OPTIMAL DESIGN OF VISCOELASTIC COMPOSITES WITH PERIODIC MICROSTRUCTURES

Transcription

1 OPTIMAL DESIN OF VISCOELASTIC COMPOSITES WITH PERIODIC MICROSTRUCTURES eong-moo 1, Sang-Hoon Park 2, and Sung-Ke oun 2 1 Space Technology Dvson, Korea Aerospace Research Insttute, usung P.O. Box 3, Taejon, ,Korea 2 Department of Mechancal Engneerng, Korea Advanced Insttute of Scene and Technology 373-1, usung-dong, usung-ku, Taejon, , Korea SUMMAR: Optmal desgn of vscoelastc compostes wth prescrbed or optmal stffness/dampng characterstcs s presented. The effectve complex modul n frequency doman are obtaned by applyng the homogenzaton method n frequency doman wth Correspondence Prncple. To fnd a vscoelastc composte wth gven consttuent materal propertes, an nverse homogenzaton problem s formulated as a topology optmzaton problem usng the homogenzaton method. An artfcal materal model s presented n order to construct a topology optmzaton problem as a densty dstrbuton problem. The senstvty wth respect to densty s formulated. Desgn varables, objectve functon and desgn constrants ncludng materal symmetry and geometrc symmetry are defned to construct the optmzaton problem, as a mcrostructure desgn problem by an nverse homogenzaton approach. Numercal desgn examples are presented wth dscussons on the optmal desgn of mcrostructures for vscoelastc compostes. KEWORDS: vscoelastc, dampng, nverse homogenzaton, optmzaton. INTRODUCTION Vscoelastc compostes have been wdely appled for the purpose of reducng nose and vbraton as well as for the purpose of ncreasng stffness to weght rato n structures. These applcatons are manly due to the fact that vscoelastc compostes have desrable dampng characterstcs and desgn flexblty,.e., tradeoffs between dampng and stffness. Polymer compostes, rubber-toughened compostes, and engneerng plastcs are the typcal examples. From the desgner s vewpont, t would be very useful f there are materals whose servce performances are optmal n the gven envronments. The servce performances may be the dampng characterstcs n the case of nose and vbraton controls. However, t s practcally mpossble to nvent new materals wth optmal propertes for each new desgn. Instead, t s more practcal to construct and use composte materals, whch have good servce performances n the gven envronments, by usng the already exstng materals.

2 The nverse homogenzaton approach emerged for desgn of materals wth prescrbed materal propertes as a specal applcaton of the homogenzaton method and the topology optmzaton [1]. It has been appled to the mcrostructure desgn of elastc materals wth extreme materal propertes such as Posson s rato close to 1 and 0.5 [2]. It has been extended to the mcrostructure desgn of thermoelastc compostes, whch have unusual or extreme materal propertes such as negatve thermal expanson coeffcent or zero thermal expanson coeffcent [3]. It has been suggested that stffness and dampng characterstcs of vscoelastc compostes may be mproved by changng the topology of the mcrostructures of the consttuent materals [4]. In the present work, an nverse homogenzaton approach for obtanng a mcrostructure of two-phase vscoelastc compostes s presented to acheve desrable stffness and dampng characterstcs of the vscoelastc compostes. HOMOENIZATION IN VISCOELASTICIT The homogenzaton method has been appled to the estmaton of the effectve materal propertes of composte materals wth perodc mcrostructures [5]. For general vscoelastc compostes, the homogenzaton process s formulated and a systematc way of obtanng the effectve relaxaton modul n tme doman s presented [4]. The effectve relaxaton modul n tme doman could be obtaned from the effectve relaxaton modul n Laplace transform doman by nverse Laplace transforms. Then the effectve complex modul n frequency doman could be readly obtaned usng smple formul. Dampng characterstcs of vscoelastc compostes are drectly related to the complex modul of the compostes. The values of complex modul are strongly dependent on frequency. Snce the present work deals wth the mcrostructure desgn problem n whch dampng characterstcs are of major concerns, the effectve relaxaton modul n tme doman need not to be computed. Instead, the effectve complex modul n frequency doman can be drectly obtaned by applyng the homogenzaton method n frequency doman wth Correspondence Prncple. The homogenzaton problem n vscoelastcty can be easly formulated n complex doman by the same procedures as the homogenzaton n elastcty wth the use of Correspondence Prncple. In general, n the homogenzaton process, two problems, the local and the global problem, are obtaned respectvely. By solvng the local problem, the homogenzed modul are obtaned. The local problem n complex doman s as follows. jpq kl p v y y y d y v (, ) χ = y d v V (, ), (1) q j The homogenzed complex modulus s obtaned as follows. j h kl j χ p χ r ( ) = pqrs( )( δ d kpδlq )( δrδ js ) y y q s (2) kl where χ p s the soluton of the local problem (1). By solvng the local problem (1) and usng (2), the effectve complex modul at a gven frequency are obtaned. The most mportant reason for usng the complex modulus n frequency doman, s to facltate the senstvty analyss, whch s an ndspensable element for optmzaton. As wll be dscussed n the later secton, the senstvtes of the object functon are readly obtaned from the effectve modul n frequency doman.

