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A study of the thermally induced large amplitude vibrations of viscoelastic plates and shallow shells
Journal of Sound and Vibration (1987) 116(2), 323-337
A STUDY OF THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATIONS OF VISCOELASTIC PLATES AND SHALLOW SHELLS D. L.
HILL AND
J.
MAZUMDAR
Department of Applied Mathematics, University of Adelaide, Adelaide, South Australia 5001, Australia (Received 6 March 1986, and in revised form 4 June 1986) A method for the study of large amplitude vibrations of viscoelastic plates and shallow shells is presented. The concept of iso-amplitude contour lines in conjunction with Galerkin's method is used and the resulting non-linear differential equations are solved for appropriate time functions. The thermally induced large amplitude vibrations of a square viscoelastic plate and of a shallow shell of equilateral triangular planform are examined. All details are explained by using graphs. 1. INTRODUCTION
In many modern applications viscoelastic plates and shells experience substantial deflections when they are induced to vibrate, in some cases the size of these deflections being of the order of the thicknesses of the structures themselves. In these cases the linear theory is inadequate to accurately describe the induced motions and must be suitably modified. There is however a heavy cost with the significantly more accurate theories and this is due to the retention of non-linear terms in the governing equations describing the motion. Exact solutions of these equations are unknown except in certain very special cases and usually one must resort to numerical procedures, for example the finite element method. In this paper a quasi-analytical method is presented for obtaining accurate approximate solutions to the non-linear governing equations which will enable one to gain a useful insight into the behaviour of the structures modelled by these equations. The method is an extension of one used earlier to study the small amplitude (linear) thermally induced vibrations of plates and shallow shells of arbitrary shape [1-3] and is shown to be applicable for the non-linear, large amplitude analysis. The method consists of using the idea of iso-amplitude contour lines to derive a pair of coupled ordinary differential equations in terms of a contour function u(x, y) and consequently a non-linear ordinary differential equation which depends on the particular viscoelastic model chosen for the material of which the structure is made. The method of iso-amplitude contour lines has been applied when the deflection on the boundary of the plate or shell is the same everywhere on that boundary, for example, in the cases of clamped, simply supported, partly clamped and partly simply supported, and movable boundaries. The cases of free boundaries and problems involving more than one distinct family of iso-amplitude contour lines require extensions of the basic theory. The method is applied to study the response of certain viscoelastic plates and shallow shells with rectilinear boundaries. The results should be useful to nuclear engineers and to workers in thermoplastics. 323 0022-460X/87/ 140323 + 15 $03.00/0
© 1987 Academic Press Limited
324
D. L. HILL AND J. MAZUMDAR
2. DERIVATION OF THE EQUATIONS FOR THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF VISCOELASTIC PLATES
The thermally induced vibration of viscoelastic plates whose induced deflection is of the order of the plate thickness is analyzed here by using the theory of non-linear plates usually associated with the name of von Karman. Consider a homogeneous, isotropic viscoelastic plate of thickness h and mass per unit area p in a non-stationary temperature field. The plate is referred to a system of rectangular Cartesian co-ordinates x, y and z such that the horizontal plane Oxy coincides with the middle surface of the plate in its undeformed state. The deflection w(x, y, 'T) of the plate, which is a function of the spatial co-ordinates x and y and of the temporal variable 'T, is taken as positive downward in the z-direction. An incremental increase in temperature, T(x, y, Z, 'T), in the plate will produce a state in which thermal stresses are evident. The stress-strain equations are given by [4] 0,2)
where Sij and Bij are the deviatoric stress and strain tensors, Ukk and e kk are the traces of the stress and strain tensors and aT is the coefficient of linear thermal expansion of the material of which the plate is composed. P', Q', P" and Q" are differential operators in the time variable T and are of the form n,
P'
n
= L: anp n,
P"=
n=O
f
cnpn,
n=O
where p = ala'T. The unknown constants can be determined from the properties of the particular plate material being studied. When the plate vibrates, at any instant of time the intersection of the deflected surface with parallels z = constant yields a set of contours which, after projection on to the Oxy plane form a family of closed, non-intersecting level curves which will be denoted by u (x, y) = constant. If the plate is subjected to clamped, or simply-supported, or a combination of clamped and simply-supported boundary conditions then its boundary will be a member of this family of curves. Denote this family by e u , 0",; U ",; u* where u = 0 is the boundary of the plate and u = u* is the maximum value attained by the function u. Consider an area n u of the plate bounded by an iso-amplitude contour line eu at any instant of time 'T. The governing equation of motion for the large amplitude vibration of a viscoelastic plate can be expressed, corresponding to the case of an elastic plate [5], in the form (3)
where Qn is the shearing force, aMnrl as is the edge rate of change of twisting moment and N x , N; and N x y are the in-plane stresses. Expressing these in-plane stresses in terms of a stress function ¢(x, y, -r) as (4)
substituting these expressions into equation (3) and using the well-known expressions for Qn and aMnJas results in the integro-differential equation
a3wf
D(p)-,
au'
CO'
azwf
R. ds+ D(p)-z
au
c"
aWf
PI ds+ D(p)-
au
CO'
G~
ds
325
LARGE VISCOELASTIC PLATE VIBRATIONS
-D(p)(l+J.L(p»
+
I [u x a,KT+ Uy aKTJ d~
Jc"
rJx ay vi 2w 2¢ 2¢ 2w a2w a a a a rPep ph-2 --2 -2 --2 -2+2-- - - - d!1 = 0, aT ax ay oy ax ayax ay ax
rPw]
If [ nu
(5)
where w is the deflection of the plate, D( p) is the viscoelastic flexural rigidity and fL (p) is the viscoelastic Poisson ratio of the plate material, and F 1 --
t=U~+U;,
t -1 / 2[3UxxU 2x + 3 UyyU 2y + Uxxu 2y + UyyU 2x
-
K T ==
P12I
+ 4UxyllxUy ] ,
h /2
aTT' z dz.
-1,/2
The functions wand ¢ must also satisfy a compatibility relation of the form V4 ¢ = hE(p)[(a 2w/ay ax)2 - (a2w/ax2)(a2w/(iy)] - hE(p)V 2S , T
(6)
where E(p) is the viscoelastic operator corresponding to the Young's Modulus of the material and lOT
1
=
h
I
hl2
cxTT'dz.
-h12
Since this equation must hold at all points in the plate it can be integrated over the region il u yielding
If [ a;
V4¢ - hE ( P) { ( a2w )2_a2~a2~}+hE(P)V2BTJ d!1=O. ayax ax ay
(7)
If in particular, the temperature variation across the thickness of the plate is of the form T'(x, y, Z, T) == T(x, y, T)G(Z)
for some function G(z) then are given by 1
C. =-h
f
h/2
eT =
C. T and K T = CKT, where the coefficients C. and C K
cxTG(z) dz,
flr/2
12
CK = h 3
--hI2
cxTG(z)z dz.
-h/2
Equations (5) and (7) are a coupled system of non-linear partial differential equations for the deflection and stress functions which describe the thermally induced large amplitude vibration of viscoelastic plates.
3. SOLUTION OF THE PLATE EQUATIONS
An approximate solution to the system of equations can be obtained by using Galerkin's method with the eigenfunctions of the corresponding linear elastic free vibration problem as the co-ordinate functions. The equation for the corresponding free vibration problem is, for the ith eigenmode [6], D
f (Qn - a~nt) ell
ds = phAT
If
W';(u) df2,
(8)
nil
where the Ai are the eigenfrequencies and the W;(u) are the eigenfunctions. D is the flexural rigidity. It is now assumed that ¢ can be written in terms of the contour function
326
D. L. HILL AND J. MAZUMDAR
u and the time variable 7"; that is, ¢ = ¢(u, 7") (see reference [7]). To proceed, a separation of variables type solution of the form cc
CD
L
w(u, 7") =
L
¢(u, T) =
W;(u)ai( T),
;=}
W;(U),Bi( T),
(9)
i=l
is assumed, where the ai (T) and ,Bi(7") are time functions to be determined. For simplicity consider only the first terms in equations (9), namely,
=
w(u, 7")
(10)
¢(u, 7") = W(u),B( 7"),
W(u)a( 7"),
where the subscripts i have been omitted. Substitution of equations (10) into equations (5) and (7) gives 3
Wf d- 3 D(p)a(T) [ du
c,
d2Wf R 1 dS+-2 du
f[
- D(p)(1 + /-L(p))
+pha"(7")
If
a;
c,
Cu
dWf 0; ds ] F1 ds+du c;
aK T
ux -
ax
aKT] ds + uy - I. ay
W(u) dfl+a(T),B(T)
vt
II
[2P
A
fl"
d2~dW+PB(dW)2J dfl=O du
du
du
(11)
and
where PA
= U;Uy y + u;ux x - 2u x yuxuy and PB = 2uxx uyy -2u;,y,
If
nil
V4W(U) dfl
= ~U~
f
ell
R] ds+
~u~lJe
and the relation
F 1 ds+
l,
~:lJell 0; ds
has been employed (see reference [8]). The integral relations
f
L"
W(u) d.Q = -
f.
