On the Investigation of Dynamical Hierarchical Models of Elastic Multi ...

saqarT velos mecn ierebaTa erovnu li ak ad em iis moam be, t. 8, #3, 20 14 BULL ET IN OF THE GEORGIAN NATIONAL ACADEM Y OF SCIENCE S, vol. 8, no. 3, 2014

Mechanics

On the Investigation of Dynamical Hierarchical Models of Elastic Multi-Structures Consisting of Three-Dimensional Body and Multilayer SubStructure Gia Avalishvili*, Mariam Avalishvili**, David Gordeziani* *

Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, Tbilisi of Informatics, Engineering and Mathematics, University of Georgia, Tbilisi

**School

(Presented by Academy Member Guram Gabrichidze)

ABSTRACT. In this paper initial-boundary value problem for multi-structure consisting of threedimensional body with general shape and multilayer part composed of plates with variable thickness is considered. A hierarchy of dynamical models defined on the union of three-dimensional and twodimensional domains for dynamical three-dimensional model for the multi-structure is constructed. The pluridimensional initial-boundary value problems corresponding to the constructed hierarchical models are investigated in suitable function spaces. The convergence of the sequence of vector functions of three space variables, restored from the solutions of the constructed initial-boundary value problems defined on the union of three-dimensional and two-dimensional domains to the solution of the original threedimensional problem is proved and under additional regularity conditions the rate of convergence is estimated. © 2014 Bull. Georg. Natl. Acad. Sci.

Key words: dynamical models of elastic multi-structures, initial-boundary value problems, hierarchical modeling, Fourier-Legendre series.

Elastic multi-structures are the bodies, which consist of several parts with different geometrical shapes. Many engineering constructions are multi-structures consisting of plates, shells, beams and other substructures and therefore mathematical modeling of them is important from practical as well as theoretical point of view. One of the first theoretical investigations of multi-structures was carried out by P.G. Ciarlet, H. Le Dret, R. Nzengwa [1]. Applying asymptotic method they constructed and investigated a mathematical model defined on the product of three-dimensional and two-dimensional domains for a multi-structure consisting of three-dimensional body with a plate clamped in it. Multi-structures consisting of plates and rods were considered by H. Le Dret [2]. Further, many works were devoted to mathematical modeling and numerical solution of problems for elastic multi-structures (see [3] and references given therein). Different approach for constructing two-dimensional models of elastic plates with variable thickness was suggested by I. Vekua [4],

© 2014  Bull. Georg. Natl. Acad. Sci.

On the Investigation of Dynamical Hierarchical Models...

21

which was based on approximation of the components of the displacement vector-function of plate by partial sums of orthogonal Fourier-Legendre series with respect to the variable of plate thickness. Note that I. Vekua’s hierarchical dimensional reduction method is one of spectral approximation methods. Moreover, classical Kirchhoff-Love and Mindlin-Reissner models can be incorporated into the hierarchy obtained by I. Vekua, and so it can be considered as an extension of the widely used engineering plate models. Later on, various investigations were devoted to the study of mathematical models constructed by I. Vekua’s dimensional reduction method and its generalizations for elastic plates, shells and rods (see [5-7] and references given therein). The present paper is devoted to the construction and investigation of hierarchical models of multistructures consisting of three-dimensional body with general shape and multilayer part composed of plates with variable thickness, which may vanish on a part of the lateral surface, applying spectral dimensional reduction method. We consider variational formulation of three-dimensional initial-boundary value problem for dynamical linear model of elastic multi-structure and construct a hierarchy of models in Sobolev spaces defined on the union of three-dimensional and two-dimensional domains, when density of surface force is given on the upper and the lower faces of multilayer substructure, on a part of its lateral boundary and on a part of the boundary of elastic three-dimensional part with general shape, and the remaining part of the boundary of the multi-structure is clamped. We investigate the existence and uniqueness of solutions of the reduced pluridimensional problems in suitable weighted Sobolev spaces. Moreover, we prove convergence of the sequence of vector-functions of three space variables restored from the solutions of the constructed problems to the solution of the original three-dimensional initial-boundary value problem and if it possesses additional regularity we estimate the rate of convergence. For any bounded domain   R p , p  1 , with Lipschitz boundary we denote by L2    the space of square-integrable functions in  in the Lebesgue sense. H k    , k  1 , is the Sobolev space of order k based on L2 () , Hk ()  ( H k ()) 3 , L2 ()  ( L2 ()) 3 and Lk (ˆ )  ( Lk (ˆ ))3 , where ˆ is a Lipschitz surface. For any Banach space X, C 0 ([0, T ]; X ) denotes the space of continuous functions on [0, T ] with values in X, L2 (0, T ; X ) is the space of such functions g : (0, T )  X that g (t )

X

 L2 (0, T ) . We denote

2 by g   dg / dt the generalized derivative of g  L (0, T ; X ) .

