Vsco-Rubber Elastc Model for Pressure Senstve Adhesve Kazuhsa Maeda, Shgenobu Okazawa, Koj Nshgch and Takash Iwamoto Abstract A materal model to descrbe large deformaton of pressure senstve adhesve (PSA s presented. A relatonshp between stress and stran of PSA ncludes vscoelastcty and rubber-elastcty. Therefore, we propose the materal model for descrbng vscoelastcty and rubber-elastcty and formulate rate form of the presented materal model for three dmensonal fnte element analyss. And we valdate the present formulaton by usng one axs tensle calculaton. Keywords: vscoelastcty, rubber-elastcty, adhesve, large deformaton, fnte element method 1 Introducton The elastc modulus of a pressure senstve adhesve (PSA s about 10 5 Pa at room temperature and ndcates extremely low compared wth other sold materals. Therefore, large deformaton behavor can be observed n conventonal PSA deformaton. Fg.1 shows the tensle Stress-Stran curve of PSA. As shown n Fg.1, the stress ncreases exponentally for large stran zone. Ths behavor s called rubber-elastcty n ths paper. PSA s generally consdered to be a vscoelastc materal. However, only vscoelastcty can not descrbe practcal behavor ncludng rubber-elastcty of PSA consstently. The generalzed Maxwell model s usually used for descrbng vscoelastcty. On the other hands, hyperelastcty s popular to smulate ncrease n stress[1]. However, there s a dffculty n use of hyperelastcty, because hyperelastcty ndepend on tme. In order to evaluate materal constants, hyperelastcty needs tme ndependent parameters wth expermental data wthout stress relaxaton. The am of ths study s the establshment of materal model descrbng vsco and rubber elastcty for PSA. The establshed materal model can ndcate rubber-elastcty wthout hyperelastc model. We formulate the above materal model and ts rate formulaton Ntto Denko Corporaton, Department of Relablty Evaluaton Technology, Toyohash, Ach, 441-3194, Japan, Tel/Fax: +81-532-41-8849/43-1853 Emal: kazuhsa maeda@gg.ntto.co.jp Hroshma Unversty, Department of Socal and Envronmental Engneerng, Hgash-Hroshma, Hroshma 739-8527, Japan, Tel/Fax: +81-82-424-7810/422-7194, Emal: okazawa@hroshmau.ac.jp (Shgenobu Okazawa nomnal stress [Pa] 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0% 100% 200% 300% 400% 500% 600% 700% 800% 900% 1000% nomnal stran Fgure 1: Stress-stran curve of PSA for fnte element analyss and then shows the valdaton by usng computatonal example. 2 Materal Model 2.1 Concept of Proposed Materal Model PSA ndcates remarkable vscoelastcty at room temperature. The generalzed Maxwell model for descrbng vscoelastcty s used n the present study. Fg.2 shows generalzed Maxwell model. Where denotes unt number of the generalzed Maxwell model. And E and η are elastc modulus of sprng and vscous coeffcent of dashpot, respectvely. Fgure 2: Generalzed Maxwell model However, the generalzed Maxwell model of Fg.2 can not descrbe rubber-elastc behavor of PSA as shown n Fg.1. The reason s that elastc modulus of the generalzed Maxwell model s constant. Then, we propose evaluaton of elastc modulus of the sprng component. Fg.3
shows the modfed generalzed Maxwell model. Ths proposed model s called Advanced Generalzed Maxwell Model. In the advanced generalzed Maxwell model, elastc modulus s functon of total stran. In addton, vscous coeffcents of dashpot s functon of stran to assume that relaxaton tme s constant lke the generalzed Maxwell model. Fgure 3: Advanced Generalzed Maxwell model 2.2 Rubber-Elastcty Elastc modul for the advanced generalzed Maxwell model are determned as the functon of total stran. Frst, we measure stress relaxaton behavor wth varous ntal strans n order to nvestgate the stran dependency of an elastc modulus. For ths experment, common acrylc PSA s used. Cylndrcal PSA sample, whose cross-secton s 2mm 2, s attached to a tensle machne so that the length of PSA s 10mm, and ntal stran s gven 100% by nomnal stran. Then, the sample s extended and keeps n the fxed stran. The stress relaxaton curve s obtaned by measurng the stress change at the measurement tme. The relaton of nomnal stress - nomnal stran s changed nto the relaton of true stress - true stran. Regresson analyss s appled to the obtaned curve usng the stress relaxaton formula of the generalzed Maxwell model as shown n eq.