Periodic excitation of a finite linear viscoelastic solid

NUCLEAR ENGINEERING AND DESIGN 30 (1974) 349-368. © NORTH-HOLLAND PUBLISHING COMPANY

PERIODIC

EXCITATION

OF A FINITE

LINEAR

VISCOELASTIC

SOLID

H . L LANGHAAR, A.P. BORESI and R.E. MILLER

Theoretical and Applied Me¢.hanicsDepartment, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Received 19 December 1973

The response of a finite three-dimensional standard linear viscoelastic solid to periodic excitation is investigated. The theory is specialized for a circular cylindrical body that is bonded to a rigid base at one end and shaken by a prescribed oscillation of a circulm" rigid plate that is bonded concentrically to the other end. A practical relevance of the theory is the design of vibration absorbers that are made of blocks of viscoelastic materials. Four types of motion of the driving plate are treated: (1) a simple harmonic translation of the plate perpendicular to its plane; (2) an angular oscillation of the plate about its normal axis of symmetry; (3) a simple harmonic translation of the plate in ;.ts plane; and (4) an angular oscillation of the plate about a diameter. The theory provides the displacements, stresses, and natural frequencies of the body for the four types of motion. In addition, the mean power input and driving force or driving moment applied to the plate are obtained. Computer programs have been written for the solutions of finite difference approximations of the governing equations. Sample results of the computations are included.

1. Introduction One purpose of this investigation is to provide computer programs that can be used in conjunction with vibration testing of blocks of plastics to ascertain the extent to which the hypothesis of first-order linear viscoelasticity is applicable to specific materials. Test specimens of this type have been conceived to simulate, on a small scale, the main rei~ion of excitation o f soil and rock caused by a periodic disturbance applied to the ground. The theory provides a basis for designing such models, it also is relevant for the design of vibration absorbers that are made of blocks of viscoelastic materials.

2. General equations The constitutive equations of the general linear isotropic viscoelastic material [ 1] are Psmn = Qemn, where Stun is the stress deviator, emn the strain deviator, and P, Q are linear differential operators of the form

ar

ar-I

a

p:arat---7+ar_la-~+..,+al-~+ao,

as

Q:bs-~+bs_t

as-I

a

a--~+...+b,~-~+b

o.

(1)

The factors a 1 and b i are constants, and t denotes time. Volumetric strain is elastic;i.e, the cubical dilatation is proportional to the first invariant of the stress tensor. For the so-called standard linear viscoelastic material,

ai = b i = 0 i f L > 1. Consequel~tly, 2pemn + 2Gemn : Smn + 2f/sin,,. Here G i s the shear modulus, # the viscosity coefficient, and ~ the relaxation constant. Their dimensions are [(7] = [ML-17"-2], [p] = [ M L - t T -l] and [~] : [7']. Dots denote derivatives with respect to t.

(2)

350

H.L. Langhaar et aL, Periodic excitation era viscoelastic solid

The linear geometric approximations of classical elasticity theory are used. Consequently, the strain tensor-is e.mn = ~(Um, n + Un,m) ,

(3)

where u m is the displacement vector and tim. n = aUm/aX n. Only rectangular Cartesian coordinates xn are considered in this article. The summation convention for subscripts is used. The stress tensor is Omn, By definition, .... : iii i:i s = .~Okk,

e = ~egk = ~ U k k ,

Omn = Stun + S6mn,

emn = emn + eSmn,

(4)

where 8mn is the Kronecker delta, The scalars 3s and 3e are called the first invariants of the stress and strain t e n sors. Since volumetric deformation is elastic, s = Ke

K = ~ Uk.k,

e

K = I - 2v

2 G ( l +v) 1 - 2v

(5)

The equilibrium ,equations are Omn, n + Pgm = PUm,

(6)

where p is the mass density and gm are the components of the gravitational vector g on the coordinate axes. By eqs (2) and (4), 2P(dmn -- dgmn) + 2G(emn - e~3mn) + S f mn + 2¢liOmn = Omn + 217fiSH,

(7)

Hence, by eqs (3)-(5), ffmn + 217dmn = I2(t~m,n + fin, m) + G(um, n + Un,m) + ~(~K - I2)~mnl~k,k + ~(.K - 2G)SmnUk, k.

(8)

Differentiating eq. (8) with respect to xn and summing over n, we get Omn, n + 2ndmn, n = lafim, n n +Gum, nn + ~(2¢/K + IJ)fin, mn + ~(K + G)~n, mn.

(9)

Consequently, by eq. (6), I~flm, nn +Gum, nn + ~(2¢/K + bl)fln, mn + ~(K + G)un.mn = pi~rn + 2np'fi'm - Pgm.

Alternatively, eq. (I O) may be written as follows: I~VZfim + GV2Um + , 2 ~ K + ~)d. m + (K + G)e, m ffi Ogim + 2rlp'fi'm - Pgm.

Ol)

Eqs (10) and (1 1) are the differential equations of motion of the standard linear viscoelastic solid. In view of the derivatives with respect to t, they axe of third order. Since periodic motion is considered; the displacement vector is taken to be of the following form: Um= Um(x) e iWt + Urn(x).

(12)

Here x stands collectively for xz, x2, xs and e i ~ t = cos cot + i sin cot. Ultimately, only the real part o f u m is retained. Because of internal damping, the motion represented by eq. (12)can be maintained only if there is an external exciting force. Equation (12) splits Um into a statical part ~/m andan oscillatingpart Ume i"n. Substituting eq. (12) into (10), we get an equation of the form (1) e iwt + (11) = 0. Since u m is to be a solution of eq. (10), (1) = 0 and (!I) = 0. The equation (11) = 0 merely determines the statical deformation Urn. Since there is no need to determine Urn, the equation (II) = 0 is disregarded. Expression (I) does not contain Um or gin. Consequently, the solution of (1) = 0 is not altered • if gin is disregarded. Therefore, in the following we setgm --"0.The equation ( D - 0 is (G + iwl~)Um, nn + ~(2i~*/K ÷ i¢o/~+ K + G ) U ~ m n + p w 2 U m + 2i~¢o3Um = 0 i~ i/~i I

(13)

H.L. Langhaaz et al.,, Pe,ffodic excitation of a viscoelastic solid

351

or

(G + ico#)\72Um + ~(G + K + ice# + 2ico17K)b~mn + 06o2(1 + 2ico17)Urn = 0.