Let us consider an elastic multi-structure, which consists of three-dimensional part with general shape and multilayer substructure attached to it consisting of plates with variable thickness, i.e. elastic body with m

initial configuration   bd 



pl k

, where   R 3 and bd  R3 are bounded Lipschitz domains,

k 1

 kpl  {( x1 , x2 , x 3 )  R 3 ; hk ( x1 , x2 )  x3  hk ( x1 , x2 ), ( x1 , x2 )  k } ,

k  R 2 , k  1, 2,..., m , are two-dimensional bounded Lipschitz domains with boundary k , bd   kpl   , k  1, 2,..., m , hk  C 0 (k )  C 0,1 (k  k ) are continuous on k and Lipschitz con-

tinuous in k and on k   , hk ( x1 , x2 )  hk ( x1 , x2 ), for ( x1 , x2 )  k  k , k  k is a Lipschitz   , pl curve, hk ( x1 , x2 )  hk ( x1, x2 ) , for ( x1 , x2 )  k \ k , k  1, 2,..., m . The interfaces bd   kpl  bd k

between the three-dimensional part bd  R 3 and plates kpl are parts of lateral surfaces of plates

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Gia Avalishvili, Mariam Avalishvili, David Gordeziani

, pl bd  {x  R 3 ; hk ( x1 , x2 )  x3  hk ( x1 , x2 ) ( x1 , x2 )   kbd , pl } ,  kbd , pl  k , k  1, 2,..., m . The upper and k

the lower faces of plate kpl , defined by equations x3  hk ( x1 , x2 ) and x3  hk ( x1 , x2 ), ( x1 , x2 )  k , we denote by  k and  k , respectively, and the lateral surface, where the thickness of kpl is positive, we denote by  k   kpl \ ( k   k )  {x  kpl ; hk ( x1 , x2 )  x3  hk ( x1 , x2 ), ( x1 , x2 )  k } , k  1, 2,..., m . The interfaces  kpl,k 1 , k  1, 2,..., m  1 , between neighbour plates kpl and kpl1 are the common parts of the upper and the lower faces of kpl and kpl1 , respectively,,  kpl, k 1   kpl   kpl1  {( x1 , x2 , x3 )  R 3 ;

x3  hk ( x1 , x2 )  hk1 ( x1 , x2 ) , when ( x1 , x2 )  k  k 1 , k  1, 2,..., m  1} . Let us assume that the three-dimensional part  bd consists of inhomogeneous anisotropic elastic material and the linear dynamical three-dimensional model of its stress-strain state is given by the following system

 bd

 2 uibd t 2

3





 ijbd (ubd ) x j

j 1

 fibd

in bd  (0, T ) ,

(1)

where i  1, 2, 3, u bd  (uibd )i31 is the displacement vector-function of the three-dimensional part,  bd  0 denotes mass density of the three-dimensional part in reference configuration, f bd  ( fibd )i31 is density of applied body force of the three-dimensional part, and σ bd  ( ijbd )i3, j 1 denotes linearized stress tensor of the three-dimensional

part

of

the

multi-structure,

which

is

given

by

3

 ijbd ( v) 

bd ijpq e pq ( v ),

a

bd e pq ( v)  (v p / xq  vq / x p ) / 2 , where aijpq (i, j , p, q  1, 2,3) are parameters

p , q 1

characterizing mechanical properties of the three-dimensional part bd . m

The remaining part



pl k

of the multistructure consists of plates kpl , k  1, 2,..., m , with variable

k 1

thickness, which consist of anisotropic inhomogeneous elastic material, and their three-dimensional models are given by the following systems

 pl , k

 2uipl ,k t

2

3



 j 1

 ijpl , k (u pl , k ) x j

 f i pl ,k in kpl  (0, T ) ,

(2)

where i  1, 2, 3, u pl , k  (uipl , k )3i 1 denotes displacement of k-th plate kpl ,  pl , k  0 is the mass density of plate kpl in reference configuration, f pl , k  ( fi pl ,k )3i 1 is density of body force for plate kpl , and 3

σ pl , k  ( ijpl ,k ) 3i, j 1 is linearized stress tensor of the plate kpl , which is given by  ijpl ,k ( v) 

pl , k ijpq e pq ( v ) ,

a

p ,q 1

pl , k i, j  1, 2,3 , where aijpq are parameters characterizing mechanical properties of the plate kpl of multi-struc-

ture.

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On the Investigation of Dynamical Hierarchical Models...

