THE IDENTIFICATION OF MATERIAL PARAMETERS IN NONLINEAR DEFORMATION MODELS OF METALLIC-PLASTIC CYLINDRICAL SHELLS UNDER PULSED LOADING

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1 Materals Physcs and Mechancs (2015) Receved: March 27, 2015 THE IDENTIFICATION OF MATERIAL PARAMETERS IN NONLINEAR DEFORMATION MODELS OF METALLIC-PLASTIC CYLINDRICAL SHELLS UNDER PULSED LOADING N.А. Abrosmov, N.А. Novosel tseva Research Insttute for Mechancs of Lobachevsky State Unversty of Nzhn Novgorod, Prospekt Gagarna (Gagarn Avenue) BLDG 6, Nzhny Novgorod, , Russa e-mal: Abstract. A method of dentfcaton of materal parameters n defnng relatons of elastc deformaton of composte materals s developed based on the mnmzaton of dsagreement between expermental data and the results of numercal analyss of the dynamc behavor of double-layered metallc-plastc cylndrcal shells under explosve loadng. 1. Introducton Any progress n desgnng modern engneerng structures s only possble wth the use novel composte materals. However, ther use s hampered by certan dffcultes connected wth the lack of comprehensve and relable data on deformatonal and strength characterstcs of such materals, requred for equppng mathematcal models used n strength analyses. One of the methods of determnng parameters of models of physcal correlatons of composte materals s drectly usng expermental nformaton obtaned from loadng structural elements made of the materals n queston. Such dentfcaton methods were used n a number of works for determnng effectve elastc characterstcs of composte materals, based on the results of computatonal-expermental analyss both for statcally [1] and dynamcally [2] loadng structural elements made of homogeneous composte materals. The present work s an extenson of these studes, amed at determnng elastc deformaton characterstcs of composte materals based on expermentally and theoretcally nvestgatng the dynamc behavor of nhomogeneous metallc-plastc structural elements under pulsed loadng. 2. Formulaton of the dentfcaton problem Let there exst a numercal soluton of an axsymmetrc problem of non-statonary deformaton of a metallc-plastc cylndrcal shell n the form of crcumferental and longtudnal deformatons on the outer surface of the shell as a functon of tme. And let there exst the correspondng deformaton tensograms obtaned n expermental tests. As the computed and expermental deformaton oscllograms are assumed to be mono-harmonc oscllatons t s possble to determne maxmal and mnmal values of computed expermental m e strans, as well as the correspondng tmes m t и m e and m t ( m 1, M ) they are reached at. In what follows, a parameterzed verson of the formulaton of the parameter dentfcaton problem for a model of elastc behavor of a composte materal of a doublelayered metallc-plastc shell s presented. The task s to fnd a complex (vector) of stffness characterstcs of defnng relatons of the composte elastc materal of the outer layer of a 2015, Insttute of Problems of Mechancal Engneerng

2 The dentfcaton of materal parameters n nonlnear deformaton models... T shell E E, E, E, G,, 11 33, for whch the numercal soluton of the problem of dynamc behavor of cylndrcal shells agrees best wth the expermental data. Here, E, G are elastcty modul,, are Posson coeffcents. As a result, a problem of mnmzaton of objectve mult-varable functon s obtaned n the form of a sum of mean-square dsagreements between the theoretcal and expermental values of the stran K M 2 m m m m 2 2 k 1 m1 1, (1) k C E e e e t t t where K s number of ponts for whch expermental nformaton about strans s known, e are maxmal values of crcumferental and longtudnal strans n the frst half-perod of vbratons, t s tme of the process. Thus, the problem n queston s reduced to a classcal problem of nonlnear programmng: to fnd the values of the vector components of controlled parameters T,,..., 1 2, to whch the mnmal value of objectve functon C E mn C E E e e e r corresponds n the regon of admssble values regon П E : e e e, j 1, r j j j D E : f E 1, E П belongng to search. The lmts of the search regon e j, e j are determned by condtons of stablty of the materal and expermental data [3]. Objectve functon (1) s computed by numercally solvng the drect problem of dynamc deformaton of a metallc-plastc cylndrcal shell under explosve loadng. The metallc-plastc shell s supposed to be formed by cross-sprally wndng a undrectonal composte materal around a metallc case. The cnematc model of deformaton of a multlayered package s based on a non-classcal theory of shells. To ths end, the components of the dsplacement vector are approxmated by fnte seres across the thckness of the multlayered package [4, 5]. The formulaton of geometrc correlatons s based on the relatons of the smplest quadratc verson of the nonlnear elastcty theory n curvlnear coordnates [5]. The correlaton between stress and stran tensors n the composte layer s determned usng Hook s law for orthotropc bodes n combnaton wth the effectve modul theory 67 3 Cje ( 1,3), Ge, (2) jj j1 where Cj, G are effectve stffness characterstcs expressed usng components of vector E. The defnng relatons for the sotropc metallc layer are formulated usng the dfferental theory of plastcty wth lnear hardenng [5]. An energy-matched resolvng system of equatons of moton of the shell s constructed usng the vrtual dsplacement prncple [5]. 3. The method of solvng the dentfcaton problem The numercal soluton of the formulated dentfcaton problem s subdvded nto three stages: 1) solvng the ntal boundary-value problem of nonlnear deformaton of metallcplastc cylndrcal shells under pulsed loadng, 2) analyzng the senstvty of the objectve functon for the sought parameters (desgn varables) 3) seekng a global mnmum of the objectve functon. The frst stage of solvng the dentfcaton problem s based on the explct varatonaldfference scheme [5].