(14)

Eq. (14) is of the form

V2Uj + AUk,ik + BU! = O;

j = 1, 2, 3

(]5)

where A alld B are complex constants. For an elastic solid,/a = 17= 0 and A and B are real. Eqs (8) and (12) yield (Omn + 2nOmn) e - i ~ t = (G + icoldXUm, n + Un, rn) + ~(K - 2G + 2ico17K - - 2ico#)brnnUk, k.

(16)

Over a fixed part of the boundary, Um = O. If the plane xn = constant is a free surface, Omn = Omn = 0 on that plane, fi)r m = 1, 2, 3. Consequently, if the plane Xn = constant is a free surface, (G + icoUXUm, n + Un, m) ÷ ~(K - 2G + 2 i c o 1 7 K

-

2icoiJ)6mnUk. k

= 0

(17)

on that plane, for m = 1, 2, 3. If a part P of the plane xn = constant has a prescribed motion u~ = Oi sin (cot - oq), we replace this by Ui e it°t, where Ui is a complex vector point function on P, such that the real part of Ui e i c ° t is equal to ~i sin (cot - oq). Thus, complex boundary conditions are obtained. By eq. (12), Ui determines u]. in the end, only the real part of ui is kept. The constants A and B for a linear viscoelastic solid may be obtained from those for an elastic solid if the elastic constants E and K are modified suitably. The requisite values of E and K are complex, and they depend on the frequency co. This modification of the elastic constants is the essence of the method of ,:omplex damping [2, 3]. The boundary conditions for a free surface (eq. 17) are compatible with the method of complex damping. Complex elastic coefficients are not used in the following, since eqs (15) and (17) are more direct. Instead of the complex form o f urn, represented by eq. (I 2), one may use the real form Um = Urn sin cot + Vm cos cot,

(18)

where Um and Vrn are real functions o f x l , x2 and x 3. Substituting eq. (18) into (10), dropping gin, and setting the coefficients of sin cot and cos cot equal to zero+, we get CO

wuV2Um + 70a + 2r17)

a2Un

+ GV2Vm +

aXmaXn

1

(G + K)

"'3

a 2 Vn

+o

2vrn + 2np 3u

=0

aXm~Xn

(19)

w 32Vn -co#V2Vm - -~ 02 + 21(17) ~Xm~Xn + GV2U, m + 1 (G + K) a2Un '

3

+ Oco2Um - 2170co3Vm

=

O.

~Xm(}Xn

Eqs (19) are equivalent to eq. (13). A periodic exciting force of the type F - Fo sin cot + F~ cos cot generally causes a response of the eq. (18) type. For an elastic body, the force and the displacement are in phase, so that 2./w W! = f

Ffidt=O.

0

This means that ~ere is no net work input. Hosever, for a viscoelastic body (Fo, Ft) are not proportional to (Urn, I'm), so WI ÷ 0. Since the differential eqs (19) are linear and homogeneous, superposition is permissible. For instance, if the solution for an oscillating ~,oint force on a viscoelastic body is known, superposition may be used to get the solution for an oscillating press,are on a finite spot.

H.L. Langham, et al., Periodic excitation o f a viscoelastic solid

352

3. Cylindrical coordinates For cases in which the body possesses axial symmetry and circular boundaries, cylindrical coordinates are useful. Eqs (2), (4) and (5) are valid for any orthogonal coordinates, except that the divergence is no longer Ou/]Oxi, Let (r, O, z) be cylindrical coordinates, and let (u, v, w) be the projections of the displacement vector on the tangents to the coordinate lines. Then [4] eo : ( . + vo)lr~

e , = u,,

1

e~ = w , ,

1(

7=r

~o= = v= + ( w o l O ,

u ve

Wr + Uz,

"r.o = (.o I.) + v, - (,,/r),

)

e = -~ (er + eo + ez) = "~ Ur + -r + --r + wz

(20)

Subscripts on u, v and w denote partial derivatives. By eq. (7), o, + 2rt6, : 2/air + 2Get + 2(nK - p)~ + (K - 2G)e,

Oo + 2060 = 2/ado + 2Geo + 20?K-/z)~ +(/(7- 2G)e,

o= + 2r/bz = 2Off= + 2G e= + 2(r/K-/~)~ + ( K - 2G)e,

rOz ÷ 2*~Oz = VTo= + GToz,

rzr + 2714"zr = P~l=r + GTzr,

(21)

fro + 2Wi'rO = la~lro + G7ro.

The equilibrium equations are afro ar

aOr+ . 1. . 87r0 . . + -07rz - + ~Or-" = p t O0 ~, 8r r ~0 az r e _

_

.

.

1 800 8re= 2 + +--*'tO = pij, r 80 Oz r -

(22)

+

~r,, + 8to= 8r r 80

.

--+--

.

+

8oz az

- -

+

I =My. 7r= r

--

Eqs (21) and (22) yield 0it 2G 0e, 0i 8e /z 8"fro + G 87r__fio+/z 8~.._~._+ G 07rz + 2 -p"er 2v ~ + 8r + 2(nK - U) ar + (K - 2G) --St+ -r -80 • 00 oz Oz r 2G 2/a G + - - er - - - eo - 2 -- eo = p/i + 2~#~ r

r

r

a'~ro G 07,o ~ 8eo G Oee 2 0~. 1 Oe / ~ - ~ r + -'~-r + 2 - r 00 +2--, 00 + -+( ~ K - v ) ' ~ +G- ( K - 2 G r ) ' ~r+ P ~ z

07ez

07oz 8z

(23) + 2 Iz ~[ro + 2 G 7rO = p6" + 2rldb, r ag,,

r aT,=

p a%=

G aT0, -+ 2" ~ f

# -8-~- + G ar + r a0 G

+-

r

r a0

÷ 2 a -~-zz + 2('~K - u) ~

+ (K - 2 a ) az

r

7,.= : ,o~ + 2rlp~.

