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Measurement of nonlinear mechanical properties of PDMS elastomer
Microelectronic Engineering 88 (2011) 1982–1985
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Measurement of nonlinear mechanical properties of PDMS elastomer Tae Kyung Kim, Jeong Koo Kim ⇑, Ok Chan Jeong ⇑ Department of Biomedical Eng., Inje University, Gimhae 621-749, South Korea
a r t i c l e
i n f o
Article history: Available online 4 January 2011 Keywords: PDMS (polydimethylsiloxane) Cyclic tension test Nonlinear mechanical model
a b s t r a c t This paper presents the measurement of the nonlinear mechanical properties of polydimethylsiloxane (PDMS) elastomer based on the mixing ratio of base polymer to curing agent. Strip-type PDMS samples with different mixing ratios were prepared using a simple coating, curing, and cutting process. A cyclic uniaxial tension test with a fixed magnitude of applied strain and a single-pull-to-failure tension test were performed with a micro-tensile tester at room temperature. Our new finding is that when the PDMS is mixed with excessive curing agent, stress softening occurs and residual strain exists in cyclic tension tests when the magnitude of the applied strain increases. For the PDMS-05 samples, in which the mixing ratio of base polymer to curing agent was 5 to 1, there were large differences in the stresses for the same strain level under loading and unloading during the first cycle with a 100% fixed strain amplitude, but the softening effect of the stress in the PDMS dropped rapidly starting from the second cycle. Nonlinear mechanical Neo-Hookean, third-order Mooney, and second-order Ogden models of three different PDMS films were computed from the stress–strain data. The results showed that all models were preferable for the small strain region of PDMS compared with other models. In the nonlinear, large strain region, only the second-order Ogden model properly described the mechanical behavior of the PDMS, while the Neo-Hookean and third-order Mooney-Rivlin models were too stiff or flexible in the measurement range. The bulk modulus of PDMS increased with the amount of curing agent in it. Therefore, the second-order Ogden model is preferable for analyzing the PDMS structure over the entire measurement range. This could provide reasonable mechanical models of PDMS for rapid computational prototyping and for designing active and passive components from PDMS. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Polydimethylsiloxane (PDMS) is used to fabricate various microdevices because of its outstanding properties, such as low price, optical transparency, biocompatibility, and flexibility [1,2]. The mechanical properties of PDMS enable the realization of pneumatic [3], electromagnetic [4], and thermal [5] actuators. Since knowledge of the mechanical properties of PDMS is necessary for calculating and simulating its deformation [6,7], there has been growing interest in measuring the isotropic mechanical properties of PDMS [8] as functions of its thickness [9] and curing temperature [10]. Although the nonlinear properties like stress softening and residual strain might exist as functions of the magnitude of the applied strain [11], previous studies report little experimental data on these effects. Therefore, another approach for quantitative study on nonlinear material property of PDMS is necessary to obtain stable experimental data on the stress–strain relationship in PDMS for MEMS applications. ⇑ Corresponding authors. E-mail addresses: [email protected] (J.K. Kim), [email protected] (O.C. Jeong). 0167-9317/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2010.12.108
Another interesting point is that the mechanical properties of PDMS are dependent on its mixing ratio [12]. Quake and his colleagues proposed this to explain the structural bonding of relatively flexible vinyl PDMS and rigid Si-H PDMS. We have fabricated many devices using their method for the structural bonding of PDMS layers [3,12], including a pneumatic particle trap [13], pump [14], and micro-finger [15]. Unfortunately, there is no reasonable material model of PDMS for computational analysis, although its mechanical properties are essential for a computational understanding of the structure of PDMS. Therefore, a quantitative study of the mechanical properties of PDMS as functions of the mixing ratio of the base polymer and curing agent is necessary. Moreover, in the case of large deformation analysis using the isotropic properties of PDMS, as for a micro-balloon actuator [15], it is very difficult to predict and evaluate the nearly incompressible mechanical behavior of PDMS structures. This paper determines the mechanical properties of PDMS according to its mixing ratio, based on a cyclic test at a fixed strain and a single-pull-to-failure tension test. Moreover, the nonlinear material properties of PDMS are modeled from the experimentally measured stress–strain curves of PDMS. This work provides the
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985
1983
