privarnikov radovskii moving load on viscoelastic multi layer base 1981 eng

8/2/2019 Privarnikov Radovskii Moving Load on Viscoelastic Multi Layer Base 1981 Eng

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ACTION OF A MOVING LOAD ON A VISCOELASTIC

MULTILAYER BASE

A. Ko Pr i var nik ov and ]3. S . Rad ovs ki i UDC539.374

On the surfa ce of a mul til aye r base ther e move s a load at a velocity which is const ant in magnitude anddirect ion, this being applied with the same intensit y over a same are a at any instant of time t r (--~, oo) .The problem is to determin e the stres ses and the displacemen ts in the base. Such a problem aris es, specific-ally, in the design of highway and airpo rt pave ments for strength with materi als which have elastic as well asviscous proper t ies .

A similar problem regarding the action of a moving load on multilayer bases was considered in earlierstudies with use of the simpl est models of viscoe lastic media [3, 6, 7] and for the case of deformation of adoub le- laye r medium [4, 5]. In the following solution there will be no cons trai nts impo sed on the mode of basedeformation, on the (finite) number of layers, and on their thickness. The viscoelastic prop erti es of the variousbase la yers can be different and are d escrib ed by the relations between str ess es and strains

where s~, e~ arc components of the deviators of respect ively the stre ss t ensors and the strain tenso rs, Ois the instantaneous shea r modulus, B is the bulk modulus, ~ is the spheri cal component of the stres s tensor,0 is the volume strain, and Fc, Fv are the relaxation kernels .

The multi layer base is a stack of n + 1 homogeneous isotropic layers. Each layer is bounded only bytwo paral lel planes. The bottom layer of the base is a half-spac e. Any two adjacent layers can be eitherfastened together or be in a smooth contact so that the conditions of coupling in such a multilayer base canapply in any sequence.

1. We consider one of the base lay ers. We will use a stationary Cartesian syste m of coordinates (x, y,z) and let the z = 0 plane in this syst em coincide with the upper bo undary plane of that layer. The Z axis ispointing downward and the X axis is pointing in the dire ction in which the load mov es. This movi ng load pro -duees in the given l aye rs st re ss es ~afi(x, y, z, t), displa ceme nts u~(x, y, z, t), and strai ns co6(x, y, z, t).

We also introduce a moving system of coordinates (x', y ' , z ') whose axes are p arallel correspon dinglyto axes x, y, z and which moves along the X axis at the veloci ty v of the load. We will dwell only on the st a-tionary state of str ess of the base. Then in the moving refere nce syste m the stress es and the displacementsin the lay er will not be functions of t ime. Let f(x' , y ' , z ') be an arbi tra ry differentiable function. Consideringthat x' = x- vt, y' = y, and z' = z, we obtain

t

~ r ( t - - ~ ) [ ( x -of ofOx" Ox '

~, y, z) dr = i f (s -- x' I f(s, y', z') d~;v jr \ v /x'

of of of of a"- f = v'- ~@ ' = @ ' Oz" = Oz ' Ot ~ Ox ~ "Using these relat ions, we write the fundamental equations for a viscoelastic layer in the moving refer encesys tem 02a~ ,~ (x , y , z ) = pv ~ ~ u~ (x , y , z ) , ~ = 1 [u~ ,,~ + u~ ,~ , ];

Dnepropetrovsk State University. State Scientif i c-Resea rch Institute of Highways. Transla ted fromPrik ladn aya Mekhanika, Vol. 17, No. 6, pp. 45-52, June, 1981. Original art icle submitte d Janua ry 29, 1979.

534 0038-529 8/81/17 06-0534 507.50 9 198_ Plenum Publishing Corporation


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[ i t ]" s - - x ( s , ! f , z ) d s "s a ~ ( x , y , z ) - = 2 6 e ~ r ( x , V , z ) - - v r ~ ~ e ~ox

( x , V , z ) - -- - B 0 ( x , y , z ) - - v r ~ o ( s , v , z ) d sx

(1.1)

w i t h t h e p r i m e s i g n s a t v a r i a b l e s x ' , y ' , z ' o m i t t e d h e r e a n d h e n c e f o r t h . T h e i n d i c e s ~ , r s t a n d f o r x , y , za n d s u m m a t i o n i s p e r f o r m e d o v e r r e p e a t i n g i n d i c e s , w h i le d i f f e r e n ti a t io n i s p e r f o r m e d w i t h r e s p e c t t o t hei n d ex b e h in d a c o m m a .

