Boundary Element Methods: Principles and Applications : Proceedings of the Third Japan-China Symposium on Boundary Element Methods, 4-7 April, 1990

387 Pages·1990·19.78 MB·English

by  Masataka Tanaka

Pergamon Titles of Related Interest
BAUTISTAetal Theory of Machines &
Mechanisms
Dl PILLO Control Applications of
Nonlinear Programming &
Optimization
GEERING&MANSOUR Large Scale Systems: Theory &
Applications 1986
GHOSH &NIKU-LARI CAD/CAM &FEM in Metal
Working
HEARN Mechanics of Materials, 2nd
edition
HORLOCK Cogeneration: Combined Heat&
Power
KANT Finite Elements in Computational
Mechanics (2-volume set)
MAZHARetal Mathematical Analysis & its
Applications
NIKU-LARI Structural Analysis Systems,
Volumes 1-5
RAO The Finite Element Method in
Engineering, 2nd edition
YANetal Mechanical Behaviour of
Materials V
Pergamon Related Journals (free copy gladly sent on request)
Acta Mechanica Solida Sinica
International Journal of Engineering Science
Computers & Fluids
International Journal of Impact Engineering
International Journal of Mechanical Sciences
Journal of Applied Mathematics & Mechanics
Mechanics Research Communications
Mechanism & Machine Theory
BOUNDARY ELEMENT METHODS
Principles and Applications
Proceedings of the Third Japan-China Symposium on
Boundary Element Methods,
4-7 April, 1990, Hachiohji, Tokyo, Japan
Editors
MASATAKA TANAKA QINGHUA DU
Department of Mechanical Engineering Department of Engineering Mechanics
Shinshu University Tsinghua University
500 Wakasato, Nagano 380, Japan Beijing, China
Sponsored by
Japan Society for Computational Methods in Engineering (JASCOME)
Beijing Society of Mechanics, China
Tsinghua University, Beijing, China
PERGAMON PRESS
Member of Maxwell Macmillan Pergamon Publishing Corporation
OXFORD · NEW YORK · BEIJING · FRANKFURT
SÄO PAULO · SYDNEY · TOKYO · TORONTO
U.K. Pergamon Press pic, Headington Hill Hall,
Oxford 0X3 OBW, England
U.S.A. Pergamon Press, Inc., Maxwell House, Fairview Park,
Elmsford, NY 10523, U.S.A.
PEOPLE'S REPUBLIC Pergamon Press, Room 4037, Qianmen Hotel, Beijing,
OF CHINA People's Republic of China
FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6,
OF GERMANY D-6242 Kronberg, Federal Republic of Germany
BRAZIL Pergamon Editora Ltda, Rua Ega de Queiros, 346,
CEP 04011, Paraiso, Säo Paulo, Brazil
AUSTRALIA Pergamon Press Australia Pty Ltd., P.O. Box 544,
Potts Point, N.S.W. 2011, Australia
JAPAN Pergamon- Press, 5th Floor, Matsuoka Central Building,
1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan
CANADA Pergamon Press Canada Ltd., Suite No. 271,
253 College Street, Toronto, Ontario, Canada M5T 1 R5
Copyright © 1990 Pergamon Press pic
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
in any form or by any means', electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers.
First edition 1990
Library of Congress Cataloging-in-Publication Data
Japan-China Symposium on Boundary Element Methods
(3rd: 1990: Tokyo, Japan)
Boundary element methods: principles and applications:
proceedings of the 3rd Japan-China Symposium on
Boundary Element Methods, April 4-7, 1990, Hachiohji/
Tokyo, Japan/editors, Masataka Tanaka, Quinghua Du:
sponsored by Japan Society for Computational Methods in
Engineering (JASCOME), Beijing Society of Mechanics,
China [and] Tsinghua University, Beijing, China.—1st ed.
p. cm.
Includes bibliographical references.
I. Boundary element methods—Congresses. 2. Mechanics,
Applied—Congresses. I. Tanaka, M. (Masataka), 1943-
II. Tu, Ch'ing-hua. III. Japan Society for Computational
Methods in Engineering. IV. Ch'ing hua ta hsüeh (Peking,
China). V. Beijing Society of Mechanics. VI. Title.
