Stress concentration due to an array or hemispherical cavities at the surface of an elastic half-space

Transcription

1 Jurnal f Elasticity 28: , Kluwer Academic Publishers. Printed in the Netherlands. Stress cncentratin due t an array r hemispherical cavities at the surface f an elastic half-space XIAOPING LUg', J.R. BARBER* and MARIA COMNINOU Department f Mechanical Engineering and Applied Mechanics, University f Michigan, Ann Arbr, MI 18109, USA (*authr fr crrespndence) Received 27 Nvember 1989; in revised frm 19 March 1991 Abstract. The paper investigates the perturbatin in an therwise unifrm stress field in an elastic half-space due t a dubly-peridic array f small hemispherical hles at the free surface. The slutin is btained using three ptential functins f duble Furier series frm in Galerkin's strain ptential slutin, the cefficients f which are determined using the cllcatin methd. The unperturbed field is taken t be ne f unifrm plane stress parallel t the free surface. Tw special cases arc studied - unifrm tensin and unifrm shear stress. Numerical results fr these cases can be generalized by superpsitin t give slutins fr a general state f biaxial plane stress. It is fund that, fr bth tensin and shear, the maximum stress cncentratin ccurs at the bttm f the hles. The stress cncentratin factr increases with the rati f hle spacing t radius, appraching the knwn slutin fr a single hemispherical hle at large ratis. 1. Intrductin Many engineering materials have varius kinds f surface defects such as blw hles, scratches and micrcracks, which tend t intensify the lcal stress field and hence reduce the lad carrying capacity f the cmpnent. Such defects can act as initiatin sites fr fatigue cracks and are therefre an imprtant factr in the experimentally bserved sensitivity f fatigue life t surface finish. Blw hles, in particular, are difficult t avid in the manufacturing f cmpnents frm resin-based cmpsite materials, because f the necessity fr mixing f the resin and its relatively high viscsity which inhibits the mbility f air bubbles in the slidifying cmpnent. Experimental evidence als indicates that surface defects influence the prpagatin f the fracture surface in such materials [ 1]. The perturbatin in a state f unifrm biaxial hydrstatic tensin due t a single hemispherical surface cavity was analysed by Eubanks [2] and results fr mre general spheridal shapes have been given by Fujita et al. [3, 4]. Hwever, the prblem f the interactin between the stress fields arund ~" Nw at Department f Mathematics, University f Wllngng, NSW 2500, Australia.

2 112 Xiaping Lu et al. several cavities has nt been addressed and frms the subject f the present paper. Specifically, we investigate the perturbatin in an therwise unifrm biaxial state f plane stress due t a peridic array f identical hemispherical cavities at the surface f an elastic half-space. Results are presented fr the cases where the unperturbed field is ne f uniaxial tensin r pure shear, since mre general ladings can then be cnstructed using superpsitin. The perturbed stress fields are represented in terms f Galerkin strain ptentials, which are chsen in the frm f duble Furier series. The peridic prperties f the series permit us t define the bundary cnditins in terms f the surface tractins in a unit cell cntaining a single hle. The cllcatin methd is used t satisfy the bundary cnditins at the free surface f the half-space and at the hle surface. 2. Methd f slutin We cnsider the elastic istrpic half-space, z > 0 in which a state f unifrm stress is perturbed by a peridic array f small hemispherical hles f radius a at the surface z = 0 as shwn in Fig. 1. The centers f the hles are situated _ Fig. I. Cnfiguratin f the system.

3 Hemispherical cavities 113 at the pints (mb, nb, 0), where m = 0, _ 1, _ ; n = 0, + 1, _ We assume that the hles d nt intersect each ther and hence that b > 2a. The bundary cnditins at infinity take the frm trxx ~ tr, tr yy, trzz, trxe, trzx ~ O, z ~ (1) fr tensile lading, and ltxy-.c.'[, ltxx, ayy, azz, ltyz, azx--~o, Z"-~ O0 (2) fr shear lading. We als require that the surface f the hles and the remaining parts f the surface z = 0 be tractin-free. These cnditins can be stated in the frm azx=trzy=azz=o, (x+mb)2+(y+_nb)2>a 2, z=0, (3) ar=trr, =a,~r=o, (x +_mb)2 +(y + nb)2 + z2=a 2, z>0 (4) where (R, 0, ~b) defines a set f spherical plar c6rdinates centered n the hle in questin. We cnstruct the slutin f the prblem in the frm IS] = [U] + [S] (5) where [U] is the unperturbed state f unifrm stress and [S] is a crrective slutin, describing the perturbatin in [U] due t the array f hles. 3. Ptential functin representatin Slutin [S] is represented in terms f the Galerkin ptential K V4F = , 1 -- v (6) -~V2X / 2 t92 \ a~,~ = 2(1 -- v)-~x + ~vv ---~x2) div F, (7). [OV2X ~V2Y~ 0 2 axy=(1- v) (-~--y + --~--x ) - ~--~y div F, (8) etc., where F = Xi + Yj + Zk is tbe Galerkin v~tr, K is the bdy frce and