experimental data on the nonlinear material property and their nonlinear models for the analysis of PDMS structure in the small and large deformation regions. 2. Tension test Fig. 1 shows a photograph of a PDMS sample during the cyclic tension test. Three different types of PDMS (Sylgard 184, Dow Corning) samples were prepared to investigate the mechanical properties of PDMS films with different mixing ratios of the base polymer and curing agent. In this paper, ‘‘AB’’ in PDMS-AB indicates the mixing ratio of the base polymer to curing agent. Samples were prepared using a simple coating and curing process. After curing, the samples were peeled off and strips were fabricated using a precision punching machine with a rectangular cutting blade (QM130, QMESYS, Korea). The thickness, width, and length of the PDMS samples were 100 lm, 4 mm, and 4 cm, respectively. The cyclic and single-pull-to-failure tension tests were performed with a micro-tensile tester (Lrxplus, Lloyd Instruments, UK). The loading speed was 1 mm/min and a 10-kN load cell was used. After installing a sample, cyclic loading was applied. The elongation and applied force during the test were recorded automatically. All samples for each group were tested at room temperature. At least three samples were averaged. 3. Stress–strain curve Fig. 2 shows the stress–strain curves of PDMS prepared with various mixing ratios of the base polymer and curing agent. Ten repetitions of the cyclic test and the single-pull-to-failure tension test were performed sequentially. First, successive cyclic tension tests with a 50% fixed percentage strain were performed, followed by 100% cyclic tests. A single-pull-to-failure tension test was performed after the 100% cyclic tension test. After all of the tension tests, the stress–strain curves of all of the samples were calculated from the automatically recorded raw data. Fig. 2(a) shows the stress–strain curves of PDMS-05. There was no noticeable change in the stress for successive 50% strain. However, for 100% strain, hysteresis was observed in the first loading and unloading paths of the stress–strain curves. After the first cycle, the hysteresis decreased and the stress–strain curves gradually converged as the number of cycles of successive loading and unloading increased. In other words, the stress decreased after repeated loading and unloading. The reduction was largest during the first and second cycles, and then decreased. In the small strain
Fig. 1. PDMS sample during the cyclic tension test.
Fig. 2. Strain–stress curves of PDMS samples made with various ratios of base polymer to curing agent: (a) PDMS-05 (mixing ratio of base polymer to curing agent of 5 to 1), (b) PDMS-10, and (c) PDMS-15.
region, the stress reduction was extremely small; however, it was noticeable in the large strain region close to 100% strain. Structural failure of the PDMS-05 film occurred when the strain reached 120%. The stress–strain curves for PDMS-10 and PDMS-15 are shown in Fig. 2(b) and (c). As with PDMS-05, hysteresis in the strain– stress curves of PDMS-10 during the first loading and unloading cycle with 100% of strain was also observed, but the difference was relatively small. Moreover, the yield stress was larger at a relatively low applied strain. The stress reduction and residual strain in all of the stress–strain curves of PDMS-15 differed little during the cyclic tension test. From these results, we concluded that the mechanical properties of PDMS are controllable. The hysteresis could be reduced, and the yield strain increased under low stress by either increasing the base polymer or decreasing the curing agent. This result can backup the previous works by Quake and colleagues [3,12], which was based on the qualitative understanding in the usage of the PDMS with the different mixing ratios of the base polymer and the curing agent. For PDMS devices, rigid PDMS with excessive curing agent is preferable for channel or chamber structures, while flexible PDMS with excess base polymer is preferred for moveable diaphragm structures to obtain mechanical actuation without hysteresis.
1984
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985
4. Nonlinear fitting Fig. 3 shows the nonlinear fitting results for various PDMS samples. The experimental data on a single-pull-to-failure tension test after the 100% cyclic tension in Fig. 2 were modeled with commercial MSC software (Marc 2007 R1). Various material models, including Neo-Hookean, third-order Mooney, and second-order Ogden models [16], were computed through 200 iterations. Only positive coefficients for the model variables were considered, since a negative constant has no physical meaning. The results showed that all models were acceptable for the small and linear strain regions. Considering the number of coefficients and calculation time required in finite element method (FEM) simulations, the NeoHookean model is preferable for the small strain region of PDMS over the other models. In the nonlinear and large strain region, the second-order Ogden model described the mechanical behavior of the PDMS elastomer well. The results were too stiff or flexible in the measurement range for the Neo-Hookean and third-order Mooney-Rivlin models. In particular, the nonlinear model of the second-order Ogden model matched the measured strain–stress curves well in Fig. 3(c). The material constants of each model for
Table 1 Material models of PDMS and their corresponding material constants obtained from experimental data. Material model
Material constantsa
PDMS-AB (base polymer: curing agent) 5:1
10:1
15:1
Neo-Hookean MooneyRivlin
C10 (MPa) C10 (MPa) C01 (MPa) C11 (MPa) l1 (MPa) l2 (MPa)
0.208978 0 0.1342 0.0889167 0.0003428 0.131615 7.7991 3.6718 1214.84
0.0705019 0.0308307 0 0.0269727 63.4885 0.041103 6.371e-10 3.81166 962
0.0929535 0.0013643 0.0878638 0.0109259 0.244339 0.0146323 1.01795 3.74094 739
Ogden
a1 a2 Bulk modulus (MPa) a
Material constants obtained experimental data for each material models [16].