W e a s s u m e t h a t n o r m a l a n d ta n g e n t i a l s t r e s s e s a t t h e b a s e s u r f a c e , a s w e l l a s t h e so u g h t q u a n t i t ie s tr ot? ,Uc~, s a t i s f y t h e c o n d i t io n s o f e x i s t e n c e o f a F o u r i e r t r a n s f o r m w i t h r e s p e c t t o v a r i a b l e s x a n d y , w h il e th e r e l a x a -t i o n k e r n e l s F c(X ) a n d Fv (X ) s a t i s f y th e c o n d i t i o n s o f e x i s t e n c e o f a o n e - d i m e n s i o n a l F o u r i e r t r a n s f o r m [1 ]. W ew i l l d e n o t e th e t w o - d i m e n s i o n a l F o u r i e r t r a n s f o r m o f f u n c t io n f (x , y ) e i t h e r w i t h th e s y m b o l F i l l o r t h e s y m b o l~ ( (, ~ ) , w h e r e } , ~) a r e t he t r a n s f o r m p a r a m e t e r s , a n d t h e o n e - d i m e n s i o n a l F o u r i e r t r a n s f o r m o f f u n c ti o n f( x)e i t h e r w i t h th e s y m b o l F x [f ] o r t h e s y m b o l f (~ ). C o n s i d e r i n g t h a t

[ O x ~ l = ( - i5) ' F i l l ; F ~ = ( - - in?~ F i l l ; F [I ] ---- Fy [F~ fi l l ,

w e o b ta i n b y a t w o - d i m e n s i o n a l F o u r i e r t r a n s f o r m a t i o n o f e x p r e s s i o n s (1 .1 )

+ d z - - =- 1 F~ = -~ - [ [u~.0] + F [u~.d] ; ~o (~, n , z) = ~0 [ l - -1% ( - - vh)] e~;

(i 2)

~ (~ , ~1, z ) = B [ 1 - - P ~ ( - - v h ) ] 6 .

W e n o w i n t r o d u c e t h e f u n c t i o n s

a * ( 5 ) = a i - t r o ( 0 e - i g v t d t ; B * ( 5 ) = B 1 ~ F , ( t ) e - i ~ O t d t .0 0

(1 .3)

W e w il l n o w p e r f o r m t h e F o u r i e r t r a n s f o r m a t i o n o n th e e q u a t io n s o f m o t i o n , t he C a u c h y re l a t i o n s , a n dH o o k e ' s l a w . I t i s n o t e w o r t h y t h a t i n t h e s p a c e o f F o u r i e r t r a n s f o r m s o n e c a n o b t a i n t h e fu n d a m e n t a l r e l a t i o n s( 1.2 ) f o r a v i s c o e l a s t i c l a y e r f r o m t h e a n a l o g o u s r e l a t i o n s f o r a n e l a s t i c l a y e r b y r e p l a c i n g i n th e l a t t e r t h em o d u l i o f e l a s t i c i t y G a n d B w i th f u n c t i o n s G * ( ( ) a n d B * (} ) r e s p e c t i v e l y . T h i s c o n c l u s i o n a p p l i e s to a llc o r o l l a r i e s o f r e l a t i o n s ( 1.2 ). A c c o r d i n g l y , h e r e a r e t h e e x p r e s s i o n s n e c e s s a r y f o r th e a n a l o g s o f t he e l a s t i c i t yp a r a m e t e r s E , v , X

E * - - 9 B ' 6 " . v * 3 B * - - 2 6 " . ) ~ * -- B * 2 ( 1. 4)-- 3B* -}- G~ " ----2 ( 3 B * + a * ) ' - - - - y G * ,

n e e d e d f o r o u r f u r t h e r a n a l y s i s .E l i m i n a t i n g f r o m t h e f i r s t t h r e e o f E q s . (1 .2 ) t h e t r a n s f o r m s o f s t r e s s e s , w i t h t h e a id o f t h e o t h e r r e l a -

t i o n s , w e a r r i v e a t th e e q u a t io n s o f m o t i o n i n t e r m s o f d i s p l a c e m e n t s , w h i c h i n f o r m a r e i d e n t i c a lt o t h e L a m ee q u a ti o n s f o r a l a y e r a f t e r t h e tw o - d i m e n s i o n a l F o u r i e r t r a n s f o r m a t i o n h a s b e e n a p p l i ed to i t . I t c a n b e d e m o n -s t r a t e d t h a t t h e l a t t e r a r e e q u i v a l e n t t o th e s i m p l e r s y s t e m o f o r d i n a r y d i f f er e n t i al e q u a t io n s

