The mixed problem for the elastic anisotropic halfplane

THE MIXEO PROBLEM FOR THE ELASTIC ANISQTROPIC HALFPLANE (SBESSANNAIA ZAKMGRA I&IA UPS~~OI ANIZOTROPNOI POLUPLOSKOSTI) PYM Yo1.26, No.5, f962, pp.

896-906

V.Z. SVEKLO (Kalininerad) (Received

1. Fornulation

of

Decenbet

the

a solution that is continuous tions of motion [l]

problem.

8,

1961)

ive seek a regular

up to second order

derivatives

solution, i.e. of the equa-

(1.1)

at points conditions

in an anisotropic

and the following

The right

sides

halfplane

y>O

under the following

initial

boundary conditions:

of these

2. The Green-Volterra we have the Green-x’olterra

equations

formula. formula

contain

given

functions.

Jn the general

case of anisotropy

anisotropic

The elastic

1355

halfplane

tll is a solution of Equations (1.1) corresponding to the uZJ vZ corresponds to the body Xl, YI, while the solution Y,; Tdenotes an arbitrary volume in the xyt space which is

Here ulr body forces

forces X,, bounded by the surface

S with interior

P (u, @) = a, cos (m)

normal n

+ z,, cos (ny)

p SOS at

-

(at) (2.2)

p $

Q (% 4 = z,, cos (ns) -f- a, cos (ny) This strong 3. tions

formula remains valid discontinuity

cos (nt)

even when one of the solutions

has a true

[21.

Fundamental solutions. We of the homogeneous Equations

shall

construct

(1.1).

some special

‘Ihese will

solu-

be nonzero within

a characteristic cone whose vertex is at the point (x0, y,,, t,) and will have the required singularity on its axis n = x,,, y = y,,. We call them fundamental

solutions.

Let us introduce

the functions

4j ----

1

Between ftj and hi there &jLZj 6j

We specify

U&O’= i

7 = as -

=

t0j’hj’

to

exists

-

incident

=

t -

(5

(: -

= a0,t + dljr

d)* ’

the relation

LG = dtlj8 + a;lr” -

0 -‘%I)

8

disturbances

+

(Y

-

90)

h

=

1 (3.2)

0

in the form

RS i r CEhj (E) 0; (&) dE,

vkoo = i

j=i

Re

i f Llj (E) o;(t)

d$

(3.3)

j=l

where O*

O&j

L*j

=

-

eh;

Fj” Fj”

=

0~ C%ja

Forming the solutions condition

fk= f, 2). ,~j** (Ojalo~jo

[31 for

+

hj B

= -

lj*OG”)

+

the reflected

that the boundary is stress

free,

“j”o*j’+

Lej -

L*I~tj” Pi”

Chj*

(3.4)

disturbances

and superposing

under the these solu-

tions on the corresponding incident solutions, we obtain the fundamental solutions uLo, vko. These are nonzero in the interior of a characteristic

cone and zero

on its

boundary and in its

exterior.

Upon passage

1356

V.Z.

Sveklo

throughthe pertinentcharacteristicsurface,lying either within the cone mentionedabove or comprisingits surface,the first derivativesof these solutionssuffer a discontinuity. However,as is easily shown, the kinematicand dynamicconditionsof compatibilityare satisfied. 4. Estimate of the fundamental solutions near the axis of the characteristic cone. To make this estimate,we show that it is sufficientto estimatethe function(3.3) for large values of %i, since only incidentdisturbanceshave singularitieson the axis of the cone. Taking into accountthe branch of hi to be chosen for c ==z a - d IIll, and writing out only the main terms, we obtain

whereby = 1,

M,M,

M12$M$+=-$,

(a - Mj’d) (d - Mj%)+

e2Mj2 = O (4.2)

From (3.2) we find 6, =

x‘ - iMjy’ q

t’,

2’ ==sz -

x0,

y’ = y -

7&j, t’ = t, -

t

For large ej we readilyobtain 02jo =t -

(4.4) IJj=a+(C-d)Mj*

(i = f, 2)

Hence lim (%jsCLQjo+hj*W*j”)= 0, and for large

ej

%j-+ 00

the main parts of the functionsekjo and ekjoo coincide.

