A STOCHASTIC INTERFACE MODEL FOR THE FRACTURE OF BARS

Transcription

1 A STOCHASTIC ITERFACE MODEL FOR THE FRACTURE OF BARS Sándor agy and Feren Kun Department of Theoretal Physs Unversty of Debreen, H-41 Debreen P. O. Box:5, Hungary ABSTRACT We study the breakng of an elast bar under quasstat and dynam loadng ondtons by means of a stohast nterfae model. The bar s omposed of two rgd bloks glued together by a thn elast nterfae whh an break when the bar s bent. The progressve damage and frature of the nterfae s analyzed subjetng the spemen to three pont bendng performed quasstatally and by mpat loadng. We show that ahead the nterfae rak a proess zone s formed where the loal falure mehansm s perolaton n a gradent feld. The rak propagates n bursts whose sze dstrbuton shows a unversal power law behavor. The largest burst has a power law dependene on the strength of dsorder. The proess zone proved to shrnk wth nreasng deformaton makng the rak tp sharper as the rak advanes. Analyzng the mpat loadng of the spemen, we demonstrate that the perforaton of the spemen ours analogously to ontnuous phase transtons when nreasng the energy of mpat. We determne the rtal exponents of the transton and reveal the onsequenes for the ballst loadng of solds. Introduton The frature of heterogeneous materals s an mportant sentf and tehnologal problem whh has attrated an ntensve researh durng the past years [1,2]. The nteraton of the heterogeneous stress feld wth the dsordered mrostruture of the materal makes the frature proess dffult to handle theoretally, hene, most of the theoretal approahes rely on large sale omputer smulatons of dsrete models [1-11]. Under quasstat loadng the damage of dsordered materals proeeds by mrorak nuleaton, rak growth and arrest gvng rse to a nosy response of the materal before atastroph falure [3]. Ths raklng nose an be reorded n the form of aoust sgnals whose analyss provdes mportant nformaton about the mrosop damage proess [3,13]. One of the most mportant theoretal approahes to the falure of heterogeneous materals are the fber bundle models (FBM) n the framework of whh the materal s dsretsed n terms of elast fbers of fnte strength, where dsorder s aptured by the dstrbuton of of strength values. In spte of ther smpltes, FBM-s apture the most mportant ngredents of materal falure and also served as the startng pont of the development of more omplex mromehanal model [3,4,5,6,7,11]. In ths paper we study the dynam frature of a sold bar by means of a stohast nterfae model. In the usual Charpy mpat test, the nothed spemen s lamped at the ends and the hammer hts t n the mddle resultng n a dynam three pont bendng [14]. Under suh boundary and loadng ondtons, the damage loalzes to a relatvely thn layer of the spemen gvng rse to a sngle growng rak startng from the noth. Dependng on the energy of the ht, the rak an ether termnate and the spemen suffers only partal falure, or t runs to the opposte boundary breakng the spemen nto two pees. For brttle spemens the onset of a rak propagaton ours n a atastroph manner preeded by a relatvely small deformaton [14]. In dutle spemens however stable rak propagaton ours whh s senstve to the detals of the loadng ondtons. Experments and theoretal alulatons showed that the stable rak propagaton s not a smooth proess, t s aompaned by flutuatons whh an be reorded by varous expermental tehnques suh as aoust emsson. In ferromagnet materals lke steel the dynam propagaton of raks gves rse to sudden hanges of the magnet feld whh an generate eletr urrent n a ol [15]. Ths so-alled magnet nose provdes addtonal nformaton on the dynams of rak propagaton ompared to the usual aoust emsson measurements [15]. Analysng the struture of magnet emsson (ME) spetra, reently, t was found expermentally that the rakng of a bar under three pont bendng proeeds n bursts whh are haraterzed by power law dstrbutons [15]. The experments showed that the exponents of the power laws are senstve to the type of frature,.e. dutle materals are haraterzed by hgher exponents than the brttle ones [15]. Motvated by the expermental fndngs, we study the damage proess of a bar

