561
Internat. J. Math. & Math. Sci.
VOL. II NO. 3 (1988) 561-574
ON THE NUMERICAL SOLUTION OF THE MULTIDIMENSIONAL
SINGULAR INTEGRALS AND INTEGRAL EQUATIONS,
USED IN THE THEORY OF LINEAR VISCOELASTICITY
E.G. LADOPOULOS
Department of Mathematics and Mechanics
The National Technical University of Athens
Athens GR- 157 73, GREECE
(Received February 3, 1987 and in revised form October ii, 1987)
ABSTRACT.
In the present report, we investigate the formulation, for the numerical
evaluation of the multidimensional singular integrals and integral equations, used
in the theory of linear viscoelasticity.
Some simple formulas are given for the
numerical solution of the general case of the multidimensional singular integrals.
Moreover a numerical technique is also established for the numerical solution of some
and three
special cases of the multidimensional singular integrals llke the two
dimensional singular integrals.
An application is given to the determination of the
fracture behaviour of a thick, hollow circular cylinder of viscoelastic material
restrained by an enclosing thin elastic ring and subjected to a uniform pressure.
KEYS WORDS AND PHRASES.
Numerical integration, Integration interval, Multidimensional
singular integrals and integral equations, Linear Viscoelasticity, 2-D and 3-D
singular integrals.
65LI0, 65R20.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
In the present report we consider the multidimensional singular integrals and
integral equations, with many applications in the theory of elasticity, plasticity
and viscoelasticity.
Tricomi
[I], [2] has
written the first important work on multi-
dimensional singular integrals and investigated double singular integrals of the
following form:
f
l(x0,Y 0)
where
f
s
w(x,y)
(x0,Y0 ,8
(I.i)
u(x,y) ds
r2
__(x-x
0)__
O) + i(y-y
r e
i8
(1.2)
and w(x,y) is the weight function for various quadrature rules.
If the density function
u(x0,Y 0)
in eqn. (i.I) is a bounded and Hider
continuous function in s and also if the characteristic function
and for a fixed pole
X(x0,Y 0)
f(x0,Y0,8)is
bounded
is continuous with respect to 8, then according to
E.G. LAIX)POULOS
562
Tricomi, the necessary and sufficient condition for the existence of the singular
integral I in the principal value sense, is that its characteristic should satisfy
the following condition:
2
f(x,y,e) de
(1.3)
0
0
The next important work on multi-dlmenslonal singular integrals was done by
Giraud
r
[3]
[5].
He investigated integrals taken over a closed Liapounov manifold
of any dimension m.
This manifold is broken up into a finite number of mutually
overlapping parts, each one of which has a one-to-one mapping on a region of an
m-dlmenslonal Euclidean space.
He investigated singular integrals of the following form:
f
K(x,y) u(y) dr
r
(1.4)
y
where the function u(y) satisfies a Lipschitz condition with positive index.
Giraud also studied the followipg singular integral equation:
f
v
u(x)
K(x,y) u(y) dr
r
.
(1.5)
f(x)
Y
with a kernel, the singular part K (x,y) of which has the following special form:
KI(x,y)--"
1
(Xa-Ya)
C (x)
a=1
a
B, =1
Ay(x-yB) (xy-yy)
-m+l
2
(1.6)
where C (x) are certain given functions.
If we operate on both sides of eqn. (1.5) with:
f
u(x) + V
H(x,y;) u(y) dr
r
Y
where H(x,y;) is any singular kernel then we shall get the following equation:
[I + t 2 (x,v)] u(x) + V
f
[H (x,y;v)
K(x,y)
r
v
H(x,z-v) K (z,y) dr
Z
u(y) dry
r
f (x)+
v
tI(x,y;v) f(y) dr
r
y
(1.8)
where @(x,v) is a function, completely determined by the kernels K and H and the
manifold
r.
Further:Investlgations were done by Mikhlin [6]
[9] who proved that a singular
operator of the type:
a u(x)
f(O)
+
E
m
r
m
u(y) dy
(1.9)
563
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
r
where:
IY
xl
0
and:
(l.lOab)
r
(Em is an Euclidean space of m dimensions)
is bounded in the Hilbert
Euclidean space
L2(Em)
if the symbol of the operator is
bounded and its norm doesn’t exceed the maximum of the modulus of the symbol.
Furthermore, Mikhlin has studied the following integral equation:
f(x,0)
f
a(x) u(x) +
r
E
m
u(y) dy
(I.ii)
m
If the symbol satisfies certain demands with regard to smoothness, then a multi-
dimensional singular integral equation permits regularization, if and only if, the
modulus of its symbol has a positive lower band.
