ing arbitrarily shaped plates containing cracks or notches ... · ing arbitrarily shaped plates containing cracks or notches of general shape and a circular elastic inclusion. Our

Versatile stress analysis of arbitrarily shaped

plates with an inclusion by body force method

H. Nisitani," A. Saimoto^

"Department of Mechanical Engineering, Kyushu Sangyo

University, 3-1-2 Matsukadai, Higashi-ku, Fukuoka 813, Japan

^Department of Mechanical Systems Engineering, Nagasaki

University, 1-14 Bunkyo-machi, Nagasaki 852, Japan

Abstract

The body force method is extended so as to be applicable to the analysisof the elastic problem of an arbitrarily shaped plate with an inclusion. Themost characteristic point of the present method is use of the elastic fielddue to an isolated force acting in an infinite medium containing a circularinclusion as the fundamental solution. By using this elastic field, we cantreat easily various problems of homogeneous, inhomogeneous, finite aridinfinite plates, only by changing the material constants in a matrix andan inclusion, or by varying a radius of the inclusion in the range of zeroto infinity. Many results relating to the problems of cracks, notches andinclusions are presented. In addition to the theoretical background of thepresent method, a short history of the progress of the body force method isalso shown.

1 Introduction

In order to predict the failure or fracture behavior of materials, an analyzingsystem which enables us to obtain highly accurate solutions easily for theinterference problems among arbitrarily cracks, notches and inclusions in ahomogeneous matrix material is indispensable.

In the present paper, the authors develop a versatile method for analyz-

Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

110 Localized Damage

ing arbitrarily shaped plates containing cracks or notches of general shapeand a circular elastic inclusion. Our numerical technique is based on theBody Force Method (BFM), which is one of the boundary type numericalmethods for stress analysis and was originally proposed by the first authorin 1967.

In order to develop a versatile and highly accurate calculation systemsfor 2-dimensional stress analysis concerning to the inhomogeneous materi-als, we use an elastic stress field due to an isolated point force acting in aninfinite plate containing a circular elastic inclusion as the fundamental solu-tion for the present BFM. By using this fundamental solution, we can easilytreat homogeneous, inhomogeneous, infinite or semi-infinite plate problems.

2 Body Force Method

BFM was originally proposed by H. Nisitanifl] in 1967. In the early stage ofthe progress of the method, it was mainly applied to the el as to-static prob-lems for determining the stress concentration factors of notches or the stressintensity factors of cracks[2]. Then, this method was extended gradually soas to be applicable to the elastic-plastic problems[3-5], elasto-dynamic prob-lems ], thermoelastic problems[7] and the interface crack problems[8,9].

BFM is a method based on the principle of superposition. In BFM,the solutions are always obtained by superposing the fundamental stressfields. That is, in an infinite plate (for 2D problems) or in an infinite body(for 3D problems), the point forces and the discrepancies are continuouslydistributed along the contour (which has the same shape of the given prob-lem but is just an imaginary boundary, not the real one) so that the givenboundary conditions specified on the real boundary are satisfied. Thus, therequired elastic field is obtained simply by superposing the fundamentalelastic fields due to the point forces and the discrepancies applied in aninfinity plate or body. Since the fundamental principle of BFM is simpleand intuitive, the various inventions for improving the accuracy of solutionsare easy to be taken in. In fact, from the beginning of the development,the BFM has been produced a lot of dependable solutions of important en-gineering problems by introducing suitable inventions which have referenceto the physical characteristics of individual problems.

BFM should be recognized as being effective, not only for a numericaltechnique but also for the theoretical method for obtaining the exact solu-tions. In general, the exact solution of an elastic problem must satisfy thethree conditions shown in Table 1. To solve the elastic problems, there arethree types of method of solution ,that is, (i) use of displacement function,(ii) use of stress function and (iii) use of fundamental solution, which isthe stress field due to a point force acting in an infinite plate or body. Inmethods (i) and (ii), we have to determine the concrete form of the speci-fied functions. Generally, in the methods (i) and (ii), the elastic problem istransformed into the form of differential equations. On the other hand, in


Localized Damage 111

the method (iii), the given problem is transformed into the simple problemof determining the density of distributed point forces acting in an infinitemedium. As supposed from above discussion, BFM (method (iii)) is themost suitable technique for numerical analysis among those three methods.Furthermore, we can obtain the closed form solution for the density of bodyforce (continuously distributed point force), as far as the problem has anexact closed form solution.