three different PDMS samples are summarized in Table 1. Another interesting point is that the bulk modulus increases with the amount of curing agent. We concluded that the second-order Ogden model is preferable for the analysis of PDMS structures over the entire measurement range. 5. Conclusions We described our new finding regarding stress softening and residual strain in cyclic tension tests of PDMS, depending on the mixing ratio of the base polymer and curing agent. The nonlinear mechanical properties of PDMS were obtained from a nonlinear fit of the experimental stress–strain curves of the PDMS samples. The experimental results showed that softening and residual strain for stresses at the same strain level under loading and unloading during the first cycle increased with the proportion of curing agent in a mixture like PDMS-05. This phenomenon was reduced rapidly after the second cycle. As the ratio of base polymer increased in PDMS, the stress softening and residual strain in the cyclic and single tension tests decreased, and for PDMS-15, hysteresis was not observed. The modeling results show that all of the models considered were suitable for the small strain region of PDMS. However, in the nonlinear and large strain regions, only the second-order Ogden model described the mechanical behavior of PDMS. The Neo-Hookean and third-order Mooney-Rivlin models were too stiff or flexible in the measurement range. Therefore, we conclude that the second-order Ogden model is preferable for analyzing the structure of PDMS over the entire measurement range. Acknowledgement This work was supported by the 2010 Inje University research grant. References
Fig. 3. Nonlinear fitting results of various PDMS samples: (a) PDMS-05, (b) PDMS10, (c) PDMS-15.
[1] G.M. Whitesides, Nature 442 (2006) 368–373. [2] F. Schneider, J. Draheim, R. Kamberger, U. Wallrabe, Sens. Actuators A 151 (2009) 95–99. [3] T. Thorsen, S.J. Maerkl, S.R. Quake, Science 298 (2002) 580–584. [4] H. Yin, Y. Huang, W. Fang, J. Hsieh, Sens. Actuators A 39 (2007) 194–202. [5] A. Werber, H. Zappe, J. Micromech. Microeng. 16 (2006) 2524–2531. [6] G.A.A. Rodriguez, C. Rossi, K. Zhang, Sens. Actuators A 141 (2008) 489–498. [7] P.J. Hung, K. Jeong, G.L. Liu, L.P. Lee, Appl. Phys. Lett. 85 (23) (2004) 6051–6053. [8] F. Schneider, T. Fellner, J. Wilde, U. Wallrabe, J. Micromech. Microeng. 18 (2008) 065008. [9] M. Liu, J. Sun, Y. Sun, C. Bock, Q. Chen, J. Micromech. Microeng. 19 (2009) 035028. [10] M. Liu, J. Sun, Q. Chen, Sens. Actuators A 151 (2009) 42–45. [11] A. Dorfmann, R.W. Ogden, Int. J. Solids Struct. 41 (2004) 1855–1878.
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985 [12] M.A. Unger, H.P. Chou, T. Thorsen, A. Scherer, S.R. Quake, Science 288 (2000) 113–116. [13] O.C. Jeong, S. Konishi, Microfluidics Nanofluidics 6 (1) (2009) 139–144. [14] O.C. Jeong, S. Konishi, J. Micromech. Microeng. 18 (8) (2008) 085017.
1985
[15] O.C. Jeong, S. Konishi, JMEMS 15 (4) (2006) 896–903. [16] www.mscsoftware.com. (The detailed nonlinear material models were described in Elastomer, Chapter 7 Material library, Vol. A: Theory and User Information, in user manual for MarcÒ 2007 R1).
