revolution subjected to axisymmetric harmonic deformation," Prikl. Mekh., 25 ... Small intrinsic vibrations of shell of revolution with respect to some stable equilib-.
3o 4.
5.
6o 7.
O. Zenkevich, The Finite-Element Method in Engineering [Russian translation], Mir, Moscow (1975). V. I. Kozlov and S. N. Yakimenko, "Thermochemical behavior of viscoelastic solids of revolution subjected to axisymmetric harmonic deformation," Prikl. Mekh., 25, No. 5, 22-28 (1989). M. Mirsaidov, "Solution of a Lamb problem by the finite-element method with the use of radiation conditions," in: Mechanics of Deformable Solids [in Russian], Tomsk (1987), pp. 126-131. V. Novatskii, Theory of Elasticity [Russian translation], Mir, Moscow (1975). L. Sutherland, Use of the Finite-Element Method [Russian translation], Mir, Moscow
(1979). 8~
9o i0° Ii.
I. K. Senchenkov, V. I. Kozlov, I. G. Rubtsova, and A. B. Oleinikov, "Stress-strain state of a half-space in the neighborhood of a rigid sphere under a normal load," Prikl. Mekh., 24, No. 4, 19-25 (1988). O. P. Chervinko and I. K. Senchenkov, "Harmonic viscoelastic waves in a layer and an infinite cylinder," ibid., 22, No. 12, 31-37 (1986). F. Medina and J. Penzien, "Infinite element for elastodynamics," Earthquake Eng. Struct. Dyn., i0, No. 5, 699-709 (1982). J. Sochacki, "Absorbing boundary conditions for the elastic wave equations," Appl. Math. Comput., 28, No. i, 1-14 (1988).
INTRINSIC VIBRATIONS OF SHELLS OF REVOLUTION MADE FROM THERMOSENSITIVE COMPOSITE .MATERIALS D. V. Babich, V. V. Vorobei, V. I. Tarasyuk, and L. P. Khoroshun
UDC 539.3
An important practical problem in the dynamics of thin-walled structures is the investigation of the eigenfrequency spectrum, as a stage in studying the dynamic behavior of such mechanical systems. The intrinsic vibrations of shells consisting of traditional constructional materials have been very thoroughly considered. The vibrational behavior of shells made from composite materials due to the specific properties of these materials has been less fully investigated. In particular, there has been little study of the intrinsic vibrations of composite plates and shells under heat treatment. There are no data on the influence of the temperature dependence of the elastic properties of polymer composite materials on the amplitude-frequency characteristics of thin-walled structures. Nevertheless, this influence is significant even with slight heating. The aim of the present work is to develop a method of calculating the eigenfrequencies of composite shells, taking account of the temperature dependence of the properties of the shell material. A shell of revolution made by cross-winding of fiber filling is investigated. The bundle structure of the laminar shell is chosen so that the shell material is orthotropic in the lines of principal curvature, i.e., each elementary unidirectional layer inclined at an angle Yi to the shell meridian must correspond to another at the angle -Yi" The assumption made is that, in tension-compression and heating, the angles between the elasticsymmetry axes of the material are not distorted. Small intrinsic vibrations of shell of revolution with respect to some stable equilibrium position are considered. The intrinsic vibrations are described by means of the Ostrogradskii-Hamilton principle, formulated for the shell as a three-dimensional elastic body occupying a volume V [3]
Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika~ Vol. 28, No. 4, pp. 8-16, April, 1992. Original article submitted November 15, 1990o
1063-7095/92/2804-0209512.50 © 1992 Plenum Publishing Corporation
209
t~
8 ~ (~. - T)dt ----O,
(1)
to
where
1
N
i = -F [ (%e,1 + a22e22 -l- u12et2 -t- a~aer~ + 0'23e23 -{- a~oen + %0~ ~2) x
(2)
X Hflxldxflx 3 is the potential energy of the shell in the perturbed state, and
T = -~- p
(3)
[ at ] H2dxldxflx3 t=1
is
the kinetic
energy of the
shell.