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Let us denote by u :   (0, T )  R 3 displacement vector-function of the entire multi-structure  , which equals to ubd : bd  (0, T )  R 3 in the three-dimensional part bd , u equals to u pl , k :  kpl  (0, T )  R3 , pl in the plate kpl of the multilayer part, k  1, 2,..., m . We assume that on the interfaces bd between the k

three-dimensional part bd and plates kpl , k  1, 2,..., m , rigid contact conditions, i.e. continuity of displacement vector-function and stress vector, are valid, 3

u bd  u pl ,k ,



3 bd bd ij (u ) n j

j 1





pl , k (u pl ,k )n j ij

, pl on bd  (0, T ) , i  1, 2, 3, k  1, 2,..., m , k

(3)

j 1

pl 3 , pl where n  ( n j ) j 1 is a unit normal vector to the surface bd . On the interfaces  k , k 1 between neighbour k

plates kpl , k  1, 2,..., m  1 , conditions of continuity of displacement vector-function and stress vector are given 3

u pl , k  u pl , k 1 ,



3

 ijpl ,k (u pl , k )n j 

j 1



pl ,k 1 (u pl , k 1 )n j ij

pl on  k , k 1  (0, T ) ,

(4)

j 1

pl where i  1, 2, 3, k  1, 2,..., m  1 , n  ( n j )3j 1 is a unit normal vector of the surface  k , k 1 .

The multi-structure  , consisting of the three-dimensional part bd and elastic plates kpl , k  1, 2,..., m , is clamped along a part  0 of its boundary, the density of the surface force is given on the remaining part of the boundary, and the initial values of the displacement and velocity are given 3



ij (u) n j

 gi

on 1 ,

u0

on  0 ,

j 1

u ( x, 0)  ( x ),

(5)

u ( x, 0)   ( x ), t

x  ,

bd bd pl , k pl , k where  ij (u)   ij (u ) in bd and  ij (u)   ij (u ) in kpl , k  1, 2,..., m i , j  1, 2,3 ,  0 is an ele-

ment of Lipschitz dissection of the boundary    of the domain  , 1   \ 0 , n  ( n j )3j 1 is a unit outward normal vector of 1 ,   (i )i31 and   ( i )i31 are initial displacement and velocity vectorfunctions, g  ( gi )i31 denotes density of surface force acting on the boundary 1 of  , which equals to bd gbd  ( gibd )i31 on the boundary of  and equals to g pl ,k  ( gipl ,k )3i 1 on the corresponding part of the

boundary of kpl . The clamped part of the three-dimensional part bd of the multi-structure we denote by bd bd bd 0 , and the remaining part, where surface force is given, is denoted by 1  1   . The clamped

parts

of

elastic

plates

are

parts

of

the

lateral

boundaries

 kpl,0  {( x1 , x2 , x3 )   kpl ; hk ( x1 , x2 )  x3  hk ( x1 , x2 ), ( x1 , x2 )   k ,0  k \  kbd , pl } , k  1, 2,..., m , and pl pl the remaining part, where surface force is given, we denote by  k ,1  1   k .

The variational formulation of the dynamical three-dimensional problem (1)-(5) for multi-structure  consisting of three-dimensional body and multilayer part is of the following form: find the unknown vector-

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24

Gia Avalishvili, Mariam Avalishvili, David Gordeziani

function u  C 0 ([0, T ]; V ()) , u  C 0 ([0, T ]; L2 ())  L2 (0, T ; V ()) , which satisfies the equation

d dt

3

3

3

   u(.)v dx    a i

ijpq e pq (u (.)) eij ( v ) dx

i

i 1 

i , j , p , q 1 



 f

3 i

vi dx 

i 1 

  g v d , i i

v  V (),

(6)

i 1 1

in the sense of distribution on (0, T ) and the initial conditions u(0)  , u (0)  ψ,

(7)

where   V ()  {v  H1 ()  ( H 1 ())3 ; tr( v )  0 on  0 } , ψ  L2 () , tr denotes the trace operator from Sobolev space H1 () to H1/ 2 ( ) , fi  f ibd in bd , fi  fi pl , k in kpl and bd  aijpq ( x ), aijpq ( x )   pl ,k  aijpq ( x),

x  bd , x   kpl , k  1, 2,..., m,

i, j, p, q  1, 2,3.

The bilinear forms in the left and right parts of the equation (6) we denote by 3

 , v )  R (w

3

3



 w i vi dx, A(w, v) 

i 1 

 

aijpq e pq (w )eij ( v)dx, L( v) 

i , j , p , q 1 



3

f i vi dx 

i 1 

  g v d , i i

i 1 1

 , v  L2 (), w, v  V(),    bd in bd ,    pl , k in kpl , k  1,..., m . where w Note, that each vector-function v  V () from the space V () can be represented as m  1 vectorfunctions v bd and v pl ,1 , v pl ,2 , . . . , v pl ,m , which are restrictions of the vector-function v on the sets bd and 1pl , 2pl , . . . , mpl , respectively. Consequently, the space V () can be considered as a space of the bd pl ,1 pl ,2 pl , m pl ,1 vector-functions ( v bd ) H1 (1pl )  ...  H1 ( mpl ) , such that v  0 on bd  0 on 0 , v N , v N , v N ,..., v N pl pl pl , pl 1,0 , v pl ,2  0 on  2,0 , . . . , v pl , m  0 on  m,0 , vbd  v pl , k on bd , k  1, 2,..., m , v pl , k  v pl , k 1 on k

 kpl,k 1 , k  1, 2,..., m  1 .