Table 1. Comparison of solution methods for elastic problem

(a) Equilibrium condition

(b) Compatibility condition

(c) Boundary condition

must be satisfied

in the elastic problem.

0)1

00

(iii)

Solution method

Jse of displacement function u[ (b) is always satisfied ]

Use of stress function y[ (a) is always satisfied ]

Use of fundamental solution(a),(b) is always satisfied ]

Express (a) byand determine

Express (b) byand determine

Determine thefundamental sc

using uu from

using ({y from

density>lutions

(c).

(c).

offrom (c).

3 Fundamental Solutions Used in Two-DimensionalStress Analysis by BFM

In this section, a short history of the progress of BFM within the rangeof two dimensional problems is explained from the standpoint of the fun-damental solutions used in the analysis from the beginning of BFM untiltoday.



Fundamental solutionsTypical problems solved by

left fundamental solutions

t t t t t t t !

TTTTTTTT(Tension of rectangular plate)

(ID

numerical integrationwas used (1967) [1]

'*•• ••*" Ktof semi-circular(Tension of semi-infinite notch was 3.0654.plate with a notch) (exact: Kt = 3.0653)

_numerical integration wasnot used (1990)[10]

Each interval except acircular arc is appoximatedby a straight line.

(specimen analysis)

numerical integration was not usedexcept the oscillation singularityat an interface crack tip (1994)[9]

(interface crack)

numerical integration was not usedexcept the oscillation singuralityat an interface crack tip (this paper)

G2-K.J

(Interfarence between crack and inclusion)

Figure 1 Short history of fundamental solutions and typical examples



4 Procedure of Analysis for 2D Problems by BFM

Based on the principle of BFM, a given problem, for example, Figure 2(a)is transformed into a problem of an infinite plate (Figure 2(b)). In problem(b), continuously distributed point forces (body force) are applied along thecurves or lines which correspond to the real boundary in problem (a) exceptthe crack boundaries. For the crack boundaries, continuously distributedpoint force doublets (body force doublet) are applied.

t t ft f

1TTTT

(a) Given problem (b) Infinite plate with body forceand body force doublet

Figure 2 Solution procedure in BFM

Usually, the magnitudes of applied body force or body force doublet aredetermined from a discretization procedure. That is, the imaginary bound-aries are divided into several intervals, the unknown magnitude of the bodyiorce or the body force doublet in each interval is assumed to be constant orchange linearly and the unknowns are determined from the given boundaryconditions. In this paper, the resultant force on each interval is used forexpressing the given boundary conditions[ll,12].

»*|3:

SingleImaginaryBoundary

f-

Z; int

>

1*_L

4-

i i \

erval i

*y

&DoubleImaginaryBoundary

t t I-

f

-kI interval jB /_!_

ir-

f

Division ofimaginary boundary

iCalculation ofinfluence coefficent

*Make up influencecoefficent matrix

4Solve simultaneousequations

4Calculation of sires;and displacement

Table 2

in Table 2

(3) in Table 2

Figure 3 Division of imaginary boundary and solution algorithm



Table 2 Detailed explanations for procedure ©,© and (3) in Figure 3in the case of stress boundary value problem

Calculation of influence coefficient

| SIB* | The influence coefficient e,-j in SIB is the resultant force of interval"j" due to the unit body force of interval "i". e-ij is obtained byusing following relation.

where, $l(z),u(z) are complex potentials expressing the fundamen-tal solution of point force (see, next section).

The influence coefficient eij in DIB is the resultant force of interval"j" due to the unit weight function of interval "i", considering thebasic density function[13] (see, section 6). €{j is obtained byusing following relation.

where, (l(z),w(z) are complex potentials expressing the fundamen-tal solution of point force doublet (see, next section).