The theorem about the regulariza-
tion can be extended also to systems of singular equations and likewise for the case
where the singular integral entering the equation is taken over any closed Liapounov
manifold.
The basic problem studied by Calderon and Zygmund [I0]
[13] are singular
integrals of the form:
r
E
m
(1.12)
u(y) dy
m
Lp(Em)
They investigated integral (1.12) in the Lebesgue-Euclidean spaces
< p <
,
p
# 2.
They proved that the integral (1.12) is bounded in L (E
P
m)
where
if f()
satisfies the Dini condition:
re(t)
t
at <
(I.13)
0
where (t) is the modulus of continuity of the characteristic
f(0).
Moreover
Calderon and Zygmund investigated the compounding of singular operators of the type:
K u
a
f()
u(x) +
E
r
m
u(y) dy
(1.14)
m
Over the past years some papers have been published by using singular integral
equation methods in elasticity, plasticity and fracture mechanics theory for
isotropic and anisotropic solids [14]
[33].
On the other hand, some scientists have studied problems of the classical theory
[39], while Rizzo and Shippy [40]
have used the Boundary Integral Equation Method (B.I.E.M.). In this report the
Singular Integral Operators Method (S.I.O.M.) which has been used by the present
author to the solution of elasticity and plasticity problems [21], [22] shall be
of viscoelasticity, followed classical lines [34]
extended to the solution of viscoelasticity problems.
2. INTRODUCTORY FORMULAE OF THE MULTIDIMENSIONAL SINGULAR INTEGRALS.
Let us consider the following multidimensional singular integral: [6]-[9]
564
E.G. LADOPOULOS
f
I(x)
u(y) dy
m
Em
where x,y are points in the space E and:
m
,IY
r
xl.
and
x
y
e
(2.2ab)
r
Furthermore, the point x is the pole, the function f(x,e) the characteristic,
u(y) the density of the singular integral (2.1).
Thus, let us consider the following assumptions:
and the function
(a) In any bounded part of the space Em the density u(x) Lip a, a > 0
(b) at infinity u(x) 0
k > 0
(c) The characteristic is bounded and for a fixed pole x is continuous with
(Ixl-k),
e.
respect to
If these three assumptions are valid, then according to Mikhlin [6], [7] for
the existence of the singular integral (2.1) it is necessary and sufficient that:
f(x,8) ds
(2.3)
0
s
where S is the unit sphere over which 8 moves.
Thus, from (2.1)
we have the following formula:
I f(x,O)rm
u(y) dy
Em
+
f
f(x,O)rm
u(y) dy
r>l
f f(x,e)m
u(x)] dy + u(x)
[u(y)
f f(x,O)m
(2.4)
dy
r
r
r<l
r<l
The first two integrals on the right-hand side converge absolutely, but for the
third integral if we introduce spherical coordinates with a centre at x, then we’ll
get:
f(x,8)
r
r<l
lim
-0
m
in
dy
J
lim
+0
f
e<r<l
f(x,e)
r
dy
m
(2.5)
f(x,e) ds
S
(2.3) is satisfied.
But if condition (2.3) is not satisfied, then the singular integral (2.1) can be
expressed in terms of absolutely convergent integrals by the formula:
if condition
f f(x,O)
Em
m
r
u
(y) dy
f
r>l
f(x,e) u (y) dy +
m
r
f f(x,e)
m
r<l
r
[u(y)
u(x)]dy
(2.6)
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
565
3. BASIC FORMULAE OF THE MULTIDIMENSIONAL SINGULAR INTEGRAL EQUATIONS.
Let us consider the following multidimensional integral equation:
f
E
A(x) u(x) +
f(x,8)
u(y) dy + B
(3.1)
0
u
r
m
The integral equation (3.1) is singular if the following assumptions are valid:
(a)
The coefficient A(x) satisfies the inequality:
A(x)
iA(y
(b)
C1
r
E
[(l+x2
2
(l+y2)
(3.2)
f(x,e) satisfies the conditions of
The characteristic
section 2 and also
the inequality:
f(x,e)J s
If(y,e)
(c)
C
2
[(l+x 2) (l+y=)] 2
r
m)
The operator B is completely continuous in L (E
.
interval 1 < p
P
(3.3)
for a certain p in the
,,
In the assumptions (3.2) and (3.3) C I, C 2,
denote positive constants.
