# Mechanical behavior of porous polysiloxane with micropores interconnected by microchannels.

INTRODUCTION

There has been a growing interest in developing interconnected porous elastomers, primarily driven by many emerging applications in the biomedical field [1-7]. Currently, salt leaching is the most widely used method for creating porous polymers including porous elastomers [8-10]. However, the structures created typically lack in controllability over interconnectivity and pore morphology. Recently, the authors [11] developed a microsphere templating process for creating open-cell elastomer with well-defined porous structures. In this process, prepolymers and catalysts of elastomer were cast into a sintered wax microsphere template. The template was then sacrificially removed after curing of the elastomer to obtain the desired material. The porous elastomer fabricated by this process had a unique cocontinuous porous structure with micropores interconnected by microchannels. Such a structural feature is expected to be particularly beneficial to cell migration in soft tissue engineering. While this fabrication method has been discussed in detail, the mechanical behavior of such porous elastomer is little understood.

Because of the unique porous structure containing micropores interconnected by microchannels, the mechanical behavior of the resulting elastomer is expected to be different from that of normal close-cell [12-17] and open-cell [18-21] materials. Figure 1 offers a possible example to illustrate this point; the open-cell elastomer with interconnected and well-defined micropores experiences deformation caused by a tensile force applying in the lateral direction. Under affine deformation, the deformation of connecting microchannels is in proportion to the bulk material. For instance, the ratio of diameter [alpha]' over [alpha] is the same as the primary principal stretch. However, if the deformation is not affine, the size change of a microchannel is not necessary to be proportional to the bulk materials and can be affected by the channel orientation. This can lead to much larger increase in diameter for channels with axis perpendicular to the tensile direction (i.e., from [alpha] to [alpha]"). For the same reason, the diameter of a parallel channel can become inproportionally smaller (e.g., from [beta] to [beta]"). One would expect that the situation should become more complicated for analyzing and predicting the mechanical behavior of such elastomeric materials.

Some related work on the mechanical behavior of open-cell elastomers in finite strain deformation is surveyed here. Migneco et al. [22] investigated the nonlinear elastic behaviors of poly(glycerol-dodecanoate) porous scaffolds for tissue engineering and were able to fit their data using the Ogden Model. An exponential model was used by Jeong and Hollister [23] to analyze the compressive stress-strain relation of poly(1,8-octanediol-co-citrate) scaffolds with different porosity during biodegradation. Mitsak et al. [24] characterized mechanical properties of poly(glycerol sebacate) scaffolds under various curing conditions and fitted their nonlinear tensile stress-strain behavior by Neo-Hookean Model. However, structure-property relations of open-cell porous elastomer, in general, are not quite known.

In the current work, the mechanical behavior of the porous elastomer having micropores interconnected by microchannels was studied. Polysiloxane was chosen as the model material. The deformation characteristics and mechanical properties of the porous material under tension and compression were then studied. The microscale deformation was subsequently examined by microscopic observations. Cyclic loading and unloading tests were also performed to study the hysteresis of the material. Finally, an attempt was further made to fit the stress-strain curve using existing hyperelastic models.

EXPERIMENTAL

Materials

White jojoba wax microspheres, obtained from Natural Sourcing, LLC (Oxford, CT), were used to form porous templates. The microspheres with diameters of 425 to 500 pm were selected from the as-received materials by two sieves with mesh sizes 35 and 40 (W.S. Tyler Industrial Group, Mentor, OH). Polysiloxane elastomers were prepared from commercially available Sylgard[R] 184 silicone elastomer (Dow Coming Corporation, Midland, MI). Acetone, 99.5% purity, acquired from VWR International, LLC (Radnor, PA) was used to extract the wax templates.

Porous Elastomer Fabrication

White jojoba wax microspheres were sintered at 66[degrees]C (close to the wax melting temperature 70[degrees]C) for 5 h to form porous wax templates. The base and catalyst for the silicone elastomer were mixed at 10:1 weight ratio and sequentially cast into the wax template. After being degassed in a vacuum oven, the cast sample was cured at 40[degrees]C for 72 h. The wax template was then extracted in an acetone bath at 50[degrees]C with 300 rpm stirring for 48 h. Solid polysiloxane elastomers as control were prepared and washed in acetone under the same condition to remove unreacted residuals.

Characterization

An Olympus BX51 optical microscope installed with an Olympus UC30 digital camera (Olympus Corporation of the Americas, Center Valley, PA) was used to observe the morphology of sintered wax templates under the reflective mode. Morphology of elastomer porous structure was captured by a Hitachi SEM S800 scanning electron microscopy (Hitachi High-Technologies Corporation, Tokyo, Japan). Sliced samples were gold-sputtered for 45 s before SEM measurement.