Now it is easy to estimatethe fundamentalsolutions.We obtain 2 @lo

=

-

~2

MjAj” Re %i, j=l

VI0= -i

J3j"Re if+, j=l

13j” = ?I’$

(4.5)

The

anisotropic

elastic

It is also necessary to estimate Equations (3.3) and (4.3) we find

1357

halfplane

the derivatives

for

large

0j’

l!sing

(4.7) ‘The same result may he obtained by straightforward differentiation of the expression for uIo with respect to x. The remaining derivatives can be estimated in an analogous manner. Omitting the details of the calculations, we give the final results of the estimates of the solutions and the corresponding stresses. a) First

Formulas (4.6)

fundamental solution.

are supplemented

by

(4-Q

b) Second fundamental solution 2

u2*

3=

-

~2

I’I’

Mj-‘Aj

r’2 ’

j=l

The stresses

v** = -

i;

j=l

J

are obtained

y’t’

j&f,-‘&’

(4.9)

in the form

(4.10) c) Third

fundamental solution

u30=-c j=l For the stresses

Mj A jO f$ I

,

~8’ = i Mj-‘Bj’ j=i

KJs

(4.11)

‘j

we obtain

(4.12) The main parts

are indicated

in all

of the formulas.

1358

V.Z.

5.

Solution of the Cauehy problem for the halfplane.

solve this solution of

problem,

we apply

the Green-Volterra

and to one of the fundamental

integration

we take a portion

face S’

of a characteristic

is at the point (see

plane

(x,,

to) and, on the other S,

t = 0, and finally,

by a portion S, of

stood of

of

that S’

is

It

is

that portion

is just

on this

mental solutions that

the latter

-

portion

to a variation

l/da < 0, < l/da. that

vanish.

all

that at present

of the aforementioned

eXpreSSiOnS

for Bk have the form

+

u$bs

(nx)

-

+

surfaces

Zxy*%OS

(ny)

into

consideration

solutions,

we obtain

(up to passage

to the limit),

that depend on E Sre used. ‘Ihe

-

&4,o pa,

cos(~t)

u

+

1

(~,,Ocos(ns)

-

hand, taking

to zero body-force

the portion

(

It

of the funda-

On the other

correspond

where the index a indicates

B*-

E that cuts to be under-

of the surface

the cone that corresponds

of 0, in the interval

of the

of the

radius

the cone.

S,

of the plane

cylindrical out the axis

Uk”, Vk”’ For the region

by the sur-

by a portion

y = 0, and a portion surface

solutions

TO

to the sought

cone whose vertex

ya,

the figure),

formula

of the space

which is bounded on the one side

side

Sveklo

(\ a, cos (nx) f

7q co9 (nx)

+

$/;cos

+

2,

(lay) -

cos (ray) -

+ 6, cos (ng) -

p =C$-eos(,t))

v-

pg

cos (R&t)) l&k0-

p g

cos flat)) VkO

(5.2)

All of the integrals in Formula (5.1) are known, with the exception of the integral over S,. lhe integral over S’s is zero as a consequence of the fact that the kinematic and dynamic compatibility conditions are satisfied for the solutions uIo and uto. The integrals over S,, and S,, are known by virtue of the initial and boundary conditions. We next show that the integral over Sa, in the lirait as a - 0, is equal to a certain linear combination of the derivatives of the unknown functions u and v. On the surface

S, we have

The elastic

setting

cos (ny) =

)

=

\

[ @XkO COS cp +

11

rf2 = 5’9 +

$2

=

&l

Gykosin

and E vanish

q(~)

cp) U +

(Zyxkocos cp+ UykoSinrp) u +

i,

+ kh COScp + Gu sin cp) &to -

Since

-$ ,

w 6

where

1359

halfplane

x’ = E cos 9, y’ = E sin Q, we obtain k-n

BkdS

$-

cos (nz) =

cos (rat) = 0,

anisotropic

cos ‘p -i- uy sin cp) vk’l

and LE is a circle

together,

the unknown solution

(h

is regular,

we have,

u (z, y, t) = u (x0 + E cos y, y. +

E

dl

of radius

I

dt

E.

for example,

sin ‘f, t) =

=a(z,,y,,t)+E-~~os~+~sinp]+... 1 ux (z, y, t) = ur (2, + 8 cos cp, y. + E sin cp, t) z-

ux (20, $!o, t) +

=

1

zsincp

E

+...

