposed in terms of small parameter method. Characteristic equation of plastic zone critical radius is obtained as a first approximation. A formula of critical angular velocity is derived which determines the stability loss of the disc according to the self-balanced form. Efficiency of the proposed method is shown by an illustrative example considered in Section 7. Values of critical angular velocity of rotation are found numerically for different parameters of the disc. Keywords: Axisymmetric Elastoplastic Problem; Method of Boundary Shape Perturbation; Rotating Inhomogeneous

Circular Disc; Stability Loss; Failure of Bearing Capacity; Critical Angular Velocity

1. Introduction

The failure of bearing capacity of a quickly rotating elas-tic disc [1-3] overloaded by centrifugal dilating forces is associated with the dynamics of its boundary shape per-turbation [4]. After the disc takes up new balanced shape due to considerable growth of plastic zones [5] the insta-bility [6] develops torrentially enough with the increase of rotation velocity [7]. This is stipulated by the response of the internal points to the disc contour reshaping and outrunning growth of variable radius of the perturbed elastoplastic boundary as compared with the variation of its current radius for a stable disc [8,9].

To study the stability loss and velocity dynamics of a rotating disc the perturbation method can be applied [10-12]. In the analysis of plane stress strain state this method was employed to obtain approximate critical values of the plastic zone dimensions and angular veloc-ity of continuous homogeneous circular discs [13,14], ring-shaped discs [15] including those loaded along the contour by additional radial forces [16], stepped discs and some arbitrary profile discs [17] as well as simplest inhomogeneous discs. This proves efficiency of the ana-lytical method of boundary shape perturbation (with the use of simplest numerical procedures at certain stage) which reduces essentially the amount of calculations and at the same time facilitates fruitful application of various

numerical techniques [18-20] for stability and strength calculation of turbine and other massive discs.

Meanwhile, there is still an open problem of estab-lishing by the small parameter method the relationship between the value of boundary shape perturbation, plas-tic zone radius and rotation velocity of unstable con- ti-nuous inhomogeneous circular disc corresponding to the indicated state of perturbed elastoplastic boundary. This is the subject of our present investigation.

2. Statement of the Problem

We consider a disc D consisting of two homogeneous and isotropic plane discs D1 and D2. Continuous circular disc D1 possesses radius which coincides with the internal radius of the ring-shaped circular disc D2. The external radius of disc D2 equals to . Discs D1 and D2 made from different materials are rigidly connected into one disc D along the circumference . We designate by 1

a

b

r a

s the material yield limit of disc D1, is the elasticity modulus, 1

1E is the density and 1 is the

Poisson coefficient. The corresponding material parame-ters of disc D2 are designated by 2s , 2 , 2E and 2 respectively. It is assumed that constant angular rotation velocity of disc D is higher than its critical velocity

* . This means the presence of perturbation of the initial contour circumference r b , perturbation of the current

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D. М. LILA, А. А. MARTYNYUK 452

radius of elastoplastic boundary 0 and, in general, perturbation of stress strain state of the whole (unstable) disc.

r r

We focus our attention on the self-balanced form of the stability loss of disc D which is little different from the circular form. The disc boundary equation up to the first order infinitesimal is presented in the form

= cosr b d n , ,constd 2,n

or

= 1 cos n , (1)

where = r b is a dimensionless current radius, is a small parameter, , n is a polar angle. On this basis, let us determine critical values 0=r r and

= which accompany reaching of the above men-tioned circumference by the perturbed elastoplas-tic boundary in the elastic zone of disc D. The critical values corresponding to reaching of the disc edge by the elastoplastic boundary, i.e. its contact with curve (1), should be specially calculated. We recall that for solution of these problems it is necessary, first of all, to analyti-cally establish the condition of contact of the elastoplas-tic boundary and a circumference of given radius, i.e. to construct characteristic equation with the parameter