2 under three pont bendng fousng on the bursts of mrosop breakng events, on the spatal dstrbuton of damage and on the ourene of full perforaton of the spemen. In order to smplfy the alulatons, we model the bar as two rgd bloks oupled together by an elast nterfae. Deformaton and damage of the bar s onentrated n the nterfae, whle the two bloks reman ntat. We determne how the strength of dsorder and the nhomogeneous stress feld determne the damage proess of the nterfae. The model We onstrut a smple model to study the loadng of an elast bar of retangular shape by an external fore exerted perpendular to the longer sde of the bar n the mddle. For smplty we assume that the deformaton and damage of the bar s onentrated n a relatvely thn layer n the mddle. Hene, n the model the bar s omposed of two rgd bloks of sde lengths a and b whh are glued together by an elast nterfae of wdth l, where l <<b holds, see Fg. 1. Fg 1: The geometr layout of the model. The two rgd bloks of sde length a and b are glued together by an nterfae of wdth l whh s dsretzed n terms of elast fbers. The spemen suffers defleton δ under the aton of the external fore F exerted n the mddle of the bar. The nterfae regon an deform and suffer breakng under defleton of the spemen whle the two rgd bloks reman ntat. Bendng of the spemen s performed suh that the two bloks undergo rgd rotaton about ther outer upper orner onentratng the deformaton n the nterfae layer. We dsretse the nterfae regon n terms of elast fbers of number and length l whh are plaed equdstantly between the two bloks. The fbers do not have bendng rgdty, they an undergo only strethng deformaton haraterzed by the same value of the Young modulus E. Durng the bendng of the spemen, the fbers an support only a fnte deformaton,.e. f the loal deformaton of fber exeeds a threshold value the fber breaks and a mrorak nuleates n the nterfae. The dsordered propertes of the materal are represented by the randomness of the breakng thresholds whh are ndependent dentally dstrbuted random varables wth a probablty densty p( ) and umulatve dstrbuton ( ) = p( x) dx. The rgdty of the two rotatng bloks mples P that the marosop deformaton of the spemen an be haraterzed by a sngle varable δ whh denotes the defleton of the mddle of the bar from the orgnal poston, see Fg. 1. It an be seen n Fg. 1 that under bendng of the spemen the nterfae opens resultng n an nreasng elongaton of fbers from top to bottom. The atual length of fber l an be expressed as a funton of δ as a 1 l = l + 2δ, (1) b 1 where l1 = l + 2( b b δ ) s the length of the fber ndex =1 at the top of the bar. It follows that also the elongaton l and the longtudnal stran of fbers nrease lnearly as a funton of ther poston a 1 l l = 2b 2 b δ + 2δ, =. (2) b 1 l

3 The torque of the external fore F atng n the mddle of the bar wth respet to the lampng ponts s equlbrated by the torque of the fore exerted by the elongated fbers of the nterfae. Durng the deformaton proess, those fbers whh exeed ther threshold value break,.e. they are removed from the nterfae. Sne 1 P( ( δ )) s the probablty that the nterfae element of ndex remaned ntat under the externally mposed deformaton δ, based on the equlbrum ondton of torques σ δ of the defleted bar an be ast n the form the onsttutve equaton ( ) σ 1 = L = 1 a 1 b 1 ( δ ) δ + b δ [ 1 P( ( δ ))] E ( δ ) where L=2b+l s the overall length of the bar and the sum goes over all the fbers. On the rght hand sde (δ ) should be substtuted from Eq. (2). The amount of dsorder of the falure thresholds has a substantal effet on the marosop response of the spemen σ ( δ ) (3) =, the falure of. In the lmtng ase of zero dsorder,.e. when all the fbers have the same breakng threshold the nterfae starts at the bottom where the strethng deformaton s the hghest and ontnues upwards as δ s nreased. It an bee seen n Fg 2 that the orrespondng onsttutve urve s sharply peaked. The rtal deformaton δ defned by the peak poston orresponds to the nstant of the frst fber break = ( δ ). Beyond the peak the stress σ rapdly dereases due to the gradual breakng of the nterfae elements as δ nreases. For the atual alulatons the parameter values l =1, a=1, b=2.5 so that the onsttutve urve n Fg. 1 ends at the deformaton δ 5 when the last fber =. breaks. Due to the lnearly elast behavor of fbers of the nterfae before the peak where no fbers break ( δ ) σ should also have a lnear shape. Slght devaton from the deal lnear form arses due to the mssng bendng rgdty of fbers. In order to study how the onsttutve behavor hanges when the amount of the dsorder s vared, we onsder a unform dstrbuton for the breakng thresholds over an nterval of wdth 2 entered at the value ( ) ( ), P = < < + 2 The strength of dsorder of the breakng thresholds s haraterzed by, whle wdth an be vared over the nterval [, ] where the lmts = and strongest dsorder, respetvely. (4) only sets the sale of thresholds. The = orresponds to zero dsorder and the Fg 2: The onsttutve urve σ (δ ) of the nterfae omposed of =1 fbers for several dfferent dstrbutons of the breakng thresholds. When all the fbers have the same frature threshold ( = ), σ (δ ) has a sharp peak whh gets rounded and develops nto a quadrat mum when s nreased.