3 INVERSE HOMOENIZATION PROBLEM Desgn varables and artfcal two-phase materal model An nverse homogenzaton problem s formulated as a topology optmzaton problem n whch the dstrbuton of two vscoelastc materal phases s to be found so as to optmze the stffness and dampng characterstcs of the vscoelastc compostes. At ths stage, an artfcal two-phase materal model [3] s ntroduced wth a concept of densty. An artfcal materal s defned for each densty ρ wth the followng propertes. ( ) = ρ (1) ( ) + (1 ρ) (2) ( ) (3) where (1) and (2) are the complex modul of phase 1 and phase 2 materal, respectvely. The densty, ρ, whch s usually ntroduced n topology optmzaton problems, means the volume fracton of phase 1 materal n the artfcal two-phase materal. ρ =1 means that t s pure phase 1 materal and ρ =0 means that t s pure phase 2 materal. Desgn doman, whch s a perodc base cell, s dscretzed wth N fnte elements. The densty s assumed to be constant n each element but t can be vared from one element to another. The densty n each element becomes the desgn varable of the optmzaton problem. Wth the settng, the nverse homogenzaton problem s defned as the problem of fndng the optmal denstes n the elements so as to optmze the objectve functon, whch wll be defned later. It also should be noted that ntermedate value of densty s allowed for artfcal materal model. Artfcal materal wth ntermedate densty can be nterpreted as a mxture of two materal phases. Senstvty analyss Senstvty analyss s requred for teratve desgn modfcatons n desgn optmzaton process. In the process, t s assumed that the change of stran feld n each element due to the change of densty n an element s not large. Ths assumpton s justfed snce a large number of elements are needed for the topology optmzaton. Under ths assumpton, approxmate senstvtes of the effectve complex modul wth respect to densty n an element can be obtaned from equaton (2) as follows. h ( ) = ρ ( y, ) ( δ kpδ ρ pqrs lq χ y kl p q )( δ δ r js j χ r y s ) d (4) Also, from Equaton (3) the followng smple relaton holds for the artfcal two-phase materal model. ( ρ, ) (1) (2) = ( ) ( ) ρ (5) Objectve functon The objectve functon n ths work s values of the components of the complex modul, or any combnaton of the components of the complex modul at gven frequences. So the followng objectve functon s defned for desgn optmzaton.

4 f = ndv = 1 w ( g ) η + w penalty [ ndv = 1 ρ (1 ρ )] η penalty (6) where g s one of the complex modulus, that s, storage modulus, the loss modulus or the loss tangent and where ρ s the densty n an element and w, η, wpenalty, andη penalty, the weghtng factors. The frst term n (6) s for stffness and/or dampng characterstcs for optmzaton. The second term s the penalty to suppress the ntermedate values of densty n elements. Desgn constrants Avalable desgn constrants n ths work can be total volume fracton of phase 1 materal, effectve complex modul, geometrc symmetres and materal symmetres. Volume constrant s defned as follows. V mn 1 ρ d V max (7) where s the doman of the base cell and V mn andvmax, the bounds of volume fracton. Constrants on the effectve complex modul can be the bounds of the storage modulus, loss modulus and loss tangent of the vsocoelastc compostes. The constrants are enforced as numercal values as descrbed n the followng numercal examples. The geometrc symmetry constrant can be appled drectly by equatng denstes of the symmetrc elements n a base cell. Orthotropc vscoelastc composte materals can be desgned by enforcng one geometrc symmetry constrant nto elements n a base cell. It s possble to desgn vscoelastc composte materals wth materal symmetry. The condton for square symmetry of a vscoelastc composte materal s as follows. = 2222 (8) The condtons for transverse sotropy of vscoelastc compostes are as follows. + = 2222 and = (9) Optmzaton problem statements Wth the defntons for desgn varables, objectve functons and desgn constrants, an nverse homogenzaton problem for optmal mcrostructure desgn can be stated as a topology optmzaton problem as follows: Fnd the densty dstrbuton n a base cell so as to Maxmze dampng or stffness at gven frequences Subject to constrants on volume fracton and/or constrants on dampng or stffness at gven frequences and/or materal symmetry and/or geometrc symmetry