@ Solving simultaneous equations

Unknown density of body force for interval "i" (when "i" is SIB) or un-known weight function of body force doublet for interval "i" (when "i" isDIB) pi is obtained by solving the simultaneous equations expressing theboundary condition in each interval "j".

W • W = %}

{pi} : vector of unknowns, {&,} : vector of boundary conditions

(3) Calculation of stress and displacement

Stress and/or displacement at a certain point in the body can be expressedin the form of the linear combination of pi already obtained in procedure

SIB means Single Imaginary Boundary shown in Figure 3.DIB means Double Imaginary Boundary shown in Figure 3.



5 Concrete Form of Fundamental Solution in PresentAnalysis

The elastic field such as stresses, displacements and resultant forces over anarc can be expressed in terms of complex potentials, say fi(z) and u>(z), byuse of the following well known relations.

' xy = 2[zfl"(z)

= Ktt(z)- l'(z) - u(z)

(1)(2)

(3)

In these equations, z is a complex number expressing the coordinateof the reference point, <T,-J(Z, j = x,y) are the components of stress at z, u,v are the components of displacement at z and P,-(z = x,y) are thecomponents of resultant force over an arbitrary curve connecting a certainfixed point to z. G is a shear modulus of material, and K is also a materialconstant defined as AC = (3 - z/)/(l + z/) for plane stress and AC = 3 - 4z/ forplane strain, where z/ is Poisson's ratio. In order to simplify the expressionof complex potential w(z) in the solution of Figure 4, we rather use another

holomorphic function %(z), defined as %(z) = —ft'(z) +w(z), in stead of

using w(z) itself. Where, a is a radius of a circular inclusion.Let us denote the matrix region z\ > a by Si in which the material

constants are GI,ACI, and the inclusion region \z\ < a by % where the ma-terial constants are %, #2, respectively. The complex potentials in Figure4 can be obtained by use of the continuation theorem of complex function.Their concrete forms for the case I in Table 3 are given as follows.

Table 3 Classification of the

present fundamental solutions

Case

Case

Case

Case

I

II

III

IV

Zo G

Zg G

ZQ G

ZQ G

Si

Si

82

and

and

and

and

z

z

z

z

G

G

G

G

Si

S2Si

82

Figure 4 Fundamental solution

used in the present analysis (Case I)



For case I: ZQ £ Si and z €

0(2) =

(5)

where MI ,7Vi and AI are defined as,

(6)

(7)

a- /3 MI MI"

and a, ft are Dunders's composite parameters.

(9)

The complex potentials for the cases II,III and IV are given in a similarmanner.

For the calculation of the crack boundary, the body force doublets areapplied along the double imaginary boundary. The complex potentials ofthe elastic field due to the standard body force doublets[13] shown in Figure5 are given by combining the differentiated results of Eqs. (5),(6).

o-x X (plane stress)

(a) Tension type (b) Shear typeFigure 5 Standard body force doublets

H= mo—»U



The magnitudes of body force doublets 71 and 7n correspond directlyto the crack surface opening and sliding displacements, respectively. Inanother words, in BFM, the crack problem is transformed into a problem ofdetermining the relative displacements between crack surfaces. Because therate of change of the relative displacement at a crack tip becomes infinite,the numerical procedure whose unknowns are the density of body forcedoublets itself can not give the satisfactory solution. In order to avoidthis difficulty and obtain the solutions with high accuracy, the density ofthe body force doublets are replaced by the weight function thorough thefollowing equation.

density of

bodv force doubletsbasic density function x _ (10)

The basic density function is defined as

for two dimensional crack problems. In which, £ is a local coordinate onthe imaginary crack line (DIB) and c is a half crack length. As the cracktip singularity is expressed exactly in the basic density function, we caneasily obtain the highly accurate solution of crack problems, even undersmall number of division of DIB.

6 Numerical Examples

Based on the principle of the present BFM, we can analyze various homo-geneous, inhomogeneous, finite, infinite region problems with a satisfactoryaccuracy. In order to demonstrate the effectiveness of the present method,four examples are shown. They are (1) the problem used for the accuracytest, (2) the edge interface crack problem, (3) the crack propagation simu-lation in a bi-material and (4) the interference problem between a circularhole and an inclusion.

