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Dynamics of shallow shells taking into account physical non-linearities
International Journal of Mechanical Sciences 42 (2000) 1971}1982
Dynamics of shallow shells taking into account physical non-linearities Tran Duc Chinh* Hanoi University of Civil Engineering, 5 Giai Phong Street, Hanoi, Viet Nam Received 20 October 1998; accepted 7 July 1999
Abstract In this paper the author, has presented a solution to the problem of dynamics of shallow shell taking into account physical non-linearities. The shell is made of non-linear material obeying a cubic stress}strain law, and energy dissipation has been included in the formulation of the problem. Solutions for cylindrical and spherical shells and numerical results for these cases are based on approximate shape functions consisting of a single term in each. Some curves of amplitude}frequency relationships are given for cylindrical and spherical shells in the case of soft-type physical non-linearity. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear forced oscillation; Soft-type physical non-linearity; Shallow shell; Damping e!ect; Compressibility; Incompressibility of material
1. Introduction Problems of calculation of shallow shells made of non-elastic materials acting under static loads were considered by many scientists, but, in most cases of study of non-linear oscillations of shallow shells, only geometrical non-linearity is taken into account, and material non-linearity, on the other hand, has not been considered. In the paper by Librescu [1] as well as in the monograph by Librescu [2], the in#uence of soft and hard non-linearities on the character of the #utter boundary was analyzed and it was shown that soft and hard non-linearities render the #utter boundary `dangerousa and `non-dangerousa, respectively. In this paper, the amplitude}frequency relationship was studied for cylindrical and spherical shells in the case of soft-type physical non-linearity. Moreover, in almost all cases, the study of mechanical energy dissipation during the shell oscillations has not been carried out (with the exception of some works based on Voight}Kelvin * Corresponding author. 0020-7403/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 6 9 - 7
1972
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
concepts, but with results that do not correspond to experimental data). In this paper, the author studied small oscillations of the shallow shell made of non-linear, quasi-elastic material based on Bock's concepts corresponding to experimental data, especially for shallow shells made of some metals or their alloys as well as some plastics [3,4].
2. Theory In this study, for the shells with small de#ection compared with the thickness of 2h the following assumptions are made: (i)
Invariability of normal linear element, where for elementary layer at a distance z from the mid-surface, the following strain-displacement relations exist:
e "e !zw , e "e !zw , e "e !2zw , e "e "0, (1) V V VV W W WW VW VW VW WX XV where e , e , e are linear strains and shearing strain of the mid-surface, determined by the relation: V W VW e "u !k w, e "v !k w, e "u #v , (2) V V W W VW W V where u, v, w are projections of the displacement vector (of a point at the mid-surface) on the main direction of the tangential plane, and on the normal line to the mid-surface, respectively; k , k * main curvatures of shell mid-surface before being deformed. Lower indices of u, v, w in formulae (1) and (2) indicate the partial derivatives as per the variables corresponding to these indices. (ii) Incompressibility of the shell material. (iii) Equality of direction cosines tensors for stresses and strains. (iv) Non-linear stress}strain relation as p "Ee !me. (3) G G G The case of physical non-linearities of the soft type only are considered in this paper. For this case, the coe$cient of m appearing in Eq. (3), is negative, while for hard type non-linearities the sign of m should be positive where E and m are constants for the material under consideration, determined experimentally in case of axial tensile strain or in case of being twisted. Relations of this type are usually seen in some metals and their alloys [5], as well as in some plastics [6]. 2.1. Kinetic potential and second kind Lagrange equation The speci"c potential energy of solids made of incompressible material is expressed by formula [7] as
'(e )" p de . G G G
(4)
Using formulae (4) and (3), we obtain '(e )"Ee!me. G G G
(5)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1973
In the case of generalized plane stress state existing in each elementary layer of the shell, deformation intensity as per Ref. [7] is calculated by the formula 2 (e#e#e e #e . e" V W V W VW G (3
(6)
If attention is paid to (1), the formula (6) can be written in the form e" (b !zb #zb ), G where
(7)