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Measurement of nonlinear mechanical properties of PDMS elastomer Tae Kyung Kim, Jeong Koo Kim ⇑, Ok Chan Jeong ⇑ Department of Biomedical Eng., Inje University, Gimhae 621-749, South Korea
a r t i c l e
i n f o
Article history: Available online 4 January 2011 Keywords: PDMS (polydimethylsiloxane) Cyclic tension test Nonlinear mechanical model
a b s t r a c t This paper presents the measurement of the nonlinear mechanical properties of polydimethylsiloxane (PDMS) elastomer based on the mixing ratio of base polymer to curing agent. Strip-type PDMS samples with different mixing ratios were prepared using a simple coating, curing, and cutting process. A cyclic uniaxial tension test with a fixed magnitude of applied strain and a single-pull-to-failure tension test were performed with a micro-tensile tester at room temperature. Our new finding is that when the PDMS is mixed with excessive curing agent, stress softening occurs and residual strain exists in cyclic tension tests when the magnitude of the applied strain increases. For the PDMS-05 samples, in which the mixing ratio of base polymer to curing agent was 5 to 1, there were large differences in the stresses for the same strain level under loading and unloading during the first cycle with a 100% fixed strain amplitude, but the softening effect of the stress in the PDMS dropped rapidly starting from the second cycle. Nonlinear mechanical Neo-Hookean, third-order Mooney, and second-order Ogden models of three different PDMS films were computed from the stress–strain data. The results showed that all models were preferable for the small strain region of PDMS compared with other models. In the nonlinear, large strain region, only the second-order Ogden model properly described the mechanical behavior of the PDMS, while the Neo-Hookean and third-order Mooney-Rivlin models were too stiff or flexible in the measurement range. The bulk modulus of PDMS increased with the amount of curing agent in it. Therefore, the second-order Ogden model is preferable for analyzing the PDMS structure over the entire measurement range. This could provide reasonable mechanical models of PDMS for rapid computational prototyping and for designing active and passive components from PDMS. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Polydimethylsiloxane (PDMS) is used to fabricate various microdevices because of its outstanding properties, such as low price, optical transparency, biocompatibility, and flexibility [1,2]. The mechanical properties of PDMS enable the realization of pneumatic [3], electromagnetic [4], and thermal [5] actuators. Since knowledge of the mechanical properties of PDMS is necessary for calculating and simulating its deformation [6,7], there has been growing interest in measuring the isotropic mechanical properties of PDMS [8] as functions of its thickness [9] and curing temperature [10]. Although the nonlinear properties like stress softening and residual strain might exist as functions of the magnitude of the applied strain [11], previous studies report little experimental data on these effects. Therefore, another approach for quantitative study on nonlinear material property of PDMS is necessary to obtain stable experimental data on the stress–strain relationship in PDMS for MEMS applications. ⇑ Corresponding authors. E-mail addresses: [email protected] (J.K. Kim), [email protected] (O.C. Jeong). 0167-9317/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2010.12.108
Another interesting point is that the mechanical properties of PDMS are dependent on its mixing ratio [12]. Quake and his colleagues proposed this to explain the structural bonding of relatively flexible vinyl PDMS and rigid Si-H PDMS. We have fabricated many devices using their method for the structural bonding of PDMS layers [3,12], including a pneumatic particle trap [13], pump [14], and micro-finger [15]. Unfortunately, there is no reasonable material model of PDMS for computational analysis, although its mechanical properties are essential for a computational understanding of the structure of PDMS. Therefore, a quantitative study of the mechanical properties of PDMS as functions of the mixing ratio of the base polymer and curing agent is necessary. Moreover, in the case of large deformation analysis using the isotropic properties of PDMS, as for a micro-balloon actuator [15], it is very difficult to predict and evaluate the nearly incompressible mechanical behavior of PDMS structures. This paper determines the mechanical properties of PDMS according to its mixing ratio, based on a cyclic test at a fixed strain and a single-pull-to-failure tension test. Moreover, the nonlinear material properties of PDMS are modeled from the experimentally measured stress–strain curves of PDMS. This work provides the
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985
1983
