Một số vấn đề ổn định và động lực học của môi trường đàn dẻo và composite

Tên đề tài Một số vấn để về ổn định và động lực học của môi trường đàn - dẻo và composite.. Mục tiêu và nội dung nghiên cứu Vấn đề ổn đinh và động lực học của các môi trường đàn hồi, dẻ

ĐẠI HOC QUỐC GIA HÀ NÒITRƯỜNG ĐẠI HỌC KHOA HỌC Tự NHIÊN

BÁO CÁO KẾT QUẢ THỰC HIỆN ĐỂ t à i h a i n ă m

2000 - 2001

1 Tên đề tài

Một số vấn để về ổn định và động lực học của

môi trường đàn - dẻo và composite.

(Some stability and dynamic problems in the eỉastoplastic

and composite media)

4 Mục tiêu và nội dung nghiên cứu

Vấn đề ổn đinh và động lực học của các môi trường đàn hồi, dẻo,

composite là một trong những lĩnh vực được nhiều nhà cơ học quan tâm, một

mặt do ý nghĩa khoa học, mặt khác do vai trò ứng dụng của chúng Do vậy đề

tài nghiên cứu các vấn đề sau đây:

• Ổn định đàn dẻo của bản dưới tác dụng của lực trượt có tính đến dạng

1

b) Hệ các phương trình ổn định của vỏ trụ đàn dẻo, vật liệu nén được đã được xây dựng, ở đây nghiên cứu phương pháp giải bài toán và tìm cách xác định lực tới hạn của kết cấu chịu quá trình tải phức tạp Các kết quả thu được

mô tả ảnh hưởng của tính nén được của vật liệu đến sự ổn định của vỏ trụ Đã giải một số bài toán cụ thể và lập trình tính toán bằng số Các kết quả phù hợp với ý nghĩa cơ học của kết cấu.

c) Sử dụng lý thuyết quá trình đàn dẻo và phương pháp biến thể nghiệm đàn hồi để khảo sát bài toán ổn định ngoài giới hạn đàn hồi của bản dưới tác dụng của lực trượt có tính đến dạng vổng thực của nó sau khi mất ổn định Đã nhận được biểu thức xác định lực tới hạn Đã thực hiện các tính toán bằng số qua đó khẳng định sự hội tụ của phương pháp.

d) Giải bài toán truyền sóng trong môi trường đàn hồi không nén được với biến dạng ban đầu trong trường hợp xấp xỉ sóng dài Đã sử dụng phương pháp thuần nhất hóa của phương trình đạo hàm riêng Đã tìm được công thức vận tốc sóng.

Các kết quả nghiên cứu thể hiện trên các công bố sau:

1) DAO HUY BICH: Modified elastic solution method in solving elastoplastic problems of structures components subjected to complex loading (Viet nam Journal of Mechanics, NCST of Vietnam Vol 22,2000, N°3 (133 - 148).

2) DAO VAN DUNG: Method of solution for stability problem of elastoplastic cylindrical shell with compressible material subjected to complex loading processes (Part I) (Báo cáo hội nghị khoa học, Khoa Toán-Cơ-Tin học, Trường Đại học Khoa học Tự nhiên, Đại học Quốc gia Hà Nội 23 - 24/11/2000).

3) PHAM CHI VINH: An application of homogenization method to the problem on wave propagations in a composite layer (Nhatrang 2000 international colloquium).

4) DAO HUY BICH : On the elastoplastic stability of a plate under shear forces, taking into account its real bending form (Vietnam Journal of Mechanics, NCST of Vietnam, Vol 23, 2001, N°1 (6 - 16).

5) DAO VAN DUNG: Solving mothod for stability problem of elastoplastic cylindrical shells with compressible material subjected to complex loading processes (Vietnam Journal of Mechanics, NCST of Vol 23,

6 Tình hình kinh phí

+ Chi cho hoạt động xẽmina 20 buổi thuyết trình

(mỗi buổi 50.000đ)

+ Chạy chương trình trên máy tính

+ In ấn tài liệu, đánh máy

1.000.000đ 1.500.000đ 1.000.000đ + Chi 5 báo cáo khoa học (mỗi báo cáo 300.000đ) 1.500.000đ

+ Bồi dưỡng chuyên môn

+ Các bài báo khoa học

+ Hội nghị và các chi phí khác

2.000.000đ 2.000.000đ 5.000.000đ

7 Nhận xét và đánh giá kết quả thực hiện đề tài

• Đã hoàn thành tốt các mục tiêu nghiên cứu, các mức dự kiến, vượt chỉ tiêu về số lượng bài báo và báo cáo khoa học.

