A numerical method for the solution of plane crack problems in finite media
Department Theotical and Applied Mechanics The National Technical University Athens K. Zografou Stet, Zografou Athens 624, Gece (ceived November 7, 1979) ABSTRACT. A general method for the solution plane isotrop eltity crack problems inside a finite medium arbitrary shape or an infinite medium wth holes arbitrary shape is psented. This method is bed on the complex potential approach plane eltity problems due to Kolosov and Muskhellshvill [i] and makes no sumption on the way loading the cracks and the other boundaries the medium. The whole problem is duced to a complex singular integral equation the cracks and the other boundaries and the values the stss intensity factors at the crack tips may be evaluated dictly and accurately from the numerical solution this equation. An application the method to a rcular medium wth a straight crack is also made. KEY WORDS AND PHS. Sing integral equations, Plane crvilinear CracK, Mumal SolOns, strs intensity factor, complex potent, Cauchy-type prinpal value integrals, Lobatto-Chebyshev numerical integration rule. P.S. THEOCARIS AND N. IOAKIMIDIS 1980 MATHEMATICS SUBJECT CLSIFICATION CODES: i. 65XX, 65R05, 73XX, 73.30, 73.45. INTRODUCTION. An effient method for solving plane eltity crack problems and estimating the stss intensity factors at crack tips is the method which duces the problem to a Cauchy type singular integral equation (called in the sequel simply a singular integral equation) either by considering a curvilinear crack composed a series edge dislocations, or by using the complex potential technique Kolosov and Muskhelishvili l] The first approach w developed in papers by Goldstein, Salganik and Savova [ 2-4]. These searchers duced the problem a curvilinear crack in an infinite medium to a system two al Cauchy type sing- ular integral equations the crack by suming that both its edges we loaded in the same way. The second approach w used by Linâkov [ 5], who used the complex potentials (z) and (z) Kolosov and Muskhelishvili [I] and sumed that the total force exerted on the curvilinear crack w vanished. Panyuk, Savrk [6], and Datsyshin, in a series papers the sults which used the mo practical complex potentials (z) and (z) we viewed in Kolosov and Muskhelishvili [I], but stricted their investigations to straight cracks, which a the most ey to solve. Finally, the psent authors, using also the complex potentials (z) and (z), solved several general plane eltity crack problems in infinite media [7] by laxing any sumptions on the shape and loading mode the cracks. In this paper the pvious methods we extended to ces finite eltic media the outer boundary or inside boundaries due to holes and other voids may interact with the cracks. The problem will be duced to a complex singular integral equation all boundaries the plane eltic medium under consideration (the cracks included). For the numerical solution this equation the numerical techniques developed by the authors in fences [14] and [15] can PLANE CRACK PROBLEMS IN FINITE MEDIA be successfully used alady made in these fences, well in fences [], [II]. Finally, an application will be made to the simple problem a rcular medium with a straight crack under constant pssu. 2. AN INFINITE MEDIUM WITH HOLES AND CRACKS We consider an infinite isotrop eltic medium under generalized plane stss or plane strain conditions containing a system arbitrary-shaped cracks , ratio well a system holes L2j (Fig.l). The material the eltic medium is characterized by the eltic modulus E and the Polsson ratio The Polsson is placed by the constant [I] defined by ,=(3-)/(I+) for generalized plane stss conditions and by =3-49 for plane strain conditions. The loading the medium is the most general consisting loading the two faces the cracks and the holes and loading the plate at infinity. The loading on the cracks andâ the holes is characterized by its normal and Shear components, Sn(t) and t(t), spectively (Fig. l), t=x+iy denotes a generic point the holes and the cracks in a Cartesian coordinate system 0xy. The loading distributions the two faces (+) and (-) the same crack may be diffent. Moover, the positive dictions each