Abstract A problem concerning a spher ical mter facia l crack is so lved by the e igenfunct ion m ethod . Theprob lem is reduc ed to a coup led system of dual-series equ ation s in terms of Leg end re func tions and then toa system of s ingular integral equations for two unknown functions. The behaviour of the solution near theedge of the spherical crack, and the s tress-intensity factors and crack-opening displacements are s tudied.The case when the crack surfaces are under normal internal pressure of constant intensity is examined.
Keywords Cavity • Compos i t e • Elas t ic • Inclusion • In ter face spher ical crack
1 Introduction
Aging and damage of materials are processes that are the causes of many dramatic events in the world.They may lead to catastrophic failures in oil and gas-storage tanks, pressure vessels , turbine-generatorrotors , s team boilers , pipelines, bridges, airplanes, railways and welded ships [1]. I t is also known thate lectronic chips somet imes become dis funct ional due to mechanical damage. Scient i f ic and engineer ingevidence shows that cracks in materials are the firs t s teps in a sequence of processes leading to theirfractu re. Inter nal cracks, in the form of break s in mater ial solidity, have be en exa min ed in the li teratu re forquite a long time [1,2]. Special attention [3,4] has been paid to linear crack problems in unbounded elasticbodies; the main results were obtained for f lat penny-shaped or elliptic cracks. A comprehensive review ofthe s tate-of-the-art can be found in [2, 5, 6]. However, according to experimental analysis of the surfacesof damaged objects , the initial surfaces of the material breaks are not f lat , being mainly of spherical or
ellipsoidal shape [5, 7]. To evaluate the s trength of a material with internal cracks, one can start from thesolution of a class of problems within the theory of elasticity for three-dimensional bodies weakened bycracks with curved surfaces. Such cracks could be modelled by cuts on a part of some surface of revolu-tion having non-ze ro curvatu re. In this case, one has the possibili ty to vary the geom etrical pa ram ete rs of the
M. A. Mar ty ne nk o • I . V . Leb edy eva ( IS!)K y iv N a t i o n a l T a r a s S h e v c h e n k o U n i v e r si t y ,01033 Kyiv , Ukra inee-mai l : l e b e d e v a i @ u k r . n e t
1 1 f t — Xvf (x) = - - / ( X ) + — / / ( r )cosec—J— dr, (29)
F ( 0
1 + 2 y 1 + 2 y
Having solved the Riemann problem, we can represent the character is t ic equat ion as
Z ( x )
1 — 4 y 2X fF(0-F(x)
- fi j-0a
Tri J Z (/) sin —d/
' T
w here
Z ( x )H — T T / Ö o - x V V Ön + ' V * , 1 , 1
= " \ 4 / I 1 3 " — ) ' = 27T ® 1
+ 2y
2x
(30)
( 3 1 )
This solution of the characteris tic equation (26) allows us to convert the s ingular equation (24) into aFredholm equation, s imilar to the regularization of the s ingular equations as described in [24].
Let us rewrite Eq. (24) as follows
/ ( * ) + -7T1
00
J sin M-(32)
-o{)
We temporarily assume that the r ight part of Eq. (32) is a known function. I t is easy to show that thefunct ion k f ( x ) is regular. Solving Eq. (32) by applying (21), we g et
Z(x)
1 - 4 y 2
0Q[
7t 1 J
Y f k f ( t ) - k f ( x )
Z it) sin ^dr
1 — 4y 2.7
o ( 0 - O (jc)
Z ( 0 s i n ^ r -d t ( 3 3 )
Thus, the s ingularity of the solution of Eq . (24) at the end of the interv al f—6q,0q] is dete rm ine d by thefunct ion Z ( t ) . H ence , f ( t ) can be represented in the form
f i t ) = Z it) Lit) , L it) = L\ it) + i L2 it), L i-t) = L ( 0 , (34)
w here L (r) does not have singularities at the ends of the interval [—0o,#oJ.By in t roducing
1 — 4 y 2 <t>ix)
«U
ni J-0{)
<P(t)-<S> jx)
Z it) sin ^ d t = F*ix), (35)
we may easily show that Eq. (33) is equivalent to the Fredholm system
L{x) +I 9{>
l - 4 y 2 /L (t) M\ (t,x) + L(t) M 2 (t,x) dt = F* (x),
M i (t, x) =
-O o
iyZ(t)
n( 1 - Ay2) / Z ( r ) — — d r + Z it) K { ( f ,x ) ,
M 2(t,x) = — / Z ( r ) — ^ d r + Z(t)K2(t,x).71 ( 1 - 4y
2) J s in ^ f
The kind and properties of the functions Z (t) and L (t) allow us to separate real and imaginary parts inEq. (36) and to obtain a system of two Fredholm integral equations for the functions L\ (0 and L2 (t) onthe interval [O,0oL
4 Results and discussions
