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The M- and M c-Integrals for Multi-Cracked Problems

in Three Dimensions

J. H. CHAN G 1 Y. C. KANG 2 L. G. CHUN G 3

Subject head ings: Fractu re, Elasticity, Stra in Energy,

Integrals, Three-dimen sional Analysis, Finite Elemen ts

Keyword s: mu ltiple cracks, M-integral (M c-integr al), thr ee dimen sions

sur face energy , mod ified sur face-indep end ence

ABSTRACT

A problem-invariant M c-integral is proposed as an energy parameter for describing

the d egradation of structural integrity caused by irreversible evolution of mu ltiple cracks

in 3-D elastic solids. The ph ysical mean ing for 3-D M c, which is related to the surface

energy corresponding to creation of the cracks, does not hold in a m anner as that for 2 -D

M c and n eeds to be properly reformulated. Also, the 3-D integration is shown to be

sur face-ind epen d ent in a mod ified sense. With this pr oper ty, by choosing a closed

surface remote from the crack fronts, the 3-D M c can then be accurately evaluated withfinite element solutions even w hen the n ear-front areas ar e not simulated with v ery fine

grids.

1

Prof. of Civil Eng., Nat ional Centr al Univ., Chun gli, Taiwan2 Grad . Res. Asst. of Civil Eng., Nat ional Centr al Univ., Chun gli, Taiwan3 Grad . Res. Asst. of Civil Eng., Nationa l Centra l Univ., Chun gli, Taiwan

ournal of Engineering Mechanics. Submitted J une 22, 2011; accepted February 13, 2013;posted ahead of print February 15, 2013. doi:10.1061/(ASCE)EM.1943-7889.0000605

Copyright 2013 by the American Society of Civil Engineers

J. Eng. Mech.
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1. INTRODUCTION

Physically, the J k -integrals evaluate the energy release rates corresponding to

translation of a singular point and have been extensively used as effective energy

par am eter s in fractu re an alysis. In p ract ice, ba sed on th e con cep t of J k , various types of

contour int egrals (in 2-D) and sur face integrals (in 3-D) hav e been dev eloped for problem s

associated w ith a single crack tip . These integrals includ e, e.g., the interaction integral

(Stern et a l. (1976), Gosz an d Moran (2002)), the M 1-integral (Chen and Shield (1977)). and

the dom ain integra l (Nikishkov and Atlu ri (1987), Eriksson (2002)), etc.. Never the less,

while proper use of energy parameters in describing the 'global' fracture state of 3-D

multi-cracked structure is of practical importance, J k and the above integrals are not

feasible for this pu rp ose du e to their 'local' natu re associated w ith a single tip.

In addition to J k , the energy conservation contour integrals derived from Noether's

theorem in elasticity also includ e the M -integral (Rice (1968), Eshelby (1970), Know les and

Sternberg (1972), Bud iansk y and Rice (1973)). For a 2-D isolated sing ular point ,

evaluation of the M-integral results in the driving work in expanding the singularity.

The M-integral has been used for 2-D problems containing a single defect with various

definitions of integration contour (e.g. Freund (1978), Herrmann and Herrmann (1981),

King and Herrmann (1981). Mar kenscoff (2006), etc.). Du e to such a flexible featu re, the

M--integral can therefore be used for multi-cracked problems provided that the

integration contours are suitably chosen.

In the last decade, the M-integral has been successfully applied as an energy

par am eter in char acter izing th e global fractu re stat e for 2-D multi-crack ed problem s. Aseries of stud ies have been presented to evaluate the m aterial damage level for u niformly -

loaded microcracking infinite elastic solids (Chen (2001), Chen and Lu (2003), Wang and

Chen (2010), etc.). Also, a problem -inva riant par am eter M c is proposed by the authors

and su ggested as an energy param eter for describing the degrad ation of material and/ or

structural integrity caused by evolution of multiple cracks in elastic solids (Chang and

Chien (2002)). An imp ortan t ph ysical interp retation is illustra ted in these works that the

M-/ M c-integral evaluates twice the surface energy associated with creation of all the

cracks in the materials/ structures. Therefore, M/ M c can be used as an energy fracture



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par am eter for assessmen t of th e d am ag e stat e in th e multi-cracked bod y (Ch an g and Liu

(2009)).