crack (defining also the (+) and (-) edges this crack) a defined arbitrarily, the positive dictions the boundaries the holes a consided counter clockwise. gards the loading at infinity, it is sumed that we know the values the prinpal stsses N and N2, the angle formed between the diction N and the Ox-axls, well the does not influence the value e rotation at infinity. Since the value e stss field, it may be sumed having an arbitrary value, being equal to zero. Then the well-known constants F and Fâ Muskhelishvili [i], to which the complex potentials (z) and (z) (z=x+iy) tend at infinity, will be given by [I] 1m(z) Z- (NI+N2)+i e== E/[2(I+)] (la) P.S. THEOCARIS AND N. IOAKIMIDIS Zo2 + Fig. I. An infinite plane medium with a set holes and cracks. im(z) Fâ (Ni-N2)exp(-2iu) (lb) is the shear modulus the eltic material. The peculiarity the problem under consideration, compad to other crack problems is the existence holes arbitrary shape inside the infinite medium which may be at any distance from the cracks. Thus, the ce solved in this paper is gat practical intest since not only the influence the external boundary may be evaluated, but also the influence voids and small discontinuities the material close to the crack may be taken into account in defining the fractu mode the substance. To maintain the general character the formulation the problem in terms the complex potentials #(z) and (z), we consider the holes L2j filled with inclusions the same shape the holes and consisting the same material PLANE CRACK PROBLEMS IN FINITE MEDIA the eltic medium. the boundaries L2j these inclusions we sume no loading existing. Thus the boundaries loaded in a known way. L2j the holes can be consided cracks Now, following the developments fences [7-11], we determine the complex potentials (z) and (z) in terms Cauchy type integrals (z) z-z (2a) P(z) (z (z) 2-- z(z) dz+râ (z-z)2 (2b) L denotes both the cracks equations the density and the boundaries L2j the holes. In these (t) is an unknown function the points t L, the function q(t) is defined by (3a) (3b) the cracks and the boundaries L2j the holes spectively, because the loading the âinclusionsâ alady sumed. It can further be seen on the bis the PlemelJ formulae [I] + â-(t)++(t) that the boundary conditions a 2-(t) + L On(t)-lo(t) -+ /ds (4) /ds s is a variable denoting the arc-length. These conditions a satisfied if the density (t) in equations (2) satisfies the complex singular integral equation z-t ,-t (z-t)2 tEL, - -2r- ddrâ+---I f q()-] (5) the new function is given by (6a) P.S. THEOCARIS AND N. IOAKIMIDIS 2p (t)ffi the On (t)+iot (t) cracks and the or the holes (6b) hole boundaries L2] spectively. Moover, the conditions slngle-valuedness displacements around the cracks L2j can be written () (T )#ffi L2j m and m a the (7a) a analogous to those obtained in fences [7-10] for cracks in an infinite isotropic elticâmedium without holes. Equations (7b) coinde with the condition single-valuedness displacements for a finite medium or an infinite medium with a hole consided in fence +I fLI L2j q(T) âââââml (7a) q(z)dz ,...,m 2 (7b) to numbers cracks and holes spectively. Equations [II]. The forms equations (7a) and (7b) have no diffence. In fact, under the psent formulation the holes have been interpted contour-shaped cracks. Equations (7) supplement the singular integral equation (5) and have to be taken into account independently to obtain the corct sults for the unknown density function (t). Although equations (5) and (7) a adequate to obtain a physically acceptable solution for the crack problem under consideration, yet they a not adequate for the single-valued determination the unknown function (t). This phenomenon is due to the psence the holes, not the cracks, and w investigated in fence [II]. Briefly speaking, although we a su, because equation (4), about the vanishing the total force and moment applied to each one these inclusions, we have no guarantee that the complex potential (z) is completely determined inside these inclusions, which psysically a completely separated from the PLANE CRACK PROBLEMS IN FINITE MEDIA eltic medium surrounding them. In fact, for finite media, a complex constant can be added to (z) without any