Deployment of a Neo-Hookean membrane: experimental and numerical analysis

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1 Int. Jnl. of Multphyscs Volume 7 Number Deployment of a Neo-Hookean membrane: expermental and numercal analyss Essam A. Al-Bahkal 1, Fouad Erchqu 2, M. Moatamed 3 and Mhamed Soul 4, * 1 Department of Mechancal Engneerng Kng Saud Unversty P.O. Box 800, Ryadh 11421, Saud Araba ebahkal@ksu.edu.sa 2 Unversté du Québec en Abtb-Témscamngue 445 boul. de l Unversté, Rouyn-Noranda QC, J9X 5E4, Canada fouad.erchqu@uqat.ca 3 Narvk Unversty College Narvk, Norway moj@hn.no 4 Laboratore de Mécanque, de Llle, UMR CNRS 8107 Vlleneuve d Ascq, France ABSTRACT The am of ths research s to assess the response of a thn membrane subjected to hgh-pressure gas loadng for nflaton. Ths procedure s appled durng the desgn process of the membrane structure to allow the product to resst hgh-pressure loadng and to further characterze the hyper-elastc materal. The smulaton n ths work consders the standard procedures used n the LS-DYNA software, whch apples such assumptons as a unform arbag pressure and temperature n addton to a more recently developed procedure that takes nto account the flud-structure nteracton between the nflaton gas source and the hyper-elastc membrane; ths approach s referred to as the Arbtrary Lagrangan Euleran (ALE) formulaton. Untl recently, to smulate the nflaton of the hyperelastc membrane, a unform pressure based on a thermodynamc model or expermental test has been appled to the structure as the boundary condtons. To conserve CPU tme, ths work combnes both methods; the flud structure couplng method s used at an earler stage n whch the flud s modeled usng full hydrodynamc equatons, and at the later stage, the unform pressure procedure s appled, and the flud mesh and analyss are removed from the computaton. Both smulatons were compared to test data, ndcatng satsfactory correlaton wth the more recently developed procedure, the ALE theory, whch showed the greatest accuracy both n terms of graphcal and schematc comparson, partcularly n the early stages of the nflaton process. As a result, the new smulaton procedure model can be appled to research on the effects of desgn changes n the new membrane. 1. INTRODUCTION The Lagrangan formulaton n whch the mesh moves wth the materal s prmarly used to solve problems n sold mechancs but can also be used for flud-structure nteracton (FSI) problems. For small deformatons, the Lagrangan formulaton s able to accurately solve the *Correspondng author. E-mal: mhamed.soul@unv-llle1.fr

2 42 Deployment of a Neo-Hookean membrane: expermental and numercal analyss flud structure nterface and materal boundares; the man lmtaton of ths formulaton s the mesh dstorton for large deformatons and movng structures because the flud s solved on a movng doman due to the structure moton. Varous approaches have been nvestgated to solve FSI problems. One of the commonly used approaches to solve these problems s the ALE formulaton, whch has been used wth success n the smulaton of fluds wth large motons, such as sloshng fuel tanks n the automotve ndustry and brd mpacts n the aeronautc ndustry. Other approaches that have been nvestgated to solve FSI problems nclude the ALE mult-materal formulaton and Lagrangan formulatons. The ALE mult-materal formulaton used n ths paper and developed n the LS-DYNA code has been valdated for several academc and ndustral applcatons. After the smulatons have been valdated by expermental testng, the results can be used as a desgn tool for the mprovement of the structure. The ALE formulaton for the smulaton of flud dynamcs problems s presented n addton to the couplng algorthm. Ths method was successfully used to model many ndustral and academc applcatons, such as the prevously mentoned sloshng tank problem. For the structure nvolved n the couplng, a rubber materal s used wth a Mooney Rvln consttutve materal. For characterzaton of the Mooney Rvln materal constants under baxal deformaton, we consder the problem of bubble nflaton descrbed n Erchqu et al. [1]. In ths technque, the appled nflaton pressure and the heght at the hemspherc pole are recorded durng testng. The bubble nflaton problem s practcal because t requres relatvely smple devces. Moreover, t provdes the best representaton of the actual deformaton that occurs durng the process. The analytcal treatment of nflaton s complcated due to nonlneartes n the materal behavor and extensve deformaton. Numercal methods can be used to smulate and predct the membrane deformaton process by solvng nonlnear ordnary dfferental equatons governng the bubble moton, and the mathematcal descrpton of the ordnary dfferental equatons s descrbed n detal n Erchqu et al. [2]. From computatonal experence, we know that the majorty of the CPU tme s consumed by the resoluton of the flud equatons because the flud mesh s much denser than the structure mesh. To reduce the CPU tme, we combne both methods n the new analyss. At the earler stages, the flud structure couplng method s appled n whch the flud s modeled usng the full hydrodynamc equatons. At the later stage, the unform pressure procedure s appled, and the flud mesh and hydrodynamc analyss are removed from the computaton. Ths method has been used successfully n the automotve ndustry for arbag deployment, as descrbed n detal n Khan et al. [3]. 2. EULERIAN FORMULATION AND COUPLING ALGORITHM 2.1. ALE AND EULERIAN FORMULATION Flud problems n whch the nterfaces between dfferent materals (gas and ambent ar) are present are more easly modeled usng a Lagrangan mesh. However, f an analyss s requred for complex tank geometry, the dstorton of the Lagrangan mesh renders ths type of method dffcult to use because many re-meshng steps are necessary for the calculatons to contnue. An alternatve method s the Euleran formulaton. The change from a Lagrangan to a Euleran formulaton, however, ntroduces two problems: the frst s the nterface trackng, and the second s the advecton phase or the advecton of flud materal across element boundares. To solve these problems, an explct fnte element method s used for the Lagrangan phase, and a fnte volume method (flux method) s appled for the advecton phase. We refer to several explct codes such as Pronto, Dyna3D and LS-DYNA; see Hallqust [4] for a full