I n Eq. ( 2 ) , e i j d e n o t e s t h e s t r e s s - t e n s o r c o m p o n e n t s d e t e r m i n e d by t h e l i n e a r components of the straln-tensor components eij;oij0 denotes the components of the stressstate tensor with respect to which the vibSations occur; eli denotes nonlinear terms of the components of the final-strain tensor in the perturbed state of the shell; u i denotes the components of the displacement vector; p is the density of the shell material; H2 i s t h e Lame c o e f f i c i e n t a s s o c i a t e d w i t h t h e a x i m u t h a l c o o r d i n a t e x2; x a , x 3 a r e c o o r d i n a t e s measured in the direction of t h e s h e l l m e r i d i a n and a l o n g t h e normal t o t h e s h e l l median surface, respectively; t is the time.
The three-dimensional problem in Eq. (i) is reduced to one-dimensional form with respect to the spatial coordinates by means of the Timoshenko kinematic hypotheses and Fourier-series expansion of the displacement-vector components
Ul
-- ~,~ u~ sin nx2;
u2 =
n=l
Vn
(S) COSnX2; U3 ---
n=l
qO1 =
qOin siI1 nX2; n=l
W~ sin nx2; n=l
q)2 - ~
(4)
(P2n COS n X 2. n=l
Multiplying the components of the displacement vector and the angles of rotation (~i, ~) by the exponential factor e -iwt and integrating over time from t o = 0 to t I = 2~/w, Eq. (i) is reduced to the form L
(5) where
]I• ---- 2 C11"'12 -t- C33v21 @ Kiw2,1 @ D:uq)~,l-Jr- D33':P~,, + A2 t + 2 (C~k~ + C~2G) u,1 + 2 ~ (C~k~ + C22k~)u - A~ (C~2k~+ C22k2)nv + ~
+ G,k~ + G - ~
"~ + 2 c~ ~
K 2 n ~ w + C~k~ + 2C~2k~k2 +
u, + G~--~ v, -- G~ A~
- - K & ( ~ + ~.0 . + c= -AT + G~-~I + Gk,~ . ~ -
[
210
&,
n
cA~.,
(
~
)]
n 2 Cn,--~A~., + C~'-~2+ 2 +Kzk~2 v~+
D ~ - ~ :p~+
i+ D2z ~'~ -
-
A~ +
-'~-)q)22 - - 2 Oi2 -":2 /
A221
2 ~
(6)
(02,1 (P2 +
12 = ~ - Tuo (w2l, -- 2kiuwj + k~u~) + (7) + T2~o ~
w2 -
~vw + ~v ~
2 Jl,
T* = - - ~
Vo ~
h
A~;
Az t
h3
2
h~
(8)
u ~ ~ hv + h ~ ~ + 7 7 q~ + - - f f ~ 1 & .
/
In Eqs. (6)-(8), the subscript n is omitted. A comma preceding subscript 1 indicates differentiation with respect to the coordinate x:; L is the shell length; h is the thickness; A 2 is the Lame coefficient of the median shell surface; kl, k 2 are the curvatures of the median surface. The reduced rigidities are defined as follows:
Cij=Euh; Dg 3 =
C33=G12h;
K2=Oi3h;
Gi~h3 Ei ; El~-- %Ei 12 ; E~ = l__viv2 l__viv2
D~j= Eijh3 • 12 ' viE~ l__viV~
(i = 1,2),
(9)
E i is the elastic modulus in the direction defined by the subscript; Gij is the shear modulus in the plane defined by the subscripts; v:, v 2 are Poisson's ratios; m is the angular eigenfrequency. These elasticity characteristics depend on the temperature in the general case. The reduced linear forces Tu0 =
h72 ~ oi~odx3
in the meridional
(i = i) and parallel (i =
--~/2 2) directions determine the basic equilibrium position around which small vibrations occur. Below, as an example, the intrinsic vibrations of rigidly fixed shells of revolution in a state heated to T k are considered. The stress state of such shells is approximately assumed to be momentless. The corresponding reduced forces in this case are
!• W 1 ~2
Ee
~+--~2]~k~C2r
3 ~ - - + 2 - - - - + o A2k~ \ E t Ei k~ T2~o -
dx',
.