In order to construct dynamical model of multi-structure  let us consider subspaces VN () and H N () of V () and L2 () , respectively,, N  ( N1pl ,1 , N 2pl ,1 , N3pl ,1 ,..., N1pl ,m , N 2pl , m , N3pl , m ) consisting of pl ,1 pl ,2 pl ,m pl , k vector-functions with m  1 components v N  ( vbd is a vector-function N , v N , v N ,..., v N ) , where v N

the i-th component of which is a polynomial of order Nipl ,k , i  1, 2,3 , k  1, 2,..., m , with respect to the variable x3 , i.e. pl ,1 pl ,2 pl , m bd 1 bd VN ()  {v N  (vNi ) i31 ; v N  ( v bd N , v N , v N ,..., v N ), vNi  H (  ), N ipl ,k

vNpli , k

1  h r pl ,k  0 k



i

vNbdi

 0 on

vNbdi  vNpli ,k

bd 0 ,

r pl ,k

i  pl ,k 1  pl , k pl ,k 1 pl r  v  i  Ni Pri pl ,k ( z )  H ( k ), 2 

ri pl ,k 1/ 2 pl , k hk vNi

 L2 (k ), 0  ri pl ,k  Nipl ,k , vNpli , k  0 on  0pl , k ,

, pl on bd , k  1,..., m, vNpli ,k 1  vNpli , k on  kpl, k 1 , i  1, 2,3, k  1,..., m  1} , k

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

On the Investigation of Dynamical Hierarchical Models...

25

Nipl ,k

H N ( )  {v N 

(vNi )i31 ; vNbdi

2

 L (

bd

), vNpli ,k

1  h r pl ,k  0 k



i

ri pl ,k 1/ 2 pl , k hk vNi pl ,k  where z

r pl ,k

i  pl , k 1  pl ,k pl , k r  v  i  Ni Pri pl ,k ( z ), 2 

 L2 (k ), 0  ri pl , k  Nipl , k , i  1, 2,3, k  1, 2,..., m} ,

x3  hk h  hk h  hk , hk  k , hk  k , k  1,..., m. hk 2 2

Since functions hk and hk ( k  1,..., m ) are Lipschitz continuous in k , then from Rademacher’s theorem [8] it follows that hk and hk are differentiable almost everywhere in k and  hk  L (k* ) , for all subdomains k* , k*  k ,   1, 2 , k  1,..., m . So, for any vector-function v Npl ,k  (vNpli ,k )3i 1  H1 () the ri pl ,k

ri pl ,k

1 * * * 1 corresponding functions v pl , k belong to H (k ) , for all k , k  k , i.e. v pl , k  H loc (k ) , Ni Ni 1 pl 0  ri pl ,k  Nipl , k , i  1, 2,3 . Moreover, the norms . H1 (kpl ) in the spaces H ( k ) define corresponding r pl ,k

norms

. k*

for

vector-functions

i  vNpl , k  (vNpli , k )

 pl ,k N1,2,3  N1pl , k  N 2pl , k  N 3pl , k  3 , such that vNpl , k

k*

 v Npl , k

from H1 (  kpl )

the

space

1 [ H loc (k )]

pl , k N1,2,3

,

, and applying properties of Legendre

polynomials we can obtain their explicit expressions  vNpl , k



2 k*

  pl , k 1       ri 2 i 1 ri pl ,k  0   3

Nipl ,k



ri pl ,k 1/ 2 pl , k hk vNi

2

2

 L2 (k )

Nipl ,k

s pl ,k

i  pl , k 1  ri pl ,k  sipl ,k )hk3/ 2 vNpli ,k  si   (1  (1) 2  s pl ,k  r pl ,k

2



i

i

Nipl ,k

 2

L (k ) s pl ,k

i  pl ,k 1  ri pl ,k  sipl ,k   hk )hk3/ 2 vNpli ,k   si   ( hk  (1) 2  s pl ,k  r pl ,k 1