3 Int. Jnl. of Multphyscs Volume 7 Number descrpton of the explct fnte element method. The advecton phase has been developed by the frst author and ncluded n the LS-DYNA code, extendng the range of applcatons that can be used wth the ALE formulaton. In the Euleran approach and the general ALE descrpton, an arbtrary referental coordnate s ntroduced n addton to the Lagrangan and Euleran coordnates. The materal dervatve wth respect to the reference coordnate s descrbed n (1). Thus, substtutng the relatonshp between the materal tme dervatve and the reference confguraton tme dervatve results n the ALE equatons. f( X, t) = f( x, t) + w f ( x, t ) t t x (1) where X s the Lagrangan coordnate, x s the Euleran coordnate, and w s the relatve velocty. Let v denote the velocty of the materal and u denote the velocty of the mesh. To smplfy the equatons, we ntroduce the relatve velocty w = v u. Therefore, the governng equatons for the ALE formulaton are gven by the followng equatons (2) for mass, momentum, and energy conservaton: ρ = v ρ ρ w t x x v ρ σ ρ ρ = + b w v j, j t x j (2) E ρ σ ρ ρ = + v bv w E j, j j t x j where ρ s the densty, σ j s the total Cauchy stress, b s the body force, and E s the nternal energy per unt mass. Note that the Euleran equatons commonly used by the computatonal flud dynamcs (CFD) communty n flud mechancs are derved by assumng that the velocty of the reference confguraton s zero and that the relatve velocty between the materal and the reference confguraton s therefore the materal velocty. The terms assocated wth the relatve velocty w n equaton (2) are usually referred to as the advectve term and accounts for the transport of the materal past the mesh. It s ths addtonal term n the equatons that makes solvng the ALE equatons much more dffcult numercally than the Lagrangan equatons, n whch the relatve velocty s zero. In the second phase (the advecton phase), the transport of mass, the nternal energy and the momentum across cell boundares are computed; ths process may be thought of as remappng the dsplaced mesh of the Lagrangan phase back to ts orgnal or arbtrary poston. For dscretzaton of the equatons n (2), one-pont ntegraton s used for effcency and for elmnaton of lockng. The resoluton s advanced n tme wth the central dfference