W(u)
te"o ~duo
(13)
and
IIn"
A(x, y) d.Q =
-
L~ fc"o A(x, y)~ duo,
(14)
where A(x, y) is an arbitrary function of x and y will now be useful. Applying Green's Theorem to the vector function A= (AI. A 2 ) one has
If
nil
V' A dJ1
=l
A· n ds.
(15)
Jell
Differentiating this equation with respect to u and using equation (14) gives f
C
u
aAI ds= d- f (A ds +aA2) - lu X + A 2u y)(ax ay v't du c, v't
(16)
327
LARGE VISCOELASTIC PLATE VIBRATIONS
since n =
-(uxlY't, uylY't), where AI and A 2 are arbitrary functions of x and y. Now, letting (17)
gives
f
Pn-ds = - d ..Jt d u
C U
f
C
u
PA -ds ..[i ,
(18)
Making use of equation (8) in equations (11) and (12), then differentiating the resulting expressions with respect to u and using the integral result (13) yields
-I:
ph).. 2 D(p)a( 7") [ -V W(u) - pha fI (7") W(u)
dS]
Jc".Ji -
f
D(p)(1 + j.L(p))
W)
ds 1 . - a( 7"){3( 7") - d [(d -du du
c" v t
2
Irc
f
ds
2
V KTVi
u
c,
dS] = 0
PA r: vt
(19)
and
fcu ~] - E(p)h feu V2B"~ d [(dW)2-1: dS] =0, -"2 E ( p )ha -(7" ) du du Jc PA.Ji
(3( r) [ -
P~
2
W(lI)
1
?
(20)
u
in which Green's Theorem has been used to establish that
~1,
au Jc
u
[II aKT+ u aKT] ds =1, ax ay Jt
x
y
Jc
V2K ds
TV!'
u
Now multiplying equations (19) and (20) throughout by W( II) and integrating them from u = 0 to u = u* gives
ph).. 2C1[D(p)1 D]a( 7") + fK (7") + ph C, a"( r ) + C 2a (r){3 (r) = 0, 2( r) = 0, (ph).. 21 D)C r)+ E(p)hf.(r)+!C 1{3(
f
II' [
0
?
W-(u)
-I:Jc Jt dS] du, u
fdr) =
ioU' W(U)[ D(p)(l+j.L(p))
f.(r) =
I"
W(II)
(22)
2E(p)ha
where C1=
(21)
C2 =
f
0
u· [
d[(dduW) 1,Jc
W(u) dll
tu V2KT~]
2
dS] ] du,
PA..[i
u
du,
[fcu V2BT~] du,
(23)
Equation (22) gives the solution for (3( r) as (3 (r)
= -( D 1 ph): 2C I )[ E (p
)hf. (r) +!C2E(p )ha 2(r)]
(24)
and substitution of this expression into equation (21) results in a non-linear ordinary differential equation for the time function a (r), 1 [ DC2 D(p) 2(7")]+fK(r) ] a"(r)+ phC - ph)..2C\a(r)[E(p)h!.(r)+!C2E(p)ha +A2---v-a(r)=0 1
(25) which can be solved numerically by any of the standard numerical integration methods. In this study the Runge-Kutta-Nystrom method is used.
328
L. HILL AND J. MAZUMDAR
D.
4. DERIVATION OF THE EQUATIONS FOR THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF VISCOELASTIC SHALLOW SHELLS
The study of shell structures has become an important field of research due to their high load carrying capabilities in comparison to their own weight [9]. Let the equation of the middle surface of the shallow shell be expressed in the form
z = (x 7-;2Rx ) + (xyj R x y ) + (y2j2R),
(26)
where R x, R,. and Rxl' are the radii of curvature of the shell and the usual assumptions for shallow shells have been made. When the shell vibrates, then at any instant of time the intersection of the deflected surface with parallels z = constant yields a set of contours which, after projection onto the baseplane z = H (H denoting the apex height), is a family of non-intersecting closed level curves u(x, y) = constant. Consider an area nl/ of the shell bounded by a level curve eu at any instant of time T [7, 8]. Summing the vertical forces over this area yields the equation of motion for the large amplitude vibration of a viscoelastic shallow shell in the form
where all terms are as previously defined. The tangential force components can be expressed in terms of a stress function ¢(x, y, T) as follows: tv." = a2¢j ay 2 , N; = a2¢jax2, N."y = -a 2 ¢ j ay ax. (28) Substitution of these expressions into equation (27) and proceeding as before yields the integra-differential equation 3
D(p) a
:
au
1
re" R
1
ds+ D(p) a2~
au
-D(p)(l+Il(p))
fa [
1
rc.. F[ ds+ D(p) auaw 1re" 0; ds
aKT aKT] -+ ds ux-+u yax ay,fi
c"
If [ n,
2¢ 2 aw 1 a aT R, ay
1 rP¢ R; ax
ph~+---2+--2
2 a2¢ a2¢ 2 w fl¢ a2w a2 ¢ a2w ] - - - - - - 2 - 2 - - 2 - 2 + 2 - - - - dn=O. R x y oy ex ax oy ay ax ayax ayax
(29)
The compatibility relation for the shallow shell problem is 2 2w 2 2wa 2w] aw a - - --+
[1
1
2
(30)
which must hold at all points in the shell and so can be integrated over the region yielding 2 V4¢ _ E ( P)h [ _1 a2W+_l rPw_~ a w 2 flu u; al R y ax R xy ev ex
If {
2w)2
+ ( -a ayax
nu
2wa2w]
a -2 ax- ay
--?
+ hE(p)V 2 8T }
dIl = O.
(31)
Again it has been assumed that the stress function ¢(x, y, r) can be expressed in terms of the iso-amplitude contour line function and the time variable. Equations (29) and (31) are a coupled system of non-linear partial differential equations for the deflection and stress functions which describe the thermally induced large amplitude vibration of viscoelastic shallow shells.