change in the stss field; only a rotation the finite medium takes place. In order to get rid this arbitrariness we impose the further conditions Im(z0j) Dj z03 a points (8) inside the inclusions S2j and D.j a arbitrary constants. These conditions define in a unique manner the imaginary constants just mentioned. Furthermo, because equation (2a), valid in the whole eltic plane, equations (8) take the form (9) JL-Z0j E. a also arbitrary constants. Moover, since the arbitrariness in the values (t) the hole boundaries is stricted only L2j not the cracks L ljâ we can place equations (9) by (ârâdâr] Im[-i fL2j-z0j the contours (10, These conditions seem simpler to tat than conditions (9). However, neglecting these conditions does not lead to any erroneous sults; Just the values (t) L2j during the numerical solution equations (5) and (7) a not convergent. On the contrary, the values (t) the cracks a completely convergent. Finally, it is convenient to incorporate conditions (7b) and () into the singular integral equation (5). This is because we thus impose the fulfilment the boundary conditions at anyone the collocation points used the contours L2j during the numerical solution the system equations (5) and (7), by plang the corsponding linear equation by the equation sulting from equation (7b) or () for the same contour L2j. P. S THEOCARIS AND N. IOAKIMIDIS By taking into account the fact that equations (4) or (5) give no sultant force or moment on the inclusions S2j we introduce additional terms in equation (5) and neglect in the sequel equations (7b) and (). Thus we have (T) [ ]LTât ] d.l [-)(T)]+6 (t)[ 2 -t (v_t) =i ILâ Clj (t-z0] 2 t_z0j L2j x+-Tql (T)]d E L -2r [râ + I q()-] -t (11) I for t E ,...,m 1 L2j ,...,m2 5(t) 0 for t E (12) and Clj and C2j a arbitrary constants, the first which h a non-zero al part. The values these constants a signed befo the numerical solution equation (II) together with conditions (Ta) and, because conditions (Tb) and (), they have theotically no influence on the values (t) determined from the numerical solution equations (11) and (7a). We can also consider that Clj-=0 and igno the corsponding term in equation (II). Then this equation w-ill have an infinite number solutions, but all them will be corct in the sense that they will satisfy all boundary and physic&l conditions the problem and will give the corct values the stss intensity factors at the crack tips. However it is not permissible to neglect the term multiplied by C2j in equation (II). This term substitutes the conditions single-valuedness displacements (7b) around the holes L2j and should be taken into account. PLANE CRACK PROBLEMS IN FINITE MEDIA 3. A FINITE MEDIUM WITH A SYSTEM CRACKS The sults the pvious section can also be dictly generalized to the problem a finite medium S with a system cracks Lj (J--l,2,...,m) shown in Fig.2. The external boundary the medium S is denoted by L and the boundary ,Z o Fig. 2. A finite plane medium with a set cracks. conditions Lj and L a sumed to be the same those consided in the outside L pvious section. In this ce we sume the infinite medium S to be also occupied by the same isotropic eltic material the medium S. Furthermo, we sume the loading distribution o (t)+io (t) (13) This is contrary to what w made for the to act on both media S and S fictitious inclusions filling the holes in the pvious section, which have been sumed unloaded, but it is pferable, since, in this way, we have q(t) (14) 0 and equation (II) is simplified this contour taking the form P.S. THEOCARIS AND N. IOAKIMIDIS e Ti t (T-t) + (t_z0)2 at I LoT-Z -t ) + t-z 0 (T) L0 (15) L d0 q(t)- E e denotes the contour surrounding the eltic medium, z is a point inside 0 or the cracks the eltic medi (not belonging to L L.)j L denotes all the and the boundaries the medium (the cracks included), that is function is defined L=LoULI2U.. .ULm for t 0 for t L N L (16) Moover, equation the function is defined by equation (6a) the cracks L. and by Also the function q(t) is defined by equation (13) the contour L 0. (3a) the cracks and vanishes, in accordance with equation (14), 0. The conditions (7a) slngle-valuedness displacements the cracks have also to be satisfied when solving equation (15) independently this equation, that is The complex potentials (z) and (z) main given by equations (2) without the psence the constants finite isotropic eltic medium. q(T) e. and ,...,m (17) râ, which have no meaning in the ce a It is now necessarY to show that equation (15) ensus the single-valuedness the unknown density function (t) L0, that is (T) (18) which is equivalent to the condition single-valuedness displacements PLANE CRACK PROBLEMS IN FINITE MEDIA the fictitious crack L formed between the teal medium S and the fictitious medium 0 well the single-valuedness the complex potential O(z), expssed he (T) fLoT-Z (19) Equation (19) is suffient, so that no arbitrary puly imaginary constant can be added to the complex potential #(z) ndering it multlvalued, since such a constant would change the value the integral in equation (19) because the formula [I] 2xi if L L0-z0 (20) is a contour surrounding a finite slmply-connected medium S and z06S happens in this ce. It is worthwhile mentioning that we may igno both conditions (18) and (19) together with the term multiplied by in equation (15). Then, this equation supplemented by conditions (17) will have not a unique solution, but this is not much importance since any approximate solution obtained by solving this modified form equation (15) will satisfy this equation that is the boundary conditions our problem. This means that if, for example, we consider the approximate values for the stss intensity factors at the crack tips obtained by solving approximately equation (15), they will converge to their corct values. Yet, the a two ons for which one wants that equation (15) should have a unique solution: The first is that is customary in plane eltity problems to make use singular integral equations possessing a unique solution: and not an infinity solutions it would be the ce, if the term multiplied by we ignod. However, a strict pro the uniqueness solution equation (15) together with conditions (17) is beyond the aims this paper and not at all straightforward. P.S. THEOCARIS AND N. IOAKIMIDIS The second on for which the complete form equation (15) h to be used is that, in this way, the numerical sults for (t) 0 will be convergent to spefic values and not to floating ones. This is sometimes useful in order to estimate the effiency the numerical technique used. We will show now that the term multiplied by in equation (15) forces the satisfaction conditions (18) and (19). If the medium S contained no cracks, this statement would be evident in view the developments fence [II]. But in the psent ce this is not evident espeally in the ces when the exist sultant forces and moments exerted on the cracks L. The sults fence [II] have to be appropriately generalized. Thus, by taking into account the boundary conditions (4), well equations (14) and (16), we can wite the singular integral equation (15) d-- Lâ(t-z 0) 2 JLT-Z0 L0 C2 t-Zo ()] (21) We will show that both conditions (18) and (19) will be satisfied when equation (15), together with conditions (17), is solved, or, equivalently, that any solution equation (21) satisfies the boundary condition (t)+(t) + [tâ(t)+(t)] (22) In equations (21) and (22) and in the sequel the symbols (t), denote the boundary values the functions (z), â â (t) and (t) (z) and (z) z lles inside S and approaches the. point t L 0. If we denote by (X,Y) the sultant force exerted the whole system cracks (fLIUL2UâââULm)â iP then, because equation (3a), we have 2fL q(T) X+iY (23) PLANE CRACK PROBLEMS IN FINITE MEDIA Moover, since the medium S is sumed in equilibrium, the sultant force exerted find should be opposite to (X,Y). Then, because equation (13), we d -iP (24) By taking also into account that (z )dz (âr)dâr [+( z) -()]d, (25a) (25b) well equation (2) yielding, by application the first formula Plemelj [1] +(t)--(t) P+(t)-P-(t) (t) (26a) d--t[2q(t)-(t)] d[(t)] (26b) and equations (17) and (23), we can find that (z)dz +I P (27a) (T )dz +/-. (27b) By multiplying both sides equation (22) by into account equations (24) and integrating and taking (27), well the fact that (28) d [-( ) (21), then, because we derive that both sides the sulting equation a identically equal to iP. This means that, if