As before, we disregard the body force and set

-re i~°*, ~v=Vei~

- w = W e i~" |

+

-

(24)

_

:

~:+-/+~+

~:-

+-

.,~i--

' :

"

-

• "

-, .... :

,

-

+.

=.:

-++,.+

H.L. Langhaar et al, Periodi~ excitation of a viscoelastic solid

r ,,2//.~_////j

z=O

353

r--o

I'

r-b

I

z:h

Fig. 1. Cylindrical body region.

where U, V and [¢ are functions of r, 0 and z. Consequently,

(G + it~o)) Urr + Ur U r --~+

,, (

+r

Uoo + Uzz ,2

1,2 J

+

(: v, o - vo + Wz, ) + 0~2(1

- -~ +

3-

(K + G + iw/~ + 2i(.or/K)

+ 2icor/)U = O,

,/

(a + i~u) (V,,. + rv,

2Uo ] 1 v + Voo r 2 + Vzz + r 2 ] + 3" (K + G + i¢o/z + 2i~r/K)

x

X ~+~'+'~+--r (G+iw/a)

(25)

- "~

Wrr + Wr + r "7

+Pw2(l + 2iwo)V=O,

+ Wzz +

"3

(K + G + iwla + 2itor/K) Uzr +

r

+ Wzz +

+ pw2(l + 2iwr/)I¢ = 0.

A finite cylindrical body with a rigid circular plate on one end (fig. 1) is considered. The cylin,der is bonded to a rigid body on the end z = h, and, except for the plate, it is free on the end z = 0. The boundary conditions are as follows: U =V=W=0

atz=h

or =fro =rrz = 0

atr=b

az=rzr=rzo=O

atz=0

(26) and

a
With the aid ofeqs (20), (21) and (24), these conditions may be expressed in terms of U, V and W.

4, Special cases for a cylindrical body

The following four cases are considered: case 1, plate translates up and down; case 2, angular oscillation of the plate about the vertical axis; case 3, horizontal oscillation of the plate; and case 4, rocking of the plate. All four cases fall within the scope of eq. (25). These equations may be written as follows: ~2U~ti + ~Uli- U+ Uoo + ~i~Uff - 2Vo + ~(1 + 2C)([2U¢~ + ~,U~ - U + ~;V¢o - V o + ~2W¢~-)+ ~22~2U = 0, ~2V~ + ~V~-- V + VoO + ~2V~t + 2Uo + ~ (1 + 2C)(~Utio + Uo + Iioo + ~Wor) + ~"~2~j2V = 0,

~2[I,'~ "I"~WI~4" WOO+ ~2W~.~.+ ~(l + 26")(~2U~•+ ~j[/~.+ ~Vof + ~2[4/I-~.)+ n2~2[4/=O,

(27)

/

H.L. Langhaaret a~, Periodic excitation o[ a viscoelasti~solid

354 where

E l + 2icoq K(I + 2icor/) 2(1 - 2~,) G + icog ffi 2(G + icop) '

C

~ (28)

and

= r]a,

t

=

z/a,

~.2 = 2(1 -- 2v)(pco2a2C/E),

(29)

in which a is the radius of the plate (fig. l). In addition, the following identities among the constants are helpful for deriving these equations: ½K - G - ico# + i~vr/K = C - 1, G + ico~

Pco2(I + 2icor/)

K + G + ico# + 2icoqK = 1 + 2C, G + ico;~

G + ico~

f~2 = ~--.

(30)

4.1. Case 1: plate translates up and down Since the body is a circular cylinder and since the plate oscillates up and down, V = 0 and a/a0 = 0. The motion of the plate is specified by w = w o cos cot. By eqs (27), ~2U~ + ~ U f - U + ~2U~T+ ~(I + 2~'~2U/i~ + ~U~- U + ~2W~g) + ~ 2 ~ 2 U - 0,

(31)

The boundary conditions are (see eqs (21) and (26)) U=0,

W--w o

at ~"= 0,

ate=h/a,

U=W=0

U=0,

Wt-0

0 ~ 1

O~b/a

O<~h/a

at ~=0,

~Ut + ~(C - l)(~Ut + U + ~Wt) = 0

ate-b/a,

Ur+ W t - 0

Wf-Ut

0 < ~ < h/a

(32)

O<~
~Wf+~(C-IX~Ut+U+~Wf)-O Ut+Wt=0

at ~ = b[a, at ~"= 0,

l<~
l<~b/a

ate'=0,

ate=b/a,

{'=0.

Equations (31) and (32) pose an elliptic mixed boundary-value problem for the rectangular region shown in fig. 2. The unknown functions U and W are continuous in the closed rectangle. However, derivatives of U and W

/ -

.

.

•

•

.

•

.

.

.

.

~ =b/o

,,~, = hlO

.

~ r

k

r ( ~ V ~ ' ~ ' ~ ' ~ ' ~ r ~ . r l ~ Z l ~ , ~ ' I . ~

~ r ~ ' ~ H ~

.

. .

H.L. Langhaar et el., Perioch'c excitation era viscoelastic solid

355

might be discontinuous at ~ = I, ~"= 0 and ~ = b/a, ~ = h/a. Since U and W are proportional to w o, the value of w o may be assigned arbitrarily - say wo = 1. Consequently, the input parameters are b/a, h/a, C, and [2. These are all dimensionless. For the elastic case,/.l = r/= 0 and C = (1 + v)/(1 - 2~,), ~22 = p~2a2/G. The viscoelastic solid differs only in that C and ~22 are complex.

4.1.1. Axial oscillations of a slender rod if the plate completely covers the top of the body, and if the body (fig. l) is long, the situation is as shown in fig. 3. Because of the Poisson ratio effect, the bar alternately expands and contracts laterally. However, this effect has little influence on the longitudinal displacements of the cross sections, except near the ends. Consequently, as in the classical theory of wave motion in a bar, we set v = 0. Then K = 2G. Also, U2 = U3 = 0, and Ul = U(x), where x'is taken as shown in fig. 3. Hence, by eq. (14), (33)

d2U/dx 2 + k2U = 0,

where k 2 = Lo~2(l + 2 i w n ) ] / [ 2 G + ~iw(p + ~G)].

