Proceedngs of the EM Annual Conference June 1-4, 009 Albuquerque New Meco UA 009 ocet for Epermental Mechancs Inc. Hperelastc Materals Characterzaton b Planar Tenson Tests and Full-feld tran Measurement G. Palmer*, G. Chappn, M. asso,. Papaln Department of Mechancs, Poltechnc Unverst of Marche va Brecce Banche, 60131, Ancona Ital * g.palmer@unvpm.t ABTRACT everal technques for full-feld stran measurement are nowadas avalable and relable, and the topcs of research should be to fnd well suted methods and procedures to use the great amount of data n an approprate and sharp wa. In ths terms, one of the best and effectve technques developed n recent ears s the Vrtual Feld Method (VFM). In ths paper a Dgtal Image Correlaton technque s appled to acqure stran felds on flat rubber specmens wth dfferent aspect ratos. The are loaded wth a standard tracton machne and the dfferent tpes of geometr generate heterogeneous tensons dstrbutons (from unaal up to more and more baal). A recursve numercal procedure s proposed n order to fnd the best ft materal dependent parameters of a partcular hperelastc model. Global load data, deformaton data of the surface of the specmen and stress dstrbuton data obtaned startng from a guess set of consttutve law parameters are used to calculate the terms and coeffcents of the VFM equatons. Two ndependent vrtual felds are nvolved n the procedure to use as man as possble avalable data. After a recursve process where the global error defned from VFM equatons s mnmzed and combnng together data from all specmen geometres, t s possble to obtan a set of materal parameters able to reproduce a heterogeneous planar tensonal state. INTRODUCTION Most known and used mechancal tests on elastomerc materals (more n general for hperelastc materals) are unaal tenson/compresson tests, equbaal tenson tests (e.g. bulge test or multaal tenson machnes), planar tenson tests (pure shear) [1-4]. The am of ths work s the development of a procedure of nverse characterzaton of the materal from epermental data referred to planar tenson tests wth specmen of dfferent aspect rato. In order to obtan dfferent tensonal states, from unaal to more and more baal, three dfferent geometres for specmens have been conceved (Fg 1). tartng from geometr a) that gves appromatel a perfect unaal dstrbuton of stress, the characterstc rato h/w decreases respectvel n b) and c) wth the effect of contrast of the lateral contracton of the specmen. Therefore transversal stress ncreases gvng a heterogeneous stress dstrbuton on the specmen. Even f a perfect equbaal tenson s not possble to reach, the focus of authors was to obtan a stress dstrbuton as much heterogeneous as possble wthout usng comple testng apparata lke multaal machnes or bulge test rgs, but onl b means of a standard unaal tensle machne. Fg 1 Aspect ratos of specmens for planar tenson tests: a) h=60 mm, w=15 mm, h/w=4; b) h=60 mm, w=60 mm, h/w=1; c) h=15 mm, w=60 mm, h/w=0.5. For all geometres thckness s t=1.7 mm.
Full feld measurement of deformaton has been performed b a DIC sstem developed b authors; the algorthm for mage correlaton s based on [5], whch s an approach nspred to the fnte element theor. The classcal method for nverse problems s the so called model updatng, especall based on FEM numercal models. Results of eperments, e.g. n terms of load-dsplacement data, are compared wht results obtaned wth the FEM model of the eperment, usng an approprate consttutve model of the materal. If numercal and epermental data are not comparable, materal dependent parameters of the consttutve law are teratvel vared untl the best set of parameters s reached. Even f ths approach to nverse problems s wdel known and adopted, the FEM model updatng shows some mportant drawbacks. At frst t usuall needs a great quantt of computaton, that means tme; nverse problems on non-lnear materals usng a standard PC and FEM model updatng ma requre weeks of computaton. The other dsadvantage s about the assumptons made about loads dstrbuton, constrants, boundar condtons, geometr, etc. n the FEM model. Usuall these assumptons are rough because of the lack of nformaton and the makes the model not full fathful to the realt. In order to avod ths knd of problems and also wth the am of usng epermental data n the most effcent wa, a novel technque known as Vrtual Felds Method (VFM) [6] has been developed n recent ears, and ts applcatons nvolve nowadas the characterzaton of a large varet of materals, from homogeneous to orthotropc and ansotropc [7] (e.g. compostes), from lnear to non-lnear (e.g. elasto-plastct [8], vsco-plastct [9], hperelastct [3]). In ths paper authors propose the applcaton of VFM to characterze the well known Ogden hperelastc consttutve model [10] usng the results of the planar tenson tests. TET The test rg for planar tenson s shown n Fg. A flat specmen of rubber s glued to L shape brackets, leavng accessble to the vew of the camera a regon where the speckle pattern s appled. Ths regon, where the stran feld wll be measured, s the onl one that wll be subjected to deformaton and ts dmensons agree respectvel wth the three geometres of Fg 1. The use of the brackets s to avod the sldng