5 For the optmzaton algorthm, the sequental lnear programmng (SLP) s used. In SLP, the optmzaton problem s lnearzed around the current desgn pont n each teraton and the next desgn s found by the lnear programmng. In ths work, the method of feasble drectons s used as the constraned lnear programmng method. The reason for usng SLP s ts robustness snce the nverse homogenzaton problem has numerous local mnma and s not a well-behaved problem. For the mplementaton of the optmzaton algorthm, DOT (Desgn Optmzaton Tool) verson 4.00 s used [6]. Introducton NUMERICAL EXAMPLES AND DISCUSSIONS In ths secton, numercal examples for the mcrostructure desgn to mprove the stffness and dampng characterstcs of vscoelastc compostes are presented. The desgn doman s dscretzed wth 12 by 12 square fnte elements n a unt base cell. The desgn optmzaton starts wth an ntal densty dstrbuton of the two-phase materal. The ntal densty n each element s randomly gven. Intal densty dstrbuton should not be unform snce the desgn changes can not occur wth unform densty dstrbuton n all elements. In the followng fgures of mcrostructures n a base cell, as shown n Fg. 1, elements n black color represent materal 1 and elements n whte color represent materal 2. Elements n gray color represent the materal of ntermedate property. Example 1 As a frst example, the mcrostructure desgn of the vscoelastc composte composed of an elastc materal phase and a vscoelastc materal phase s consdered. Although the materal propertes are non-dmensonal, ther relatve magntudes are taken from the approxmate materal propertes of the typcal glass for the elastc materal and the typcal epoxy for the vscoelastc materal. For the vscoelastc materal, the vscoelastcty s modeled by the standard lnear sold [7] and the relaxaton tme s artfcally assumed. The materal propertes are as follows. E = 70 Pa, ν = 0.22 for phase 1 materal (10) t E( t) = e Pa, ν = 0.35 for phase 2 materal () Suppose we are to desgn mcrostructures of a vscoelastc composte whch s composed of the above two materals wth 50% volume fracton for each materal. Ths composte s to be used n the vbratng structures and the operatng frequency s artfcally assumed to be 0.5 as the relaxaton tme s assumed. A smple way for constructng mcrostructures s to use a square unt cell wth a crcular ncluson. Let us defne two knds of compostes wth ths confguraton of the mcrostructure for later references. For the composte 1, the elastc materal s used for the crcular ncluson and the vscoelastc materal for the matrx. For the composte 2, the roles of the two materals are nterchanged. Table 1 shows the effectve complex modul of these two compostes at the operatng frequency. In many engneerng applcatons, certan degrees of lower bounds are requred for both stffness and dampng n compostes. For example, to reduce vbratons effectvely by use of a damper, stffness of a damper should not be too low. Suppose that we need a composte whose stffness, = 22, should be greater than 15 and the correspondng loss tangent should be as large as possble at any gven frequency. Wth a smple