• Các vấn đề nghiên cứu có tính thời sự và có khả năng ứng dụng trong thực tiễn.

• Sinh hoạt học thuật đều đặn thông qua các xêmina khoa học góp

phần nâng cao chuyên môn cũng như đào tạo cao học, NCS Góp

phần phát triển ngành cơ học Trường Đại học Khoa học Tự nhiên, Đại học Quốc gia Hà Nội.

PGS.TS Đào Văn Dung

HuLt Co >\£ ị

XÁC NHẬN CỦA NHÀ TRƯỜNG

SOME STABILITY AND DYNAMIC PROBLEMS IN THE ELASTOPLASTIC

AND COMPOSITE MEDIA DAO VAN DUNG, DAO HUY BICH, PHAM CHI VINH

In this project, our staff have studied the following topics

1 Modified elastic solution method in solving elastoplastic problems

of structures components subjected to complex loading

2 Solving method for stability problem of elastoplastic cylindrical shells with compressible material subjected to complex loading prosesses.

3 On the eỉastoplastic stability of a plate under shear forces, taking into account its real bending form.

4 An application of homogenization method to the problem on wave propagations in a composite layer.

Mechanics and 2 research reports presented in the national and International conferences.

V ie tn a m J o u rn a l of M echanics, N C ST of V ietn am Vol 2 2 , 2000, No 3 (133 - 148)

M ODIFIED ELASTIC SOLUTION M ETHOD IN SOLVING ELASTOPLASTIC PROBLEMS OF

STRUCTURE COMPONENTS SUBJECTED

TO COMPLEX LOADING

D a o H u y B ic h

Vi et nam National University, Hanoi

SU M M A R Y Modified elastic solution method in the eiastoplastic process theory has been proposed by the author [2j and was applied in solving some 2D and 3D elastoplastic problems of structure components subjected to complex loading The method makes use

of an algorithm in which a step is made in the loading process and iterations are carried out on this step The performance of the method was fulfilled and the convergence of the method was considered numerically In this paper the other performance of this method is presented and the convergence of the method is proven theoretically in the general case of

a hardening body which obeys the elastoplastic process theory The more complicated 3D problem of bodies of revolution subjected to non-axially symmetric load is investigated.

1 B o u n d a ry v alu e p ro b le m of th e e la s to p la s tic p ro cess th e o ry a n d

m o d ifie d e la s tic so lu tio n m e th o d

T h e form ulation of the boundary value problem of the ela sto p la stic process

th eory and analysis o f the existence and uniqueness theorem s have b een carried out in [3, 4|.

Let K t ( x , t) and be external volum e and surface forces th a t act on

th e b o d y and let p i ( x , t ) be displacement on th e b o d y ’s surface It is necessary to find d isp lacem ents u t ( x ,i) , strain tensor and stress tensor Ơt j ( x i ) , where

t - th e loading param eter, th a t satisfy the following equations

and the boundary conditions

and for the process theory with average curvature

\ _ _ ( n ) ( 0 ) , V - " ' A _ ( m ) _ ( n — ! ) , A ( n )

msl

j r , ( x , í „ ) = ft-,*"1 F , ( z , t „ ) = FỈn\ <ữ,(x,t„) = v ỉ

n>-A t each ste p n = 1 , 2 , , N of the change in the above - m en tioned q uantities,

from (1.1) - (1-6) and talking into account (1.9) we set up the follow ing sy stem of

eq u a tio n s

135

x e n ,

(1.12)

where { Htj k t ^ n^ is an average quantity of Ht j ki in the interval ( i n_ i , i n ) which