etc. The omitted terms are of order ~~ and higher. Substituting Equation (5.3) and taking into account that dl = E+, we obtain

\\

Bk ds =

“in e {u (zo,yo, t) 7(G~COSq + ii

Sk

+

v

(%

Yo,

tl

p

(%~~cos

cp +

2n az” cl

+ au8

aY0

u,isincp)

dcp +

0

824

+

rxy~sin cp) dq +

6

(%k°COs2cp

i-

zxyo

sin cpcoscp) dcp +

s

0 2x s

(&,“sin

cp cos cp + %xykosin2cp) dq +

0 2x

i- -$y 8 \ (%~~~Cos~cp -k uykosin tp cos cp) dq 0 276 + au 8 s (%,xkosin cp cos cp+ uyhosin’ cp) dq aYkl 0

+

-

into

(5.3)

1360

-

V. 2.

8

\

Sveklo

[@x0COS Cp + ?x,,‘SinCp)ukO + (rxll COS Cp+ Uyosin

Ii G

0

=%(h,yO

cp) ufi’j dq] dt

tj,...

Terms which vanish when E vanishes need the integrals

(5.4)

have been omitted.

In the sequel,

we shall 2% l

s 0

sin rpcosqdq~ r.2 N

2x

* sin cpcoscpctg,

=

\

‘jG

0

2%

sin rpco9 qr

ZZZ c

4

2x

dq =

c

‘ii,

i;

6

cos cpsins g,

drp = 0

‘Z

2s 5

cos4

fp -

2n

sin2 4 g, ~09 9,

Mj2

da,=

” (1

‘jv

0

+

MjP

sin2cpCO@9 - Mi2 sin4q (5.5)

dT=

r.4 3’p

s 0

It is easy to convince oneself that in Equation (5.4) the coefficients of n(x,,, yo, t) and v(no, y,,, t) are equal to zero for all of the fundamental solutions. In addition, as a consequence of Fquation (5.5)‘ the following integrals are zero for the first and second f~d~ental solutions 2% 0

s

(a,,“sin Q,cos fp + zwk‘sin2 tp) drp = ‘{ (z~~~%os~ cp + uv&%in cp cos cp) dcp = 0 (k =U*,

Conversely, solution.

the following

2)

integral

vanishes

for

the third

fundamental

an

s(~x*ocos2 rxy, 9 +

0

“sin q coscp) dcp = 1 (zyx3%in cp costp + q,,0sin2 cp)dtp = 0 U

Let us carry out the calculation of the first fundamental solution more detail. Ibnoting the known quantities by Dls, we write

in

The

On the circle

elastic

1361

halfplane

anisotropic

L, we have

(5.7)

dli

j,f.n.A.e2 a I 1 1 x?4’= F j=l From Hcsoke’s law it

follows

sin4, ~0s rp&I - 0 4

‘j9

that

ox(5a,Yort)=u~~+(c-d)~~, 59 (rip y,, t) = G, (G Y,, tf = d (& + &) We indicate

next the values

eiy the use of Equations

Analogous

calculations

2x

K

2

=

s6

E2

0

,,%A

(5.5)

(c- 4

E$_Ua& (5.8)

of ulo and ulo on L,

and (5.7)

we obtain

give

2

‘M .-wjAj”

cp cos cp+ q,,* sin2 ‘p) drp = 2nd ~--J-

j+

’ + Mj

(43- t)

(5.11)

1362

V,Z.

Sveklo

We turn now to the third and fourth terms of Equation (5.6). We separate and compute the coefficients of &,&I, and &Jay,, which are contained in these terms. On the basis of Equations (5.51, WC obtailt

(5.12)

Equation

(5.6)

can

be rewritten

in

the

form

We have

diIj J.fj2d +

a-

= Zi-c~dui j=~‘-

-h

n.