(10)
E~ k22 dxt
ki k~ Ttto-
Here Tk
Clr = h .I [Eu (Tk) cq (T) + El~ (T~) ~2 (T)] dT; o Tk C2r = h ~ [Et~ (Th) ~((T) + E~ 2 (Th) c% (T)I dT. 6
(ll)
211
The elastic moduli E i (i = i, 2), shear moduli Gij, and linear temperature-expansion coefficients ~i are arbitrary functions of the temperature T, in the general case. This problem of intrinsic shell vibrations is solved by the method of discrete approximation of the variational-equation function [i]. In solving specific problems of the deformation of heat-sensitive composite shells, the effective thermoelastic characteristics of the laminar bundle must be known as a function of the temperature. The most reliable method of determining such characteristics is by means of experimental measurements on samples simulating the properties of the shell bundles. However, this approach is impractical and expensive. For preliminary design workups and check calculations, theoretical data on the effective thermoelastic characteristics of the bundles are acceptable. Despite its obvious shortcomings - associated with disregarding the influence of technological factors on the mechanical properties of the composite - this approach is widely used to predict the physicomechanical properties of composites of various structures, since it avoids an enormous quantity of experimental work in analyzing reinforcement schemes by adopting simpler basic elements: models of continuous filler and binder media. The method of [7] is used to determine the effective thermoelastic characteristics of the laminar-fibrous bundle of the composite shell. In the first stage, the thermoelastic characteristics of a unidirectionally reinforced fibrous material (a monolayer) is calculated from the known properties of the fiber and binder, taking account of their dependence on the temperature (T). It is known [8] that, for many materials used as component elements in creating modern composites, the temperature dependence of the elastic modulus (El*), Poisson's ratio (vi*), and temperature coefficient of linear expansion (~i*) may be described by a linear function in a definite temperature range for each component [2]
E ~ ( T ) = a I + a~T;
~,~(T)=bf + b~T;
cz~(T) = c~ + c~T
(12)
(i = 1, 2).
Here the subscripts i = i, 2 denote parameters of the filler and binder. The constants ak i, bki, cki (k = i, 2) are determined by the corresponding values of the thermoelastic characteristics at the two temperature values T I and T 2
E;(TOT~--E](T2)T~
E;(T~)--E~(T~)
a, =
T2__T '
; a~=
bl =
v*, ( T , ) T 2 - v T ( T 2 ) Tt Tl - - T2
.
C~=
5"~ ( T t ) T ~ - - % (*T 2 ) T ,
. Ci
T~--T,
O~=
Ti__T 2 vi (TI) - - v i (T2) T l - - T~
(i3)
5"~(TO - - 5 * ~(T2)
T,--T~
2=
'
,
Note that the temperature dependence of the thermoelastic characteristics may be more general, without introducing fundamental changes in the procedure for taking account of the thermosensitivity of the material in shell-eigenfrequency problems. The effective elastic characteristics of the monolayer are calculated using expressions obtained by the conditional-moment method [7]
A =
. '
~
=
4AG23 2c
3z; +
. A
'
(Gi G~)~ -G,2 = -G,3 = clGt + c~G2-- tic2 c,G2 + c~Gi + m -
212
-
( G ~ - G~)~ (l + 3m)
(14)
D=a = c~G~ + c~G~ - - c~c2 ~ciG= + c2G~) (l + 3m) + m (l + m) ;
~;
-
~
in Eqs. (12). E~, E2 are the elastic moduli in the direction of fiber winding and the transverse direction, respective!y; G~2, G~s, G=s are the shear moduli in the reinforcement plane and the normal planes; vz, v2 are the longitudinal and transverse Poisson' s ratios; and