An energy fracture parameter termed M c is proposed in this present stud y for use in

3-D multi-cracked pr oblems. Based on the concept of the 3-D M-integral, M c is defined by su itably choosin g a closed su rface an d ta king th e in tegr at ion with respect to th e

geom etric center of all the enclosed cracks. The integrat ion is shown to be sur face-

independ ent in a modified sense. The physical meaning of M c is illustr ated . The

feasibility of using M c as a 3-D fracture energy parameter is demonstrated in the

num erical examp les.

2. THE M c-INTEGRAL IN THREE DIMENSION S

2.1 A single crack

Consider a crack embedd ed in a three -dimensional homogeneou s elastic body (Figure

1), wh ere the front is of arbitrary curved shape. A coordinate system originating at an

arbitrarily chosen point O is introduced and, with no loss of generality, the crack parallel

to the x 1-x3 plan e. The geometr ic center of the crack center C is pos itioned by . When

the body is subjected to a system of external loads (but no body forces), the M-integral

with r espect to O is d efined as

uk M [Wn ix i Tk ( )xi ] da (1)D xi

wh ere D is a closed surface around the w hole crack and consists of three parts as S 1+S i+S 2,

Si is a curved tubular surface around the crack front, S 1 and S 2 are the planar surfaces

par allel to th e u pper an d low er crack surfaces, an d a is the a rea over th e surface. Also, W

is the strain energy d ensity, T is the traction vector, n is the outw ard u nit vector norm al to

D, s is the a rc length, x is the position vector, and u is the disp lacemen t vector. Note that ,

by d efin ition, t he in tegr at ion is car ried ou t by taking th e lim iting case wher e S i is shrunk

onto th e crack front, and S 1 and S 2 are lying on the crack sur faces (this limiting case is not

show n in Figures 1).

As indicated by Eq (1), the value of M appears to vary with the selection of origin O.

Never th eless, by loca ting th e or igin at th e cen ter of th e crack , i.e. by taking =(0,0,0), we

then d efine a problem -invariant pa rameter M c as



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M c M | (2)=(0,0,0) Further, it can be shown that the quantities in Eq (1) are independent of the orientation of

the coord inate system. Such a character istic imp lies that M, and M c, remain unchanged

when they are evaluated with respect to an arbitrarily oriented system, e.g. x 1'-x 2'-x 3'

d epicted in Figure 1.

2.2 Mult ip le cracks

Consider the 3-D homogeneous body containing N distributed cracks, each of

arbitra ry shape, orientation, and location, as show n in Figure 2. The geomet ric center of

the r-th crack is positioned by r . The geomet ric center of all the cracks, den oted C, is

position ed by . The M-integral associated with all the cracks is then defined as

N uk M [Wn ix i Tk ( )x i ] da (3)r=1 Dr x i

where D r =S 1r +S ir +S 2r is the closed sur face associated w ith the r-th crack. Still, by

definition, the integration is conducted by taking the limiting case in which D r is shrunk

onto the crack front and lying over the r-th crack surfaces. Note that the value of M

var ies with r espect to d ifferent selections of origin O. Also, by locating the origin at thegeometric center C, a pr oblem-invariant par ameter M c can be defined .

3. MOD IFIED SURFACE-IND EPENDENCE

3.1 A single crack

We first take an outer closed surface S o (Figure 1) that can be arbitrarily chosen

(except for the requirements to be inside the body, enclose the whole crack, and contain

no other singu larity in it). Then, an arbitrar y tubu lar cut ting surface CS connecting S o

and S 1 is introd uced, w ith its rad ius van ishing asym ptot ically. Next, by delimiting the

closed su rface D c=D+CS So , Eq (1) can thu s be rew ritten a s

uk uk M = [Wn ixi Tk ( )xi ] da + [Wn ix i Tk ( )x i ] da (4)Dc xi So xi We further consider th e dom ain enclosed by D c, where is simp ly-connected d ue

to introdu ction of the cut surface CS. By applying diverg ence theorem , the first

integrat ion on the RHS of Eq (4) becomes



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W uk,i kj uk [( kj ) xi ( uk,i) xi + (W ii ki ) ] dv (5) xi x j x j xi where is the stress tensor, is the Kronecker delta tensor, and dv is the infinitesimal

integrat ion volum e. The first and second integran d s of Eq (5) vanish when the material

enclosed in is uniform in th e three x k -directions and und er the state of equilibrium . As

to the th ird integr and , while it van ishes in two d imensions (Chang and Chien (2002)), it is

observed that th is term is actually equal to W and needs to be taken into account in three

dimensions.