.Moreover, let us consider the following singular integral equation in simplest form:
f
A(x) u(x) +
I(x,y,8)
f(x,e)
(3.4)
u(y) dy
Thus, by using some series of m-dimensional spherical functions, of order n,
one obtains:
r.
f(x,e)
(x,e)
A
n,m
n=
I;
A
n=
n,m
(x e
e2
era_
By using (3.5), equation (3.4) takes the following form:
A(x) +
I(x,y,e)
z
Bn,m An,m (x,e)
n=l
(3.6)
m
where:
i
Bn,m
n
2 r()
(3.7)
By using Stirling’s formula we obtain the following assumption:
m
IBn,m
=<
Cn
2
(3.8)
where C is a constant.
Moreover, let us consider the more simplest form of multidimensional singular
integral equation:
I
A u(x) +
(2)
m
2
f
Em
m
r
u(y) dy
(3.9)
E.G. LADOPOULOS
566
A simplest method for the numerical evaluation of eqn (3.9) is by using the
Fourier transform:
I=F -I @Fu
(3.10)
where F is a Fourier transform, of the singular operator with symbol
(0).
In (3.10), (x,0) is given by the following formula:
r.
(x,e)
a
m=]
where A
4.
n
(k) (x)A
n,
(3.11)
m(k) (e)
(k) (e) are
linearly Independent spherlcal functions of order n.
SOME SPECIAL CASES FOR THE MULTI-DIMENSIONAL SINGULAR INTEGRALS.
The 2
Dimensional Singular Integral
Let us consider the following two-dimensional singular integral: [22], [31],
[32], [33]
4.1.
w(x,y)
(R)(x0’Y0)
S
(x-x 0) + i
in which:
(y-y0)
g(x0,Y0,B)
r e
u
(x,y) dS,
(x0,Y0)
e S
iB
(4.1)
(4.2)
If we assume that the boundary of S is described by the following equation:
R(e),
R
0 <
e
(4.3)
< 2’
then eqn (4.1) may also be written as follows:
f
S
#
g(e)
in which:
g(e)
u
(4.4)
(r,e) ds
r2
go (Xo’Yo’e)
(4.5)
(x0,Y0)
(4.6)
u(r,O)
u
0
By using the trapezoidal rule with m abscissae, then it can be concluded that:
"J
*
G(o)
de
m-1
z-!
Z
m
i=O
0
G
-’2__
Ill
(4.7)
R(e)
g(e)
A u (R (e)
k=l
where
i.
u(r,O) dr
0
n
):
f
g(e)
G(e)
where:
k
Pk,e)
+
u (p,e) in
IR(e)I]
(4.8)
Pk and Ak are the abscissae and weights.
Furthermore let us use the following numerlcal integration rules:
The Gauss-Legendre rule.
By using the Gauss-Legendre numerical integration rule then the weight function
has the following form:
w
(x,y)
1
(4.9)
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
567
In this case the singular integral (4.1) may be numerically evaluated as follows:
lea (x0’Y0’Xm)- CB (x0’Y0’xm)]
m
Am
Z
k=l
2
x
XO
m
(4.10)
(A(x0,Y0) -B(x0,Y0))Hn (x 0)
where:
g(Xo,Yo,x,Yk)
n
0
A
X A
k
k=l
(Xo,Yo,X)
2g
(Yo # Yk’
z
k=l
(x,y O) Hn
(yo)
g(x0,Y0,x,yk)
Ak
(4.11)
n)
I, 2,
k
u
CB
u
(x,yk)
u
Yk-Y0
u
Yk-Y0
(x,yk)
k6"m
d[g(xo,Yo,X,y)
+Am
_
u(x,y)
Y=Yo 2g(xo,Yo,X) u(x,Yo)Xn (Yo)
dy
(Y0 Ym’
(4.12)
n)
1, 2
k
ii. The Gauss-Radau rule.
By using the Gauss-Radau numerical integration rule the singular integral
(4.2) can be numerically evaluated as follows:
2_.
m
m-i
,2i,
z
g--fi--
i=0
+
u
w.
n
z
j=l
(0, 2i
in
x .+i
_I__
2
__I_ [u
x .+I
3
2i
--)
m
)]
u(O, 2i
m
2i
[R
(4.13)
where x. are the zeros of the Legendre polynomials of order n and w.
the weights
of the classical Gauss-Legendre quadrature equation.
iii.
The Gauss-Lobatto rule.
In the case of the Gauss-Lobatto numerical integration rule, the integral (4. i)
can be evaluated as follows:
m-I
n
x.+l
wj
2r
Xj-l u
2i
2i
Y"
Z
g 2i
u (I’m-(O’)- -)z
m
m
j=l 1-x
i=O
x .+I
+
where
u
wj, xj
nomials
(--2 2W__im )]
+
u (i,
.