The porosity of fabricated porous elastomers was measured by a micro computed tomography (Micro-CT) system (model: 1172; Bruker Corporation, Billerica, MA). Binary images generated from two dimensional (2D) scanning images were used to reconstruct 3D models. The porosity was obtained by calculating void volume fraction in the 3D model.

To observe the in situ deformation of the pore structure during tension, a thin porous elastomer sheet was installed on a customized device with two ends fixed at mobile clamps. The sample was deformed with tension under the Olympus BX51 optical microscope. Selected pores were dyed black by a marker pen and their shape and position during deformation were tracked.

An Instron 5667 universal testing machine (Instron Corporation, Norwood, MA) with a 100 N load cell was used to conduct tensile and compressive tests. For tensile tests, both porous and solid elastomer sheets, 2.0 mm thick, were cut into dumbbell testing samples by an ASTM D412-C cutting die [25], For compressive tests, the elastomer sheets were cut into cubic testing samples with an approximate dimension of 6.0 X 6.0 X 6.0 mm. Before compressive testing, an approximately 0.01 N preload was applied to ensure a complete contact between the sample and the surface of mounting cylinder on the machine. Both tensile and compressive tests were operated at room temperature in the air. Each representative stress-strain curve was obtained by averaging the results from at least 10 tests.

RESULTS AND DISCUSSION

Porous Elastomer Fabricated

Jojoba wax microspheres were sintered at 66[degrees]C for 5 h to prepare porous templates. The sintering condition was determined based on the results from the previous work [11]. The morphology of wax templates obtained from optical microscopy is shown in Fig. 2A. Sintering necks were formed between wax microspheres. These necks later served as the templates for the formation of microchannels interconnecting neighboring micropores. This is confirmed by the SEM image of the elastomer porous structure as shown in Fig. 2B. Microchannels with gradually enlarged outlets were found to connect adjacent micropores, indicating the formation of a cocontinuous porous structure. Additionally, the pores were round with a diameter between 400 and 500 pm, reflecting the size and shape of jojoba wax microspheres in the template. The overall structure observed is quite similar to the schematic one shown in Fig. 1A.

Tensile Mechanical Behaviors

Uniaxial tensile test was used to study the influence of the porous structure on the mechanical strengths and extensibility of the porous elastomer. The solid elastomer was studied first as control. Figure 3A shows representative stress-strain curves of solid elastomers under five different crosshead speeds. The standard derivative curves were found to be very close to the mean curve. As all the testing samples had the same dimension, the crosshead speed was considered to be proportional to the engineering strain rate. Therefore, the uniaxial tensile properties of solid polysiloxane in this crosshead speed range can be considered to be independent of strain rate. For this reason, the tensile test of the porous elastomer was performed at a constant crosshead speed of 30 mm/min. The uniaxial tensile stress-strain curves of the porous are shown in Fig. 3B. For comparison, the stress-strain curve for the solid elastomer is also included. Both solid and porous elastomer demonstrated a nonlinear stress-strain relation. Table 1 summarizes the mechanical properties of the two types of elastomers under uniaxial tension. The ultimate tensile strength, the tensile modulus, and the ultimate tensile strain of the porous elastomer were 0.34 [+ or -] 0.01 MPa, 0.46 [+ or -] 0.02 MPa, and 55.8 [+ or -] 0.9%, respectively, whereas, for the solid elastomer, these values were 6.64 [+ or -] 0.16 MPa, 1.96 [+ or -] 0.04 MPa, and 160.8 [+ or -] 2.8%. One can notice that the mechanical strengths and extensibility of the porous elastomer were considerably lower than those of the solid elastomer. This can be attributed to the existence of porous structure.

The modulus and strength of the porous elastomer were also estimated using an additive rule based on the affine deformation assumption. This was achieved by multiplying the modulus or strength of the solid elastomer by the solid volume fraction (equal to one minus porosity). The tensile modulus of the porous elastomer with 56% porosity was calculated to be 0.86 MPa. Likewise, the strength was calculated to be 0.43 MPa. One can notice that the calculated values are considerably larger than the experimental ones, indicating the porous elastomer did not follow the affine deformation. The actual difference between the calculated and experimental stress-strain curves are presented in Fig. 3B. The calculated one largely deviated from the experimental one, and this deviation started from a very small strain that was less than 0.5% (see inset graph of Fig. 3B).

