Internattonal Appl ied Mechanics. VoL 35, No. 8. 1999

N O N S T A T I O N A R Y D E F O R M A T I O N P R O C E S S E S IN A M U L T I L A Y E R C Y L I N D E R

E. G. Yanyutin and L. P. Dzyubak UDC 539.3

The problem of the pulsed loading of a multilayer cylinder has been solved. The simulation is based on the linear elastic theory for homogeneous and isotropic substances. The manner in which the problem is solved guarantees exact fulfillment of motion equations and systems of initial and boundary conditions. The numerical results presented show the wave pattern of the deformation of the inside surface of a multilayer cylinder, and describe the strain state of this cylinder.

Numerical-analytical methods for the solution of dynamic equations of the theory of elasticity have undergone

vigorous development in recent years; this is ,associated with the need to research nonstationary wave processes in structural

components of various geometric form. Individual questions concerning tiffs problem have been brought to light, for example,

in 12, 3, 6]. The present paper is devoted to description of the solution of the problem concerning tile pulsed loading of a

multilayer cylinder.

1. Let us examine the nonstationary elastic defonnation of m coaxial cylindrical layers of finite length/, which are

closed one in the other; the layer material is characterized by different elastic constants ,and densities. It is assumed that the interlayer contact is not broken during deformation.

Let us number the layers in the direction of increasing radial coordinates (i = 1, 2 ..... m). Let RO, i and RI, i be the radii of the inside and outside surfaces of the ith layer of the cylinder, respectively.

The system of equations describing the nonstatiomu'y axisy mmetnc deformation of an elastic medium in a cylindrical

coordinate system in the form of displacements [5] will assume the form

0 2 OA~ c3o) 0 , u r ' i - ( k i + 2 ~ , ) +2 " "

O t z - f i r ~ Oz "

P i - - 0 2 Uz,, O A i 2 tx, O (r o)o , i )

= (X, + 2 ~t, ) - (1.1) Ot 2 Oz r Or

for the ith layer, where k i ,and P-i ,are L,'un6 elastic moduli, Pi is tile density of tile material, ur. i and u:. i ,are file radial ,and ,axial

displacements, respectively. A, = O r + - O z is the volumetric exp~msion, and o)0, , = ~ a z Or is the

projection of the vector of the rotation in the direction of the coordinate semi-axis 0.

The following system of wave equations for fimclions Ai (r, z, t ) and o)0, ~ (r, z, t ) is equivalent to Eqs. ( 1.1):

V2 I ] 1 C720~0,i -- ~ o30 , i - bi 2 O ti 2 ;

k _ _ _ _

A. M. Podgornyi Institute of Machine-Building Problems, Nalional Academy of Sciences of Ukraine, Khar'kov, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 10-17, August, 1999. Original ,article submitted July 21, 1998.

1063-7095/99/3508-0759522.00 �9 1999 Khlwer Academic /Plenum Publishers 759

V2 Ai_ 1 02 Ai (1.2) ai 2 0 t 2 "

where V 2 - 02 1 0 O 2 O r 2 + - - + ai = ( ' ~ i + ~ ) / G and b, = ~,'~/9~ are the spread velocit ies o f file longitudinal and r Or OZ 2

transverse defonna t ion waves in file layer materials, respectively.

Let us a s s u m e that the mechanical system under considerat ion was at rest at the initial time. Let boundary conditions

in the fonn of s t resses

(yr,1 (R0,1 z , t ) = ( y ~ �9 r , l ( Z , t ) ;

(yz.1 (Ro. 1 - z , t ) = (yz~ (z, t )"

G.,,, (RI.m - z . t ) = (y,'?m (z . t )

cvz.,,, (Rl.,n . z " t)=a0z.,,, (z . t ) (1.3)

be assigned to file inside and outside cylindric,'fl surfaces of file multilayer cylinder. Here, (y o (z t ) (y o (z t ) (yr~ (z , t ), r, I ' , z, 1 , , ,

and (yz?m (z, t ) are assigned functions of time and the a,,dal coordinate. The following kinematic and force conditions correspond

to the rigid c lamping o f the layers: the radial Urd told urd+l and ,~xial Uz, i and Uz,i+ 1 displacements, as well as file nonrual (yr,i and