the same procedu is used for equation equation (20) too, we must have C2 (z)dz (29) LO P.S. THEOCARIS AND N. IOAKIMIDIS or since C2#0 by sumption, condition (18) is satisfied. In another wording, the addition the term multiplied by the constant C 2 to the boundary condition (22) to obtain the boundary condition (21), sus the satisfaction the boundary condition (22). Furthermo, if we denote by exerted on the system cracks L (Mx,My) the sultant moment the forces then the opposite moment should sult which is exerted on the boundary L by the loading applied this boundary since the medium S w sumed in the state equilibrium. In this ce, it is ey to see that | p() -M M +iM x y (30) (31) Furthermo, by taking into account equation (26b), it is possible to show that [ L (z)dz -M and, further, to prove that the term multiplied by the constant C I in equation (2 I) should vanish. Hence, the addition this term to the boundary condition (22) does not pvent this condition from being satisfied. In this way, the pro that equation (15), equivalent to (21), sus the fulfilment both conditions (18) and (19) and the satisfaction the boundary condition (22) h been completed. 4. THE NUMERIC.AL TECKNIQU.E The numerical solution the singular integral equations derived in this paper can be achieved by dung such an equation to a system linear equations. This can be achieved by approximating the integrals through the use appropriate numerical integration rules and applying the sulting equation at appropriately selected collocation points. To illustrate this technique, we consider the simple problem a finite isotropic eltic medium S surrounded by a smooth contour L and containing a PLANE CRACK PROBLEMS IN FINITE MEDIA simple smooth crack L equations at L In this ce we sume that we know the parametric z0+iY 0 Xc+iY c and 0(s0)+iy 0(s o 0(s o c(s c) and (32) and at L c(s C )+iy c(s C (33) s Sc a al variables varying spectively. Without loss generality we can sume that s and Sc vary in the intervals [0,2] and [-i,i] spectively. In the opposite ce this linear variable tranformation the from can eily be achieved through s" +b (34) a and b a appropriate constants. Moover, we sume to know the normal and shear loading components 0 and L P0(t) Pc(t) On(S0)+iot(s0) On(Sc)+io t(sc (35) T E Pc(t) is exerted (36) suming, for simplity, that the same loading distribution both edges (+) and (-) the crack L Finally, the constant z 0 in equation (15) is sumed to have an appropriate value inside S, but not on the crack L z0=c such that the point z and C lies Similarly, the constants C in this equation a sumed having concte values. For numerical integrations the contour L 0 we can use the well-known trapezoidal rule with. n collocation points absssae. By defining the absssae T0i and the 0 by T0i z0(0i) 0i 2i/n0+00 ,n o (37) z0(s0k) S0k (2k-1)/n0+O 0 (38) P.S. THEOCARIS AND N. IOAKIMIDIS is an arbitrary constant, we have h0(T,) i=l A0ih0(Tâ0i,) (39) the weghts A01 determined by A0i 2z6(o0i)/n 0 ,n o (4O) the numerical integration rule (39) is valid both for gular and Cauchy-type integrals provided that the points a determined by equation (38). Moover, for the numerical integrations the crack L c we can use the Lobatto-Chebyshev numerical integration rule 12] with n absssae. By defining the absssae Cl and the collocation points t ck by (O) cos[=(i-l)/n-i] c , (41) we have Zc(Sck) Sck cos[(k-0.5)/(nc-l)] nc-I (42) (l-s c 2)-1/2h c ( )dY c1 i=l A h c (c iâ t ck (43) the weights A determined by i c zâ(O)/ (nc- I) 2,3 ânc-I %i (n0+nc) and 1/2 l,n c (44) In this way, the exist the values (t) on L complex unknowns in equations (15) and (17), X(t)=(t). (l-S2c)1/2 on at the absssae used 0i and and spectively, or, equivalently, 2(n0+n c) al unknowns in these equations. Furthermo, we obtain n o complex linear equations by applying equation (15) at the collocation points and (nc-l) complex linear equations by PLANE CRACK PROBLEMS IN FINITE MEDIA applying the same equation at the collocation points . One mo linear equation is obtained from the condition slngle-valuedness displacements (17). In this way, we have finally (n0+n c) complex linear equations, or 2(n0+n c) al linear equations with an equal number unknowns. By solving this system equations we obtain the approximate values the unknown function (t) at the absssae T01 L0, well the values X(t)=(t).