A = ;~; + '%3- - 2x2; -- c~c~ (~,~ +
G~ -- ~,~ -- O2) ~ z];
- - X~-- G=) z;
'%~
2 [q (~ + O~) + c= (X~+ G2) --
X*2 = ci~,~ + c~,2 -- c~c= (~,~ -- ~,2) (~,~ + Oi --
~'~ = c~ ( ~ + GO + c~ ( ~ + O~) - - c~c~ (X~ - - ~)2 z;
z = [c~ (~2 --b G2) + c~ ( ~ + GO + m] -~ ;
• 2 K~ = %~-/- -3- Q;
m = G~G= (c~G= - - caG~)-~;
~
(1
=
2
l = K~K~ (c~K~. + c~K~)-~ ----~ m;
+
(15)
E; . Gi = 2 (1 + '~;) '
v'E* i i ~,~) (1 - - 2v~)
cl, c= are the volume contents of filler and binder, respectively; the constants Ei* and vi* for thermosensitive components are defined by the corresponding expressions in Eqs. (12). Using Eqs. (14) the elastic characteristics for composites may be calculated when K2* < KI*; G 2 < G~. The temperature-expansion coefficients are given by the expressions
~i--
p~%~ -- 2~)~ A
~2 ;
~ ----
-- ~;~)~ A
(16) '
where -
(8,-
~ = c~i + e ~ -- c~c2el(~ + G~) + c~ (~ ÷ G~) + m
(17) 2
(~,, + G, - - ~,~ - - 03) (P, - - P~) ciPi + c~p~-- e~c~ ci (~2 + G2) + c2 (~l + GO + m
Note that Eqs. (14)-(17) are obtained for isothermal states, under the assumption that the characteristic scale of variation in the external thermoforce loads is considerably greater than the dimensions of the elementary macrovolume of composite. Within these constraints, the use of Eqso (14)-(17) to determine the effective thermoelastic characteristics for a unidirectional composite, taking account of the thermosensitivity of the composites, is correct in the case of steady temperature fields varying smoothly in the spatial measurements. Thus, the effective thermoelastic characteristics of a monolayer, taking account of the thermosensitivity of the components for a specified temperature field, may be calculated from Eqs. (14)-(17) in combination with Eqs. (12) and (13), which determine the character of the temperature dependence of the filler and binder properties. The reduced thermoelastic characteristics of the layer bundle of a shell consisting of cross-wound armatures at angles of ±y may be determined within the framework of a model of a macrohomogeneous material by means of conversion formulas from the known values of the monolayer characteristics [6]
213
TABLE 1 c:t=0,353; ~='l" 15°
Elastic constant E,.10 -~, MPa E~-10 -~, MPa O,~. 10-4, MPa G,~. I0,-~, MPa
Yq ",'2
[6] I [41117] I [5]
cx=0,343;~ :
c~0,385; y= ~3i° [61
[41
[71
[51
:t:45°
[61t[41t[71115]
0,740 0,740 0,750 0,710 0,210 0,270 0,280 0,310 0,076 0,076 0,085 0,084 0,490 0,570 0,620 0,620 0,660 0,530 0,520 0,660 0,760 0,760 0,850 0,780 0,760 0,760 0,780 0,801 1,602 2,012 2,022 2,042 2,340 2,360 2,360 2,040 0,210 0,210 0,210 0,300 0,220 0,210 0,220 0,2,50 0,180 0,230 0,290 0,270 1,29 1,17 1,09 t,091 1,52I 1,6711,6111,5200,8800,8500,8300,890 0,086 0,089 0,090 0,700 0,370 0,320 0,330 0,370 0,880 0,850 0,830 0,870
TABLE 2 Cohstant determining the elastxcproperties ol the components
Boron fiber (i=1)
Binder (i=2)
a~, MPa a~, MPaI°C b~ b~ c~, (°C)-1
4,0.10 ~ 0,0 0,2 0,0 0,5 "10-~
0,370.10 ~ 0,113"10~ 0,400 0,560- I0 -~ 0,595" 10~
c~, (°c)- ~
0,o
o,180.10 -=
Eli = E , , cos ~ 'l' + E'2~ sin~ ? -+" 2 (Eiiv~ + 2Cal2) sin ~ "l' cos~ ?;
e.~ = 7~,, s~n~'¢ + e-~ cos~ v + 2 (E~,,,-I + 2g,~) sin s V co~~ ~'; (18) G,2 =
(f.
+ e-==- ~ . ~ )
sin= v co~ v + ~ cos= 2v;
6~a = 6 ~ cos ~ V + O=a sin~ V; O2~ = Giz sin~ ? + O~z c°s~ Y; vi = EI~/E22;
~t = ~i cos~ 7 + ~2 sin2 ?;
v 2 = Et2/Ett;
o~z --- ~zxsin s y + cz2 cos ~ ?.
