Numerical simulation of wooden beams fracture under impact

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IOP Conferene Seres: Maerals Sene and ngneerng OPN ACCSS Numeral smulaon of wooden beams fraure under mpa To e hs arle: P Radheno e al 5 IOP Conf. Ser.: Maer. S. ng. 7 39 Relaed onen - Insulaon Teser vershed and Vgnoles Ld.

1 IOP Conferene Seres: Maerals Sene and ngneerng OPN ACCSS Numeral smulaon of wooden beams fraure under mpa To e hs arle: P Radheno e al 5 IOP Conf. Ser.: Maer. S. ng Relaed onen - Insulaon Teser vershed and Vgnoles Ld. - A proposal for dynam albraon of brae eser Paulo L S Ferrera, Paulo R G Couo, Luz C Cabral e al. - A porable wheel eser for yre-road fron and rollng ressane deermnaon J Pya, P Budzys, P Tarows e al. Vew he arle onlne for updaes and enhanemens. Ths onen was downloaded from IP address on /7/8 a 8:43

2 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 Numeral smulaon of wooden beams fraure under mpa P Radheno, S Bauev and A Radheno 3 Toms Sae Unversy of Arheure and Buldng, Deparmen of Appled Mahemas, Toms, 6343, Russa Toms Sae Unversy of Arheure and Buldng, Deparmen of Geonformas and Cadasre,Toms, 6343, Russa 3 Toms Sae Unversy of Arheure and Buldng, Insue of Cadasre, onoms and ngneerng Sysems n Consruon, Toms, 6343, Russa -mal: andrey-radheno@lve.ru Absra. In hs paper he fraure of wooden (pne) beams under low-speed mpa of seel ram eser s numerally nvesgaed, onsdered speed s rangng beween -3 m/s. The behavor of wood s desrbed whn phenomenologal approah ang no aoun ansoropy of elas and srengh properes. The behavor of he seel ram eser s modelled by he elasoplas medum. Numeral modelng s arred ou by a fne elemens mehod n hree-dmensonal saemen. Fraure propagaon n wood s under sudy.. Inroduon Impa aon of sold bodes n a wde range of nema and geomer ondons s a omplex as of mehans. Dffules onneed wh heoreal sudes of he proess of maerals desruon and deformaon under mpa aon usng analyal approahes maes one o nrodue a number of smplfyng hypoheses ha anno always refle he real suaon. Therefore should be onsdered ha he leadng role n sudyng evens onneed wh hgh-speed neraon of sold bodes s urrenly gven o expermenal and numeral sudes. Researh n maerals desruon whle mpa aons demonsrae ha due o hange n mpa ondons he fraure mehansms are also hanged []. xpermens demonsrae ha n number of ases fnal desruon s defned by a ombnaon of several mehansms. However he expermens do no allow rang he sequene, me of mpa and onrbuon of he dfferen fraure mehansms. Furhermore, fraures obaned a he nal sages of proess anno always be denfed whle analyss of he fnal maerals desruon. Therefore numeral smulaon s relevan when sudyng mpa neraon [].. Major equaons of mahemaal model The sysem of equaons desrbng he me-dependen adaba moon of a ompressble medum n an arbrary oordnae sysem ( =,,3) nludes he followng equaons: - equaon of onnuy - equaon of moon dv () Conen from hs wor may be used under he erms of he Creave Commons Arbuon 3. lene. Any furher dsrbuon of hs wor mus manan arbuon o he auhor(s) and he le of he wor, journal aon and DOI. Publshed under lene by Ld

3 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 where a a F, () m m, m m (4) - equaon of energy d e, (5) where - densy of he meda - veloy veor a - omponens of aeleraon veor F - omponens of he nodal fores veor - Chrsoffel symbols (3) - onravaran omponens of symmer sress ensor - spef nernal energy e - omponens of he symmer sran veloy ensor: e j j. (6) 3. Consuve equaons of sorop meal maerals The behavor of sorop meal maerals s onsdered whn elasoplas model. The sress ensor s represened n he form of he sum of devaor S and spheral par P : P S, where - Kroneer symbol. Assumng ha for meda rue s he prnple of mnmum wor of rue sresses on nremens of plas deformaons, he onneon of ensor omponen of sran rae and sress devaor wll be reorded n he form of: DS Ge e S, 3 D. (7) Here DS ds j S j S D,, G - shear modulus. where j j Parameer under elas deformaon and under plas deformaon ( ) deermned by he help of a ondon of Mses: S S d d dynam yeld sress. Pressure n he maeral 3, was alulaed from he Me-Grünesen equaon as a funon of he spef nernal energy and densy : P 3 n where, K, K, K, K3 - maeral onsans. n V Kn V K, (8)

4 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 4. Consuve equaons of ansorop maerals To desrbe he behavor of ansorop (orhorop) body he followng ondons and resrons are aeped:. The body s sold (onnuous meda). The ross dmensons of sruural (renforng) elemens are small n omparson wh he body szes,.e. he medum s quas-homogeneous 3. Relaonshps beween nremens of sress ensor omponens and ensor omponens of sran rae s lnear. Therefore he relaons beween omponens of sress ensor and sran rae ensor before fraure o orhorop body an be gven as: d e d Ce Ce C3 33, C44e, (9) d e d3 Ce Ce C3 33, C66e3, () d 33 e d 3 C3e C3e C33 33, C55e3, () where, C elas onsans. las onsans Posson's raos : C an be expressed by means of shear modulus C C A 3 3A C3 C33 A 3 A C C 3A 3 A 3 C G C G C G3 G, oung's modulus: and æ A= -n n 3 n 3 - n 3 - n - ö 3 ç n 3 3 è 3 ø (3) Knowng nne szes of ehnal onsans n he man axes of symmery of orhorop maeral, one an deermne he value of any onsan n any dreon. 5. Fraure model of ansorop maeral under dynam loadngs Defnng he ranson of he barrer maeral o omplee fraure he Hoffman reron was used [3] ha aes no aoun he dfferen srengh sruural properes of he maeral n ompresson and ensle aons: where oeffens f ( ) C ( ) C ( ) C ( C C C C C C C are deermned from he followng equaons: 9 ) (), (4) 3