In approximation we take {H xjià) lj ■ the system of eq u ation s ( 1 1 2 ) car be considered as a system of equations for a certain inhom ogeneous an isotrop ic elastic body with additional volume and surface forces This sy stem of eq u ation s

is solved step by step, beginning from the first step n = 1 At the n -th step,

E*"- 1 ', are known functions, which have been determ ined at the (n - 1 )-th step, the problem leads to determine and At each step in the loading the problem generally is nonlinear, so we w ill solve it by using an iterative method - a modified elastic solution m ethod |2, 3] - w h ich is analogous to the elastic solution method in the deformation theory of p la sticity

i 1, 5, 8 , Non-linearity of the problem is expressed in the co n stitu tiv e eq u a tio n s, i.e the third relation of (1.12) The procedure of the m odified elastic so lu tio n method on this relation is written as follows

where k 1 , 2 , IS the number of iteration on the n-th step of the ch ange in the

loading parameter In the result at n-th step and k-th iteration, we can w rite th e

system of equations in the form

can be taken as ị { H ị j k i l) +

Hịịkt)-(n,k) = ( n —1)

+ p K \ n) = 0 , i e n ,

(1.14)

T h e sy ste m o f equation (1.14) and the boundary conditions (1.15) represent

a b ou ndary value problem for an elastic body w ith the sam e ela sticity constants

Exjkt 35 th e initial bod y but w ith changed volum e and surface forces.

A fter th e sy stem o f equations has been solved, i.e Au-"^ known, th e d isplace­

m ent is represented as = u ị n ^ + A u -n^ The strains axe determ ined from th e C auchy equations, these strains are then su b stitu ted into the co n stitu tiv e

eq u ation s (1 3 ), from where are obtained.

2 O n th e convergence of th e m odified elastic s o lu tio n m e th o d

T h e m odified elastic solution m ethod was applied in considering stress and strain sta tes o f som e 2D and 3D bodies subjected to com plex loading [3, 9, 10, 11] From obtain ed num erical results, we can talk about the convergence o f the

m eth o d Generally, results of the third and fourth iterations are already closer to each other; th ey diSer from each other with sm all errors.

N ow w e introduce the proof of the convergence of the m ethod theoretically.

For th is aim we bring into use the functional Hilbert space H ( n ) w ith the

norm

n Let A v be any sm ooth vector function such that

I ( £ , , « - =

n

Analogously (3], we can show that the expression on the left hand sid e is a linear and continuous functional on t f ( n ) It follows from R iesz’s th eorem th a t there

exists an operator A : H{ n ) - H ' ( n ), where B ‘ {n) is the dual fu n ctio n a l space

of H { n ) , such that

(Au , v ) f ỉ = Ị (Eijki ~ Hi i ki ) A£i j { u) AEi j [ v ) dr i

n

Let known function ữ\" £ L 2 and Ki E Lp (p > 6 / 5), Ft € Lq {q > 4 / 3 ) , then

the expression at the right hand side of (2.2) is also a linear contin uous fu nctional

on H { n ), and there exists an operator L : E —*• H ' such th at

In the case, when iterations are carried out, we pu t Hi j ki = Using

the inequality (1.11) we can prove that, the operator equation (2.3) has a unique solution (similarly [3, 7]).

Now consider the convergence of the above m entioned m eth od B ecause of the scalar product in # ( n )

th e eq u ation (2.3) is rew ritten in another form

U sing th e inequality (1.11) into the last equation, it follows

n

B y p u ttin g u = the obtained result reduces to

< ( f 2GAeiy( u ^ _ u^ - ') ) A tl](uW - u ^ ) d n + 9 K I Ac2(u(k) - u ^ ) d n

X ( f 2GAemn(u(fc+1> - u ^ ')A emri(ti^+^ - u (fc)) ^

Since max(u*2) < 1, the operator Q is compressible, from (2.7) one can lead the

is equivalent to 4>'{s) > 0, i.e the material must be hardening.

3 P e rfo rm a n c e of th e modified elastic so lu tio n m e th o d for th e p ro b ­ lem in c u rv ilin e ar coordinates

On curvilinear coordinates the system of equations (1.14) is of the form

or

convergence of the iterative method The condition m a x u 2 = m ax ( 1 — < 1

(3.1)

(3.2)

and boundary conditions

(3.3)

w here

E i’ kt = \ g ' ’ g kt + G ( i ‘ V * + s ’ V ‘ ) ,

V j d en otes th e covariant derivative w ith respect to X1;

g tJ - m etric tensor o f curvilinear coordinate.

