R-functions method in electroelasticity problems Текст научной статьи по специальности «Математика»

UDC 539.3

R-FUNCTIONS METHOD IN ELECTROELASTICITY PROBLEMS

SHEVCHENKO O.M., SINEKOP M.S.

The general statement of the electro-elasticity problem is considered. The algorithm of reducing of dynamic problem to a quasi-static one and also the scheme of separating of the connected system to a set of independent sub-problems are cited. The plan of obtaining the general structure of solution is adduced. Computational model is written out. For further explorations classical numerical methods are suggested.

The general statement of the electro-elasticity problem is considered. The given problem was reduced to a system of elliptic type for the mechanical components and for electrostatic potential by means of perturbation and difference methods.

When the direction of preliminary polarization vector coincides with the positive direction of axis Oz, connected equations of motion for piezoelectric body Q with boundary dQ can be written out in the form [1]

cE _ сE сE _ сE

cfj grad div u —11 ^ 12 rot rot u + (cE4 —11 ^ 12) x

X (It + kV2uz) + (C44 + cE3 - C11 ; °12) xp (k div u +

dz2 2 dz

+ gradu-) + k(cE3 + cE1 - 4cE4 - 2cE3)= p-?tr -

oz2 dt2

d

- Є15 k V 2y- (Є31 + e^)—grad у- (Є33 - Є31 -

OZ

- 2e,s)kff (1)

oz2

З 2ш

eS1 V2 у + (eS3 - eS1 bf = e15 V2 uz + (e31 + e15 ) X

3z2

3 3 2u

X — divu + (e33 -e31 -2e15^——, (2)

oz oz2

where u = (ux,uy,uz) is a displacement vector, Tis an electrostatic potential, k is a basis vector of axis Oz; c1E1,cE2,cE4,cE3,c1E3 are the modules of elasticity for piezoelectric materials; e15,e31,e33,e15 are the values of

piezo-modules, s1S1, є|3 are the values of permittivities,

t є [0, T] is time, p is the density of the environment.

The displacement vector and the electrostatic potential satisfy the following boundary conditions:

{cE2 ndivu + (cE -cE2)[(n• grad)u + -2 (nx rotu)] +

3u

+ (cf3 - cf2 )(n—z + knz div u) + (cf3 + c1E1 - 4cE4 -dz

- 2cE3 )k nz + [cE4 - -2 (c1E1 - c1E2 )][nz (^ +

dz 2 dz

du

+ graduz) + k(n• graduz + n—)]}эп =-{(e33 -e31 -

Oz

РИ, 2000, № 3

9y

- 2e15)knz—=- + e31 n—h + e15[nzgrad у +

dz

dz

+ k(n • grad y)]}3Q,

=±у>(а

(3)

(4)

(9Q1 is a union of surface parts dQ, which are covered with electrodes)

{e1S1 (n• grady) + nz(e|, -S1S1)2Z-nze31divu-

dz

9uz

-nz(e33 -e31 -2e15^-----e15[(n• graduz) +

dz

+ (n )]}3Q\3Q1 - 0,

Oz

(5)

where n = (nx, ny, nz) is an external normal vector to

the surface dQ of the body Q, V0 (t) is a prescribed function.

Initial conditions for the mechanical variables are

= 0 (6)

I ° da

h=° ’ st

t=0

Carry out time sampling [4] and separating of connected determining system (1)-(6). For that purpose divide the time interval [0,T] into N parts:

T

T-N tl = iT^

I = 1,..., N.

Denote u1 (x, y, z) = u(x, y, z, ix), у1 (x, y, z) = y(x, y, z, ix).

Substitute the time derivative at time layer i for difference operator

d2u m-2 - 2ui-i + u1

dt2 x2

On the basis of the constants analysis choose a perturbation -2- in (1),(3),(5) which will be substituted

cE

11

by є —E-. Further, we’ll consider є as a perturbation in

cE1

terms of which the solution is expanded in the form

u1 = u0 +Є uj +Є 2u12 +...

уi =y 0 +syi +є2 у 2 +... , i =1,..., N.

Thus, problem (1), (2) is reduced to a successive search of solutions

uj, yj (j = 0,1, ,2,...; i = 1,..., N)

of simpler subproblems.