(19)
InEqs. (18), Eii = gi/(I - ~i~i ) (i = i, 2). The mean elastic moduli are determined on the basis of the first two expressions in Eqs. (18) E t =Eit--
E~2
E22 -- E i i ( 1 - - v , v 2 ) ;
E~ = E~2
EI~
E~I
=
E2~ (1
- -
~).
(20)
Table 1 compares the theoretical results for the effective thermoelastic characteristics of cross-reinforced coal-based plastics obtained by the conditional-moment method Eqs. (14) and (18) - and the methods of [4, 6, 7] with the experimental data of [5] for such samples based on ETF binder (E 2 = 0.31-104 MPa; v 2 = 0.35) and carbon fibers (E I = 2.6.10 s MPa; v I = 0.25). Analysis of these results indicates satisfactory agreement of the experimental and theoretical elastic constants. The conditional-moment method gives the best agreement with experiment. Therefore, this method is used below to calculate the thermoelastic constants of temperature-sensitive composites. To illustrate the temperature dependence of the effective thermoelastic characteristics of cross-reinforced composites, consider the example of boroplastic. The thermoelastic properties of the components of boroplastic are determined from the parameters in Table 2. The results of calculating the effective thermoelastic characteristics by the conditionalmoment method are shown in Table 3. Analysis of the results shows that, with a linear temperature dependence of the binder properties, the temperature dependence of the effective thermoelastic characteristics of a monolayer and cross-reinforced composites is also nearlinear.
214
TABLE 3 c1=0,5; ~ = 0 °
Thermoelastic constant
~
9 ~
~
=0
Ez'10 -~, ~ E2"I0 -s, Gt=.10-4, btPa G13.t0-4, MPa G2~.10-4, MPa v~ v~ aj.10 ~, (oC)- 1 a2.104, (oC)- 1
cz=0,5; ~7=30*
2,010 0,125 0.474
2,006 0,t14 0,433 0,474 0,433 0,409 0,374 0,018 0,116 0,283 0,284 0,608 0,649 0,609 0,894
~
¢~t=0,5; ~.~----45°
9 oo
?
P°
P
2,005 0,095 0,356 0,356 0,308 0,117
0,576 0,553 0,492 0,174 0,119 0,I09 0,090 0 , 1 7 4 4,020 3,990 3,950 5,210 0,457 0,418 0,345 0,441 0,425 0,388 0,320 0,441 0,326 0,326 0,327 0,844 0,284 1,582 1,646 1,784 0,844 0,699 1,978 2,717 4,064 3 , 3 5 2 1,419 0,473 0,687 1,082 3,352
?
0,t60 0,t60 5,t90 0,403 0,403 0,855 0,855
0,I34 0,134 5,1.60 0,323 0,323 0,877
41781
7 438
0,877 4.781 7,438
TABLE 4
~.0o T. °C
0 23 50
80 96 T=23+ +76 s i n ( ~ )
?=30 °
";,HZ 0 --0,273 --0,613 --1,009 --I,220
89,2 76,4 66,9 46,8
--0,859
50,8
0
:e~o, mN/m I ~, Hz
0 23 50 80 96 T=23--}-
0 --0.,694.10 -a --0,164 --0,274 --0,331 X1
-}-76 sin(~u--)
--0,239
T11e, ~/m
~,Hz
0 --0,~04.10-! 0,325.10 -l 0,470.10 -I 0,119
145,2 142,5 141,7 140,4 140,3
0 --0,374.10-~ --0,668.10-: --0,788.d0 -I --0,759.I0 -1
142,4 139,7 137,5 134,6 133,1
0,342
144,6
--0,703-10-~
136,1
~;=60 o
r~o, mN/m
?=45 °
~=75 °
t
v, Hz
r~o, mN/m
?=90 °
I
v, Hz
l
r,,0, mN/m v, Hz
I36,0 0 1 3 4 , 6 --0,907.10 -I 130,1 --0,223 125,4 --0,387 123,8 --0,475
136,0 131,6 126,9 120,5 1t7,2
0 --0,103 --0,258 --0,452 --0,558
111,0 108,0 102,7 98,8 92,5
126,9
123,8
--0,394
98,8
--0,337
As an illustration of the method, the intrinsic vibrations of cross-reinforced boroplastic cylindrical shells with fixed ends which satisfy the following conditions are considered: u=v=w=~:=~O (x~=0, xt=L).