4 44 Deployment of a Neo-Hookean membrane: expermental and numercal analyss method, whch provdes second-order accuracy n tme usng an explct method n tme. For each node, the velocty and dsplacement are updated usng second-order tme ntegraton n equaton (3): u ( ) n t u M F ext = + + F n+ 12 / 12 / nt (3) where F ext s the nodal external force, F nt s the nodal nternal force, and M s the nodal mass. In the second phase (the transport of mass), the momentum and the nternal energy across the element boundares are computed. Ths phase may be consdered as a re-mappng phase as well. The dsplaced mesh from the Lagrangan phase s remapped nto the ntal mesh for the Euleran formulaton or nto an arbtrary undstorted mesh for an ALE formulaton. In the advecton phase, we solve a hyperbolc problem or a transport problem, where the varables are the densty, the momentum and the nternal energy per unt volume. Detals of the numercal method used to solve the equatons are descrbed n detal n Aquelet et al. [5], whch uses the Donor Cell algorthm, a frst-order advecton method, and the Van Leer algorthm, a second-order advecton method. As an example, the equaton for mass conservaton s: ρ +.( ρu) t = 0 (4) The objectve of ths paper s not to descrbe the dfferent algorthms used to solve equaton (3); these algorthms have been prevously descrbed n detal n Benson [6]. 3. CHARACTERIZATION OF THE CONSTITUTIVE MATERIAL AND THE EXPERIMENTAL SETUP 3.1. CHARACTERIZATION OF THE CONSTITUTIVE MATERIAL Frst, we characterze the materal data usng a bubble nflaton technque from Erchqu et al. [1] to ft the consttutve model and obtan the materal property constants of the membrane structure. The expermental setup has been descrbed n detal n Erchqu et al. [2] and s llustrated n Fgure 1. The hyperelastc constants were determned by baxal characterzaton: the membrane s subjected to free nflaton (wth a controlled ar flow rate), and the formaton pressure and bubble heght are measured. A hyperelastc materal s defned as an elastc materal n whch the stress at each pont can be derved from a scalar functon W(F) known as the stran energy functon. To meet the requrements of objectvty, the stran energy functon must be nvarant under changes n the observer frame of reference. It s well known that the Cauchy Green deformaton tensor s nvarant under changes n the observer frame of reference. Thus, f the stran energy functon can be wrtten as a functon of the rght Cauchy Green tensor C, t wll therefore meet the objectvty prncple. The general stress stran relatonshp s gven by the formula (5): S j = 2 W C j (5)

5 Int. Jnl. of Multphyscs Volume 7 Number Membrane 15 cm 13 cm 19 cm Inlet gas 2 cm Fgure 1 Descrpton of the expermental Setup. where S s the Pola Krchhoff stress tensor. The dfferent models n the lterature defne the stran energy as a functon of the stran feld. We consder Rvln s theory [7] of sotropc materals, whch descrbes the energy as a functon of the three Cauchy stran nvarants I 1, I 2, and I 3. Assumng materal ncompressblty, the stress can be obtaned from the followng Neo-Hookean form of the Mooney-Rvln materal model gven by: ( ) W = C I (6) I = λ + λ + λ (7) where C 10 s a materal constant, I 1 s the frst nvarant of the left Cauchy-Green deformaton tensor, and λ are the Lamé s frst parameters EXPERIMENTAL AND NUMERICAL IDENTIFICATION Erchqu et al. [1] expermentally nvestgated the nflaton behavor n a thermoplastc membrane under the combned effects of appled stress and temperature usng bubble nflaton tests. The results were used to dentfy the materal constants embedded n the consttutve models of the polymerc materals n ths work. The polymerc materal, acrylontrle-butadene-styrene (ABS), was tested under baxal deformaton usng the bubble nflaton technque. The ntal ABS sheet thckness was 1.57 mm. The expermental setup s descrbed n Fgure 1, n whch crcular membranes were mounted between two metal dsks contanng a crcular aperture and subsequently clamped onto a support. The exposed crcular area of dameter D = 15 cm was heated n an nfrared heatng chamber to the softenng pont. When the temperature became unform over the flat sheet, the crcular area was nflated usng compressed ar under a controlled flow rate. The appled nflaton pressure, whch was unform over the membrane, was measured wth a pressure sensor durng the testng. The heght at the hemspherc pole (or the center of the membrane) and the tme were recorded smultaneously usng a vdeo camera and a data acquston system.

6 46 Deployment of a Neo-Hookean membrane: expermental and numercal analyss Fgure 2 Inflaton of a flat crcular ABS membrane Pressure (Pa) Mooney-rvln model Data expermental Bubble heght (cm) Fgure 3 at 145 C. Expermental and predcted pressure versus bubble heght for ABS For most nflaton tests, the testng ends when the bubble bursts. Fgure 2 shows the nflaton profle of a crcular plane membrane, and Fgure 3 shows the expermental and ftted bubble heght versus tme data at the pole of the membrane for ABS at 145 C. Fgure 3 compares the predcted pressure versus the bubble heght data at the pole of the bubble usng the Mooney Rvln model for the expermental ABS measurements. The measurement error s less than 5% durng the pressure rse. However, the numercal model for the ABS provdes a better ft to the expermental data, as shown n Fgure 3. It should be noted that at the maxmum pressure, the bubble heght s of the order of the ntal membrane radus. Beyond ths crtcal pont, the heght ncreases rapdly whle the pressure falls. Ths unstable behavor contnues untl the bubble bursts. 4. COUPLING ALGORITHM Several couplng methods for CFD and structural dynamc solvers have been developed to solve couplng problems. Classcal mplct and explct couplng are descrbed n detal n Aquelet et al. [8], where hydrodynamc forces from the flud solver are passed to the