(yr, ~ + L and tangential ~z r, i and (yz r, ~ + 1 stresses are in agreement on the ~mating surfaces, i.e.,

Ur. t ( R l a , z . l ) = u r ,1 + 1 ( R o , t + I 2 , t )

u:., ( R h , . z . t ) = u : . , + I (Ro. ,+ I z . t )

( Y r , , ( R I , , ' Z ' I ) = ( Y r . / + 1 ( R o , i + l Z , l ) "

C y z r a ( R L i , z , t ) = C y z r , ~ + l ( R o , t + i z , t ) " i = 1 , 2 ... . . m - 1 . (1.4)

Let us a s sume that there are no radial displacements and nonnal stresses on the end surfaces

G a ( r . z , t ) = O when z = 0 . 1 ,

c y z . , ( r . z . t ) = O when z = O . I . (1.5)

2. To separale the spatial variables, all fimciions used in the description of the nonstat ionary defonnat ion process

are expanded into a Fourier series in tenns of trigonometric fimctions o f the variable z so as to satisfy boundary condit ions

(1.5)

oO

U r a ( r , z , t ) = Z u n ( , t ) s i n X n Z ;

n = l

oO

cyz . , ( r , z t ) = ~ _ c ~ n i ( r . t ) s i n X n z ~.n_ n ~ (2.1) �9 - " l

n = l

The vo lumet r ic expansion A r the projection of the vector o f rotation o)0., in direction 0, the axial displacement Uz, ~,

and the componen t s of the stress tensor CYr. ,, cy z r , , and Cyo,/, ,as well as the fimclions (yr.~ (z, t ), (yz,,0 (z, t ), err, m (z, t ), and

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0 (z, t ), which reflect the pattern of stress variation on the boundary, are expanded into appropriate series by analogy with z , m

(2.1). When the orthogonality of the corresponding trigonometric functions is considered, the indicated expansions make

it possible to derive differential relationships linking the functions of only two variables r and t on the basis of Eqs. (1.1) and (1.2).

Use of expansions into Fourier series ensures the separation of the ,%xial coordinate under boundary and contact

conditions. Applying the Laplace transform in terms of the variable l to systems (1.1) and (1.2) with allowance for zero initial

conditions, and also focusing attention on the above-mentioned expansions into Fourier series, the following two systems of equations can be derived in ilmage space:

u n L ( r , s ) = a i 2 dAi nL 2bi2Lno) nL. r,, 2 dr 2 0 ,s

S S

_ o , i ) ( 2 . 2 ) n L (r, s ) = ai2 ~'n Atn L 2 bi 2 d (r 03 n L Uza ~s r s 2 d r '

d 2_ AinL + _ - + = 0 " dr 2 r -dr At

a 2 c o . L + l ,2' co.L ( ~ s z dr 2 O,i d r 0 , , - +--+X2n "]03nL J 0 , l = 0 (2.3)

bi2

Here, s is a parmneter of the Laplace transform, the superscript L indicates the Laplace transform, and the subscript n denotes the number of the coefficient in the expansion of the function into the corresponding Fourier series.

The general solution of Eqs (2.3) is represented using modified Bessei fimctions [1]

03nL(r s ) = c ~ L ( s ) K , r ~ s 2 + L ~ b , 2 +-4., ( s ) I ! "~/s2+~.ffb, 2 0 , i , ,

C nL where ,j (s) U = 1, 4 ) are mbitrary functions of fl~e p~anmter s, as subsequently defined from file boundary conditions, and

li(x ) and Ki(x ) (i = 0, 1) are the Bessei function of file imagixmry argument and a MacDonald function of ith order, respectively. C nL Turning attention to ,arbitrary functions j./ (s) (/= 1, 4 ), the expressions for the coefficients of the expansion of

the displacements can be represented in the following form on the basis of (2.2) and (2.4) in image space:

u n L ( r ' s ) -ai2 d 1,, (s)OSR~ 2 I

)} ~., Io G2 + z.ff ~2

2bi2)~-nfcnL [r Xls2 2 ] / q s 2 + ) v 2 b 2 - s 1 3., C,) 0 "R0"h,,% b, + X, b, =

c n L ( s ) O - s R , / h ' ( r ]/.X/s 2 + L 2 } + -4., It ~, Xls 2+;~]b, 2 bi 2 "

