Generalized Solutions of Boundary Value Problems of

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Let u(x,t) be the solution of the system of hyperbolic equations which ... x =(x/,..., xN), x E S~, S is the boundary of elastic body S~ E RN and belongs to the class ...

Generalized Solutions of Boundary Value Problems of Dynamics of Anisotropic Elastic Media L.A. Alexeyeva, Institute 480100y

Almaty,

Pushkin

of

Mathematics

street,

125,

Republic

[email protected]

of

Kazakhstan

of

Kazakhstan

kz

G.K. Zakiryanova Institute 480100y

Almaty,

Pushkin

of

Mathematics

street,

125,

Republic

[email protected]

1. SUMMARY The method of boundary integral equations (BIE) for the solution of non-stationary boundary value problems (BVP) of dynamics of anisotropic elastic mediums is elaborated. The central moment of this method is constructing the fundamental solutions of equations system, kernels of BIE. Here the fundamental solutions in two- and three-dimensional cases (N, M=2,3) are considered and their properties are studied. In the space of generalized functions the solutions of initial BVP are obtained and their integral representations, regular inside a range of definition are given. Generalizing the Green and the Gauss formulas for the generalized solutions of these equations, singular integral equations for the solution of non-stationary BVP are constructed. The uniqueness theorem of the solutions, including for the class of shockwaves, is presented.

2. STATEMENT OF NONSTATIONARY BOUNDARY VALUE PROBLEMS Let

u(x,t)

be the solution of the system of hyperbolic equations which describes dynamics of anisotropic

elastic mediums. We consider it in the cases of plane (N=2) and space (N=3) deformation:

k j i P x A ) «

Lij [ d

x

j

M

+ C

i

M

A ) = C tij f d j um b uil - Зуд*, 'ml _ f^lm ij ~ ij

_ ~

сml ji '

^

(x,t)GR

i, j = l , M ,

N + ]

(2.1)

m , l = l , N

(2.2)

259

Boundary Value Problems of Dynamics of Anisotropic Elastic Media

Vol. 16, Nos. 4-5, 2005

Here С™ is the matrix of elastic constants satisfying to the condition of strong hyperbolicity

v V > О V пф0, v * 0

W(n,v) = Cfnmn{

x =(x/,..., xN), x E S~,

S is the boundary of elastic body S~ E RN and belongs to the class of Lyapunov's

surfaces with continuous exterior normal n,

||я|| = 1, (x,t)e D~,

D~ = S

x(0, oo),

D/~ = S'~x(0, t),

D= S~ x(0, oo), DT = S x(0, t). Everywhere summation is carried over like indexes in the indicated limits. It is supposed that ueC(D~

u Д), GeC(D~

kjD) and G

>0, t -> +°o,

VxeS~.

Further и is twice differentiated vector function almost everywhere by exception characteristic surfaces wavefronts Fh on which following conditions on gaps are executed /1,2/: (2.3)

[«/(*>=0

[иыщ+сиы]р

(2.4)

=0

(2.5)

Here " c " is the speed of a wavefront motion, which instituted from the solution of the characteristic equation of the system (2.1):

where v=(vi,..., с = -v, /(v,vi)u2,

vl) is the vector of characteristic normal: с = ±ck(v),

0 0, Итии-> 0, xeS~ then I-++CO ' t-t-KO ' \ Gi(x,t)uiMdV 0,

Vjc e S" then it is unique.

261

Boundary Value Problems of Dynamics of Anisotropic Elastic Media

Vol. 16, Nos. 4-5, 2005

4. THE GENERALIZED SOLUTIONS OF BOUNDARY VALUE PROBLEMS. ANALOGY OF THE KIRCHHOFF AND GREEN FORMULAS Here D'M(RN+]) DM(RN+])

is the space of generalized vector functions / ( * , 0

=

(/I>--->/A/)

on

s

Pace

of base vector-functions cp(x,t) = {(Р\,—,(Рм} > V ^ G D ( R N ) /3/. For regular f we have

Cf (x,i)M*,t)>

] d r -oo

\/cpeDM(RN+l)

S /;(х,ф;(х,т)с1У(х), r

N

For И which is determined on D~ we introduce a generalized function u(x,t)=H(t)Hs(x)

u(x,t)

(4.1)

Here H(t) is Heaviside's function, H$(x) is the characteristic function of the set S~ which is equal to 1 for jte S~, to 0,5 for jte S and to 0 for jce RN \ (S U S~). Analogously to (4.1) we have G k(x,t)= H(t)Hs(x)

Gk(x,t). к

Green's matrix t/ z (x,t) is the fundamental solution of Eq. (2.1), corresponding to G

(x,t)=SikS(x,t)

and Ukit (x, 0) = 0

U\ (jc, 0) = 0,

JC * 0

Regarding the construction for U see /4/. Also we use the fundamental solution of Eq. (2.1) by G (x,t)= Sik8(x)H(t)

as a convolution

t Theorem 4.1. If the function и is the classical solution of first (or second) BVP then u(x,t) can be represented over convolutions: л и i(x, t) +(t/£,(x,