, ~ [ d u ~ S ) - * [ d z S p z S ) - ~ - - p v 2 5 2 S ;+ u

535


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w h e r e

(~* -~" G*) ( d'zuz dS \ *[ d~uz 0d ~ T [ ~ P v ~ ~- F

p~ ---- ~ -1- Xl~; S = t~u= + i ~ ; T = i~lu~ - - i[u~. (1.6 )

T h e t r a n s f o r m s o f t h e s o u g h t d i s p l a c e m e n t s a n d s t r e s s e s i n t h e l a y e r a r e r e l a t e d t o t h e f un c t io n s S , T ,a n d u z a s f o l l o w s :

~ = = X * [ d u~ S ) , - 2\ d z -- --2G[p ( [ S + q T ) ; u ~ - ~ - - p - e ( i [ S + i ~ ] T ) ;% ~ - ~ ~ * ( d ~ff~ - - S ) - - 2 G * * lp -2 0 1 S - - [ T ) ; u y = - - p - 2 ( i q S - - i [T ) ;

- d h ~ - ( 1 . 7 )%z = (X* -{- 2G*) ~ -- L*S; ( ~g ~ - O*p ~ [([~ -- ~ ) T -- 2[qS];

T h e c o e f f i c i e n t s i n th e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s ( 1.5 ) a r e n o t fu n c t i o n s o f t h e v a r i a b l e z a n d ,t h e r e f o r e , i t s g e n e r a l s o l u t i o n is e a s i l y f o u n d t o b e

S - -~ p~p-~ ' ( - - C ,e p " -F Q e ~z ) - t - P~ ( - - Ca -"~ + CteP~);T = C~e ~'~ + Ceer~; u, = C1 -p 'z -~- C~e 'z --~ C~e p~z -I- C tep~ . (1 .8)

H e r e C . . . . . . C s a r e a r b i t r a r y f u n c t i o n s o f t h e v a r i a b l e s ~ , ~ t o b e d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i ti o n sf o r th e l a y e r , a n d

Pv ~ Pv~'~2 ( 1 . 9 )' 6*

H e r e X * a n d G * a r e c o m p l e x fu n c t i o n s o f t h e r e a l v a r i a b l e ~ s o t h a t p l a n d p 2 a r e f u n c t io n s o f t h e s a m ek i n d ( t w o - v a l u e d f u n c ti o n s ) . H e n c e f o r t h Pl a n d P 2 w i l l r e f e r t o t h o s e v a l u e s o f t h e s e f u n c t i o n s w h ic h h a v e

R e p l y 0 ; R e p 2 ~ 0 . (1 .1 0)2 . W e w il l n o w c o n s i d e r a v i s c o e l a s t i c m u l t i l a y e r b a s e d i s r e g a r d i n g t he f o r c e s o f i n e r t i a , i . e . , a s s u m i n g

t h a t a ll l a y e r s h a v e a d e n s i t y p = 0 . T h e n t h e s o l u t io n t o t h e p r o b l e m o f d e t e r m i n i n g t h e s t r e s s e s a n d t h e d i s -p l a c e m e n t s i n s u c h a v i s c o e l a s t i c m u l t i l a y e r b a s e u n d e r a m o v i n g l o a d c a n b e o b ta i n e d , i n t e r m s o f t w o - d i m e n -s i o n al F o u r i e r t r a n s f o r m s , f r o m t h e s o l ut i o n to t he c o r r e s p o n d i n g s t a t i c p r o b l e m i n t h e th e o r y o f e l a s t i c i tyt h r o u g h t h e s u b s t i t u t i o n s E ~ E * , p ~ p * w i t h G ~ G * , B ~ B * , X ~ ~ * f o r e a c h l a y e r a n d w i t h e q u a l i t i e s( 1 . 3 ) , ( 1 . 4 ) t a k e n i n t o a c c o u n t .

