I nternat. J. Math. & Math. Sci.
Vol. 4 No. 4 (1981) 795-803
795
ELASTODYNAMIC SINGULAR POINTS AND THEIR APPLICATIONS
A.D. ALAWNEH
Department of Mathematics
Jordan Universtiy
Amman, Jordan
R.P. KANWAL
Department of Mathematics
Pennsylvania State University
University Park, PA 16801
(Received October 13, 1980)
ABSTRACT.
Suitable singularites such as a dynamical Kelvin quadropole are defined
to study the dynamical displacements set up in an infinite homogeneous and iso-
tropic elastic medium.
Approximate solutions are presented up to terms which are
of higher order than those known so far.
KEY WORDS AND PHRASES.
Fundamental
ton spaceme’nt feld,
soon, Doublet, quarople, centre of
distribution
1980 MATHEMATICS SUEC CLASSFICATIUN C@DES.
i.
rota-
of singulares, force, spheroid.
73D30, 46F.
INTRODUCTION.
In the recent studies in the displacement type boundary value problems the
solutions have been obtained by matched asymptotic expansions
tion techniques [2] and singularity methods [3-5].
[i], integral equa-
The singularity methods are
based on distributing the fundamental solutions and multipoles, and centres of
rotation on suitable lines, curves and surfaces
[6,7].
By these methods it is
possible to allow various dimensionless parameters inherent in the problem to
vary independently.
Furthermore, these methods enable us to obtain the solutions
ot asymmetrical boundary value problems as well
[8].
However, all the dynamical
solutions presented so far are derived up to the first order of approximation.
796
A. D. ALAWNEH AND R. P. KANWAL
Our aim in this paper is to introduce the steady-state dynamic singularities and
apply them to obtain the solutions for the rectilinear oscillations of inclusions
embedded in an infinite homogeneous isotropic medium up to a higher order accuracy.
2.
DYNAMIC SINGULAR POINTS.
-
The dynamical equations of elasticity are
u
grad div
medium, 0 is the density of the medium and
Let us assume that
f
f0
e
it
u
u
0
+
0,
f
(2.1)
f
is the body force per unit volume.
it
e
p
are Lam$’s constants of the elastic
and
is the displacement vector,
where
S2u
V2u-
+
Substituting these values in (2.1)
and dropping the zero subscript, we have
V2u + 02u + f
(I + ) grad (div u) +
0,
(2.2)
or
where m
n2u
m
2
+ f
0,
(2.3)
2
p /D, and m is the Poisson ratio.
2
dk
The fundamental solution U
corresponding to the force
fdk(x)
where
+
grad div u
2v
i
x
(Xl,X2,X3)
,
4 (x)
(x)
is the field point,
(2.4)
is the Dirac delta function,
is a
constant vector and the superscripts d and k signify the words dynamic and Kelvin
respectively, located at the origin of the coordinate system is
~U
dk
Ix
where R
and
~
e
(x;)
R
2
+
V(-V)
m).
e
R
m
2m)/2(i
(i
e
R
This solution will be called the
dynamical Kelvin solution.
The net dynamical force F experienced by a control surface S enclosing the
singular point is given by
F
n
T
V
ds
T
V
s
4D +
m
2
fdk
dV
V
I
V
[ dk (x,)dV,
dV
+
(x;)dV
m D
V
(2.6)
797
ELASTODYNAMIC SINGULAR POINTS
n is the unit outward normal to S, T is the stress tensor and V is the con-
where
trol volume enclosed by S.
M
The net moment
is clearly zero.
In free space a derivative of (2.5) of any order in arbitrary direction is
also a solution of (2.3).
fdk
The corresponding forcing function is the derivative of
of the same order in the same direction.
dk
easily by expanding
about x.
(x_;),
These derivatives can be obtained
is a given vector, in Taylor series
where
This expansion is
udk(x-;) udk (x,)
udk (x;)
(.V)
+
i
(.V)
2
udk
(x;) +
(2.7)
The interpretation of various terms on the right hand side of (2.7) is as
The first term is the dynamical Kelvin solution discussed above, the
follows.
second term represents the dynamical Kelvin doublet characterized by the vectors
,
and
the third term gives the corresponding quadrople and so on.