6 mcrostructure confguraton such as the composte 1 and the composte 2, we cannot obtan the satsfactory desgn. Even f we ncrease the volume fracton of the elastc ncluson n the composte 1 up to 70%, = 22 have a value of only In that case, the dameter of the crcular ncluson becomes f the unt cell sze s 1. If we use the composte 2 to get the suffcent stffness, then the dampng characterstcs becomes worse. The frst desgn example has been selected wth the above observatons n mnd. The objectve of ths example s to fnd a mcrostructure whch yelds as large dampng as possble whle the desred stffness of the mcrostructure s to be acheved at an operatng frequency,. The exact descrpton of the objectve functons and the constrants are as follows. Objectve functon: tan δ ( ) = tan δ 22( ), where = 0. 5 Constrants: ( ) = 22( ) v f = 0.5 for each materal eometrc symmetry about X-axs and -axs. Where tan δ ( ) means loss tangent n 1-1 (x) drecton, ( ) means the storage modulus n 1-1(x) drecton and v f represents the volume fracton of materal 1. X-axs and -axs mean the horzontal centerlne and the vertcal centerlne of the base cell, respectvely. Also, the penalty term has been added as n (6), to reduce the regons of ntermedate densty. Fg. 1 shows a unt cell wth the optmal mcrostructure, for whch the soluton converged after 148 functon calls n desgn optmzaton routnes. Fg. 1 also shows the same mcrostructures n a coarse 3 by 3 cell for easer understandng of the desgned mcrostructure. The effectve complex modul are lsted n Table 1. Frequency Table 1: Effectve complex modul of example 1 Storage Modulus ( ) Loss Modulus ( ) Loss Tangent tan δ ( ) Materal Materal Composte Composte Optmal Composte It can be concluded that the newly desgned mcrostructure shows the successful tradeoff between stffness and dampng. Some nterpretatons are possble for the desgned mcrostructure. The elastc phase materals are nearly connected so as to provde suffcent stffness. At the same tme, the elastc phase materals form mechansm-lke structures n some regon and lumps of elastc and vscoelastc materals exst around the lnkage. Ths mechansm-lke structures n the elastc phase materals result n suffcent deformatons n the vscoelastc phase materals.

7 Fg. 1: Mcrostructures and perodc mcrostructures for maxmum dampng n example 1 These two facts show the tradeoff between stffness and dampng n the compostes and show that the desgn gves as large dampng as possble whle mantanng the stffness at the desred level. The mechansm-lke structures are also found n the mcrostructures wth extreme propertes such as negatve Posson's rato and t has been ponted out that the mechansm-lke structures have a crucal role n such extreme mcrostructures [2]. At ths pont, t should be mentoned that the nverse homogenzaton problems have such numercal dffcultes as mesh dependency and numerous local mnma manly because of the characterstcs of ther loadng and boundary condtons. In fact, the loadng and the boundary condtons n a unt cell are very unform, especally at the ntal stages of the optmzaton process, compared to the typcal structural optmzaton problems. The more unform the loadng and the boundary condtons are, the more the local mnma are present. Also, mesh dependency of the convergence can be ncreased because of the exstence of the numerous local mnma. In other words, the objectve functon surface or the response surface s very complex for the nverse homogenzaton problems. As a result, the optmzaton processes tend to proceed towards a local mnmum and the optmzaton steps need large number of teratons n such cases. Some technques such as employng low-pass flters have been suggested to escape from these numercal problems. Thus, t s recommended that the ntal densty dstrbutons be ncely chosen, whch can be done ether by the desgner's ntutons or by the already exstng nearly optmal solutons [3]. If these are not avalable, the next choce s the tral and error method. Although the desgned solutons,.e. the mcrostructures, can be dfferent wth dfferent ntal guesses, the common characterstcs of the optmal mcrostructures are the same as descrbed before. Also, the values of the objectve functons for the dfferent optmal mcrostructures gve lttle dfference n the numercal values. Example 2 As a second example, the mcrostructure desgn of the vscoelastc composte composed of an elastc materal and a vscoelastc materal s consdered. The materal propertes are as follows. E = 70 Pa, ν = 0.22 for phase 1 materal (12) t E( t) = e Pa, ν = 0.35 for phase 2 materal (13)

