DETERMINATION OF COMPOSITE EFFECTIVE CHARACTERISTICS AND COMPLEX MODULES FOR SHELL STRUCTURES FROM THE SOLUTION OF INVERSE PROBLEM

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1 DETERMINATION OF COMPOSITE EFFECTIVE CHARACTERISTICS AND COMPLEX MODULES FOR SHELL STRUCTURES FROM THE SOLUTION OF INVERSE PROBLEM V. Matveyenko, N. Yurlova, D. Grachev Insttute of Contnuous Meda mechancs UB RAS Acad. Korolev St., Perm 6403 Russa SUMMARY: In ths paper a numercal - expermental method s proposed to dentfy the effectve mechancal constants for a composte materal n shell structures. As expermental data, the results obtaned by analyzng the stran state of a materal structure under dfferent statc loadng condtons or the spectrum of natural frequences and vbraton modes are used. Numercal part of the method s based on the soluton to the nverse elastcty problem nvolvng the estmaton of the model parameters to descrbe the stran state recorded expermentally. Two models - of elastc ansotropc body subject to statc loadng and wth complex dynamc modul n the case of vbratons - are consdered. In ths treatment, concurrent wth the soluton to the nverse problem, the senstvty analyss technques are appled to choose, for example, the requred experments, and the wanted data from a partcular experment. The potental of the method proposed s llustrated numercally. KEYWORDS: mechancal characterstcs of composte, nverse problem, shells INTRODUCTION A dstngushng feature of fber- renforced compostes s smultaneous producton of the materal and structure by contnuous flament framework wndng followed by soldfcaton of a polymer matrx. As a result, the materal structure s able to vary from pont to pont whch mples the dependence of the structure materal propertes on the manufacturng technology, the sze of an artcle, the dsorentaton of fbers, the bendng and tenson of a framework, and the ntal mcro-and macroscopc stresses. Therefore, the standard test data on the specmens wth a fxed structure or cut from an artcle workpece may provde an nadequate assessment of the materal behavor wthn a structure. One way to defne the effectve mechancal characterstcs wthout recourse to a reference specmen s to use an expermental evdence for shells under load. Complex studes n these composte structures requre the development of structure-senstve technques of determnaton and control of the materal behavor. For ths reason, much attenton s gven today to the dentfcaton methods (to the soluton of the nverse problem) allowng the refnement of the materal characterstcs for a partcular structure under calculaton and the predcton of ts behavor n operatng regmes.

2 In ths work, we develop further the above-lsted nvestgatons by constructng the algorthm for shells of complex geometry under dfferent boundary condtons. The observaton s supported by the data on dfferent realstc statc loadng and on the resonance regmes (the spectrum of natural frequences and vbraton modes). The algorthm should be capable to estmate the test data n the context of ther nformatve value needed to fnd the requred mechancal characterstcs. Mathematcal problem formulaton We use here the concept of the nverse problem suggestng that the equaton coeffcents are determned by the vector ( ) p = p p2 pq a a, a,!, a Τ, () where q s the number of the parameters to be found, n partcular, the mechancal characterstcs whch provde the best agreement of the mathematcal model for the structure stran state wth the expermental observatons. Assume that, for the structure under T h load, we have the data on the dsplacement u k, k strans ε j and resonance frequences ω n ( n =, 2,... ). Here, the superscrpts and k desgnate the expermental results at T h loadng For numercal mplementaton, a parameterzed formulaton of the nverse problem s suggested [, 2], where the vector of coeffcents of the state equatons a 0, provdng wthn rk rk r the assumed norm a mnmal dstance between the calculated u, ε, ω and expermental u k k, ε j, n ω results, must be found. As a norm, we take the functonal beng the sum of mean K 3 ( p) k α ( k rk ) Fa = u u + k = = V ( rk 2 N ) dv + n( n n r ) βj k εj k εj γ ω ω = j= n= j n 2, (2) Where N s the number of natural frequences known from experments, and α k, βj k, γn are the weght coeffcents. So, we arrve at the functonal mnmum problem Eqn.(2): ( ) F( a p ) F a 0 = mn. (3) provded that the vectors a p are n the range of admssble values. As a rule, the expermental data on the dsplacement and deformaton felds may be obtaned at separate ponts of the regon V occuped by the body consdered. In ths case, functonal Eqn.(2) becomes the functon of several varables:

