The equations of refined geometrically nonlinear theory of dynamic deformation of flexible with respect to
transversal shear and compression ofplates and shells are proposed. The solutions for problems of transversal
nonlinear vibrations of hinge fixedplate-strip are obtained. Numerical results are compared with the known in
literature

Kushnir, R. M.

Marchuk, M. V.

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

Proceedings of the 5th International Conference on Nonlinear DynamicsND-KhPI2016September 27-30, 2016, Kharkov, UkraineN o n lin e a r  D y n a m ics  o f  C o m p o site  P la te s  an d  S h ellsR.M. Kushnir1’ 2, M.V. Marchuk1’ 2*AbstractThe equations o f refined geometrically nonlinear theory o f dynamic deformation o f flexible with respect to transversal shear and compression ofplates and shells are proposed. The solutions for problems o f transversal nonlinear vibrations o f hinge fixedplate-strip are obtained. Numerical results are compared with the known in literature.KeywordsNonlinear dynamics, composite plates-strips, pliability to transversal shear and compression1Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, L’viv, Ukraine ^National University “L’viv Polytechnic”, L’viv, Ukraine* Corresponding author: mv_marchuk@ukr.netIntroductionThe plate and shell elements are widely used in various constructions and technical means because of their rational materials consumption. These constructions and engineering structures are exposed to intensive static and dynamic loadings. The loads can cause substantial bending commensurate with the thickness of thin-wall elements and hence there occurs the geometrically nonlinear stress state. The majority of studies on nonlinear statics and dynamics of thin-wall structure elements are based on the quadratic Karman theory being the generalization of classical linear theory [1-14] by Kirchhoff-Love. In some works the relations of nonlinear technical theory based on the Timoshenko shear model [15-20] were used. However the theories grounded on the hypotheses of these authors do not consider fully the peculiarities of deformation of the composites used for modern constructions [16, 21-23]. In view of the above we can mark the necessity of building mathematical model of geometrically nonlinear deformation of plates and shells which would take into account such specific peculiarities of composites as pliability to transversal shear and compression. Of course, they must also satisfy the condition of possibility of practical application. In the present article the method of series expansion of characteristics of the stress-strain state of thin-wall structure elements by Legendre polynomials from a normal to median surface of the coordinate [23, 24] is proposed. The method is developed for the case of geometrically nonlinear deformation. The boundary conditions in stresses on the face surface are satisfied exactly. This enabled us to obtain the solvable system of the equations of dynamic elasticity of plates and shells in geometrically nonlinear statement taking into account the material pliability to transversal shear and compression. On this basis an analytic solution for problem about nonlinear vibrations of plates-strips was obtained. A good coincidence of separate results with solutions known in literature is noted.1. The problem statementA thin curved anisotropic elastic layer with thickness 2h and superficial dynamic forces is exposed of volumetric f 1 ( i = 1,2,3). This layer is related to a mixed system of coordinates Q ( i = 1,2,3) on the medial surface = 0. Then its stressed and strained state is characterized by dynamic elastic displacement vector u = uie ieJ = uieie1, finite strain tensor Te = eije , e J = e iJe tej329R.M. Kushnir, M. V. Marchukand symmetric stress tensor by Piola Ta = a i]- e 1 e J = <r1J ei e j , where e, , e1, i = 1,2,3  are covariant andcontravariant basis vectors of system £  , i = 1, 2, 3. The components of strain tensor are determined via the components of the vector elastic displacement of layer points by relationshipseij = 2 ( viuj +v j ui + V iuk v j uk ) , (1)where V j means a covariant differentiating with respect to coordinate £,1 [2].If for the material of layer the generalized Hooke's law is executed, then the following dependences are valid^  = A iJkmekm. (2)In this case the motion equations of a layer points have the formd 2uj du^V S  + f j - P ~ u ^ - n d u = 0, j  = 1,2,3 , (3)dt2 dtwhere S 1j are components of the nonsymmetrical Kirchhoff’s