7 Int. Jnl. of Multphyscs Volume 7 Number structure solver for stress and dsplacement computaton. In ths paper, a couplng method based on the contact algorthm s used, and the couplng algorthm s descrbed n detal n Aquelet et al. [5]. Because the couplng method descrbed n ths work s based on the penalty method for contact algorthms, the contact approach s a good ntroducton to ths method. In contact algorthms, a contact force s computed proportonal to the penetraton vector, whch quantfes the constrant volaton. In an explct FEM method, contact algorthms compute the nterface forces due to mpact of the structure on the flud, and these forces are appled to the flud and structure nodes that are n contact to prevent a node from passng through the contact nterface. One surface s desgnated as a slave surface and the second as a master surface n contact algorthms. The nodes that le on both surfaces are also referred to as slave and master nodes accordngly. The penalty method mposes a resstng force on the slave node proportonal to ts penetraton through the master segment. Ths force s appled to both the slave node and the nodes of the master segment n opposte drectons to satsfy equlbrum. The penalty couplng behaves lke a sprng system, and the penalty forces are calculated proportonally to the penetraton depth and sprng stffness (see Fgure 4). The head of the sprng s attached to the structure or slave node, and the tal of the sprng s attached to the master node wthn a flud element that s ntercepted by the structure. Smlar to the penalty contact algorthm, a couplng force proportonal to the penetraton depth s appled to both the flud and the structure nodes. The penalty force that s appled to both the flud and the structure nodes n opposte drectons s gven by: F = kd (8) where k s the sprng constant, and d s the relatve dsplacement between the flud and the structure nodes. The accuracy of the couplng also depends on the number of couplng ponts used n addton to the element nodes. Addtonal couplng ponts can be postoned on the arbag membrane structure, as shown n Fgure 2. A compromse must be made between too many addtonal couplng ponts, whch leads to long computatonal tmes, or too few (f there are no addtonal couplng ponts), resultng n the effect of artfcal leakage Flud node k d Structure node Fgure 4 Descrpton of the Penalty Contact.

8 48 Deployment of a Neo-Hookean membrane: expermental and numercal analyss through the membrane arbag fabrc. Ths leakage s caused by the nflaton gas source falng to connect wth the couplng ponts and passng through the membrane arbag structure. Another possble factor s that the drecton of the nflaton gas source flow dffers from the drecton of advecton, whch results n mass adverted through the structure membrane. 5. NUMERICAL SIMULATIONS To valdate the formulaton descrbed above, we frst consder the problem of free nflaton of a vscoelastc membrane. The geometry of the structure s modeled usng a quartersymmetry model and meshed wth four-node shell elements (2733 nodes and 132 shell elements) as shown n Fgure 5. The measured pressure s nterpolated by a polynomal functon, as shown n Fgure 6, and used n the fnte element program (Lagrangan formulaton) to compute the tme Pole Fgure 5 Mesh and shape of the hyper-elastc membrane Pressure (Pa) Tme (sec) Pressure nterpolaton Expermental data Fgure 6 Pressure nterpolaton versus expermental data.