3 [ ] K M 3 k k rk ( p) = α ( m) ( m) Fa u x u x k = m= = N ( n n r ) + γ ω ω n= n 2 + rk [ ( x ) ( xl )] L 3 3 βj k ε j k 2 l ε l = = j= j + 2, (4) and the functonal mnmum problem reduces to searchng for the mnmum of the functon of several varables. Here, ( x m ) and ( x l ) are the ponts for whch the expermental data on dsplacements and deformatons are known. In the general case, orented composte materals are ansotropc nonlnear vscoelastc materals whch, are, however, mostly exploted at temperature below the glass transton temperature. Therefore, n the frst approxmaton these materals can be consdered elastc solds, partcularly, at small strans, and ther mechancal behavor can be descrbed by the model of ansotropc elastc body wth elastc constants σ = C ε (5) j jkl kl The vscoelastc materal behavour can be descrbed based on the model of vscoelastc body whch, for sotropc case, s t σj σδj = 2G0 εj ϑ δj R( t τ) εj ( τ) ϑ ( τ) δj dτ, t σ K0 ϑ T t τ ϑτ dτ. (6) 0 where = ( ) ( ) When the body s subjected to dynamc loadng, the use can be made of the materal model, where the dsspatve propertes are descrbed by the complex dynamc modul. For sotropc body, the physcal equatons have the form σ = Kε δ + 2G ε ϑ δ 3 j j j j j, (7) Where G, K are the complex modul, defned through relaxaton kernels R and T [3]. Numercal realzaton methods of ths nverse problem are based on the data obtaned n solvng the drect statc or dynamc deformaton problems whch nvolve the followng three specal problems [3]:. Natural vbratons of elastc bodes. Under unform boundary condtons, we fnd the soluton of the form: ( ) ( ) u x, t =ξ x cos ω t, (8) Where ω s the natural frequency, and ξ ( x ) s the vbraton egenform.

4 2. Natural dampng vbratons of vscoelastc bodes. Dampng vbratons of a vscoelastc body can be represented as or n a complex form [ R R ] ωi t c s ( ) = () + () u x, t e u x cosω t u x sn ω t (9) (, ) = () u x t u x e ω t (0) Where u () x are the complex components for the dsplacement vector ampltudes, ω = ωr + ωi s the complex frequency wth the actual part beng the frequency and wth the magne part beng the dampng rato of natural dampng vbratons. In ths case, under unform boundary condtons, we fnd the soluton of the form: ( ) = ξ () u x, t x e, ω t () Where ξ () x s the complex egenform for shell vbratons. 3. Steady-state forced vbratons. We fnd here a perodc n tme moton wth the perod equal the perod of external acton: ( ) = () u x, t u x e., (2) pt In the analyss of shell structures, we use the relatons from the momentum theory of shells based on the Krchgoff-Love hypothess [4]. Our study s concentrated on the shells of revoluton. A numercal analyss of represents a FEM sem-analytcal varant [5], accordng to whch the components of dsplacement and loadng vectors, and the stress and stran tensors are gven n Fourer seres along the crcumferental coordnate. As fnte elements, the element suggested by Zyenkevch O. has been used beng the frustum of a cone wth the lnear approxmaton (axal and crcumferental) and cubc approxmaton (normal) components of the dsplacement vector for each harmonc of the Fourer seres. The formulaton proposed reduces to classcal problem of nonlnear mathematcal programmng suggestng the mnmzaton of functon Eqn.(4) subject to constrants n the form of equaltes and nequaltes. The last defne the range of admssble values of the vector a p and result from the known constrants mposed on the ansotropc materal parameters, see for example [6]. In decdng on a partcular method to solve the problem on nonlnear mathematcal programmng (the optmzaton method), we take nto account, among other factors, the senstvty of the optmzaton methods to hndrance (for example, to measurement errors). The methods, appled to construct the non-local approxmaton of the object functon by ts values at a number of ponts (of smplex search type or of barycentrc coordnate method type), proved to be least senstve to measurement errors.