   1

i

hk1/ 2 

i

ri pl ,k vNpli , k  (ri pl , k

 1)hk3/ 2  hk

ri pl ,k vNpli ,k

2

  , 2 L (k )  

where we assume that the sum with the upper limit less than the lower one equals to zero.  pl ,k  N pl ,k 1 Note, that for vector-function vNpl ,k  [ H loc (k )] 1,2,3 , which satisfies the conditon vN

k*

  , we can

define the trace on  k ,0 . Indeed, the corresponding vector-function v Npl ,k  (vNpli , k )3i 1 of three space variables belongs to H1 (kpl ) and hence the trace tr (vNpli ,k ) of vNpli , k on kpl belongs to H 1/ 2 (kpl ) (i  1, 2,3) . Therefore, for components

ri pl ,k vNpli , k

be defined as follows

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

 of vector-function vNpl , k the trace operator tr k ,0 on  k ,0 can

26

Gia Avalishvili, Mariam Avalishvili, David Gordeziani

tr k ,0

ri pl ,k (vNpli , k )

h pl , k Ni ) | pl

 tr (v



k ,0

Pr pl ,k ( z pl , k )dx3 , i

0  ri pl , k  N ipl , k , i  1, 2,3.

h

On the subspaces VN () and H N () from the original three-dimensional problem we obtain the following variational problem: find w N  C 0 ([0, T ]; VN ()), w N  C 0 ([0, T ]; H N ()) , which satisfies equation d dt

3

3



 wN i (.)vNi dx 

i 1 

 a

ijpq ( x) e pq ( w N (.)) eij ( v N ) dx



i , j , p ,q 1 

(8) 3



3

 f v

i Ni dx 

i 1 

  g tr

i 1 (vNi )d ,

v N  VN (),

i 1 1

in the sense of distributions on (0, T ) and the initial conditions w N (0)  ψ N ,

w N (0)   N ,

(9)

where  N  VN () , ψ N  H N () . Since the unknown vector-function w N of the problem (8), (9) in the multilayer part of multi-structure is of the following form Nipl ,k

wN 

( wNi )i31 ,

wNi 

wNpli , k

1  h r pl ,k  0 k



i

r pl ,k

i  pl ,k 1  pl , k   wNi Pr pl ,k ( z pl , k )  ri i 2 

in  kpl , i  1, 2,3, k  1,..., m ,

hence in the sub-structures kpl the vector-function w N is defined by functions of two space variables ri pl ,k wNpli ,k

. Therefore, problem (8), (9) is equivalent to the following problem defined on the union of three  dimensional and two-dimensional domains: find the unknown vector-function wN  C 0 ([0, T ];VN (bd , pl )) ,   wN  C 0 ([0, T ]; H N (bd , pl )) , which satisfies equation d      RN ( wN (.), vN )  AN ( wN (.), vN )  LN ( vN ), dt

  vN  VN (bd , pl ),

(10)

in sense of distributions on (0, T ) and the initial conditions     wN (0)  N , wN (0)   N ,     pl ,1  pl ,m where  N  VN (bd , pl )  {vN  ( vbd )  [ H 1 (bd )]3  N , vN ,..., vN

(11) m

[H

pl ,k

N1,2,3 1 ; loc (k )]

bd vbd N  (vNi ),

k 1

trbd (vNbdi )  0, v pl , k 0 N

trbd , pl k

 Nipl ,k 1   pl ,k hk  ri 0

Nipl ,k 1



r pl ,k

k*

r pl ,k

i i   , vNpl , k  (vNpli , k ), tr k ,0 (vNpli , k )  0, ri pl , k  0,..., N ipl , k , trbd , pl (vNbdi )  k

ri pl ,k   pl , k 1  pl ,k   vNi Pr pl ,k ( z pl , k )  , k  1, 2,..., m,  ri i  2   ri pl ,k 1

Nipl ,k



ri pl ,k  0

1 hk

ri pl ,k

 pl ,k 1  pl , k   vNi   ri 2 

pl , k 1 1  pl , k 1 1  pl , k 1   vNi (1)ri in k  k 1 , i  1, 2,3, k  1, 2,..., m  1} ,  ri h  2 r pl ,k 1  0 k 1



i

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

On the Investigation of Dynamical Hierarchical Models...

27

     pl ,k N1,2,3  N1pl , k  N 2pl , k  N 3pl , k  3 , k  1, 2,..., m ,  N  H N (bd , pl )  {vN  (vNbdi , vNpl ,1 , vNpl ,2 ,...,  vNpl , m )  [ L2 (bd )]3

m 2

[L ( )]

k N1,2,3

k

 ; vN

2  H N ( bd , pl )





(|| vNbdi

||2L2 ( bd )





ri pl ,k 1/ 2 pl ,k || hk vNi

k 1 ri pl ,k  0

i 1

k 1

Nipl ,k

m

3

||2L2 ( ) )  }, k

     N and  N correspond to  N  VN () and ψ N  H N () , respectively. The bilinear RN ( yN , vN ) ,    AN ( yN , vN ) and linear LN (vN ) forms are defined by corresponding forms R(.,.) , A(.,.) and L(.) , and applying properties of Legendre polynomials we obtain their explicit expressions m