From Eqs (4) and (5), the M -integral then becomes

uk M = [Wn ix i Tk ( )xi ] da + W dv (6)

So xi As show n in Eq (6), in ad dition to th e sur face integrals over S o, an extra dom ain integral in

mu st be includ ed for evalua tion of M. With this, the idea of sur face-indep end ence for

M-integral is thus mod ified by including this add itional domain integral. As a

consequence, the M c-integral is also surface-independent in the same modified sense.

With this add itional dom ain integral, the a symptotic singular behavior is thus inevitably

involved in the integration. Cau tious investigation is ther efore necessary in evalua tion

of the M/ M c-integral.

3.2 Mult ip le cracks

As indicated by Eqs (3) and (6), the M -integr al for the N distribu ted cracks (Figure 2)

can be written as N uk

M [Wn ixi Tk ( )x i ] da + W dv (7)r=1 Sor xi r where S or is an arbitrarily chosen closed surface that encloses the r-th crack and no other cracks, and r is the region enclosed by S or (Figure 3). This imp lies that , for mu ltiple

cracks, M is sur face-ind epen den t in a mod ified sense.

Alternat ively, Eq (7) can be r ewr itten asuk

M [Wn ix i Tk ( )xi ] da + W dv (8)So xi

where S o is an arbitrarily chosen outer surface that enclose the N cracks, and is the

region enclosed by S o , as shown in Figu re 3.



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Eqs (7) and (8) show that, for multiple cracks, the integration surface for M can be

selected either locally (i.e., S or , r=1,2,...,N) or globally (i.e., S o , as a whole). As a

consequence, the M c-integral is also surface-ind epend ent in the same m odified man ner.

4. ORIGIN-IND EPENDENCE (SPECIAL CONDITION)

The value of M in general depen d s up on the location of origin O. Nev erth eless, for

the special cond ition w hen th e cracks are em bedd ed in an infinite body an d subjected to a

far-field uniform loading system, it can be show n th at the result of M is independ ent of

the origin O. To this end, we first rewrite the M-integral in Eq (3) as

uk M M c + i { [Wn i Tk ( ) ] da]} (9)

So x i The last term of Eq (9) appears to dep end up on the comp onents of the position vector .

This term a ctually accounts for the origin-dep end ent featur e of the M-integral.

For the special cond ition w hen the bod y is an infinite med ium a nd subjected to a far -

field uniform loading system , we furth er take a r emote closed cubic surface S that is

far from the cracked region. It is thu s observed tha t, over the surface S , the stresses, the

derivatives of the displacements u i,j, and the strain energy density W are uniformly

d istributed . This indicates that the last term of Eq (9) vanishes an d, in such a case, Eq (9)

becom es

M M c (10)

Eq (10) shows that, under this special condition, the result of M is equivalent to M c an d

independ ent of the origin O.

5. PHYSICAL INTERPRETATIONIn this section, the ph ysical mean ing of the M c-integr al is ana lytically illustr ated . We

first consider a penny-shaped crack of radius R subjected to far-field uniform loads and

the crack sur faces are traction-free. By taking Eqs (1) and (2) w ith =(0,0,0), we can have

M c related to the p ointwise J k -integr als as

2 M c = R Jm( ) R d = 2A R JR (11)0

where J m( ) is the pointwise J k -integral along m , JR is the averaged v alue of J m( ) over the

whole crack front, and AR is the area en closed by th e crack front.



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Further, we consider evolution of the crack as a continuously varying process of 'self-

expansion' starting originally from its geometric center C and an intermediate state with

radius a is selected. At this state, the energy release rate G d ue to increase of a unit ar ea of

the crack surface is evaluated by averaging the p ointwise energy release rate G m over the

wh ole crack front an d can be expressed as

2 Gm( ) a d d 0 G(A) = = (12) 2 a dA

The value of G(A) is equal to J R at th e final stat e of crack exp ansion (i.e. A=AR ).