329
LARGE VISCOELASTIC PLATE VIBRATIONS
5. SOLUTION OF THE SHALLOW SHELL EQUATIONS
The system of equations can be solved approximately by using Galerkin's method with the eigenfunctions of the corresponding linear elastic free vibration problem as the co-ordinate functions. These equations are [10, 11]
D
If (
AMnr) , f (Q" -----;;;- ds = pli):i
a;
ell
IIn"
4
V 4>i dfl = Eh
dcflif
\.Vi u) dfl- du
K I ds,
ell
~~j fc" K
I
ds,
(32)
where the Ai are the eigenfrequencies and the Wj(u) and lPi(u) are the eigenfunctions. A separation of variables type solution of the form co
L
w(u, 7")=
(33)
Wj(u)aiCr), ;=1
;=1
is used where the cd 7") and (3j(T) are time functions to be determined. For simplicity only the first terms in equations (33) are used, the subscripts being omitted. Hence w(u, r) = W(u)a( T),
¢( u, or) = cI>(u)f3( r),
(34)
and substitution of equations (34) into equations (29) and (31) yields 2W 3 i dW i d Wi d D(p)a(T) [ du 3 Te" R I ds+ du2 Teu F I dS+d;Te" 0; ds
J
If
f[
ds ux -aK T+ Uy -aKrJ r:+ pha"( r) W(u) dil . Ax By y t flu 2WdlP 2 dcI> i d dW( d tP dcI»] j +/3(r) du Te K d s + a ( r ){3 ( r ) n PA du 2 ~+ct; PAdu2+Pnd;;" dfl=O
-D(p)(l + J.L(p))
c;
If [
II
II
(35)
and 3 d tP i d2tPi dlPi [ (3(T) du3 Te R 1 ds+ du2 Te" F I ds+ du
re" a; ds ]
u
- E(p)ha(r)
~: fcu K
+E(p)ha 2(r)
ds+ E(p)h
I
Ifa"
2
V e T dfl
If [
d2WdW P (dW)2] P A--, -+~ dfl=O. du" du 2 du
a;
(36)
Substitution of equations (32) into equations (35) and (36) gives 2
D(p)a(r) [ -PhA D
If
a;
-D(p)(1 + J.L(p))
d
1dcI>f
W(u) dil-- D du
f[ c,
AK T u.~-+ Ax
+(3(r)-d K I ds+a(r){3(r) u c;
K I ds
]
Cu
aK T ] ds r:+ pha"
Uy -
If
W(u) dfl
Ay
yt
n,
d d 4J2+PndcI»] PA--2 - +dW( - PA dfl =0 du du du du du (37)
If [
2WdlP
a;
2
330
D . L. HILL AND J . MAZUMDAR
and
(38) Now, differentiating equations (37) and (38) with respect to u gives
ph): 2 D (p)a(r) [ -DW(u)
lJe
1 dud [dct;7Je
ds
Ji- D
1
- pha"W(u) Jc
K. ds
]] -D(P)(l+p,(P))Jeu 1 V K ds 2
u
u
ds d[difJ 1 ] Ji+ f3( r ) du ~Je K, ds -
II
TJi
l
d[dWd
II
(39) and
f3(r)~[dW1, KldS]_E(P)a(r)~[dW1, - . du
du
_ E(p) E
Je
1,
Jc
E
u
V2ST
du
du Jeu
ds _~ E(p) a2(r)~[(dW)2l PA dS] =0. 2 E du du Jeu Ji
Jt
u
(40)
Then multiplying equation (40) throughout by
B1 =
f
0
u·
d[dW 1,] du Jc K ds du,
€/Jeu) du
1
U
f~ (r)=
f"· o
B2 =
Jor
d[(dct;"W)
II"
f
2
d] du , Je PAlL
1,
«
V
elf
2ETJt dS]
duo
(42)
Equation (41) can be rearranged to give
f3(r) =.f!(T) + E(p) a(r)+ B 2 E(p) a2(r). B.
E
2B(
E
(43)
Now, multiplying equation (39) by W(u) and integrating from u =0 to u = u* gives 2 D(p) D (p) ph): B3------v-a (r ) + B 4----rJ" a( T) +[« (r) + phB3a"( r) - B 4f3 (r) + Bsa( r )f3( T) = 0,
where
(44)
331
LARGE VISCOELASTIC PLATE VIBRATIONS
and consequently equation (44) yields a second order non-linear ordinary differential equation for a ( T), of the form a"(T)=_>..2
D
( P) a(T)-~ D(p) a(r) phB) D
D
- -1- { fK(r)-(B 4 - B sa(r)) phB)
[.f:(T) E(p) B, E(p) ]} --+--a(r)+-~- - a 2(T) B1
E
2B.
(46)
E
and this equation can also be solved numerically by any of the usual numerical integration methods. In this study the Runge-Kutta- Nystrom method is again used.
6. THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF A CLAMPED SQUARE PLATE
Consider the thermally induced vibration of a clamped square plate which is initially at rest and at a constant temperature To. At time zero the boundary of the plate is instantaneously set to zero temperature, the plate being composed of material which can be modelled as an "incompressible" Kelvin material which is represented by the equation D(p)/ D = 1 + 'YP,
(47)
where 'Y is a material constant. To simplify calculations the Poisson's ratio of the plate material is also taken to be a constant and hence D(p)jD=E(p)/E.
(48)
The method of iso-deflection contour lines depends on being able to find a good approximation to the actual contour line function. Some work has been carried out on this problem [12] and various forms for the contour line function have been used (see, for example, reference [13]). Using the equation of the plate boundary to obtain a contour line function has proved useful in the past and this approach will be used here. Thus the equation of the iso-amplitude contour lines is taken to be (49)
where I is the length of a side of the square boundary and in fact the contour line u = 0 is the equation of the boundary of the square. For a clamped plate the boundary conditions are
wi u=o =0 '
(JwjiJull/=o = O.
(50)
A method for solving heat conduction problems of this type was proposed by Mazumdar [14] and leads to the partial differential equation
-aTf Jl ds + 21 au
c"
C
If
-erdfl = 0,
a; aT
(51)
where c 2 is a material parameter and the temperature T is expressed as a function of u and r: that is, T = T(u, 1'), following references [1-3]. It is assumed, for simplicity, that the temperature variation across the thickness of the plate is linear, that is, T' = zT(x, y, 1'), and so f:T = 0 and K T = aTT. Equation (51) is solved by using the method of separation of variables, seeking a solution of the form (52)
332
D. L. HILL AND J. MAZUMDAR
After differentiating equation (51) with respect to u and using the mean value theorem to evaluate approximately the contour integrals involved yields the equation (53) where w 2 = [2TJ/4c 2 , and after introducing the new variablej? == 1 - u this equation becomes the Bessel differential equation of zeroth order d 2T* /df+ (II!) dT*/df + w 2 T * = 0,
(54)
the complete solution of which is given by T*(f) = AJo(wf) + BYo(wf),
(55)
where J o and Yo are the Bessel functions of the first and second kinds respectively. Since the temperature must be a finite quantity at all points in the plate then B must be zero and the edge condition requires that W be a root of the equation (56) The initial uniform temperature To is now expressed in the Fourier-Bessel series To =
l:
(57)
A"Jo(wnf),
n=l
where W n is a root of equation (56) and it is assumed that the series can be integrated term by term. It can be shown that the coefficients An are given by (58) and hence
T(l, or) =2To
co
Jo(wnf) J ( ) exp
l: n=1 W
n
1
co;
(4c --1 2 2
2 ) WIlT
(59)
is the solution to this temperature distribution problem. Equation (8) gives the eigenfunctions of the corresponding elastic problem and this equation is solved by assuming a power series of the form W(u) =l:~=l QiU i, where N is a pre-determined number of terms to be considered and the Q j are unknowns which are found by using the well-known method of collocation. The collocation nodes were chosen so that they were evenly spaced over the region of integration and it was found that seven terms were enough to ensure reasonable convergence to the required eigenfunctions and variation in the chosen nodes did not alter the results significantly. The unknown time function aCT) is obtained by solving equation (25) approximately by using the Runge-Kutta-Nystrom method. Since the plate is initially at rest the initial conditions of the problem are o (O) = 0 and a/CO) = O. Equation (10) is then used to find w(u, or). For illustrative examples parameter values were chosen based on those given in reference [9]. These are [== 150 mm, E = 1640 kg/rnm", h = 3 mm, J.L == 0·3 and p == 1·19 X 10- 3 kg/em? is the mass per unit volume. The values aT == 1O-4;oC, To = lOOO°C and c 2 == 1 m 2 / s were also used. The effect of the non-linear terms in the preceeding analysis, although negligible for very small time values, becomes quite significant as time advances. This is seen in Figure 1 which shows a graph of non-dimensional deflection against non-dimensional time for 'Y = O. The non-dimensional parameters are defined as (60)
LARGE VISCOELASTIC PLATE VIBRATIONS
333
4,.------------,
*:3
a
0,04
0·08
.,.*
0·12
0·16
Figure 1. Comparison of linear and non-linear deflections for the square plate. 2,.--------------,
'3
0 t-==------ - - - \ A - - - - j
-1
o
0'04
0·12
0·16
Figure 2. Effect of changing the viscoelastic parameter y for the square plate.
In Figure 2 the effect of varying the viscoelastic parameter y on the large amplitude vibrations is demonstrated. It can be seen that increasing the value of 'Y causes the motion of the plate to be damped, which is to be expected. 7. THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF A CLAMPED SHALLOW SHELL OF EQUILATERAL TRIANGULAR PLANFORM
Consider the thermally induced vibration of a clamped shallow shell of equilateral triangular planform, initially at rest, which is acted upon by a hot spot of strength q at its apex and whose boundaries are held at zero temperature for all time, the shell being composed of the same material as that of the plate of the previous section. The equation for the iso-arnplitude contour line function for the triangular shallow shell is chosen to be u(x, y) = x 3 -3 xy 2- ax 2 - ay2+4a3/27, (61) where a is the perpendicular height of the equilateral triangular boundary and the contour line u = 0 is the equation of the boundary of the shell planform.