4 Fg. 2 shows that nreasng the strength of dsorder of the nterfae, the peak of the onsttutve urve gets more and more rounded and develops nto a quadrat mum. The mum value of σ (δ ) s preeded by a longer and longer nonlnear regme due to the breakng of fbers, so that for δ the lnear behavor prevals only for small deformatons. On the mrolevel ths proess s aompaned by the randomzaton of the breakng sequene of fbers along the nterfae,.e fbers do not smply break n the dereasng order of ther ndex. Bursts of fber breaks The omplete onsttutve urve of the system presented n Fg. 2 an only be reovered by deformaton ontrolled loadng. When δ s ontrolled externally, the loal load on the nterfae elements s solely determned by the externally mposed deformaton, whh mples that there s no load redstrbuton after the breakng of fbers. Under stress ontrolled loadng of the bar, after a fber breakng the load of the broken fber has to be redstrbuted over the ntat ones. Due to the wedge shape of the deformed nterfae, at a gven external load σ the load on the fbers lnearly nreases from top to bottom. It has the onsequene that n spte of the rgdty of the two sold bloks, the load redstrbuton followng fber falure dffers from the usual equal load sharng (ELS) approxmaton ommonly used for the study of parallel bundles of fbers. The rgd surfaes, however, ensure that the load s redstrbuted globally n suh a way that the exess load reeved by an ntat fber depends on the fbers' poston along the nterfae but not on ts dstane from the faled one. Ths mples that no stress enhanement arses n the vnty of the faled fbers as n the ase of loal load sharng approxmaton (LLS), suh that t an be expeted that the system stll remans n the mean feld unversalty lass. In order to analyze the mrosop damage mehansm of the nterfae, we worked out an effent smulaton tehnque for a spemen where the nterfae s omposed of fbers wth breakng thresholds, = 1 sampled from the probablty,..., dstrbuton Eq. (4) Wth the breakng thresholds, we an determne the value of the marosop deformaton parameter δ = δ (, ), = 1,..., at whh the fbers break. Of ourse, δ s a funton of both the poston of the fber along the nterfae and the loal breakng threshold. Durng the loadng proess the fbers break n the nreasng order of ther rtal marosop deformaton δ, whh an be a randomzed sequene of the fbers' poston. The omputer smulaton of the loadng proess proeeds as follows: after generatng the breakng thresholds of fbers we alulate the orrespondng rtal defleton δ and sort them nto nreasng order. The onsttutve urve of the sample an be smply obtaned by alulatng the load needed to aheve the deformaton δ after the breakng of the frst -1 fber remanng only only nt =-(- 1) ntat elements. Between the breakng of the (-1)th and the th fbers, the onsttutve equaton of the system takes the form 2 E 2 4a 1 ' 2δa ' 2 = 2δ [ b ] nt + δ 2ab 2a ( j 1) + ( 1), j ll b 1 1 ( 1) 1 j= b j= σ (5) wth = b δ and the prme ndates that the summaton s restrted to ndes of ntat fbers, whh are not neessarly onseutve ntegers. ote that n Eq. (5) the value of δ falls n the rangeδ 1 < δ < δ. Performng stress ontrolled experments, after the breakng of a fber the deformaton of the spemen an freely hange resultng n a redstrbuton of load over the ntat fbers. The exess load taken up by the ntat fber an gve rse to further fber falures whh may trgger an entre avalanhe of falure events. Ths avalanhe ether stops and the bar beomes stable under the externally mposed load, or t spans the entre nterfae and the spemen breaks nto two pees. In order to study numerally ths mrosop breakng proess, n the smulatons frst we nrease the deformaton δ suh that a sngle fber breaks,.e. δ = δ 1 s taken whh orresponds to the fber of ndex 1. Then the load needed to mantan ths deformaton σ s alulated from Eq. (5) for nt = fbers. After the breakng of fber 1 ts load has to be redstrbuted over the remanng -1 fbers. In order to determne the load of ntat fbers after the removal of the broken one, we remove fber 1 on the rght hand sde of Eq. (5) and nvert the equaton for (σ ) δ keepng the load σ fxed. The fbers wth threshold values δ < δ (σ ) break as a onsequene of load redstrbuton. Ths teraton has to be repeated under a fxed external load σ untl the breakng sequene stops or all the fbers break resultng n a marosop falure of the system. In order to haraterze the damage proess of the loaded bar under stress ontrolled ondtons, we determned the dstrbuton of burst szes s of fber breaks for the dsorder dstrbuton Eq. (4) varyng the wdth of the dstrbuton.

