Dynamic analysis of fibre breakage in singleand multiple-fibre composites

Transcription

1 JOURNAL OF MATERIALS SCIENCE 31 (1996) Dynamc analyss of fbre breakage n sngleand multple-fbre compostes M.L. ACCORSI, A. PEGORETTI**, A.T. DIBENEDETTO * Department of Cvl Engneerng, and ~ Department of Chemcal Engneerng and Insttute of Materals Scence, Unversty of Connectcut, Storrs, CT 6269, USA The dynamc effects assocated wth fbre breakage n sngle- and multple-fbre compostes were nvestgated usng the fnte element method. The goal of ths work was to determne f the dynamc stresses dffered sgnfcantly from the statc stresses and, consequently, f dynamc effects were mportant n the predcton of subsequent fracture of the composte. For sngle-fbre compostes, dynamc and statc analyses were performed over a range of modulus values for epoxy matrces wth a glass fbre to establsh a range of the dynamc effects. The maxmum dynamc stresses, spatal dstrbuton of dynamc stresses, and correspondng tmes were evaluated. For multple-fbre compostes, the dynamc effects assocated wth sngle- and double-fbre breakage were nvestgated for a typcal epoxy/glass composte and compared to the correspondng statc problems. 1. Introducton In contnuous fbre-renforced compostes, ndvdual fbre flaments can fracture at stresses less than 5% of the ultmate strength of the composte. As the stress ncreases, fbre flaments contnue to break at locatons throughout the materal untl enough breaks occur cumulatvely at one locaton to produce a defect large enough to cause catastrophc falure. Varaton n flament strengths tends to offset the effects of stress concentratons at fbre breaks, thus favourng random fbre fracture rather than cumulatve ones at the ste of an ntal fracture (.e. co-ordnated fbre fracture). It has been found expermentally, however, that a coordnated number of fbre fractures can occur at nterfbre spacngs of up to approxmately sx to eght fbre dameters, wth strong co-ordnaton at nterfbre spacngs less than four [-1, 2]. Such patterns appear to be dependent on the fbre volume fracton and the nature of the localzed matrx fracture and nterface debondng at the ste of the flament fracture. At the nstant a contnuous flament s fractured, the broken fbre end loses load and retracts n the matrx. A compressve stress wave moves through the fbre and a multaxal stress wave moves out nto the matrx, creatng a sgnfcant overshootng of the equlbrum stresses at the flament fracture ste. The stored energy lost n the regon around the fbre end generates fractures n the matrx and/or along the nterface as well as localzed stress concentratons n adjacent fbres. In a recent study, a lnear fnte element method (LFEM) has been used to calculate the ntaton and arrest stran energy release rates for these dfferent modes of propagaton [3]. For an S- glass fbre/epoxy matrx mcrocomposte wth the same consttuent propertes as used n ths study, stran energy release rates n the range of J m- 2 for nterface debondng and 8-1 J m -2 for matrx crackng were obtaned. In another study [4], t was shown that the statc equlbrum stresses n adjacent fbres, generated by the fracture of a sngle flament, sgnfcantly ncreased the probablty of a co-ordnated pattern of sequental fbre fractures. An LFEM analyss was used to smulate the fracture of a sngle flament n a contnuous E-glass fbre-renforced epoxy composte. It was shown that at constant stran, subsequent falure of the adjacent fbre s constraned to wthn about three to fve fbre dameters dstance from the broken end [4]. The computer smulatons showed that even under statc condtons, the probablty of subsequent falure of an adjacent flament could attan levels of the order of 3%-4% at a stress equal to the average breakng strength of a sngle flament. Expermentally, t was observed that n E-glass/epoxy multple fbre mcrocompostes the subsequent falures occurred nstantaneously, thus