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nL a2 n ,.L s , , Uza ( r , s ) = - - - - - ~ . S IJ (S)O K 0

-~R, ,~, , (~_ 2 z)} C "L (s) 0 xls2 + + 3,i 0 [ a ~'n ai

- - ; -~ e~ " 3:

c n L (s ) O - s R l / b , l l ( b ' ~ S 2 + ~ 2 bi2 ) / ' ~ s 2 + ~ 2 bt 2 (2.5) + r "43

Additional cofactors were introduced to realize the inversion operation in expressions (2.5); this made it possible to

use tabulated relationships 14] and obtain the following relationships:

II ,i t2'i t ur,n( r , t ) = a i 2 H ( t l . i ) f C l n i ( x ) p n a 2 , i ( r , q , i - x ) d x +g(t2,i)~ Cn2,i ( x ) P n a l , , ( r , t z , , - x ) d x

o o

f t3 a t4,i [ - 2 b , 2 ~ . n H ( t 3 . , ) ~ C n ( x ) G ~ 2 . , ( r - x ) d x + " ( 1 4 . t ) ~ c 4 n l ( T ) G b l , i ( r , 1 4 , i - ' ~ ) d ' ~ " 3.t , t3.1

0 0

"i' t f, } u:. n, ( r . t ) = a, 2 ~-n H ( t l . , ) ( ' in, (x ) Irn~., ( r . t l . , - z ) d z + H (t2. , ) C2n., (z ) FnaL, ( r . t2a - x ) d ' c

0 0

13: 14,i

2bi2 t ' I ( , 3 . , ) ~ C 3 n l ( ~ ) Q b 2 t ( r t3. I T ) d ' r . + n ( t 4 , i ) f C 4 n d ( ' g ) _ n l , t ,14.i , - 0 b ( r - z ) d x r

o o

Tile following notation was adopted in Eqs. (2.6)

(2.6)

t l . ~ = t - (r - Ro. , ) / a , �9 t2. , = t - (R1. , - r ) / a , "

t3. , = t - (r - R0. , ) / b , " t4. i = t - (RI, , - r ) / b , �9

f2. i -- "t

o

I i, i -- ,~

Fna2.i ( r . t la - "t ) = ~ f~n2a (r , z l ) d Xl " o

14. t -- .~ . b

( ' , ,L,(r.:4.,-~) : ~ g,)l.,(r-~l Idol" 0

13, t -- Z G b ( r , " t ) ~ b n2j t3a - = gn2a (r , "t 1 ) d x 1

0

762

panl,i (r , t2,1- x )= -~rO Fanl,i ( r , t 2 , i - x )"

O F a (r t l , i x ) " P . ~ . i ( r , t l , i - T ) = ~ r nz , i , - ,

O~ (r - x ) = 0 ( r ) -nl,i ,t4,, O r Gbl" (r " t4"i - x ) "

0 ( r G f 2 ( r . t 3 a _ x ) ) Qf2 (r . t3, , - "t ) =

)=1 H(2---r-r - t ) co sh (ai ~.n ~l t (2 r / a i - t ) ) / ~ l t (2 r / a i - , ) "

7 t ~ a i

. ' ) = cos (a, (2

g n l . , ( r , t ) = - - - H - t silfll Xn~/r I l ) " 7"[ r A n h i - i -

g b (r 1 (b~ Xn n2a , t ) = r-~n sin ~t (2 r / b I + t ) ) .

The components of the tensor of stresses developed in each layer of the cylinder are calculated on the basis of expressions (2.6) cited for the components of the displacement vector.