5 Fg 3: Dstrbuton of burst szes D(s) at several values of. A straght lne of slope 2.5 s drawn as a referene. The nset presents the average value of the largest bursts s as a funton of. The avalanhe sze dstrbutons D(s) are presented n Fg. 3, where the normalzaton was arred out suh that the ntegral of the urves s equal to the total number of avalanhes ourred. It an be observed n Fg. 3 that D(s) shows a power law form τ D ( s) ~ s (6) at any. Smulatons revealed that the exponentτ s unversal,.e. the value of τ s nsenstve to the strength of dsorder, t ondes wth the exponent of the lassal parallel bundle of fbers wth equal load sharng τ 2. 5 [3,5]. However, t an be observed n Fg. (3) that the sze of the largest burst s have a strong dependene on the value of. To obtan a quanttatve haraterzaton of ths effet, the nset of Fg. 3 shows separately the average sze of the largest bursts s as a funton of, where a power law dependene s evdened α s ~. (7) For the value of the exponent α =.7 ±. 2 was obtaned over the range.5 < / <. 6. For >. 6 the sze of the largest avalanhe levels off. Spatal evoluton of damage Besdes the temporal orrelaton of mrosop falure events, also a very nterestng spatal orrelaton emerges. In a bundle of fbers loaded between two parallel rgd plates due to the equal load sharng after fber breakng, the breakng of a fber s solely determned by ts breakng threshold. Hene, fbers break n a ompletely random sequene wthout any spatal orrelaton. In our system, however, durng the loadng proess the fbers break n the nreasng order of ther rtal marosop deformaton δ (, ) whh depends both on the loal breakng thresholds and on the spatal poston of = the rtal deformaton δ ( ) fbers. In the lmtng ase of zero dsorder,.e. = and, s a monotonally dereasng funton of, whh mples that the fbers break one-by-one startng from the bottom. Ths breakng sequene generates a rak penetratng the nterfae upward suh that below the rak tp all the fbers are broken whle above t the fbers are ntat but deformed. Under stran ontrolled loadng stable rak propagaton s obtaned gradually breakng the fbers by the external stran nrements. Controllng the external load, however, the onset of rak propagaton ours n an unstable manner resultng n mmedate atastroph falure as t s expeted for ompletely brttle materals. Inreasng the strength of dsorder, the breakng sequene of fbers determned by δ beomes spatally randomzed. At a gven deformaton δ the fbers wth δ > δ have already faled. If the dsorder s not too strong, an nterestng spatal dstrbuton of broken fbers emerges: startng from the bottom of the nterfae a ontnuous regon of faled fbers develops formng a rak. On the opposte sde, startng from the top of the nterfae a ontnuous regon of ntat, elongated fbers an be observed. The two regmes are separated by a proess zone, whh s a sparse sequene of ntat and broken elements.