of the specmen between the clamps, and also to constran the upper and lower sde of the regon of nterest n the drecton transversal to the load. Durng the test load and dsplacement are measured respectvel b a 5 kn load cell and b the cross head of the tensle machne. A CMO camera (Pelnk B471F) wth a resoluton of 180104 s used to grab frames requred for stran feld measurement b the DIC technque. Fg Test rg for planar tenson tests: specmen, tensle machne and vson sstem. tran maps of the three dfferent specmens are shown n Fg 3; prncpal deformaton on plane are gven at three dfferent values of load. As t s possble to see the stran feld becomes more and more planar gong from geometr a) to geometr c). In the last one the transversal prncpal stran s near to zero n all the mddle part of the specmen
Fg 3 tran maps of the three dfferent specmen. Prncpal strans on plane are plotted for three dfferent values of load. VFM BAIC EQUATION The VFM s based on the prncple of vrtual feld, that can be epressed as follows: σ dv + V T u d + f u dv V (1) V where σ, T, f are respectvel the nternal stresses, the eternal forces and the nternal forces of the bod of volume V and surface V, whle u* s a knematcall admssble vrtual feld of dsplacements and * s the assocated vrtual feld of strans. Applng the followng hpothess: a. no nternal forces are actng on the sold (f=0), b. the stress dstrbuton on the surface of the specmen s the same n all sectons trough the thckness (plane stress), c. stresses can be epressed as functons of strans and materal dependent parameters X (σ = g(,x)),
equaton (1) reduces to: t g X d + t Tu dl = (, ) 0 () where t s the thckness. If the consttutve model g(,x) s non-lnear, consderng equaton () at dfferent tme steps t s possble to obtan a seres of lnearl ndependent equatons where X are unknowns, even f onl one vrtual feld s consdered. Nevertheless n ths work two dfferent vrtual felds have been nvolved n the procedure of materal parameters estmaton, n order to use both strans along load drecton an transversal drecton. More n detals the frst vrtual feld s that one represented n Fg 4a; for the reason that onl deformatons of the central part of the area covered wth speckle are measured, the surface of the specmen s dvded n three. 1 and 3, where strans are unknown, are consdered rgd n the vrtual feld, whle s subjected to dsplacements. The feld of vrtual dsplacements can be epressed lke: u u u 1 u = 3 u u = h (3) that leads to the followng assocated vrtual deformatons: 1 = 1 3 (4) For ths partcular feld of vrtual dsplacements t s eas to note that the second term of () reduces to P h flterng an assumpton about the real load dstrbuton: g (, X ) d P h ( vf 1) = t (5) P 3 w w P h3 3 h3 3 3 h h h1 1 1 h1 1 1 a) b) Fg 4 a) Frst vrtual feld, b) second vrtual feld. The second vrtual feld s that represented n Fg 4b, where 1 and 3 are agan rgd regons and s subjected to transversal dsplacements. Analtcall t can be epressed lke: u u = ( h ) u 1 3 u u u (6) and the assocated vrtual deformatons are:
= ( h ) 1 3 = ( 1 h ) (7) In ths case the load P s not subjected to an vrtual dsplacement, and the second term of () s zero: ( vf ) g (, X ) d (8) Among the consttutve models avalable n lterature for hperelastct, n ths paper we focused on Ogden [10] model that s one of the most relable for large deformatons. Lke ever model for hperelastct t s based on the defnton of a partcular stran energ functon W. For an sotropc and ncompressble materal W can be epressed as a functon of the n plane prncpal stretches: (, ) W = W λ λ 1 (9) from whch t s possble to calculate the true stresses wth the followng epressons: W σ1 = λ1 λ 1 W σ = λ λ (10) Beng the thckness much smaller than the other two dmensons t s possble to sa that σ 3. The partcular stran energ functon proposed b Ogden, accordng to the notaton proposed nsde the FEM code Abaqus, s: W N α (, ) ( 3 ) p α p α p α λ λ µ λ λ λ = + + λ p / α 1 p 1 1 p = 1 p (11) where, choosng a value of N=3, µ p and α p are s materal dependent parameters and represents the unknowns of our problem. CHARACTERIZATION PROCEDURE AND REULT To obtan a set of parameters of the Ogden model (N=3) able to descrbe the mechancal behavor of the rubber under a general planar tensonal state, a Matlab procedure has been conceved n order to mnmze the dscrepanc between epermental and numercal, usng equatons of VFM descrbed n the prevous secton. More n detals, equatons (5) and (8) are coupled to obtan the objectve functon to mnmze: ( vf 1) P h ( vf ) g (, X ) d g (, X ) d t OBJ = + ( vf ) g geom a) b ) c ) n steptme ( vf 1) P h = = g (, g ) (, X guess ) d X guess d t n g (1) In ths functon the errors obtaned wth a set of parameters X are dvded b the errors obtaned at the begnnng of the procedure wth the guess soluton X guess n order to obtan two normalzed errors for the two vrtual felds; these errors are both equal to one at the begnnng of the mnmzaton. The guess soluton s chosen wth the onl requrement of stablt for unaal and baal tenson. tartng from ths values of the parameters, that gve a global normalzed error equal to, the teratve procedure s able to reduce the error to 0.6, etng wth the values wrtten n Tab1.