stress tensor TS = S v eiej = S je 'e -1[8], which are connected with the components of the symmetrical Piola’s stress tensor by relationsS 1 = o ik (8( + v u ). (4)The remain of the quantities in (3) has the following meaning: p is  density of material layer; rjj is coefficient of damping in the direction £ J ; t is time variable.Equations (1) -  (4) are closed system for determining twenty four unknown functions characterizing the three-dimensional dynamic geometric nonlinear stress-strain state of curvilinear anisotropic layer.The constants in general integrals of differential equations are determining from the conditions3at boundary surfaces Q+, Q- of layer, that are described by equations £, = ±h , on the side surface Q = Qa + Qu, and from an initial conditions. Boundary conditions are written in the formS3 (£1,£ 2, ± h,t) = X±t(£1,£ 2,t) , i = 1,2,3, (£1,£ 2) e Q ±, (5) S im(£1,£ 2, £3,t )nt = p m(£1,£ 2,£3, t ) ,i = 1,2,3, m = 1,2,3, (£1,£ 2,£3) e Q 5 , (6)u (£1,£ 2, £3,t ) = v 1 (£1,£ 2,£3,t ), i = 1,2,3, (£1,£ 2,£ 3)e Q u , (7)where ni are covariant components of the vector of unit normal to the lateral surface Qa of theundeformed layer; p m are the components of forces vector on the lateral surface of deformed layer. Itis related to the unit area of the lateral surface of undeformed layer; v1 are the components of the vector of elastic displacement of points on the lateral surface Qu .The conditions at the initial time moment t = t0 have the form330R.M. Kushnir, M. V. Marchuku '\ = g  (E},E2,E3) ; It=t„dudt= g2 tf1^ 2^ 3), i = 1,2,3, (8 )t=t„Here g ‘k , k = 1,2 are given functions.2. The construction of two-dimensional equationsAccording to the approach proposed in the papers [24, 25] tensors stress TS and strain Te for physical components the vectors of elastic displacement u and volumetric forces f  components are3accepted as the following approximation expressions as regards coordinate a  = £, / h :ut = ufP0(a )  + u1p(a), f  = f  P0 (a) + f ^ P ^ a )  , i = 1,2,3; (9)SV = S0P0(a) + S fP ^ a )  , Si 3  = ZS*P k(a), i = 1,2; j  = 1,2;k= 0S 3 3  = Z S 3k%  (a), (1 0 )k= 0ej = ejP0(a) + e jp ^ a ) , i = 1,2; j  = 1,2,3;e33 = e303P0(a), (1 1 )where Pk(a), k  = 0,1,2,..., - Legendre polynomial of the k  -th order relatively of variables a  .By the boundary conditions (5) on the surfaces a  = +1 we haveSf = X ;  -  S3, S43i = X+ -  S ^  -  S 3 , i = 1,2;S3 3  = X3+- S33, S33 3  = X3- - S33, (12)where X + = (X+ + X3- )/2.Substituting (9), (10) in equation (3), using the procedure by Bubnov-Galerkin [8 ] and considering normal coordinate a  to the median surface of layer at the system of basic functions P0  and P1, we get the following two-dimensional equations of motion:- y^( A2 S,1) + ^ J  (A, S21)-dA2 SG2 2  + dA 1  S 12 + d £ l d £ 2 d ^ 1 ° d £ 2 °+ A1A2(k1S0 + T X1 ) + A1A2(f0 -  p u1 - ^1u1 ) = 0; h(a 2s12)+  - dy  (A1 s 022) - M -  s "  + dAf  s g21 + d£ l d£2 d£,2 o d ^ 1 o331R.M. Kushnir, M. V. Marchuk+ A1 A2(k2S03 + T X  2 ) + A A2(fo -  Pu2 ~^2 u2.) -  0 ; h(A2S;1) + - ^  (A t S,21) - M l  S,22 + M -  S,12 + ag1 ag2 ag1 ag2 13 3+ A, A2 (£1S11 3 - -  S. 1 + -  X + ) + h h+ A1A2 ( fT -  pu\ -  Vl ul ) = 0 ;T( A S '2) + - | r (A 'Sf2) - M .  s ' '  + a A l S,2' +an rl (a 2s 1 ) + „ r 2  ^ | |  ^ n r2 ' n r, ag' ag2 ag2 ag'+ A1A2 (k2 S12 3  - TS0 2  + TX 2  ) + h h+ A1A2( f12 -  p u 2 - ^2u2) = 0 ;~ T (A2ST3) + (A,S23) + A,A2(T X3- -  k'Sl1 -  ag ag h-  k2S(32) + A1 A2( f 0 -  pu3 - ^3U0) = 2 ;(A2S|3) + - ^ 2 (A,S,23) + A,A2(-3 X3+ - h S 033 -  ag' ag2 h  h-  k1S1 1 -  k2S122) + A1 A2(f13 -  pu3 - ^3 u |)  = 0 . (13)In (13) A,,A2 and k,,k2 are coefficients of the first quadratic form and the main curvatures of the middle surface of layer, and dot over the notation of functions means derivative with respect to time coordinates.Since the strainsS ij = O  , i, j  -  1,2,3,are infinitesimals then the order of approximations for components of the Piola tensor will be the same as in expressions (10).From the condition of a minimum of quadratic functionalR  -  i  f  £  £  [Sj  - o * (S'k + V kU ) ]  2} d a ,  ( 14)-1 Li=' j-1 Jwhere as S iJ , 0 k and u are taken their approximations, we obtain expressions for the coefficients ofapproximations S'k , that are available in the equations of equilibrium (13):e '1  _ „ - i in _ ._ o 0 \_ L .  A 2 _ 1 2  0 , 1 13 0 .S 0 -  O 0 (1 +  s 11)  +  7 T CT0 s 12 +~TO0 s 13 +A 1 A 1+ 3 ( 0 %  + A °1^^12) ,332R.M. Kushnir, M. V. Marchuko 12 _  A 1 ,^11  0 . 