TA347.B69J36 1990 620'.001 '51535—dc20
90-6844
British Library Cataloguing in Publication Data
Japan-China Symposium on Boundary Element Methods
(3rd: 1990: Hachiohji/Tokyo (Japan))
Boundary element methods: proceedings of the 3rd Japan-
China Symposium on Boundary Element Methods, April
4-7, 1990, Hachiohji/Tokyo, Japan.
1. Mathematics. Boundary element methods
I. Title. II. Tanaka, Masataka. III. Du, Qinghua
515.35
ISBN 0-08-040200-3
Printed in Great Britain by BPCC Wheatons Ltd, Exeter
Preface
The integral equation formulations of the initial-boundary value problems are the
foundation of the boundary element method (BEM), and have a long history in
applied mechanics. Significant advances in the BEM, however, have been made in
the last two decades.
The rapid developments in boundary element research in the last ten years have been
mainly due to the innovation of efficient computational techniques by introducing
boundary elements for discretization of the boundary integral equations resulting
from the so-called direct formulation. Owing to many efforts in this respect, the
BEM has been widely recognized as a powerful alternative to the domain-type
numerical methods of solution such as the finite element method. In fact, for most
linear problems the BEM is more efficient and more accurate than the finite element
method, mainly because only the boundary surface is to be discretized in the BEM.
From this point of view the BEM can provide an efficient tool for optimal design and
other inverse problems, and hence CAD/CAM/CAE rather than the finite element
method, as can be seen in this book.
Further efforts continue to be made to innovate and develop more efficient solution
procedures based on the BEM for both linear and nonlinear problems. The impact
of the advanced computer technology including super and parallel computers is a
major element in further extensions and applications of the BEM.
The series of the joint Japan-China Symposia on BEM was first planned in 1985 to
bring together active researchers in the two countries working for computational
algorithms and applications of the BEM. The first Joint Japan/China Symposium
was held on 1-5 June 1987 in Karuizawa, Japan, and the second one on 11-15
October 1988 in Beijing, China. This third Symposium, to be held again in Japan
on 5-7 April 1990, will provide good opportunities to discuss and exchange new
research ideas for further developments of the BEM. Moreover, we hope that this
book, published as the Proceedings of the 3rd Symposium will convey useful
information to the boundary element people, not only in the two countries but also
throughout the world.
December 1989
Masataka TANAKA
Shinshu University
Japan
Qinghua DU
Tsinghua University
P.R. China
TDBKM—A·
IX
COMMITTEES
ORGANIZING COMMITTEE
Professor Masataka Tanaka (Chairman)
Shinshu University, Japan
Professor Qinghua Du (Co-Chairman)
Tsinghua University, China
LOCAL ORGANIZING COMMITTEE
Professor M. Tanaka Professor N. Kamiya
Shinshu University (Chairman) Nagoya University
Professor T. Aizawa Professor J. Kihara
University of Tokyo University of Tokyo
Professor S. Aoki Professor K. Kishimoto
Tokyo Institute of Technology Tokyo Institute of Technology
Professor T. Fukui Professor M. Kitahara
Fukui University Tokai University
Professor H. Hasegawa Professor S. Kobayashi
Meiji University Kyoto University
Professor T. Honma Professor K. Onishi
Hokkaido University Fukuoka University
Professor A. Ishiyama Professor N. Tosaka
Waseda University Nihon University
Professor Y. Kagawa Professor R. Yuuki
Toy a ma University University of Tokyo
xi
BOUNDARY ELEMENT ANALYSIS OF THREE DIMENSIONAL
ANISOTROPIC BODY WITH A CRACK
H. Ishikawa^ and H. Takagi2>
DDept. of Mechanical and Control Engineering, Univ. of
Electro-Communications, Chofu, Tokyo, Japan
2)Bridgestone Corporation, Kodaira, Tokyo, Japan
ABSTRACT
Three dimensional anisotropic bodies are analysed by the boundary element
method with the fundamental solutions of Wilson and Cruse. The effects of
the accuracy of the fundamental solutions obtained by the numerical
integration on the boundary element analysis are discussed. Some crack
problems are calculated by the present boundary element method, and the
stress intensity factors for the cracks are obtained with fairly good
accuracy.