As an aside, it is noted that G app ears to be directly pr oportional to the rad ius of the

crack a (e.g Hwa ng et al, 2001) and can then be expressed a s the following separ able form

G(A) = 1 a = A 1/2 (13)

where 1 an d are functions of material properties and the ap plied loads.

By considering Eqs (11), (12) and taking the integration of Eq (13) throughout the

p rocess of cr ack evolu tion , with th e geometric and load ing cond ition s (excep t the cr ack size)

remaining unchanged, we then have

M c = 3 (14) As illustrated in Eq (14), the M c-integral can be interpreted as equal to th ree times of

the surface energy required for creation of the whole crack, i.e. the energy change due to

creation of the entire crack sur faces. For multiple cracks, the validity of this phy sical

interpretation w ill be d emonstrated nu merically in the next section.

6. NUMERICAL EXAMPLES

6.1 Example problems

In the following calculations, quadratic finite elements are used for interpolation of

the displacement field. No particular singular elements are used through out the stud y.

Problem 1

Two specimens are consider ed in this problem. In the first specimen, an inclined

pen ny-shap ed crack of rad iu s R is em bedd ed in a cylindrical solid su bjected to unifor m

tensile stress (Case I, Figure 4(1)). The crack plane is inclined at an angle to the

horizontal plane. In the second specimen, a horizontal penn y-shaped crack is embedd ed



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in a cylindrical solid under a nonuniformly-stressed condition induced by the inclined

boundar y load (Case II, Figure 4(2)).

The convergence study is performed by using two FE meshes, where four and eight

layers of elements are used in the radial direction within 0.1R of the local near-front arearesp ectively. In Table I, the results of M evaluated from the two meshes, along with the

analytical solution (by considering the results of J k given by Hwang et al, 2001) for a

special condition of Case I, show very good convergency and consistent well with the

ana lytic solution, with d eviation less than 1%.

Three integration surfaces are used to verify the property of modified surface-

ind epen den ce. The results of M show n in Table II ap pea r to be very similar. The

dom ain integral over makes rather significant contribution to the comp utation and thus

accoun ts for the 'mod ified' sense of sur face-indep end ence.

Further, we consider the specimen at its uncracked state under the same loading

condition. The sur face energy du e to creation of the crack is obtained by evaluating the

d ifference of potential energy . The valu es of M c, along w ith thr ee times of the surface

energy -3 , are show n in Table III. The validity of the p hysical meaning of M c as

associated w ith the surface energy is thus w ell verified.

Figure 5 shows the variation of the normalized results of M c versus for both cases,

where M c,max correspond s to the cond itions of =0 o (Case I) and , app roximately, 45 o (Case

II) resp ectively. For Case I, it is observed th at the sur face energy is maximum w hen the

crack plane is perpendicular to the loading direction, then decreases gradually as

increases, and finally vanishes when the crack plane becomes parallel to the loading

d irection, as anticipated . As to Case II, the surface energy reaches its maximum at 45o

and then decreases when the loading direction moves toward either perpendicular or

par allel to th e cr ack plane.

Problem 2

We consider tw o collinear p enny-shaped cracks, both of radius R and separated by a

distance 1.5R, embedded in a large cylindrical specimen and subjected to nonuniform

loads ( , (1- ) ) on its top, as shown in Figu re 6.



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The surface energy due to creation of the cracks is obtained by evaluating the

difference of potential energy associated with the cracked and uncracked configurations.

With the va lues of M c show n in Table IV, along w ith thr ee times of the surface energy

-3 , the validity of the ph ysical mean ing of M c is well eviden t.Figu re 7 show s the variation of the norma lized results of M c versus the loading ratio ,

where M c,ref correspond s to the condition of =0. It is observed that the sur face energy is

minimum when =0, and increases gradually as increases. When the loads are

uniformly distributed (i.e. =0.5), the surface energy r eaches its maximum .

Problem 3

We consider a specimen containing N parallel elliptic cracks of equal size and

separated by a constant distance d from each other (Figur e 8). The specimen is relatively

large compared with the size of the cracks and subjected to uniform far-field tensile stress

in the x 2-direction.