1) grad div u‘ -

c11 c12 2cE1

rot rot u‘ +

'(c44

pry — p '-'I 1 ^

12

d 2ui 1

"izr+kV2-"zi)++ cE -

1

c

2

53

СЕ ^ cE Я 1

11 12-)—(kdivui + gradu1-) + k—(cE3 + cfi -

2 dz

- 4cf4 - 2сЗз)

52uZj __p_u.

3z2 c31 x2

= Fj

u71 = 0, u0 = 0 ;

(7)

(8)

c 12 cE .1

Hf n div u i + (1 —E.)[(n • grad)u j + - (n x rotu j)] +

зз

11 11

cE - cE ^u1 • 1

+ ( 13 E 12 )(n—^ + knzdivui)+ —(cE + cE1 -

c

11

-.E 1/^E

dz

E

11

Suz 1 1

_ 4c44 _ 2ci3 )k nz ~Z h _“[c44 _T(cn _ ci2)]>

dz cE1 2

9u‘

X [nz (--• + graduz,) + k(n • graduz, +

dz zj zj

duj

+ n^r—)]}sq = fj,

dz J

(9)

where Fj, f і are the following functions

FL =

p u12 - 2u11

J=0

cn т

- e15kV2^i_1 - (ез1 + Є15) Jz(grad ^1_1)

s2^i_1

- (e33 - e31 - 2e15)k-+

5z2

p u1"2 - 2u1_1

ЧИ j 2 j )» 1

L = 0,

Vj _1

f j = і {-(e33 - e31 - 2e15)knz J dz

'e31n

^ -1

dz

- e15[nzgrad Vj _1 + k(n • grad_1)]}sq, j> 1;

2)

dwj

^2Ф‘ + (B3S3 -sS) = ®j,

V j =91j,

j

{3i1 (n • grad ці j) + nz(s S3 —e„)^—}3q\sq1 = J dz

-Ф \\ ,

2j l5Q\5Q1

where Фj,9jj,92j are the following functions:

(10)

(11)

(12)

• • d •

фі = e^uz + (Є31 + Є15)— divuj + (Є33 -Є31 -

52uzj

- 2Є15)- zj

dz2

0

|+ V0 (ix),

I0,

j = 0 j £ 1,

zj

ф2 = nze31divuj + nz(e33 -Є31 -2Є15) —+

J J dz

duj

+ Є15 [(n• gradu‘zj) + (n•-—)].

J dz

The calculating model obtained supposes done in turn search of solutions of the problems given in the following sequence:

1) problem (7)-(9) when F0 = 0;

2) problem (10)-(12) with right sides which are determined from the solution of the previous equation.

Then the search of approximations is carried out in the following order

u0, ф 0>u1> v1>->uj> t j>-

For exploration of the computing model obtained R-functions method (RFM) [2] and variational method [3] are suggested. They give an opportunity to reduce the initial problem to a system of linear algebraic equations and to get a numerical solution.

Then a scheme of obtaining of general structure of solution (GSS) for the electric component is cited.

Let ю1 = 0, ю2 = 0, a = 0 are equations of 5Q1, dQ \ 3Q1, and dQ respectively. They are normalized to the first order.

(юь ю2, ю > 0 in Q ;

3ю1 _ 1 da 2 _ 1 Зю _ 1

dn ’ dn ’ da

on dQ1, dQ\ 5Q1, and dQ respectively).

The solution of the problem is found in the form of expansion

T1j=9jj +®1P1j +roP2j, (13)

where Pq , P2j are some unknown functions. They are continuous and differentiable in the domain q so many times as needed. Every optional collection of these functions guarantees the fulfilment of condition (11). Using the following relationships

da2

da7

da2

nx =--

nz = --

dx ’ dy dz

the right sides of which have sense everywhere in the domain Q , continue the boundary conditions inside the domain Q . After a number of transformations we obtain: _ •

S • 4 , S S 4 da 2 ^Фп

eS1(Vro2, V9jj) + (єS3 '

dz dz

+ єІ1(У(Ю1Ри), Vro2) + (sS3 -є®1)

da 2 d

(ra 1 p1j) +

dz dz

+ P2j(Bf1 + (8 3S3 -Bf1)( ^-^)) + ffl(Bf1(Vffl2, VP2j) + dz dz

+ (E3 _ES)^2^

+ (Є 33 811) dz dz

) = -ф2 +ю 2 Sj,

(14)

54

РИ, 2000, № 3

where £j are some optional functions.