suggestng that dynamc processes were operatve. Prelmnary LFEM smulatons showed that the dynamc stress waves generated at the nstant of flament fracture caused sgnfcant overshootng of the equlbrum stress concentraton factors n the adjacent fbres. Because the probablty of falure of a glass fbre depends exponentally on the level of stress, t s apparent that the nature of the dynamc stress waves could have a sgnfcant effect on co-ordnated patterns of fbre fracture and, thus, on the generaton of crtcal defects n contnuous fbre-renforced compostes. In ths study, we used the MARC fnte element code to examne the dynamc stress patterns generated *Permanent address: Department of Materals Engneerng, Unversty of Trento, va Mesano 77, 385 Trento, Italy Chapman & Hall 4181

2 wthn the frst 1 ns of a fracture event n both sngle and multple glass fbre mcrocompostes. It was shown that the dynamc stress waves have a sgnfcant effect on the probablty of localzed falure of adjacent fbres n the regon of the ntal fracture ste. 2. Sngle-fbre composte In ths secton, the dynamc effects assocated wth a fbre break n a sngle-fbre composte are nvestgated. The model consdered conssts of an nfnte epoxy matrx contanng a sngle glass fbre, as shown n Fg. la. An axsymmetrc fnte element (FE) model of plane ABCDEF, as shown n Fg. lb, s used for the analyss. The dmensons of the FE model are chosen large enough so that the waves produced by fbre breakage do not mpnge upon the boundares durng the tme perod consdered. These dmensons are dependent upon the wave speeds n the fbre and matrx. For the materals consdered, values of dr = 55 ~m and d= = 3 gm were used. Approxmately 26 axsymmetrc sold elements were used n the FE model. It s desred to load the FE model quas-statcally up to fbre breakage, then ntroduce a break n the fbre along edge BC. Because ths problem s lnear, t s suffcent to break the fbre at 1% stran and scale these results by the actual fbre breakage stran. Approprate boundary condtons are requred along the four edges of the FE model to smulate the problem. Edge AD s a free edge. As prevously mentoned, dr s chosen suffcently large so that the stresses along AD correspond to unform stresses n an nfnte matrx materal. Along edge CF, the radal dsplacement s prescrbed to be zero. A unform dsplacement n the z drecton s prescrbed along edge DEF. To smulate the expermental procedure, the dsplacement s ncreased quas-statcally up to fbre breakage, then held constant. Pror to fbre breakage, the dsplacement n the z drecton along edge ABC s zero. At fbre breakage, ths dsplacement boundary condton s released along edge BC and retaned along edge AB. A FE model D The baselne materals are chosen to be a stff or soft epoxy matrx contanng an S-glass or E-glass fbre. The followng materal propertes are used E31 : stff epoxy: E = 2.9 GPa, v =.35, P = 1.23 gcm-3; soft epoxy: E--1.6GPa, v=.35, p=1.23gcm-3; S- glass: E=86.9GPa, v=.22, P=2.46gcm -3, rf = 5 gn; E-glass: E = 72. GPa, v =.22, p = 2.46 gcm -3, rf = 6.5 Ixm. The axal and shear stresses (~z and ~z) at the crack tp (pont B) as a functon of tme followng fbre breakage are shown n Fgs 2 and 3, respectvely, for an S-glass fbre n an epoxy A matrx. For comparson, the values from a statc analyss of the model wth a broken fbre are shown. The dynamc stresses start at the statc values for an unbroken fbre, ncrease rapdly and obtan ther maxmum values, then oscllate around the statc values for a broken fbre. Ths behavour s expected, because no dampng s ncluded n the analyss. The maxmum dynamc stresses at the crack tp and correspondng tmes are gven n Table I. The dynamc amplfcaton factor (DAF) s defned as the rato of the maxmum dynamc stress dvded by the statc stress. From Table I t s seen that the DAF and tmes are approxmately the same for all four components of stress. The spatal dstrbutons of the axal and shear stresses along the fbre-matrx nterface (BE n Fg. 1) are shown n Fgs 4 and 5, respectvely. The spatal dstrbutons of these stresses along the matrx (BA n O_ N ! l Fgure 2 Sngle-fbre composte: axal stress versus tme at the crack tp (pont B n Fg. 1) followng fbre breakage: (--) dynamc, (---) statc. II ~!N~I~ ~!