W e w i l l a p p l y th e m e t h o d o f t h e c o m p l i a n c e f u n c t io n , t h o r o u g h l y d e s c r i b e d i n a n e a r l i e r s t u d y [2 ] i n c o n -n e c t i o n w i th b o u n d a r y - v a l u e p r o b l e m s p e r t a i n i n g t o e l a s t i c m u l t i l a y e r m e d i a , a n d , w i t h o u t g o i n g i n to th e d e t a i l so f o b t a i n i n g th e s o l u t i o n f o r t h e s t a t i c e l a s t i c c a s e , w e w i ll g i v e th e f i n a l a l g o r i t h m o f s o l v in g t h e p r o b l e m o fd e t e r m i n i n g t h e a c t i o n o f a m o v i n g l o a d o n a v i s c o e l a s t i c m u l t i l a y e r b a s e . S o a s n o t t o c l u t t e r u p t h e t e x t w i t hm a t h e m a t i c a l e x p r e s s i o n s , w e w i ll o nl y c o n s i d e r t h e c a s e w h e r e a l l l a y e r s o f t h e v i s c o e l a s t i c b a s e a r e b o n d e d .W e s e t up t he co m pl i a nc e f un c t i o ns A k (~ , q ), B ~ (~ , q ), B ~k (~ , q ), C~k (~ , ~l) an d t he au x i l i a r y f un c t i o ns

a~ (~, q) -~ (1 - - Ah) e2Ph~; b~ (~, ~l) ~- [(1 - - 2v~) 2- : ( t -- ~;) - : - - S~] e2Vh*; (2 .1)b ~ ( ~ , q ) = ( 1 - - B ~ k ) e ~ P h ~ ; c , k ( ~ , ~ ) - - - - [ ( 1 - - ~ i ) - ~ - - C ~ k ] e ~ k ( k = n . . . . . 1 )

536


4/7

T A B L E 1 .o , m / s e e "

T e = T ~ " P ' k g ] m ~ 0 0 , 2 7 8 2 , 7 8 5 5 , 5 6 I 1 1 , 1

1 0 ,01 ,00 ,1

022002o00

o2200

4 , 3 8 74 , 3 8 74 , 3 8 74 , 3 8 74 ,3 8 74 , 3 8 7

1 ,9 5 41,9542,6732,6734 , 1 4 44 ,1 4 4

1 ,8 2 91 ,8 2 91 ,9 5 41 ,9 5 42 , 6 7 32 , 6 7 3

1 3 ,8 9 2 7 , 7 81 ,8 2 0 1 ,8 2 01 ,823 1 ,8311 ,841 1 ,8281 ,844 1 ,8382 ,0 8 2 1 ,9 5 42 ,0 8 6 1 ,9 6 7

1 ,8 2 01 ,8 6 91 ,8 2 31 ,8 7 31 ,8 8 31,934

1 , 8 2 02 , 0 8 41 , 8 2 02 , 0 6 51 , 8 4 72 , 0 9 4

w i t h th e a i d o f t h e r e c u r r e n c e r e l a t i o n sD,r = 2 - 2 ( ! - - v ~ ) - 2 [ 0 - - 4 v ~) sc -- u] + ha (Ak+~c + B,k+,s ~) +

+ g~sk+~ [0 - 2 ,~) so - - u ] + G ( sc + u ) ;2 ( I - - v k ) D k B,~ .~_ 2 - 2I * - - v~)-~" [ ( 3 - - 4 v ~ ) (1 - - 2v,~)*s ~ - - p k u] +

ha+ a ~ A k + ~ [ ( 1 - - 2 v ~ ) sc + u] + A u B , k + t [ (1 - - 2 v ~) sc - - u] + . -~--Bk+, X [ ( l - 2 ~ ,; )~ ( s ~ + c ~ ) + ( 3 - 4 . ~ - - 2 p g ) e a ] + G [ 0 - - 2 , ~ ) s ~ + p , u ] ;D~.B~ - ~ 2 - 2 ( 1 - - v*~) 2 [ ( 3 - - 4 v ~) sc + u] + A~ (Ak+iS + B~i+ td) +

+ a a B ~ + ~ [ (1 - - 2 ~ , h s c + u l + H ~ (s c - - u ) ;c , . = [ a a (1 - ~;) G,+~c + s l ( 1 - - v ; ) - ~ [ A a ( I - - v ~ ) C.~+,s + c ] - ' ;

2O~,a. = 2 - ~ ( 1 - - v ; ) - - 0 [ ( 3 - - 4 v ; ) ( 1 + e h ) + 2 p a (1 + p ~ ) + 2 ( i - - 2 v ~ ) ~ ] - -

X ( 1 - - e ~ ) - - 2 p a ( 1 + p ~ ) ] ~ H a [ 1 + 2 p ~ ( 1 + P t 3 - - G ] ;Dabn = 2 - - 0 ( 1 - - v ~ ) - ' 2 [ 2 ( 1 - - 2 v ~ ) ( 1 - - v ~ ) + P~ I - - G p a (A~+, - - B ~ + ~ ) +2D,,bvr = 2 - - 0 ( 1 - - v 2 ) ~ [ ( 3 - - 4 v ~) ( 1 + e a ) - - 2 p ~ ( 1 - - p h ) + 2 ( 1 - - 2 v ~) ~ ] +

X ( 1 - - e ~) + 2 p n ( t - - P a ) ] - - g ~ [ 1 - - P a ( 1 - - P a ) - - e n ];c . a = [ 1 - - A a ( 1 - - v : ) C ~ + , ] ( I - - v ; ) - ' [ A ~ ( I - - v ; ) C~+ , s + c ] - ' .