1 11 1 . „ .1 2 n  . „ 0  n ,S 0 a 0 s 21 + ~ a l  s 21 + a 0 (1 +  s 22) +A -  3l  12 1 l  13 0+  T a l S 22 + ~ T a  S 23 ,3  a 2SQ3 -  4 ( ^ 3 0  + _3'a 1 lfi31) + A2 (a O2^02 ++  p a 12^  +  a 03(1 +  4 s  KC l 1 - „ - 1 1 n _ l _ , . 0 \  , A 2 ^ ! 2 r,0 . 1 13 0 .S 1 -  a 1 (1 + £ 11)  +  —  a 1 S 21 +  —  a 1 s 13 +A 1 A 111 1 A o 12 1+  a 0 S 11 + ~ T a 0  s 1 2 ,A 1S12 - ■A i a j1s2?i + a 11e!°1 + aj12(1 + s - )  +A212 1 1 13 0+  a 0 S 22 + - T a 1 S 23 ,A 2Sj3 -  A i(a(j1s ;1i + a ^ j )  + A2(a^j2s^2 ++  a 024 )  +  a 03(1 +  S 3 3 ) ,S031 -  a 01(1 + Sl01) + ^ V i  + ^ ( a f ^ i  ++  3 a i 2 3 s 0 2 )  +  A  a 3^ ,  (1 o  2 )S 033 =  A 1( a ^3 s 301 +  ^ ^ S 0^  +  A 2 ( a 023S302 ++  ^ l 23^  +  a 0 3(1 +  s 303 ) . ( 1 5 )In (15) the following designations are acceptedk 1 Su0 1 SA  k 7 kSk = ------- V + ----------- j  u2 + k u k11 Aj d£} AjA2 13k _  1 Su2 1 SA0 k S 21 = -------- 'l--------------- T uj ;Aj d ^ 1 AlA2u k s* = ^ _  _ kjujk, k  = 0,1; 1 o  2;,k _ _ 1 Su ^ h „  A  d ^ 1s °3 = u1 / h , i = 1,2,3 .Substituting the approximate expressions for ut of (9) and eij  of (11) in (1) and applying theprocedure by Bubnov-Galerkin we get a two-dimensional deformation relationships in the same way as the equilibrium equations0 1 / 0 0 \ J  0 0 , 1 1 1 \ 1 1 1 , 1 \ , 1 / 1 0 , 0 i \ / " '  ■ i'>\eij =~ (sij + s ji) + Z  (— skjski + T SkjSki) , eij = T (sij + Sji)  + T  Z (skjski + skjski) , ( i , J = j , 2 )  ;2 k=1 2 6 2  2  k=1e i3 =  ~ (sB +  s 30i )  +  ~  Z Sk3Ski , e j3 =  ~ sQ3 +  Z  S°3Ski-, ( i  =  j , 2 )  ; e33 =  s 33 +  “  Z ( s °3 )2 . ( 1 6 )2  2  k=1 2  k =1 2  k=1333R.M. Kushnir, M. V. MarchukConsidering analogous to (14) concerning the functional dependencies (2), we obtain two­dimensional elasticity relationships3^ _  AVkl p m n -,,a„  _  A nm eki (17)Substituting the corresponding of approximations to boundary conditions (6 ), (7) we get($"n +^kum ) n L _ pm ( ^ 2,£3, t), m _ 0,1; n _ 1,2,3, ( £ \£ 2,£3) eQ „  ;IL2ctukmI _ g km(Z \Z 2,Z3, t), m _ 1,2,3; k _ 0,1. (18)The initial conditions of relatively the generalized displacements u k are written asuk\ _ g p ^ u 2,^),_ g(2)k( ^ 2,^ 3). (19)du.~8tl0kPerforming sequentially substitution (16) ^ (17) ^ (15) ^ (13), we obtain six nonlinear for two spatial coordinates and linear by time the equations in partial derivatives relatively the generalized displacements, which together with the corresponding boundary and initial conditions constitute a complete system of equations of nonlinear elastic dynamic deformation, pliable to transversal shear and compression, plates and shells (at k1 _ k2  _ 0 ). Moreover, the initial boundary value problems are formulated in the initial undeformed system of coordinates.3. Numerical resultsFor example, consider free nonlinear transversal vibrations of composite plate-strip with elastic characteristics:E E- 1  _ 40, —— _ 66,67, v1 3 _ 0,25.E3 G1 3 1 3Using the technique from [1] for amplitude-frequency dependence we obtained formula1 + 3KA2  , (20)where mNL is natural frequency of nonlinear vibrations, mL is natural frequency of linear vibrations,A _ wax, k _ 3 + hf l  1^1-1+^, « _ ( 1  + v 1 )(v3 ) 2  j l ,  k '_  14/15, h  _ 0,014 | a J k  'I G1 3 J 1 -v1v3 1 -v - 2 v1v3 E3t _t334R.M. Kushnir, M. V. Marchuk Table 1. Comparison of the results obtained by different techniqueswmax 0}m lc0Lh [26] PerturbationmethodKirchhoff-LoveRefinedtheory0,2 1,0313 1,0401 1,04403 1,044570,4 1,1198 1,1214 1,16619 1,168120,6 1,2536 1,2695 1,34536 1,349130,8 1,4199 1,4418 1,56205 1,567821,0 1,6086 1,6588 1,80278 1,810581,2 1,8127 1,8627 2,05913 2,06896Table 1 shows the dependence of the dimensionless frequency values mNL / a>L on thewdimensionless amplitude max obtained by different techniques. The first column contains results hfrom the work [26]. The second column shows results obtained by M.V. Marchuk, T.V. Horiachko, V.S. Pakosh and O.F. Lesyk using the perturbation method. The third column contains dimensionless frequencies calculated for the amplitude-frequency dependence obtained in [1] using the classical plates theory by Kirchhoff-Love. In the fourth column are the results calculated by the formula (20).ConclusionsThe table shows the qualitative coincidence of the results obtained by different approaches. A numerical difference between the first two and last two columns is existing. This is a consequence of the use of the third and fourth cases of the amplitude-frequency dependency averaging method in the oscillation period [1].A good coincidence of results in the first and second cases is the result of direct numerical searches these dependencies: 1. -  using refined theory; 2 . -  using spatial relationships of geometrically nonlinear dynamic elasticity.In the future, similar research should be carried out for wide range of change physical and mechanical parameters and at different boundary conditions.References[1] Vol’mir A.S. Nonlinear dynamics plates and shells. M.: Nauka, 1972 /in Russian/.