For in vestigation o f th e body of revolution subjected to com plex loading, we usually consider th e problem in a cylindrical coordinate

{r,<p,z) : g 11 = 1 , g22 = 4 , g33 = 1 , glj = 0 (t Ỷ j).

r

D en o te A u r = A u , Au^, = A ư, A u r = A w, the strain increment com ponents are

d eterm ined by Cauchy equations (3.2):

-R*, J2‘ , 7?^, have similar forms to (3.6), where

a't] = 2G w 1(A e 1-y - + 2 G(u >2 - m ) 5ijSfc*A g - ■ { i j = r, <p, z) (3.7)

By com bining with boundary' conditions, we have the solving equation s C onse­ quently, at each iteration it is necessary to solve a problem of linear e la sticity w ith new supplem entary volume and surface forces.

4 P a r tic u la r case N um erical exam ple

A short cylinder of radius R and length L rotates w ith angle v e lo city u ( t ) and is subjected to axesymmetrically with respect to axis z tan gen tial and norm al

Crz and radial displacement - Ru axial displacem ent - Rw.

T he system of solving equations (3.5) becomes

£ * ( n,k 1 )^ £•(»>* 1) have g^niiaj forms to (4.2) where

1 — ỉ/ d A u j ( n'k) ư ( d ù í u(n,ki Au^n,;cK Fj"* ơ z i n,lz~ L^

1 - 2i/ d ị + 1 - 2 iA dj, 1 "ạ Ì ~ ~2G 2G 4 2G ’

w here Ft (i = 1 , 2 ) - norm al and tangential forces acting on cylinder surface / 7 = 1 ;

F, (i = 3 — 6) - forces acting on butt-ends of cylinder; ^ - known values of

stress co m p o n en ts at preceding step (n — 1-th step); ơ ‘ì j n,k ^ - also known values

at considering step (n -th step) but on preceding iteration (k - 1-th iteration).

Follow ing [6Ị in order to solve the hom ogeneous system of equations (4.1) we

express surface forces F [ n\ F ị n^ and

we express them into series

^ { gi 0 } + 5 Z G! i ) j o(AJp ) | sin fcif,

ỹ = L E c! ? Ji(aj/,) cosìc‘í ’

( 4 .

i = l j = l where

J , (A,■)=<>,

J t- - B essel fu n ctio n of i-degree.

A particu lar solu tion with respect to these forces has the form

T he general solution (4.7) contains undetermined constants, w hich can sa tis _

boundary conditions on p = 1, taking into account expansion (4.6) w ith k x -r 0

But with Jfc, = 0 it must be to seek a part of the solution of hom ogeneous (4.1) m elem entary functions

Au^s ^ = Ai

P-Hence, the general solution of the system (4.1) as follows

Ail> = &W{T) -p Au>(5) + Au;(n) + A u;(K ), 12J

Au = A + Au<s) + Ati<n) + A u (ỉf), '

■which contains 4 arbitrary constants C i , C 2 , A \ , A 2 - They are sufficient to satisfy

exactly boundary conditions at p = 1 and integrally boundary conditions at b u tt-

ends of the cylinder.

Numerical calculation was carried out for cylinder made of steel 15X 18H 12C 4T IO

where 7 2 (i), Pz {t), T4 (i) may depend arbitrary on a param eter t It m ean s th at

the loading process may be complicated.

Calculations in solving the problem have been fulfilled on PC w ith P A S C A L

programme |9]- Subdivide loading parameter t into steps, increasing from 0 to 40

and solve th e problem step by step At each step 4 iterations w ere carried out From the results it can be seen that

a) T he error between two successive approximations decreases w hen th e num ber

of iterations increases, i.e the condition (2.7) is satisfied It is show n th a t th e

m odified elastic solution method can be applied to this problem and its convergence Las been proved.

b) When the cylinder IS subjected to th6 sains loading process, if w e su b d iv id e the loading process into smaller steps, the error between two iteration s of all q u a n tities

is also sm aller, i.e the error decreases.

c) With the same value of load, the plastic deformation region in the cylinder appears differently depending on the character of the loading process w hich reaches

th a t value: th e loading process is more com plicated, so the plastic region is more enlarging U nder com plex loading the body works more weakly.

d) E sta b lish ed calcu lation s may give a picture o f elasto-plastic sta tes of th e cylinder under axesym m etrical loads Further we can consider elasto-p lastic problem s of

th e cylind er under non-axesym m etrical loads by th e above m entioned m eth od

5 C o n c lu sio n s

a) A noth er perform ance of the modified elastic solution m ethod in theory of ela sto p la stic process is presented.

b) T h e convergence o f this m ethod is proved theoretically in th e general case

of a hardening b od y w ith a supplementary assum ption in approxim ation.

c) T he app lication o f the m ethod to the more com plicated 3D problem of bod ies o f revolution is considered.