6.1 Accuracy Test — Uniform tension of center cracked plateTable 4 shows the calculation result of Figure 6. In this example, thecalculation for each crack length was continued until the results convergein the 6 digits. It was found that Isida's solutions in the range of a/b < 0.8are in good agreement with ours.



N division(A-B)A

IFigure 6 Uniform tension of center cracked plate and

division method of the imaginary boundaries

Table 4 Non-dimensional SIF (Fi = KI in Figure 6

I - _ _; _

MOroifiCN

§00

CN005?

mm®

r -VDOO

§

P:inp

$CNP

1

| Isida[14]

1

1

TtocTt

rr

CNcroc

oc

Tto>If)p

I/Ic

CN

If?

I

[£

omt -«nCN

«/">r-ooo

£r--rr

1

r-»nr-

i

o•np

c

1

Koiter[16]

0ooI/I04

300

I/ICN3

ooONCN

OOCNOC

!

sIf)p

?(Np

ON

F

3



6.2 Tension of an Edge Interface Cracked PlateSIF of an edge interface crack in the dissimilar half plate specimen (Figure7) was calculated. And our solutions were compared with those of Yuukiet. als[lS] and Miyazaki et. als[19}.

0G

HB

I I I

#— •<

G,,v,

MA

Gj.v,

« w *

1 i -

'!

E ;

ia

N=5(F-G)

2* — -node

Single ImaginaryBoundary

(plane stress)

N=15(A-B)

DoubleImaginaryBoundary

:N=5(C-D)

i \(N: Number of division)

(a) Specimen geometry (b) Division modelFigure 7 Tension of single edge interface crack specimen

division number of imaginary boundary, z/i = r/% = 0.3, T =

Table 5 SIFs of Figure 7(a)1

r c/ M/

| 0.050.10.2

1 0.32.0 0.4

0.50.60.70.050.1

| 0.2! 0.3

4.0 0.4| 0.5| 0.6! 0.7

" 0.050.10.20.3

10.0 i 0.4i 0.5' 0 6

0.7

FPresentAnalysis1.1601.1951 ,3681.6592.1092.8204.0236,3371.1981.2091.3681.6552.1012.8074.00]6.2911 .2521 .2291.3691 .6492.0912.7893.9716.231

= A'i/(7o\/7Yuuki

1.1401.1881.3661.6572.1082.8204.0246,3481.1751.2011.3871.6532.1002.8074.0006.2801.2301/>201.3671.6462.0882.7883.9666.229

~ _iMiyazaki

1.195] .3681.6592.110 12.8224.0316,353_ — y

1.2091 .3681.6542.1012.8074.006 I6.304

1.229 !1 .369 j1 .648 12.090 i2.7893.9746.241 i

FPresentAnalysis0.1310.1290.1370.1580.1970.2670.3960.6710.2480.2390.2500.2880.3580.4830.7)61.2100,3610,3400.3490.3990.4940.6630.9791.651

2 =K2/"o\/;Yuuki

tt a/s{l8]0.1300.1280.1370.1570.1980.2680.3980.6730.2470.2380.2540.2880.3590.4830.7161.2090.3590.3380.3490,3980.4950.6640.9801 .65 1

~Miyazaki

0.1290.1370.1580.1980.2670.3970.670

0.2390.2500.2880.3590.4830.7)61.208

0.3400.349 i0.3990.4940.6630.9781 .648

(Definition of the complex SIF of an interface crackis based on the Yuuki e/ %A^ one [18])

Table 5



6.3 Crack Propagation Simulation from a 45° Slant Initial CrackFigure 8 shows the result of crack propagation simulation from a 45° ini-tial edge crack emanating the crossing point of the free boundary and aninterface. In Figure 7, the initial length % for the 45° slant crack wasset to be a$ — 0.1414W(W: width of the specimen). In order to calcu-late the crack extension path, the crack length was increased step by step(GO + n • Aa —> a$ + (n + 1) • Aa)in the direction of the maximum hoopstress appears at the crack tip one step before. The crack extension lengthin each step was assumed to be A a — 0.25«o-