[2] K. Galimov Z. Fundamentals o f  nonlinear theory o f  thin shells, Kazan: Kazan University, 1975 /in Russian/.[3] Grigorenko Ya.M., Bahramova S.N., Sudavtsova G.K. Numerical solution of boundary value problems of large bending of a long cylindrical panel, Prikl. Mechanika, 1978, 14 (10), p. 47-51 /in Russian/.[4] Grigorenko Ya.M., Kryukov N.N. On numerical solutions of nonlinear boundary value problems of flexible circular plates. Vychislitelnaya i Prikl. Mechanika, Kiev: KGU, 1980, Vol. 41, p. 46-52 /in Russian/.[5] Grigorenko Ya.M., Mukoed A.P. Solution o f  Nonlinear Problems o f  the Shells Theory on Computers, Kiev: Vyshcha Shkola, 1983 /in Russian/.[6] Kornishyn M.S., Isambaeva F.S. Flexible Plates and Panels, M.: Nauka, 1968 /in Russian/.[7] Kubenko V.D., Kovalchuk P.S., Podchasov N.P. Nonlinear Vibrations o f  Cylindrical Shells, K.: Vyshcha Shkola, 1989 /in Russian/.[8] Morozov N.F. Selected two-dimensional problem o f  elasticity theory, L.: Leningrad Univerditet, 1978 /in Russian/.[9] Mushtari H.M., Galimov K.Z. Nonlinear Theory o f  Elastic Shells, Kazan': Tatknigoizdat, 1957 /in Russian/.[10] Bielski W., Telega I. I. On existence of solutions for geometrically nonlinear shells and plates. Z  Angew. Math. undMech., 1988, Vol. 68 (4), p. 155-157.335R.M. Kushnir, M. V. Marchuk[11] Lopez S. Geometrically nonlinear analysis of plates and cylindrical shells by a predictor - corrector method. Int. J. Somput. and Struct., 2001, Vol. 79 (15), p. 1405-1415.[12] Niyogi A. K. Nonlinear bending of rectangular orthotropic plates. Int. J. Solids Struct., 1973, No.9, p. 1133-1139.[13] Ohga Mitao, Murata Motozi. Large deformation dynamic analysis thin plates by FEIM method. Mem. Fac. Eng. Ehme Univ., 1988, Vol. 11 (3), p. 201-210.[14] Tsao C. H. Strain-displacement in large displacement theory of shells. AIAA J., 1964, Vol. 2(11), p. 236-238.[15] Vanin G.A., Zhukov N.B., Semenyuk N.P. On nonlinear theory of Tymoshenko-type shells in quadratic and more high approximations. Tr. XIV Vsesojuzn. Conf. po Teorii Plastin i Obolochek. Kutaisi, 1987, p. 267-272 /in Russian/.[16] Grigorenko Ya.M. Some approaches to numerical solution of linear and nonlinear problems of the shells theory in classical and refined statement. Prikl. Mechanika, 1996, Vol. 32 (6), p. 3-31 /in Russian/.[17] Krys’ko V.A. Nonlinear Statics and Dynamics o f Inhomogeneous Shells, Saratov: Saratovslii Universitet, 1976 /in Russian/.[18] Semeniuk N.P., Zhukova N.B. On equations of nonlinear theory of Tymoshenko-type plates at large displacements. Dop. NAN Ukrainy, 2002, N° 10, p. 48-55 /in Russian/.[19] Shul’ga N.A., Mejci V.F., Hatrenko Ju.A. Nonlinear axially symmetric vibrations of three-layer shells of revolutions of pulse load. Teoret. iPrikl. Mechanica, 2001, N° 33, p. 159-163 /in Russian/.[20] Di Sciuva M., Librescu L. Contribution to the nonlinear theory of multilayered composite shells featuring damaged interfaces Compos. B., 2001, Vol. 32 (3), p. 219-227.[21] Galimov K.Z. On construction of nonlinear theory of thin shells of complex form considering transversal displacements and compression. Issledovanija po Theorii Plastin i Obolochek: in 2 ps. Kazan’: Kazanskii Universitet, 1984, Vol. 17, Part 1, p. 70-109 /in Russian/.[22] Piskunov V.G., Verizhenko V.E. Linear and nonlinear problems o f calculation o f laminated constructions, Kiev: Budivel’nyk, 1986 /in Russian/.[23] Ciarlet P.G., Roquefort A. Justification of a two-dimensional nonlinear shell model of Koiter's type. Chinese Ann. Math. Ser. B 22, 2001, Vol. 2, p. 129-144.[24] Pelekh B.L., Sukhorol’skii M.A. On one new approach to construction of the shell theory considering boundary conditions on the surfaces. Dop. AN URSR. Ser. A, 1978, N° 5, p. 441-444 /in Ukrainian/.[25] Pelekh B.L., Sukhorol’skii M.A. Contact Problems o f the Theory o f Elastic Anisotropic Shells, Kiev: Naukova Dumka, 1980 /in Russian/.[26] Kurpa L.V., Budnikov N.A. Investigation of forced nonlinear vibrations of laminated shallow shells using a multimode approximation. Bulletin o f Donetsk Univ. Series A. Natural Sciences, 2013, Vol. 1, p. 55-60.336