T h is paper is com pleted w ith financial support from the N ation al B asic Re­ search P rogram in N atural Sciences.

REFERENCES

1 Birger I A Som e general methods for solving problems in p la sticity theory.

J A ppl M ath & Mech V ol 15, No 6, 1951, 765-770.

2 D ao Huy Bich T he modified m ethod of elastic solution in the elastoplas- tic deform ation process theory Journal of Sciences, Hanoi U niversity, series

M ath, and P hys No 2, 1985, pp 1-6, in V ietnam ese.

3 D ao Huy B ich Research on boundary value problem of the local theory of ela sto p la stic deform ation processes Dr Sc T hesis T he M oscow State

U niversity 1988, in Russian.

4 D ao Huy Bich A boundary value problem of the elasto-p lastic deform ation theory: E xisten ce and uniqueness theorem s J of A ustralian M ath Society, Vol 35 part 4, 1994, pp 506-524.

5 Ilyushin A A Plasticity Gostechizdat, M oscow 1998, in R ussian.

6 Lurie A I Space problems of elasticity theory G ostechizdat, M oscow 1965,

in R ussian.

7 P ob ed rya B E Num erical m ethods in the theory of ela sticity and plasticity

P u b l H ouse of M oscow State University 1995, in R ussian.

3 Vorovich I I., Krasovsky Yu p On the m eth od of elastic solu tion s Dokl.

147

Acad Nauk SSSR, Vol 126, No 4, 1959, pp 740-743, in Russian.

9 Vu Khar Bay Elastoplastic state of a short cylinder under complexAloadmg

J o f M echanics, Vol 17, No 1, 1995, pp 1-8, in V ietnam ese.

10 Vu Khac Bay Investigation of elastoplastic states of som e structures su b jected

to com plex loading by the modified elastic solution m ethod Ph D T h esis, Hanoi 1996 in V ietnam ese.

11 Vu Khac Bay, D ao Huy Bich Investigation of e l a s t o - p l a s t i c state o f cy lin ­ drical] shell subjected to complex loading by the m odified m ethod of elastic solution Proceeding of th e NCST of Vietnam, Vol 8, No 9, 1996, pp 3-14.

Hội nghị khoa học trường Đại học Khoa học Tự nhiên - Đại học Quốc gia Hà Nội

A D S T R A C T T h e syst em of stability equations of elasto plastic cylinfitic.ll shrll m.vlr

of compressi bl e material was established in work Ị3 1 Ill the present paprr, VVI- st udy the solution and met hods for determining critic.il load of structures snbjectrrl to rnmpl ex loading processes T h e results obtained describe the influence of the comprrssiliilily of

Assume that a material is compressible and a stress state in the plructurp is

<1 mrmbrĩine plane stress state We consider the shell being acted by the rxternal

forcos P n , p 1 2 , P 22 which depend arbitrarily on some loading parameter t One of

the main aims of the stability problem is to find the moment when the instability

of structure happens and respectively the critical loads í 4, p't] = Suppose that the unloading does not happen in the structure We use f.he r-rilrrion of bifurcation of equilibrium state to study the proposed problem.

An analysis of the elastoplastic stability problem is always made ill two parts: pre-buckling process and post-buckling process.

1.1 P r e -b tick lin g p rocoss

Suppose that at any moment t there exists a membrane plane stress slntr in

thp shell

St) that

ơị 1 f- Ơ22 Pi 1 t- P22

Ơ — -

ơ« - y ° n - ƠMƠ22 I- ƠJ2 + 3ơ?2 = \ / p Ĩ i — P11P22 + P22 + 3P|2 •

The components of deformation velocity tensor determined according to thf* theory

of elaatoplastic processes Ịl| are of the form

TIic coefficients Dij in (1.6) are calculated as follows ị3|

In order to solve the stability problem of shell, we suppose that the kinematic boundary condition is simply supported at the planes Ij = 0 and I | = Ỉ J

count the existence of non-trivial solution i.e Amn r 0, we receive the expression