~N~ ~:,~N \ B E \ N Fbre break / - 5 (a) I : c Fbre I I~ TM d~ d ~1 (b) F re, Fgure 1 (a) Sngle-fbre composte. (b) Axsymmetrc fnte element model.?- dr b Fgure 3 Sngle-fbre composte: shear stress versus tme at the crack tp (pont B n Fg. 1) followng fbre breakage: ( ) dynamc, (---) statc. 4182

3 T A B LE I Dynamc results for stff epoxy/s-glass Stress Statc Dynamc DAF Tme (MPa) (MPa) (ns) cr %~ or %~ Q ~ F t ~ ~... ~ \ r dstance along matrx (pm) ~... ~...,... "~ 4...,~... T... Fgure 6 Sngle-fbre composte: axal stress along the matrx (sde BA n Fg. 1): (--) statc, ([]) DAF* statc, (---) maxmum dynamc. x ' " [ ~..~ z dstance along fbre (pro) -5 Fgure 4 Sngle-fbre composte: axal stress along the fbre-matrx nterface (sde BE n. Fg. 1) : (--) statc, (V3) DAF* statc, (---) maxmum dynamc '......!... ~... c~... / ~... ~ ~... ~,~... /~ / /! ': r,,, r, I r z dstance along fbre (lxm) Fgure 5 Sngle-fbre composte: shear stress along the fbre-matrx nterface (sde BE n Fg. 1) : (--) statc, ([]) DAE* statc, (---) maxmum dynamc. ~ ,,,,,,,, t ] rdstance along matrx (gn) Fgure 7 Sngle-fbre composte: shear stress along the matrx (sde BA n Fg. 1) : (--) statc, (V]) DAF* statc, (---) maxmum dynamc. TABLE II Dynamc results for S-glass Case E m Cf/C m DAF Tme (GPa) (ns) Fg. 1) are shown n Fgs 6 and 7. In these fgures, the maxmum dynamc stresses, statc stresses wth a fbre break, and the statc stresses scaled by the DAF are shown. It s seen that the spatal dstrbuton of the statc and dynamc stresses are smlar, and f the statc stresses are scaled by the DAF, they agree closely wth the dynamc stresses. Therefore, f the statc stresses are known, the dynamc effects can be characterzed by a DAF. The range of values for the DAF was evaluated for S-glass and E-glass fbres embedded n dfferent epoxy matrces. For ths evaluaton, only the value of Young's modulus of the matrx, Era, was changed. Tables II and III summarze the matrx modulus values and gve results for the DAF and tme for the maxmum stress for S-glass and E-glass fbres, respectvely. The rato of the longtudnal wave speeds for the fbre and matrx (Cf/Cm) s also gven. The DAF TABLE III Dynamc results for E-glass Case E m Cf/C m DAF Tme (GPa) (ns) and tme for maxmum stress are plotted versus wave speed rato n Fgs 8 and 9, respectvely. The results for S-glass and E-glass overlap, suggestng that the wave speed rato can be used to characterze these results. The DAF ranges from for glass fbres embedded n these epoxy matrces. The axal stress at 4183

4 u [] !... ~ "'~ '~=~ Y~ C Fbre B D x C+l C~ Fgure 8 Sngle-fbre composte: dynamc amplfcaton factor versus rato of wave speeds. (-~-) S-glass, (--[2--) E-glass. - v E '"~" ~.,... ~ ~ C+I Cm Fgure 9 Sngle-fbre composte: tme for maxmum stress versus the rato of wave speeds. Gl-) S-glass, (-- [~--) E-glass. D- 8 : "~, ; ~ '-..! : ~ ~: L C+I Cm Fgure 1 Sngle-fbre composte: axal stress at the crack tp versus the rato of wave speeds. S-glass: (-~-) dynamc, (- t--) statc. E-glass: (-[~,) dynamc, (--C)-~ statc. the crack tp s plotted versus wave speed rato n Fg. 1. It s seen that the largest DAF occurs for the smallest wave speed rato whch corresponds to the case wth the hghest statc stress. Therefore, the dynamc effects are most sgnfcant for ths case. 3. Multple-fbre composte In ths secton, the dynamc effects of fbre breakage are examned for a two-dmensonal (plane stran) composte materal 9 multple fbres, as shown n Fg. 1 la. The fnte element model (ABCD) s (a) T Y~ A Rule of mxtures ~.. pont 2 Fbre 3 ] dv e~.