(1, and we get the 5 sets of relaxaton tme, τ, and elastc modulus of the sprng component, E. σ = n =1 E exp( t τ (1 where σ, E, τ, and denote true stress, elastc modulus of a sprng component, relaxaton tme, and the unt number respectvely. Then, the stress relaxaton measurement wth ntal stran 200, 300, 400, 500, and 600% s measured, and we get the sets of the relaxaton tme and elastc modulus at each ntal stran. Table.1 shows relaxaton tmes and elastc modul of each stran. The strans n Table.1 are converted nto true stran. Then, elastc modulus E s depends on the total stran and s consdered about each case of τ. The case of τ = 100[sec] s consdered for an example. Fg.4 shows elastc modulus, whch depends on stran. Fgure 4: stran Relatonshp between elastc modulus and When the exponental functon of stran s used as an approxmated curve to ths curve, the correlaton coeffcent s very hgh. Ths s the same for the cases of other relaxaton tmes. So, we decde to use the exponental functon of stran as a functon of an elastc modulus as follows. E = A exp(b ε (2 where A and B denotes materal parameter, and ε does stran. 3 Consttutve Formulaton Although the elastc modulus of the sprng component of the advanced model s defned by eq.(2, t s necessary to dstngush scalar, vector, and tensor strctly n the case of dealng wth three dmensons. So, the elastc modulus of a sprng component s replaced wth eq.(3. Here, an elastc modulus uses shear elastc modulus G. G = A exp(b ˆε (3 where ˆε denotes scalar of stran. And ˆε s defned by eq.(4 usng the small stran tensor, ε. 2 ˆε = 3 ε : ε (4 It s assumed that vscoelastcty s n a devatorc component n the model. So, a consttutve equaton of devatorc and volumetrc component s respectvely formulated, and then the consttutve equaton of whole component s formulated. 3.1 Davatorc Component The sprng of the th unt s assumed to be an ncompressble lnear elastc materal. Here shear elastc modulus and small stran tensor of the sprng of th unt s expressed as G and ε sp, respectvely, and the devatorc stress tensor of th sprng unt s σ = 2G ε sp, (5 The prme of a rght shoulder shows a devatorc component. The materal tme dervatve of both sdes of eq.(5
Table 1: Relaxaton tmes and elastc modul of each stran Relaxaton tme [s] True stran 0.6931 1.0986 1.3863 1.6094 1.7918 1.9459 1.00 10 0 0.1541 0.3637 0.5694 1.2057 2.0546 3.8663 1.00 10 1 0.0326 0.0832 0.1368 0.3548 0.5107 0.7711 1.00 10 2 0.0363 0.0665 0.1237 0.1808 0.2078 0.3701 1.00 10 3 0.0250 0.0519 0.0761 0.1040 0.1549 0.1522 1.00 10 10 0.0639 0.1326 0.2150 0.3279 0.4608 0.5936 gves Dσ = 2GD sp, + DG σ G (6 where D denotes rate of stran tensor. The left sde of eq.(6 does not have objectvty. So the consttutve equaton s not objectve. Therefore object stress rate s assumed ( Dσ ( = 2GD sp, + DG σ G (7 where ( wth the lower rght means arbtrary objectve stress rate. Eq.(7 s used as a consttutve equaton of a sprng. Next, the dashpot of th unt s consdered. The shear vscous coeffcent and the stran rate tensor of th dashpot unt s expressed as η and D dp, respectvely. And dashpot s assumed to be ncompressble Newtonan flud. The consttutve equaton of th dashpot unt s gven as σ = 2η D dp, (8 The model property nssts that the small stran tensor of th unt can be assumed to be equal to the stran of the whole model. That s, ε = ε = ε sp, + ε dp,. The materal tme dervatve of ths equaton gves ( D = D sp, + D dp, (9 From eqs.(7(8(9, the consttutve equaton of the th unt s devatorc component s derved to ( Dσ = 2GD sp, + DG σ G σ τ (10 The materal tme dervatve of eq.(3 gves DG = B G Dˆε (11 And eq.(10 s gven as, wth eq.(11 ( Dσ = 2GD sp, + B Dˆε σ σ τ (12 ( Snce the stress of the whole model s derved from summaton of the stress of each unt, the consttutve equaton of the whole model s descrbed to ( Dσ = ( 2GD sp, + B Dˆε ( σ σ τ (13 3.2 Volumetrc Component Here t s assumed that the volumerc component s a compressble lnear elastc materal. Pressure, p, s gven as p = K v trε (14 where Kv denotes the coeffcent of volumetrc elastcty derved from eq.