(21) The parameters determining the thermoelastic properties of the boron fibers and binder are shown in Table 2. In calculating the effective thermoelastic characteristics of the composites, the reinforcement coefficient is assumed to be c z = 0°5. The results of calculating the fundamental frequency for shells with relative geometric dimensions L:R:h = 2:1:0.01 are shown in Table 4. As well as the fundamental frequencies, Table 4 gives values of the preliminary linear tangential forces due to the restriction of the temperature elongation of the shell.
Analysis of these results shows that there are two factors associated with heat treatment which may significantly influence the eigenfrequencies of the thermosensitive shells. The first is due to the temperature dependence of the thermoelastic properties of the material; the second is due to the character of the basic stress-strain state of the shell with respect to which the intrinsic vibrations are considered. In the latter case, the determining role is played by the type of fixing of the shell ends and the method of reinforcement of the composite.
215
Thus, in comparison with shells of traditional materials, composite shells offer broader scope for control of the dynamic characteristics. LITERATURE CITED i*
2. 3. 4.
5.
.
7.
D. V. Babich, "Method of discrete approximation of the functional in stability problems of shells of revolution," Prikl. Mekh., 19, No. 2, 38-44 (1983). L. G. Belozerov and V. A. Kireev, "Influence of reinforcement schemes on shell stability in heating," Mekh. Kompozit. Mater., No. I, 49-86 (1984). V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilayer Structures [in Russian], Mashinostroenie, Moscow (1980). G. A. Vanin, Micromechanics of Composite Materials [in Russian], Naukova Dumka, Kiev (1985). G. M. Gunyaev, I. G. Zhigun, M. I. Dushin, et al., "Dependence of the elastic and strength characteristics of high-modular composites on the reinforcement scheme," Mekh. Polim., No. 6, 1019-1027 (1974), A. K. Molmeister, V. P. Tamuzh, and G. A. Teters, Resistance of Polymer and Composite Materials [in Russian], Zinatne, Riga (1980). A. N. Guz', L. P. Khoroshun, G. A. Banin, et al., Mechanics of Composite Materials and Structural Elements, Vol. i, Mechanics of Materials [in Russian], Naukova Dumka, Kiev
(1982). 8.
J. Lubin, ed., Handbook on Composite Materials [Russian translation], Vol. i, Mashinostronie, Moscow (1988).
AXISYMMETRIC DEFORMATION OF TOROIDAL SHELLS WITH STRONG FLEXURE A. V. Korovaitsev and A. Yu. Evkin
UDC 539.3
There have been many studies of the axisymmetric deformed state of toroidal shells [1-5, 8, 9], focusing on their stability and nonlinear behavior in moderate flexure. In the present work, strong flexure of a toroidal shell is investigated by two methods: numerical and asymptotic. Numerical calculation is by the method of continuous extension of the solution with respect to a universal parameter [4], although an implicit algorithm is used for this process, in contrast to [4], on account of the large variation in the displacement in axisymmetric deformation [7]. On the basis of these calculations, a form of the resolving equations which is more convenient and facilitates the analysis of the results is proposed. In the asymptotic integration of the equations of strong flexure of nonshallow shells of revolution, the small parameter adopted is that characterizing the thinness of the shell walls, which is proportional to the ratio of the shell thickness and the radius of principal curvature. In the basic approximation, formulas describing the stress-strain state and the dependence of the pressure on the amplitude of shell flexure in the equilibrium state are obtained. These relations are a generalization of the results of Pogorelov geometric theory to the case of nonshallow shells of revolution in strong flexure. Comparison of the results obtained by the two methods shows that asymptotic integration of the equations is effective in the whole range of axisymmetric deformation, except for the initial stage. I. The axissnnmetric behavior of a shell of revolution is considered here in one of the technical versions of nonlinear shell theory [i0]. For convenience of numerical realization, the resolving system of equations is written in the form
dS~
dA2 F~+A~ dSt Rl Ni+A~p~=O;
Moscow Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 28, No. 4, pp. 16-23, April, 1992. Original article submitted October 22, 1990.
216
1063-7095/92/2804-0216512.50 © 1992 Plenum Publishing Corporation