4 114 Xiaping Lu et al. v is Pissn's rati fr the material. The remaining stress cmpnents can be btained frm equatins (7, 8) by cyclic permutatin f X, Y, Z and x, y, z. Since there is n bdy frce in the present prblem, equatin (6) reduces t V4F = 0, (9) and hence F is required t be a biharmnic vectr functin. Westergaard [5 Art. 70] argues that the cmpnent Z f the Galerkin vectr can always be arbitrarily set t zer withut lss f generality. Hwever, his prf depends upn the cnstructin f a biharmnic functin f such that Ox Oz - Z, (10) and there are restrictins n the dmains fr which this can be dne. This difficulty was first nted by Sklnikff [6] in the related prblem f cmpleteness f the Papcvich-Neuber slutin when ne cmpnent f the crrespnding vectr functin is set t zer. A mre detailed investigatin f the questin is given by Eubanks and Sternberg [7] wh shw that the crrespnding integratin can be perfrmed as lng as every line parallel t the z axis intersects the bundary f the bdy in at mst tw pints. This cnditin is satisfied in the present prblem, thus permitting us t set Z t zer. A further simplificatin in the representatin can be achieved by writing X=f~+zf2; Y=fa+zf4 (11) where the functins f. are harmnic [8]. It then fllws that the harmnic parts f~,f3 f X, Y can be replaced by a single harmnic strain ptential ~A A (12) ~ = Ox Oy in terms f which tr~,~ = Ox2, trx~ =Ox coy' etc. (13) (see [5, Art. 67]). In summary, the slutin [S] is represented by superpsing a Galerkin vectr F = z(if2 +il) (14) and a strain ptential 4~, where f2,f4, 4~ are harmnic functins f x, y, z and

5 Hemispherical cavities 115 the crrespnding stress cmpnents are defined thrugh equatins (7, 8) fr F0 and (13) fr ~b. We ntice that this representatin gives us three independent harmnic functins and permits us in general t satisfy three tractin bundary cnditins at the free surface. The dubly peridic nature f the prblem and the requirement that the perturbed field decay as z-~ shws that the apprpriate frm fr the functins ~b,f2,f4 is i=j~ A.e -a '~,/;~-~z sin(i~x) sin(j'~y) (15) where ). = 2~z/b, Aij are a set f arbitrary cnstants t be determined frm the bundary cnditins and the chice f sine and csine in the varius series depends upn the symmetry f the applied lading. These functins are harmnic fr all A;j and the chice f a negative argument fr the expnentials ensures that the perturbed field decays as z ~ ~ as required by (1, 2). It is als easily verified that the use f functins like (15) in (7, 8, 13, 14) generates stress fields which are dubly peridic, as required, l The peridicity and symmetry f the prblem and the representatin permits us t satisfy the bundary cnditins by applying (3, 4) inside the unit cell 0 < x < b/2, 0 < y < b/2. We als nte that n the hemispherical surface x2+y2+z2=a 2, x>0, it is mre cnvenient t replace the bundary cnditins (4) by the equivalent cnditins tr x~,nx + 17xyny + a~,~nz = 0] tr yxn x "+" Tyyny "]- ayzn z = O ( ~ a zx nx + ffzy ny + azz n z = 0 J (16) where nx =-x/a, ny =-y/a, nz =-z/a are the directin csines f the spherical surface. By impsing the bundary cnditins (3, 16) at a suitable set f cllcatin pints, we btain a set f algebraic equatins fr the unknwn cefficients in the series (15), frm the slutin f which the stress cmpnents can be btained, using equatins (7, 8, 13, 14). 4. Uniaxial tensile lading When the applied lading cnsists f uniaxial tensin (equatin (1)), the prblem is symmetrical abut the planes x = O, y = 0 and the apprpriate m This wuld nt be true if fr example the Papcvich-Neuber slutin were used, which is the reasn fr preferring the present representatin.