(l-s2) 1/2 at the absssae T C1 We can also mention that it is very advantageous to use complex arlthmet in the computer for the evaluation all quid quantities and espeally the matrix coeffients the system linear equations to be solved. Moover, it is evident that the factor / in equation (15) can eily be computed at a collocation point L or [/]t=-- zâ(s0k)/Zâ(S0k [/d-]t=tc zâ(s0k)/Zâ(Sck) Finally, the complex stss intensity factors K (45) and at the tips the crack L corsponding to s =-I and s --+I, can eily be computed if one takes C C into account the formula 2.21/2exp --I1m[ z /z-+c (z-c)1/2(z) (46) c is the value the complex variable z=x+iy corsponding to the crack tip under consideration and % the angle between the Ox-axis and the tangent to the crack at the crack tip z=c in the diction extension the crack. Then we can find that K A KIA-IKIIA ilz(-1) 11/2x(z (-1)) KIB-IKII B -ilz(+1) 11/2X(Z c(+l)) (47a) (47b) P.S. THEOCARIS AND N. IOAKIMIDIS Since the Lobatto-Chebyshev numerical integration rule for the numerical integrations the crack L contains among the absssae used, equations (41), the end-points o =+i the integration interval, the evaluation the stss c intensity factors at the crack tips, by using equations (47), is straight-forward. Moover, we can mention that it is also possible to tat the ces cracks complicated shape like branched cracks or cruform cracks by applying techniques psented in fence [14] and fences [13] and [15] spectively. Similarly, it is possible to apply the psent technique to the ce edge cracks, made in fence [16] for the ce a crack terminating at the boundary a half-plane. In this ce, no condition single-valuedness displacements h to be taken into accoun the cracks. Furthermo, in the ce when a crack tip approaches another crack tip or a boundary the eltic medium, the modification the Lobatto-Chebyshev method proposed in fence [17] can be successfully used. Also in the ce semi-infinite straight or curvilinear cracks, the Lobatto-Chebyshev method numerical integration the cracks should be placed by the methods proposed in fences [18] and [19]. Several mo numerical techniques for the solution singular integral equations, which may be applied to the solution the singular integral derived in this paper, a viewed in fence [20]. 5. AN APPLICATION an application we consider the problem a rcular plane isotropic eltic medium with a straight crack one its diameters sumed for simplity to coinde with the Ox-axis, shown in Fig.3. The crack is sumed to be loaded by a pssu constant intensity o, the rcumfence the rcular medium is sumed unloaded. The radius the rcular medium is denoted by R, the length the crack by 2a and the distance the middle-point M the crack from the cent 0 the rcle by d. PLANE CRACK PROBLEMS IN FINITE MEDIA x-a x=b Fig. 3. A rcular disc with a crack one its diameters. Then, in accordance with the notation the pvious section we will have Zo(S) xp(is) Zc(S) d Po(S) Pl(S) (48) The singular integral equation (15), together with the condition singlevaluedness displacements (17), w solved by using the numerical technique proposed in the pvious section. In Table I we psent the dimensionless values the stss intensity factors KA/(oa) and /( oa1/2) at the crack tips A and B (Fig.3), for the ce when a/d=2.4 and d/R=0.2, for several values the numbers and nc the absssae used. These sults we obtained for =C2=I and z0=c=0.5i. factors K The influence these constants on the values the stss intensity and is not gat. Moover, from the sults Table I we see that, the numbers n and nc absssae used ince, the values the numerical sults obtained converge. Finally, the values for KA/(a1/2) and KB/(oa1/2) psented Table I we seen to be in accordance with expected values these factors psented in fence [21] for the same geometry and loading conditions (Table 3.3.1). P.S. THEOCARIS AND N. IOAKIMIDIS 6. CONCLUSIONS By using the method complex potentials Kolosov and Muskhellshvill it w seen that every plane eltity crack problem inside a finite or infinite medium with or without holes can eily be duced to a complex