6 It an be observed that the proess zone sharpens,.e. ts wdth dereases as the deformaton δ nreases. It an be obtaned analytally that at a gven defleton δ the wdth w of the proess zone also depends both on the strength of dsorder and on the geometral extensons a,b of the spemen ( δ ) 2b + b δ b w = 1, 2δ a (8) b w = ( 1), δ a (9) w + 2b + b δ 2δ b, a = (1) for δ values where > +. Due to the nterplay of the threshold dsorder and the global load redstrbuton n the form of the lnearly nreasng load along the nterfae, the mehansm of the progressve damage n the proess zone s perolaton n a gradent feld [12]. A smlar mehansm was obtaned n [6]. It also follows from Eq. (1) that from the vewpont of the damage proess, the strength of dsorder of the breakng thresholds s determned by the wth of the dsorder dstrbuton together wth the geometral setup of the spemen Energets of the loadng proess The dynam frature proess an also be studed by means of our stohast nterfae model. A ommonly used method to ntate hgh speed rak propagaton n a spemen s the dynamally ndued three pont bendng test. In the wdely used Charpy mpat test method the load s exerted by a heavy pendulum whh s rased to an ntal heght and released, strkng and breakng the spemen at the bottom of ts swng [14]. The two ends of the spemen are lamped and the pendulum hts t n the mddle, whh results n a damage loalzed to a thn layer gvng rse to a sngle growng rak. Experments showed that dependng on the energy E of the ht on the spemen, the outome of the mpat proess an be lassfed nto two states: at low values of the mpat energy the propagatng rak termnates and the spemen suffers only a partal falure. However, exeedng a rtal energy value E the rak passes through the entre system resultng n a global falure breakng the spemen nto two pees [14]. The results an be nterpreted suh that the rtal energy separates two phases of the system, n the damaged phase below E the spemen keeps ts ntegrty, whle above E the spemen breaks nto two pees, whh defnes the fratured phase. At the rtal energy a transton ours between the two phases. Based on our model, the energets of the mpat proess an smply be studed. In order to smplfy the mpat proess, we assume that the mpatng body httng the spemen n the mddle wth an ntal knet energy E stays together wth the spemen for the entre duraton of the loadng proess. Furthermore, we neglet the energy stored by the rgd moton of the spemen so that the knet energy of the mpator E k s transformed nto the elast energy E el of the elongated fbers and s dsspated by the fber breakngs E ds. Hene, the energy balane of the system reads as E = E ( δ ( t) ) + E ( δ ( t) ) + E ( δ ( t) ) k el ds (11) whh means that at any tme t of the mpat proess the sum of the knet energy of the mpator E k, of the elast energy stored n the elongated fbers E el, and of the energy E ds dsspated by the fbers faled untl tme t must be equal to the ntal nput energy E. Durng the defleton proess, the energy gets stored by the elongated fbers whh s then dsspated by breakng resultng n a propagatng rak. If the mpat energy E s not too hgh, the proess stops at a fnte defleton at whh the mpator has no knet energy E k=. In ths ase the entre mpat energy s stored n deformaton of the ntat fbers and s partly dsspated by fber breakng E = E el ( δ ) + Eds ( δ ). In the framework of our smple model, the rtal mpat energy E an be defned as the mpat energy entrely dsspated by fber breakng at the stoppng of the hammer,.e. E ds ( δ ) = E and E el ( δ ) =. In order to haraterze the fnal state of the mpat proess at dfferent mpat energes, we defne the order parameter nt / of the system as the fraton of ntat fbers nt (E ) survved the mpat and the total number of fbers of the nterfae. The order parameter takes non-zero values n the damaged phase below the rtal energy E <E, and monotonally dereases when E E, whle t s zero n the fratured phase for any energes above the rtal pont E >E. To reveal the nature of the transton from the damaged to the fratured state when nreasng the mpat energy, we alulated the order parameter as a funton of the mpat energy for several dfferent strength of dsorder of the breakng thresholds, whh s plotted n Fg 4. For the lmtng ase of zero dsorder = t an be shown analytally that the order parameter has a power law dependene on the dstane from the rtal pont δ

7 ( E ) β nt / ~ E, (12) where the exponent β has the value β = 1/ 2. Computer smulatons performed at varous dfferent values of showed that the funtonal form Eq. (14) of the order parameter s also vald for dsordered nterfaes, however, the value of the exponent β s not unversal, t proved to depend on the strength of dsorder. The hange of the exponent β s apparent wth nreasng to the largest meanngful value of β approahes 2/3 when s nreased. n the nset of Fg. 4. Startng from 1/2 the numerally determned value Fg 4: The order parameter nt / of the system as a funton of the mpat energy E for several dfferent values of the strength of dsorder. Furthermore, we determned numerally the suseptblty of the system to the hange of the ontrol parameter as the dervatve of the bendng δ wth respet to the mpat energy E. The suseptblty also shows a power law behavour. The exponent γ haraterzng ts salng behavour depends only on the strength of dsorder. Smulatons revealed that the exponents γ and β are equal. Summary In ths paper we nvestgated the breakng of an elast bar under quasstat and dynam loadng usng a stohast nterfae model. We fous on statstal physal aspets of the breakng proess and desrbe the breakng of the bar as a phase transton. We showed that the amount of dsorder of the falure thresholds has a substantal effet on the marosop response of the spemen. The frature of the bar was found to proeed n bursts whose dstrbuton proved to be a power law wth the exponent 2.5 ndependently of the strength of dsorder. Furthermore, we showed that ahead the rak tp a frature proess zone s formed where the loal falure mehansm s perolaton n a gradent feld. When the load s appled dynamally, alulatons revealed that there exsts a rtal value of the mpat energy below whh partal falure of the nterfae ours (damaged phase), whle above t the rak passes through the entre spemen resultng n global falure (fratured phase). Computer smulatons showed that the system approahes the rtal pont wth salng behavour haraterzed by rtal exponents. The results mply that the perforaton of the bar under nreasng loadng ours analogously to ontnous phase transstons. Aknowledgments Ths work was supported by OTKA M41537, T4929 and KFP-3A/43/4. F. Kun aknowledges support of the György Békés Foundaton of the Hungaran Aademy of Senes.

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