3

  RN ( yN , vN ) 



bd

yNbdi vNbdi dx 

i 1 bd

ri pl ,k ri pl ,k pl ,k

where 

Nipl ,k

3

r pl ,k r pl ,k r pl ,k

  k 1

r pl ,k

i i i i  pl ,k 1   pl ,k 1  1    ri    pl ,k yNpli ,k vNpli ,k d k ,  ri 2 2  2  hk i 1 r pl ,k ,r pl ,k  0   i



i

k

hk





pl ,k

Pr pl ,k ( zk ) Pr pl ,k ( zk )dx3 , 0  ri pl , k , ri pl , k  Nipl , k , i  1, 2,3 , k  1, 2,..., m , the bilinear i i

hk

  form AN ( yN , vN ) is given by   AN ( yN , vN ) 

3 bd bd bd ijpq e pq ( y N ) eij ( v N ) dx 

 a

i , j , p , q 1 bd pl , k N max

m

 pl ,k 1  pl , k 1  1    r   r 2  2  hk2 k 1 r pl ,k ,r pl ,k  0  

 



k

r pl ,k r pl ,k r pl ,k r pl ,k   pl ,k k aijpq e pq ( yN ) eijk (vN )d k ,

3

 i , j , p ,q 1

pl ,k  max{N ipl , k } , where N max 1 i  3

r  eijk (vN )

r

1    i (vNplj , k )   j (vNpli ,k )  2  r

r

dirpl ,k  

r 1 1 i hk , displ , k   hk hk

for s  r , and

ri pl ,k vNpli , k



ri pl ,k yNpli ,k

r

pl , k Nmax

 (d

r pl , k is

s r

s s r  vNplj , k  d jspl , k vNpli , k )  , r  N  {0} ,  

1 1   r s   s  2  (i hk  (1) i hk )  h   k

 0 , for

Nipl ,k

 ri

pl ,k



pl ,k Nmax

,

1  r  s (i  1)(i  2) ,  s  2  (1  (1) ) 2  

r pl ,k r pl ,k pl ,k aijpq

hk



pl , k ijpq Pr pl ,k

a

( zk ) Pr pl ,k ( zk )dx3 ,

hk

 pl , k i, j, p, q  1, 2,3, 0  r pl , k , r pl , k  N max , k  1, 2,..., m . The linear form LN (vN ) is of the following form  LN (vN ) 

3



i 1 bd

m



  k 1

bd i 1 (vNi )d  

  g tr i 1 bd 1

ri pl ,k  ri pl ,k  pl , k  pl ,k 1   1 pl ,k  pl , k   vNi f i  gipl , k ,  k ,  gipl , k ,  k ,  (1) ri  ri  2   hk i 1 ri pl , k  0   k 3

N ipl ,k

3

f i vNbdi dx 



Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

 ri pl ,k ri pl , k  1  d k  vNpli ,k gipl ,k d  k ,1  ,   h  k ,1 k  



28

Gia Avalishvili, Mariam Avalishvili, David Gordeziani

where  k ,1 

k \  kbd , pl

1  (1hk )2

  k ,0 , k ,  

ri pl ,k f i pl , k

 ( 2 hk )2 ,

hk



f

pl ,k i

Pr pl ,k ( zk )dx3 , gipl ,k ,  ( x1 , x2 ) i

hk

 gi ( x1 , x2 , x3 ), for ( x1 , x2 , x3 )   k  1   and gipl , k , ( x1 , x2 )  0, for ( x1 , x2 , x3 )   k \ 1 ,

gipl , k ,  ( x1 , x2 )  gi ( x1 , x2 , x3 ), for ( x1 , x2 , x3 )   k  1   and gipl , k , ( x1 , x2 )  0, for hk

ri

pl ,k

( x1 , x2 , x3 )   k \ 1 , g pl ,k  i

g

pl , k Pr pl ,k i i

( zk )dx3 ,

hk

ri pl , k  0,..., Nipl , k , i  1, 2, 3, k  1, 2,..., m .

The obtained hierarchical models (10), (11) are pluridimensional models written in variational form of multistructure consisting of three-dimensional body with general shape and multilayer part composed of plates, and for corresponding initial-boundary value problems the following existence and uniqueness theorem is valid. Theorem 1. Let the three-dimensional part bd with general shape and plates kpl , k  1, 2,..., m , are such that the domain  occupied by multi-structure is a Lipschitz domain, the parameters characterizing bd  L ( bd ) , i, j, p, q  1, 2,3, satisfy symmetry and mechanical properties of three-dimensional part aijpq

positive definiteness conditions 3 bd bd aijpq ( x )  abd jipq ( x)  a pqij ( x ),



3 bd aijpq ( x ) ij  pq  cabd

i , j , p , q 1

 (

ij )