2__im
--’I=--
2
+
u
(0,
2___im
[n R
2W__im )-i]
(4.14)
are the weights and zeros corresponding to the set of orthogonal poly-
(p (i
0(x
)j
).
E.G. LADOPOULOS
568
4.2.
The 3
Dimensional Singular Integral.
The following integral is a three-dimensional singular integral defined on a
I 1)
three-dimensional finite region V, containing the third-order pole
Z
(xl,y
whose boundary is a closed Lyapounov surface S: [33]
I(x
I,
(xl yl
f
1
z )=
y
r
V
I
g( x 1
z
1
*) u(x, y, z)dv
0,
1
0, )
+ (y-y I)2 + (z-z I)
[(x-x )
y
I
3
z
u
3/2
(x,y,z) dv
(4.15)
Furthermore let us introduce the following system of spherical coordinates:
x
x
y
y
z
z
1
+
r sin 0 cos
1
+
r sin 0 sin
(0 S
=< )
1
+
r cos 0
(0 S
=<
(x-x I)
r2
2
+
(y-yl)
2
(4.16)
)
+ (z-z 1) 2
Then, from eqn (4.16) we obtain:
2 R(O,)
I--iim
0
0
f 0f f
sinO
f(O,,)u(r,O,)
dO
Thus, if we integrate eqn (4.17) with respect to O and
d,
dr
(4.17)
by using the trapezoidal
rule with abscissae G, D, then we’ll get the following formula:
G
D
I--
sin
G---
0
i
(0 i,
i
(4.18)
j=l
i=l
in which:
i
(i
I)
(4.19)
Cj=
2 (j- I)
D
R(O,)
(0 )
and
g(O )
J
dr
(4.20)
,0
For the numerical evaluation of the integral in (4.20) let us use the following
numerical integration rule:
R(O,)
0
where
f
u(r,O,) dr
r
Pk
L
Z A u [R(O,)
k
k=l
pk,O,] +
u(0,O,) In[R(O,)]
(4.21)
are the abscissas and A the weights for the integration interval [0,I].
k
569
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
5. ASIC FORMIYAE FOR THE THEORY OF LINEAR VISCOELASTICITY.
Let us consider a linear isotropic viscoelastic solid with the following stress
field:
Oij
2
Oil
3
0
where
oj
field, V
V2(F,t-T)
0
(F’0+)
J
with:
f V I(F,t-T)
and
--
(F,O+)
(5.1)
(F,z) dT
+
V2(F,t eli
(F,O+)
(5.2)
(F,t)
(5.3ab)
ii (F’0+)
(F,t),
t/01im eli
are, respectively, deviatoric components of the stress and strain
Eij
and V
t/01im ij
ej
(F,T) dT + V l(F,t)
ST
are relaxation functions in shear and isotropic compression,
2
respectively, t is the time and F denotes a point in the infinite elastic space where
the viscoelastic solid belongs.
By taking the Laplace transform of eqs (5.1) and (5.2), one obtains:
%j
2s V
ii *
3s
(s)
V2* (s)
eli
(5.4)
ii *
(5.5)
Furthermore, the Laplace transformed equation of equilibrium is as following:
,
(5.6)
0
oij,j
Let us use Hooke’s law for a linear isotropic, viscoelastic solid:
o
,
ij
(F)
X
,
Un,n
(F)
61j
+
,
U
[ul, j
,
where u
A*
denotes the boundary displacements,
i
constants and
6ij
(F) +
,
uj,i (F)]
,*
and
(5.7)
the additional Lame elastic
is the delta function of Kronecker.
By applying the Betti-Rayleigh theorem the boundary displacement yields:
u
where
t.*
3
i
* (F)
L
uj * (H)Tij
(H,F) dH +
f tj*(H) Uij(H,F)
dH
(5.8)
L
are the boundary tractions on the contour L and H,F are points on the
union of the contour L.
In (5.8) the fundamental solutions for the displacements and tractions are
given by the following relations:
Uij
+ *
*
8* (X*+2),
3
I
I*
*
+ ,)
6ij+(’X ,+3
r
i
(5.9)
r,j
, ,
Tij
4
(2*+X*)
n
+
*
r
,i
r
,j
njr
i+ni r ,J |
(5. I0)
E.G. LADOPOULOS
570
By combining eqs (5.4), (5.5) and (5.7) one obtains:
},
-2/3 s V
(s) + s V 2 (s)
(5.11)
s
v
(s)
Finally, for small strain, the transformed components of strain and displacement
are related by the formula:
2
ij
ulj,
i
+ uji,
(5.12)
i
is given by (5.8).
i
Thus, the components of the stress field for a linear, isotropic, viscoelastic
where u
solid can be evaluated by using the numerical technique of Section 4.1.