5 To numercally mplement the optmzaton problem wth constrants, we use the smplex method of a sldng admttance based on the Nelder-Md method [7]. To predct an nformatve value of the expermental data used to dentfy the effectve materal constants and to estmate the possblty of defnng any materal constants from the same experment by changng the form of the objectve functon (for example, on nsertng the weght ratos), the senstvty analyss technques has been used [2, 8]. Results Determnaton of elastc constants n terms of statc deformaton data for shells Numercal calculatons have been done for cylndrcal, cone and semsphercal orthotropc shells under dfferent boundary condtons on the shell edges. Statc loadng has been produced by twstng or tensle forces, and by the nternal pressure. In the numercal nvestgatons made, nstead of the full-scale experments we have normally used the numercal and analytcal results obtaned at the prescrbed values of mechancal characterstcs. The calculated dsplacements and strans have smulated the measurement data from the approprate experment whereas the values of mechancal characterstcs have served as a relablty crteron for the nverse problem. The calculaton procedure nvolves the prelmnary analyss of the senstvty coeffcents. As would be expected, when the consdered shells are loaded by twstng forces, only the senstvty vector component Ψ G dffers from zero. Therefore, n solvng the nverse 2 problem such data have to provde the way of fndng only the modulus G 2, as has been calculated. For the analyss, t s more convenent to use a normalzed senstvty vector. Calculatons have been made for dfferent values of the ntal approach n the problem on the functon mnmum (4). Consderaton has been gven to the followng patterns of shell loadng: a) one shell end s fxed and the other s acted upon by the tensle force; b) the shell s subjected to nternal pressure, and no stresses are observed on ts both edges; c) the shell s subjected to nternal pressure, and ts both edges are fxed. By comparng the senstvty coeffcents, we can evaluate the possblty of searchng for the correspondng elastc constants. For example, the senstvty coeffcent l 2, obtaned when the shell was loaded accordng to the pattern a, s small compared to the other coeffcents. From the approprate calculatons followed that when the search for the elastc constants was made based on the shell tensle data, the values E and ν 2 could be found suffcently accurate, and the value E 2 was found provded that a proper choce of the ntal approach would be gven. In ths problem, the senstvty analyss technques can be appled to fnd the most nformatve experments to dentfy elastc constants. Consder as an example the shell wth free stress ends subjected nternal pressure. When ths shell s renforced at ts mdpart by the external rgd rng, the senstvty coeffcents prove to be more commensurable.