(34)

Since U(0) = 0, the solution of eq. (33) is U= Csin/~,

(35)

where C is an arbitrery complex constant. It may be represented as C = Co e ia, where Co and oe are real constants. Consequently, u = CoR[ei(Wt+a)sin/~],

(36)

where R denotes the real part. if there is no damping, rl - tt - 0 and k 2 = pco2/2G. In this case, U

(37)

= Co sin k-~ cos (wt + a). ~e

i~t

~ffff~i~Jf~fl

t' !

/ l

Fig. 3, Slender bar subjected to ax/al oscillation.

356

H.L. Langham" et al., Periodic excitation o f a viscoelastic solid

In this case, the amplitude of oscillation of the cross section with abscissa x is a sine function of.~. If o~LLo/2G)~/2 > ~r, there are nodes in the bar; i.e. there are some cross sections, other than the luted ead, that do not move. All cross sections vibrate in phase with each other. If ;t ~ 0 or ~/9 O, k is a complex constant, Set k - a ÷ ib. Then, sin/o'~ = sin ax cosh bx + i cos a~" sinh bx and -

Isin k~l = (cosh 2 bx - cos 2 ax) ~/2. Consequently, sin kx ÷ 0 if.~ > O; i.e. there are no nodes in the bar, other than the luted end, if damping exists. Evidently,

(38)

sin kx = (cosh 2 b,~ - cos 2 ax) ~/2 e i0, where tan 0 = cot ax tanh bx.

(39)

Consequently, by eq. (36),

(40)

u = Co(cosh 2 bx - cos 2 a.x) ~/2 cos ( ~ t + 0 + a).

Since 0 depends on x, the phase is a function of x; i.e. the various cross sections do not vibrate in phase with each other. For large £, u is almost proportional to cosh b£. The ratio of the amplitude at section x to that at the free end x = h is A = [(cosh 2 bx - cos 2 ax)/(¢osh 2 bh - cos 2 ah)] ~/2.

(41)

If b ~ 0 and x is large, A ~- e-~O-£).

4. Z Case 2: angular oscillation of the plate about the vertical axis For this case (fig. 4), because of the symmetry, U, V and W are independent of 0. Consequently, by eq. (27),

~2U~ + ~Uf - U + ~2Uff + ~(1 + 2C)(~2U~ + ~U~ - U + ~2W~f) + ~2~2U = O, ~2V~ + ~V/~- V + ~2V[-l- + ~'~2~2V = O,

(42)

~2W~ + ~W~ + ~2W~-t- + ~(l + 2 C ' ~ 2 U ~ - +/~Uf + ~2[¥1-~-) + ~'~2~2W = O. The boundary conditions are U=0,

V= a¢~,

U=V=W=O

v=o,

W=0

ate=h/a,

v=o,

w =o

at [" = 0,

0<

w =o atE=b/a, ~V~-V=0

O<

ate=b/a,

O<~
at ~ = b/a,

0 < ~ < h/a

(43)

< h/a

O<[
~W£+~(C- I)(~U~+U+~W[)=O UI.+W~/=0,


at~ffi0,

~U~ + ~ ( C - 1X~U~ + U + ~w~.) = 0

0~<~<1

ate'=0,

V~=O a t [ ' - - 0 ,

l<~
1 <~
Because of the axial symmetry of the problem, U = V = 0 at ~ = 0. The condition W~ = 0 at ~ = 0 is not obvious. It follows from the fact that 7~r= Oat r = 9. -

.

.

.

.

.

-

.

~

-

H.L. Langhaar et aL, Periodic excitation era viscoelastic solid

'

~

x

357

r

-z=O

~'////////f/~ll

r/ ////-////t

1

gp

i

z

T

[

PX

-*-T

r - Constant-~

i 1

p/fIFrl

z=h

Fig. 4. Energy flow.

It happens that the solution for this case is such that U = I¢ - 0 everywhere. Then the first and third of eqs (42) are satisfied automatically. The second of eqs (42) remains. In eq. (43), only the boundary conditions pertaining to V remain, since the boundary conditions for U and I¢ are satisfied automatically. It is to be noted that V is independent of 0, but it depends oi~ ~ and ~'; i.e. on r and z.

4.3. Case 3: horizontal oscillation of the plate For the case now considered, the plate capping the cylinder (fig. 4) oscillates horizontally in the x direction. Consequently, a solution of the following form is sought: U = 0 cos 0,

V - V"sin 0,

I¢ = g' cos 0.

(44)

Here, 0, F and g' are functions of ~ and ~'. Eq. (27) yields

~20~/~ + ~0~-- 2U+ ~20ff

2V+~22~20+ ~(1 + 2C)(~201~ + ~U/~- 0 + ~V~-- V+ ~2~/t~,)- o ,

~ 2 ~ + ~-/~ _ 2V + ~2~.~_ 2U + ~,,~2~2~r ~2~

~(i + 2C)(~Uf + 0 + V"+/~g'~.) = O.

+ ~gts/- ~, + ~2g,tt + f~2~2g, + ~(1 + 2C)(~20t~r + ~0 r + ~Vf + ~2~/~.~.)= O.

(45)

358

H.L. Langhaar et at.,.Periodicexcitation of a viscoelastU:solid

Following eqs (20), (21), and (24), we obtain the boundary conditions: U=uo,

W=O

V=-uo,

O= V= gt=O at[=h/a, ¢0= + t ( c -

at ~"= O,

O
O<~
IX¢O~, U + P + ewr) = o

at ¢ = b/a,

0 < r < h/=

ate=b/a, O<~
~ P ~ . O - V=O

~Pg.- W= 0 at ~"= O, l
(46)

O<[
The constant Uo is the amplitude of oscillation of the plate. The preceding boundary conditions express the facts that the part of the cylinder in contact with the oscillating plate moves with the plate, that the lower end of the cylinder is fLxed to a rigid foundation, that the cylindrical b 3undary is free of stress, and that the part of the upper end outside of the plate is free of stress. The conditions at ~ = 0 follow from the fact that er is converted to e0 and Tzr is converted to %o by a 90 ° rotation about the z axis, if r = O. More explicitly,

