Numerical simulation of damage in glass subjected to static indentation

Transcription

1 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007 Numerial simulaion of damage in glass subjeed o sai indenaion Jewan Ismail, Fahmi Zaïri, Moussa Naï-Abdelaziz & Ziouni Azari Laboraoire de Méanique de Lille (UMR CNRS 807), USTL, Polyeh Lille, Avenue P. Langevin, Villeneuve d Asq Cedex, Frane Laboraoire de Fiabilié Méanique de Mez, ENIM, Ile de Sauly, Mez Cedex, Frane Absra: The presen paper is foused on he numerial simulaion of a glass plae subjeed o sai indenaion by a spherial indener. An axisymmeri finie elemen model of he problem was designed and validaed agains analyial soluions. A oninuum damage mehanis (CDM) based onsiuive model wih an anisoropi damage ensor was seleed and implemened ino a finie elemen ode. The numerial resuls were analysed from he disribuion of damage omponens. Key-words: Glass; Indenaion; Coninuum damage mehanis Inroduion Nowadays, glass is widely used in many engineering appliaions (ivil onsruions, vehiles and airrafs, eleroni appliaions). The mos appliaions of his maerial have he shape of panels wih imporan areas. Beause of he brileness of glass, he sudy of he ona problem wih exernal objes is of prime imporane. A pariular ase of he elasi ona s heory of Herz is ha onerning he ona beween a spherial obje and a fla surfae. When a riial load is ahieved, a sysem of raks is iniiaed a he maerial surfae and in is bulk. Signifian experimenal sudies were performed for deermining he sysem of raks in glasses as well as during he proess of indenaion ha during an impa. As shown in figure in he ase of a rigid indener, he main ypes of raks are one, half-penny, laeral, median and radial raks (Knigh e al., 977; Cook and Pharr, 990). FIG. Morphologies of differen raks in glass indued by indenaion: C) one, H) halfpenny, L) laeral, M) median and R) radial raks (Cook and Pharr, 990). The iniiaion and propagaion of eah rak vary aording o ype of glass (Arora e al., 979). In pariular, for soda-lime glass, he proess of fraure regarding he applied load sars firsly by he formaion of one hen median raks when he indener is sill in he sage of loading, hough radial and laeral raks develop laer in he sage of unloading. As shown in figure, radial and median raks propagae perpendiularly o he surfae; when hese wo ypes of raks inerse eah oher hey form half-penny raks. While laeral rak propagaes in a parallel manner o he surfae, and as soon as i inerses wih he surfae an amoun of maerial is removed. These raks are generaed by prinipal sresses, i.e. mode I rak opening. Anoher deformaion observed in glass during an indenaion is a permanen deformaion named abusively as plasi deformaion (Arora e al., 979). Tha ours due eiher o he signifian ompaion (densifiaion) of he siliae glass sruure or o loal shearing (figure ). The

2 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007 appariion of his deformaion (mode II or sliding mode) is relaed o shear and ompressive hydrosai sresses. FIG. Seion views for region beneah an indenaion of a) soda-lime glass wih deformaion by shear flow and b) silia glass wih deformaion by densifiaion (Arora e al., 979). Pioneering works foused on he experimenal desripion of he damage mehanisms in glass. However, he numerial modelling of he damage behaviour of glass during an indenaion or an impa is less repored. This modelling is neverheless essenial for a beer undersanding of damage mehanisms and he design proess of glass omponens. The hermodynamial framework of Coninuum Damage Mehanis (CDM) is used in his sudy. CDM onep was firs inrodued by Kahanov (958) and generalized laer by Lemaire and Chabohe (985). This onep was suessfully applied o differen lasses of maerials suh as meals, onree and polymers. Reenly, i was applied o glasses by Sun and Khaleel (004). In his sudy, he damage following an indenaion on a glass plae by a spherial indener was analysed by finie elemen (FE) simulaions. For his aim, a CDM based onsiuive model was used and implemened ino a FE ode o desribe he damage disribuion in he plae. The paper is organized as follows. In seion, he FE model is deailed. Seion is devoed o he desripion of he CDM model. Seion 4 presens he numerial resuls obained from FE simulaions. Finally, onluding remarks are given in seion 5. FE modelling of indenaion FE modelling was used o simulae he indenaion proess by a rigid sphere in normal ona wih a fla plae of glass. The ommerial FE ode MSC.Mar was used o arry ou he simulaions. FIG. FE model used for he numerial simulaions of indenaion: omplee mesh and region lose o ona. Due o he axisymmeri haraer of spherial shaped indenaion proess, he problem was analysed in a wo dimensional axisymmeri ross-seional model (figure ). The plae was meshed wih a oal of 90l axisymmeri four-node isoparameri elemens, while he indener was onsidered as infiniely rigid. The blok of glass was modelled as semi-infinie spae in order o he numerial resoluion approahes he analyial soluions of he elasi ona s problem of Herz. No friion beween he indener and he plae was onsidered. The mehanial properies of he plae used in his sudy are hose of a soda-lime glass. The Young s modulus is aken equal o 7000MPa and he Poisson s raio o 0.5. Figure 4 illusraes a good