experimental data on the nonlinear material property and their nonlinear models for the analysis of PDMS structure in the small and large deformation regions. 2. Tension test Fig. 1 shows a photograph of a PDMS sample during the cyclic tension test. Three different types of PDMS (Sylgard 184, Dow Corning) samples were prepared to investigate the mechanical properties of PDMS films with different mixing ratios of the base polymer and curing agent. In this paper, ‘‘AB’’ in PDMS-AB indicates the mixing ratio of the base polymer to curing agent. Samples were prepared using a simple coating and curing process. After curing, the samples were peeled off and strips were fabricated using a precision punching machine with a rectangular cutting blade (QM130, QMESYS, Korea). The thickness, width, and length of the PDMS samples were 100 lm, 4 mm, and 4 cm, respectively. The cyclic and single-pull-to-failure tension tests were performed with a micro-tensile tester (Lrxplus, Lloyd Instruments, UK). The loading speed was 1 mm/min and a 10-kN load cell was used. After installing a sample, cyclic loading was applied. The elongation and applied force during the test were recorded automatically. All samples for each group were tested at room temperature. At least three samples were averaged. 3. Stress–strain curve Fig. 2 shows the stress–strain curves of PDMS prepared with various mixing ratios of the base polymer and curing agent. Ten repetitions of the cyclic test and the single-pull-to-failure tension test were performed sequentially. First, successive cyclic tension tests with a 50% fixed percentage strain were performed, followed by 100% cyclic tests. A single-pull-to-failure tension test was performed after the 100% cyclic tension test. After all of the tension tests, the stress–strain curves of all of the samples were calculated from the automatically recorded raw data. Fig. 2(a) shows the stress–strain curves of PDMS-05. There was no noticeable change in the stress for successive 50% strain. However, for 100% strain, hysteresis was observed in the first loading and unloading paths of the stress–strain curves. After the first cycle, the hysteresis decreased and the stress–strain curves gradually converged as the number of cycles of successive loading and unloading increased. In other words, the stress decreased after repeated loading and unloading. The reduction was largest during the first and second cycles, and then decreased. In the small strain
Fig. 1. PDMS sample during the cyclic tension test.
Fig. 2. Strain–stress curves of PDMS samples made with various ratios of base polymer to curing agent: (a) PDMS-05 (mixing ratio of base polymer to curing agent of 5 to 1), (b) PDMS-10, and (c) PDMS-15.
region, the stress reduction was extremely small; however, it was noticeable in the large strain region close to 100% strain. Structural failure of the PDMS-05 film occurred when the strain reached 120%. The stress–strain curves for PDMS-10 and PDMS-15 are shown in Fig. 2(b) and (c). As with PDMS-05, hysteresis in the strain– stress curves of PDMS-10 during the first loading and unloading cycle with 100% of strain was also observed, but the difference was relatively small. Moreover, the yield stress was larger at a relatively low applied strain. The stress reduction and residual strain in all of the stress–strain curves of PDMS-15 differed little during the cyclic tension test. From these results, we concluded that the mechanical properties of PDMS are controllable. The hysteresis could be reduced, and the yield strain increased under low stress by either increasing the base polymer or decreasing the curing agent. This result can backup the previous works by Quake and colleagues [3,12], which was based on the qualitative understanding in the usage of the PDMS with the different mixing ratios of the base polymer and the curing agent. For PDMS devices, rigid PDMS with excessive curing agent is preferable for channel or chamber structures, while flexible PDMS with excess base polymer is preferred for moveable diaphragm structures to obtain mechanical actuation without hysteresis.
1984
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985
4. Nonlinear fitting Fig. 3 shows the nonlinear fitting results for various PDMS samples. The experimental data on a single-pull-to-failure tension test after the 100% cyclic tension in Fig. 2 were modeled with commercial MSC software (Marc 2007 R1). Various material models, including Neo-Hookean, third-order Mooney, and second-order Ogden models [16], were computed through 200 iterations. Only positive coefficients for the model variables were considered, since a negative constant has no physical meaning. The results showed that all models were acceptable for the small and linear strain regions. Considering the number of coefficients and calculation time required in finite element method (FEM) simulations, the NeoHookean model is preferable for the small strain region of PDMS over the other models. In the nonlinear and large strain region, the second-order Ogden model described the mechanical behavior of the PDMS elastomer well. The results were too stiff or flexible in the measurement range for the Neo-Hookean and third-order Mooney-Rivlin models. In particular, the nonlinear model of the second-order Ogden model matched the measured strain–stress curves well in Fig. 3(c). The material constants of each model for
Table 1 Material models of PDMS and their corresponding material constants obtained from experimental data. Material model
Material constantsa
PDMS-AB (base polymer: curing agent) 5:1
10:1
15:1
Neo-Hookean MooneyRivlin
C10 (MPa) C10 (MPa) C01 (MPa) C11 (MPa) l1 (MPa) l2 (MPa)
0.208978 0 0.1342 0.0889167 0.0003428 0.131615 7.7991 3.6718 1214.84
0.0705019 0.0308307 0 0.0269727 63.4885 0.041103 6.371e-10 3.81166 962
0.0929535 0.0013643 0.0878638 0.0109259 0.244339 0.0146323 1.01795 3.74094 739
Ogden
a1 a2 Bulk modulus (MPa) a
Material constants obtained experimental data for each material models [16].