The strain components are expressed in terms of (u, v and w) by eq. (20). Consequently, eq. (441 yields - e l'at . cos 0, er = Ur

e0 = ~O + V ei~Ot cos 0,

3'z0 =

ei'"tsin 0,

-

7zr (~/r+ Uz) eiWt cos O, =

r

Accordingly, eq. (47) yields

O+P cos 0 = - U r m

~in O,

r

Therefore, m

~

U+V

lira ~

r-.O

r

= O,

(48)

iim Ur = O,

(49)

r'-*O

o,t o

r"*O

A sum 0 : + Vz has been cancelled from eq. (50) because of eq. (48). Equations (48)--(50) are three boundary conditions for the line r = O. However, they yield four conditions if the functions U, V and I¢ can be expanded in power series in r, since, then

,

:

,

+(0~)+ V~))+ (0~o+ V~)~)+. ~

""

and W 7-

~,:Wo

, - ~ Wr ; + " "

where primes, denote derivatives with respect to r, and subscript 0 denotes values at r = O. Consequently, by eqs

(48)-(50), U+ V=O,

0 , = O, .

V,=O,

W=O

at r = O.

.

(51)

tt.L. Langhaar et aL, Periodic excitation o f a visco~lestic solid

359

Since r = a~, these are equivalent to conditions stated in eq. (46), Eqs (51) ensure that 7,0 and coz vanish o n t h e line r = 0, where c~z is the z component of rotation of the medium. Also, they ensure that the differential eqs (45) are satisfied at the liner = 0. There appears to be adifficulry because there are three boundary conditions for each of the lines z = 0, z = h, and r =b, but there:are four bounda~conditions for the l i n e r - 0. Whether or not a solution to this boundaryvalue problem exists is an open question. Problems concerning the existence of solutions of boundary-value problems for systems of differential equations are not easy. If a solution does notexist, the initial assumption (44) must b e wrong, or else tb.e steps leading from eqs (48), (49) ~ d (50) to eq. (5 I) are invafid. The latter situation might arise if the functions U, V, and W were such that the indicated series expansions did not exist. Even if the boundary-value problem, as stated by eqs (45) and (46), is consistent, the fi~ite diiference approximation of ~ i s problem might lead to inconsis~.ent algebraic equations. To avoid such a situation, the boundary condition U+ V= 0 at r = 0 is waived. Since the boundary conditions on the lines z = 0 and z = h ensure that U + V = 0 at the points r = O, z = 0 and r = O, z = h, it is likely that U ÷ V i~ small on the line r = O.

4.4. C~w 4: rocking o f the plate in this case~ eqs (44) and (45) are again applicable. The boundary conditions at the plate are

~/=aO~

U = i?= 0,

at ~'= 0,

0 ~ I

(52)

where the consta~t ~ is the amplitude of angular oscillationof the plate. The other boundary conditions are the same as in eq. (46). At ~thecorner ~ = b/a, ~ = 0, there are five boundary conditions, arising from the requirements ar = 0, Oz = 0, 7",z= 0, ~'z0= 0, 1"r0= 0. However, these five boundary condltions lead to an overdetermined set of algebraic equations ff the finitedifference method is used and the corner is a net point. This difficulty arises in all cases. One way to avoid it is to arrange the net so that the corner is not a net point, but this procedure complicates the finite difference approximations of the boundary conditions. Consequently, we arbitrarilyred~,ce the corner boundary conditions to three in some way - say to or = a:, 7"rz = 0, TrO -- 7zO. Then,

U~- Wg =0,

U~ + W~ =0,

~(I/'~- Vi.) - V = g ( U -

(53)

14/) atf,-b/a,

~'=0

where for case I, g = V = O; for case 2, g = O; and for cases 3 and 4, r = I.

5. Stresses, energy flux, driving forces, and Jresonance The stresses and the flux of energy within the body are now coP.sidered. Also, the amplitude of the force required to cause the prescribed motion of the plate is derived. Integration determines the energy input per cycle. A graph of the amplitude of the exciting force versus the frequency shows dips whose abscissas a~e the resonant frequencies. Attention is confined to a circular cylindrical body with a concentric circular plate bonded to one end (fig. 4). Consequently, cylindrical coordinates r, 0 e~ndz are used.

5.1. ,Stresses in the body The strains are determined byeqs (20). Consequently, with eq. (24), er = Ur ©

,

e0 =

'yzr=(Wr'÷Uz)e iwt,

U + rt • _iwt

vOjV

,

ez =Wz eiWt,

7roffil Uo÷Vr-

eiwt

7or e=~-

r1 Ur÷--------lr

eiWt '

(54) .

H.L.Langhaar et aL, Periodic excitation era viscoelastic solid

360 Eq. (21) #yes o, + 2n , =

+ it~/l)Ur e i°~t + 1_. [X -- 2G + 3

-

2(nK - u)i.,]

~ U + Vo i. Wz e i°~t,

(a)

r

and similar eft ~ations for o0, ~r0, etc. The general solution of eq. (a) is

Or = CI e-tl2n

4

2(G + i~/~)/Jr e i°Jt + K - 2G + 2io.'07K - / z ) 1 + 2k,n/ 3(1 + 2 i ~ )

(Ur -i U + Ire + W ~ e i°~' __ ""--~ z] •

(b) o

The term Ct e -t/2n attenuates, and eventually the periodic stress remains. By definition, the periodic stresses are

Or = Sr eit°t,

oo = So eit~t,

Oz = Sz e i~t,

l"Oz= Toz eiWt,

Tzr = Tzr e i~t,

7re = Tro eiWt-

(55) Consequently, by eqs (b), (28)--(30), and the corresponding relationships for the other stress components,

K[ l (Ur V+Vo S r = -~ Ur + -~ (C - l ) +~

Wz)]

Sz=~

Wz ,

Wz + - ~ ( C - 1) Ur+~+r

•

+ v.).

r'°=2-c

'

K [ U + V o + _1( C _ I ) ( V

S ° = -C

r

3

, +U+Vo ~ +

r

Wz)] ,

(56)

r

The real stresses are the real parts of the expressions in eq. (55). Since the functions U, V and W are known only at mesh points, it is necessary to use numerical differentiation to get the derivatives in eq. (56).