334
D. L. HILL AND J. MAZUMDAR
To find the temperature distribution under these conditions the method of Laplace transforms is employed. If the shell is reasonably shallow the temperature distribution within it can be approximated by that in the corresponding plate problem (R, = R; = R x y = (0). Again, following reference [14] yields the equation
d2~1
du Jc
u
v'ids+dT..i-l vtds- s2(T-To)1 ds=o, du du Jc c Jc" vt
(62)
u
where the temperature is again assumed to be a function of u and T and the transform of T( u, T) is denoted by 7'(u, s), s being the Laplace transform parameter and To = 0 the initial temperature of the shell. It is desirable to be able to calculate the contour integrals analytically because numerical inversion of the Laplace transform is a non-trivial problem, the results of which can be unstable (see reference [15]). To get an analytical solution to the temperature distribution problem the contour integrals involved are calculated approximately by using the mean value theorem and the following equation is obtained (upon remembering that To is zero);
(63) where w 2 = s/ ac' and j? = u* - u, where u* is the maximum value obtained by the function u. Equation (63) is a modified Bessel's equation of zeroth order and has as its complete solution (64)
where 10 and K, are modified Bessel functions. The unknown functions C 1 and C 2 are to be determined from the initial condition of zero temperature except at the hot spot, the conservation of heat generated, and the given boundary condition of the problem: that is, (1) T = 0 at T = 0 everywhere except at the centre U = u*, (2) lim
II~O
°
f
Cu
aT q -ds = - 2 B(T), an c
(65)
= on the boundary u = 0 for T;:' 0, where B( T) is the Dirac delta function. Taking Laplace transforms of conditions (2) and (3) above, yields
(3) T
limf d7'/df= -q/ BC
2
f ...1
and
7'=
°
onf= 1,
(66,67)
where B = 2aA o/ u* and A o is the area of the planform of the shallow shell. Use of these 2 conditions in equation (64) gives C 1 = - ( q/ BC )Kv(w )/ Io(w ) and C 2 = q/ ec2 • Hence T(f, s) = -!L [Io(w ) Ko(wf) - Ko(w )Io(Wf)] , BC 2 Io(w)
(68)
and use of the inversion theorem for Laplace transforms yields the result
7TqU* T(f, T) = - Au
00
I
"=1
Jo(W,,j) 2 2 2( ) exp (-ac WilT), J1
(69)
W"
where the W n are the roots of J o( w,,) = 0. The boundary conditions for this problem are
wlu=o=O,
aw/aulu_o=o,
A*a2¢/au 2+B*a¢/ou=o atu=O,
(70)
where
A*= t,
(71)
LA RGE VI SCO E LA STI C P LAT E VI B RATIO NS
335
6r----------------,. 4 13
2
o Figure 3. Comparison of linear and non-linear deflections for th e shallow sh ell of equilateral triangular pla n form .
4 r - - - - - -- - - - - - - ,
o
0 ·1
0 ·2 'f
0 ·4
0 '3
Figure 4. Effect of changing the viscoe last ic parameter y for the shallow shell o f equilateral triang ular planform.
2-0
1·6
~
1·2 13
0 ·8
0·4
/
a
0 ·06
0·08
0 '10
T Figure S. Effect of varying th e radi i of curvat ure in the shallow shell problem. R.", =ro. • , R., = R ,. =600; x , R.< = R,. = 700 ; +, R , = R,. = 1000; 0 , linear plate solution . . .
336
D. L. HILL AND J. MAZUMDAR
and /-L has again been taken to be constant. The third of these conditions is a statement of the requirement of vanishing circumferential strain [8]. The eigenfunctions of the corresponding elastic problem are obtained by solving equations (32) and this is done by assuming power series of the form N
W(u)
= L
giui,
cP(u) =
"-1
N
L hu', n=l
where N is the predetermined number of terms to be used and the gi and hi are the unknowns which are again solved for by using the method of collocation. The collocation nodes were evenly spaced over the region of integration and again seven terms were sufficient for convergence. The time function a( 7") is obtained by solving equation (46) with the initial conditions a (0) = 0 and a/CO) = 0, and the deflection w( II, T) is found by substituting into equation (34). For illustrative purposes the shell is assumed to be made of the same material as the plate of the previous section. The values R; = Ry = 1000 mm, Rxy = 00 and q = 108 mm" °C were also used. The non-linear terms in the analysis again have a marked effect on the size of the thermally induced vibration and Figure 3 demonstrates this, the non-dimensionalized parameters in this case being
w=wa 2/haTQ,
f=.JE/p(T/a),
(72)
and l' was taken to be zero. Figure 4 shows the effect of altering the viscoelastic parameter. It is interesting to note that if the radii of curvature are allowed to go to infinity in the shallow shell analysis then the governing system of equations reduces to those of the corresponding linear viscoelastic plate problem, this problem being solved by ignoring the non-linear terms in the equations of section 2 while assuming the appropriate temperature distribution. In this case f3(T) = 0 and equation (43) is no longer needed. This fact provides a useful check on the accuracy of the methods being used. Figure 5 shows one such example, produced with l' = O. 8. CONCLUSION
A method has been presented of obtaining approximate solutions to problems involving the large amplitude thermally induced vibrations of viscoelastic plates and shallow shells and the method has been shown to be applicable to relevant problems in modern high temperature technology. It is seen that unless the amplitudes of the vibrations are very small when compared to the thicknesses of the plates or shallow shells then the non-linear terms in the basic equations which decribe those motions cannot be ignored. When these terms are taken into account however, the set of equations becomes considerably more difficult to obtain solutions of, and only for very special cases can exact solutions be obtained. The method employed in this work is one based on a combination of the method of constant deflection contour lines and the Galerkin approximation technique. Application of the method results in a non-linear ordinary differential equation in one variable and this can be solved accurately and quickly by using such methods as the Runge-KuttaNystrom method. The approach should be immediately useful to researchers and technologists in fields related to high temperature solid mechanics. REFERENCES 1. J. MAZUMDAR, D. HILL and D. L. CLEMENTS 1980 Journal ofSound and Vibration 73,31-39.
Thermally induced vibrations of a viscoelastic plate.
LARGE VISCOELASTIC PLATE VIBRATIONS
337
2. D. HILL, J. MAZUMDAR and D. L. CLEMENTS 1982 International Journal of Solids and Structures 18, 937-945. Dynamic response of viscoelastic plates of arbitrary shape to rapid heating. 3. J. MAZUMDAR and D. HILL 1984 Journal of Sound and Vibration 93, 189-200. Thermal1y induced vibrations of viscoelastic shallow shells. 4. R. MUKI and E. STERNBERG 1961 Journal of Applied Mechanics 28, 193-207. On transient thermal stresses in viscoelastic materials with temperature dependent properties. 5. J. R. COLEBY and J. MAZUMDAR 1982 Journal ofSound and Vibration 80, 193-201. Non-linear vibrations of clastic plates subjected to transient pressure loadings. 6. J. MAZUMDAR 1971 Journal ofSound and Vibration 18,147-155. Transverse vibration of elastic plates by the method of constant deflection lines. 7. J. MAZUMDAR and D. Bucco 1978 Journal of Sound and Vibration 57,323-331. Transverse vibrations of viscoelastic shallow shells. 8. R. JONES and J. MAZUMDAR 1974 Jot/mal of the Acoustical Society of America 56, 1487-1492. Transverse vibrations of shallow shells by the method of constant deflection contours. 9. M. SUNAKAWA and N. KUNAI Institute ofSpace and Aeronautical Science, University of Tokyo, Report No. 521. Response of viscoelastic shallow spherical shells to dynamic pressures. 10. J. MAZUMDAR and R. JONES 1977 Journal of Sound and Vibration SO, 389-397. A simplified approach to the large amplitude vibration of plates and membranes. 11. J. S. HEWIlT and J. MAZUMDAR 1974 Journal of the Engineering Mechanics Division, Proceedings of the American Society ofCivi I Engineers 100, 1143-1148. Vib rations of triangular viscoelastic plates. 12. R. JONES, J. MAZUMDAR and FU-I'EN CHIANG 1975 International Journal of Engineering Science 13, 423-443. Further studies in the application of the method of constant deflection lines to plate bending problems. 13. D. Bucco and 1. MAZUMDAR 1984 Journal of the Australian Mathematical Society Series B 26, 77-91. Buckling analysis of plates of arbitrary shape. 14. J. MAZUMDAR 1974 Nuclear Engineering and Design 31, 383-390. A method for the study of transient heat conduction in plates of arbitrary cross section. 15. R. E. BELLMAN, R. E. KALABA and J. A. LOCKElT 1966 in Modern Analytical and Computational Methods in Science and Mathematics. New York: American Elsevier Publishing Company.