8 Suppose we are to desgn a mcrostructure of a vscoelastc composte, whch s composed of the above two materals. Ths composte s to be used n the vbratng structure and the operatng frequency s assumed to be 0.5 same as n example 1. The structure experences hgher vbraton levels n one drecton. For example, the payloads mount structure for nstallng electronc devces n a launch vehcle, experences hgher vbraton levels n longtudnal drecton than n lateral drecton. So we want to desgn a vbraton damper made of vscoelastc composte materals, whch should have a good dampng characterstcs n that drecton as an ant-vbraton damper. To reduce vbratons effectvely by use of a damper, stffness of a damper should not be too low. Suppose that we need a composte whose stffness, = 22, should be greater than 15 and the correspondng loss tangent should be as large as possble at a gven frequency n a specfed drecton. Wth such a smple mcrostructure confguraton as the composte 1 and the composte 2 n example 1, we cannot obtan the satsfactory desgn as stated n example 1. The second desgn example has been selected wth the above observatons n mnd. The objectve of ths example s to fnd a mcrostructure that yelds as large dampng as possble n a specfed drecton whle the desred stffness of the mcrostructure s mantaned. The exact descrpton of the objectve functons and the constrants are as follows. Objectve functon: tan δ ( ), where = 0. 5 Constrants : ( ) = 22 ( ) eometrc symmetry about X-axs. The constrant on volume fracton s not gven n ths example. Fg. 2 shows a unt cell wth the resultng optmal mcrostructure. Fg. 2 also shows the same mcrostructures n a 3 by 3 cell for easer understandng of the desgned mcrostructure. The effectve complex modul are lsted n Table 2. It can be concluded that the newly desgned mcrostructure shows the successful tradeoff between stffness and dampng. Some nterpretatons are possble for the desgned mcrostructure as n example 1. The elastc phase materals are lumped together so as to provde suffcent stffness as n example 1. At the same tme, the elastc phase materal forms mechansm-lke structures and lumps of elastc materal surround the vscoelastc materals. The vscoelastc phase materals form a wave-lke fashon along the -axs n the perodc mcrostructures as shown n Fg. 2. Ths mcrostructure wth wave-lke vscoelastc phase materals and lumps of elastc phase materals result n suffcent deformatons n the vscoelastc phase materals n the specfed drecton. Frequency Table 2: Effectve complex modul of example 2 Storage Modulus ( ) Loss Modulus ( ) Loss Tangent tan δ ( ) Materal Materal Composte Composte Optmal Composte

9 Fg. 2: Mcrostructures and perodc mcrostructures for maxmum dampng n example 2 These facts show the tradeoff between stffness and dampng n the compostes and show that the mcrostructure desgn gves as large dampng as possble n the specfed drecton whle mantanng the stffness at the desred level. CONCLUSIONS Optmal desgn of mcrostructures of vscoelastc compostes s presented usng the nverse homogenzaton approach. The numercal examples show that the mprovements n dampng characterstcs of the vscoelastc compostes can be acheved by the nverse homogenzaton approach based on the gven materals nstead of desgnng new materals. Dampng capacty of a vscoelastc composte, whch s represented by the value of loss tangent at specfed frequences, s optmzed by desgnng the topology of the mcrostructure n the composte whle mantanng ts stffness wthn the specfed bounds. The desgned mcrostructures show the tradeoff between stffness and dampng n the vscoelastc compostes. More crtcal problem resdes n manufacturng of the mcrostructures. Advanced manufacturng technology should be developed so that the desgn and manufacturng of mcrostructures n vscoelastc compostes are guaranteed n the vewpont of costs and manufacturng abltes. Newly developed technologes such as mcromachnng and fabrcaton of molecular compostes by mcro-network may be the possble canddates. For example, works on fabrcatng molecular compostes at the molecular level s n progress now [8]. REFERENCES 1. Sgmund, O., Materals wth prescrbed constutve parameters: An nverse homogenzaton problem, Int. J. Solds. Struct. 31, 1994, pp Sgmund, O., Talorng materals wth prescrbed elastc propertes, Mechancs Of Materals 20, 1995, pp Sgmund, O., Desgn of materals wth extreme thermal expanson usng a threephase topology optmzaton method, J. Mech. Phys. Solds. 45, 1997, pp

10 4.,.M., Park, S.H. and oun, S.K., Calculaton of effectve modul of vscoelastc compostes wth perodc mcrostructures usng a homogenzaton method, Eleventh Internatonal Conference on Composte Materals, old Coast, Queensland, Australa, July 14-18, Sanchez-Palenca E., Non-Homogeneous Meda and Vbraton Theory, Lecture Notes n Physcs 127, Sprnger-Verlag, Berln, VMA Engneerng, Dot Users Manual, Verson 4.00, Vanderplaats, Mura & Assocates Inc., CA., Chrstensen, R.M., Theory of Vscoelastcty: An Introducton, 2nd Ed., Academc Press, New ork, Sperlng, L.H., Introducton to Physcal Polymer Scence, John Wley & Sons Inc., New ork, 1992.