b "e#e#e e #e , V W V W VW b "2e w #2e w #e w #e w #e w , (8) V VV W WW V WW W VV VW VW b "w #w #w w #w . VV WW VV WW VW The strain energy of the shell is determined by summing up Eq. (5) on the entire volume of the shell, and Eq. (7) is taken into account:
%"
? @ F
1 ? @ '(e ) dx dy dz" (Bb #Db ) dx dy G 2 \F 1 ? @ ! [B b #B (b #2b b )#D b ] dx dy, 4
(9)
where (10) B" Eh, D" Eh, B " mh, B " mh, D " mh. The change in potential energy % of the external load q is related to the change in its position. By O X overlooking the rotatory inertia of the elements, kinetic energy K of the systems may be expressed by the formula
1 ? @ ? @ K" k(u#v#w) dx dy and % "! q w dx dy, (11) R R R O X 2 where k is the shell's mass density, taking into account the useful load that is taken on a unit area of the mid-surface and t is the time. The kinetic potential of the system can be determined by the algebraic sum of Eqs. (9) and (11):
¸"K!%!% "! O
? @
F dx dy,
(12)
where B D B B D k F" b # b ! b # (b #2b b )! b ! (u#v#w)!q w. X 2 2 4 4 4 2
(13)
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
For an approximate solution of the problem, it is usually assumed that u" u u ; v" v t ; w" w s . (14) H H H H H H H H H Here, u , t , s are functions of coordinates x and y, selected depending on the form of the problem, H H H and the form of boundary condition, and u , v , w are functions depending on time, that must be H H H found. Sequentially using formulae (2), (8), (9) and(11), the relation of kinetic potential ¸ may be written in the form of an explicit function for displacement of the mid-surface. This allows one to establish a di!erential equation set for the motion of the shell made of ideal non-linear elastic materials on the basis of variational calculuses. However, the best way is to substitute Eq. (14) directly into the relation of the dynamic potential ¸ through the mid-surface displacement, instead of integrating the above equation set. If Eq. (14) is limited only by the "rst terms of the series, the expression of kinetic potential ¸ can be expressed in the form !¸"B(J u#J uv#J v!J uw!J vw)#(BJ #DJ )w ! B (J u#2J uv#J uv#2J uv#J v!2J uw ! 2J uvw!2J uvw!2J vw#J uw#2J uvw#J vw ! 2J uw!2J vw#J w)!B (J uw#2J uvw#J vw ! 2J uw!2J vw!(B J !D J )w !k(J u #J v #J w )!Qw. (15) The points ( ) ) in Eq. (15) indicate derivatives as per time-dependent variables, and J expresses ) constants which are dependent on the form of approximate shape function and dimensions of the shell. On the basis of Eq. (15), 2nd kind Lagrange equations can be given as
d *¸ *¸ ! "0. (16) dt *XQ *X H H Non-linear partial di!erential equations can be obtained to express the motion of the shell made of ideal non-linear elastic materials. 2.2. Principal stresses and other form of equations of shell motion dissipation function When calculating structures, attention is paid not only to oscillation amplitudes but also to stress, because in some cases, dynamic stresses play an important role. Therefore, in this study, another method is used to establish the equations of motion that allow one to "nd out the principal stresses and internal forces of the shell. Here, besides the above assumptions, we further assume that non-elastic damping (in Sorokin's sense [1]) for oscillations can be described by the formula c p " Ee , S p
(17)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
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where c is the internal non-elastic damping factor of the material, and p the oscillation frequency. In order to establish the formulae for principal stresses, the following relations [7] are used: p 1 p p 1 p ! p " Ge ; p ! p " Ge ; q " G e . W 2 V e W VW 3e VW V 2 W e V G G G Using Eqs. (18) and (1), we obtain
(18)
4p 1 1 p " G e # e !z w # w , V 3e V 2 W VV 2 WW G p (19) q " G (e !2zw ). VW VW 3e VW G Here, p is not cited, because it is easy to obtain it by permuting indices of 6 in Eq. (19). We will do W V so later to simplify the way of writing. From formulae (3) and (7) we obtain 4 p G "E! m(b !zb #zb ). (20) 3 e G Now, non-elastic damping stresses are brought into Eq. (19) corresponding to Eq. (17). At the same time, it should be noted that in this case, generalized plane stress state of each separate elementary layer is equidistant from the shell mid-surface. In addition, we use Eq. (20), and obtain formulae of the principal stresses in the following form: 4 cE * 16 4 ) (eH!ziH)! mb eH p " E(eH!ziH)# ) V V V V 3 V 3 p *t V 9 #
16 16 16 mz(b iH#b eH)! mz(b iH#b eH)# mzb iH; V V V V V 9 9 9
}}}}}}}}}}}}}}}}}}}} cE * 4 1 q " E(e !2zw )# ) (e !2zw )! mb e VW VW V 3 VW 3p *t VW 9 VW 4 8 4 # mz(2b w #b e )! mz(2b w #b e )# mzb w , VW VW VW VW VW 9 9 9
(21)
where 1 1 eH"e # e ,2, iH"w # w . V V 2 W V VV 2 WW
(22)
Using formulae of shell theory we can obtain expressions of principal stresses corresponding to Eq. (21). Substituting expressions of principal stresses just obtained in equations of motion of
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Fig. 1. In"nitely small element of the shell's mid-surface.