A STUDY OF THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATIONS OF VISCOELASTIC PLATES AND SHALLOW SHELLS D. L.
HILL AND
J.
MAZUMDAR
Department of Applied Mathematics, University of Adelaide, Adelaide, South Australia 5001, Australia (Received 6 March 1986, and in revised form 4 June 1986) A method for the study of large amplitude vibrations of viscoelastic plates and shallow shells is presented. The concept of iso-amplitude contour lines in conjunction with Galerkin's method is used and the resulting non-linear differential equations are solved for appropriate time functions. The thermally induced large amplitude vibrations of a square viscoelastic plate and of a shallow shell of equilateral triangular planform are examined. All details are explained by using graphs. 1. INTRODUCTION
In many modern applications viscoelastic plates and shells experience substantial deflections when they are induced to vibrate, in some cases the size of these deflections being of the order of the thicknesses of the structures themselves. In these cases the linear theory is inadequate to accurately describe the induced motions and must be suitably modified. There is however a heavy cost with the significantly more accurate theories and this is due to the retention of non-linear terms in the governing equations describing the motion. Exact solutions of these equations are unknown except in certain very special cases and usually one must resort to numerical procedures, for example the finite element method. In this paper a quasi-analytical method is presented for obtaining accurate approximate solutions to the non-linear governing equations which will enable one to gain a useful insight into the behaviour of the structures modelled by these equations. The method is an extension of one used earlier to study the small amplitude (linear) thermally induced vibrations of plates and shallow shells of arbitrary shape [1-3] and is shown to be applicable for the non-linear, large amplitude analysis. The method consists of using the idea of iso-amplitude contour lines to derive a pair of coupled ordinary differential equations in terms of a contour function u(x, y) and consequently a non-linear ordinary differential equation which depends on the particular viscoelastic model chosen for the material of which the structure is made. The method of iso-amplitude contour lines has been applied when the deflection on the boundary of the plate or shell is the same everywhere on that boundary, for example, in the cases of clamped, simply supported, partly clamped and partly simply supported, and movable boundaries. The cases of free boundaries and problems involving more than one distinct family of iso-amplitude contour lines require extensions of the basic theory. The method is applied to study the response of certain viscoelastic plates and shallow shells with rectilinear boundaries. The results should be useful to nuclear engineers and to workers in thermoplastics. 323 0022-460X/87/ 140323 + 15 $03.00/0
© 1987 Academic Press Limited
324
D. L. HILL AND J. MAZUMDAR
2. DERIVATION OF THE EQUATIONS FOR THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF VISCOELASTIC PLATES
The thermally induced vibration of viscoelastic plates whose induced deflection is of the order of the plate thickness is analyzed here by using the theory of non-linear plates usually associated with the name of von Karman. Consider a homogeneous, isotropic viscoelastic plate of thickness h and mass per unit area p in a non-stationary temperature field. The plate is referred to a system of rectangular Cartesian co-ordinates x, y and z such that the horizontal plane Oxy coincides with the middle surface of the plate in its undeformed state. The deflection w(x, y, 'T) of the plate, which is a function of the spatial co-ordinates x and y and of the temporal variable 'T, is taken as positive downward in the z-direction. An incremental increase in temperature, T(x, y, Z, 'T), in the plate will produce a state in which thermal stresses are evident. The stress-strain equations are given by [4] 0,2)
where Sij and Bij are the deviatoric stress and strain tensors, Ukk and e kk are the traces of the stress and strain tensors and aT is the coefficient of linear thermal expansion of the material of which the plate is composed. P', Q', P" and Q" are differential operators in the time variable T and are of the form n,
P'
n
= L: anp n,
P"=
n=O
f
cnpn,
n=O
where p = ala'T. The unknown constants can be determined from the properties of the particular plate material being studied. When the plate vibrates, at any instant of time the intersection of the deflected surface with parallels z = constant yields a set of contours which, after projection on to the Oxy plane form a family of closed, non-intersecting level curves which will be denoted by u (x, y) = constant. If the plate is subjected to clamped, or simply-supported, or a combination of clamped and simply-supported boundary conditions then its boundary will be a member of this family of curves. Denote this family by e u , 0",; U ",; u* where u = 0 is the boundary of the plate and u = u* is the maximum value attained by the function u. Consider an area n u of the plate bounded by an iso-amplitude contour line eu at any instant of time 'T. The governing equation of motion for the large amplitude vibration of a viscoelastic plate can be expressed, corresponding to the case of an elastic plate [5], in the form (3)
where Qn is the shearing force, aMnrl as is the edge rate of change of twisting moment and N x , N; and N x y are the in-plane stresses. Expressing these in-plane stresses in terms of a stress function ¢(x, y, -r) as (4)
substituting these expressions into equation (3) and using the well-known expressions for Qn and aMnJas results in the integro-differential equation
a3wf
D(p)-,
au'
CO'
azwf
R. ds+ D(p)-z
au
c"
aWf
PI ds+ D(p)-
au
CO'
G~
ds
325
LARGE VISCOELASTIC PLATE VIBRATIONS
-D(p)(l+J.L(p»
+
I [u x a,KT+ Uy aKTJ d~
Jc"
rJx ay vi 2w 2¢ 2¢ 2w a2w a a a a rPep ph-2 --2 -2 --2 -2+2-- - - - d!1 = 0, aT ax ay oy ax ayax ay ax
rPw]
If [ nu
(5)
where w is the deflection of the plate, D( p) is the viscoelastic flexural rigidity and fL (p) is the viscoelastic Poisson ratio of the plate material, and F 1 --
t=U~+U;,
t -1 / 2[3UxxU 2x + 3 UyyU 2y + Uxxu 2y + UyyU 2x
-
K T ==
P12I
+ 4UxyllxUy ] ,
h /2
aTT' z dz.
-1,/2
The functions wand ¢ must also satisfy a compatibility relation of the form V4 ¢ = hE(p)[(a 2w/ay ax)2 - (a2w/ax2)(a2w/(iy)] - hE(p)V 2S , T
(6)
where E(p) is the viscoelastic operator corresponding to the Young's Modulus of the material and lOT
1
=
h
I
hl2
cxTT'dz.
-h12
Since this equation must hold at all points in the plate it can be integrated over the region il u yielding
If [ a;
V4¢ - hE ( P) { ( a2w )2_a2~a2~}+hE(P)V2BTJ d!1=O. ayax ax ay
(7)
If in particular, the temperature variation across the thickness of the plate is of the form T'(x, y, Z, T) == T(x, y, T)G(Z)
for some function G(z) then are given by 1
C. =-h
f
h/2
eT =
C. T and K T = CKT, where the coefficients C. and C K
cxTG(z) dz,
flr/2
12
CK = h 3
--hI2
cxTG(z)z dz.
-h/2
Equations (5) and (7) are a coupled system of non-linear partial differential equations for the deflection and stress functions which describe the thermally induced large amplitude vibration of viscoelastic plates.
3. SOLUTION OF THE PLATE EQUATIONS
An approximate solution to the system of equations can be obtained by using Galerkin's method with the eigenfunctions of the corresponding linear elastic free vibration problem as the co-ordinate functions. The equation for the corresponding free vibration problem is, for the ith eigenmode [6], D
f (Qn - a~nt) ell
ds = phAT
If
W';(u) df2,
(8)
nil
where the Ai are the eigenfrequencies and the W;(u) are the eigenfunctions. D is the flexural rigidity. It is now assumed that ¢ can be written in terms of the contour function
326
D. L. HILL AND J. MAZUMDAR
u and the time variable 7"; that is, ¢ = ¢(u, 7") (see reference [7]). To proceed, a separation of variables type solution of the form cc
CD
L
w(u, 7") =
L
¢(u, T) =
W;(u)ai( T),
;=}
W;(U),Bi( T),
(9)
i=l
is assumed, where the ai (T) and ,Bi(7") are time functions to be determined. For simplicity consider only the first terms in equations (9), namely,
=
w(u, 7")
(10)
¢(u, 7") = W(u),B( 7"),
W(u)a( 7"),
where the subscripts i have been omitted. Substitution of equations (10) into equations (5) and (7) gives 3
Wf d- 3 D(p)a(T) [ du
c,
d2Wf R 1 dS+-2 du
f[
- D(p)(1 + /-L(p))
+pha"(7")
If
a;
c,
Cu
dWf 0; ds ] F1 ds+du c;
aK T
ux -
ax
aKT] ds + uy - I. ay
W(u) dfl+a(T),B(T)
vt
II
[2P
A
fl"
d2~dW+PB(dW)2J dfl=O du
du
du
(11)
and
where PA
= U;Uy y + u;ux x - 2u x yuxuy and PB = 2uxx uyy -2u;,y,
If
nil
V4W(U) dfl
= ~U~
f
ell
R] ds+
~u~lJe
and the relation
F 1 ds+
l,
~:lJell 0; ds
has been employed (see reference [8]). The integral relations
f
L"
W(u) d.Q = -
f.