5. 2. Energy flux Let S be a closed surface within a vibrating solid, and let R be the region within S. The stress exerted by the material outside of surface S upon the contiguous material within surface S ispi = oklnk, where nk is the outwarddirected unit normal to S, and ola is the stress tensor. For the present, rectangular coordinates are considered, if body forces are absent, the rate at which external forces perform work cn the material in R is

we=ff$

ff$

Propagation of waves is usually considered to be an adiabatic process. Consequently, by the first law of thermodynamics, gee is equal to the rate of increase of the sum o f the internal energy and the kinetic energy of material in R. Accordingly, I¢0 is interpreted [5] as the rate at which energy flows inward through surface S. Therefore, if dS is any fixed surface element with unit normal na, and ff the material on the positive side of dS (the side toward which n1 is directed) exerts a stress Pt on the material on the negative side of dS, the rate at which energy flows through dS in th~ sense ofni is -pgt~i dS = -o~ngfii clS. If the process is cyclic:, we may integrate with respect to time over the period of oscillation to get the energy that flows through dS during one cycle. Dividing by the period of oscillation, we get the mean rate of energy flux through dS. Referring to fig, 4,_we See that the energy that flows through the surface z = Oin the range r > a is zero, since there is nostress on that surface, Also, the flux ofenergy~thr0ugh the surface z = h is Zero, since there is no movement there, Consequently, the flux of energy through ~ cy~dricalsurface, r = constant > a, withinthe solid deter-

-

H.L. Langhaar etal., Periodicexcitation era viscoelastic solid

361

mines the rate at which energy escapes from that cylinder. The instantaneous rate of flux of energy outward through an element d S o f this surface is denoted by T dS. In view of the preceding remarks,

T=

+

+

(57)

To apply eq. (57); we must use the real stresses (or, fro and rrz) and the real displacements (u, v and w), since the product of the real parts of two complex variables is not equal to the real part of the product of the variables. This can be done conveniently withcomplex conjugates, since the real part of a complex variable z is ½(z + z*), where z* is the conjugate of z. Consequently, by eqs (20) and (55),

R(or) = ½(St eit''t + S* e-itat),

R(/i) = ½iw(V ei~t- U* e-itat),

where R denotes 'real part'. Therefore, 21r/¢o

R(or)R0J) = ~iw[USr e 2itat- S'U* e -2liar- U*Sr ~ US*l,and f

R(or)R(~) dt = ½rri(VS* - U*Sr) = ird(US*),

o

where I denotes 'imaginary part'. Consequently,

(58)

Tm = -½iw[I(US*) + I(VT~ro)+ l(WT~rz)],

where Tm dS is the mean rate of flux of energy through a surface element dS on a surface r = constant. Integrating Tm over the cylinder r = constant, we get the mean rate of flux of energy through the entire cylindrical surface r = constant; namely, h

2~r

(59)

E r f - ½ i w r f dz ~ II(US*)+ l(VT~ro)+ l(ICT~rz)l dO. 0

o

For axially symmetric motion of the plate, the integral with respect to 0 in eq. (59) is simply 2rt times the integrand. If dS is an element of area of the bonded side of the plate, the rate of flux of ener[~y through dS is T d8 = -(rz,i

+ rzof, + oz ) dS.

Consequently, the rate at which the plate delivers energy to the body is a 2~r

-

S~ 0

(r,~ + rzOV+ Oztb)r dr dO.

0

The mean rate of energy flux through dS is Tm dS, where

(60)

Tra = -½iw[l(UT~zr) + l(VT~zo) + I(WS*). Consequently, the mean rate at which the plate delivers energy into the body (mean power input) is a 2~

(61)

E p - -½iw f f [l(UT~zr)+ l(V~zo) + l(W$*)lr dr dO. 0

0

The difference Ep - E r is the mean rate at which energy is dissipated into heat in the cylinder, r

5.3. Driving forces Four different motions of the driving plate (designated as cases 1-4) are considered separately.

=

constant.

362

H.Lo Langhaar et aL, Periodic excitation of a viscoelastic~solid

5.3.1. Case 1: plate tnmslates up and down

In this case, U and W ate independent of 0, and V = 0. Consequently, Toz = Tro = 0. The other stresses in eq. (56) may be simplified accordingly. On the bond between the plate and the cylinder, U = V = 0 and It' = Wo, Con, sequently, by eq. (56), S , = 5 o = ~ - ~ ( C - l)Wz,

S, = ~-~ (C+ 2)Wz,

Tzr =0,

onz =0,

O~r~a.

(62)

Therefore, by eq. (61), the mean power input i~ a

/ _ , .

'~

Ep = ~rKic~w°l(C~21W:rdr --

z=0,

(63)

O~r~a.

0

The force that the viscoelastic cylinder exerts upon the plate is a

F = 27t [ to, dr,

(64)

z = O.

rip

0

The positive sense of F is the same as that of z. The force that the plate exerts on the cylinder is - F . In complex form, a

F = 2n ei'~t f rSz dr,

z = O.

0

Consequently, by eq. (62), a

F = -2nK ~ (C + 2) e i ' t f rWz dr,

(65)

z = O.

0

Because of the inertia of the plate, the force - F is not the same as the force that must be applied to drive the plate. The driving force for the plate is F] = - F - m ~ 2 w o e |''t,

(66)

where m is the mass of the plate. Only the real parts of F and Fi are to be retained. The positive senses of F and F! are alike. 5. 3. 2. Case 2: angular oscillation o f the plate about the vertical axis

in thi~ case, U = W = 0 everywhere, and V is independent of O. Consequently, by eq. (56),

s,-so=s:

=o,

d v,,

r,,-o,

v,-

.

On the bond between the plate and the cylinder, V= tO. Consequently, on the bonded area, Tro - O. By eqs (61) and (67), the mean power input is ti

Ep=-~a" rito~ f l(VzlC)r 2 dr,

z = 0.