5 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 C [( ) ( ) ( ) ]/ C [( ) ( ) ( ) ]/ C [( ) ( ) ( ) ]/ 3 C ( ) C S 4 7 yz C ( ) C S 5 8 zx C ( ) C S. 6 9 xy (5) Here, - ulmae ensle and ulmae ompressve srengh along he -axs respevely,, - ulmae ensle and ulmae ompressve srengh along he axs,, - ulmae ensle and ulmae ompressve srengh along he axs, S, S, S - ulmae shear srengh along xy yz zx he orrespondng axes. I s supposed ha fraure of ansorop maerals n he ondons of nensve dynam loadngs ours as follows [4]:. If falure reron s broen under ompresson e, s onsdered ha maeral loses properes of ansoropy, and s behavor s desrbed by hydrodynam model, and he maeral reans only ompressve srengh sress ensor beomes n hs ase spheral P and s value s deermned from he relaon [5]: V V P exp 4b V 4b, (6) where - he nal densy, and b - he oeffens of he sho adaba D bu m, u m - mass veloy, V and V - nal and urren spef volumes.. If falure reron s broen under enson e, he maeral s onsdered ompleely desrued, and omponens of sress ensor are se equal o zero. Numeral smulaon s arred ou by fne elemen mehod [6]. 6. Resuls and dsusson The proess of neraon of seel ( =7 g/m 3, d =3 MPa) ram eser s onsdered, orrespondng 4g n wegh, speed range of -3 m/s ( =7 g/m 3, d =3 MPa) wh a barrer from glued wood (pne) hness of 6 mm. The densy of he barrer maeral =58 g/m 3, he elas and srengh properes have he followng values: MPa, MPa, =84 MPa, S yz =3 MPa, =46 MPa, S =3 MPa. zx =3 MPa, =46 MPa, x =7 MPa, =3 MPa, y =5 MPa, = 46 MPa, z =6 S =3 xy Fgures - show he alulaed onfguraon of he ram eser and he barrer and gradaon of grey olor show values of relave level of fraure V r /V n barrer. The values V r and V are gven o he nodes of he ompuaonal grd, V r s a volume of he elemens onneng n node n whh he 4

6 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 ondon of fraure (4) was sasfed, V he oal amoun of he elemens formng hs node. Value Vr /V = orresponds o omplee fraure of maeral n he node of he ompuaonal grd. Resulng from he mpa aon on he barrer annular area of fraure ours on a fron surfae along he permeer of he ram eser. Ths desruon area spreads o he ba surfae along he barrer hness. Fgure 3 demonsraes hanges n he me of speed of he mass ener of ram eser for varous nal speeds of neraon whh llusrae nensy of ram eser brang. Peneraon of a barrer sn' observed up o speed 3m/s =6 мкс, = мкс Fgure. Confguraon of he projele and arge, solnes of relave volume of fraure n barrer. =5 m/s =6 мкс, = мкс Fgure. Confguraon of ram eser and barrer, solnes of relave volume of fraure n barrer = m/s. 5

7 TSUAB4 IOP Conf. Seres: Maerals Sene and ngneerng 7 (5) 39 do:.88/ /7//39 Fgure 3. Change n me of veloy of he ener of mass of ram eser. for nal veloy m/s, 5 m/s, 3 m/s, 4 5 m/s. 7. Conluson The developed model desrbes behavor of brle-fraured orhorop maeral, ransferrng no model of brle-fraured ransrop or sorop maeral a equaly of properes n he dreons and, or n all dreons. I allows nvesgang he behavor of a wde lass of maerals wh varous properes usng he same model. Wood and sruural elemens obaned from an also be desrbed well by he suggesed model hus allowng s broad applaon n sudyng he behavor of smlar maerals under dynam loadng. The model does no resr he reron of srengh, s hoe s defned by appropraeness and gven srengh properes for parular maeral. Referenes [] Канель Г И, Разоренов С В, Уткин А В и Фортов B Е 996 Ударноволновые явления в конденсированных средах (Москва: Янус-К) 47 [] Фомин В М, Гулидов А И, Сапожников Г А и др. 999 Высокоскоростное взаимодействие тел (Новосибирск: Издательство СО РАН) 6 [3] Ву Э 978 Феноменологические критерии разрушения анизотропных сред Механика композиционных материалов 4 [4] Radheno A and Radheno P Numeral modelng of developmen of fraure n ansorop ompose maerals a low-veloy loadng J. Ma. S [5] Канель Г и Щербань В 98 Пластическая деформация и откольное разрушение железа "Армко" в ударной волне ФГВ 6 93 [6] Johnson G 977 J. Appl. Meh

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1)

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1) Pendulum Dynams Consder a smple pendulum wh a massless arm of lengh L and a pon mass, m, a he end of he arm. Assumng ha he fron n he sysem s proporonal o he negave of he angenal veloy, Newon s seond law