=r h

with respect to having solved first the system of lin-ear equations

0r

0d= 0 for = ,

drr

rr u r br

0 0 d= 0 for = ,

drr

r

ur b

b

0= 0 for = ,rr r r

0= 0 for = ,r r r

0

0=u rd

= 0 ford

rr

with respect to and arbitrary constants found in the expressions for stress and displacement components

rr

0u r

, r , and determining perturbed stress strain state of the rotating disc D. The above mentioned linearized perturbations of the first order of smallness satisfy differential balance equations of plane problem and partial differential equations of relationship between stresses and displacements [1]. Unperturbed stress state (designated by the upper index 0) is determined by ordi-nary differential equations of quasistatic equilibrium and constraint equations in the elastic zone or by the yield Saint-Venant condition [5] in the plastic zone.

u

In view of the instability development mechanism of the inhomogeneous disc under consideration the stated problem will be solved for each of the four cases: (a) D1peD2e (Figure 1(a)); (s2) D1pD2pe (Figure 1(b)); (b)

D1eD2pe (Figure 1(c)); (c) D1peD2pe (Figure 1(d)).

3. Solution in the Case D1peD2e

In order to use boundary and conjugation conditions

1 = 0 for = 1,e eAu (2)

2 = 0 for = 1,e

e duA

d

(3)

0= 0 for = ,e (4)

0= 0 for = ,e (5)

3 = 0 for = ,e eA u 0 (6)

for perturbations of the first order of smallness e ,

e and e

of radial, contact and tangential stresses related to the yield limit 2s we recall that in D2e (Figure 1(a))

2 1

2 1

= , ,

, , c

eI II

III IV

a a a a

a b a b

os ,n

2 1

2 1

= ( , ) ( , )

( , ) ( , ) sin ,

eI II

III IV

c a c a

c b c b

n

and in D1e

0 01 0

0 01 0

= , ,

, ,

eI II

III IV

a a a a

a b a b

cos ,n

0 01 0

0 01 0

= , ,

, ,

eI II

III IV

b a b a

b b b b

cos ,n

0 01 0

0 01 0

= , ,

, ,

eI II

III IV

c a c a

c b c b

sin ,n

and, besides,

0 1 2 2 1 3 2 4 1

0 5 2 6 1 7 2 8 1

= ,

= .

a q a q a q b q b

b q a q a q b q b

Here 0 01= r b , = a b , , 1 , 2 and 1b are indefinite coefficients and 1 are the coefficients expressed via ,

2a, ,q

a

8qb

n 0 , , 1 , 2 and 2E 1= E ;

, ,I IVa a , , ,I IVb b , , ,I IVc c are

known functions [16]. Moreover,

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D. М. LILA, А. А. MARTYNYUK

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453

b

e e

a h

p

*

01r

b

e

a

h

p

*

02r

p

(a) (b)

b

e e

a

h

p

*

02r

b

e e

a

h

p

*

02r

p

*

01r

(c) (d)

Figure 1. Perturbed elastoplastic boundary reaching given circumference r = h.

1 2 2 1

3 0 1

= 2 6 3 , = 24 ,

= 8 3 1 ,

A C x A A

A x

x (7)

where

12 2 2 21 0 0= 3 1 2 1C s m x

,

2 2 2 22

2 2 2 21 0 0

12 2 2

1 0

= 24 = 1 1

1 3 1 2 1

3 1 2 ,

x q s k

m

l k k

2

0

11 2 1 2 2 2 2 2 1= , = , = , =s s ss q b k ,

2 ,

22 1 1

22 1 2

=

3 3 3 1 3 3 1

= 3 3 3 3 .

l

k

m

For = 1 perturbation of the first order of smallness

of the radial displacement related to is known from (1)

b

1 = cos .eu n

Therefore,

d 1= sin

d

eun n .

D. М. LILA, А. А. MARTYNYUK 454

Conditions (2)-(6) in the extended form [16] are

2 1 = 0,a A

2 2 = 0,b nA

1 2 2 1 3 2 4 1 = 0,q a q a q b q b

5 2 6 1 7 2 8 1 = 0,q a q a q b q b

1 2 2 1 3 2 4 1

3 0

cos

= 0,e

w a w a w b w b n

A u

where

1 1 5

2 2

= , , ,

= , , ,

II IV

I II IV

w b q b q

w b b q b q

6 ,

3 3 7

4 4

0

= , , ,

= , , , ,

= .