W(u)
te"o ~duo
(13)
and
IIn"
A(x, y) d.Q =
-
L~ fc"o A(x, y)~ duo,
(14)
where A(x, y) is an arbitrary function of x and y will now be useful. Applying Green's Theorem to the vector function A= (AI. A 2 ) one has
If
nil
V' A dJ1
=l
A· n ds.
(15)
Jell
Differentiating this equation with respect to u and using equation (14) gives f
C
u
aAI ds= d- f (A ds +aA2) - lu X + A 2u y)(ax ay v't du c, v't
(16)
327
LARGE VISCOELASTIC PLATE VIBRATIONS
since n =
-(uxlY't, uylY't), where AI and A 2 are arbitrary functions of x and y. Now, letting (17)
gives
f
Pn-ds = - d ..Jt d u
C U
f
C
u
PA -ds ..[i ,
(18)
Making use of equation (8) in equations (11) and (12), then differentiating the resulting expressions with respect to u and using the integral result (13) yields
-I:
ph).. 2 D(p)a( 7") [ -V W(u) - pha fI (7") W(u)
dS]
Jc".Ji -
f
D(p)(1 + j.L(p))
W)
ds 1 . - a( 7"){3( 7") - d [(d -du du
c" v t
2
Irc
f
ds
2
V KTVi
u
c,
dS] = 0
PA r: vt
(19)
and
fcu ~] - E(p)h feu V2B"~ d [(dW)2-1: dS] =0, -"2 E ( p )ha -(7" ) du du Jc PA.Ji
(3( r) [ -
P~
2
W(lI)
1
?
(20)
u
in which Green's Theorem has been used to establish that
~1,
au Jc
u
[II aKT+ u aKT] ds =1, ax ay Jt
x
y
Jc
V2K ds
TV!'
u
Now multiplying equations (19) and (20) throughout by W( II) and integrating them from u = 0 to u = u* gives
ph).. 2C1[D(p)1 D]a( 7") + fK (7") + ph C, a"( r ) + C 2a (r){3 (r) = 0, 2( r) = 0, (ph).. 21 D)C r)+ E(p)hf.(r)+!C 1{3(
f
II' [
0
?
W-(u)
-I:Jc Jt dS] du, u
fdr) =
ioU' W(U)[ D(p)(l+j.L(p))
f.(r) =
I"
W(II)
(22)
2E(p)ha
where C1=
(21)
C2 =
f
0
u· [
d[(dduW) 1,Jc
W(u) dll
tu V2KT~]
2
dS] ] du,
PA..[i
u
du,
[fcu V2BT~] du,
(23)
Equation (22) gives the solution for (3( r) as (3 (r)
= -( D 1 ph): 2C I )[ E (p
)hf. (r) +!C2E(p )ha 2(r)]
(24)
and substitution of this expression into equation (21) results in a non-linear ordinary differential equation for the time function a (r), 1 [ DC2 D(p) 2(7")]+fK(r) ] a"(r)+ phC - ph)..2C\a(r)[E(p)h!.(r)+!C2E(p)ha +A2---v-a(r)=0 1
(25) which can be solved numerically by any of the standard numerical integration methods. In this study the Runge-Kutta-Nystrom method is used.
328
L. HILL AND J. MAZUMDAR
D.
4. DERIVATION OF THE EQUATIONS FOR THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF VISCOELASTIC SHALLOW SHELLS
The study of shell structures has become an important field of research due to their high load carrying capabilities in comparison to their own weight [9]. Let the equation of the middle surface of the shallow shell be expressed in the form
z = (x 7-;2Rx ) + (xyj R x y ) + (y2j2R),
(26)
where R x, R,. and Rxl' are the radii of curvature of the shell and the usual assumptions for shallow shells have been made. When the shell vibrates, then at any instant of time the intersection of the deflected surface with parallels z = constant yields a set of contours which, after projection onto the baseplane z = H (H denoting the apex height), is a family of non-intersecting closed level curves u(x, y) = constant. Consider an area nl/ of the shell bounded by a level curve eu at any instant of time T [7, 8]. Summing the vertical forces over this area yields the equation of motion for the large amplitude vibration of a viscoelastic shallow shell in the form
where all terms are as previously defined. The tangential force components can be expressed in terms of a stress function ¢(x, y, T) as follows: tv." = a2¢j ay 2 , N; = a2¢jax2, N."y = -a 2 ¢ j ay ax. (28) Substitution of these expressions into equation (27) and proceeding as before yields the integra-differential equation 3
D(p) a
:
au
1
re" R
1
ds+ D(p) a2~
au
-D(p)(l+Il(p))
fa [
1
rc.. F[ ds+ D(p) auaw 1re" 0; ds
aKT aKT] -+ ds ux-+u yax ay,fi
c"
If [ n,
2¢ 2 aw 1 a aT R, ay
1 rP¢ R; ax
ph~+---2+--2
2 a2¢ a2¢ 2 w fl¢ a2w a2 ¢ a2w ] - - - - - - 2 - 2 - - 2 - 2 + 2 - - - - dn=O. R x y oy ex ax oy ay ax ayax ayax
(29)
The compatibility relation for the shallow shell problem is 2 2w 2 2wa 2w] aw a - - --+
[1
1
2
(30)
which must hold at all points in the shell and so can be integrated over the region yielding 2 V4¢ _ E ( P)h [ _1 a2W+_l rPw_~ a w 2 flu u; al R y ax R xy ev ex
If {
2w)2
+ ( -a ayax
nu
2wa2w]
a -2 ax- ay
--?
+ hE(p)V 2 8T }
dIl = O.
(31)
Again it has been assumed that the stress function ¢(x, y, r) can be expressed in terms of the iso-amplitude contour line function and the time variable. Equations (29) and (31) are a coupled system of non-linear partial differential equations for the deflection and stress functions which describe the thermally induced large amplitude vibration of viscoelastic shallow shells.
329
LARGE VISCOELASTIC PLATE VIBRATIONS
5. SOLUTION OF THE SHALLOW SHELL EQUATIONS
The system of equations can be solved approximately by using Galerkin's method with the eigenfunctions of the corresponding linear elastic free vibration problem as the co-ordinate functions. These equations are [10, 11]
D
If (
AMnr) , f (Q" -----;;;- ds = pli):i
a;
ell
IIn"
4
V 4>i dfl = Eh
dcflif
\.Vi u) dfl- du
K I ds,
ell
~~j fc" K
I
ds,
(32)
where the Ai are the eigenfrequencies and the Wj(u) and lPi(u) are the eigenfunctions. A separation of variables type solution of the form co
L
w(u, 7")=
(33)
Wj(u)aiCr), ;=1
;=1
is used where the cd 7") and (3j(T) are time functions to be determined. For simplicity only the first terms in equations (33) are used, the subscripts being omitted. Hence w(u, r) = W(u)a( T),
¢( u, or) = cI>(u)f3( r),
(34)
and substitution of equations (34) into equations (29) and (31) yields 2W 3 i dW i d Wi d D(p)a(T) [ du 3 Te" R I ds+ du2 Teu F I dS+d;Te" 0; ds
J
If
f[
ds ux -aK T+ Uy -aKrJ r:+ pha"( r) W(u) dil . Ax By y t flu 2WdlP 2 dcI> i d dW( d tP dcI»] j +/3(r) du Te K d s + a ( r ){3 ( r ) n PA du 2 ~+ct; PAdu2+Pnd;;" dfl=O
-D(p)(l + J.L(p))
c;
If [
II
II
(35)
and 3 d tP i d2tPi dlPi [ (3(T) du3 Te R 1 ds+ du2 Te" F I ds+ du
re" a; ds ]
u
- E(p)ha(r)
~: fcu K
+E(p)ha 2(r)
ds+ E(p)h
I
Ifa"
2
V e T dfl
If [
d2WdW P (dW)2] P A--, -+~ dfl=O. du" du 2 du
a;
(36)
Substitution of equations (32) into equations (35) and (36) gives 2
D(p)a(r) [ -PhA D
If
a;
-D(p)(1 + J.L(p))
d
1dcI>f
W(u) dil-- D du
f[ c,
AK T u.~-+ Ax
+(3(r)-d K I ds+a(r){3(r) u c;
K I ds
]
Cu
aK T ] ds r:+ pha"
Uy -
If
W(u) dfl
Ay
yt
n,
d d 4J2+PndcI»] PA--2 - +dW( - PA dfl =0 du du du du du (37)
If [
2WdlP
a;
2
330
D . L. HILL AND J . MAZUMDAR
and
(38) Now, differentiating equations (37) and (38) with respect to u gives
ph): 2 D (p)a(r) [ -DW(u)
lJe
1 dud [dct;7Je
ds
Ji- D
1
- pha"W(u) Jc
K. ds
]] -D(P)(l+p,(P))Jeu 1 V K ds 2
u
u
ds d[difJ 1 ] Ji+ f3( r ) du ~Je K, ds -
II
TJi
l
d[dWd
II
(39) and
f3(r)~[dW1, KldS]_E(P)a(r)~[dW1, - . du
du
_ E(p) E
Je
1,
Jc
E
u
V2ST
du
du Jeu
ds _~ E(p) a2(r)~[(dW)2l PA dS] =0. 2 E du du Jeu Ji
Jt
u
(40)
Then multiplying equation (40) throughout by
B1 =
f
0
u·
d[dW 1,] du Jc K ds du,
€/Jeu) du
1
U
f~ (r)=
f"· o
B2 =
Jor
d[(dct;"W)
II"
f
2
d] du , Je PAlL
1,
«
V
elf
2ETJt dS]
duo
(42)
Equation (41) can be rearranged to give
f3(r) =.f!(T) + E(p) a(r)+ B 2 E(p) a2(r). B.