in"nitely small element (Fig. 1) of the shell's mid-surface, we obtain equations of motion of the shell written in deformations of the mid-surface as follows:
* c *eH BeH# B V !B b eH!B (b iH#b eH) V p *t V V V *x #
1 * c *e Be # B VW !B b e !B (2b w #b e ) !ku #q "0; VW p *t VW VW VW RR V 4 *y
}}}}}}}}}}}}}}}}}}}} c * * ! D w! D w# [B (b iH#b eH)#D b iH] V V p *t *x V #
1 * * B b w # b e #D b w # [B (b iH#b eH)# D b iH] VW W W *y W *x*y VW 2 VW
c *eH #k BeH# B V !B b eH!B (b iH#b eH) V p *t V V V
c *eH # k BeH# B W !B b eH!B (b iH#b eH) #kw #q "0. W p *t W W W RR X
(23)
In order to solve the three partial di!erential equations (23), we use Eq. (14) and the known Galerkin's method. If only the "rst terms of the series in Eq. (14) are limited, we will obtain three ordinary non-linear di!erential equations for functions of time, and u, v, w must be found. Here we cite the above equation set in the case, when load acts only perpendicularly on cylindrical panel (considering u"0): 1 c 1c kJ v #BJ v! BJ w# BJ v ! BJ w !B J v RR 2 p R 2p R 1 1 3 # B J vw! (B J #B J )vw# (B J #B J )w"0, 2 2 2
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
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1 1c c ! BJ v#kJ w #(BJ #DJ )w! BJ v # (BJ #DJ )w RR R R 2 2p p 1 1 3 # B J v! (B J #B J )vw# (B J #B J )vw 2 2 2 !(B J #B J #D J )w"Q, (24) where Q is work of external force on displacements is determined by function s(x, y). In case, the spherical shell is equally situated on the edges of a square boundary, it can be considered that u"v, (this can be valid only if the additional condition of the lateral load featuring identical variations in both x- and y-directions is ful"lled). As a result, the equation in the form of Eq. (24) is obtained only for other expressions of invariable coe$cients. Before solving equation set (24), we introduce into Eq. (24) the function in the form of c c 2RH" (J u#J u v #J v!J u w !J v w )# (BJ #DJ )w, R R R R R R R R p p R
(25)
where RH is the Rayleigh's dissipation function of the shallow shell, assuming that law (17) is existing. The second kind Lagrange's equation for non-conservative system takes the form
d *¸ *¸ *RH ! # "0, (26) dt *x *x *x R where x is any function among functions u, v, w. It is true that using relations (15), (25) and (26) we can obtain equation set (24) when u"0. For convenience, we introduce symbols BJ BJ B J 2B J , g " , g " , u" , g " T kJ 2kJ kJ 2kJ B J #B J BJ #DJ B J #B J , u" , , g " g " T 2kJ kJ 2J k BJ B J B J #B J , g " , g " , g " 2kJ 2kJ 2kJ 3(B J #B J ) B J #B J #D J , g " . g " 2kJ kJ In this case equation set (24) must be rewritten as
(27)
c c v #uv!g w"g v!g vw#g vw!g w! uv # g w , RR T p T R p R c Q c . !g v#w #u w"!g v#g vw!g vw#g w# g v ! u w # RR U p R p U R kJ
(28)
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
In order to obtain usable results, we change Eqs. (28) into dimensionless magnitudes by introducing into relations u q u p u v" U y, w" U z, t" , j" T , u" . p u u (g (g U U As a result, we obtain the equation uy #jy!j z"e(j y!j yz#j yz#j z!cjy #jr z ), OO O O !j y#uz #z"e(!j y#j yz#z#cj y !cz #H cos q#H sin q), OO O O A Q where
(29)
(30)
g g g g g j " , j " , j " , j " , j " , u g g g g U g g g g Q(g . j " , j " , j " , j " , H" (31) u g g g kJ u U U In order to use the method of small parameter, we introduce a small parameter e into the right-handside of Eqs. (30), instead of a multiplier. The solution that must be found is expressed in the form y"y #ey #2, z"z #ez #2, u"a #ea #ea #2. (32) Substituting Eq. (32) into Eqs. (30) we obtain an ordinary linear di!erential equation. Here, we cite the solution of Eqs. (30) in the second approximation: y"oa cos q#ec a cos 3q#2, z"a cos q#ee a cos 3q#2, 3 N (j!j )H N M u"a !e ) a $ !c !e a #2, 4 N N a N N H "H#H, A Q where 1 1 j , a " (j#1)! (j!1)#j j , o" 2 4 j!a (1!9a )A #j A (j!9a )A #j A , 4c " , 4e " (j!9a )(1!9a )!j j (j!9a )(1!9a )!j j N "j o#j!a , N "j A #(j!a )A , N "j (jo!j )#(j!a )(1!j o), M "j G #(j !a )G , A "j o!j o#j o!j , A "!j o#j o!j o#1, G "(3j oc #2j oc !j oe #2j oe #j c !3j e ), G "(!3j oc #2j oc !j oe #2j oe #j c !3e ).
(33)
(34)
(35) (36) (37) (38) (39)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1979
The sequence according to the formulae obtained above is calculated as follows. First, based on Eq. (10), calculating the shell's rigidity after using the expression of functions u, t, s, we calculate J . Formulae (27), (29) and (31) are used to calculate all coe$cients of Eq. (30). I Alternatively, using formulae (35)}(39) to calculate coe$cients contained in the solutions of Eqs. (33) and (34). It should be realized that for spherical, the calculation process will be carried out similarly, but formula (27) will take a di!erent form.
3. Numerical examples Consider spherical and cylindrical shallow shells situated on a motionless simply supported square boundary. Approximate shape functions are selected in the form of u"sin
py px 2py 2px py 2px sin , t"sin sin , s"sin sin . b b b b b b
Boundary condition for motionless simply supported edges is u"v"w"M "0 (on all boundaries). The calculated results are expressed by the curve of amplitude}frequency relationship given in Fig. 2 (for cylindrical shell), and in Fig. 3 (for spherical shell). These are curves of system with soft-type physical non-linearity of material. During the calculation process, assuming that (R/b)"2 and c"0.01, maximum values of a are calculated by equating null to the expression under the radical of Eq. (34).