By comparing the results of M c (Figures 9(1)) and 3 M ref (Figure 9(2)) versus the

nu mber of cracks N (for d / a=0.2, 0.5, 1, and 2), the validity of the physical meaning of M c

as associated with the sur face energy is well eviden t. It is also observed tha t the value of M c decrease as the spacing distance d d ecreases. This is anticipated becau se the

asymp totic near-front stresses app ear to decrease as d decreases due to interaction am ong

the cracks (Chen, Y. Z., et al. (2009)). In par ticular , for d / a=2, the v alues of M c turn out to

be alm ost equ ivalen t to th e su m of th ose from each ind ivid ual crack . This means th at,

under this spacing distance, these cracks are apart enough so that the effect of interaction

can be neglected.6.2 Discussions

The unique nature of the 3-D M c-integral is well demonstrated and verified in the

above calculations. First, the prop erty of mod ified sur face-ind epen d ence is eviden t

(Table II) so that it is not necessary to have a very fine near-front mesh to achieve good

accur acy (Table I). Furth er, the validity of the corresp ond en ce between M c and the

sur face energy is nu mer ically sup por ted by the results (Tables III and IV, Figur e 9). It is

observed that M c essentially characterizes the effect due to different crack orientation,



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loading direction, and load ing d istribution (Figur es 5,7). This ph ysical mean ing holds for

p roblem s conta ining str on gly interact ing cr ack s (Figure 9).

In order to numerically verify the physical meaning of M c, the surface energy d ue to

creation of the cracks was also alternatively determined by calculating the difference of poten tial en ergy in the above nu mer ical examp les. Indeed , M c can be easily

implemented in the post processor of finite element codes and then be accurately

calculated w ith almost no extra compu tational cost. On the other hand , direct calculation

of requires finite element solutions of both the uncracked and cracked configurations

and so double comp utational costs will be in need. Further, can only be roughly

app roximated because it is extracted from the nu merical solutions of p otential energy that

are relatively large in magnitude than itself. By compa ring the above nu mer ical

aspects of these two approaches, M c appears to be more efficient as a computational

device and thu s more straightforward for u se in p ractice.

7. CONCLUSIONS

The analytic procedure in formulation of 3-D M c is in genera l more comp licated than

its 2-D coun terp art. In ord er to extend to 3-D ap plications, all the associated geom etric

and state variables need to be deliberately reinterpreted and generalized. Also, it is

importan t to note that two of the ma in features for 2-D and 3-D M c-integrals are d ifferent.

First, while 2-D M c evaluates twice the surface energy associated with creation of the

cracks, it is illustrated in the present study that the result of 3-D M c is equivalent to three

times of the surface energy req uired for creation of the cracks. Secondly, while 2-D M c-

integral is path-independent, 3-D Mc

is shown to be surface-independent in a modified

sense. This indicates that , for 3-D M c, in addition to the surface integral over an

arbitrarily chosen outer closed surface that encloses all the cracks, an extra domain

integral m ust also be includ ed in th e calculation.

The computation procedure for 3-D finite element calculations for multi-cracked

problem s is generally far mor e complicated th an 2-D an alysis. Nev er th eless, sin ce th e

integration is surface-independent in a modified sense, the closed surface can always be

chosen to be remote from the crack tips. In this regard, the calculation thus app ears to be

very insensitive to th e local finite elemen t mod els in th e crack front r egion so that it is not



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necessary to have very fine near-front grids in order to achieve an acceptable value of 3-D

M c.



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Chang, J. H., Liu, D. Y. (2009). Damage Assessment for 2-D Multi-Cracked

Materials/ Structures by Using Mc-integral. ASCE J. Eng. M ech., 135 , 1100-1108

Chen , F. H. K., Shield , R. T. (1977). Conservat ion law s in elast icity of th e J-integra l typ e. J.

Appl. Math. Phys. (ZAMP), 28, 1-22.

Chen, Y. H. (2001). M-integral analysis for two-dimensional solids with strongly

interacting m icrocracks. Part I: in an infinite br ittle solid . Int. J. Sol. Struct. , 38, 3193-

3212.

Chen, Y. H., Lu, T. J. (2003). Recent developments and applications of invariant integrals.

Appl. Mech. Rev., 56, 515-552.

Chen, Y. Z., Lin, X. Y., Wang, Z. X. (2009). Evaluation of the stress intensity factors and

the T-stress in per iodic crack pr oblem. Int. J. Fract. , 156 , 203-216.