E
2B(
E
(43)
Now, multiplying equation (39) by W(u) and integrating from u =0 to u = u* gives 2 D(p) D (p) ph): B3------v-a (r ) + B 4----rJ" a( T) +[« (r) + phB3a"( r) - B 4f3 (r) + Bsa( r )f3( T) = 0,
where
(44)
331
LARGE VISCOELASTIC PLATE VIBRATIONS
and consequently equation (44) yields a second order non-linear ordinary differential equation for a ( T), of the form a"(T)=_>..2
D
( P) a(T)-~ D(p) a(r) phB) D
D
- -1- { fK(r)-(B 4 - B sa(r)) phB)
[.f:(T) E(p) B, E(p) ]} --+--a(r)+-~- - a 2(T) B1
E
2B.
(46)
E
and this equation can also be solved numerically by any of the usual numerical integration methods. In this study the Runge-Kutta- Nystrom method is again used.
6. THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF A CLAMPED SQUARE PLATE
Consider the thermally induced vibration of a clamped square plate which is initially at rest and at a constant temperature To. At time zero the boundary of the plate is instantaneously set to zero temperature, the plate being composed of material which can be modelled as an "incompressible" Kelvin material which is represented by the equation D(p)/ D = 1 + 'YP,
(47)
where 'Y is a material constant. To simplify calculations the Poisson's ratio of the plate material is also taken to be a constant and hence D(p)jD=E(p)/E.
(48)
The method of iso-deflection contour lines depends on being able to find a good approximation to the actual contour line function. Some work has been carried out on this problem [12] and various forms for the contour line function have been used (see, for example, reference [13]). Using the equation of the plate boundary to obtain a contour line function has proved useful in the past and this approach will be used here. Thus the equation of the iso-amplitude contour lines is taken to be (49)
where I is the length of a side of the square boundary and in fact the contour line u = 0 is the equation of the boundary of the square. For a clamped plate the boundary conditions are
wi u=o =0 '
(JwjiJull/=o = O.
(50)
A method for solving heat conduction problems of this type was proposed by Mazumdar [14] and leads to the partial differential equation
-aTf Jl ds + 21 au
c"
C
If
-erdfl = 0,
a; aT
(51)
where c 2 is a material parameter and the temperature T is expressed as a function of u and r: that is, T = T(u, 1'), following references [1-3]. It is assumed, for simplicity, that the temperature variation across the thickness of the plate is linear, that is, T' = zT(x, y, 1'), and so f:T = 0 and K T = aTT. Equation (51) is solved by using the method of separation of variables, seeking a solution of the form (52)
332
D. L. HILL AND J. MAZUMDAR
After differentiating equation (51) with respect to u and using the mean value theorem to evaluate approximately the contour integrals involved yields the equation (53) where w 2 = [2TJ/4c 2 , and after introducing the new variablej? == 1 - u this equation becomes the Bessel differential equation of zeroth order d 2T* /df+ (II!) dT*/df + w 2 T * = 0,
(54)
the complete solution of which is given by T*(f) = AJo(wf) + BYo(wf),
(55)
where J o and Yo are the Bessel functions of the first and second kinds respectively. Since the temperature must be a finite quantity at all points in the plate then B must be zero and the edge condition requires that W be a root of the equation (56) The initial uniform temperature To is now expressed in the Fourier-Bessel series To =
l:
(57)
A"Jo(wnf),
n=l
where W n is a root of equation (56) and it is assumed that the series can be integrated term by term. It can be shown that the coefficients An are given by (58) and hence
T(l, or) =2To
co
Jo(wnf) J ( ) exp
l: n=1 W
n
1
co;
(4c --1 2 2
2 ) WIlT
(59)
is the solution to this temperature distribution problem. Equation (8) gives the eigenfunctions of the corresponding elastic problem and this equation is solved by assuming a power series of the form W(u) =l:~=l QiU i, where N is a pre-determined number of terms to be considered and the Q j are unknowns which are found by using the well-known method of collocation. The collocation nodes were chosen so that they were evenly spaced over the region of integration and it was found that seven terms were enough to ensure reasonable convergence to the required eigenfunctions and variation in the chosen nodes did not alter the results significantly. The unknown time function aCT) is obtained by solving equation (25) approximately by using the Runge-Kutta-Nystrom method. Since the plate is initially at rest the initial conditions of the problem are o (O) = 0 and a/CO) = O. Equation (10) is then used to find w(u, or). For illustrative examples parameter values were chosen based on those given in reference [9]. These are [== 150 mm, E = 1640 kg/rnm", h = 3 mm, J.L == 0·3 and p == 1·19 X 10- 3 kg/em? is the mass per unit volume. The values aT == 1O-4;oC, To = lOOO°C and c 2 == 1 m 2 / s were also used. The effect of the non-linear terms in the preceeding analysis, although negligible for very small time values, becomes quite significant as time advances. This is seen in Figure 1 which shows a graph of non-dimensional deflection against non-dimensional time for 'Y = O. The non-dimensional parameters are defined as (60)
LARGE VISCOELASTIC PLATE VIBRATIONS
333
4,.------------,
*:3
a
0,04
0·08
.,.*
0·12
0·16
Figure 1. Comparison of linear and non-linear deflections for the square plate. 2,.--------------,
'3
0 t-==------ - - - \ A - - - - j
-1
o
0'04
0·12
0·16
Figure 2. Effect of changing the viscoelastic parameter y for the square plate.
In Figure 2 the effect of varying the viscoelastic parameter y on the large amplitude vibrations is demonstrated. It can be seen that increasing the value of 'Y causes the motion of the plate to be damped, which is to be expected. 7. THE THERMALLY INDUCED LARGE AMPLITUDE VIBRATION OF A CLAMPED SHALLOW SHELL OF EQUILATERAL TRIANGULAR PLANFORM
Consider the thermally induced vibration of a clamped shallow shell of equilateral triangular planform, initially at rest, which is acted upon by a hot spot of strength q at its apex and whose boundaries are held at zero temperature for all time, the shell being composed of the same material as that of the plate of the previous section. The equation for the iso-arnplitude contour line function for the triangular shallow shell is chosen to be u(x, y) = x 3 -3 xy 2- ax 2 - ay2+4a3/27, (61) where a is the perpendicular height of the equilateral triangular boundary and the contour line u = 0 is the equation of the boundary of the shell planform.