6.
AN APPLICATION OF LINEAR VISCOELASTICITY.
As an application of the previous theory, let us consider a thick, hollow
circular cylinder of viscoelastic material restrained by an enclosing thin elastic
ring and subjected to a uniform pressure p, applied as a step in time at t
0
(see:Figure I).
The same problem has been previously solved by Ting [39] by using an exact
analytical solution and by Rizzo and Shippy [40] by using the Boundary Integral
A comparison will be made between the new Singular
Integral Operators Method (S.I.O.M.) introduced in the present report, the theoretical solution [39] and the B.I.E.M. [40].
Equation Method (B.I.E.M.).
Vt ;c, octc;t Lc,
Figure I:
A thick, hollow circular cylinder of viscoelastic solid,
restrained by an enclosing thin elastic ring and subjected
to a uniform pressure p.
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
The geometrical sizes of the cylinder are as follows:
571
Ratio of inner to outer
R2/R
--0.3 (see:Figure I).
Moreover we consider that the viscoelastic solid behaves elastically in bulk
and as a standard linear solid in shear, so that the relaxation functions V and V
radii of the cylinder is
2
are given by the formulas:
V1(t)
V [g
V2(t)
N
+ (l-g)
e
-x’]
(6.2)
where g, l, V and N are constants and we assume that g
Thus, the transformed relaxation functions V
*
0.5 and V
and V
2
0.6 N.
are given by the follow-
ing relations:
,
V
V
2
(V/s) (s + gl) / (s + X)
(6.3)
N / s
(6.4)
Furthermore the thin, elastic ring is characterized by the relation:
C
where E is the
(Fig. I).
E d / (I
9) R 2
Young’s modulus, v
(6.5)
Polsson’s ratio, and d its thickness
Also we assume the value of C/N to be unity: C/N
I.
is the
Figures 2 and 3 show the radial
gr/p
and transverse stress
gs/p,
respectively,
as functions of time.
TL’m
0
0
2
3
k+/-
7
6
5
4
mm,mmmmmmm
.o
//,=
,ia =0:825
mmmm(
’R/R, =0.650
0.2 ,-,,_., ,..=.., ,_._ mX
R/P,
=10:4
75
x
S.I.O.M.
0
B.I.E.M.O]
R/R, =0.3’
Figure 2:
Radial stresses
Figure I.
r/p
as functions of time for the cylinder of
572
E.G. LADOPOULOS
x
S.I.O.M.
0.6
0.4
0.2
’=-’,=,-.>,===)_=,,=.=).,=_,).== ==)
-
).=,)_=,
=’="()=..)====
0
0
Figure 3:
)==,===_
2
-=.,,=I()--=.--()
)--===,..()--==-,,.=(
4
3
Ti.’m M:
Transverse stresses
of Figure 1.
o0/p
5
75
R/R=0.650
R/R,=O.B25
6
7
as functions of time for the cylinder
Finally, as it is easily seen from Figures:2 and 3, the numerical results of the
S.I.O.M. coincide very well with the theoretical results [39] and the additional
numerical results of the B.I.E.M. [40].
7.
CONCLUSIONS.
A new numerical technique has been investigated for the numerical evaluation of
the multidimensional singular integrals and integral equations used in many fields
of mathematical physics.
Especially for the surface singular integrals, some formulas are derived by
using the Gauss-Legendre, Lobatto and Radau numerical integration rules.
For the
construction of such a cubature formula, the two-dlmensional singular integral was
considered as an iterated one, and the second-order pole involved in this integral
was analyzed into a pair of complex poles.
Thus, the methods of numerical integra-
tion, valid for one-dlmenslonal singular integrals, were extended to the case of the
two-dlmenslonal singular integrals.
Also, a complete analysis for the numerical
evaluation of the three-dlmenslonal singular integrals was also presented.
The technique for the numerical evaluation of the surface singular integrals
described in section 4.1 has been used for the determination of the fracture
behavior of a linear, viscoelastic, isotroplc solid.
An application was given to the
determination of the radial and transverse stresses in a thick, hollow circular
cylinder of viscoelastic solid restrained by an enclosing thin elastic ring and
subjected to a uniform pressure.
NUMERICAL SOLUTION OF SINGULAR INTEGRALS AND INTEGRAL EQUATIONS
573
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