3. Numerical realization of the developed procedure can be mmlyzed in a special ex,'unple of the plane deformation of an infinitely long two-layer cylinder. In that case. the displacement vector and stresses are calculated from the following equations:

,f ] -, n a Ur.ln (r , t ) = a, 2 H (t l . , ) ('ln.l(l:)Pna2.,(r.tl. , - z ) d z + H (t2. I) ( 2 j ( ' t )Pn l , l ( r ' t 2 , , - x ) d x "

0 0

Pn2a ( r , t 1,1 - ~ ) = a r n2,i ( r t l,i - ~ )"

/2,n - "~

h~Ul, ( r . t2. , - z ) = J" f~al j ( r . ' t I ) d ' t I o

Ii, t --

F~a2., ( r . tl. 1 - "t ) = J" fnanl.i ( r . z I ) d x I �9 o

tl , , = t - (r - Ro, i ) / a i �9 t2a = t - (R 1,1 - r ) / a i �9

f~nl j ( r , t ) = 1 H ( 2 ~ r - t t / ' ~ t (2 r / a i - t ) a i

/fin2, (r. t )= l / ~ t ( 2 r / a l + t )

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n ai2 (~r,i ---- H (tl, i ) I Cln, i ( ' r ) Pn2,, ( r , t ld - "r ) + (Xi + 2 I.t i ) O r n2,i ,

0

+ H (t2, i ) I C2n,, ( x ) p,al, i ( r , t2, i - 0

0 a ] "r Ovi + 2 ~i ) ~ T Pnl,i (r , t2,, - x ) d x

( '! ~on, i =a, 2 H ( t l,i) CI n , (x ) ' +--r2p'' a 0 p a (r - x ) ] d x - - Pn2,i ( r , t 1 , i - x ) + )v i O-r n2,, " tl,, J

+ H (t2. i ) I C2n.i ( x ) ~ + 2 ~t i a r P n l a ( r , t 2 , i - ' c ) + X, p n a l , i ( r , t 2 d - x ) d x ; 0

t,f I "~ oz. n, = ai 2 L, H (q . , ) Cln., (x ) Pna2.i (r , tL, - x ) + -~r n2.i

0

f2,1

+ H ( t 2 . i ) ~ ( 2 . , ( x [ t2, , " r )+ .n ) pnOl,, (r , _ 0

~-r t nl,, ( r , t2 , i - x ) d ' t . (3.1)

The unknown functions Cln, ( 'r) and C�89 i (x) (i = 1, 2 ), which enter into relationslfips (3.1), can be found by fulfilling

boundary conditions (1.3) ,and (1.4), according to which the stress field for the case in question is assigned as a function o f

time on the inside surface of the first layer, and on the outside surface of the second layer, and the radial displacements and

normal stresses concur on the contact surface, i.e.,

c;rnl (RO.l . t ) = oO1 n (l ) "

o o n ( t ) . C;r, 2 (RI.2, t ) = r,2

t~ ?1 u,. l ( R l , l - t ) = u , . .2 ( R 0 , 2 . t ) "

n ?./ .r/ ~r , l (RI.I " I ) = C~r. 2 (R0,2 , t ) . (3.2)

Tile writing developed for relationships (3.2) is a system of integral Volterra equations o f first order in time in terms -, F/ of functions Cln,, (-r) and ( 2,i ( ' t) (i = 1, 2 ).

4. The numerical method used to solve the system of integral equations is based on replacement of the integrals in

the Voiterra equations by the sum of the integrals over the partitioned sections on conven ing to discrete time

I I = ( J - - 1 ) A t (A t = TN/ (N - 1 ), where T N is a fixed integration limit, N is the number of nodes, and i = 1, 2 ..... N) . Functions

Clnl ( t ) and C n = -22 ( t ) (l 1, 2 ) are assumed constant for each section and equal to the value itself at the midpoint o f the section.

Let us describe the investigation o f the nonstaliouaD' dcfonnat ion of a two-layer cylinder with geometr ic parameters

R o l = 0 . 0 9 m " R l l = R 0 2 = 0 . 1 0 5 1 n R 1 2 = 0 . 1 2 m .

We will examine two alternate groupings of tile cyl inder layers, tile characteristics o f which are described in

Table 1. The first alternate scheme corresponds to a "steel-aluminum" system, and the second to ,an " 'aluminum-steel" system.