334
D. L. HILL AND J. MAZUMDAR
To find the temperature distribution under these conditions the method of Laplace transforms is employed. If the shell is reasonably shallow the temperature distribution within it can be approximated by that in the corresponding plate problem (R, = R; = R x y = (0). Again, following reference [14] yields the equation
d2~1
du Jc
u
v'ids+dT..i-l vtds- s2(T-To)1 ds=o, du du Jc c Jc" vt
(62)
u
where the temperature is again assumed to be a function of u and T and the transform of T( u, T) is denoted by 7'(u, s), s being the Laplace transform parameter and To = 0 the initial temperature of the shell. It is desirable to be able to calculate the contour integrals analytically because numerical inversion of the Laplace transform is a non-trivial problem, the results of which can be unstable (see reference [15]). To get an analytical solution to the temperature distribution problem the contour integrals involved are calculated approximately by using the mean value theorem and the following equation is obtained (upon remembering that To is zero);
(63) where w 2 = s/ ac' and j? = u* - u, where u* is the maximum value obtained by the function u. Equation (63) is a modified Bessel's equation of zeroth order and has as its complete solution (64)
where 10 and K, are modified Bessel functions. The unknown functions C 1 and C 2 are to be determined from the initial condition of zero temperature except at the hot spot, the conservation of heat generated, and the given boundary condition of the problem: that is, (1) T = 0 at T = 0 everywhere except at the centre U = u*, (2) lim
II~O
°
f
Cu
aT q -ds = - 2 B(T), an c
(65)
= on the boundary u = 0 for T;:' 0, where B( T) is the Dirac delta function. Taking Laplace transforms of conditions (2) and (3) above, yields
(3) T
limf d7'/df= -q/ BC
2
f ...1
and
7'=
°
onf= 1,
(66,67)
where B = 2aA o/ u* and A o is the area of the planform of the shallow shell. Use of these 2 conditions in equation (64) gives C 1 = - ( q/ BC )Kv(w )/ Io(w ) and C 2 = q/ ec2 • Hence T(f, s) = -!L [Io(w ) Ko(wf) - Ko(w )Io(Wf)] , BC 2 Io(w)
(68)
and use of the inversion theorem for Laplace transforms yields the result
7TqU* T(f, T) = - Au
00
I
"=1
Jo(W,,j) 2 2 2( ) exp (-ac WilT), J1
(69)
W"
where the W n are the roots of J o( w,,) = 0. The boundary conditions for this problem are
wlu=o=O,
aw/aulu_o=o,
A*a2¢/au 2+B*a¢/ou=o atu=O,
(70)
where
A*= t,
(71)
LA RGE VI SCO E LA STI C P LAT E VI B RATIO NS
335
6r----------------,. 4 13
2
o Figure 3. Comparison of linear and non-linear deflections for th e shallow sh ell of equilateral triangular pla n form .
4 r - - - - - -- - - - - - - ,
o
0 ·1
0 ·2 'f
0 ·4
0 '3
Figure 4. Effect of changing the viscoe last ic parameter y for the shallow shell o f equilateral triang ular planform.
2-0
1·6
~
1·2 13
0 ·8
0·4
/
a
0 ·06
0·08
0 '10
T Figure S. Effect of varying th e radi i of curvat ure in the shallow shell problem. R.", =ro. • , R., = R ,. =600; x , R.< = R,. = 700 ; +, R , = R,. = 1000; 0 , linear plate solution . . .
336
D. L. HILL AND J. MAZUMDAR
and /-L has again been taken to be constant. The third of these conditions is a statement of the requirement of vanishing circumferential strain [8]. The eigenfunctions of the corresponding elastic problem are obtained by solving equations (32) and this is done by assuming power series of the form N
W(u)
= L
giui,
cP(u) =
"-1
N
L hu', n=l
where N is the predetermined number of terms to be used and the gi and hi are the unknowns which are again solved for by using the method of collocation. The collocation nodes were evenly spaced over the region of integration and again seven terms were sufficient for convergence. The time function a( 7") is obtained by solving equation (46) with the initial conditions a (0) = 0 and a/CO) = 0, and the deflection w( II, T) is found by substituting into equation (34). For illustrative purposes the shell is assumed to be made of the same material as the plate of the previous section. The values R; = Ry = 1000 mm, Rxy = 00 and q = 108 mm" °C were also used. The non-linear terms in the analysis again have a marked effect on the size of the thermally induced vibration and Figure 3 demonstrates this, the non-dimensionalized parameters in this case being
w=wa 2/haTQ,
f=.JE/p(T/a),
(72)
and l' was taken to be zero. Figure 4 shows the effect of altering the viscoelastic parameter. It is interesting to note that if the radii of curvature are allowed to go to infinity in the shallow shell analysis then the governing system of equations reduces to those of the corresponding linear viscoelastic plate problem, this problem being solved by ignoring the non-linear terms in the equations of section 2 while assuming the appropriate temperature distribution. In this case f3(T) = 0 and equation (43) is no longer needed. This fact provides a useful check on the accuracy of the methods being used. Figure 5 shows one such example, produced with l' = O. 8. CONCLUSION
A method has been presented of obtaining approximate solutions to problems involving the large amplitude thermally induced vibrations of viscoelastic plates and shallow shells and the method has been shown to be applicable to relevant problems in modern high temperature technology. It is seen that unless the amplitudes of the vibrations are very small when compared to the thicknesses of the plates or shallow shells then the non-linear terms in the basic equations which decribe those motions cannot be ignored. When these terms are taken into account however, the set of equations becomes considerably more difficult to obtain solutions of, and only for very special cases can exact solutions be obtained. The method employed in this work is one based on a combination of the method of constant deflection contour lines and the Galerkin approximation technique. Application of the method results in a non-linear ordinary differential equation in one variable and this can be solved accurately and quickly by using such methods as the Runge-KuttaNystrom method. The approach should be immediately useful to researchers and technologists in fields related to high temperature solid mechanics. REFERENCES 1. J. MAZUMDAR, D. HILL and D. L. CLEMENTS 1980 Journal ofSound and Vibration 73,31-39.
Thermally induced vibrations of a viscoelastic plate.
LARGE VISCOELASTIC PLATE VIBRATIONS
337
2. D. HILL, J. MAZUMDAR and D. L. CLEMENTS 1982 International Journal of Solids and Structures 18, 937-945. Dynamic response of viscoelastic plates of arbitrary shape to rapid heating. 3. J. MAZUMDAR and D. HILL 1984 Journal of Sound and Vibration 93, 189-200. Thermal1y induced vibrations of viscoelastic shallow shells. 4. R. MUKI and E. STERNBERG 1961 Journal of Applied Mechanics 28, 193-207. On transient thermal stresses in viscoelastic materials with temperature dependent properties. 5. J. R. COLEBY and J. MAZUMDAR 1982 Journal ofSound and Vibration 80, 193-201. Non-linear vibrations of clastic plates subjected to transient pressure loadings. 6. J. MAZUMDAR 1971 Journal ofSound and Vibration 18,147-155. Transverse vibration of elastic plates by the method of constant deflection lines. 7. J. MAZUMDAR and D. Bucco 1978 Journal of Sound and Vibration 57,323-331. Transverse vibrations of viscoelastic shallow shells. 8. R. JONES and J. MAZUMDAR 1974 Jot/mal of the Acoustical Society of America 56, 1487-1492. Transverse vibrations of shallow shells by the method of constant deflection contours. 9. M. SUNAKAWA and N. KUNAI Institute ofSpace and Aeronautical Science, University of Tokyo, Report No. 521. Response of viscoelastic shallow spherical shells to dynamic pressures. 10. J. MAZUMDAR and R. JONES 1977 Journal of Sound and Vibration SO, 389-397. A simplified approach to the large amplitude vibration of plates and membranes. 11. J. S. HEWIlT and J. MAZUMDAR 1974 Journal of the Engineering Mechanics Division, Proceedings of the American Society ofCivi I Engineers 100, 1143-1148. Vib rations of triangular viscoelastic plates. 12. R. JONES, J. MAZUMDAR and FU-I'EN CHIANG 1975 International Journal of Engineering Science 13, 423-443. Further studies in the application of the method of constant deflection lines to plate bending problems. 13. D. Bucco and 1. MAZUMDAR 1984 Journal of the Australian Mathematical Society Series B 26, 77-91. Buckling analysis of plates of arbitrary shape. 14. J. MAZUMDAR 1974 Nuclear Engineering and Design 31, 383-390. A method for the study of transient heat conduction in plates of arbitrary cross section. 15. R. E. BELLMAN, R. E. KALABA and J. A. LOCKElT 1966 in Modern Analytical and Computational Methods in Science and Mathematics. New York: American Elsevier Publishing Company.