H e r e

p ~ = p h k , p - = V ' ~ + ~ I* ', e k = e " -2 pk , U = p k e h , C = 0 , 5 ( 1 + e a ) , s = l ~ c ;*2

Hk = h~ (Ak+,B,~+, - - B ~ + t ) ;D k = 2 - 2 ( 1 - - ~ ,; ~) -2 [ ( 3 - - 4 v k ) c + u p~ + ( l 2 v ~ )~ e ~ ] + A ~ A k + I ( s c - - u ) +

+ & B , ~ + ~ ( sc + u ) + akBk+~ [ (1 - - 2v~) s ~ + u P d + G ( s ~ - - u p a );a n d h k i s t h e t h i c k n e s s o f t h e k - t h l a y e r .

W e s ta r t f r o m t h e c o m p l i a n c e f u n c t i o n s f o r t h e h a l f - s p a c e= 2 v ,, + ,) 2 ( l - - v , , + 3 ; G , ~ + l = ( l - - v , ~ + , ) - ., + ~ l ; B ~ , , + I = I ; B , , + ~ ( 1 - - " - l * - I * l

W e d e t e r m i n e t h e a u x i l i a r y f u n c t i o n s ~ , (~ , ~ ) , 6~ (~ , , ]) , ~ ( L q ) f r o m t h e s t r e s s e s g i v e n a t t h e u p p e r b o u n d a r yo f t h e b a s e

5 3 7


5/7

1

r

-p-- [ine~:~ - - zNu ~] Iz=o; f (~, n) ----- f (x, y) er--me

W e f ind the va l ue s o f func t io ns ~a(~, ~1), a n (~, ~1), ~ (~, ~ ) fo r the l a y e r w hos e s t a t e o f s t r e s s i s t o bed e t e r m i n e d , w i t h th e a i d o f t h e r e c u r r e n c e r e l a t i o n s

ah + i = e~ {an [1 + aas -- p~ (ahc -- bns -- 1)] - - an [bns q- Pn (bt~s -- bnc + 1)]};~+ ~ = en {(zn [ - - b~,s + p~ (a : -- b: + 1)] + (~n [1 + b~: + p~ (b,~c -- b :- - I)]};

He re e k = e -p hk , c = 0 .5 (1 + e ~) , s = 0 .5 (1 - e ~( ), a nd func t io ns p , Pk , a k , bk , bTk , C ~k a r e the s a m e a s de f i ne db y e x p r e s s i o n s ( 2. 1) .

T h e t r a n s f o r m s o f s t r e s s e s a n d d i s p l a c e m e n t s i n t h e l a y e r w e d e t e r m i n e a c c o r d i n g t o r e l a t i o n s ( 1. 7) .T h e r e

2 6 * p u z k ( ~ , TI, z ) = ( r - - 8~b~) [2 (1 - - ~ *k ) C - - p z D ] - - ( ~ z n b k - - 6 ~ b ~ h ) X2 "[(1-- v~) D -- pzC] ~ (cr ~ 8~) (1 -- 2v~ q- pz) e-Pz--cche-Pz; 2G*S (~, rl, z ) =

= - - (~nan -- ~ubn) [(1 -- 2~ ) D -t- pzC ] -+. (r -- 8nb,k [2 (1 ~ v*~) C +pz D] ---- (an ~ ~h) ( l ~ 2v ~ pz ) e -p~+She-Pz; 2G*T (~, ~, z )=28h [( l - -v~) c~kC~ e-pz],

w h e r e p , a k , b k , b T k , c T k a r e f u n c t i o n s a s d e f i n e d i n ( 2 .1 ) ; C = 0 . 5 [ e p ( " h k - z ) + e - p ( Zh k + z )] ; D = e- p ( Z h k " z ) - C .T h e s t r e s s e s a n d t h e d i s p l a c e m e n t w e n ow d e t e r m i n e t h r o u g h an i n v e r s e t r a n s f o r m a t i o n

T h e s t r e s s e s a n d t h e d i s p l a c e m e n t s a r e t h u s o b t a in e d in t h e m o v i n g r e f e r e n c e s y s t e m o f c o o r d i n a t e s .R e p l a c e m e n t o f x w i t h x - v t i n t h e s e e x p r e s s i o n s i s e q u i v a l e n t t o a c h a n g e to t h e s t a t i o n a r y r e f e r e n c e s y s t e mo f c o o r d i n a t e s , i n w h i c h t h e l o a d m o v e s a t t h e v e l o c i t y v a lo n g t h e X a x i s .