Eriksson, K. (2002). A domain independent integral expression for the crack extension

force of a curved crack in thr ee dimen sions, J. Mech. Phys. Sol., 50, 381-403.

Eshelby, J. D. (1970) The Energy Momentum Tensor in Continuum Mechanics, Inelastic

Behav ior of Solid s. McGraw -Hill, New York.

Freund, L.B. (1978). Stress intensity factor calculations based on a conservation integral.

Int . J. Solids Struct. , 14, 241-250.

Gosz, M., Moran, B. (2002). An interaction energy integral method for computation of

mixed-mode stress intensity factors along non-planar crack fronts in three

dimensions, Eng. Fract. Mech. , 69, 299-319.

Herrm ann , A. G., Her rm ann , G. (1981). On en ergy r elease rates for a p lane crack. ASM E J.

Appl. Mech., 48, 525-528.

Hwang, C. G., Wawrzynek, P.A., Ingraffea, A.R. (2001). On the virtual crack extension

meth od for calculating th e der ivatives of energy release rates for a 3D planar crack of

arbitrary shape und er mod e-I loading. Eng. Fract. Mech. , 68, 925-947.



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King, R.B., Herrmann, G. (1981). Nondestructive evaluation of the J -and M-integrals.

ASM E J. Appl. Mech. 48, 83-87.

Knowles, J. K., Sternberg, E. (1972). On a class of conservation laws in linearized and

finite elastostatics. Arch. Rat. Mech. Anal., 44, 187-211.Markenscoff, X. (2006). Eshelby generalization for the dynamic J, L, M integrals. C. R.

Mecaniq., 334 , 701-706.

N ikish kov, G. P., Atlu ri, S. N . (1987). Calcu lation of fractu re mech anics par am eter s for an

arbitrary three-dimensional crack, by the equ ivalent d omain integral method , Int . J.

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Figu re 1 An arbitrarily shap ed crack in a 3-D hom ogeneou s elastic body .

Figu re 2 A hom ogeneou s 3-D elastic body containing N distributed cracks

(N=4 in th is figu re).

Figu re 3 Closed sur faces for mu ltiple cracks.

Figu re 4 A penny -shaped crack of rad ius R in a cylindrical solid (Problem 1).

(1) The crack is inclined and the solid is uniform ly-stressed (Case I).

(2) The crack is horizontal and the solid is non un iform ly-stressed (Case II).

Figu re 5 The norm alized results of M c versus for both cases (Problem 1).

Figu re 6 Two collinear pen ny-shap ed cracks in a large cylind er and

subjected to nonu niform loads ( , (1- ) ) on its top (Problem 2).

Figu re 7 The norm alized results of M c versus the loading ratio (Problem 2),

Figu re 8 N par allel elliptic cracks (Problem 3).

Figu re 9 The values of M and 3 versus the num ber of cracks und er various

spacing distance d (b/ a=0.5, Problem 3).



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Table I The results of M from tw o FE mod els for Problem 1. Case I Case II( 10 2)( =0 o) ( =45 o)

mesh 1 (4 layers) 7.210 3.956

mesh 2 (8 layers) 7.230 3.954Analytic 7.280

(Note: R/ w=1/ 20, E=1, =0.3, = 1, =(0.0,0))

Table II Mod ified sur face-indep end ence for Problem 1.

Eq(6) = M = MSo So (Case I, =0 o) (Case II, =45 o) 10 2

su rface 1 485.594 478.384 7.210 51.302 47.346 3.956su rface 2 68.551 61.351 7.200 29.974 26.006 3.968su rface 3 35.245 28.048 7.197 7.258 3.331 3.927

(Note: R/ w=1/ 20, E=1, =0.3, = 1, =(0.0,0))

Table III The resu lts for Prob lem 1. Case I Case II ( 10 2)( =0 o) ( =45 o)

M c 7.210 3.956 3( ) 7.168 3.968

(Note: R/ w=1/ 20, E=1, =0.3, = 1, =(0.0,0))

Table IV The resu lts for Prob lem 2. 0.25 0.4 0.5

M c 2.972 3.245 3.423

3( ) 3.001 3.290 3.388 (Note: R/ w=1/ 20, E=1, =0.3, =(0.0,0))

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