(to-

t) = 2na (to -

t)

(5.14)

= 2n (c + d) (to -

t)

(5.15)

j

Analogously

=-2&i

rIj(C,--a) j=l

Equation

(5.13)

takes on the form

where qI(s) and E vanish si~ltan~~l~. tain in the limit the first auxiliary fundamental solution

&zf’ SC

u~+(c+d$&--t)dt=4

0

Here

Letting E tend to zero, we obequation corresponding to the first

(5.16)

The elastic

anirotropic

1363

halfplane

(5.17) In an analogous manner, by applying the Green-Volterra required solution and the remaining fundamental solution, second and third auxiliary relationships

2n*’ (c+ d)&+ S[

formula to the we obtain the

a a$] (t, - t) dt = Da

0

(5.18) 0

ulO, where D, and D are obtained from D, by a change of the functions d and u30, vSo, respectively. v* To complete the problem we O to uzo, VI rewrite Equations (1.1) in the form

Differentiating Fquations (5.17) and (5.18) with respect to the corresponding arguments, collecting terms, and integrating by parts, we obtain, with the aid of Equations (5.19)

u @09 Yo* to) = uo(% Yo)+ uo’ (501 Yo)to+

+5x (50,yo, t) (to -

t) dt +

0

+2&(as+$) vh

Yo* 43)=

vo6% Yo) + v’o(209Yo) to +

$ \Y(501Yo, q (to -

(5.20) t) dt +

0 +

‘(EL2)

2nP

allo

These formulas give the solution of the problem that has been posed in closed form. Furthermore, they generalize the known result of Sobolev k?] relating to th e isotropic body. Analogous results may also be obtained in a more general case of anisotropy - for example, in the case of four elastic constants [3] .

1364

Svek 1o

V. Z.

6.

Effect

of

point

sources.

the

effect

of

various

the

action

of

an instantaneous

Fe assume

that

sorts

up to

the

of

onset

We turn

in accordance

with

the

of

impulse

on an unbounded

of

the

oscillation,

disturbance

uo ($1 Y) = vo (x9Y) = 09 Then,

now to

sources

investigation

of

in particular, anisotropic

the medium is

to plane.

at rest

u’o (x9Y) = 8’0(4 Y) = 0 we obtain

(S. 20),

for

an unbounded

(6.1) plane

where

Dk = -

sss

vko = vkoO(6.3)

ulr’== uk“‘,

(Xuko + Yvko) dxdy dt,

T

I!ere

T is

a portion

of

the xyt

bounded

space,

by the

large

surface

of

at the point (n,, yO, t,), and the the characteristic cone, constructed plane t = 0. Because of the singularities uko, vkoj It is not possible in

(6.2)

tion

to directly

sign

in

insert

the differentiation

the calculation

and yO. IIowever,

in order

of

the

sign

derivatives

to compute

these

of

under

nk with

derivatives,

the integrarespect

to x0

we may represent

D, in the form

\\s(

Xuko + Yv,$‘) dxdydt

& = -

(A-Q” + Y2)h-O) dx dy dt

-

(6.4)

T’LT’,

where

T’ E is

in the

first

a circular integral,

cylinder of the interior

radius E and height t0 - q(~) . I!ence, boundary of the region of integra-

tion does not depend on x,, and yO. (3n the other ary of this region can also be made independent ments

by virtue

in the

of

the properties

differentiation

of

the

first

y0 the derivative may be inserted be rewritten in t!le form

of

the

fundamental

term of under

the

‘”

u (20, yo, to) =

*x $1 0

+($$+f$)Y]dr-&

hand, the external boundof the indicated argu-

(x0, yo, t) (to - t) dTY-

(6.4) integral

solutions.

with

respect

sign

Hence, to x0 and

and (6.2)

2&-ss\ [(tg + c$) x T-T’,

a Xu,“da+&\\Xu,“ds+ azo. Q at GE

may

4

The

elastic

t.

u (x0, y,, t,) = +

5Y (x0,

Y,,

1365

halfplane

anisotropic

t) (to- t) dt -

(6.5)

sss K- -

aua~ aus0

&

0

1x +

x

aY0

T-T,

We calculate next the terms containing an integral over uE. For small E the functions uko, vko can be replaced by the estimates (4.6), (4.9), and (4.11). This allows one to use certain results from the theory of the logarithmic potential in the calculations. ‘Ihe functions x’ I r’2

I’

are

harmonic

on the

a

ax

y; / r?,

yi'

plane

where

xyj,

=

ss

a

aYjo a22"

sider

yj

= Mjy.