Nonlinear Dynamics of Composite Plates and Shells

Library of National Technical University "Kharkiv Polytechnic Institute"

Proceedings of the 5th International Conference on Nonlinear Dynamics
ND-KhPI2016
September 27-30, 2016, Kharkov, Ukraine
N o n lin e a r  D y n a m ics  o f  C o m p o site  P la te s  an d  S h ells
R.M. Kushnir1’ 2, M.V. Marchuk1’ 2*
Abstract
The equations o f refined geometrically nonlinear theory o f dynamic deformation o f flexible with respect to 
transversal shear and compression ofplates and shells are proposed. The solutions for problems o f transversal 
nonlinear vibrations o f hinge fixedplate-strip are obtained. Numerical results are compared with the known in 
literature.
Keywords
Nonlinear dynamics, composite plates-strips, pliability to transversal shear and compression
1Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, L’viv, Ukraine 
^National University “L’viv Polytechnic”, L’viv, Ukraine
* Corresponding author: mv_marchuk@ukr.net
Introduction
The plate and shell elements are widely used in various constructions and technical means 
because of their rational materials consumption. These constructions and engineering structures are 
exposed to intensive static and dynamic loadings. The loads can cause substantial bending 
commensurate with the thickness of thin-wall elements and hence there occurs the geometrically 
nonlinear stress state. The majority of studies on nonlinear statics and dynamics of thin-wall structure 
elements are based on the quadratic Karman theory being the generalization of classical linear theory 
[1-14] by Kirchhoff-Love. In some works the relations of nonlinear technical theory based on the 
Timoshenko shear model [15-20] were used. However the theories grounded on the hypotheses of 
these authors do not consider fully the peculiarities of deformation of the composites used for modern 
constructions [16, 21-23]. In view of the above we can mark the necessity of building mathematical 
model of geometrically nonlinear deformation of plates and shells which would take into account such 
specific peculiarities of composites as pliability to transversal shear and compression. Of course, they 
must also satisfy the condition of possibility of practical application. In the present article the method 
of series expansion of characteristics of the stress-strain state of thin-wall structure elements by 
Legendre polynomials from a normal to median surface of the coordinate [23, 24] is proposed. The 
method is developed for the case of geometrically nonlinear deformation. The boundary conditions in 
stresses on the face surface are satisfied exactly. This enabled us to obtain the solvable system of the 
equations of dynamic elasticity of plates and shells in geometrically nonlinear statement taking into 
account the material pliability to transversal shear and compression. On this basis an analytic solution 
for problem about nonlinear vibrations of plates-strips was obtained. A good coincidence of separate 
results with solutions known in literature is noted.
1. The problem statement
A thin curved anisotropic elastic layer with thickness 2h and superficial dynamic forces is 
exposed of volumetric f 1 ( i = 1,2,3). This layer is related to a mixed system of coordinates Q 
( i = 1,2,3) on the medial surface = 0. Then its stressed and strained state is characterized by 
dynamic elastic displacement vector u = uie ieJ = uieie1, finite strain tensor Te = eije , e J = e iJe tej
329
R.M. Kushnir, M. V. Marchuk
and symmetric stress tensor by Piola Ta = a i]- e 1 e J = <r1J ei e j , where e, , e1, i = 1,2,3  are covariant and
contravariant basis vectors of system £  , i = 1, 2, 3. The components of strain tensor are determined 
via the components of the vector elastic displacement of layer points by relationships
eij = 2 ( viuj +v j ui + V iuk v j uk ) , (1)
where V j means a covariant differentiating with respect to coordinate £,1 [2].
If for the material of layer the generalized Hooke's law is executed, then the following 
dependences are valid
^  = A iJkmekm. (2)
In this case the motion equations of a layer points have the form
d 2uj du^
V S  + f j - P ~ u ^ - n d u = 0, j  = 1,2,3 , (3)
dt2 dt
where S 1j are components of the nonsymmetrical Kirchhoff’s stress tensor TS = S v eiej = S je 'e -1
[8], which are connected with the components of the symmetrical Piola’s stress tensor by relations
S 1 = o ik (8( + v u ). (4)
The remain of the quantities in (3) has the following meaning: p is  density of material layer; 
rjj is coefficient of damping in the direction £ J ; t is time variable.
Equations (1) -  (4) are closed system for determining twenty four unknown functions 