II IV

III II IV

w b q b q

w b b q b q

8

,n

Hence,

0 0= coseu U

where

0 1 2 2 1 3 2 4 1= ,U w a w a w b w b 3A

and, besides,

2 1 2 2

1

1 4 5 2 7 2 8 1 2 3 2 2 8 4 6

1

1 6 1 2 3 2 2 5 2 7 2 2 8 4 6

= , = ,

= ,

= .

a A b nA

a q q a q b q q a q b q q q q

b q q a q b q q a q b q q q q

As a consequence, the characteristic equation with re-spect to the plastic zone radius corresponding to the mo-ment of contact of the perturbed elastoplastic boundary and the mentioned circumference = =h b becomes

0 0 = 0.U (8)

The critical value of angular velocity corre-sponding to the critical value of radius of the plastic do-main 0 0= , 0 0 ,

( 0 is a critical radius of the plastic zone D1p for which the disc loses its stabil-ity) is obtained in terms of (7).

4. Solution in the Case D1pD2pe

Now, in contrast to Section 3, the elastic domain is ho-mogeneous and represents zone D2e (Figure 1(b)), therefore [14,15]

2 2

=

2 2 co

e

n n n nnA nB n C n D n

s ,

2 2

=

2 2 c

e

n n n nnA nB n C n D n

os ,

in ,2 2= se n n n nnA nB nC nD n

A , B , Cwhere and are indefinite coefficients. Correlations (7) are written as

D

2

= 2 6 3 , = 24 ,

1 2 2 1

33 0 0= 2 6 3 1 ,

A C x A A x

where

A C x (9)

2 22 2 0 0= 1 3 3 3 1 1C x

1,

1 20 0

2 22 2 0 0

12 3 1 2

2 2 0 0 0

0 02

= 2x 1 1

3 3 3 1 1

3 3 3 1 8 1 1 ,

= .

s

r b

Therefore, the system of equations for determination

of

0( )eu has the form

12 2 = 0,nB n C n D A

2 = 0,A D A

nA

B C

2 20 0 0 02 2n n n nn A n B n C n D = 0,

2 20 0 0 0 = 0,n n n nA B C D

2 20 0 0 0

3 0

2 2 c

= 0.

n n n n

e

n A n B n C n D n

A u

os

Hence,

0 0= coseu U ,n

where

0

2 20 0 0 0 32 2 ,n n n n

U

nB n C n D A

and, moreover,

= nA

2 21 0 0

2 22 0

20

= 1

2 1 2 2 ,

n

n

A A n n

A n n n n N

2 21 0 0

2 2 22 0 0

= 1

2 1 2

n

n

B A n n

2 ,A n n n n N

2 21 0 0

2 22 0 0

= ( 1)

( 2) ( 1) 2

n

n

C A n n

,A n n n N

2 21 0 0

2 22 0 0

= 1

2 1

n

n

D A n n

2 ,A n n n N

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D. М. LILA, А. А. MARTYNYUK

Copyright © 2012 SciRes. AM

455

Thus, characteristic Equation (8) is constructed. The characteristic equation with a parameter with re-

spect to critical radius of the plastic domain which re- ached the external edge of the disc D reads [9]

2 10 0

2 22 2 0

13 2

1 0 0

0 02

= 0,5 1 1

3 3 3 1 2

4 3 1 1

= .

x k k

k

k

r b

2 2 2 2 2 20 0 0 0= 2 1 .n nN n n

0

1 ,

0 01 1 = 0,U (10)

where 0 0U for 0

. 6. Solution in the Case D1peD2pe

In order to follow the perturbation dynamics of elasto-plastic boundary between D2p and D2e (Figure 1(d)) we will take into account (see Section 4) the fact that

5. Solution in the Case D1eD2pe

compared stress states (unperturbed and per-of the disc cases (s1) and (b) (Figure 1(c))

bout the neces-ction 4 having

Havingturbed) D in

12 22 2 0 0

0 02

= 1 3 3 3 1 1 ,

= ,

C x

r b

of instability development we conclude asity to preserve here all calculations of Sechanged the expression for 2 2

2= 24x q

2 2

1=

3 1x

k k

2 21 01 01 1 1

,2 3 3 1 3

s k

k

where

2 201 2 2 1 3 1= 4c c c c c 2 ,

0

21 1

2 10

= 3 1

1 2 1 2 ,

c

s k k

2 1

2 10 0

= 2 3 1

1 1 2

c

k k

,

23 1 1

2 10 0

2 21 0

1 20 2 2 0 0

= 3 3 1 3

1 1 2 1

8 3 3 1

2 3 3 3 1 2

c k

s s

2 .

k

Similarly to Section 5 the rest of calculations including general form of characteristic Equations (8) and (10) coincide with the results presented in Section 4.