6 116 Xiaping Lu et al. duble series frms derived frm equatins (14, 15) are X= ~ ~ A,,z e-a~zsin(iax)cs(jay), i=lj=o (17) Y= ~ ~ B~z e-x a'cw~cs(iax)sin(jay), (18) i~jffi 1 ~b = ~ ~ C/~ e -~ '~'fic-~z cs(/ax) cs(jay), (19) i~oj=o where Aij, Be, Cq are the unknwn cnstants t be determined frm the bundary cnditins (3, 16). T btain a numerical slutin, we truncate the series at i,j = N, giving a ttal f N(3N + 4) unknwn cnstants. We therefre require an equal number f algebraic equatins, which we btain by enfrcing cnditins (3, 16) at a set f cllcatin pints chsen as the zers f the first neglected trignmetric functins in the duble series [9] which fall within the unit cell 0<x <b/2,0<y <b/2, i.e. sin A(N + 1)xt cs A(N + 1)yt = 0, l = 1, 2,..., N(N + 1), (20) cs A(N + 1)Xm sin A(N + 1)y,~ = 0, m = 1, 2... N(N + 1), (21) cs A(N + 1)x. cs A(N + 1)y. = 0, n = 1, 2... (N + 1) 2, (22) 5. Shear lading When the unperturbed stress field cnsists f pure shear (equatin (2)), the prblem is antisymmetric abut x = 0, y = 0 and the fllwing ptential functins are apprpriate: X = ~'. ~ aiyz e -a~z cs(iax) sin(jay); (23) iffi0j=l Y = ~ ~ Bijz e -a~z sin(iax) cs(jay); (24) i=lj=0 ~b = ~, ~ C e -a '~,/a--~* sin(/rx) sin(jay). (25) i=lj=l Truncating the series at i,j = N, we btain N(3N + 2) unknwn cnstants, which are determined by impsing bundary cnditins (3, 16) at the cllcatin pints cs 2(N + 1)X 1 sin 2(N + 1)yt = 0, l = 1, 2... N(N + 1); (26)

7 Hemispherical cavities 117 sin 2(N + 1)Xm cs 2(N + 1)ym = 0, sin 2(N + l)xn sin 2(N + 1)yn = 0, m = 1,2,...,N(N+ 1); (27) n = 1, 2... N 2. (28) 6. Results and discussin Numerical calculatins have been perfrmed fr several values f the rati b/a in the tw cases f applied uniaxial tensin and pure shear. Slutins were btained fr varius values f the number f terms N in the truncated series and the results shwed less than 1% change in the stress cmpnents fr increase in N beynd a value between 4 and 10, depending n the type f lading and the rati b/a. Ntice incidentally that a value f N = 10 results in a system f 340 unknwns and equatins in the case f uniaxial tensin. A value f Pissn's rati f 0.3 was used thrughut the calculatins. Figure 2 shws the distributin f maximum principal stress at the plane z = a fr the case f unifrm tensile lading in the x directin and b/a = 4. The stress is greatest near the hle, reaching a maximum f 2.08tr at the bttm f the hle (0, 0, a). At this pint, the stress cmpnents a~,x, ttyy are principal stresses (as required by symmetry), the frmer being large and tensile, whilst the later is small and cmpressive. The stress cncentratin factr trmax/tr is shwn in Fig. 3 and varies frm 1.44 t 2.20 as b/a increases frm 2.4 t 5. Over the same range, the dimensinless rthgnal stress ayy is cmpressive, varying mntnically frm t At b/a = 2, the edges f adjacent hles wuld just tuch. The present methd is nt well-suited t this limiting case, but the trend f the results indicates that the stress cncentratin factr wuld fall t clse t unity and the rthgnal stress t zer- a result which culd be predicted frm cnsideratins f frce flw, since the remaining 'islands' between the hles wuld then have little influence n the stress field in the material belw the plane z = a. At the ther extreme, as b/a -~ ~, the prblem reduces t that f the single hle in a field f uniaxial tensin. Eubanks [2] btained a stress cncentratin factr f 2.23 fr the related prblem f a single hle in biaxial hydrstatic tensin, but Fujita et al. claim that a mre accurate result is The present results can be cmpared with these slutins by superpsing equal unixial tensile fields in the tw rthgnal directins x, y, and predict a stress cncentratin factr f 2.00 at b/a = 5. Figure 4 shws the largest principal stress trxx alng the axis f symmetry f the hle, x = y = 0, fr b/a = 4. The stress cncentratin diminishes rapidly with increasing z falling t the unperturbed value at abut z = 2a. Similar results were fund fr ther ratis b/a.