The dynamical
Kelvin doublet can be written as
udkd(x;,)
(.V) U
e
(.v)
dk
(x;)
-imR
v(g.v)(.v)
R
m
[
e
-imR
e
R
-imR]
(2.8/
while the corresponding forcing function is
dkd
4[’V(x)]
4(.V) 6(x)
(2.9)
where the superscript dkd stands for the dynamical Kelvin doublet.
.
From relation (2.8) it follows that a dynamical Kelvin doublet is not symmetric
with respect b the changing of
and
Its symmetric and antisymmetric parts
yiled another important fundamental solution.
i
U
dkd
(x;,8)-
U
dkd
(x;,)
i
-f
The antisymmetric part is
e
.-v.
~_~.V_
R
R
-imR
Vx
2
R
We shall call it the dynamical center of rotation and denote it as
]
()/2,
dr
If we set
the above relation becomes
U
dr
(x,%)
(2.10)
R
where the superscript dr stands for the dynamical center of rotation.
The
798
A. D. ALAWNEH AND R. P. KANWAL
corresponding forcing function is
fdkd
(x;2,)
(x,,)
dr
]
4p V x
[a(x)].
-imR
U has only a vector potential % e /R and no scalar
F vanishes while the torque M experienced by the control
Since
net force
Ifdkd
1
fdr
(2.11)
potential, the
volume V con-
taining the singular point and bounded by S is
M
Ix
I
x(n- T)ds
(V-
T)x
x
dV
V
S
x
_x
I
m2 p Udr
+
dV
V
(x;,)
x
_x dV
V
8V% +
I
m2v Udr
(x;Z,)
x
(2.12)
dV
V
The symmetric part of the dynamical Kelvin doublet gives us another fundamental solution which will be called the dynamical stresslet.
The corresponding dis-
placement field and the forcing function respectively are
U
dks
(x;,)
1
2
i
E
[udkd (x;a
[(J-V) + g(a-Vl]
dkd
j) + U
e
-imR
I__ V(g.V) (.V)
2
R
m
<
e
-imR
R
e
-imTR
R
)
(2.13)
and
fdks
2=p
{j-
$()}a + {a
V
(2.14)
V(x)}j
This fundamental solution represents a self equilibrating system and accordingly
contributes neither a net force nor a moment to the medium.
Another useful fundamental solution in the present investigation is the potential doublet which is characterized by a scalar function and is also useful in
discussing vibrations in Stokes flow [7].
The displacement field due to this
potential is
U
dd
(x;)
V(V
e
-imR
R
2
) + m
e
-imR
R
a.
(z.i5)
ELASTODYNAMIC SINGULAR POINTS
799
In the next section we demonstrate that these fundamental solutions are very
effective in solving boundary value problems in elastodynamics.
To fix the ideas
we consider the case of a spheroid.
TRANSLATION OF A PROLATE SPHEROID.
3.
--
Let the rigid spheroid S,
x
2
r
+
2
I,
r
2
y
2
+
z
2
c
2
(a
2
2
2 2
b )= a e
(3.1)
b
a
where e(O _< e _< i) is the eccentricity and 2c is the focal length, be embedded in
It is excited by a periodic force
an isotropic and homogeneous elastic medium.
with period 2/m acting in the direction of its axis of symmetry so that the dis-
placement of the points on S is
U=U X’
where
(32)
is the unit vector along x-axis and U is a constant.
Our aim is to find the displacement field in the elastic medium so that equation (2.3) and the boundary condition (3.2) as well as the far-field radiation
condition are satisfied.
For this purpose we apply the technique of distributing
the appropriate singularities in reference [3].
In the present situation we have
the following line distributions between the focii x
C
C
I
u(x)
-c and x
FI(E)U
dk
ex)dX
(x-,
+
--C
I F2(E)(c
2- E 2)
udd(x-
c,
, L)dE,
x
S,
--C
(3.3)
E $
where
The first integral in (3.3) represents the line distribution of
dynamical Kelvin solutions (2.5) of variable strength
FI(E)
while the second inte-
gral is the line distribution of potential doublets (2.15) with the strength
(c
E2
2
as
151
F
2
(E).