,

7. Examples and Concluding Remarks

For inhomogeneous disc with the parameters = 2n= 0.9 , = 0.93 , 1 = 0.31 , 2 = 0.3 , = 1 , = 0.99 , = 0.99s and 2 2 = 0.01s E , which loses its

stability according to case (a) for 0 = 0.7148 and

2 = 1.6825q the values of critical radius 0 of

nd rela otation velocity plastic zone D1p a tive critical r*

2q are presented in Table 1. Table 2 presents characteristic critical values obtained

tion of characteristic Equation (10) for in terms of solu

the disc with parameters , = 2n = 0.1 , 1 = 0.4 ,

2 = 0.3 , = 0.8s, = 1 , = 1 and 2 2 = 0.01Es which loses its stability according to case (s2) for

01 = = 0.1 , 0 = 0.7167 and 2 = 1.6674q . The same problem is solved for the disc with = 2n ,

, = 1.2 , = 1.1s and , 1 = 0.3 , 2 = 0.2 = 0.52 2 = 0.01s E whose instability develops according to

(b) (Table 3) for = 0.8 , = 0.7106 and 0

2 = 1.7310q and according to (c) (Tab ) le 4 for = 1 ,

01 = 0.1889 , 0 = 0.7112 and 2 = 1.7262q .

o s and r eloTab c-ity

δ 10

le 1. Values f critical radiu elative critical v depending on δ.

–7 10–6 10–5 10–4 10–3

*

0 0.8999 0.8996 0.8967 0.8770 0.8031

*

2q 1.7228 1.7227 1. 7046 1.7223 7193 1.

Table 2. Values of critical radius and relative critical veloc-ity depending on δ.

δ 10–7 10–6 10–5 10–4 10–3

*

0 0.9932 0.9853 0.9682 0.9310 0.8474

*

2q 1.7146 1.7144 1.7138 1.7109 1.6984

Table 3. Values of critical radius and relative critical veloc-ity depending on δ.

δ 10–7 10–6 10–5 10–4 10–3

0.9927 0.9842 0.9658 0.9255 0.8339 *

0

*

2q 1.7733 1.7732 1.7725 1.7696 1.7567

D. М. LILA, А. А. MARTYNYUK 456

Table 4. Values of critical radius and relative critical veloc-

–6 10–5 10–4 10–3

ity depending on δ.

δ 10–7 10

*

0 0 0 0 0 0.9926 .9842 .9658 .9254 .8338

*

2q 1.7779 1.7778 1.7769 1.7733 1.7572

rturbation, location and type ofturbed elastoplastic boundary and rotation velo ofunstable continuous ci disc allow quantitative conclusi e bo lithe disc su h-s T lic

e obtained results enables us to forecast the develop-

The relationships established between the value of boundary shape pe per-

city rcular qualitative and

ons to b made a ut pecu arities of per-hig peed dynamics. he app ation of

thment of unstable state and to calculate possible loss of stability and failure of bearing capacity of rotating discs.

It should be noted that basic equations of stability the-ory of spatial deformable bodies derived by linearization of nonlinear equations contain terms specified via the components of unperturbed ground state. This causes some difficulties in the problem on loss of stability, since a loading parameter associated with the critical efforts enters the basic equations. Application of the approxi-mated approach presented in the paper for stability inves-tigation of spatial elastic bodies simplifies the problem because both the perturbations ij satisfy the initial balance equations and the loading parameter is intro-duced into boundary conditions on the perturbed initial surface of the body. The loading parameter is determined by essentially more simple characteristic equations.

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