4. Discussion and some important remarks We know that the assumption on incompressibility of material will be more suitable to reality, in the case of large deformation of shell. Therefore, in the case of small deformation of shell, the assumption mentioned above has no base. However, when the shell is subjected to dynamic load of periodical type, deformation in value should be limited, because this is related to fatigue
Fig. 2. Curves of amplitude}frequency relationship given for cylindrical shell.
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
Fig. 3. Curves of amplitude}frequency relationship given for spherical shell.
phenomenon of material. Therefore, in fact, we often have dealings with the cases of shell with relatively small dynamic deformation. Therefore, we may overlook the assumption on incompressibility of the material mentioned above, that is replaced by another assumption: elongation in the direction of the shell thickness plays a minor role when calculating linear elastic shells. It can be determined by l e "! (e #e ), 8 W 1!l V
(40)
stemmed from Hooke's generalized law on three-dimensional elastic bodies, when reciprocal pressure p does not exist beween elementary layers equidistant from the mid-surface. In Eq. (40) 8 and the following formulae l is the Poisson's ratio corresponding to small deformations. The speci"c potential energy of a solid made of compressible material according to Ref. [7] can be determined by the formula
CG 1 '(h, e )" Kh# p de , G G G 2 where K is the three-dimensional compression modulus, and is calculated as follows: E K" 3(1!2l)
(41)
(42)
and volume deformation 1!2l (e #e ). h"e #e #e " W V W X 1!l V
(43)
Squaring the deformation intensity (in case of spatial deformation) in the form
3 2 e" (e !e )#(e !e )#(e !e )# (e #e #e ) , W W R X V WX XV G 2 VW 9 V
(44)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1981
Using Eq. (40) and two "nal relations of Eq. (1), we obtain
4 1 e" l (e#e)#l 2e e # e . G W V W 4 VW 3 V
(45)
Here, we introduce symbols
l 2l 1 1 #1 , l " !1 . l " 3 (1!l) 3 (1!l)
(46)
By assuming that l"0.5, formula (45) will return to formula (6) known for the case of generalized plane stress state of incompressible material [7]. Using Eq. (1) we can perform square of deformation intensity in the form of Eq. (7), where bH"l (e #e)#l e e #e , bH"l (w #w )#l w w #w , VV WW VV WW VW V W V W VW bH"2l (e w #e w )#l (e w #e w )#e w . V VV W WW V WW W VV VW VW The speci"c work ' of the deformation can be expressed through shell deformation as
1 1 '" E (e#e)#2E e e # E e W V W 4 VW 2 V
(47)
4 1 ! m l e #2l l e e #(2l )e e#2l l e e#l e# l ee V W V W V W W 2 V VW 9 1 1 1 # l ee # l e e e # e 2 W VW 2 V W VW 16 VW
(48)
where
1!2l 4 1 K, E "l E #K , E " l E #K , E " E, K " 3 2 1!l
(49)
Strains of the shell is determined by the formula
? @ 1 [B(bH#cd )#D(bH#cd )] 2 1 ! [B bH#B (bH#2bHbH)#D bH] dx dy, 4
%"
where c"(1!2l)/[4(1!l)]; d "(e #e ); d "(w #w ). V W VV WW The stresses in the shell are determined on the basis of Eq. (48) and relations *' *' p " ,2,q " , V *e VW *e 2 V VW
(51)
(52)
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
These stresses are in the form of p "E e #E e !m[4l e#6l l e!e #2(2l #l )e e V V W V W V V W #2l l e#l e e #l e e ], W V VW W VW }}}}}}}}}}}}}}}}}}}} (53) q "E e !m(l ee #l ee #l e e e #e ). W VW V W VW VW VW VW V VW We can add non-elastic damping component to Eq. (53) that is carried out as above to obtain the expression of kinetic potential in the form of Eq. (15) and dissipation function in the form of Eq. (25), but with coe$cient j written in another form. Finally, we can also obtain equations of I motion of shell made of compressible materials in the form of Eq. (23) which is somewhat di!erent. By using Galerkin's method, we will obtain equations similar to Eq. (24), but with di!erent coe$cients j . All algorithms of the problem for oscillation of the shell made of incompressible I material can be used for the case of shell made of compressible material.
Acknowledgements This publication is completed with partial "nancial support from National Basic Research Program in Natural Sciences.
References [1] Librescu L. Aeroelastic stability of orthotropic heterogeneous thin panels in the vicinity of the #utter critical boundary (II). Journal de Mecanique 1967;6(1):133}152. [2] Librescu L. Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Leyden: Noordho!, 1975. p. 46}52, 158}64. [3] Tseytlin AI, Kusainov AA. Method of internal damping modeling for dynamic structural design. Alma-Ata: Nauka, 1987. [4] Volmir AS. Non-linear dynamics of plates and shells. Moscow: Nauka, 1972 (in Russian). [5] Vinson JR. The behaviour of shells composed of isotropic and composite materials. Dordrecht: Kluwer Academic Publishers, 1993. [6] Tarnopol'sky YuM. Thickwalled rolling constructions made of "bre-composite. Journal of Mechanics of Composite Materials (Riga) 1975;1:134}144 (in Russian). [7] Iliouchine A. A plasticite. Paris: Eyrolles, 1956.