2

, x  bd ,  ij  R,  ij   ji , (12)

i , j 1

i, j, p, q  1,2,3 , cabd  const  0 , and  bd  L (bd ) ,  bd  c  const  0 , and coefficients r pl ,k r pl ,k pl , k aijpq / hk

pl , k  L (k ) , r pl , k , r pl ,k  0,..., N max , k  1, 2,..., m , defining two-dimensional models of plates

satisfy the following conditions r pl ,k r pl ,k pl ,k aijpq



r pl ,k r pl ,k pl , k a jipq

pl ,k N max

3







r pl ,k r pl ,k pl , k aijqp



pl , k pl , k r pl ,k r pl ,k r r pl , k aijpq ,  pl , k

r pl ,k r pl ,k pl , k

 

r pl ,k r pl ,k

pl ,k , i, j , p, q  1, 2,3, r pl , k , r pl ,k  0,..., N max , N pl ,k

3

max 1  pl , k 1   pl ,k 1  pl ,k r r  pl , k 1  r r    r   aijpq  ij  pq      ij  ij , r r h  2  2 2  r pl ,k ,r pl ,k  0 i , j , p , q 1 k r pl ,k  0 i , j 1

r pl ,k

pl , k

pl , k

pl , k

pl ,k



r pl ,k

in k ,

r pl ,k r pl ,k

r pl ,k

for all  ij  R ,  ij   ji , i, j, p, q  1,2,3 ,   const  0 ,  pl , k / hk  L (k ) , r pl ,k r pl ,k pl ,k



/ hk  c  const  0 , r

pl ,k

    bd , pl pl , k ),  N  H N (bd , pl ) , r pl , k  0,..., N max , k  1, 2,..., m . If  N  VN (

and given functions f i , gi ,

ri pl ,k fi pl ,k ,

gipl , k ,

and

ri pl ,k gipl , k

are such that

fi  L2 (0, T ; L2 (bd )) , gi , gi  L2 (0, T ; L4 / 3 (1bd )) , i  1, 2,3 , hk1/ 2

ri pl ,k fi pl ,k

 L2 (0, T ; L2 (k )), k3/,4 g ipl , k ,  , k3/, 4 ( g ipl ,k ,  )  L2 (0, T ; L4 / 3 (k )),

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

On the Investigation of Dynamical Hierarchical Models...

hk1/ 4

29

ri pl ,k ri pl ,k pl , k 1/ 4 g i , hk ( g ipl ,k )  L2 (0, T ; L4 / 3 ( k ,1 )), ri pl , k

 0,..., N ipl , k , i  1, 2,3, k  1,..., m,

then initial-boundary value problem (10), (11) defined on the union of three-dimensional and two-dimensional domains possesses a unique solution. For the considered multi-structure consisting of three-dimensional part with general shape and multilayer substructure composed of plates with variable thickness the relationship between the original three-dimensional model and the constructed hierarchy of pluridimensional ones is investigated. In order to formulate the corresponding theorem let us define the following anisotropic weighted Sobolev spaces s H 0,0, (kpl )  {v; hkp  3p v  L2 (kpl ), p  0,1,..., s}, h

s  N,

k

s H1,1, ( kpl )  {v;  3p 1 v  H1 ( kpl ), p  1,..., s,  hk  3r v  L2 (  kpl ),   1, 2, r  1, s, min{2, s}}, s  N, h k

where k  1, 2,..., m, which are Hilbert spaces with respect to the following norms v

s H 0,0, ( kpl )  hk

v

s H1,1, (  kpl ) hk

  hk  3s v

2 L2 ( kpl )

 s   p 1 v  p 1 3 





1/ 2

  pl 2 L (k )   2

2

2 1

H

( kpl )

2

  hk  3s v

 s  h p p v  p 0 k 3 

L2 ( kpl )



   h  v     k 3

1

  hk  3min{2, s} v

,

2 2

L

( kpl )

2 L2 ( kpl )

  hk  3 v

2 L2 ( kpl )

  hk  3min{2, s} v

2

 1/ 2

  L2 (  kpl )  

.

The following theorem is valid. Theorem 2. Let the parts bd and kpl , k  1, 2,..., m , of the multi-structure are such that  is a Lipschitz domain, the parameters characterizing mechanical properties of the general three-dimensional bd pl , k  L ( bd ) , aijpq  L ( kpl ) , i , j , p, q  1, 2,3, satisfy symmetry and positive and multilayer parts aijpq

definiteness conditions (12) and 3 pl , k ,k pl ,k aijpq ( x)  a pl jipq ( x)  a pqij ( x),



3 pl , k aijpq ( x) ij  pq  capl , k

i , j , p ,q 1

 pl , k  c  const  0 ,

ij )