(68)

0

The moment that the cylinder exerts on the plate is M = 2n ~ r2r, o dr, 0

(69)

z-- O. "

"

K L . Langhaar et ~ , Periodic excitation o f a viscoelastic.solid

363

The positive sense of M is the same as that of 0. The moment that the plate exerts on the cylinder i~ -M. In complex form,

M =~ e

r2Vz dr,

z = 0.

(70)

0

Because of the inertia of the plate, the couple - M is not the same as the moment that must be applied to drive ~he elate. The driving couple is MI = - M - Iw2¢ eiwt,

(71)

where ~ is the amplitude of oscillation of the plate (radians). Only the real parts of M and M~ are to be retained. The positive senses of M and M~ are alike. The factor I is the moment of inertia of the plate about the z axis. If the plate is a flat disk, I = ½ma2.

5.Z3. Case 3: horizontal oscillation of the plate In this case, eq. (44), U = 0 cos 0,

V = V sin 0,

(72)

W = ~ / c o s 0,

where 0, V, and W are functions o f r and z. On the bond between the plate and the cylinder,

U=uo~

V = - u o,

W=0;

(73)

O
z=0,

Consequently, by eq. (56), =

K

K Tzo = ~-~ F'z sin 0

osO,

on z = 0,

O
(74)

Therefore, by eq. (61), the mean power input is a /i'

Ep = "~ iwuoK ~

l[(Uz

-

Vz)/C]r dr.

(75)

0

The force F that the cylinder exerts on the p~ate acts along the x axis; its positive sense is that ofx. It is seen that a

2~

F = f ~ (TzrcosO-TzosinO)rdrdO. 0

0

Therefore, in complex form, eq. (74) yields a

F = ~K2--'~ e i~t f (Uz

Vz)rdr,

z = O.

(76)

0

Because of the inertia of the plate, the force - F is not the same as the force that must be applied to drive the plate. The driving force for the plate is

.1;'1= - F ' mw2uo ei ' t . The positive senses o f F and F l are alike.

(77)

364

H.L. L anghaar et aL, Periodic excitation of a viscoelastic solid

5. 3.4. C.'ase4: rocking o1"the plate in this case, eq. (72) again applies. The bounda~ conditions at the plate are, eq. (52), 0 = V=O,

~l=r¢

atz=O,

O~r*~a.

(78)

Consequently, by eq. (56), K

S== ~ ( C +

z=O,

2)WzcosO,

O<~rga.

(79)

Therefore, by eq. (61), the mean power input is It

Ep = ~ k,.~X

"

I

i[(C + 2)~/=/C]r 2 dr,

z = O.

(80)

0

The moment that the viscoelastic cylinder exerts upon the plate is a

M =f 0

2~r

(81)

f r2ozcosO drdO. 0

The moment M is positive if it tends to rotate the plate in the positive sense of 0. Eqs (79) and (8 l) yield a

ng (C + 2) ei''t M= ~-

f r2W, dr,

z

= O.

(82)

0

Because of the inertia of the plate, the couple - M is not the same as the moment that must be applied to drive the plate. The driving couple for the plate is M ! = --M

-

(83)

Jw2¢ e i'°t,

where J is the moment of inertia of the plate about a diameter. If the plate is a uniform disk. J ~ ¼ma2. The positive senses of M and Mi are the same as that of¢. 5.4. Resonance Equations (65), (70), (76), and (82) Ove the complex forces and couples that the viscoelastic cylinder exerts on the plate in the four cases considered. These complex quantities may be regarded as vectors of constant magnitude rotating with angular velocity t~ in the complex plane. The absolute values of these vectors are the amplitudes of the real harmonic forces and moments. In view of eqs (65), (70), (76), and (82), these amplitudes are

IFI=yK

1+

•

rW=dr,

z=0

(84)

o

Iml = ~K ~

- o

I Y l = 5~x-

• o (0,-

(85)

~,)rdr,

i I- 3 I +'d " : ~ , o ,

z=o

(86)

(87)

z=o

in which the vertical bars denote absolute values of the complex quantities.

"

:~:i

i!~

" ....

, .

H.L. Langhaar et aL, l',,.riodic excitation o f a viscoelastic solid

365

or

l.I I

I i I

!

!

....

Wl

I

I

~2

w3

Fig. 5. Dz iving force-frequency curve. Ii

fll[

j~

0

3

M"I

C I= J,O "I" QOi ~ ~2 I=0.0001 ÷O.OJ

,1

II¢ f"J / J ' J / f J f J f J f f ~

I

2

3

4

Fig. 6. Elastic displacement pattern.

Since fixed amplitudes of oscillation of the pltte have been prescribed, a curve of IFI or IMI versus co shows dips, as indicated by fig. 5. The abscissas of thes( minima are resonant frequencies. Because of viscosity, only a finite number of resonant frequencies exist.

6. Numerical computations With few exceptions, the finite solid calls for a numerical treatment. For the cyliadricai solid, the azimuth coordinate has been eliminated in each of the four cases treated in section 4. Consequently, the problem is reduced to two dimensions, and a network in a single radial plar e suffices (fig. 2). For numerical work, it is advisable to multiply eq. (25) by r 2 to avoid the indeterminate form ([0 which otherwise occurs at r = 0. Also, it is advisable to divide the equation by G. Then dimensionless paramer;rs colt[G, v, and cot/are obtained. Each of the four cases treated in section 4 pr ;sents an elliptic mixed-boundary-value problem for the functions /~, V, W or U, V, W in the rectan~lar region sho ~n in fig. 2. A finite difference method has been used to solve this problem. The rectangle in fig. 2 is covered with a rectangular net with constant mesh dimensions A/j and A~'. The values of U, Vand Wat the net point (m, n) are denoted by Umn, Vmn and Wren. In all cases, second order finite difference approximatiom of the differential e,ts (31), (42), or (45) were programmed for all interior net points, and finite difference approximations of she boundary conditions ((32), (43), or (46)) were programmed for all boundary points. Thus, linear algebraic e tuations were obtained. Since C and ~22 are generally complex, the program was adapted to handle complex algebr:t. For axial motion of the capping plate about the vertical axis