Dynamics of shallow shells taking into account physical non-linearities Tran Duc Chinh* Hanoi University of Civil Engineering, 5 Giai Phong Street, Hanoi, Viet Nam Received 20 October 1998; accepted 7 July 1999
Abstract In this paper the author, has presented a solution to the problem of dynamics of shallow shell taking into account physical non-linearities. The shell is made of non-linear material obeying a cubic stress}strain law, and energy dissipation has been included in the formulation of the problem. Solutions for cylindrical and spherical shells and numerical results for these cases are based on approximate shape functions consisting of a single term in each. Some curves of amplitude}frequency relationships are given for cylindrical and spherical shells in the case of soft-type physical non-linearity. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear forced oscillation; Soft-type physical non-linearity; Shallow shell; Damping e!ect; Compressibility; Incompressibility of material
1. Introduction Problems of calculation of shallow shells made of non-elastic materials acting under static loads were considered by many scientists, but, in most cases of study of non-linear oscillations of shallow shells, only geometrical non-linearity is taken into account, and material non-linearity, on the other hand, has not been considered. In the paper by Librescu [1] as well as in the monograph by Librescu [2], the in#uence of soft and hard non-linearities on the character of the #utter boundary was analyzed and it was shown that soft and hard non-linearities render the #utter boundary `dangerousa and `non-dangerousa, respectively. In this paper, the amplitude}frequency relationship was studied for cylindrical and spherical shells in the case of soft-type physical non-linearity. Moreover, in almost all cases, the study of mechanical energy dissipation during the shell oscillations has not been carried out (with the exception of some works based on Voight}Kelvin * Corresponding author. 0020-7403/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 6 9 - 7
1972
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
concepts, but with results that do not correspond to experimental data). In this paper, the author studied small oscillations of the shallow shell made of non-linear, quasi-elastic material based on Bock's concepts corresponding to experimental data, especially for shallow shells made of some metals or their alloys as well as some plastics [3,4].
2. Theory In this study, for the shells with small de#ection compared with the thickness of 2h the following assumptions are made: (i)
Invariability of normal linear element, where for elementary layer at a distance z from the mid-surface, the following strain-displacement relations exist:
e "e !zw , e "e !zw , e "e !2zw , e "e "0, (1) V V VV W W WW VW VW VW WX XV where e , e , e are linear strains and shearing strain of the mid-surface, determined by the relation: V W VW e "u !k w, e "v !k w, e "u #v , (2) V V W W VW W V where u, v, w are projections of the displacement vector (of a point at the mid-surface) on the main direction of the tangential plane, and on the normal line to the mid-surface, respectively; k , k * main curvatures of shell mid-surface before being deformed. Lower indices of u, v, w in formulae (1) and (2) indicate the partial derivatives as per the variables corresponding to these indices. (ii) Incompressibility of the shell material. (iii) Equality of direction cosines tensors for stresses and strains. (iv) Non-linear stress}strain relation as p "Ee !me. (3) G G G The case of physical non-linearities of the soft type only are considered in this paper. For this case, the coe$cient of m appearing in Eq. (3), is negative, while for hard type non-linearities the sign of m should be positive where E and m are constants for the material under consideration, determined experimentally in case of axial tensile strain or in case of being twisted. Relations of this type are usually seen in some metals and their alloys [5], as well as in some plastics [6]. 2.1. Kinetic potential and second kind Lagrange equation The speci"c potential energy of solids made of incompressible material is expressed by formula [7] as
'(e )" p de . G G G
(4)
Using formulae (4) and (3), we obtain '(e )"Ee!me. G G G
(5)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1973
In the case of generalized plane stress state existing in each elementary layer of the shell, deformation intensity as per Ref. [7] is calculated by the formula 2 (e#e#e e #e . e" V W V W VW G (3
(6)
If attention is paid to (1), the formula (6) can be written in the form e" (b !zb #zb ), G where
(7)
b "e#e#e e #e , V W V W VW b "2e w #2e w #e w #e w #e w , (8) V VV W WW V WW W VV VW VW b "w #w #w w #w . VV WW VV WW VW The strain energy of the shell is determined by summing up Eq. (5) on the entire volume of the shell, and Eq. (7) is taken into account:
%"
? @ F
1 ? @ '(e ) dx dy dz" (Bb #Db ) dx dy G 2 \F 1 ? @ ! [B b #B (b #2b b )#D b ] dx dy, 4
(9)
where (10) B" Eh, D" Eh, B " mh, B " mh, D " mh. The change in potential energy % of the external load q is related to the change in its position. By O X overlooking the rotatory inertia of the elements, kinetic energy K of the systems may be expressed by the formula
1 ? @ ? @ K" k(u#v#w) dx dy and % "! q w dx dy, (11) R R R O X 2 where k is the shell's mass density, taking into account the useful load that is taken on a unit area of the mid-surface and t is the time. The kinetic potential of the system can be determined by the algebraic sum of Eqs. (9) and (11):
¸"K!%!% "! O
? @
F dx dy,
(12)
where B D B B D k F" b # b ! b # (b #2b b )! b ! (u#v#w)!q w. X 2 2 4 4 4 2
(13)
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For an approximate solution of the problem, it is usually assumed that u" u u ; v" v t ; w" w s . (14) H H H H H H H H H Here, u , t , s are functions of coordinates x and y, selected depending on the form of the problem, H H H and the form of boundary condition, and u , v , w are functions depending on time, that must be H H H found. Sequentially using formulae (2), (8), (9) and(11), the relation of kinetic potential ¸ may be written in the form of an explicit function for displacement of the mid-surface. This allows one to establish a di!erential equation set for the motion of the shell made of ideal non-linear elastic materials on the basis of variational calculuses. However, the best way is to substitute Eq. (14) directly into the relation of the dynamic potential ¸ through the mid-surface displacement, instead of integrating the above equation set. If Eq. (14) is limited only by the "rst terms of the series, the expression of kinetic potential ¸ can be expressed in the form !¸"B(J u#J uv#J v!J uw!J vw)#(BJ #DJ )w ! B (J u#2J uv#J uv#2J uv#J v!2J uw ! 2J uvw!2J uvw!2J vw#J uw#2J uvw#J vw ! 2J uw!2J vw#J w)!B (J uw#2J uvw#J vw ! 2J uw!2J vw!(B J !D J )w !k(J u #J v #J w )!Qw. (15) The points ( ) ) in Eq. (15) indicate derivatives as per time-dependent variables, and J expresses ) constants which are dependent on the form of approximate shape function and dimensions of the shell. On the basis of Eq. (15), 2nd kind Lagrange equations can be given as
d *¸ *¸ ! "0. (16) dt *XQ *X H H Non-linear partial di!erential equations can be obtained to express the motion of the shell made of ideal non-linear elastic materials. 2.2. Principal stresses and other form of equations of shell motion dissipation function When calculating structures, attention is paid not only to oscillation amplitudes but also to stress, because in some cases, dynamic stresses play an important role. Therefore, in this study, another method is used to establish the equations of motion that allow one to "nd out the principal stresses and internal forces of the shell. Here, besides the above assumptions, we further assume that non-elastic damping (in Sorokin's sense [1]) for oscillations can be described by the formula c p " Ee , S p
(17)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1975
where c is the internal non-elastic damping factor of the material, and p the oscillation frequency. In order to establish the formulae for principal stresses, the following relations [7] are used: p 1 p p 1 p ! p " Ge ; p ! p " Ge ; q " G e . W 2 V e W VW 3e VW V 2 W e V G G G Using Eqs. (18) and (1), we obtain
(18)
4p 1 1 p " G e # e !z w # w , V 3e V 2 W VV 2 WW G p (19) q " G (e !2zw ). VW VW 3e VW G Here, p is not cited, because it is easy to obtain it by permuting indices of 6 in Eq. (19). We will do W V so later to simplify the way of writing. From formulae (3) and (7) we obtain 4 p G "E! m(b !zb #zb ). (20) 3 e G Now, non-elastic damping stresses are brought into Eq. (19) corresponding to Eq. (17). At the same time, it should be noted that in this case, generalized plane stress state of each separate elementary layer is equidistant from the shell mid-surface. In addition, we use Eq. (20), and obtain formulae of the principal stresses in the following form: 4 cE * 16 4 ) (eH!ziH)! mb eH p " E(eH!ziH)# ) V V V V 3 V 3 p *t V 9 #
16 16 16 mz(b iH#b eH)! mz(b iH#b eH)# mzb iH; V V V V V 9 9 9
}}}}}}}}}}}}}}}}}}}} cE * 4 1 q " E(e !2zw )# ) (e !2zw )! mb e VW VW V 3 VW 3p *t VW 9 VW 4 8 4 # mz(2b w #b e )! mz(2b w #b e )# mzb w , VW VW VW VW VW 9 9 9
(21)
where 1 1 eH"e # e ,2, iH"w # w . V V 2 W V VV 2 WW
(22)
Using formulae of shell theory we can obtain expressions of principal stresses corresponding to Eq. (21). Substituting expressions of principal stresses just obtained in equations of motion of
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
Fig. 1. In"nitely small element of the shell's mid-surface.