-w-

Initial crack

Crack path

A a

(plane stress)

0.25

(a) Calculation model (b) Crack propagation pathFigure 8 Crack propagation simulation from a 45° slant edge crack

Stai

Aa4p =3/lip =

pt (i - 0)

i o-Oao cos 4 5°OQ sin 4 5"

— 4 ; — »•

Calculate SIFs K* , K^for the crack

whose tip position is*l£ =^ip + Aacos^yltp =yjjp + Aasin^

i + +

Calculate crack extensiondirection in the nextiteration #*+* from

K\ , A'J'Jusing cr^rnax criterion

1

Figure 9 Algorithm used in the present simulation



6.4 Tension of an Infinite Plate With a Circular Hole anda Circular Inclusion

Figure 10 Interference between a circular hole and an inclusion

. 4*—-*-~"*~rr:r-"

5 6 7 8 9

Figure 11 (a) b = a (T = 1/1 = 1/2 = 0.0, plane stress)

6.0t t t

5 6 7 8Pitch p

Figure 11 (b) b = 2a (T =

3 4 5 6 7 8

= ^2 = 0.0, plane stress)

9 10



Figure 11 (c) 6 = 0.5a (F = G = 0.0, plane stress)

Table 6 Stress concentration factor of Figure 10for the case of b — a (v± = //2, plane stress)

pitchP2.53.03.54.04.55.05.56.06.57.08.09.010.02.53.03.54.04.55.05.56.06.57.08.09.010.0

«2Gl

0.0

0.5

_ < ^ °<? ' xAF~

2.58642.62322.66292.70262.73962.77242.80082.82542.84572.86322.89092.91132.92672.90882.91362.92122.93002.93852.94632.95302.95872.96362.96782.97442.97922.9828

= 0® xB

2.58642.62322.66292.70262.73962.77242.80082.82542.84572.86322.89092.91132.92671.23111.25521.28321 .30941.33201.35111.36691.38001 .39081.3999.4140,42411.4317

v?*£.

4.03393.26413.07733.02023.00082.99422.99232.99222.99272.99352.99492.99612.99703.41843.14793.06953.03913.02483.01723.01273.00983.00783.00643.00463.00353.0028

= 0

^y^4.03393.26413.07733.02023.00082.99422.99232.99222.99272.99352.99492.99612.99702.09511.70181.58511.53751.51411.50081.49261.48711 .48331.48041.47671.47431.4727

^

2.0

10.0

"2°® xA

3.07253.07013.06473.05783.05073.04433.03873.03403.02993.02653.02113.01713.01413.15693.15383.14333.12853.11303.09883.08633.07563.06663.05893.04683.03793.0313

= 0& xB

0.53330.53410.54300.55330.56280.57080.57750.58300.58760.59140.59730.60150.60470.14270.13830.13850.14020.14210.14390.14540.14670.14780.14870.15010.15110.1519

<?r^yC_

2.58022.83182.91332.94762.96492.97472.98082.98492.98782.98992.99272.99452.99561.96992.56382.76802.85692.90272.92922.94602.95732.96542.97122.97922.98422.9876

= 0VyD

0.99730.79610.71930.68310.66340.65150.64380.63860.63480.63210.62830.62600.62450.27360.22010.19390.18040.17280.16810.16510.16300.16150.16040.15890.15800.1574

In this numerical example, the curved boundary element was applied forthe SIB corresponding to the circular hole, and the number of the ele-ments of the SIB was set to be 24. Although there exist very few publishedsolutions which enable us to compare with our numerical results in thepresent example, it may possible to know the numerical accuracy of thepresent solution by comparing it to the Ling's solutions[20] solved for thecollinear circular holes. For example, Ling's solution for the case of p — 4.0is cr%c/cr 3.020, while our solution is 3.0202.