H.L. Langhaar et 41., Periodic excitation o f a viscoelastic solid

366

i ~

"1 [I

¢ - 2 ~

+ 0 . 4 ~ 8 x !0"4i

I

+

N,,I

I

' "'-"!,

I

'

I_

'

I-0

1

. . . . t slr~ --'--'--t - • M-i

2

3

5-1~lIR

4

Fig. 7. Viscoelasticdisplacement pattern. (case 2), U -- W = O. Typical examples of the displacements for an elastic body and for a viscoelastic body, in the case of axial motion of the plate, are shown in figs 6 and 7. For axial oscillation of a slender bar (fig. 3), the numerical solution agrees closely with the exact solution (eqs (37) and (40)). With Umn, Vmn'and Wren determined, the stresses, the mean power input, and the driving force (or moment) may be computed by previous equations, depending on which case is considered (eqs (62), (63), and (65); (67), (68), and (70); (74), (75), and (76); or (79), (80), and (81)). For these computations, first derivatives of U, V, or W with respect to z are required at certain points. These values were computed by means of a second order numerical differentiation formula. A few definite integralsinare involved; they were computed by Simpson's rule. As remarked in section 4, the boundary condition U + V - 0 at r -'- 0 was waived in case 3. In case 4, the finite difference equations are the same as in case 3, except that for ~ -- 0, 0 ~ ~ ~ 1, Umn - O, Vmn = 0, and Wren - aCe,n, in which ¢ denotes the amplitude of the rocking oscillation of the plate. The natural frequencies of the body are determined by the theory presented in section 5. They conespond to the values ~oI, co2, ~o3,. •. of the dips in the force-frequency curve of fig. 5. For the computation of the forcefrequency curves for cases 1-4, the integrals in eqs (84)-(87) are required. After (U, V, and W) and their first derivatives with respect to z are computed for node points (m, n), numerical integration yields the corresponding force-frequency relations. The critical frequencies (~1, ¢a2, ~ s , • • • are the abscissas of the minima on the curves. In the computer program, an adaptation of the method of interval halving [6] was used to determine these minimum values of force, and, hence, the natural frequencies. The resonance calculations require considerable computation time.

Acknowledgement The problems treated herein were conceived and propounded in general outline by personnel of the Construction Engineering Research Laboratory (CERL), Department of the US Army, Champaign, Illinois, USA. Dr. W.E. Fisher, Chief of the Structural Mechanics Branch, CERL, and Dr. J.D. Prendergast of that organization provided helpful advice and direction in the study. The work was sponsored by the Department of the US Army.

Nomenclature Summation convention is used. Dots denote time derivatives.+i = V/- 1. x~, x2, and x3 are rectangular coordinates. = displacement vector, Um.n = = strain tensor, eq. (3) = stress tensor

Um emn O,ln +

.

.-

-

.

.

.

aUm/aXn

+

-

,

,

"

.

.

++

+

.

"

•

+

H.L. Langhaar e t aL Periodic excitation o f a viscoelastic solid

mn

$ e emn

Smti P gm # G P

E K 03

Um r,O,z U,V, W

= Kronecker delta = invariant of the stress tensor, eq. (4) = invariantof the strain tensor, eq. (4) = strain deviator, eq. i(4) = stress deviator, eq. (4) = mass density = components of the downward-d rected gravitational vector g on the axis Xm -viscosity coefficient, eq. (2) = relaxation coefficient, eq. (2) = shear modulus, eq. (2) = Poisson's ratio = Young's modulus, eq: (5) = bulk moduS.us, eq. (5) = angular frequency, eqs (12)and (24) = factor in u m that is independent of time, eq. (12) = cylindrical coordinates = projections of the displacemenl vector on the tangents to the coordinate lines in cylindrical coordinates. Ur = au/ar, etc.

er, eo, ez, 70z, 7=r, 7re t = R,I =

u,v,w h b (i

367

strain cc,mponents in cylindric~,l coordinates, eq. (20)

time symbols for the real and imagi:iary parts of a complex quantity symbo~ for the conjugate of a :orr4plex quantity = factors in u, v, and w that are ndependent of time, eq. (24) = length of a cylindrical body (tig. 1) = radiu~; of a cylindrical body (fg. 1) -'- radius of plate bonded to cyli ldrical body (fig. 1) =

r/a

= C X

k u,v,w WO

U0

-- a

complex dimensionless con' tant defined by eq. (28) a complex dimensionless frequency defined by eq. (29) an axial coordinate for a rod ~fig. 3) = a dimensionless constant deft ned by eq. (34) = factors in U, V, and W that a~e independent of 0, eq. (44) amplitude of oscillation of tte end plate, eq. (32) = amplitude of angular oscillat on of the end plate, eq. (42) = amplitude of horizontal oscillation of the end plate, eq. (46)

Oz, ~oz, ~zr, ~'rO S,,So, S=, To=, T,r, Tm Or, oO,

stress components in cylindr ical coordinates factors in =

F

--

M

=

Or,. • . , fro

that ar~ independent of time, eq. (55)

raean power input of oscfila :ing plate tbrce that the viscoelastic c3 linder exerts on the driving plate moment that the viscoelasti,: cylinder exerts on the driving plate. Amplitudes of F and M are given by eqs (84)-(87).

H.L. Langlmar et aL, Periedie excitation o f a viscoelastic solid

368

References [i] B.A. Boley and J.H. Weiner, Theory of Thermal Stresses, John Wiley, New York (1960) 461,462. [2] T.M. Lee, Surface dynamic Ioadings on a viscoelastic half space, unpublished manuscript, School of Engineering, San Fernando Valley State CoUege. [3] D.R. Bland, The Theory of Linear Viscoelasticity, International Series of Monographs on Pure & Applied Mathematics, voL I0, Pergamon, New York (1960). [4] A,P. Boresi and P.P. Lynn, Elasticity in Engineering Mechanics, 2nd Edition, Prentice Hall, Englewood Gifts, N.J. (1974) Appendix 2A. [ 51 R.B. Lindsay, Mechanical Radiation, McGraw-Hill, New York (I 960). [6] R. Beckett and J. Hurt, Numerical Calculations and Algorithms, McGraw-HiU, New York (1967) 46-52.

t

i

i

.

if:

•