9 Int. Jnl. of Multphyscs Volume 7 Number evoluton of the bubble heght at the center pole. The predctons obtaned wth the Mooney- Rvln model gve almost dentcal results and are n agreement wth the expermental data n Fgure 6 at a settng temperature of 145 C. We note that the Mooney-Rvln model predcts a smlar bubble heght at the pole, whch s close to that of the expermental measurements. However, the model gves dfferent results that are also close to the expermental measurements when the bubble pressure level s close to the maxmum pressure reached durng the experments. Ths phenomenon s related to the models used and s the subject of several studes. Indeed, for these models, there exsts a crtcal pressure beyond whch the system dverges (unstable branch). The work as presented does not descrbe the unstable regon of the soluton. To valdate the numercal results and to prove the ablty of the ALE flud structure couplng method to solve problems n cases where the classcal pressure boundary condtons cannot, we use the two types of materal wth dfferent Moony-Rvln coeffcents, C 10 = Pa and C 10 = Pa. For the frst materal wth C 10 = Pa, both formulatons gve the same results and show good correlaton wth the expermental data. The correlaton can be observed n Fgure 7 n whch the evoluton of the bubble heght at the center of the membrane structure versus tme s plotted for the expermental test and the numercal smulaton. For the softer materal wth C 10 = Pa, the soluton from the smulaton usng the pressure boundary condton and no flud becomes unstable after t = 0.6 seconds. Ths nstablty has not been shown n our expermental tests when usng ths materal and s due to the lack of dampng that can be provded by the flud. The mssng flud n the pressure boundary condton smulaton creates a lack of dampng n the system that leads the structure to numercal nstablty. Ths nstablty dd not occur n the smulaton usng the flud structure couplng because the flud nvolved n the smulaton has a dampng effect durng nflaton. For the materal wth C 10 = Pa, the tme evoluton of the bubble heght for both smulatons s plotted n Fgure 8. In ths fgure, the nstablty s evdent after t = 0.6 seconds of the smulaton usng the unform pressure boundary condton Heght bubble (m) Pressure nterpolaton Expermental data Wth couplng Tme (sec) Fgure 7 Bubble heght tme evoluton wth C 10 = Pa.

10 50 Deployment of a Neo-Hookean membrane: expermental and numercal analyss 0.05 A: Couplng method B: Unform pressure Heght bubble (m) B A B A B A B A B A Tme (sec) Fgure 8 Bubble heght tme evoluton wth C 10 = CONCLUSION In ths work, we present the applcaton of a Euleran approach for smulatng the response of an sotropc and ncompressble thermoplastc materal. The expermental valdaton s performed for an ABS materal n the case of free nflaton of a crcular membrane subject to a dynamc pressure load dstrbuton. An outlne of the newly developed smulaton procedures for membrane nflaton has been detaled and explaned n ths paper. The most commonly used methodology for membrane nflaton smulaton s the unform pressure method n whch a unform pressure s appled to the structure, and several assumptons are made regardng the nflaton gas source. To conserve CPU tme, the approach used n ths paper combnes both methods. At an early stage, the flud structure couplng method s appled n whch the flud s modeled usng the full hydrodynamc equatons; at the later stage, the unform pressure procedure s appled, and the flud mesh and analyss are removed from the computaton. The greatest advantage of ALE s ts ablty to accurately model the nflaton gas source, the effects of the flud flow and the nteracton wth the structural parts of the model, whch s especally useful n the early stages of the nflaton process. Because the unform pressure method s not accurate enough to model these stuatons, ALE can be used to accurately depct the ntal pressure dstrbuton nsde the nflatng structure. At present, smulatons for the above model are run usng the ALE methodology descrbed n ths artcle and wll be publshed at a later date to compare the results wth the Control Volume (CV) method and the test data shown n ths work. REFERENCES [1] Erchqu, F., Derdour, A., Gakwaya, A., and Verron, E., Analyse expérmentale et numérque en soufflage lbre d une membrane thermoplastque, Entrope, 2001, 37(235/236), pp [2] Erchqu, F., Soul, M., Ben Yedder, R., Nonsothermal fnte-element analyss of thermoformng of polyethylene terephthalate sheet: Incomplete effect of the formng stage. Polymer Engneerng and Scence, 2007, 47(12), pp

11 Int. Jnl. of Multphyscs Volume 7 Number [3] Khan, M., Moatamed, M., Soul, M., Zeguer, T., Multphyscs out of poston arbag smulaton, Internatonal Journal of Crashworthness, 2008, 13(2), pp [4] Hallqust, J.O., LS-DYNA theoretcal manual, 1998, Lvermore Software Technology Company Calforna, USA. [5] Aquelet, N., Soul, M., Olovson, O., Euler Lagrange couplng wth dampng effects: Applcaton to slammng problems, Computer Methods n Appled Mechancs and Engneerng, 2005, 195, pp [6] Benso, D. J., Computatonal Methods n Lagrangan and Euleran Hydrocodes, Computer Methods n Appled Mechancs and Engneerng, 1992, 99, pp [7] Rvln, R. S., Large Elastc Deformaton of Isotropc Materals-IV: Further Developments of the General Theory, Phl. Trans. R. Soc., 1948, A241, pp [8] N.Aquelet, M.Soul and J.Gabrys, ALE Formulaton for fuel Slosh Analyss, Structural Engneerng and Mechancs, an Internatonal Journal, 2003, Vol. 16, No. 4, pp

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