6 Usng the spectrum of natural frequency values, we fnd those frequences by whch the approprate elastc constants can be determned n terms of the analyss of senstvty coeffcents. Numercal calculatons lead us to the followng conclusons. The expermental data on twstng, extenson and nternal pressure loadng allows us wth the ad of the approach proposed to defne the elastc characterstcs for the orthotropc shells of complex confguraton. The results on shell extenson can be used not only n solvng the optmzaton problem but n checkng calculatons. To dentfy the mechancal characterstcs, whch vary lnearly along the length of the shell, a set of the expermental data on twstng and nternal pressure loadng of the shell renforced wth a rgd rng frame can be used. To fnd the elastc mechancal characterstcs n terms of the natural frequency spectrums, we requre the nformaton about one of torsonal vbraton modes and one or two frequences of untorsonal vbraton modes. The last were taken by estmatng the senstvty coeffcents. Selected nformaton allows to determnate components of complex dynamc modules. From the results on modelng the expermental error follows that the data on dsplacements allow one to fnd the elastc modul wth an error commensurable wth the expermental error, and that for Posson's ratos s, on the average, twce as large. When the stran values are used as the expermental data, the error arsng n defnng the elastc constants s less than the measurement error. To llustrate the above, let us consder the problem of defnng the effectve elastc constants for a multlayer cylndrcal shell obtaned by contnuous wndng. In [9], the expermental data for a shell subjected to nternal pressure are gven. One end of the shell s fxed, and the other s under tensle force. Two varants of tensle forces were mplemented expermentally: T = pr 2 and T = pr ( R 0 ) 2. The second corresponds to the case when the nonfxed end cap has the hole of radus R 0. The shell geometrcal parameters are: h = 0.23 mm, R = 00 mm, L = 350 mm, R 0 = 25 mm, where h s the thckness of a sngle layer. The shell s composed of 28 layers, of whch crcular 6 and spral 6. The wndng angle s The shell s subjected by nternal pressure, p = 0.85 MPa. The dsplacements u r and u ϕ were measured n the four cross-sectons. To measure dsplacements, 6 gauges were mounted on each of these cross-sectons. The nverse problem was solved based on the data obtaned at the four ponts along the structure length as the arthmetc mean of the gauge records for the secton consdered. The elastc characterstcs gven n ths work for the undrectonal layer ( E = 7.0 MPa, E 2 = MPa, ν 2 = 0.23, G 2 = 0.96 MPa ) allow the calculaton of the effectve elastc characterstcs for the mult-layer shell materal by the formulae commonly used to calculate

7 the mean (effectve) elastc characterstcs of an arbtrarly renforced composte, see for example [0]. The dffer from the values found by solvng the nverse problem s no more than 2 %. Table : Effectve elastc characterstcs E MPa E 2 MPa ν 2 G MPa calculated from the values of parameters 2, , ,3 0,7 0 4 of undrectonal layer found by solvng nverse problem, , ,27 0, Determnaton of complex dynamc modul n terms of shell vbraton data We defne here the complex dynamc modulus components n the framework of the approach proposed. The body model wth the complex dynamc modulus, allows takng nto account the materal dampng propertes whch can manfest themselves n dfferent ways dependng on the moton regme. Under forced vbratons, the dsspatve propertes show themselves n the values of fnte ampltudes (dsplacements, deformatons, and stresses) at the resonance frequences. The dampng propertes can be also observed when the free vbratons attenuate n a fnte tme owng to dampng. To smulate the dampng vbraton rate, nstead of classcal problem wth ntal condtons on free vbratons we suggest a new mechancal spectral problem on natural vbratons n vscoelastc bodes Ths problem seems to be more effcent for numercal mplementaton of the nverse problems. The frst varant of defnng the dynamcal modul relates to the use of the expermental data on the ampltude-frequency characterstcs of dsplacements or deformatons at some shell ponts. The above method s verfed based on the numercal results (taken as expermental data) from the soluton to the drect problem. The role of the last s played by the problem on steady-state forced vbratons n vsco-elastc bodes. Consder a shell wth the geometrcal dmensons L / R = 2, h / R = Assume that the mechancal behavor of the materal s descrbed by the relatons from a lnear heredtary theory, and no rheologcal propertes show themselves under volume deformaton. The objectve functon can be wrtten as M N 3 p [ j ( m) j( m) ] F = u z u z m= j= = 2 (3) p Here u ( z ), u ( z ) j m j m are the calculated and expermental values of dsplacements at z m ponts of the shell at the frst N resonance frequences