3 . L e t a n o r m a l l o a d o f i n t e n s i t y q u n i f o r m l y d i s t r i b u t e d o v e r t h e a r e a o f a c i r c l e o f r a d i u s a m o v e a ta c o n s t a n t v e l o c i t y v o n t h e s u r f a c e o f a v i s c o e l a s t i c i s o t r o p i c h o m o g e n e o u s h a l f - s p a c e w i t h i n e r t i a . L e t u sd e t e r m i n e t h e d i s p l a c e m e n t U z a t t h e c e n t e r o f t h e c i r c l e .

F r o m t h e b o u n d a r y c o n d i t i o n s a t z = 0 w e f i n d t h a t ~ I,= 0= ~ ]z=o = O, -(r Iz=o = 2 na qp =~ Jt (pa ), w h e r eJ l (x ) i s t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d .

C o n s i d e r i n g t h e b o u n d a r y c o n d i t i o n s a n d t h a t U x, U y , U z ~ 0 a s z ~ o o, w e d e t e r m i n e t h e f u n c t i o n sC l ( ~, ~) . . . . . C 6( ~, ~/) f r o m r e l a t i o n s ( 1. 7) a n d ( Z .8 ). I n s e r t i n g e x p r e s s i o n s ( 1. 8) in t o t h e s e f u n c t i o n s , w e o b t a i nt h e t r a n s f o r m U z ]z = 0. A n i n v e r s e F o u r i e r t r a n s f o r m a t i o n y i e l d s t h e s o u g h t q u a n ti t y

aq :~ (P~ -- p2) PlJ~ (pa) ~d~la z (0, O, O) = ~ ,) j G~p [4p~ pxp ~-- (p~ q-pg)~]' (3.1)

w i t h P l , P2 d e fi n e d b y e x p r e s s i o n s (1.9) and (1.10), G * a n d ~ * d e f i n e d b y e x p r e s s i o n s (1.3) a nd (1 .4 ) , a ndP---- V ~ - F .q2.

N u m e r i c a l r e s u l t s h a v e b e en o b t a in e d f o r th e r e l a x a t i o n k e r n e l st t

r~ (0 = ~ 1 - e , Fo (0 = 1 - e ,

w h e r e G a n d G ~ a r e t h e m o m e n t a r y a n d t h e l o n g - t e r m s h e a r m o d u l i , B a n d Boo a r e t h e m o m e n t a r y a n d t h el o n g - t e r m b u l k d e f o r m a t i o n m o d u l i , a n d T c , T v a r e t h e r e l a x a t i o n t i m e i n s h e a r a nd in b u l k d e f o r m a t i o nr e s p e c t i v e l y . W i t h t h e a i d o f e x p r e s s i o n s ( 1. 3) a n d ( 1. 4) w e f i n d

538


6/7

~ I v ~u z I O )

0 2~ ~J 5g 8~ ,: k m / mF i g . i

G * ~- i~ v T~G q - G= .i ~ v T ~ + 1~ , = i ~v ( 3 T ~ B - - 2 G T ~ ) + 3 B % - - 2 G =

3 ( i~vV~ - -} - 1)

I t i s e a s i l y a s c e r t a i n e d t h a t t h e i m p r o p e r i n t e g r a l i n e x p r e s s i o n ( 3.1 ) c o n v e r g e s a t a ny v a l u e s o f v .W h e n v > v 0 = ~ , m o r e o v e r , t h e r e e x i s t f o u r p a t h s o f a p p r o a c h to t h e p o i n t } = 0 , 7 = 0 a l o n g w h i c ht h e i n t e g r a n d f u n c t i o n i s F ( } , ~ /) = 0 ( p -2 ) . F o r t h e r e m a i n i n g p a t h s a n d a l s o w h e r e v < v 0 a t p ~ 0 , F ( } , 0 =0 ( p - i ) . A t t h e o t h e r p o i n t s i n t h e } , V p l a n e t h e f u n c t i o n F ( ~ , 0 i s b o u n d e d . T h e q u a n t i t y ~ i s th e s m a l l e s tp o s i t i v e r o o t o f t h e e q u a t i o n