(15. K v = E 3 (1 2ν (15 where ν denotes Posson rato. The materal tme dervatve of eq.(14 gves where Dp = DK v p K v trd (16 K v K v = DK v = 2 (1 + ν 3 (1 2ν 2 (1 + ν 3 (1 2ν G (17 DG Substtutng eq.(11 nto eq.(18 derves to DK v = 2 (1 + ν 3 (1 2ν B G Dˆε (18 (19 Therefore, the consttutve equaton of volumetrc component s Dp = pdˆε B G K v trd (20 G 3.3 Whole Component Here the Jaumann rate s used as objectve stress rate. The materal tme dervatve of Cauchy stress and ts Jaumann rate are connected wth eq.(21. ( Dσ (J = Dσ + W σ σ W (21 where W denotes spn tensor, and the lower rght (J shows the Jaumann rate. Here the Cauchy stress s dvded nto devatorc and volumetrc component, σ = σ pi (22
where I denotes unt tensor. Substtutng eq.(22 nto eq.(21 derves to ( ( Dσ Dσ = Dp I (23 (J Therefore, substtutng consttutve equaton of devatorc and volumetrc component nto eq.(23, the consttutve equaton of the whole component s derved to ( Dσ (J = (J ( 2G D + B Dˆε σ σ τ I ( p Dˆε B G K v trd G 4 Explct Fnte Element Method (24 The present study employs an explct fnte element method[2] to calculate the followng computatonal example. The explct fnte element method s computatonally robust because of no teratons. 4.1 Dscrete equlbrum equaton The equlbrum equaton gnorng the body force s, ρa = σ x (25 where ρ s the materal densty, a s the spatal acceleraton, and σ s the Cauchy stress. We can derve the vrtual work equaton by multplyng both sdes of eq.(25 by the arbtrary vrtual dsplacement δu wth the Gauss dvergence theorem of volume V. ρa δudv + σ : (δεdv = t δuds (26 V V where t s the external surface force on the boundary area V, and ε s the lnear stran as follows, ε = 1 2 [( u + x V ( u T ] x (27 The dscrete equlbrum equaton can be derved usng the fnte element as follows; Ma + F nt = F ext (28 where M s the mass matrx, F nt and F ext are the nternal and external force vectors respectvely. For the numercal ntegraton of the soparametrc element n the plane stran state, the selectve reduced ntegraton method s used to avod volumetrc lockng [3]. 4.2 Central dfference method To advance the tme of the dscrete equlbrum eq.(28, we select the central dfference method. Let t s the tme ncrement from tme t n to t n+1. The current tme s t n and any propertes of the materal at tme t n+1 wll be explctly calculated wth the central dfference method. The materal coordnates x at t n+1 s evaluated wth the materal velocty v at the central ncremental tme t n+ 1 2. x n+1 = x n + v n+ 1 2 t (29 where the materal velocty v at tme t n+ 1 2 v n+ 1 2 = v n 1 2 + a n t (30 and the spatal acceleraton a at tme t n s solved as follows wth eq.(28. a n = M 1 (F n ext F n nt (31 Eq.(31 requres no soluton of the smultaneous equatons by usng the dagonal lumped mass matrx for M. 5 Computatonal Results Here, n order to verfy an above-mentoned technque, one axs tensle measurement of PSA s analyzed. 5.1 Materal Constants The materal constants whch must be defned n the consttutve equaton are A, B and τ. These values are calculated from the expermental data of one axs elongaton measurement. Frst, the method of one axs elongaton measurement s explaned. It s measured at room temperature. Cylndrcal PSA sample whose cross-secton s 2mm 2 s attached to the tensle machne so that the length of PSA s 10mm, the sample s elongated at the predetermned rate. The rate s 10, 50, 300 mm/mn. Snce the data from the measurement gves nomnal stress and nomnal stran, the Stress-Stran curve changes nto true stress - true stran s made. Consttutve equaton of the advanced model calculated by one dmenson s dσ dt = A exp ( B ε dε dt ( 1 dε B σ (32 τ dt Eq.(32 s appled to the Stress-Stran curve wth nonlnear least squares method, and the materal constants are determned. Approxmate curve s calculated so that all Stress-Stran curve wth dfferent three elongaton rate are satsfed. The result s shown n Table.2. 5.2 Comparson wth Experment Analyss model s shown n Fg.5. The analyss object s cubc PSA whose length of one sde s 1cm. The densty of PSA uses 1000kg/m 3. s