Let us investigate the s tress and displacement f ields near the edge of the spherical cut. I t is known that,while calculating these characteris tics near the contour of the penny-shaped crack placed on the boundaryof the division of two diffe rent materials , mathem atical difficulties in estimating asymptotically th e num berof integrals arise. I t can be proved that near the boundary circumference of the spherical cut, that is , for0 0q — (s 1), the diffe renc e of the displacem ent in the firs t appro xim ation is repr esen ted by the
integral
+ i A w * 2 r ° 1z ( W ) d r
= W l ~ 4 y 2 ? / t a n V V * L^
d t( 3 S )
Uo 1 ur ~ G\G 2 J s / 2 c o s 0 — 2 c o s t GiG2 J I tan && ) V2cos<9 - 2 co sr 'e e x 4 7
w here
Au r = - 42)
= Y/MPnCcosO),
00 i 1 /o
Applying the method of asymptotic integration to the integral (38), where 0 ~ we get
2r 0Lm V l - 4 y 2 j tan*?)1 r IK r<l + iA)
co s ' r |3 1A uo + \A ur * n ^ ^ T ' 7 ^ — » (39 )
where P() is the Gamma function. I t should be noted that (39) can be derived from (38) by differentmethods. But, obviously, the most rational one is the substitution
s in = s in | + 5 ( s in ^ - s in 0 . ( 4 0)
The stress f ields on the surface of the sphere outside the cut are defined by the right parts of the equalit ies
(12) when 0 > 6q. An analysis of the series (12) prove s that the main stress com pon ents fo r 0 ^ 6o+e (s 1)are found f rom the sums
, = £ [ a o / P W f J ( « + (cos 0) ,
2
1 , ( * + £ ) / ^ ( c o s f l )*« = Z + ^ - ' (41)
/1=1
Considering correlations for the Legendre functions [23], asymptotic expressions for the s tress f ields canbe represented in the form
3 82 J E n g M a t h ( 2 0 0 6 ) 5 6 : 3 7 1 - 3 8 4
1 d
sinl? d0
0 0 r „ i ( « + ^ ^ ( C O S Ö )
n= 1
Ä Z r ^ + 2 «o/f/z=0
n (n + 1)
( cos 0 ) , (0 > 0 0 ) .
(42)
If /„ \ 4 2 ) in (42) are subs t i tu ted by the in tegral ope rato rs (8) and the order of sum ma tion and in tegrat ion
is changed, we have
2a {) d / sin tip (0 dr _ d f $(t)dto r ^ — / , ~ 2<7Q — / , (43)
sin Ode J y/2 co s J — 2 cos 0 d0 J V 2 c o s 7 - 2 c o s 0^ tf
O n the basis of (25), (31), (34) the last relationsh ips, wh ere 0 ~ 0o, can be pre sen ted in the followin g
equ iva len t fo rm
/ - r V t a n V V " df* - 2 « 0 s in 0„L (flo) v / l - 4 y 2 / ^ - = = = = . 44)v J \ t a n / v / 2 c o s r - 2 c o s 0
After asymptot ic in tegrat ion we wil l get
- o s / £ / 0CTrt) + i<7r% (/C2 + i / C i ) r 0
u o I s i n - - s i n y J - l s i n - + s i n y j , (45)
where the normal K\ and tangent ia l K2 SIF are determined f rom
/ r (1 + iX) r (i - a)K2 + iK X = -400^/1 - 4y
2y^L (0O) y-f ^
0
(46)
The form ula s for /Ci and could be redu ced to the resul ts for a penn y-sha ped crack betw een d iss imilarhalf-sp aces ( the pro ble m was stud ied by Willis [27]) in the limit ro 00. O ne of the auth ors of thispaper has car r ied out such an operat ion for penny-shaped and spher ical cracks in homogeneous space[28]. This dem ons tra te s the re l iabi li ty of the meth od p resen ted here . The characte r of the asym ptot ic s t ressf ie ld near the boundary c i rcumference of the spher ical cut on the in ter face boundary is the same as nearthe edge of a penny-shaped crack between two diss imilar half -spaces . I f the osci l la tory character in th isproblem is determined by the mult ip l iers ( s in 0 /2 — sin 0 o / 2 ) " 1 / 2 + i \ then , near the penny-shaped crack, i ti s determined by the mult ip l iers ( r —
We shal l now cons ider the case of an external uniform expans ion when the cut sur faces are loaded by
normal in ternal pressure of in tens i ty q. The boundary condi t ions (2) have the fo l lowing form
^ ( 0 _ „(2) „CD _ ,,0 ) - „(2) „(1) _ „(2) ( _ . 0 *oy — ar , ar0 = ar0 , ur — ur , u0 — , ( r — ro,tfo < V < jt) , ^
a r( 1 ) = o>® = /t (0) - -(7 , a ™ = <r® = /2 (0) = 0, (r - r 0 ,0 < 0 < 0 O ) ,
The right parts of system (36) are easy to solve. Note that the solution of the system of integral equations(36) depends on the e las t ic character is t ics of the mater ia l v \ , G i,G2, ge om etry of the cut ro,0o and thecondi t ion of loading on the surface of the cut . Each of the ment ioned parameters has a ra ther wide range
of var ia t ion . In th is ar t ic le we wil l provide conclus ions and numer ical resul ts which could be of pract icalvalue .