Tor defining critical loads

From here, we get

■2 = + <*Ĩ0:' + Q3/72 4- ữẠ0 + q8) ( j M * + / M 3 + M 2 + 1 1 1 )- 0 5)

: ( p n / ? 2 + 2 p i 2/? + P2 2) (/31/?4 4- 0 2 0 3 + 03024- 0 4 0 + 0 s ) ỷ - N 0 A

( 2 . 6 )

[Jccause of the complication of loading process passing the loading parameter t ,

the quantities in the expression (2.6) are the functions of t and 5 Besides, the arc- IciiRth of the strain trajectory 5 from (1.3) is the function of f, too So, wc have

to solve simultaneously the equations (1.3) and (2.6) by the loading parameter method |l | After determining the critical value L , we can find the critical loads

as follows

p 11 = / M i ( í * ) i p 12 = Pi 2 (i *)» P22 = P22 (^*) ■

Hereafter, we will solve some concrete problems by applying the above presented method and the general formula (2.6).

3 Some concrete problem s

3 1 C y lin d r ic a l sh e ll su b jected to com p ression a lo n g th e g e n e r a tr ix

In this case, we get

Ơ11 = - p , <712 = Ơ22 = Ot ƠU = k i l l ,

Substituting this value into (3.3), we get

2 4yv2 (a >ơ + “ 3 + - f i

t = p2 í m + 0 3 + £ IJy taking into account (3.2), from this expression, we obtain

, AN2 ( ( 1 zậ' ệ' \ _ / 1 l ộ ' \ _ 1 i>' ' ~ ~pĩ~ ỉ ( ĩ c + I N C + 9K c ) + V1 + c ~ 9K c ) + c + 9 / 7 c

c + 9 K C

N N

V ~ỹ + 9K y

(3.5)

* If the material is incompressible i.e K —* -foo, c = 1, 0 = — - then

v 1 + 3 £

(3.-1) and (3.5) return to the results in Ị1, 4, 5|.

* The obtained result in (3.4) coincides with the one given in [3|.

3 2 C y lin d r ic a l sh e ll su b jected to e x te rn a l p ressu re

The pre-buckling process is of the form

1 Theory of elastoplastic processes can be applied to the stability problem

of cylindrical shell when both pre-buckling and post-buckling processes are com­ plicated.

2 A compressibility of material has influence on the stability of structure.

3 Applying loading parameter method, the proposed method and form of solution give a way to solve efficiently series stability problems of shell.

REFERENCES

1 Dao IIuy I3ich Theory of elastoplastic processes Vietnam National Univer­ sity Publishing House, Hanoi 1999 (in Vietnamese).

2 Volmir A s stability of deformable systems Moscow 1963 (in Russian).

3 Dao Van Dung Stability of cylindrical shells with conpressible material in the theory of elastoplastic processes Proceedings of the Fifth National Con­ ference on Solid Mechanics, Hanoi, November 29-^30, 1996, pp 152-159 (in Vietnamese).

A Dao Van Dung Stability problem outside elastic limit according to the the­

ory of elastoplastic deformation processes Ph D Thesis, Hanoi 1993 (in Vietnamese).

r> Dao Van Dung Stability of thin shell subjected to complex loading .Journal

of Mechanics, Vol 1 0 , No I, 1988, pp 8-16 (in Vietnamese).

PHƯƠNG PHÁP GIẢI DÀI TOÁN Ổn ĐỊNH CỦA v ỏ TRỰ ĐÀN DẺO

VỞI V Â T LIÊU NF,N Đ ư ơ c CHỊU QUÁ TRÌNH TẢI PHỨC T A P (Phần 1)

Bài này nhầm nghicn cửu lời giải và các phương pháp dể xác (linh tải tới liạn cùa kct cấu chịu quá trinh tài phức tạp Các kết quả thu (lươc clio tliiíy ành luicVng cùa tínli nén dược cùa vật liệu lên sự ổn dịnh cùa vổ trụ Khi vậl liệu là kliông nén dược, ta nhân dược các kết quả đã có trước đây.