Clearly, (3.3) satisfies the differential equation and vanishes
FI(E)
and
/
To find
F2(E)
we apply the boundary condition (3.2) on the surface
S of the spheroid and get
C
U $
FI(E)
U
dk
(x-
x
C
dE +
F2(E)
(c
E 2)
(x-
ex)
dE
x
S
(3.4)
A. D. ALAWNEH AND R. P. KANWAL
800
It is obvious that we cannot easily find a closed form solution of quation (3.4).
Accordingly, we use a perturbation technique for small values of the parameter m.
For this purpose, we expand all the functions in (3.4) in powers of m,
FI()
+
fll()
-
mfl2()
f21 () + mf22 ()
F2()
U dk(x
i,
+
udd(x_
L
i
ex)
(3
4(i-)
+
T
4)
+
0(m3),
(3.5a)
+ m 2 f23 () +
0(m3)’
(3.5b)
+
2
f13()
, ex)
uk(x
lx-l
8
m
+
d (x- , x
, x)
2+T
T4)(x
(i
x
i
8 (x
+
3
)(x_- )
m
)
+
(x-) (x-S)
+
[x- 1
2
3
0 (m ), (3.5c)
m2 + 0(m3),
(3.5d)
where
4)
(3
k
U (x,)
R
+
(
x)x
R
dd
U0
3
-(x,
R
+
3
x)x
3(
R
5
are the Kelvin solution and potential doublets respectively for the elastostatic
3 ].
field
When we substitute the expansions (3.5) in (3.4) and equate the equal powers
of m we get the following system of equations
c
(x-
fll
4(1- )
, x)d
C
+
(c
2
$2)f21 U0
ex)
(x-
~x
(3.6)
C
C
4(_,)?
f
k
u (- g, gla+
i(2 + "r
3
~x
(c
f
o
(-
’
3
f
(3.7)
d
Ii
--C
C
C
4 (I
f13
)
Uk~
(
--C
’
x )d
I fll [
2)f23 U0
--C
C
--C
(c2
+
(I
4
8
Ix- l -
4)(X
S lx-
1(
I
)
]
d
801
ELASTODYNAMIC SINGULAR POINTS
I
i
(c
2 )f21 IX--ex I
2
(x-) (x-i)
+
I3
Ix
i(2 +
d +
3
T
xe
fl2d"
-C
(3.8)
The solutions of equations (3.6) and (3.7) are available in reference [3] so that
I
4(1
v)
fll
D)
f12
i
4(i
2e
I
2
e
2e
f21
2
f22
I
2e
2i(2 +
%3)
Ue
2
e
i
2
2
21-1
+ (i + 3e 2
fll a
3
4Ye )L
-2e
e
+
(i
(3.9)
-I
+ 3e 2
(3.10)
+ e)/(l
in [(i
where L
e)].
Next we substitute (3.9) and (3.10) in (3.8) and obtain
C
C
i
4(1
f
)
U
13
k
(x
+
ex) d
i,
(c
i X2)
+
(@0
i ex)d
(x
+
r’
@2 xr
(B.11)
2
a f
40
-
Ii
I
42
[e(l +
4
(y
r
(3.11) we
%4)
+ (i
e
2
)L]
a
2
2iae (2
(-2e + L) +
f21
+
%3)
3
f12’
(3.12a)
fll
i
and e
2)f23 U0
--C
--C
where
2
[e(l +
I
T
%4
+
f2114e
(2
e
2
)L],
(3.12b)
4
f
4
y
2
e )L]
(i
z
z
ii
+ f 21 (-2e + L)
e
(3.12c)
)/r is the unit radial vector in the yz
plane.