-pont 1 Fbre 2 Fbre 1 B 1- dx -I (b) C D I drom Fgure 11 (a) Multple-fbre composte. (b) Plane stran fnte element model. shown n Fg. 11b. In ths model, three fbres and two matrx regons are explctly modelled and the propertes of the composte obtaned from a rule of mxtures are used n the outer regon. The overall dmensons are, agan, chosen suffcently large to ensure that the waves caused by fbre breakage do not mpnge a boundary (dx = 6 ~tm, dy = 13 ~tm). The boundary condtons for ths model are smlar to those descrbed for the sngle-fbre model. Approxmately 1 8 plane stran elements were used n the fnte element model. For the case consdered, the spacng of the fbres s equal to the fbre dameter (dn = df = 14 gm) whch gves a fbre volume fracton of vf =.556. A stff epoxy matrx and E-glass fbres are used. The followng propertes for the composte regon (drom) are obtaned usng the rule of mxtures: Eaa = 41.3 GPa, E22 = 1.8 GPa, Glz = 3.67 GPa, G31 = 3.67 GPa, G23 = 3.6 GPa, v12 =.278, v31 =.727, v23 =.5. The stresses at ponts 1 and 2 n fbres 2 and 3, shown n Fg. 11b, were montored over tme for two cases. These ponts are slghtly nsde the lower left corner of the fbres. In the frst case (dynamc 1), only fbre 1 s broken. In the second case (dynamc 2), fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1. Both fbres are broken at edge AB by releasng the dsplacement boundary condton on the fbre along that edge. The correspondng statc analyses (statc 1 and 2) were also performed for comparson. All cases correspond to an ntal stran of 1%. The axal stress (cyxx) at ponts 1 and 2 versus tme followng fbre 1 breakage are shown n Fgs 12 and 13, respectvely, for both cases. For case 1, the axal stress at pont 1 ncreases from the statc value wth no X 4184

5 13- t TABLE IV Results for multple-fbre composte Statc(MPa) Dynamc(MPa) DAF 1 2 t=44ns t=88ns 1 2 %= at pont %= at pont Fgure 12 Multple-fbre composte: axal stress at pont 1 of Fg. 11 versus tme for case 1 (only fbre 1 s broken) and case 2 (fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1). (---) Statc 1, (--) statc 2, (- -) dynamc 1, (--) dynamc O =D... ~... =..~... ~... ~ "k * ~, ~! ".... ~ r ~...!... ~ t xdstance along fbre 2 (gn) 12, Fgure 14 Multple fbre composte: axal stress along fbre 2 for case 1 (only fbre 1 s broken). (---) Statc 1; (E3) DAF* statc; (--) dynamc, t = 44 ns Fgure 13 Multple-fbre composte: axal stress at pont 2 of Fg. 11 versus tme for case 1 (only fbre 1 s broken) and case 2 (fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1). (---) Statc 1, (--) statc 2, (--Q dynamc 1, (--) dynamc 2. fbre breakage to ts maxmum value at t = 44 ns, as seen n Fg. 12. In fbre 3 at pont 2, the maxmum stress due to breakage of fbre 1 s not reached untl a later tme (t = 8 ns) because pont 2 s further from the break, as seen n Fg. 13. For case 2, t s assumed that fbre 2 breaks when t reaches ts maxmum stress at t = 44 ns. When fbre 2 breaks, the stress at pont 1 drops to approxmately zero, as seen n Fg. 12. The stress s then transferred dynamcally to fbre 3. As seen n Fg. 13, the stress n fbre 3 ncreases dramatcally after fbre 2 breaks and reaches a maxmum at t = 88 ns. Numercal results are gven n Table IV. The dynamc effects assocated wth breakage of fbre 1 ncrease the stress at fbre 2 by a factor of over the correspondng statc value. The dynamc effects assocated wth breakage of fbre 2 ncrease the stress at fbre 3 by a factor of over the correspondng statc value. The axal stress along fbres 2 and 3 at ther maxmum dynamc stress state (t = 44 and 88 ns, respectvely), the correspondng statc values, and the statc values scaled by the DAF are shown n Fgs 14 and 15, respectvely. From these plots, t s seen that the spatal dstrbuton of the dynamc stress dffers sgnfcantly from the correspondng statc dstrbuton :t-~... --m... "- c3... 