For the purpose of satisfying the boundary conditions let’s choose a function out of the set of arbitrary

functions P2j such that it would promote fulfilment of condition (12). (14) gives an expression for the function P2j which we substitute in (13). Finally we have the

general structure of solution (GSS) for the electric component

ФІ =ФІ, +®ipij -

S _ S S ч 5ю2 dm

4 + (є 3з -4) 2

• [ф 2, +

dz dz

+ Bfi(V(ffliPij), Vro2) + (є3з"4)^ 5(ЮяіРі,)]. ( }

J dz dz

Practical utilization of formula (15) can cause some troubles by possible conversion of the denominator

(4 >єІз):

S _ S S ч dm 2 dm

4 + (є Із-efi^ 2

(16)

dz dz

into zero. Let’s transform it. Using the properties of the functions ю 2 and ю on the part SQ \ 5Qi we have

dm 2 dm dm 2 dm dm 2 5юч

--- — = i - (--2----+ —2-----)

dz dz dx dx dy dy '

Then (16) can be substituted with the expression

S _ S S 4 ,3ro2 dm dm2 dm

s|3 + (ef, -є33)(----- — +-----2 —

33 ii 33 dx dx dy dy

)

which is a positive value everywhere in Q because the fulfilment of the condition

.dm2 dm dm2 5юч

( 2----+ —2----) > 0

dx dx dy dy

can always be provided while building the functions ю 2 and ю.

Then the given boundary problem can be reduced to a variational one. By means of Rits method [3] we get a system of linear algebraic equations which has solutions determining completely the solutions of the boundary problem.

Let’s write down not fixed component Pi, of GSS in form of approximate expansion

M

Pij = £ aqTq(x,y,z), j = 0, i, 2,... , (17)

r=i

where Trj (x, y, z) is an element of some complete sequence offunctions (Chebyshev’s polynomials, splines, etc.).

Denote A the operator of the kind

A = 4 v 2 +(e І3

It’s possible to prove that the operator A is positive defined and, hence, Rits method [3] can be used for obtaining the solution. The problem

РИ, 2000, № 3

Ay J = Ф J

is equivalent to one on building an element in the energetic space which realizes a minimum of functional

I(yJ) = (Ay1, yr) - 2(yr, Ф1) (18)

in operator’s A energetic space where the scalar product of elements u and v is defined by equality

(u, v) = J uvdQ

Q

(18) gives the following expression for energy functional:

. 5yi dy j dy.

I(T|) = -sfi f((—4)2 + (—2)2 + H4)2)dU-

J ox dy

dz

- (є33 -eSi)J0^z|-)2dQ-2Jyj(eisV2ujz| + (e3i +

Q 2z Q

d . 5 2ujz,

+ eis)—divuj + (Є33 -e3i -2eis)—-2}dQ +

oz J oz2

+ Jy1j912jd(SQ \ SQi), j = 0, i, 2,.. (i9)

dQ\5Qi

Representation (17) gives an opportunity to write out expression (15) of y‘ as follows

M

Ф j=@ 0j +Z arj ® j (20)

r=i

where

0 0j =ф1|

є s + (є s

ii 33

4

"4)

dm 2 dz

dm ’ dz

j = 0,i, 2,..,

©r, =®iTr, --

ms si (Vro 2, V(roiTrj))

4 + (є33 4)

dm 2 dm dz dz

TsSi(Vro 2, V(roiTq)) + (8 3S3-є Si) 2

(ro i Trj)]

dz dz

j = 0, i, 2,... r = i,...,M.

Taking into account (20) in expression (19) of the energy functional we have

^ 1) -Si I S' 4 + |a.j42 +' 4 +

M d©q S©0, m з©г,

+^arj^)2 + (4T+garj4T)2}dQ-

- (£3S3~4) J (^ + Iaq^)2dD-

Q dz r=i

M

_ 2 J (^0j +Xarj®rj){ei5 V2uz, + (e3i Q r=i

d . 5 2u‘.