The present fundamental solution covers most of

fundamental solutions used in the previous plate problems

(V)

Present paper

+

(IV)/

1

(m>o'

(I) (H)

Figure 12 Relationship between the present fundamental solution andthe fundamental solutions used in the previous 2D analysis by BFM

Conclusion

A method for calculating the various problems of notches, cracks and acircular inclusion in the homogeneous plate of arbitrary shape was developedbased on the BFM. The most characteristic point of the present BFM isuse of the elastic field due to an isolated force acting in an infinite platecontaining a circular inclusion as the fundamental solution. Most of thefundamental solutions used for the previous BFM analysis were covered bythe present fundamental solution, as illustrated in Figure 12.

In addition to using this fundamental solution, the present BFM anal-ysis contains no numerical integration not only for the straight imaginaryboundaries but also for the curved imaginary boundaries, except for thecase of the oscillation singularity appearing at the crack tips in the inter-face crack problem of a bi-material. Therefore, the present method saves



much calculation time compared with the usual BEM, and the obtainedresult contains no error relating to the numerical integrations except for theinterface crack problem.

Furthermore, by increasing the division number of the imaginary bound-aries regularly, we can obtain the practically exact solution conversing to aconstant value, as shown in Table 4.

References

[1] Nisitani, H., Journal of the Japan Society of Mechanical Engineers, 70, No.580, pp. 627 (1967).

[2] Nisitani, H., Mechanics of Fracture, 5, (Sih,G.C.,ed.), Sijthoff & Noordhoff,Chapter 1, pp. 1 (1978).

[3] Chen, D.-H. & Nisitani, H., Transactions of the Japan Society of Mechan-ical Engineers, Vol. A 51-462, pp. 571 (1985).

[4] Nisitani, H., & Chen, D.-H., Transactions of the Japan Society of Mechan-ical Engineers, Vol. A 51-465, pp. 1471 (1985).

[5] Nisitani, H., & Chen, D.-H., Transactions of the Japan Society of Mechan-ical Engineers, Vol. A 52-474, pp. 544 (1986).

[6] Chen, D.-H., Nisitani, H. & Noguchi, H., Transactions of the Japan Societyof Mechanical Engineers, Vol. A 55-515, pp. 1485 (1989).

[7] Nisitani, H., Saimoto, A., Noguchi, H. & Chen D.-H., Transactions of theJapan Society of Mechanical Engineers, Vol. A 57-542, pp. 2561 (1991).

[8] Nisitani, H., Saimoto, A. & Noguchi, H., Transactions of the Japan Societyof Mechanical Engineers, Vol. A 59-557, pp. 68 (1993).

[9] Saimoto,A., Nisitani, H. & Chen D.-H., Boundary Element Method XVI,(Brebbia,C.A.,ed.), Computational Mechanics Publications, pp. 461 (1994).

[10] Nisitani, H., Saimoto, A. & Noguchi, H., Transactions of the Japan Societyof Mechanical Engineers, Vol. A 56-530, pp. 2123 (1990).

[11] Isida, M., International Journal of Fracture Mechanics, 7-3, pp. 301 (1971).[12] Newmann, J. C., NASA TN D-6376, pp. 1 (1971).[13] Nisitani, H. & Chen, D.-H., Body Force Method, Baifukan publication, pp.

95 (1987).[14] Isida, M., Engineering Fracture Mechanics, 5-3, pp. 647 (1973).[15] Feddersen, R. E., ASTM STP-410, pp. 77 (1966).[16] Koiter, W. T., University of Technology; Report No. 314, Delft, Nether-

lands (1965).[17] Tada, H., Engineering Fracture Mechanics, 3-3, pp. 345 (1971).[18] Yuuki, R. & Cho, S. B., Transactions of the Japan Society of Mechanical

Engineers, Vol. A,55, pp. 340 (1989).[19] Miyazaki, N., Ikeda, T., Soda, T. & Munakata, T., Transactions of the

Japan Society of Mechanical Engineers, Vol. A, 57, pp. 2903 (1991).[20] Ling, C.B., Journal of Applied Physics, 19, pp. 77 (1948).


Documents

ing arbitrarily shaped plates containing cracks or notches ... · ing arbitrarily shaped plates containing cracks or notches of general shape and a circular elastic inclusion. Our