!~r162 --I

T h e r e s u l t s o f a n u m e r i c a l e v a l u a t i o n o f t h e q u a n t i t y w = - 2 (1 + v) x G u z ( 0, 0. 0 )/ a q , n a m e l y i t s v a l u e sc o r r e s p o n d i n g t o a r a d i u s a = 0 . 16 5 m o f t h e a r e a o n w h i c h t h e l o a d i s a p p l i e d , a r e s h o w n i n T a b l e i f o r G =9 4 .2 9 M P a , G oo = 3 6 .3 2 M P a , B = 2 0 4 . 3 0 M P a , B oo = 1 0 8 .9 6 M P a , a n d w i th T c = T v . T h e s e v a l u e s c o r r e s p o n dt o ~ = 0 . 8 7 4 2 a n d v = 1 2 1 . 3 m / s e c a t a d e n s i t y p = 2 . 2 - 1 0 a k g / m 3.

A t a d e n s i t y p = 0 , a c c o r d i n g t o t h e s e d a t a , a n i n c r e a s e o f t h e v e l o c i t y v c a u s e s t h e d e f l e c t i o n u z o f t h es u r f a c e o f t h e h a l f - s p a c e u n d e r t h e c e n t e r o f t h e m o v i n g l o a d t o d e c r e a s e m o n o t o n i c a l l y . T h e l o n g e r th e r e -l a x a t i o n t i m e i s , m o r e o v e r , t h e f a s t e r d o e s t h i s d e f l e c t i o n d e c r e a s e . W h e n p ~ 0, o n t h e o t h e r h a n d , t h e n t h ed e f l e c t i o n i s n o t a m o n o t o n i c f u n c t i o n o f v . I t t h e n h a s i t s m i n i m u m w i t h i n t h e 0 < v < v 0 i n t e r v a l . W e n o t et h a t t h e v a l u e s o f G , G ~ , B , a n d Boo u s e d i n t h e s e c a l c u l a t i o n s a r e c l o s e t o t y p i c a l v a l u e s f o r h i g h w a y a n da i r p o r t p a v e m e n t s , w h i l e th e s i z e o f t h e l o a d i n g a r e a i s e q u a l t o t h e s i z e o f i m p r i n t s m a d e b y t r u c k s o n t h eh i g h w a y p a v e m e n t . A n i n t e r e s t i n g c o n c l u s i o n c a n b e d r a w n h e r e , n a m e l y t h a t a t u s u a l c a r s p e e d s w i t h i n t h ev -- 0 . 2 5 v 0 ~ 1 0 0 k m / h l i m i t t h e f o r c e s o f i n e r t i a h a v e a n e g l i g i b l e e f f e c t o n t h e d e f l e c t i o n o f t h e h a l f - s p a c e .

T h e g r a p h i n F i g . 1 d e p i c t s t h e r a t i o o f d e f l e c t io n a t v e l o c i t y v t o d e f l e c t io n u n d e r a s t a t i o n a r y l o a d ,t h i s r a t i o b e i n g p l o t t e d a l o n g t h e a x i s o f o r d i n a t e s . T h e s h a d e d r e g i o n c o v e r s m e a s u r e d d e f l e c t i o n s o f a na s p h a l t - c o n c r e t e h i g h w a y p a v e m e n t w i t h v a r i o u s t y p e s o f b a s e , a t v a r i o u s p a v e m e n t t e m p e r a t u r e s , a n d v a r i o u sm o i s t u r e I e v e l s i n a l o a m y s o i l u n d e r a c a r w h e e l w i t h a l o a d o f 2 0 k N . T h e d a s h - l i n e c u r v e s r e p r e s e n t r e s u l t so f c a l c u l a t i o n s a c c o r d i n g t o e x p r e s s i o n ( 3. 1) f o r p = 2 2 0 0 k g / m 3, G = 9 4 . 29 M P a , G oo = 34 .5 3 M P a , B = B ~ =2 0 4 . 3 0 M P a ( i . e . , k e r n e l F v (t ) = 0 ) , a n d v a r i o u s l e n g t h s o f t h e r e l a x a t i o n t i m e T c : 1 ) 0 .0 1 s e e , 2 ) 0 . t s e c , 3)1.0 s e e .

A comparison of the data indicates a satisfactory agreement between the theoretically determined andthe experimentally determined dependence of the pavement deflection on the velocity of the load.

On the basis of these results, one can expect this algorithm of solving boundary-value problems for aviscoelastic multilayer base to rather completely cover all features of the behavior of highway and airportpavements under moving loads.