8 118 Xiaping Lu et al. MAXIMUM PRINCIPAL STRESS TENSILE LOADING I I Fig. 2. Cntur plt f maximum principal stress n z - a fr uniaxial tensile lading (b/a = 4). Crrespnding results fr the case f pure shear lading are shwn in Figs Figure 5 shws the maximum principal stress n the plane z = a fr b/a The maximum stress again ccurs at the bttm f the hle (0, 0, a) and cnsists f a state f pure shear, with,;,y the nly nn-zer stress cmpnent. The stress cncentratin factr (ax~,)ma,,/z is shwn in Fig. 6 as a functin f b/a. As in the case f tensin, the results increase mntnically with b/a, and suggest a value f 3.10 fr the single hle (b/a-,.). At the ther extreme, they apprach an intuitive limit f unity as b/a -~ 2. Ntice incidentally that the stress cncentratin in shear fr the single hle can be btained by superpsing rthgnal states f uniaxial tensin and cmpressin, but the

9 Hemispherical cavities II 1'4 II II X 2 X E b _ 0 I I Fig. 3. Maximum principal stress at the bttm f the hle (0, O, a) vs. b/a, fr uniaxial tensin. 3 II ~ >~ ~! 0. I I I 1 2 ~ 4./ Fig. 4. Decay f xx alng the z axis fr uniaxial tensile lading (b/a = 4).

10 120 MAXIMUM PRINCIPAL STRESS SHEAR LOADING,O I i I Fig. 5. Cntur plt f maximum principal stresses n z = a fr shear lading (b/a = 4). 4 X O ~ I 0!! b/. Fig. 6. Shear stress at ~e bttm f ~e ~le (0, O, a) vs. b/a, fr sh~r la~ng.

11 Hemispherical cavities II II x 3 2 X, 0 I I I ~ :~,~ 5 Fig. 7. Decay f tr,,y alng the z axis fr shear lading (b/a = 4). same superpsitin cannt be used fr finite b/a, since the array f hles renders the bdy gemetrically anistrpic. Finally Fig. 7 shws the decay f axy # alng the line x = y = 0 fr b/a = Cnclusins The results shw that the maximum stress cncentratin due t the array f hles always ccurs at the bttm f the hle and the magnitude f the stress cncentratin factr decreases mntnically frm the single hle value when b/a is large t a value near unity (n stress cncentratin) when the hles nearly tuch. Previusly published results fr a single hle in a biaxial hydrstatic tensile field give a gd apprximatin (better than 10%) when b/a > 5- i.e. when the space between the hles is greater 1.25 times their diameter. In all cases, the stresses decay rapidly with depth t the applied unifrm value.

12 122 Xiaping Lu et al. References 1. Maria Cmninu, An verview f interface cracks. Eng. Frac. Mech. 37 (1990) R.A. Eubanks, Stress cncentratin due t a hemispherical pit at a free surface. ASME J. Appl. Mech. 21 (1954) T. Fujita, E. Tsuchida and I. Nakahara, Stress cncentratin due t a hemiprtate spheridal pit at a free surface f a semi-infinite bdy under all-arund tensin. Bull. Japan Sc. Mech. Eng. 23 (1980) T. Fujita, E. Tsuchida and I. Nakahara, On the stress cncentratin arund a hemi-blate spheridal pit in a semi-infinite bdy. Bull. Japan Sc. Mech. Eng. 24 (1981) H.M. Westergaard, Thery f Elasticity and Plasticity. New Yrk: Dver (1952), Chapter I.S. Sklnikff, Mathematical Thery f Elasticity. New Yrk: McGraw-Hill, 2nd. edn. (1965). 7. R.A. Eubanks and E. Sternberg, On the cmpleteness f the Bussinesq-Papcvich stress functins. Rat. Mech. Anal. 5 (1956) Y.C. Fung, Fundatins f Slid Mechanics. Englewd Cliffs: Prentice-Hall, 2nd. edn. (1965), pp J.P. Byd, Private cmmunicatin.