To solve
set
fl3(X)
A
0
A2 x2
+
f23 (x)
C
O
+
C2 x2
xl
< c
(3.13ab)
The substitution of relations (3.13) in (3.11) and simplification yields
2
4(1
)
(A0
+
+
c
l-e
+
-x
2
(A2%I
+
2%2
x
[0A0
+
(A2@I + 2@2
2
(C0
+
c C
Nx2
2)
a
IA2 2CoL + @2C2
xr
~r
(0 +
~r
2 2
ex
"X
’I x2) @x + @2 xr r’
(3.14)
A. D. ALAWNEH AND R. P. KANWAL
802
where
I
)0
2a
2
2
2(1
+L
)e
2
[6e- (3- 2e )L],
4)i
e-
=a
(i
e
2)(3
4 (i
21))
(14- 12) e- (7 -6- 3e
4(i
)
I1
]
L
))
2
(3.15ab)
|
+
2ve2)L
(3.15cd)
2
2
-36e + 2(9
3e )L,
2e+L
@i
@2
)
2(1
8e
16e +
12L
2
(3.15efg)
Equation (3.14) is satisfied if we choose
i
4(1
))
"--O(A-
q0A0
+
c2A2)
2e
2
e
i
2
(CO
+ c 2 C 2)
(3.16)
2LC0 + 2C2 0
+ qiA2
(3.17)
@IA2 + @2C2
2
(3.18)
IIA2 + 12C2
I
(3.19)
This is a linear system of equations which has the following solution
A
C
2
2
12@ 2 @2U21
1182 1281
(3.20)
(3.21)
1181 1281
C0
i- e
+ 2L
2
8a2 (l-v)e3
2
i- e
2
C
0 + (ql- c2q0)A2
]
(3.22)
2
8e (i
A0
i
-2e
(C
O
+ c 2 C 2)
c
2A2
(3.23)
The net force F experienced by the spheroid can be computed by superposition
of (2.6)
c
F
4Zl/x
I [fll()
+
mfl2()
+
m2f13() + 0(m3) ]
-c
where the volume integral is taken over the spheroid.
d +
l/m2
j"
V
u-dv
When we substitute the values
803
ELASTODYNAMIC SINGULAR POINTS
of
f
fll’
F
12’
f13
P0
i
and
+
u
in the above formula, we have
P0i(T
3
+ 2)
am
12a U
J
3P0
(e
Ia
3A0
+ 8ae
-I
+
4e
+
A2e
2
+
5re)L]
f21
I
[2
)e
4(1
(am)2_]
2
0(am)3,
ex +
(3.24)
where
P0
Relation
32a (i
)Ue3[-2e
2
+ (i + 3e
4ve2)L] -I
(3.24) agrees with Kanwal’s formula [1,2] up to order a m.
order 0(am)
2
(3.25)
The term of
appears to be new.
REFERENCES
I.
KANWAL, R. P.
2.
KANWAL, R. P.
3.
KANWAL, R. P. and D. L. Sharma. Singularity Methods of elastostatics, J.
Elasticity 6 (1976), pp. 405-418.
4.
DATTA, S. and R. P. Kanwal.
5.
DATTA, S. and R. P. Kanwal.
6.
ALAWNEH, A. D. and R. P. Kanwal. Singularity methods in mathematical physics,
SlAM REVIEW i0 (1977), pp. 437-471.
7.
ALAWNEH, A. D. and N. T. Swagfeh.
8.
KANWAL, R. P. Approximations and short cuts based on generalized functions,
Proceedings Second Int. Symp. Num. Analy. in Appl. Sci 4 Engng, University
Press of Virginia, (1980), pp. 531-542.
Dynamical displacements in an infinite elastic space and
matched asymptotic expansions, J. Math. and Phys. 47 (1965), pp. 273-283.
Integral equation formulation of classical elasticity, Quart.
Appl. Math. 27 (1969), pp. 57-65.
Rectilinear oscillations of a rigid spheroid in
an elastic medium, Quart. Appl. Math. 37 (1979), pp. 86-91.
Slow torsional oscillations of a spheroidal
inclusion in an elastic medium, Utilitas Math. 16 (1979), pp. 111-122.
Singularity method for longitudinal
oscillations of axially symmetric bodies in Stokes flow, DIRASAT, The
Jordan University, to appear.