Cauchy-type singular integral equation both the cracks and the boundaries the medium. These singular integral equations can be effectively solved by dung them to systems linear equations. In his way, it is possible to determine numerically the values the stss intensity factors at crack tips for almost any geometry the crack and the whole eltic medium and under arbitrary loading conditions on the cracks and the boundaries the medium. Even the ce when the loading distributions a not the same on both edges the cracks and the exists a sultant force, well a sultant moment on each crack can be tated. gards the accuracy the numerical sults obtained, this can be made good we want at the expense computer time and the use appropriate numerical integration rules. FENCES I. N.I. Muskhellshvili, Some Bic Problems the Mathematical Theory Eltity, 4th Edn. P. Noordhf, Groningen (1963). 2. R.V. Golâdshteyn and R.L. Salganik, Planar problem curvillnear cracks in an eltic solid. Mech. Solid 5, 54 (1970). 3. R.V. Golâdshtein and L.N. Savona, Determination crack opening stss intensity coeffients for a smooth curvilinear crack in an eltic plane. Mech. Solids 7, 64 (1972). 4. R.V. Golâdstein and R.L. Salganik, Brittle fractu solids with arbitrary cracks. Int. J. Fract. , 507 (1974). PLANE CRACK PROBLEMS IN FINITE MEDIA 5. A.M. Linâkov, Integral equations in the theory eltity for a plane with cuts, loaded by a balanced system forces. Sov. Phys. Dokl. 19, 718 (1975). 6. V.V. Panyuk, M.P. Savruk and A.P. Datsyshyn, A general method solution two-dimensional problems in the theory cracks. Enr.act. Mech. 9, 481 (1977). 7. N.I. loakimidis and P.S. Theocaris, Array periodic curvilinear cracks in an infinite isotropic medium. Acta Mech. 28, 239 (1977). 8. P.S. Theocaris and N.I. loakimidis, A star-shaped array curvilinears cracks in an infinite isotropic eltic medium. J. Appl. Mech. 44, 619 (1977). 9. N.I. loakimidis and P.S. Theocaris, Doubly-periodic array cracks in an infinite isotropic medium. J. Elt. 8, 157 (1978). . P.S. Theocaris and N.I. loakimidis, The problem interaction between a misfitting inclusion and a crack in an infinite eltic medium. J. Elst. 9, 97 (1979). II. N.I. loakimidis and P.S. Theocaris, On a method numerical solution a plane eltity problem. Strojn. C. 29, 448, (1978). 12. P.S. Theocaris and N.I. loakimidis, Numerical Integration methods for the solution Singular integral equations. Quart. Appl. Math. 35, 173 (1977). 13. P.S. Theocaris and N.I. Ioakimidis, The Inclusion problem in plane eltity. Quart. J. Mech. Appl. Math. 30, 437 (1977). 14. P.S. Theocaris and N.I. loakimidis, The symmetrically branched crack in an infinite eltic medium. Zeit. A. Math. Phys. (ZP) 27, 801 (1976). 15. P.S. Theocaris and N.I. loakimidis, A method solution the problem the unsymmetric cruform crack in an infinite plane isotropic eltic medium. Acta Mech. 29, 127 (1978). 16. N.I. loakimidis and P.S. Theocaris, A system curvilinear cracks in an isotropic eltic half-plane. Int. J. Fract. 15, No.4, 299 (1979). P.S. THEOCARIS AND N. IOAKIMIDIS 17. P.S. Theocaris and N.I. loakimidis, Stss intensity factors at crack tip near boundaries or other geometrical discontinuities. Int. J. Fract. 15 No 5, 419 (1979). 18. N.I. loakimidis and P.S. Theocaris, The Gauss-Hermite numerical integration method for the solution the plane eltic problem semi-infinite periodic cracks. Int. J. Engng S. 15, 271 (1977). 19. N.I. loakimidis and P.S. Theocaris, The practical evaluation stss intensity factors at semi-infinite crack tips. Engng Fract. Mech. (to appear). 20. N.I. loakimidis and P.S. Theocaris, The numerical evaluation a cls generalized stss intensity factors by use the Lobatto-Jacobi numerical integration rule. Int. J. Fract. 14, 469 (1978). 21. D.P. Rooke and D.J. Cartwright, Compendium Stss Intensity Factors, Ist Edn. Her Majestyâs Stationary fice, London (1976). TABLE I Dimensionless values the stss intensity factors and KA/(oa1/2) and KB/(oa1/2) at the tips the crack numerical technique Fi. (for a/d=2, g and d/R=0.2) obtained by using the (with this paper C1=C2=1 Zo=C=O. 5i) for several values n o and nc. KZ/(ca1/2) I. 3269 KB/(ca1/2) I. 3800 ,5 I. 3495
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