Table 2: Materal constants Relaxaton tmes [s] A [Pa] B 1.00 10 0 1.28 10 5 1.17 1.00 10 1 8.93 10 1 4.40 1.00 10 2 3.71 10 2 3.67 1.00 10 3 1.30 10 5 0.67 1.00 10 6 7.13 10 1 3.85 the correlaton worsens at the area of large stran s to estmate the Posson rato to be low, and that good correlaton s acqured f the near ncompressblty can be descrbed. 6 Conclusons and Future Work Ths paper has treated the materal model whch can descrbe the deformaton of PSA. Our results ndcate the followng. 1. The present advanced generalzed Maxwell model can descrbe vsco and rubber elastcty. Fgure 5: Boundary condtons The analyss s calculated wth one element model for the effcency of analyss tme. PSA shows near ncompressblty. However, the analyss usng 0.49 for the Posson rato s stopped before completng calculaton. The reason s consdered to be lockng. Snce the purpose was verfcaton of the model ths tme, 0.3 s used as Posson rato. The elongaton rate used for analyss s 5, 50, and 500 mm/mn. The materal constants use the values shown n Table.2. The result s shown n Fg.6. In Fg.6, the rate, for example 5mm/mn, shows elongatng rate. Fg.6 shows that the computatonal result can descrbe the expermental data well. And t shows that the elongatng rate dependablty orgnatng n the vscoelastcty can be descrbed well. Therefore, t s thought that ths model s approprate as a materal model descrbng deformaton of PSA. However, some dfference between computatonal result and expermental data can be observed n the area of large stran and at hgh elongatng rate. It s thought that the reason for ths dfference s usng 0.3 for Posson rato. 5.3 Investgaton of Posson Rato In order to study the reason for bad correlaton at hgh elongatng rate, we analyze at varous Posson rato. The result s shown n Fg.7. Fg.7 s the result of the analyss at 500mm/mn of elongatng rate and the area of nomnal stran 400-600% s magnfed. The correlaton between computatonal result and expermental data s so good that the Posson rato used for calculaton s close to 0.5. The curve of the Posson rato 0.4 shows the strange behavor near 550% of nomnal stran. Ths s because the analyss does not progress accordng to the lockng phenomenon. From above mentoned, t s thought that the reason why 2. We have formulzed the three-dmensonal consttutve equaton of the advanced generalzed Maxwell model. 3. We have valdated the proposed advanced generalzed Maxwell model wth the one axs tensle analyss. The remaned subjects to smulate practcal PSA behavor wth large deformaton are as follows. 1. The present code uses dynamc explct method. Therefore, computatonal tme step sze s extremely small because of the requrement of the Courant condton. It s necessary to consder the soluton method, whch can use large tme step sze, for sutable analyss of PSA deformaton. 2. The hghly dstorted Lagrangan fnte elements cannot retan numercal accuracy. The present formulaton should be extended to an Euleran formulaton[4], whch s attractve for large deformaton problem lke PSA. References [1] Smo, J.C. and Hughes, T.J.R., Computatonal Inelastcty, Sprnger, 1973. [2] Okazawa, S., Kashyama, K. and Kaneko, K., Large deformaton dynamc sold analyss by Euleran soluton based on stablzed fnte element method, Internatonal Journal for Numercal Methods n Engneerng, 72, pp.1544 1559, 2007. [3] Hughes, T.J.R. Generalzaton of selectve ntegraton procedures to ansotropc and nonlnear meda. Internatonal Journal for Numercal Methods n Engneerng, 15, pp.1413 1418, 1980. [4] Benson, D.J., Okazawa, S., Contact n a multmateral Euleran fnte element formulaton, Computer Methods n Appled Mechancs and Engneerng, 193, pp. 4277-4295, 2004.
0.9 0.8 0.7 experment computaton 500 [mm/mn] nomnal stress [MPa] 0.6 0.5 0.4 0.3 0.2 50 [mm/mn] 5 [mm/mn] 0.1 0 0% 100% 200% 300% 400% 500% 600% 700% 800% 900% 1000% nomnal stran Fgure 6: Computatonal and expermental stress-stran curves nomnal stress [MPa] 0.4 0.38 0.36 0.34 0.32 0.3 0.28 0.26 experment comp.1 comp.2 comp.3 0.24 0.22 0.2 400% 450% 500% 550% 600% nomnal stran Fgure 7: Stress-stran curves wth dfferent Posson ratos. Comp.1, 2 and 3 are computatonal solutons n case of Posson rato 0.3, 0.35 and 0.4 respectvely.