8 Three varants are consdered to mplement the forced steady-state vbratons of the shell. Fgure presents the schemes, whch llustrate the boundary dsturbng dsplacements, changng n vew of the harmonc law wth the ampltude A 0 u u u a b c Fg.: Schematcal representaton of exctng dsplacements appled to edges Ampltude-frequency curves (AFC) for shell dsplacements and deformatons result from the soluton to the drect problems smulatng the test on a vbraton machne. For calculaton, the followng model materal parameters have been gven: G R / G M = 0.2, K / G R = 24.7, ρ =.0. Moreover, two types of polyurethane have been consdered: G R = kgs / cm 2, G M = kgs / cm 2, K = kgs / cm 2, ρ = kgs / cm 2 ; and G R = kgs / sm 2, G M = 84.3 kgs / sm 2, K = kgs / sm 2, ρ = kgs / sm 2. From numercal results t follows that the volume modulus and the complex dynamc modulus components can be found wth an accuracy of 0.0 and, for ths, we only requre the nformaton on ampltude values of dsplacement or deformaton of one or two frequences at a sngle harmonc (possbly, non-resonance). Suppose that we have the test results on tme varaton of deformaton and dsplacement produced by the ntal mpulse at dfferent structure ponts. These data have been treated to dstngush from the general pcture the harmonc components whch, by vrtue of dampng, demonstrate attenuaton. When the dampng s descrbed by the complex dynamc modulus, every approprate egenform can be wrtten as: (, ) = ( ) = ( ) ω t ω t ω t R uxt uxe uxe e I (4) The problem on dampng natural vbratons n the context of the consdered approach appled to solvng the nverse problems of materal mechancal characterstcs can be consdered when the objectve functon has the form: F N = ( p I I ) ( p R R ) = ω ω + ω ω 2 2 (5) Calculatons have been made based on the expermental data obtaned by solvng the drect problem the role of whch s played here by the problem on dampng natural vbratons of a cylndrcal vscoelastc shell wth one fxed and the other free ends. Dmensons and

9 propertes for the shell materal are: L = 0.6 cm, R =.65 cm, h = 0.2 cm, G R = kgs / cm 2, G M = kgs / cm 2, K = kgs / cm 2, ρ = kgs / cm 2. Complex natural frequences (sec - ) of the shell vbratons are shown n Table 2. Table. Complex natural frequences of the shell harmonc frequency number number 0 ω I ω R ω I ω R The obtaned complex vbraton frequences have smulated the expermental values ω R and ω I. Numercal results showed that the volume modulus and the complex dynamc modulus components can be found wth an accuracy of 0.0 and, for ths, we only need the nformaton on two natural frequency at a sngle harmonc. The dependence of volumetrc modulus and components of shear dynamc modul was assumed havng suffcently complex arbtrary form. CONCLUSIONS The approach proposed opens the new possbltes for dentfcaton of complex dynamc modules of composte materals of shell structures manufactured by contnuous flament framework wndng. REFERENCES Matveyenko V. P., Yurlova N. A., An nverse problem for dentfyng mechancal characterstcs of composte materals, Proceedngs of the 3RD nternatonal conference on Modern practce n stress and vbraton analyss, Dubln, Ireland, 997, pр Matveyenko V. P., Yurlova N. A., Identfcaton of effectve elastc constants of composte shells on the bass of statc and dynamc experments., MTT, No.3, 998, pp Matveyenko V. P., Klgman E. P., Numercal Vbraton Problem of Vscoelastc Solds as Appled to Optmzaton of Dsspatve Propertes of Constructons, Int.J. of Vbr. and Control, No.2, 997, pр Bderman V.L. Mechancs of thnshelled structures. Moscow., Mashnostroyene, 977, 488р. 5. Zenkevch O., FEM n engneerng. Moscow., Mr., 975, 542p. 6. Abramchuk S.S., Buldakov V.P., Admssble Values of Posson's rato for ansotropc materals. Mech. Comp. Mater., No. 2, 979, pp

10 7. Hmmelblau D. Appled nonlnear programmng, Moscow, Mr., 975, 534p. 8. Houg E.G., Cho K., Komkov V. Senstvty analyss technques n structure desgn, Moscow, Mr, 988, 428p. 9. Protasov V.D., Fllpenko A.A. Momentless lamnate cylndrc shells wth varable elastc parameters obtaned by contnuous wndng, Mech. Comp. Mater., No.3, 984, pp Obraztsov I.F., Vaslyev V.V., Bunakov V.A. Optmal renforcement of composte revoluton shells, Moscow., Mashnostroyene, 977, 44p.