2 .

L I T E R A T U R E C I T E DV . A . D i t k in a n d A . P . P r u d n i k o v , I n t e g r a l T r a n s f o r m a t i o n s a n d O p e r a t i o n a l C a l c u l u s [ in R u s s i a n ] ,F i z m a t g i z , M o s c o w ( 19 61 ).A . K . P r i v a r n i k o v , " T h r e e - d i m e n s i o n a l d e f o r m a t i o n o f a m u l t i l a y e r b a s e , " i n : S t a b i l i t y a n d S t r e n g t ho f S t r u c t u r a l C o m p o n e n t s [ in R u s s i a n ] , I z d . D n e p r o p e t r o v s k . U n i v . , D n e p r o p e t r o v s k ( 19 7 3) , p p . 2 7 -4 5 .

5 3 9


7/7

3.

4.5.6.7.

B . S . R a d o v s k i i , " B e h a v i o r o f a h i g h w a y s t r u ct ~ a r e a s a v i s c o e l a s t i c m e d i u m u n d e r a m o v i n g l o a d s , ~I z v . V y s s h . U c h e b n . Z a v e d . , S t r o i t . A r k h i t e k t . , N o . 4 , 1 41 - 1 4 6 ( 19 7 5) .A . S. S e m e n o v , " A v i s c o e l a s t i c l a y e r u n d e r t h e a c t io n o f a m o v i n g l o a d , " Z h . P r i k l . M e k h . T e k h . F i z . ,No. 3, 177-180 (1975).V. I. Tsepordei, "Action of moving loads on viscoe lastic multilayer media,' Prikl. Mat. Program mim v,,Kishinev--Shtiintsa (1974), No. 11, pp. 99-109.Y. T. Chou and H. G. Larew, "Stresses and displacements in v iscoelastic pavement system s under amoving load," Highway Res . Rev ., No. 282, 25-40 (1969).S. Mandal, "A plate on a viscoe lastic foundation under a moving load," Pure App. Geophys., 93, No. 1,55-59 (1972).

COMMENT ON THE ART ICLE "WAVE PROPAG ATION INA CYLIN DRICA L SHE LL CONTAINING A VISCOUSCOM PRES SIBLE LIQUID " BY A. N. GUZ'*

A . N . G u z '

I n t h e f i r s t p a r a g r a p h o n p a g e 8 45 o f t h a t a r t i c l e , t h e s e c o n d a s s u m p t i o n w a s o m i tt e d . T h e t h r e e a s s u m p -t i o n s i n t h i s p a r a g r a p h s h o u l d c o r r e c t l y r e a d a s f o l l o w s : " I f w e l e t a 0 - ~ , i n e x p r e s s i o n s ( 1. 11 ) a n d ( 2 .8 ) ,t h e n w e h a v e t h e c a s e o f a v i s c o u s i n c o m p r e s s i b l e f l ui d . I f x , ~ z 2 = 0 a n d w - -= k '= v '= 0 , i n e x p r e s s i o n s ( 1 .1 1 ), ( 1. 1 2) ,a n d ( 2 . 8) , t h e n w e h a v e t h e c a s e o f a n i d e a l c o m p r e s s i b l e f l ui d . I n t h e l a t t e r c a s e i t i s n o t n e c e s s a r y t o a l s os a t i s f y t h e l a s t t w o k i n e m a t i c c o n d i ti o n s ( 2. 2) . ~ F u r t h e r m o r e , t h e a s y m p t o t i c r e p r e s e n t a t i o n s o f t h e q u a n t it i e sf it 2 and 833 in exp re ss io n (4 .10) on pag e 848 shou ld be

. . . . n l -+- i . ' l - -Fi s .8 . ~ - - z , ~ - - ~ - ~ c ~J ; P3 s ~ 2 , ' ~ J .

a n d t h e f i r s t o f E q s . ( 4. 15 ) o n p a g e 8 4 8 s h o u l d r e a d[( 4 v ' a ) . 1 0 '1

'+ 3 at - 0wJ

*Prikl. Mekh., I__66 No. i0 , 10-20 (1980) [SovietApplied Mechanics, i_.66, No. I0, pp. 842-850 (1980)].Transla ted fro m Prikladnaya Mekhanika, Vol. 17, No. 6, p. 52, June , 1981.

540 0038-5298/81/1706-0540507.509 1981 Plenum Publishing Corporation

privarnikov radovskii moving load on viscoelastic multi layer base 1981 eng

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