in"nitely small element (Fig. 1) of the shell's mid-surface, we obtain equations of motion of the shell written in deformations of the mid-surface as follows:
* c *eH BeH# B V !B b eH!B (b iH#b eH) V p *t V V V *x #
1 * c *e Be # B VW !B b e !B (2b w #b e ) !ku #q "0; VW p *t VW VW VW RR V 4 *y
}}}}}}}}}}}}}}}}}}}} c * * ! D w! D w# [B (b iH#b eH)#D b iH] V V p *t *x V #
1 * * B b w # b e #D b w # [B (b iH#b eH)# D b iH] VW W W *y W *x*y VW 2 VW
c *eH #k BeH# B V !B b eH!B (b iH#b eH) V p *t V V V
c *eH # k BeH# B W !B b eH!B (b iH#b eH) #kw #q "0. W p *t W W W RR X
(23)
In order to solve the three partial di!erential equations (23), we use Eq. (14) and the known Galerkin's method. If only the "rst terms of the series in Eq. (14) are limited, we will obtain three ordinary non-linear di!erential equations for functions of time, and u, v, w must be found. Here we cite the above equation set in the case, when load acts only perpendicularly on cylindrical panel (considering u"0): 1 c 1c kJ v #BJ v! BJ w# BJ v ! BJ w !B J v RR 2 p R 2p R 1 1 3 # B J vw! (B J #B J )vw# (B J #B J )w"0, 2 2 2
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1977
1 1c c ! BJ v#kJ w #(BJ #DJ )w! BJ v # (BJ #DJ )w RR R R 2 2p p 1 1 3 # B J v! (B J #B J )vw# (B J #B J )vw 2 2 2 !(B J #B J #D J )w"Q, (24) where Q is work of external force on displacements is determined by function s(x, y). In case, the spherical shell is equally situated on the edges of a square boundary, it can be considered that u"v, (this can be valid only if the additional condition of the lateral load featuring identical variations in both x- and y-directions is ful"lled). As a result, the equation in the form of Eq. (24) is obtained only for other expressions of invariable coe$cients. Before solving equation set (24), we introduce into Eq. (24) the function in the form of c c 2RH" (J u#J u v #J v!J u w !J v w )# (BJ #DJ )w, R R R R R R R R p p R
(25)
where RH is the Rayleigh's dissipation function of the shallow shell, assuming that law (17) is existing. The second kind Lagrange's equation for non-conservative system takes the form
d *¸ *¸ *RH ! # "0, (26) dt *x *x *x R where x is any function among functions u, v, w. It is true that using relations (15), (25) and (26) we can obtain equation set (24) when u"0. For convenience, we introduce symbols BJ BJ B J 2B J , g " , g " , u" , g " T kJ 2kJ kJ 2kJ B J #B J BJ #DJ B J #B J , u" , , g " g " T 2kJ kJ 2J k BJ B J B J #B J , g " , g " , g " 2kJ 2kJ 2kJ 3(B J #B J ) B J #B J #D J , g " . g " 2kJ kJ In this case equation set (24) must be rewritten as
(27)
c c v #uv!g w"g v!g vw#g vw!g w! uv # g w , RR T p T R p R c Q c . !g v#w #u w"!g v#g vw!g vw#g w# g v ! u w # RR U p R p U R kJ
(28)
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
In order to obtain usable results, we change Eqs. (28) into dimensionless magnitudes by introducing into relations u q u p u v" U y, w" U z, t" , j" T , u" . p u u (g (g U U As a result, we obtain the equation uy #jy!j z"e(j y!j yz#j yz#j z!cjy #jr z ), OO O O !j y#uz #z"e(!j y#j yz#z#cj y !cz #H cos q#H sin q), OO O O A Q where
(29)
(30)
g g g g g j " , j " , j " , j " , j " , u g g g g U g g g g Q(g . j " , j " , j " , j " , H" (31) u g g g kJ u U U In order to use the method of small parameter, we introduce a small parameter e into the right-handside of Eqs. (30), instead of a multiplier. The solution that must be found is expressed in the form y"y #ey #2, z"z #ez #2, u"a #ea #ea #2. (32) Substituting Eq. (32) into Eqs. (30) we obtain an ordinary linear di!erential equation. Here, we cite the solution of Eqs. (30) in the second approximation: y"oa cos q#ec a cos 3q#2, z"a cos q#ee a cos 3q#2, 3 N (j!j )H N M u"a !e ) a $ !c !e a #2, 4 N N a N N H "H#H, A Q where 1 1 j , a " (j#1)! (j!1)#j j , o" 2 4 j!a (1!9a )A #j A (j!9a )A #j A , 4c " , 4e " (j!9a )(1!9a )!j j (j!9a )(1!9a )!j j N "j o#j!a , N "j A #(j!a )A , N "j (jo!j )#(j!a )(1!j o), M "j G #(j !a )G , A "j o!j o#j o!j , A "!j o#j o!j o#1, G "(3j oc #2j oc !j oe #2j oe #j c !3j e ), G "(!3j oc #2j oc !j oe #2j oe #j c !3e ).
(33)
(34)
(35) (36) (37) (38) (39)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1979
The sequence according to the formulae obtained above is calculated as follows. First, based on Eq. (10), calculating the shell's rigidity after using the expression of functions u, t, s, we calculate J . Formulae (27), (29) and (31) are used to calculate all coe$cients of Eq. (30). I Alternatively, using formulae (35)}(39) to calculate coe$cients contained in the solutions of Eqs. (33) and (34). It should be realized that for spherical, the calculation process will be carried out similarly, but formula (27) will take a di!erent form.
3. Numerical examples Consider spherical and cylindrical shallow shells situated on a motionless simply supported square boundary. Approximate shape functions are selected in the form of u"sin
py px 2py 2px py 2px sin , t"sin sin , s"sin sin . b b b b b b
Boundary condition for motionless simply supported edges is u"v"w"M "0 (on all boundaries). The calculated results are expressed by the curve of amplitude}frequency relationship given in Fig. 2 (for cylindrical shell), and in Fig. 3 (for spherical shell). These are curves of system with soft-type physical non-linearity of material. During the calculation process, assuming that (R/b)"2 and c"0.01, maximum values of a are calculated by equating null to the expression under the radical of Eq. (34).