3 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007 agreemen beween he analyial soluions of Huber (904) and numerial resuls for he prinipal sresses. [MPa] σ σ σ z [mm] FE analyial FE analyial FE analyial FIG. 4 Comparison beween FE and analyial (Huber, 904) soluions of he prinipal sresses for a spherial indener of mm diameer and an applied load of 500N. Boh prinipal sress σ a in he plan of symmery, while σ akes aion in he direion of θ (figure ). σ is ompressive everywhere wihin he plae, whereas σ are ompressive beneah he ona zone bu hey beome ensile far from his zone. Tha is illusraed in he onour plos of hese sresses shown in figure 5 (Noe ha whie zones in hese onour plos are in ompression). () FIG. 5 Sresses iso-values obained by FE simulaion in he axisymmeri ross-seional under an indenaion load of 500N by a sphere of mm diameer: a) σ, b) σ, ) σ. CDM model The effe of damage on he deformaion proess is aken ino aoun by inroduing a damage variable ino he onsiuive equaion of he glass maerial. The damage variable has values beween 0.0 (virgin sae) and.0 (fully damaged sae or raking sae). This variable is alulaed from a linear evoluion law and is inrodued ino an anisoropi damage marix D whih models he maerial nonlineariy, arising from he deformaion proess. This anisoropi marix is added ino he onsiuive equaion of he virgin maerial whih is onsidered isoropi. The onsiuive equaion of he glass maerial is defined by (Sun and Khaleel, 004): e d { K K }.{ ε } σ = + () kl kl kl e d where K kl denoes he siffness marix for he elasi maerial and K kl represens he added damage influene. The full expressions of hese ensors omponens are given by: K ( ) ( δ δ ) ( δ δ ) e kl = λδδ kl + µ δikδ jl + δilδkj d kl = kl + kl + jk il + il j k K C D D C D D where δ is he Kroneker-dela symbol, λ and µ are he Lame s onsans for he glass, C and C are he damage parameers. ()

4 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007. Damage omponen due o normal prinipal sresses (mode I) This omponen is due o normal prinipal sresses. Is value is deermined aording o a simple linear damage evoluion law so ha, diagonal omponens of damage are linearly relaed o he orresponding ensile prinipal sress omponens by: 0 σ σ D σ σ = σ < σ < σ i ii i σ σ i i σ σ where σ are he riial and hreshold sresses. σ orresponds o he sress below whih no damage is visible orresponds o he sress above whih he maerial is fully damaged.. Damage omponen due o shear sresses (mode II) As menioned above, wihin a zone undergoing a ombinaion sae of shear and ompressive sresses whih exeed erain limis τ and τ (due o he diffiuly of heir deerminaion, hey are seleed o be he same han in mode I (Sun and Khaleel, 004)), sliding mode (mode II) an be aivaed. The damage omponens are also given by a linear evoluion law, so in our axisymmeri ase hey are formulaed as a funion of shear sress in he plan of symmery. The general form is: 0 σ τ or max ( σ i ) > 0 i j σ τ D = τ < σ < τ and max ( σ i ) < 0 (4) τ τ σ τ and max ( σ ) < 0 i () FIG. 6 Shemai desripion of σ wih orresponding rak propagaion. C and C are deermined suh as he axial sress equals o zero when he damage omponen D approahes.0 in a uniaxial ension es. In his sudy, he parameers values given by Sun and Khaleel (004) were used: C =8800MPa, C =-4000MPa, σ =6MPa =94MPa. The damaged onsiuive equaions were implemened in he ommerial FE ode MSC.Mar. 4 Resuls and disussion I is known ha eah omponen of prinipal sresses is responsible of orresponding rak propagaion normal o is direion. Sine he direion (normal o he mesh plan) is a prinipal direion, so one an direly expe ha prinipal sress σ onribues o propagae raks whih an be, aording o figure, median or radial raks. Moreover on he indened surfae, he prinipal sress σ, aing radially as ensile sress, may ause he propagaion of a ring 4