Guess parameters Optmzed parameters µ 1 α 1 µ α µ 3 α 3 Normalzed error vf1 Normalzed error vf 1-1 -1-1 1 1 1 1-1.394 1.6 0.331-3.885.1 1.306 0.138 0.17 0.60 OBJ Tab 1 Values of Ogden parameters of guess and optmzed soluton Values of µ are n MPa, other values are dmensonless. The best fttngs obtaned usng the optmzed parameters of Tab 1 are plotted n Fg 5. In ths graphs epermental data are compared wth the Ogden numercal model respectvel for specmen geometres a), b) and c) and t s possble to see a good agreement for all the cases. 70 60 Numercal model Epermental data 50 Numercal model Epermental data 50 00 Force [N] 40 30 Force [N] 150 100 0 10 50 0 0 Test Tme Test Tme a) b) 50 Numercal model Epermental data 00 Force [N] 150 100 50 0 Test Tme c) Fg 5 Comparson between numercal an epermental values of force respectvel for geometres a), b), c). CONCLUION In ths paper authors focused on the development of a sharp procedure able to use data from full feld stran measurement n the most effcent wa. Whle for full feld measurement a tpcal DIC sstem s used, the Vrtual Felds Method seems to be a ver effectve technque to process the huge quantt of data n a smple and accurate wa, flterng out problems due to unknown loads dstrbutons and boundar condtons. Usng smultaneousl two dfferent vrtual felds n order to nvolve n the procedure data of both longtudnal and
transversal strans, t s possble to defne a mnmzaton problem that leads to an optmzed soluton of a partcular hperelastc model, Ogden n ths case. Results show a good agreement between the numercal model and the epermental data for all geometres of the specmen. The am of ths work was also to obtan a tensonal state as much heterogeneous as possble usng flat specmens and a standard tensle machne. Even f tensonal states lke the equbaal are far from those presented n ths paper, an accurate and general model has been obtaned, wth nstablt problems onl at hgh levels of deformaton n equbaal tenson. Further developments wll regard the stud of new geometres and clampng sstems of the specmen, n order to reach hgher levels of transversal stress and a bgger heterogenet of tensons. Further on, the procedure proposed n ths paper wll be adapted to other hperelastc consttutve models, e.g. Moone-Rvln and polnomal. REFERENCE [1] Pearson I., Pckerng M. The determnaton of a hghl elastc adhesve s materal propertes and ther representaton n fnte element analss. Fnte elements n analss and desgn 37 (001) 1-3. [] asso M., Palmer G., Chappn G., Amodo D. Characterzaton of hperelastc rubber-lke materals b baal and unaal stretchng tests based on optcal methods. Polmer Testng 7 (008) 995-1004. [3] Promma N., Raka B., Grédac M., Toussant E., Le Cam J.-B., Balandraud X., Hld F. Applcaton of the vrtual felds method to mechancal characterzaton of elastomerc materals. Internatonal Journal of olds and tructures 46 (009) 698-715. [4] elvadura P.. Deflectons of a rubber membrane. Journal of the Mechancs and Phscs of olds 54 (006) 1093-1119. [5] Amodo D., Broggato G.B., Campana F., Newaz G.M. Dgtal peckle Correlaton for tran Measurement b Image Analss. Epermental Mechancs 43 (003) 396-40. [6] Grédac M., Perron F., Avrl., Toussant E. The vrtual felds method for etractng consttutve parameters from full-feld measurements : a revew. tran 4 (006) 33-53. [7] Grédac M., Toussant E., Perron F. pecal vrtual felds for the drect determnaton of materal parameters wth the vrtual felds method. 3- Applcaton to the bendng rgdtes of ansotropc plates. Internatonal Journal of olds and tructures 40 n 10 (003) 401-419. [8] Grédac M., Perron F. Applng the Vrtual Felds Method to the dentfcaton of elasto-plastc consttutve parameters. Internatonal Journal of Plastct (006) 60-67. [9] Avrl., Perron F., Yan J., utton M. Identfcaton of vscoplastc parameters and characterzaton of Lüders behavour usng Dgtal Image Correlaton and the Vrtual Felds Method. Mechancs of Materals 40 (008) 79-74. [10] Ogden R.W., Non-lnear elastc deformaton. Ells-Horwood. Chchester (1984).