2

, x   kpl ,  ij  R ,  ij   ji ,

i , j 1

capl , k  const  0 ,

i , j , p, q  1, 2,3 ,

 (

 bd  L (bd ) ,

 pl , k  L (kpl ) ,

 bd  c  const  0 ,

f  L2 (0, T ; L2 ()), g, g  L2 (0, T ; L4 / 3 (1 )),   V () ,   bd , pl ) ψ  L2 () . If vector-functions  N  VN () and ψ N  H N (), which correspond to  N  VN (   and  N  H N (bd , pl ) tend to  and ψ in the spaces H1 () and L2 () , respectively,, as N min 

min

k  1,..., m ,

and

{Nipl , k }   or hmax  max{hk ( x1 , x2 );( x1 , x2 )   k , k  1,..., m}  0 , then the sequence

1k m, 1i 3

of vector-functions w N (t )  ( wNi (t ))i31  C 0 ([0, T ]; VN ()) restored from the solutions   wN  C 0 ([0, T ];VN (bd , pl )) of the constructed problems (10), (11) defined on the union of three-dimensional and two-dimensional domains converges to the solution u(t ) of the three-dimensional problem (6), (7),

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

30

Gia Avalishvili, Mariam Avalishvili, David Gordeziani

w N (t )  u(t )

in V (),

w N (t )  u(t )

in L2 (),

t  [0, T ], as Nmin   or hmax  0 .

1,1, s Moreover, if u satisfies additional regularity conditions d r (u | pl ) / dt r  L2 (0, T ; H h  r ( kpl )) , r  0,1 , k

2

2

2

d (u | pl ) / dt  L (0, T ; k

s0  2 , ψ

s2 H 0,0, hk

| pl  H1,1,1 ( kpl ) , hk k

(  kpl ))

k

s0 , s0 , s1 , s2  N , s0  2 , s1  2 ,  |kpl  H1,1, ( kpl ) , s0  N , h k

  k  1, 2,..., m, then for suitable initial conditions  N and  N of the initial-

boundary value problems corresponding to the hierarchical models the following estimate is valid 2s

u(t )  w N (t )

2 L2 (  )

 u(t )  w N (t )

2 H1 ( )

h    max   (T , ,  0 , N), t  [0, T ],  N min 

where s  min{s2 , s1  1, s0  1, s0  1} and  (T , ,  0 , N)  0 , as Nmin   . Acknowledgement. The present research is supported by State Science Grant in applied research of Shota Rustaveli National Science Foundation (Contract No. 30/28).

meqanika

samganzomilebiani sxeulisa da fenovani qvestruqturisagan Semdgari drekadi multistruqturebis dinamikuri ierarqiuli modelebis gamokvlevis Sesaxeb g. avaliSvili*, m. avaliSvili**, d. gordeziani* * i. javaxiSvilis sax. Tbilisis saxelmwifo universiteti, zust da sabunebismetyvelo mecnierebaTa fakulteti, Tbilisi **saqarTvelos universiteti, informatikis, inJineriisa da maTematikis skola, Tbilisi (warmodgenilia akademiis wevris g. gabriCiZis mier)

naSromSi ganxilulia sawyis-sasazRvro amocana multistruqturisaTvis, romelic Sedgeba zogadi formis mqone samganzomilebiani sxeulisa da cvalebadi sisqis firfitebisagan Sedgenili mravalfeniani nawilisagan. agebulia samganzomilebiani da organzomilebiani areebis erTobliobaze gansazRvrul dinamikur modelTa ierarqia. agebuli ierarqiuli modelebis Sesabamisi pluriganzomilebiani sawyis-sasazRvro amocanebi gamokvleulia saTanado funqcionalur sivrceebSi. damtkicebulia agebuli samganzomilebiani da organzomilebiani areebis erTobliobaze gansazRvruli sawyis-

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014

On the Investigation of Dynamical Hierarchical Models...

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sasazRvro amocanebis amonaxsnebidan aRdgenili sami sivrciTi cvladis veqtor-funqciebis mimdevrobis krebadoba sawyisi samganzomilebiani amocanis amonaxsnisaken da damatebiT regularobis pirobebSi Sefasebulia krebadobis rigi.

REFERENCES 1. P.G. Ciarlet, H. Le Dret, R. Nzengwa (1989) J. Math. Pures Appl., 68: 261-295. 2. H. Le Dret (1991) Problèmes Variationnels dans les Multi-Domaines: Modélisation des Jonctions et Applications, Masson, Paris (French). 3. P.G. Ciarlet (1990) Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson, Paris. 4. I.N. Vekua (1955) Trudi Tbil. Matem. Inst., 21: 191-259 (Russian). 5. G . Avalishvili, M. Avalishvili, D. Gordeziani, B. Miara (2010) Anal. Appl., 8: 125-159. 6. M. Dauge, E. Faou, Z. Yosibash (2004) Encyclopedia of Computational Mechanics, 1: 199-236. 7. G.V. Jaiani (2001) Z. Angew. Math. Mech., 81, 3: 147-173. 8. H. Whitney (1957) Geometric integration theory: Princeton University Press, Princeton.

Received July, 2014

Bull. Georg. Natl. Acad. Sci., vol. 8, no. 3, 2014