characterizing the three-dimensional dynamic geometric nonlinear stress-strain state of curvilinear 
anisotropic layer.
The constants in general integrals of differential equations are determining from the conditions
3
at boundary surfaces Q+, Q- of layer, that are described by equations £, = ±h , on the side surface 
Q = Qa + Qu, and from an initial conditions. Boundary conditions are written in the form
S3 (£1,£ 2, ± h,t) = X±t(£1,£ 2,t) , i = 1,2,3, (£1,£ 2) e Q ±, (5) 
S im(£1,£ 2, £3,t )nt = p m(£1,£ 2,£3, t ) ,
i = 1,2,3, m = 1,2,3, (£1,£ 2,£3) e Q 5 , (6)
u (£1,£ 2, £3,t ) = v 1 (£1,£ 2,£3,t ), i = 1,2,3, (£1,£ 2,£ 3)e Q u , (7)
where ni are covariant components of the vector of unit normal to the lateral surface Qa of the
undeformed layer; p m are the components of forces vector on the lateral surface of deformed layer. It
is related to the unit area of the lateral surface of undeformed layer; v1 are the components of the 
vector of elastic displacement of points on the lateral surface Qu .
The conditions at the initial time moment t = t0 have the form
330
R.M. Kushnir, M. V. Marchuk
u '\ = g  (E},E2,E3) ; 
It=t„
du
dt
= g2 tf1^ 2^ 3), i = 1,2,3, (8 )
t=t„
Here g ‘k , k = 1,2 are given functions.
2. The construction of two-dimensional equations
According to the approach proposed in the papers [24, 25] tensors stress TS and strain Te for 
physical components the vectors of elastic displacement u and volumetric forces f  components are
3accepted as the following approximation expressions as regards coordinate a  = £, / h :
ut = ufP0(a )  + u1p(a), f  = f  P0 (a) + f ^ P ^ a )  , i = 1,2,3; (9)
SV = S0P0(a) + S fP ^ a )  , Si 3  = ZS*P k(a), i = 1,2; j  = 1,2;
k= 0
S 3 3  = Z S 3k%  (a), (1 0 )
k= 0
ej = ejP0(a) + e jp ^ a ) , i = 1,2; j  = 1,2,3;
e33 = e303P0(a), (1 1 )
where Pk(a), k  = 0,1,2,..., - Legendre polynomial of the k  -th order relatively of variables a  .
By the boundary conditions (5) on the surfaces a  = +1 we have
Sf = X ;  -  S3, S43i = X+ -  S ^  -  S 3 , i = 1,2;
S3 3  = X3+- S33, S33 3  = X3- - S33, (12)
where X + = (X+ + X3- )/2.
Substituting (9), (10) in equation (3), using the procedure by Bubnov-Galerkin [8 ] and 
considering normal coordinate a  to the median surface of layer at the system of basic functions P0  
and P1, we get the following two-dimensional equations of motion:
- y^( A2 S,1) + ^ J  (A, S21)-dA2 SG2 2  + dA 1  S 12 + 
d £ l d £ 2 d ^ 1 ° d £ 2 °
+ A1A2(k1S0 + T X1 ) + A1A2(f0 -  p u1 - ^1u1 ) = 0; h
(a 2s12)+  - dy  (A1 s 022) - M -  s "  + dAf  s g21 + 
d£ l d£2 d£,2 o d ^ 1 o
331
R.M. Kushnir, M. V. Marchuk
+ A1 A2(k2S03 + T X  2 ) + A A2(fo -  Pu2 ~^2 u2.) -  0 ; h
(A2S;1) + - ^  (A t S,21) - M l  S,22 + M -  S,12 + 
ag1 ag2 ag1 ag2 1
3 3+ A, A2 (£1S11 3 - -  S. 1 + -  X + ) + 
h h
+ A1A2 ( fT -  pu\ -  Vl ul ) = 0 ;
T( A S '2) + - | r (A 'Sf2) - M .  s ' '  + a A l S,2' +
a
n rl (a 2s 1 ) + „ r 2  ^ | |  ^ n r2 ' n r, ag' ag2 ag2 ag'
+ A1A2 (k2 S12 3  - TS0 2  + TX 2  ) + h h
+ A1A2( f12 -  p u 2 - ^2u2) = 0 ;
~ T (A2ST3) + (A,S23) + A,A2(T X3- -  k'Sl1 -  
ag ag h
-  k2S(32) + A1 A2( f 0 -  pu3 - ^3U0) = 2 ;
(A2S|3) + - ^ 2 (A,S,23) + A,A2(-3 X3+ - h S 033 -  
ag' ag2 h  h
-  k1S1 1 -  k2S122) + A1 A2(f13 -  pu3 - ^3 u |)  = 0 . (13)
In (13) A,,A2 and k,,k2 are coefficients of the first quadratic form and the main curvatures of 
the middle surface of layer, and dot over the notation of functions means derivative with respect to 
time coordinates.
Since the strains
S ij = O  , i, j  -  1,2,3,
are infinitesimals then the order of approximations for components of the Piola tensor will be the 
same as in expressions (10).
From the condition of a minimum of quadratic functional
R  -  i  f  £  £  [Sj  - o * (S'k + V kU ) ]  2} d a ,  ( 14)
-1 Li=' j-1 J
where as S iJ , 0 k and u are taken their approximations, we obtain expressions for the coefficients of
approximations S'k , that are available in the equations of equilibrium (13):
e '1  _ „ - i in _ ._ o 0 \_ L .  A 2 _ 1 2  0 , 1 13 0 .
S 0 -  O 0 (1 +  s 11)  +  7 T CT0 s 12 +~TO0 s 13 +
A 1 A 1
+ 3 ( 0 %  + A °1^^12) ,
332
R.M. Kushnir, M. V. Marchuk
o 12 _  A 1 ,^11  0 . 1 11 1 . „ .1 2 n  . „ 0  n ,
S 0 a 0 s 21 + ~ a l  s 21 + a 0 (1 +  s 22) +
A -  3
l  12 1 l  13 0
+  T a l S 22 + ~ T a  S 23 ,
3  a 2
SQ3 -  4 ( ^ 3 0  + _3'a 1 lfi31) + A2 (a O2^02 +
+  p a 12^  +  a 03(1 +  4 s  K
C l 1 - „ - 1 1 n _ l _ , . 0 \  , A 2 ^ ! 2 r,0 . 1 13 0 .
S 1 -  a 1 (1 + £ 11)  +  —  a 1 S 21 +  —  a 1 s 13 +
A 1 A 1
11 1 A o 12 1
+  a 0 S 11 + ~ T a 0  s 1 2 ,
A 1
S12 - ■A i a j1s2?i + a 11e!°1 + aj12(1 + s - )  +
A2
12 1 1 13 0
+  a 0 S 22 + - T a 1 S 23 ,
A 2
Sj3 -  A i(a(j1s ;1i + a ^ j )  + A2(a^j2s^2 +
+  a 024 )  +  a 03(1 +  S 3 3 ) ,