Vietnam National University

Facility of Mathematics, Mechanics and Informatics

.134 Nguyen Trai Street Hanoi, Vietnam

9

Nliỉi ĩ mnỊĩ 2001) Inlcmnlioiiĩil Colloquium - Colloquc hilcnmlion;ii NI ki 1'iann 2<KH>

AIM A P P L IC A T IO N O F M O M O CICNIZ/VriO N M PÍTIIO I)

(■( IIIIỊ )| )^il (• l;i\ci I >f ini'i )UI| III ssil ill' cliishe m;il criiils tt'il 11 mil i;il ill'll >1 '11 i;i t ii HIS III I In1

c;is<- III l( Mil’ w ; i \ c l c i i ^ h l ;i|)|>i " s i l i i i i l i o n Ill n i i l n I n i nv est i u,: 111' ||||' I Ml »1 »1«-111 \ v r M-'-

Si >1 ii n| M mi l Ị 7 1 ( I !)!)ri ) S o l i n 11" " il l IS HIM I Si I nil >I( 11)(>11 If IS ị I ()| I I !)!'.! I I I I : 111 • t M

v e n i r 11 |( >1 ( 1 !)!)(i) R n u / I N< >11 :11111 S:m< lifi >1 < I |!)| ( I ( )”,! let! : 11111 l ' >1 I I ■]I >111 1 -I

( I <J<)7 ) I n Ị 11 17 1 I I ( ) | ' VI'1'!' '.I III 1 i< 11 11II- \ v; 1 ■ I <11 <1 >;i I i o n s i l l ; m I -1:1 ■ t Ir- >!' ! M>M.

< )|isisi I I I " " I I 'VI > l i i i l l |>;n IN I ; i \'< 1 ()\ i t I y Í i ! ” ;i 1 1 ; 1 1 1 > 1 ;i I ;i \ I ! I il; I ' l l I I I - I

I \V( > 11; I 11 I I ; 11 I n s I >1 I I i v r l V 'V III I f I I i r I i;i I 'I 'I |!)| I lev I >1 I ' I I 1(1 I I II- w : i \ I - I II I I| »;t>- ' M I’

N k i Tf;inn'2(HW lnicrn;ilional Colloquium - Collogue InicniniioiiiiL Nha Tr.m[j 2<H)(I

T i n - |>;i|HT will ill-ill with rliis |)I(>I)I(111 Ill o n l i T rn solve this Ii-iii \ \v II.M' I Ilf l lmnt>nni iz; itini i mcrl ind (sec Ị l Ị ) lor Ị>iirtiiil il iHcrruriii l ccjiiiit ums

1,1-1 u.s r t M i M t l r r 1 lO in jK is iic Iiiiif c iiiil liivcr \v j111 ỊN T Ìo đ ic iillv l i i y r m l i s t n i c r u i r :

it m i l l iiin.s M ỊHTÌod.s a n d Ciieli period consists PJ (liHVienr layers of iiiCdiiiprcssihk*

I -SI 11 I I > p I * ■ I - I i i > i i r 111:111 ‘l ' i i I Irs ( N - a i l i i t L ' i r v i n r c ^ i T m t ' i i i l x T A > 2 )

I l i i i r ĩ I !*• m a t r i i i i l l i i w i ' i i a r c M i l ) j e l l I ' l l t o [>1111 ' l i o i n o L ^ r i K ’d i i ^ -st I ' i i i i i .111(1 I m i n l c i l rtluiiji; l l i c i r n i m i i i i i i i [ i l i i uc l i i i mi i l i i i ic> ill Mill’ll 1 U'iiv iliilt till- I>1111- I'ipiil I li 11 | | i( II |> til MiLJiiu ill!' one ilinvi mil ln'inj^ lit >r 11 lit 1 III |||C t( >11111 ] (Ml

1 Jt 11111 • 1,11 iiv,.

W e (1«-111111 - I »v 1 1 , ( / = 1 2 \ ) 11)1' I l i i c k n c h s < >f i - f li l i i y r r i l l t l i e I )1 1- ^1 n\ s M*i l

> I ill I • li, 1 In' ll 1 1 — / / 1 -ị- h > • t- 1 1 \ is r l i e I l i i ( ' ki i (' *y (ll o n e p e r i l XI 1 1 1 11 M h in

I l i e I lilt kiii'M.s ()l t lit' r o m p o M t c l ;i \ <T ill D,

A I VC I .'111 n 11 111 r c ill 11‘siiin II )< >1 iliua I I’ M '*tvm I): Ị , ,: I is I’111 I )1( )\ Cl I W'ltli IIM 1 X 1-1