3 68 N.А. Abrosmov, N.А. Novosel tseva At the second stage of solvng the dentfcaton problem, senstvty of the objectve functon s analyzed for the desgn varables to assess the possblty of fndng the sought parameters of the defnng relatons n ths problem. To ths end, the theory of global senstvty ndcators s appled, used n studyng nonlnear mathematcal models [6]. In what follows, one-dmensonal senstvty ndcators S 1, r are used whch make t possble to rank varables e : the bgger S, the more substantal s the effect of varable e. At the thrd stage, to solve the problem of fndng a global mnmum of objectve functon (1), the evolutonary probablstc global optmzaton method s used based on the genetc algorthm that conssts n subsequently matchng, combnng and varyng the sought parameters, usng mechansms smlar to bologcal evoluton [7]. 4. Testng the method and the results of solvng the dentfcaton problem In the benchmark problem, the results of solvng a drect problem of dynamc deformaton of a metallc-plastc cylndrcal shell wth assgned parameters of the model of the defnng relatons of the materals was used as expermental nformaton. Fg. 1. Oscllograms of the crcumferental strans on the outer surface of the cylndrcal shell. The results of solvng a drect problem of elastc deformaton of a cylndrcal shell wth the radus of R =10 сm, thckness H h1 h2, h 1 =0,175 сm, h 2 =1,6 сm ( h 1 s thckness of the metallc layer, h 2 s thckness of the composte layer) and the length of L =40 сm, loaded by a pulse of pressure resultng from explodng n ts geometrcal centre an explosve charge of mass m =0,062 kg, are presented n Fg. 1 as tme hstores of the crcumferental strans n sx ponts dsplaced relatve to the symmetry plane of the shell by a value = 0; 4,2; 8,3; 12,5; 16,7; 20 cm, respectvely. The materal parameters of the physcal correlatons of the shell were as follows: composte layer E 11= E 33 =19 GPа; E =33 GPа; G =3,69 GPа; =0,427; = 0,4; =1900 кg/m 3 ; sotropc layer E =210 GPа; =0,3, =0,35 GPа; g =0,5 GPа; =7800 кg/m 3. Based on the above relatons, an objectve functon was constructed for solvng dentfcaton problems of an elastc model of the behavor of the composte layer of the shell under the followng constrants for the sought parameters: 10GPa E11 25GPa ; 25GPa E 50GPa ; 10GPa E33 25GPa ; 2,5GPa G 5GPa ; 0, 2 0,5 ; 0, 2 0,5. To control the stablty of the soluton of the dentfcaton problem aganst possble naccuraces n determnng the ntal oscllograms, alongsde wth determnstc relatons presented n Fg. 1, ther nosy analogues were used, obtaned wth the account of random 5 %-scatter of the characterstc values of the ntal oscllograms. The materal parameters of defnng relatons of elastc deformaton of the composte layer of the metallc-plastc shell, as well as ther senstvty ndcators, obtaned by solvng

4 The dentfcaton of materal parameters n nonlnear deformaton models... the dentfcaton problems are shown n Table 1 ( S are total one-dmensonal senstvty ndcators; e e e ( ) / 100% are devatons of dentfed parameters e of the model from preset values e ). It follows from Table 1 that the crcumferental elastcty modulus s the most meanngful parameter of the elastc model. The senstve parameters of the model are determned wth a hgher accuracy as compared wth the less senstve ones. Inaccuracy n assgnng ntal oscllograms has no consderable effect on the results of determnng the senstve parameters of the model of defnng relatons. Table 1. One-dmensonal ndcators of senstvty and devatons of the dentfed parameters of elastc deformaton of the composte layer. Model S, %, % parameter DО NО DО NО E 0,30 0,37 0,46 0,53 11 E 99,10 98,99 1,66 1,82 E 0,17 0,18 28,44 30,55 33 G 0,01 0,01 4,04 28,92 0,09 0,01 0,19 0,47 0,03 0,05 1,91 2,61 DО determnstc oscllogram. NО nosy oscllogram. Further on, the problem of dentfcaton based on the results of expermentally studyng the dynamc behavor of metallc-plastc cylndrcal shells under local pulsed loadng was consdered [8]. Maxmal values of the crcumferental stran n the central cross-secton of the shell and the tme to reach t, as well as the radal oscllaton perod, gven n [8] were used as ntal data. The shell geometry and loadng condtons are the same as n the above consdered example. The results of numercal computaton wth the dentfed materal parameters (denomnator) and of expermental data (numerator) for maxmal values of crcumferental stran e n the central cross-secton of the shell on ts outer surface and for radal oscllaton perod T are as follows: 1,10 e, 1, T. 204 One can see that the maxmal naccuracy for the oscllaton ampltude s 3,6 %, and for the oscllaton perod 9 %. It s to be noted that naccuracy n the expermental measurements amounted to 10 % [8]. 5. Conclusons A computatonal-expermental method s presented that makes t possble, wth a requred accuracy, to determne materal parameters of defnng correlatons of elastc deformaton of composte materals and to adequately descrbe the dynamc behavor of metallc-plastc cylndrcal shells. Acknowledgements The authors express ther thanks for the fnancal support from Russan Scence Foundaton, project No /

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