S031 -  a 01(1 + Sl01) + ^ V i  + ^ ( a f ^ i  +
+  3 a i 2 3 s 0 2 )  +  A  a 3^ ,  (1 o  2 )
S 033 =  A 1( a ^3 s 301 +  ^ ^ S 0^  +  A 2 ( a 023S302 +
+  ^ l 23^  +  a 0 3(1 +  s 303 ) . ( 1 5 )
In (15) the following designations are accepted
k 1 Su0 1 SA  k 7 k
Sk = ------- V + ----------- j  u2 + k u k
11 Aj d£} AjA2 13
k _  1 Su2 1 SA0 k 
S 21 = -------- 'l--------------- T uj ;
Aj d ^ 1 AlA2
u k 
s* = ^ _  _ kjujk, k  = 0,1; 1 o  2;,k _ _ 1 Su ^ h „  
A  d ^ 1
s °3 = u1 / h , i = 1,2,3 .
Substituting the approximate expressions for ut of (9) and eij  of (11) in (1) and applying the
procedure by Bubnov-Galerkin we get a two-dimensional deformation relationships in the same way 
as the equilibrium equations
0 1 / 0 0 \ J  0 0 , 1 1 1 \ 1 1 1 , 1 \ , 1 / 1 0 , 0 i \ / " '  ■ i'>\
eij =~ (sij + s ji) + Z  (— skjski + T SkjSki) , eij = T (sij + Sji)  + T  Z (skjski + skjski) , ( i , J = j , 2 )  ;2 k=1 2 6 2  2  k=1
e i3 =  ~ (sB +  s 30i )  +  ~  Z Sk3Ski , e j3 =  ~ sQ3 +  Z  S°3Ski-, ( i  =  j , 2 )  ; e33 =  s 33 +  “  Z ( s °3 )2 . ( 1 6 )
2  2  k=1 2  k =1 2  k=1
333
R.M. Kushnir, M. V. Marchuk
Considering analogous to (14) concerning the functional dependencies (2), we obtain two­
dimensional elasticity relationships
3^ _  AVkl p m n -,,a„  _  A nm eki (17)
Substituting the corresponding of approximations to boundary conditions (6 ), (7) we get
($"n +^kum ) n L _ pm ( ^ 2,£3, t), m _ 0,1; n _ 1,2,3, ( £ \£ 2,£3) eQ „  ;IL2ct
ukmI _ g km(Z \Z 2,Z3, t), m _ 1,2,3; k _ 0,1. (18)
The initial conditions of relatively the generalized displacements u k are written as
uk\ _ g p ^ u 2,^),
_ g(2)k( ^ 2,^ 3). (19)du.
~8t
l0
k
Performing sequentially substitution (16) ^ (17) ^ (15) ^ (13), we obtain six nonlinear for 
two spatial coordinates and linear by time the equations in partial derivatives relatively the 
generalized displacements, which together with the corresponding boundary and initial conditions 
constitute a complete system of equations of nonlinear elastic dynamic deformation, pliable to 
transversal shear and compression, plates and shells (at k1 _ k2  _ 0 ). Moreover, the initial boundary 
value problems are formulated in the initial undeformed system of coordinates.
3. Numerical results
For example, consider free nonlinear transversal vibrations of composite plate-strip with elastic 
characteristics:
E E
- 1  _ 40, —— _ 66,67, v1 3 _ 0,25.
E3 G1 3 1 3
Using the technique from [1] for amplitude-frequency dependence we obtained formula
1 + 3KA2  , (20)
where mNL is natural frequency of nonlinear vibrations, mL is natural frequency of linear vibrations,
A _ wax, k _ 3 + hf l  1^1-1+^, « _ ( 1  + v 1 )(v3 ) 2  j l ,  k '_  14/15, h  _ 0,014 | a J k  'I G1 3 J 1 -v1v3 1 -v - 2 v1v3 E3
t _t
334
R.M. Kushnir, M. V. Marchuk 
Table 1. Comparison of the results obtained by different techniques
wmax 0}m lc0L
h [26] Perturbation
method
Kirchhoff-
Love
Refined
theory
0,2 1,0313 1,0401 1,04403 1,04457
0,4 1,1198 1,1214 1,16619 1,16812
0,6 1,2536 1,2695 1,34536 1,34913
0,8 1,4199 1,4418 1,56205 1,56782
1,0 1,6086 1,6588 1,80278 1,81058
1,2 1,8127 1,8627 2,05913 2,06896
Table 1 shows the dependence of the dimensionless frequency values mNL / a>L on the
wdimensionless amplitude max obtained by different techniques. The first column contains results 
h
from the work [26]. The second column shows results obtained by M.V. Marchuk, T.V. Horiachko, 
V.S. Pakosh and O.F. Lesyk using the perturbation method. The third column contains dimensionless 
frequencies calculated for the amplitude-frequency dependence obtained in [1] using the classical 
plates theory by Kirchhoff-Love. In the fourth column are the results calculated by the formula (20).
Conclusions
The table shows the qualitative coincidence of the results obtained by different approaches. A 
numerical difference between the first two and last two columns is existing. This is a consequence of 
the use of the third and fourth cases of the amplitude-frequency dependency averaging method in the 
oscillation period [1].
A good coincidence of results in the first and second cases is the result of direct numerical 
searches these dependencies: 1. -  using refined theory; 2 . -  using spatial relationships of 
geometrically nonlinear dynamic elasticity.
In the future, similar research should be carried out for wide range of change physical and 
mechanical parameters and at different boundary conditions.
References
[1] Vol’mir A.S. Nonlinear dynamics plates and shells. M.: Nauka, 1972 /in Russian/.
[2] K. Galimov Z. Fundamentals o f  nonlinear theory o f  thin shells, Kazan: Kazan University, 1975 
/in Russian/.
[3] Grigorenko Ya.M., Bahramova S.N., Sudavtsova G.K. Numerical solution of boundary value 