F irs t of a l l , i t i s wor th not ing that the dependence of the SIF on Poisson ' s ra t io , having their valuesin the interval 0-2 < v\,v2 < 0-42, is insignificant. For example, with v\ = V2 = 1/3 and v\ = 1/3 , v2 =
1/4 (Gi = G2) S IF K\ and K2 are, respectively, equal to 0.457 and 0.462.a Spr inger
3 8 3 J E n g M a t h ( 2 0 0 6 ) 5 6 : 3 7 1 - 3 8 4
F i g . 3 D e p e n d e n c e o fn o r m a l K i a n d t a n g e n t i a lK2 SIF on the r a t i o ofs h e a r m o d u l e s
p = G 1 / C 2 i n c l u s i o n a n dm a t r i x f o r d i f f e r e n t v a l u e sof t he angle of t he cut(Oo = l 2; 24)
0 . 4
0.2
- 0 , 1
—02
_ — — — — —
// . •
tfo = 12 °
_1 10 201
ê 0 = 2 4 °
ß
1U
Figure 3 shows the SIF behaviour depending on the ra t io of shear modul i p = G1/G2 inclus ion andmatrix for different values of the angle of the cut (0q — 12; 24). Obviously, the m ain c hang es of SIF K\ andK2 take p lace when p increases to values of p » 5 . This me ans that a fur th er increase of the ra t io of shearmoduli of inclusion and matrix should not lead to s ignificant changes in the behaviour of the crack on thein ter face in compos i te mater ia ls .
These resul ts a l low us to compare SIF for compos i te ( f t > 1) and homogeneous (p — 1) mater ia ls . Thecase 0 < p < 1 is not of practical interest. Figure 3 shows that, with for increased values of p, SIF increase
and exceed the corresponding values of SIF for a homogeneous mater ia l . I f we in t roduce the parametersS\ = K^ /K.f\S2 — K^ (upper ind ices co r r espond to compo s i t e and hom ogen eous m a te r i a l s ) , t hen ,with 1 < p < 60, the parameters 5/ ( / = 1,2) are in the interval 1 < 5/ < 1-5.
5 Concluding remarks
The class of problems linked to the interaction of matrix and inclusion is interesting from the point of viewof the mechanics of compos i te mater ia ls re inforced by hard par t ic les . Cracks on the in ter face boundary ofmatr ix and inclusion in such mater ia ls could app ear du e to mec hanical coercion or enviro nm ental inf luence.Several sc ient i fic works pre sent mos t ly num er ical m etho ds for the so lu t ion of th is class of problem s , whichis expla ined by mathe ma tical d i f f icul t ies ar is ing in the analy t ical appro ach . How ever , only an a naly t icalsolution is capable of covering all the s ingularit ies of the problem, and giving a general picture of themechanical condi t ions of the sys tem dependency on changes in the problem parameters , such as externalloading, geometry of the crack, e las t ic cons tants of matr ix and inclus ion , e tc . , and, thus , to predic t thebehaviour of the cut when these parameters change. The resul ts of th is work show the advantages of such
_ an approach. In par t icular , we obta ined analy t ical express ions for the components of the s t ress tensor andthe~SlF nea r the edge of the spher ical crack on the in ter fa ce boundary . W hen the surfaces of the crack areunde r a norm al in terna l pressure of cons tan t in tensi ty , the depe nde ncie s of the SIF on the ra t io of the shea rmodul i p inclus ion and the matr ix are shown. These dependencies demons tra te that the main changes of
' the crack ' s beha viou r take p lace for p e [1 ,5] . We also compared the values of the SIF for compos i te andhomogeneous ma te r i a l s .
Acknowledgements The authors are grateful to the referees for comments and fruitful discussions.
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