2: ~ ~ , 1, r r J I r x dstance along fbre 3 (pro) Fgure 15 Multple fbre composte: axal stress along fbre 3 for case 2 (fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1). (---) Statc 2; ([~) DAF* statc; (--) dynamc, t = 88 ns. In a prevous analyss on ths same mcrocomposte system, the statc stresses and strans on matrx and fbres adjacent to a sngle flament break were smulated and then used to calculate the probablty of fracture of the adjacent fbre as a functon of the dstance along ts axs [4]. Because the extent of nterphase and/or matrx crackng at the ntal fbre break nfluences the magntude of stresses on adjacent fbres, the smulatons were carred out as a functon of the extent of precrackng at the orgnal fbre break. To calculate the probablty of fracture of the adjacent fbre, t was necessary to defne a fbre strength dstrbuton as a functon of fbre length. Fbre strength dstrbutons for varous E-glass fbres used n the authors' laboratory have been reported n other publcatons 1-5-8]. A representatve Webull functon for the length of E-glass fbres used n the LFEM 4185

6 analyses s G(cy) = 1- expl- L(;~I (1) where G(~) s the cumulatve probablty of falure, L = 1.21 mm, ~ = 2.78 and 13 = 1989 MPamm TM. In the smulaton of fracture of a sngle flament, the fbre s broken at an axal stress of %. The adjacent fbres havng been loaded to the same stress, but not broken, have been mechancally proof tested to c~p. Because t s clear that they have zero probablty of falng below ths level of stress, ther statstcal strength propertes wll be descrbed by a truncated cumulatve dstrbuton functon. Assumng that the survvng fbres were unaffected by the proof test, one can show that the approprate form of a truncated Webull dstrbuton functon, Gp(~), s related to the orgnal dstrbuton, G(c~), by the followng expresson [93 G((s) - G(~p) Gp((~) ~--- ~(-~p) (c~ > ~p) (2) G~(o) = (~ < C~l~) (3) where cyp s the proof test stress. By calculatng the stress concentratons on adjacent fbres caused by a fbre break, one can then use the truncated dstrbuton to calculate the probablty of falure at a pont on an adjacent fbre. Axal stress concentraton factors along the near sde of fbres 2 and 3 at ther maxmum dynamc stress states (t = 44 and 88 ns, respectvely), along wth the correspondng statc values, are shown n Fgs 16 and 17, respectvely. The results are scaled to a stran of 2.1% at whch pont the central flament s fractured at approxmately ts average strength of 16 MPa (.e. G(cy) =.481). The choce of boundary condton upon release of the fbre mesh nodes mposes an nterfacal crack equvalent to one mesh unt at the broken fbre end. The correspondng probabltes of falure as a functon of the dstance from the broken fbre end are shown n Fgs 18 and 19. From Fg. 18 one can see that at constant stran the probablty of fracture of the nearest adjacent fbre at pont 1 ncreases to Gp (216 MPa) =.53 based on the maxmum level of statc stress and Gp (2391 MPa) =.74 based on the maxmum level of dynamc stress. From Fg. 19 one can see that f the nearest adjacent fbre fractures at the peak of ts dynamc wave (.e. at 44 ns), the probablty of havng a thrd fbre break at 88 ns, n the same plane as the other two, has ncreased to G v (2411 MPa) =.75 based on the maxmum level of 1.7 P ~ 1.4 ID ~, 1.3 o xdstance along fbre 3 (gm) Fgure17 Multple-fbre composte: axal stress concentraton factor along fbre 3 for case 2 (fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1). (-[Z-) Statc stress/16 MPa; (-I-) dynamc stress at 88 ns/16 MPa. = t~ r~.8,7 ].6.5 I rl, f,... f x dstance along fbre 2 (gm) Fgure 18 Multple-fbre composte: probablty of fbre falure along fbre 2 for case 1 (only fbre 1 s broken). (-~-) Statc, (-I-) dynamc at 44 ns E.8 = ~_.7.~ ; #.6 Q o 1.2 t- O 1.1 o J... L... [......! l 1 15 l 2, 25 l l 3 35 l l l l r l r l, r l l l l l l l, l l l xdstance along fbre 2 (~m) Fgure16 Multple-fbre composte: axal stress concentraton factor along fbre 2 for case 1 (only fbre 1 s broken). (-Z]-) Statc stress/16 MPa; (-I) dynamc stress at 44 ns/16 MPa. 1 _~ o.5 2 Q xdstance along fbre 3 (grn) Fgure 19 Multple fbre composte: probablty of fbre falure along fbre 3 for case 2 (fbre 2 s also broken after reachng ts maxmum dynamc stress values caused by the breakng of fbre 1). ([Z-) Statc, (-I-) dynamc at 88 ns. 4186