problems of large bending of a long cylindrical panel, Prikl. Mechanika, 1978, 14 (10), p. 47-51 /in 
Russian/.
[4] Grigorenko Ya.M., Kryukov N.N. On numerical solutions of nonlinear boundary value problems 
of flexible circular plates. Vychislitelnaya i Prikl. Mechanika, Kiev: KGU, 1980, Vol. 41, p. 46-52 /in 
Russian/.
[5] Grigorenko Ya.M., Mukoed A.P. Solution o f  Nonlinear Problems o f  the Shells Theory on 
Computers, Kiev: Vyshcha Shkola, 1983 /in Russian/.
[6] Kornishyn M.S., Isambaeva F.S. Flexible Plates and Panels, M.: Nauka, 1968 /in Russian/.
[7] Kubenko V.D., Kovalchuk P.S., Podchasov N.P. Nonlinear Vibrations o f  Cylindrical Shells, K.: 
Vyshcha Shkola, 1989 /in Russian/.
[8] Morozov N.F. Selected two-dimensional problem o f  elasticity theory, L.: Leningrad Univerditet, 
1978 /in Russian/.
[9] Mushtari H.M., Galimov K.Z. Nonlinear Theory o f  Elastic Shells, Kazan': Tatknigoizdat, 1957 /in 
Russian/.
[10] Bielski W., Telega I. I. On existence of solutions for geometrically nonlinear shells and plates. 
Z  Angew. Math. undMech., 1988, Vol. 68 (4), p. 155-157.
335
R.M. Kushnir, M. V. Marchuk
[11] Lopez S. Geometrically nonlinear analysis of plates and cylindrical shells by a predictor - 
corrector method. Int. J. Somput. and Struct., 2001, Vol. 79 (15), p. 1405-1415.
[12] Niyogi A. K. Nonlinear bending of rectangular orthotropic plates. Int. J. Solids Struct., 1973, 
No.9, p. 1133-1139.
[13] Ohga Mitao, Murata Motozi. Large deformation dynamic analysis thin plates by FEIM method. 
Mem. Fac. Eng. Ehme Univ., 1988, Vol. 11 (3), p. 201-210.
[14] Tsao C. H. Strain-displacement in large displacement theory of shells. AIAA J., 1964, Vol. 2
(11), p. 236-238.
[15] Vanin G.A., Zhukov N.B., Semenyuk N.P. On nonlinear theory of Tymoshenko-type shells in 
quadratic and more high approximations. Tr. XIV Vsesojuzn. Conf. po Teorii Plastin i Obolochek. 
Kutaisi, 1987, p. 267-272 /in Russian/.
[16] Grigorenko Ya.M. Some approaches to numerical solution of linear and nonlinear problems of 
the shells theory in classical and refined statement. Prikl. Mechanika, 1996, Vol. 32 (6), p. 3-31 /in 
Russian/.
[17] Krys’ko V.A. Nonlinear Statics and Dynamics o f Inhomogeneous Shells, Saratov: Saratovslii 
Universitet, 1976 /in Russian/.
[18] Semeniuk N.P., Zhukova N.B. On equations of nonlinear theory of Tymoshenko-type plates at 
large displacements. Dop. NAN Ukrainy, 2002, N° 10, p. 48-55 /in Russian/.
[19] Shul’ga N.A., Mejci V.F., Hatrenko Ju.A. Nonlinear axially symmetric vibrations of three-layer 
shells of revolutions of pulse load. Teoret. iPrikl. Mechanica, 2001, N° 33, p. 159-163 /in Russian/.
[20] Di Sciuva M., Librescu L. Contribution to the nonlinear theory of multilayered composite shells 
featuring damaged interfaces Compos. B., 2001, Vol. 32 (3), p. 219-227.
[21] Galimov K.Z. On construction of nonlinear theory of thin shells of complex form considering 
transversal displacements and compression. Issledovanija po Theorii Plastin i Obolochek: in 2 ps. 
Kazan’: Kazanskii Universitet, 1984, Vol. 17, Part 1, p. 70-109 /in Russian/.
[22] Piskunov V.G., Verizhenko V.E. Linear and nonlinear problems o f calculation o f laminated 
constructions, Kiev: Budivel’nyk, 1986 /in Russian/.
[23] Ciarlet P.G., Roquefort A. Justification of a two-dimensional nonlinear shell model of Koiter's 
type. Chinese Ann. Math. Ser. B 22, 2001, Vol. 2, p. 129-144.
[24] Pelekh B.L., Sukhorol’skii M.A. On one new approach to construction of the shell theory 
considering boundary conditions on the surfaces. Dop. AN URSR. Ser. A, 1978, N° 5, p. 441-444 /in 
Ukrainian/.
[25] Pelekh B.L., Sukhorol’skii M.A. Contact Problems o f the Theory o f Elastic Anisotropic Shells, 
Kiev: Naukova Dumka, 1980 /in Russian/.
[26] Kurpa L.V., Budnikov N.A. Investigation of forced nonlinear vibrations of laminated shallow 
shells using a multimode approximation. Bulletin o f Donetsk Univ. Series A. Natural Sciences, 2013, 
Vol. 1, p. 55-60.
336


http://repository.kpi.kharkov.ua/bitstream/KhPI-Press/24792/1/%d0%92%d0%b2%d0%b5%d0%b4.%20Kushnir_Nonlinear_Dynamics_2016.pdf

Nonlinear Dynamics of Composite Plates and Shells

Abstract

Similar works

Full text

Available Versions

Electronic National Technical University "Kharkiv Polytechnic Institute" Institutional Repository (eNTUKhPIIR)

Library of National Technical University "Kharkiv Polytechnic Institute"

Nonlinear Dynamics of Composite Plates and Shells

Abstract

Similar works

Full text

Available Versions

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

Library of National Technical University "Kharkiv Polytechnic Institute"

Electronic National Technical University "Kharkiv Polytechnic Institute" Institutional Repository (eNTUKhPIIR)