KEYWORDS
Three dimensional anisotropic materials; elasticity; stress intensity
factor
INTRODUCTION
Recently, various kinds of composite materials have been developed and used
for structural components. In the progress of the developmenet, the
analysis of the mechanical behaviour of the composite is essential to the
design of the composite structures. Some composite materials, such as fiber
reinforced plastics, show the anisotropic elasticity. The analysis of
the anisotropic bodies by the boundary element method have been carried
out by some researchers (Cruse et al.,1971, Ishikawa et αί.,1986, Ishikawa
et ai.,1988, Nishimura et αί.,1983, Rizzo et crf.,1970, Snyder et αί.,1975,
Wilson et αί.,1978). In the present study, a basic investigation on the
accuracy of the fundamental solution is carried out to analyse the three-
dimensional anisotropic, elastic body. The fundamental solutions of
traction force and displacement that are presented by Wilson and Cruse
(1978) are used in the present paper. The fundamental solutions are
obtained by the numerical integration. The effect of the numerical
integration on the boundary element analysts is discussed. Based on the
discussion, some crack problems are analysed, and their fracture
parameter, the stress intensity factors, are obtained.
3
PROCEDURE OF ANALYSIS
We will begin by stating the boundary integral equation for the linear
elastostatics in its general form. The integral equation for the boundary
point of the domain under consideration is
cUj=J(Uj i'ti-Tj i*Ui )ds (1)
5
In the case of the 8-nodes isoparametric boundary element used in the
present paper, the fundamental solutions are obtained by the numerical
integration. In order to discuss the effect of the numerical integration on
the boundary element analysis, the process of the derivation of the
fundamental solutions is briefly introduced. By using the summation
convention, the equaibrium equation of the three-dimensional elastic body
can be expressed as
Ci J kmUl k. j m=öi 1 δ(Χ,Υ) (2)
where Cijkm(i,j,k,m=l,2,3) is the elastic constant, U« is
t
thedisplacement.the subscripts, 1 and k, mean the directions of the unit
force and the displacement, respectively, and Uik,jm is the partial
derivative of the displacement, Ui k, with respect to Xj and x . X and Y
m
mean the observation point and the source point of the unit force,
respectively, δ; ι is the Kronecker delta, and δ(Χ,Υ) is the Dirac's delta
function. The solution of Eq.(2) is expressed as (Wilson and Cruse 1978):
Uij (X) = J6(RP-Q)Kij"! ds/βπ2 (3)
s
where P and Q are difined as the vectors from the center (origin) to the
surface of the unit sphere in the § space and given by P=X/|X| and Q=l/I,
respectively. R=|X| and Ki j is given by
Ki j =Ci 1 j m §1 §m (4)
By using the characteristics of the delta function, the area
integration,Eq. (3), over the sphere surface S can be reduced to the line
integration as
Ui j (X) = JKi j-i<to/8ir2R (5)
c
The line integration path C is the unit circle which lies on the plane
perpendicular to X. Equation (5) is the fundamental solution of the
displacement of the anisotropic body which has the elastic constant,
Ci 1 j m .
Equation (5) cannot generally be evaluated in closed form. For an Isotropie
material, it reduces analytically to the well-known Kelvin's solution. For
an anisotropic material, one of the methods to evaluate the functions, Uij ,
is a direct numerical method. In the present research, the numerical
integartion by the trapezoidal rule is used.
Equation (5) is a function of the directions, βι and 02 , of the vector X
and the angle φ as shown in Figs.l and 2. φ is defined on the plane
perpendicular to X. The starting line for measuring φ can be arbitrary.
The coordinate at the arbitrary point on the integration path is related to
the angles, 0i ,02 and φ, as
4
[§i I [cos02cos0i -sin02" cos<I>
(6)
§2 = Sin02COS0l COS02 βίηΦ
l§] l -sinöi 0
3
Ki j can be obtained from Eqs.(4) and (6). Then, taking the inverse
matrix,Kij_1 , by Cramer's formula, the fundamental solution can be
evaluated by the numerical integration and the distance, R, from the source
point of the point load to the observation point. The representation for
the traction point load solution, T, j , can be derived from the derivative
of the fundamental solution of the displacement, Ui j , and the elastic
constant.