Oje

‘I;

= 2’2 f

On this

plane

ss

CT. is

l\p’(x, -1

p%?Tdxdyi+-$

10 ss

I

a region t h’e sum

bounded

by the

of

(4.6)

and (4.11)

we have

(6.7)

yj) ydxdyj 3 (x0, yj0)

J

(6.8)

x2 + yj2/hfj 2 = e2. We con-

ellipse

Xu,“dxdy %

(6.6)

dx dyi

p” In 7, dz dy, = - 2xp”

ajr

y’;

3

pis

-I

On the basis

yio,

Ojr

yj)ln+dxdyj=

l\p’(x,

Oj.5

where

-

P” (‘9 Yj) In $ dx dyi = \ 1 P” (X, yi) x2

Ojz

a=

yj

+ & \ \ Xu3’dx dy . . as

we have

(6.9)

V.Z.

Svcklo

+-$\\X(x,~,t)ln~dxdyj]=2neX(zo,Yo,

~iO=2Jw%~Yo*t)

t,j

Ojc

We next compute the

(to -

(6.10) sum

t)Nr’

=&\\Yvl”dxdy w OL

(6.11)

+ &llYv,‘dxdy %

We have

Mj-‘Bc[&SSY

Intdxdyj-&[[Y

j=l

lntdxdyj]=O *jc

Ojc

(6.12) Hence, by means of the results a passage

to the limit

Analogously

2,(x0,Yo,

that have been obtained,

we find

after

that

we obtain

to) = -

aus0 w-$$)X+(!$-g)Y]dxdydt s&SSSK T

(6.13) The problem of an applied

concentrated

impulse

is now solved

in the

same way as in the case of an isotropic medium. We consider the sequence from zero in a certain small of functions X,, and Y,. These are different

T, whose dimensions go to zero with increasing

region further

that

US X,

for

arbitrary

(x, y, t) dxdydt

n. b require n there should occur the equalities =

Q

(6.14j

wn

-r,

where P and Q are independent sequence

un(x0,yo,4y)= -

Y, (x, y, t) dxdydt

= P,

of n. Correspondingly,

we have a second

&4,o kpQSK azoff?$)X,a+($$+z)Y”]dxdydt ‘T*

The

anisotropic

elastic

1367

halfplane

(6.15)

Making use of the theorem of the mean and letting n tend to infinity, in the limit, the solution of the problem we obtain,

where

If,

u (50,Yo, to)=

-w

zJ (50’

-

Yo, GJ) =

P &lo -+$.J-&(Z+f$) 2np ( ax0 P &

(

&k1o aye _?g)__&!&!w)

w6)

all of the functions on the right are to be taken at n = y = t = 0. for example, Q = 0, then by using the values of uko, vVkowe find

iRe

i&A.

u (Zor Yo9 to) =

25p

*

. 1=1

where one should take into &j

=

to

-

BjSo

+

i&J..

2 Re + j=l

consideration hj (ej) y* =

-

hj02;‘)

(6.17)

2

V (20, ~0, to) = - &

(OjOl;”

(Ajo,; + W3;‘)

j

the equations

0,

6’J =

-

20 +

X’j (ej)

YO

It is easy to verify directly that the functions (6.17) satisfy the equations of motion in the absence of body forces. Hence they give the solution of the problem of the action of an instantaneous impulse applied to an anisotropic plane along the direction of the x-axis. That is, Formulas (6.16) give the solution of the problem of the action of an instantaneous impulse, with components P and Q, applied at the origin of the reference axes on an anisotropic plane. It is easy to show that the inequality c < a - d is not essential. However, these constants should satisfy the inequality c < a t d. ‘he latter is the condition of hyperbolicity of the system of Equations (1.1). Setting c = a - d, we arrive at the solution for the isotropic medium which was found by Sobolev.

BIBLIOGRAPHY 1.

V. A., K resheniiu dinamicheskich gosti dlia anizotropnogo tela (On the in the plane theory of elasticity for Vol. 25, No. 5, 1961.

Sveklo.

zadach ploskoi teorii uprusolution of dynamic problems an anisotropic medium). PMM

V.Z.

1368

2.

Frank,

F. and Mises,

ratematicheskoi Mathematical

3.

Sveklo, tions 1949.

R.,

fiziki Physics).

Sveklo

Differentsial’nye (Differential

i and

integral’nye

Integral

uravneniia

Equations

of

ONTI, 193’7.

V. A. , Uprugie kolebaniia of an anisotropic body).

anizotropnogo Uchen.

zap.

tela

(Elastic

Leningr.

un-ta,

Translated

vibra1’7,

No.

by

E.E. 2.