7 statc stress and to Gp (269 MPa) =.88 based on the maxmum level of dynamc stress. Because the dynamc processes are occurrng at longtudnal wave speeds, t s expected that both the glassy polymer matrces and the glass fbres wll experence lnear elastc behavour durng ths 1 ns perod. Furthermore, subsequent falures of adjacent fbres should contnue wth an ncreasng probablty untl varous dampng mechansms put an end to the damage. Expermentally, t s observed that n the mcrocompostes the subsequent falures occur almost mmedately, and, when the fbre spacng s less than three to four fbre dameters, n a coordnated planar pattern. Thus, the computer smulaton appears to descrbe accurately the expermental results. 4. Concluson Dynamc effects assocated wth fbre breakage n sngle- and multple-fbre mcrocompostes have been nvestgated usng the lnear elastc fnte element method. The dynamc stresses occurrng n both cases are shown to be sgnfcantly dfferent from the calculated statc stresses and, consequently, have an mportant effect on the predcton of subsequent fbre falure n the composte. For sngle-fbre compostes, the dynamc stresses generated by a sngle flament fracture ncrease rapdly to a maxmum and then decrease and oscllate around the calculated values of the statc stresses. The amplfcaton of all four components of the stress s approxmately equal at a specfc pont along the fbre/matrx nterface and thus the dynamc stress state can be characterzed by a sngle dynamc amplfcaton factor (DAF). For the snglefbre glass/epoxy mcrocompostes nvestgated, the DAF ranges from approxmately and s ndependent of the poston along the fbre/matrx nterface. The tme to reach maxmum dynamc stress adjacent to the crack tp ranges from approxmately 15-5ns. Both quanttes decrease monotoncally wth the rato of the longtudnal wave speeds n the fbre and matrx, respectvely, whch appears to be the prmary varable characterzng the dynamc process. For the multple fbre compostes wth 55.6% by volume of fbres, smlar values for the DAF are obtaned. Although a sngle value of the DAF can be used to descrbe the spatal dstrbuton of the four components of the dynamc stress at a specfc pont n an adjacent fbre, ts value vares wth dstance along the fbre axs. Thus, the spatal dstrbuton of the dynamc stresses dffers sgnfcantly from the correspondng statc dstrbuton. Usng a truncated Webull functon to relate the probablty of falure of adjacent fbres to the maxmum dynamc stress experenced because of a fbre break, one can show that the dynamc processes can substantally ncrease the probablty of falure and can potentally lead to a hgher probablty of a cascadng effect that wll result n a co-ordnated pattern of fbre fracture n the mcrocomposte. The results of these smulatons are consstent wth expermental observatons reported n the lterature. References 1. K.D. JONESandA. T. DIBENEDETTO, Compos. Sc. Technol. gl (1994) P.W.J. VAN DEN HEUVEL, Y. J. W. VAN DER BRUG- GER and T. PEIJS, Adv. Compos. Lett. 3 (6) (1994) A. PEGORETTI, M. L. ACCORSI and A. T. DIBENEDETTO, J. Mater. Se. 31 (1996) K.D. JONES and A. T. DIBENEDETTO, "IPCM 95", Endhoven, September 1995, J. Compos., n press. 5. A. DIANSELMO, M. L. ACCORSI and A. T. DIBENEDETTO, Compos. Se. Teehnol. 44 (1992) K.D. JONES and A. T. DIBENEDETTO n "Proceedngs of the 3rd Internatonal Conference on Deformaton and Fracture of Compostes", March 1995, pp W.A. FRASER, F. H. ANCKER, A. T. DIBENEDETTO and B. ELBERLI, Polym. Compos. 4 (1983) A.T. DIBENEDETTO and P. J. LEX, Polym. Eng. Sc. 29(8) (1989) W. WEIBULL, J. AppI. Mech. (ASME) 18 (1951) 293. Receved 3 January and accepted 18 March