NUMERICAL INTEGRATION OF THE FUNDAMENTAL SOLUTIONS
The fundamental solutions for the three-dimensional anisotropic body are
not directly computable. As mentioned previously, the solutions can be
numerically obtained. In the present paper, the trapezoidal rule for the
numerical integration is used. To discuss the accuracy of the integration,
an example of the observation point is adopted. The example point has the
coordinates of Xi =lmm, X2=2mm, X3=3mm and the normal directions of
ni =n2=n3=1/73. The effects of the numbers, n, of the integartion points on
the accuracy are shown in Table 1 for the isotropic and anisotropic bodies.
The isotropic case has Young's modulus of 206 GPa and Poisson's ratio of
0.3. The anisotropic case has the different Young's modulus, Et ,E2 and E3,
in each directiopn of the coordinate axes, xi,X2,X3,, that is, Ei of 9.8
GPa, E of 2Ei , and E3 of 3Ei .
2
From Table 1, for the isotropic case, much less number, n, brings the good
convergency of the fundamental solutions of displacement and traction
4
force. However, for the anisotropic case, more number, n, is necessary. The
fundamental solutions are fairly converged with the number of 30. In the
present research, the numerical integrations are carried out with the
number of 40.
On the other hand, the numerical value of the fundamental solutions depends
on the geometry of the given problem and the element mesh pattern for the
problem. Then, if we do the numerical integration for all the calculation
to get the value of the fundamental solutions, even the case of one mesh
pattern of the given problem requires the huge amount of the calculation.
So, in the present paper, we use an interporation technique. A table of the
numerical values of the fundamental solutions for the actual observation
points are obtained by the interporation from the values in the table.
Since the numerical integartion is carried out for the angles, 0i and 02 ,
of Eq.(5), determining the position of the observation point, the 2-
dimensional table for the discrete values of θι and 02 with the equal
interval of Δ0ι=Δ02=Δ0 is made. Lagrange's interporation tecnique is used
to obtain the value of the fundamental solutions. Namely, the values at the
actual observation point is interporated from the values at 16 points
around the observation point in the table as shown in Fig.3.
To discuss the effect of Δ0 on the accurcy of the boundary element
calculation, the five kinds of Δ0 (10* , 15° , 22.5° ,30° , 45° ) are used. By
using the Δ0, the following example problems are solved;
(a) an uniform tension problem of a square rod, shown in Fig.4,
(b) an uniform tension problem of a square rod with a center through-
crack, shown in Fig.6.
The materials of the example problems are the isotropic materials that have
Young's modulus of 206 GPa and Poisson's ratio of 0.3. These problems have
5
the geometrical symmetries. So, one eighth region of the body is analysed.
An example of the boundary element mesh for each problem is shown in Figs.
5 and 7. The element type used is the 8-nodes isoparametric element. In the
problem (a), the displacement at the loading point is calculated, and its
error, En , is discussed by comparing with the analytical value. In the
problem (b), the stress intensity factor at a point on the crack front is
calculated, and its error, Er , is discussed by comparing with the
2
numerical results(Yagawa et αί.,1977) of the finite element method. These
results are shown in Table 2. From Table 2, it is found that the results
are almostly converged with ΔΘ of 15° . In the present study, the ΔΘ of 10°
is used for the boundary element calculation of the anisotropic body.
NUMERICAL EXAMPLES
In order to discuss the accuracy of the numerical results obtained under
the present consideration, the following crack problems are analysed and
their stress intensity factors are calculated.
(a) an uniform tension problem of a transversely isotropic square rod with
a center through-crack,
(b) an uniform tension problem of an anisotropic cylindrical rod with a
penny-shaped crack.
In these calculation, to improve the accuracy of the results, two types of
crack element showing the stress singularity of the crack is adopted
around the crack tip. One is the well-known quarter element, A, as shown in
Fig. 8. Another crack element, B, shown in Fig. 8, is located at the
neighbor of the crack elements, A. In the element, The position of the
middle-point node is moved to the location defined by the equation (Lynn et
al., 1978)
r = (7r, +7r )2/4
2 3
where n (1=1,2,3) is defined in Fig. 8.
A Center Through-Crack in a Transversely Isotropic Square Rod
A square rod under uniform tension, σ, is solved. The material is the
transversely isotropic material that is one of the general anisotropic
three-dimensional materials. Namely, the xi-X2 plane, shown in Fig. 6, is
the isotropic one, and the X3 direction shows the anisotropic elasticity.