three different PDMS samples are summarized in Table 1. Another interesting point is that the bulk modulus increases with the amount of curing agent. We concluded that the second-order Ogden model is preferable for the analysis of PDMS structures over the entire measurement range. 5. Conclusions We described our new finding regarding stress softening and residual strain in cyclic tension tests of PDMS, depending on the mixing ratio of the base polymer and curing agent. The nonlinear mechanical properties of PDMS were obtained from a nonlinear fit of the experimental stress–strain curves of the PDMS samples. The experimental results showed that softening and residual strain for stresses at the same strain level under loading and unloading during the first cycle increased with the proportion of curing agent in a mixture like PDMS-05. This phenomenon was reduced rapidly after the second cycle. As the ratio of base polymer increased in PDMS, the stress softening and residual strain in the cyclic and single tension tests decreased, and for PDMS-15, hysteresis was not observed. The modeling results show that all of the models considered were suitable for the small strain region of PDMS. However, in the nonlinear and large strain regions, only the second-order Ogden model described the mechanical behavior of PDMS. The Neo-Hookean and third-order Mooney-Rivlin models were too stiff or flexible in the measurement range. Therefore, we conclude that the second-order Ogden model is preferable for analyzing the structure of PDMS over the entire measurement range. Acknowledgement This work was supported by the 2010 Inje University research grant. References
Fig. 3. Nonlinear fitting results of various PDMS samples: (a) PDMS-05, (b) PDMS10, (c) PDMS-15.
[1] G.M. Whitesides, Nature 442 (2006) 368–373. [2] F. Schneider, J. Draheim, R. Kamberger, U. Wallrabe, Sens. Actuators A 151 (2009) 95–99. [3] T. Thorsen, S.J. Maerkl, S.R. Quake, Science 298 (2002) 580–584. [4] H. Yin, Y. Huang, W. Fang, J. Hsieh, Sens. Actuators A 39 (2007) 194–202. [5] A. Werber, H. Zappe, J. Micromech. Microeng. 16 (2006) 2524–2531. [6] G.A.A. Rodriguez, C. Rossi, K. Zhang, Sens. Actuators A 141 (2008) 489–498. [7] P.J. Hung, K. Jeong, G.L. Liu, L.P. Lee, Appl. Phys. Lett. 85 (23) (2004) 6051–6053. [8] F. Schneider, T. Fellner, J. Wilde, U. Wallrabe, J. Micromech. Microeng. 18 (2008) 065008. [9] M. Liu, J. Sun, Y. Sun, C. Bock, Q. Chen, J. Micromech. Microeng. 19 (2009) 035028. [10] M. Liu, J. Sun, Q. Chen, Sens. Actuators A 151 (2009) 42–45. [11] A. Dorfmann, R.W. Ogden, Int. J. Solids Struct. 41 (2004) 1855–1878.
T.K. Kim et al. / Microelectronic Engineering 88 (2011) 1982–1985 [12] M.A. Unger, H.P. Chou, T. Thorsen, A. Scherer, S.R. Quake, Science 288 (2000) 113–116. [13] O.C. Jeong, S. Konishi, Microfluidics Nanofluidics 6 (1) (2009) 139–144. [14] O.C. Jeong, S. Konishi, J. Micromech. Microeng. 18 (8) (2008) 085017.
1985
[15] O.C. Jeong, S. Konishi, JMEMS 15 (4) (2006) 896–903. [16] www.mscsoftware.com. (The detailed nonlinear material models were described in Elastomer, Chapter 7 Material library, Vol. A: Theory and User Information, in user manual for MarcÒ 2007 R1).