4. Discussion and some important remarks We know that the assumption on incompressibility of material will be more suitable to reality, in the case of large deformation of shell. Therefore, in the case of small deformation of shell, the assumption mentioned above has no base. However, when the shell is subjected to dynamic load of periodical type, deformation in value should be limited, because this is related to fatigue
Fig. 2. Curves of amplitude}frequency relationship given for cylindrical shell.
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
Fig. 3. Curves of amplitude}frequency relationship given for spherical shell.
phenomenon of material. Therefore, in fact, we often have dealings with the cases of shell with relatively small dynamic deformation. Therefore, we may overlook the assumption on incompressibility of the material mentioned above, that is replaced by another assumption: elongation in the direction of the shell thickness plays a minor role when calculating linear elastic shells. It can be determined by l e "! (e #e ), 8 W 1!l V
(40)
stemmed from Hooke's generalized law on three-dimensional elastic bodies, when reciprocal pressure p does not exist beween elementary layers equidistant from the mid-surface. In Eq. (40) 8 and the following formulae l is the Poisson's ratio corresponding to small deformations. The speci"c potential energy of a solid made of compressible material according to Ref. [7] can be determined by the formula
CG 1 '(h, e )" Kh# p de , G G G 2 where K is the three-dimensional compression modulus, and is calculated as follows: E K" 3(1!2l)
(41)
(42)
and volume deformation 1!2l (e #e ). h"e #e #e " W V W X 1!l V
(43)
Squaring the deformation intensity (in case of spatial deformation) in the form
3 2 e" (e !e )#(e !e )#(e !e )# (e #e #e ) , W W R X V WX XV G 2 VW 9 V
(44)
T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
1981
Using Eq. (40) and two "nal relations of Eq. (1), we obtain
4 1 e" l (e#e)#l 2e e # e . G W V W 4 VW 3 V
(45)
Here, we introduce symbols
l 2l 1 1 #1 , l " !1 . l " 3 (1!l) 3 (1!l)
(46)
By assuming that l"0.5, formula (45) will return to formula (6) known for the case of generalized plane stress state of incompressible material [7]. Using Eq. (1) we can perform square of deformation intensity in the form of Eq. (7), where bH"l (e #e)#l e e #e , bH"l (w #w )#l w w #w , VV WW VV WW VW V W V W VW bH"2l (e w #e w )#l (e w #e w )#e w . V VV W WW V WW W VV VW VW The speci"c work ' of the deformation can be expressed through shell deformation as
1 1 '" E (e#e)#2E e e # E e W V W 4 VW 2 V
(47)
4 1 ! m l e #2l l e e #(2l )e e#2l l e e#l e# l ee V W V W V W W 2 V VW 9 1 1 1 # l ee # l e e e # e 2 W VW 2 V W VW 16 VW
(48)
where
1!2l 4 1 K, E "l E #K , E " l E #K , E " E, K " 3 2 1!l
(49)
Strains of the shell is determined by the formula
? @ 1 [B(bH#cd )#D(bH#cd )] 2 1 ! [B bH#B (bH#2bHbH)#D bH] dx dy, 4
%"
where c"(1!2l)/[4(1!l)]; d "(e #e ); d "(w #w ). V W VV WW The stresses in the shell are determined on the basis of Eq. (48) and relations *' *' p " ,2,q " , V *e VW *e 2 V VW
(51)
(52)
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T.D. Chinh / International Journal of Mechanical Sciences 42 (2000) 1971}1982
These stresses are in the form of p "E e #E e !m[4l e#6l l e!e #2(2l #l )e e V V W V W V V W #2l l e#l e e #l e e ], W V VW W VW }}}}}}}}}}}}}}}}}}}} (53) q "E e !m(l ee #l ee #l e e e #e ). W VW V W VW VW VW VW V VW We can add non-elastic damping component to Eq. (53) that is carried out as above to obtain the expression of kinetic potential in the form of Eq. (15) and dissipation function in the form of Eq. (25), but with coe$cient j written in another form. Finally, we can also obtain equations of I motion of shell made of compressible materials in the form of Eq. (23) which is somewhat di!erent. By using Galerkin's method, we will obtain equations similar to Eq. (24), but with di!erent coe$cients j . All algorithms of the problem for oscillation of the shell made of incompressible I material can be used for the case of shell made of compressible material.
Acknowledgements This publication is completed with partial "nancial support from National Basic Research Program in Natural Sciences.
References [1] Librescu L. Aeroelastic stability of orthotropic heterogeneous thin panels in the vicinity of the #utter critical boundary (II). Journal de Mecanique 1967;6(1):133}152. [2] Librescu L. Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Leyden: Noordho!, 1975. p. 46}52, 158}64. [3] Tseytlin AI, Kusainov AA. Method of internal damping modeling for dynamic structural design. Alma-Ata: Nauka, 1987. [4] Volmir AS. Non-linear dynamics of plates and shells. Moscow: Nauka, 1972 (in Russian). [5] Vinson JR. The behaviour of shells composed of isotropic and composite materials. Dordrecht: Kluwer Academic Publishers, 1993. [6] Tarnopol'sky YuM. Thickwalled rolling constructions made of "bre-composite. Journal of Mechanics of Composite Materials (Riga) 1975;1:134}144 (in Russian). [7] Iliouchine A. A plasticite. Paris: Eyrolles, 1956.