The material used is an unidirectional glass-fiber reinforced epoxy, GFRP,
the elastic properties of which are shown in Table 3. Because of the
symmetry of the problem, one eighth region of the cracked body is analysed.
An example of the number of elements is 58 for a case of crack length
ratio, a/b (2a: crack length, 2w: square rod width) of 0.5. The ratio used
is 0.4, 0.5, and 0.6.
The analytical method to evaluate the stress intensity factor, Ki , is the
so-called displacement method. Thus, Ki is obtained from the relationship
between the Ki and the crack tip opening displacement, v, given by Eq.(7).
ν=7(2Γ/π)Κι B Im[-(|üti +μ )/μι μ ] (7)
33 2 2
In Eq.(7), r is the distance from the crack tip, μι< (k=l,2) is the complex
root of the characteristic equation of the anisotropic matetrial, and B33
is one of the compliance coefficients. An example of the relationship
between v and r is shown in Fig. 9. From the slop of the straight line In
Fig. 9 and Eq.(7), Ki is obtained. The results are shown in Table 4. In the
6
table, the normalized value, Fi , of Ki by aVira is used. Fi * is the results
given by using the analytical fundamental solution of the transversely
isotropic material. Comparing these results, the difference between them is
less than 0.6 %. Then, it is said that the present calculation is carried
out with fairly good accuracy.
A Penny-Shaped Crack in a 3 Dimensional Anisotropie Square Rod
An uniform tension problem of a cylindrical rod with a penny-shaped crack
(Fig. 10) is analysed. The crack radius ratio, a/b (a: radius of the panny-
shaped crack, b: radius of the cylinder), is 0.5. The distribution of the
normalized stress intensity factors, Fi, along the crack front is
obtained. The position to determine the Fi is defined by the angle, 0,
measured from the xi-axis, and shown in Table 5. Because of the symmetry of
the problem, one eighth region of the body is analysed. The mechanical
properties of the material used are shown in Table 6. The numerical results
are given in Table 5. From this table, it is found that the value of the
non-dimensional stress intensity factor, Fi , is increased in the counter-
clockwise direction from the xi-axis.
CONCLUDING REMARKS
Three-dimensional anisotropic bodies are analysed by the boundary element
method with the fundamental solutions of Wilson and Cruse. The trapezoidal
rule for the numerical integration is one of the useful methods to estimate
numerically the fundamental solutions. Some three-dimensional crack
problems are calculated by the present boundary element method, and the
stress intensity factors for the cracks are obtained with fairly good
accuracy.
REFERENCES
Cruse, T.A. and Swedlow, J.L. (1971). Air force Rep., No. AFML-TR-71-268.
Ishikawa, H. and Yamagata, H.(1986). Boundary element analysis of an
orthotropic plate with a circular hole or crack. Trans. Jap. Soc.
Mech. Engineers, 52, 2646-2651.
Ishikawa,H. and Takagi, H. (1988). Three dimensional boundary element
analysis of transversely isotropic elastic body. Trans. Jap. Soc.
Mech. Engineers, 54, 471-475.
Lynn, P.P. and Ingraffea, A.R. (1978). Transition elements to be used with
quarter-point crack- tip elements. Int. J. Numer. Methods Eng, 12,
1031-1036.
Nishimura, N. and Kobayashi, S.(1983). A boundary integral equation
formulation for three dimensional anisotropic elasto statics. Proc.
5th int. conf. Boundary element, 345-354.
Rizzo, F.J. and Snippy D.J. (1970). A method for stress detemination in
plane anisotropic elastic bodies. J. Composite Materials., 4, 36-
61.
Snyder, M.D. and Cruse, T.A. (1975). Boundary-integral equation analysis
of cracked anisotropic plates. Int. J. Fract., U, 315-328.
Wilson, R.B. and Cruse, T.A. (1978). Efficient implementation of
anisotropic three dimensional boundary-integral equation stress analy
sis. Int. J. Num. Meth. Engng., 12, 1383-1397.
Yagawa, G., Ichinomiya, M. andAndo, Y. (1977). An analytical method of
stress intensity factor bt the discritization error of finite
element method (in Japanese). Preprint Jap. Soc. Mech.
Engineers, No.770-1, 29-36.
7