< o i 11«' i 1 1 i IJ u, \ \ i 111 i l l * |»I i i u i p a l t l i r c i i i o n ' ' o l t l i r [ H i r e s i i i i i n f i l l - t i l iu,i 11 l ( ) < ; i i c ( l I

\ I n I > I I , , u ,ii I- > I >ui !>«<ticiit.s III t III ■ r 11 )| II ( ) | ) I I ; 11 (• 1( )U1111 I )t 11( 'I c | ; i > i I< It V 1 ( I I M II I |c

I i l l 11 l.v I I I I I I I |'"i| ! >’ IS 1 I II I l f ( l c | ) < i i ( l c i i i p r e s s u r e i n c i Cl m- 1 1 1 /I I' I I l f ill I I M I V

, r I 1 1 1- I i i i i i f i i;il 1 1 1 ( 1 1 r o l l II ni l ni l 111 :i ilill'fi I'Ilf I ari l >11 w i r l i r t ^ Ị i c t i (I d i e l i u p l i t (i

11,11 I ; 11 I m i l i l i i i i l ' I m i i p o i H i i t II I / I i y ;i M i | ) c r | ) t ) s ( ' ( l ( l o t d e n o t e d [ > 1I I 1 11 l i i i c i

-( III I; 11 Hill \ V' 11 ) I 11 ■'i [>***" t 1 1111H ‘ t.

Nli:i Trillin '2IHM) liilcni:ilion:il Colloquium - Colloqiic liiic(ii:ilion:il Nli;i Tmiiịi’2(>*M)

I \v<> u o n / n o t I';i«' t i« > 11 ĩ I II'] t-II1«-111 s al 111« - > — riMisl : I I «• I Jit'll 1.111 11 !■'

MI<• I >1 m u h i r :

1 I / ỉ • I ; I I I I I f- ( l ỉ 11 I > I l>)ll I I r ) i* II I £ I I I I I I I ị l ỉ > / > > I I ' I I I > ' I ' I I ' I

w i l l ' l l ’ /> (It'Miil i i i £ ;i s t í i l i r ] > i i - * s u i r i n l ỉ ,

Fm < Irt ;ii!s «-«>11«- IIIĨI1ỊÌ 1 lu* (Iciiv'Mhi >11 1)1 ( l 1) ;||||| ( I ‘2) ill'' I • I * I* 1 ir- 11 l( I 11 11

lo l l i c |>;i|u IS Itv I’ H ;ui(l H o f v i s o n |ri| ; 111 f l |{di;i'rs«iii ;i 1111 S:nnlil'>t' I |!)|

2 W /W rcs IN TIIF, C O M P O S IT E I.AYKIỈ

1,1't Ils innsi( i«• 1 \ v; i\ ts I >1 I>|>;%;i I i 11'J ;il(ilijị[ 111 I lir lỉivrrs ill 111» I III I <• II'II I' I

f '(II! I S|)( >ll< I i 11" III l l ic s c Wiivrs till1 soldi inns lit <'<[Uiil IIHIS ( I I ) :ui- MiM'lil ill 'III

NIlii Tni»t;'20(H> Inlcnmiioniil Colloquium Collogue lnicm;iiioiul Nl\;i Tr;iny'2<X)0

I lie IM >1 II It lilt y conditions at the pliiiics : = M.h liavr tin* from :

lilt- 111 )|||I >;:( Iii/I‘11 Ii|iial mil* Ini ilii.1 uv ml 1( II lure two new unknown limct ion.'t

I / 1 I / • 11 I i11< -< 1 ;i>> || >11( IVV'N

N iu Tr;ing'2t)Q0 1 nlcmational Colloquium - Collogue lnlcnintioiiĩil Nh;i Trill lit '2ỊỊỊIII

It follows from (2.3) ill 1(1 (2.1 4) :

Nha Traiij;’2IH)(l lnicnuiiioiial Colloquium - Colloqiic Iiilcmnuonal Nh;i TniMỊỊ'2()()()

ll ' l u u i lil lie uoU-<l lliiir llic Mihilion uí (2.22) (Icpi-Iids oil ĩ IIc iic c when wr

; I i 11 In >1 I I f llỉil u v sllilll write 1 III , 1/jjji IJ11 f j i t ill striicl ol ) II! Ũ 2 , y I , IJ1,