. =

uf (x, t) * Gk (x, t) + uf (x, t) * u\ (хЩ t) * и(*)•+,uf u

262

j,tnm

(x,it) * gk (x)H(t)

(л:) +

(x,t)Ss(x)H(t\~ C^jVij

(x,0*uj

(x)nm

M

(42>

wamics of uc Media

LA. Alexeyeva and G.K. Zakiryanova

Journal of the Mechanical Behavior of Materials

Here giix,t)Ss(x)H{t) is the simple layer on cylinder D\ sign "*" means full convolution on (JC,0; sign "JC" or

under an asterisk corresponds to convolution only over x or t accordingly. The formula of this theorem is the generalized solution of Problem I.

ic space have 5. MATRICES OF THE FUNDAMENTAL SOLUTIONS V, T, W

New matrices are considered:

Sg(x.O 4(x,t,n)

(4.1)

to 1

k

=c f u

j t

Гki{x,t,ri)

,

= S^knm

= - i f ( * , t , n ) = ~cfnmUkj

vf (x,t) =Uj{x,t)*

SmkS(x)H(t)

{(x,t)

=Uk(x,t)*

Я - (x,t,n) = T™(x, t,n) * SmkS(x)H(t)

H(t)

= r f (x,t,n)*H(t)

Some properties of symmetry of these matrices are Ь м ) I / f (x, l) = uf (-X, t), v f ( x , t ) = vf(rx,t),

£ 1 ) by

(x,t) =

uf (x, t) = U\ (x, t), vf(x,t)=

(-x, t),

Wf (x, t,n) = -wf

vl(x,t)

(x, t,ri) = - i f (-x,t, n) = -7f (x, t, -n)

(-x, t, n) = -wf

(x, t, -n)

к

Theorem 5.1. Multipole matrix T^ (x,t,n)at can be

applicable Gi(x,0

=

nmCS!'S,l(x,t).

Let us introduce U^s\x)

Ц(дх,

(4.2)

fixed "A" is the fundamental solution of the system (2.1),

T«s\x,n)

as the Green matrix of static equations (2.1) (when dfu - 0 ):

0) u f \ x ) + SlkS{x) = 0, = -C%lnmuf

? , Т^\х,п)

uf^(x)

-> 0, IMI

= -T^\-x,n)

=

-T«s\x,-n)

263

Vol. 16, Nos. 4-5, 2005

Lemma 5.1. 7 i s

Boundary Value Problems of Dynamics of Anisotropic Elastic Media

the fundamental solution of static equations

Ьц(дх,0)Тfs\x,n)

- nmC?klS„

(x) = 0.

It is easy to see that this system is of an elliptical type. The following representations are valid.

Theorem 5.2. Vj*(x,Q = UpXx)H{t)

+

W,k (x, t) =

+

(x)H(t)

Vi(d\x,t) (x, t),

H(t) are regular functions at jc^O. At \\x\\ - » 0

where l / p > « ~ln\\x\\A»(ex)

Т«*\х,п)

U p \ x ) ~\\хГ"+2АЦ(ех)

Here ex = x/\\x\\9

Н

И

T^\x,n)~\\xrN+lB»(ex),

Г

N

=

2

N>2

(e x ) are continuous and restricted on the sphere |H|=1 functions; V ^ d \

(e x )

are regular functions, continuous at Jt=0, t>0. For anyone N: V^d\x,t)

=0 and W^d)(x,t)

=0 by

||x||> max cM ( e ) t , and for uneven N these equalities are performed and for ||jc||< min q ( e ) t . ИИ Nl=1

6. BOUNDARY INTEGRAL EQUATIONS Lemma 6.1 (analogue of Gauss formula). If S is any closed Lyapunov's surface in J T^s\y-x,n(y))dS(y)

then we have (6.1)

= SkiH~s(x)

s By xeS integral is singular, it is calculated in the sense of Value Principle. When М-1 and Ь],{дх,0)= djdj = A, this formula complies with the Gauss formula for potential of double layer of Laplace equations /3/. Notice that formula (4.2) formally can be presented in the form: \(Tj-(x-y,n(y),t-T)Ui D

s~

264

M

+ и*к(х-у^-т)&(у9т) x* , then lim uk (x, t) - uk (x*, t) = lim x-+x* x->x* s +\W*d\x s

(у, t)dS(y)

-

-yMУ\t)u{Xy)dS{y)-

t -JdS{y)\{U[{x*-y,t-T)gi

s

(x - у, п(у))щ

w£d\x*-y,n(y),t-T)uu(y,T))i

{y,T) +

о

+ \u'k(x'

-y,t)u)(y)dS~(y)+

j(U'k(x*

-y,t)u°(y))„dS~(y)

+

\Ulk{x'-y^-^G^ryiD-^T)

+

The limit on the right part can be, by means of lemmas 6.1, converted to type

jT'"(x--y,My))(u,

(y.O-u,

(*\й(у)

+ », ( x ' M

-

< = У.Р.\Т^\х--y,n(y))Ui

( y , t ) d S ( y ) - U i ( x \ t ) V.P.\T«s\x*

s = V.P.\T«s\x*-yMy))DS(y)

-у,п(у))

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