+ eis^—drvuj + (e33 - e3i - 2ei5^——2}dQ +

oz J oz2

M

+ j (@0j +Zaq0ri^^dU\dQi).

3Q\3Q, r=i

Find

5I(TIj) —

-----—, m = i, M

5an

lmj

(21)

55

Taking into account the fact that operator A is a positive defined one the condition that partial derivatives (21) are equal to zero is not only a necessary condition of minimum of functional I but also a sufficient one. Thus, unknowns amj, m = 1, M found from the condition

SI(yj) ----

—= 0, m = 1, M

3amj

realize minimum of the energy functional:

SI

50,

M s©r, s©„

Sa

h=-s" 1 «if Aa,^)

r=1

+ 2(

5@k + ;ga..5@a' 50

j)

mi

Sx dx

500

+ 1

Sy r=i

M S© r, . S© n

+ 2(-

0j

dz

rj

r=1

M d® rj, d®

(e3S3 -ef1) j2(

GZ GZ J

5©0j

Sz

+ X arj^^^>dn-2 l®mj(e1sV2u-j +

r=1

S . 32и-:

+ (Є31 + e^)—divuj + (Є33 -Є31 -2e15^—^}dQ +

GZ GZ2

+ J©mj92jd(SQ \ SQ1) = 0, m = 1, M . (22)

dQ \ 5Q1

After some transformations of (22) we obtained a system of M linear algebraic equations with unknowns

r = 1, M , Sj • Aj = Qk, where A. =

a

a1j

VaM! у

is an

unknown vector, Sj = {smr}Mr=1 is a matrix that consists of the elements

sm = jes1(

S©rj S©mj | S©rj S©mJ +5©rj S©mj

q dx dx dy dy dz dz

+ (e^3 -e?1)-

)+

,s _0S )5@mj )dQ ,

dz dz

where indexes m and r are pointing to a number of a

line and a column in the matrix respectfully, j is an

approximation number in the perturbation method.

( ai ^ qj,1

Q1 =

aj,M

is a vector of absolute terms such that

a

rj

d

aj,m = j@mj{e15V^ + (Є31 + Є15 )— divuj +

Q ^z

S

+ (Є33 - Є31 - 2e^) —-}dQ-

Sz2

2 j©mj9j2jd(SQ\SQ1)-sS1 {(^ki^L +

2 3Q\3Q1 Q

dx dx

+ 9©^ S©mL +5©0j_ _^^mL)dQ_ (E s3 _eS1);

Sy Sy

■d&0j 5Qmj , Sz Sz

Sz Sz

dQ, m = 1, M.

Thus, problem (10)-(12) is reduced to a system oflinear algebraic equations. Note that matrix S j remains the same on every time layer. To obtain a system on time layer i it is necessary only to carry out some computations

concerning Qj that is a column of absolute terms of the system. The fact cited simplifies significantly the computations and retrenches considerably the time expenditure while program realizing of computations by means of PC.

Literature: 1. Тринченко B.T., Улитко А.Ф., Шульга H.A. Механика связанных полей в элементах конструкций. Электроупругость. К.: Наук. думка, 1990.Т5.216с. 2. РвачёвB. Л., СинекопН. С.. Метод R-функций в задачах теории упругости и пластичности. Киев:Наук. думка, 1990.216с. 3. Михлин С. Т. Вариационные методы в математической физике. М.:Наука, 1970.512с. 4. Самарский А. А., Андреев B. Б. Разностные методы для эллиптических уравнений. М.:Наука, 1976.351с.

Поступила в редколлегию 25.04.2000

Рецензент: д-р техн. наук, проф. Янютин Е. Г.

Olena Mykolajivna Shevchenko, assistant of professor, teacher at the Chair of Higher Mathematics in Kharkov State Academy of Food Technology. Address: 333, Klochkovskaja St., 61051, Kharkov, Ukraine, tel. 36-40-89.

Sinekop Mykola Sergijovych, Dr.Sc., professor, head of the Chair of Higher Mathematics in Kharkov State Academy of Food Technology. Address: 333, Klochkovskaja St., 61051, Kharkov, Ukraine, tel. 37-38-74.

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РИ, 2000, № 3