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T A B L E 1

Layer alternate scheme 1 alternate scheme 2

v E, N/m ~ p, kg/m 3 v E. N/m ~ p, kg/m ~

1 0.3 2.058 - 10 I~ 7.8 �9 103 0.22 6 . 10 ~0 2.5 �9 103

6 �9 10 ]o 2.5 �9 10 3 0.3 0.22 2 .058 .10~] 7.8 �9 10 3

(Yr/~O r

0.9[ ~ II

0.3 h l-a, • jr ll~ e -.0.31-[ L I ~ ~ ~1.., r ~ / I ~Xr, I

I I I I I I

(~O/Cr o 0 .2 -

0 - I - - ' I r - ~ i ~ L _ ~ d~ -0.2"1 ~--~J '1 t, , ~ 7 " ~ . , ~ __0.6 t - ' d ' - . - ~ - f y ' ~ o ~ % ~ ,

u (~ : / (~ 0

~ ?t ,q -0.2 " ~

'

0 40 80 120 160 m

~r/~O , 2 r , fi o.4t )X

0 h - - 1 r-'~ r"hL'~ ,-~ ~ h ,~

I , i j F L J L I t;d v .~ l l mV

' V T , V (3"0/O" 0

0.2 r ,Ol o 1 - . - - , _ .-J. ' i ~

-o tL_ 4

~ a 0.2,-" II iI

oh--, . . aq , -4}.6 t L-J t J " ~ ~ ' ' "

/ / I I I I I

0 40 80 ,20 160 m

Fig. 1 Fig. 2

Let us assume that a pulsed loading of the form P ( t ) = - a o H ( t ) is applied to the inside surface o f the first layer,

and the outside surface o f the second layer is load-free, i.e., the fol lowing boundary conditions are realized:

% ( R o l , t ) = - cr o H ( t ) :

or,. (/~12, ! ) = 0 .

The points located at tile midpoint o f each of the layers r I = 0.0975 m and r 2 = 0.1125 m were selected as

computational points. The calculations were performed for a time interval A t = 1.5 �9 10 -7 sec.

Figure 1 shows results o f the calculation of the dimensionless radial o r, circumferential a 0, and axial a z stresses

(referred to the quantity ao), which are realized in the first and second layer, respectively. The number of time intervals m is

shown along the horizontal axes in the figures. The solid lines indicate the stresses in the first layer, and the broken lines the

stresses in the second. The grouping of layers corresponds to the first alternate scheme.

The pattern o f stress variation in the case of the second alternate scheme of layer grouping is shown in Fig. 2.

The graphical material presented reflects the wave pattern of stress variation in a composi te cylinder and corresponds

to five runs o f defommtion waves across its thickness. The jumps in the stress values are caused by the superposition o f

forward and reflected deformation waves propagating from four reflecting surfaces.

765

It must be pointed out that in the case of the second alternate layer-grouping scheme, the absolute magnitude o f the

radial stress cy r is higher titan that in the case of the first scheme. This is true for both layers. As for the circumferential c 0

and axial cy z stresses, negligible quantitative differences in their values are observed in the first layer for the different schemes.

In the second layer, however, the absolute magnitude of the circumferential c~ 0 and axial cr z stresses for the second alternate

layer-grouping scheme increases significantly ,as compared with the first, despite qualitative identity.

REFERENCES

1. G.N. Watson, Theory of Bessel Functions [in Russian], Izdatel'stvo Inostrannoi Litemtuly, Moscow (1949).

2. A.N. Guz', V. D. Kubenko, and A. 1~. Babaev, Hydroelasticity of Shell Systems [in Russian], Vyshcha Shkola, Kiev

(1984).

3. L.P. Dzyubak and E. G. Yanyutin, "Nonstationary deformation of an inclined cylinder of finite length," lzv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 123-131 (1997).

4. V.A. Ditkin and A. P. Prudnikov, Handbook of Operational Calculus lin Russianl, Vysshaya Sl~kola, Moscow (1965).

5. A. Love, Mathematical Theorv ~fElasticitv l in Russian I, Gosteklfizdat, Moscow-Leningrad (1935).

6. E .G. Yanyutin, "'Nonstationary wave processes of deformation in a closed cylindrical layer," Prikl. Mekh., 26, No. 10, 21-26 (1990).

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