5 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007 rak whih preedes he following developmen of a one rak downward. Figure 6 shows a represenaive volume elemen V on he indened surfae, in whih he aing direions of prinipal sresses σ are defined. The effe of hese sresses in he propagaion of mode I raks opening is also shown. () FIG. 7 Conour plos of damage omponens under an indenaion load of 500N by a sphere of mm diameer: a) D, b) D, ) D. The iso-value onours of predied damage omponens are shown in figure 7. They are obained in he las inremen of alulaion, orresponding o a normal sai loading of 500N. Conerning he evoluion of he damage omponen D, i was noied ha as soon as he ona beween he sphere and he plae sars, here was a hin damaged zone around he ouside edge of he indener wih values less han.0. However, wih he progress of he indener in he glass bulk is value inreased unil.0 and i ook a irular shape around he sphere. This damaged zone refles he iniiaion of a one rak. Considering he onour plos of he damage omponens D and D, here is an imporan damaged zone on he symmery s axis wih relaively low damage values. Above his laer zone, ours anoher damaged zone on he symmery s axis whih is approximaely siuaed a a disane equal o he radius of ona. This zone exhibis high damage values. I appears beause he shear effe (mode II) was aking ino aoun in he modelling. This zone seems o be aording o Lawn and Wilshaw (975) as he boom of he sheared zone where raks propagae during he unloading sage beause of he mismah beween he sheared zone and he elasi zone surrounding i. Aording o he damage map D, we an foresee ha his zone is siuaed jus below he boom of shear damaged zone. σ [MPa] virgin sae damaged sae r [mm] σ [MPa] virgin sae damaged sae n [mm] Fig. 8 Disribuion of maximal prinipal sress following: a) he radial direion and b) he normal o esimaed one rak direion. We paid aenion on he effe of damage on he disribuion of sresses in glass bulk a erain zones afer he indenaion proess. As invesigaed by Cook and Pharr (990), he one rak iniiaes from he ring rak whih seems o be he mos damaged zone a he side of he indener (figure 7a), and i propagaes in a direion normal o he maximal prinipal sress. A radial disribuion of his sress a a deph of 0.05mm beneah he indened surfae, and anoher 5

6 8 ème Congrès Français de Méanique Grenoble, 7- aoû 007 disribuion along a direion normal o esimaed one rak ( +90 ) are ploed in figure 8 for he virgin and damaged ases. A reduion of he maximal prinipal sress an be learly seen by aking ino aoun he damage in he modelling. Also, he disribuion of sresses in anoher region was examined. The damage in his region was regisered in boh onour plo of D and D, whih is labelled by X in figure 7. We sudied he disribuion of all omponens of prinipal sresses in his region. Indeed, here is a onribuion of more han one prinipal omponen whih seems o be he responsible of damage in his zone. We ploed in figure 9 he disribuion of prinipal sress σ along wo orhogonal pahs, lose o his region, following radial and axial direions. There is a loal effe of predied damage on his prinipal sress, whih leads o a reduion of is posiive values. σ [MPa] Conlusions 0 virgin sae damaged sae z [mm] 6 σ [MPa] virgin sae damaged sae r [mm] Fig. 9 Damage effe on he disribuion of prinipal sress σ in wo pahs: a) axial and b) radial, lose o he region X. The numerial analysis of a glass plae subjeed o sai indenaion by a spherial indener was presened. From a CDM based onsiuive modelling, he anisoropi damage mehanisms developed in he plae were examined hrough he prinipal (mode I) and shear sresses (mode II). As resuls, many riial zones were highlighed underneah he sie of indenaion or lose o he edge of indener. The effe of damage on he disribuion of sresses around riial zones was sudied. I would be ineresing o deermine he propagaion plan of raks a every riial zone, sine CDM approah does no predi i. Furhermore, i is now of prime imporane o ondu our own experimenal program o validae he numerial approah adoped in his work. Referenes Arora, A., Marshall, D.B., Lawn, B.R. & Swain, M.V. 979 Indenaion deformaion/fraure of normal and anomalous glass. J. Non-Crys. Solid, Cook, R.F. & Pharr, G.M. 990 Dire observaion and analysis of indenaion raking in glasses and eramis. J. Am. Ceram. So. 7, Huber, M.T. 904 Zur heorie der beruhrung feser elasiher korper. Ann. Phys. 4, 5-6. Kahanov, L.M. 958 Rupure ime under reep ondiions. Izv. Aad. Nauk 8, 6-. Knigh, C.G., Swain, M.V. & Chaudhri, M.M. 977 Impa of small seel spheres on glass surfaes. J. Ma. Si., Lawn, B. & Wilshaw, R. 975 Review indenaion fraure: priniples and appliaions. J. Maer. Si. 0, Lemaire, J. & Chabohe, J.L. 985 Méanique des maériaux solides. Dunod, Paris. Sun, X. & Khaleel, M.A. 004 Modeling of glass fraure damage using oninuum damage mehanis sai spherial indenaion. In. J. Damage Meh., 6-85.