Optical microscopy was used to study in situ morphological changes of the porous elastomer during tensile test. The results are shown in Fig. 4. The tensile force was applied along the lateral direction of the figure. The micrograph in Fig. 4A shows the morphology at the onset of the test at zero strain. The micropore at the center of the micrograph was tracked and highlighted by the white circle. The micropore was round with microchannels connecting to neighboring micropores. Three microchannels are shown in the figure, one with axis nearly aligned along the tensile direction and the other two aligned in the perpendicular direction. From Fig. 4A-E, it can be observed that, as the strain increased, the micropore was elongated along the tension direction and became elliptic. One can also notice that the microchannels with axes perpendicular to the force direction were widened but the third one did not undergo obvious changes. The different deformation modes observed for the micropore and the microchannels seem to conform the schematic shown in Fig. 1. For example, the micropore was elongated by 30% from Fig. 4A-C and by another 25% from Fig. 4C-E, whereas, for the two perpendicular microchannels, these two increments were approximately 66 and 45%. These inhomogeneous changes in the porous structure indicated a nonaffine deformation nature of the porous elastomer. Additionally, the results in Fig. 4 revealed high resilience of the porous elastomer; the recovered structure (Fig. 4F) was almost identical to undeformed one (Fig. 4A).

Cyclic loading-unloading tensile test was performed to study the hysteresis of the porous elastomer under tension. Representative stress-strain curves of the solid elastomer with different crosshead speeds are shown in Fig. 5A. The prestrain was fixed at 0.3 mm/mm. Only small deviation was observed between the standard derivative curve and the mean curve, indicating the hysteresis was insensitive to the change of strain rate. The results in Fig. 5B illustrate the cyclic loading-unloading behavior of the porous elastomer with varied crosshead speeds. Slight strain rate hardening was observed. However, the strain rate seemed to have no influence over the permanent sets after unloading. The effects of prestrain on the hysteresis of the solid and porous elastomers are shown in Fig. 5C and D, respectively. As expected, with increase of preset strain, the maximum tensile stress and the permanent set both increased. The loading-unloading curves were found to be repeatable. Examples of multiple cycles are shown in the inset graphs in Fig. 5C and D. For comparison purposes, the loading-unloading curves for both solid and porous elastomers are plotted in the same figure, Fig. 6. The calculated stress-strain curve from the affine deformation approximation is also shown in the figure. A large deviation was observed between the calculated curve and experimental curve of the porous elastomer. This agreed with the uniaxial tension data previously discussed.

For more quantitative investigation, the hysteresis, H, was represented as the normalized strain energy density difference between the loading and unloading processes:

H = [integral]([[sigma].sub.1] - [[sigma].sub.u])d[epsilon]/[integral][[sigma].sub.1]d[epsilon] (1)

where [[sigma].sub.1] and [[sigma].sub.u] denote loading and unloading stresses, and e is tensile strain. Effectively, the integral on the numerator represents the area enclosed by the loading and unloading curves, and the integral on the denominator stands for the area underneath the loading curve. Table 2 summarizes the results. Generally, the hysteresis of the porous elastomer was considerably higher than that of the solid elastomer. The hysteresis data are also plotted in Fig. 7. From Fig. 7A, it can be seen that the hysteresis of the porous elastomer was quite sensitive to the change of the crosshead speed, whereas the hysteresis of the solid elastomer was almost independent of the crosshead speed. Figure 7B shows that the hysteresis of the porous elastomer was considerably larger than that of the solid elastomer at all preset strains used in the experiment.

Compressive Mechanical Behaviors

Uniaxial compressive test was used to study the compressive mechanical behaviors of the porous elastomer. Figure 8A shows the representative stress-strain curves of the solid elastomer with varied crosshead speeds. Similar to the tensile testing results, the compressive stress-strain curves were nearly independent of strain rate. A fixed crosshead speed (3 mm/min) was, therefore, chosen for determining the stress-strain curves. Figure 8B compares the compressive stress-strain curves of the solid and porous elastomers. Table 3 summarizes the mechanical properties obtained from compressive test. As expected, the compressive properties of the porous elastomer were much lower than those of the solid one. The compressive stress at a preset strain of 0.6 mm/mm and the compressive modulus of the porous elastomer were 0.45 [+ or -] 0.02 and 0.39 [+ or -] 0.01 MPa, respectively, whereas for solid elastomer, these values were 9.46 [+ or -] 0.28 and 1.79 [+ or -] 0.01 MPa. The calculated compressive stress and modulus from the affine deformation approximation were 4.16 and 0.79 MPa, respectively. Comparing these calculated values with the experimental results, one can find that the deformation process under uniaxial compression for the porous elastomer was not affine either. For comparison, the experimental and calculated stress-strain curves were plotted in a single chart, as shown in Fig. 8B. The calculated curve considerably deviated from the experimental one. From the inset figure in Fig. 8B, it can further be seen that the large difference between the two curves existed even at very small strain where linear behavior was expected. Overall, the compressive test showed similar nonaffine behavior as the tensile test. Both indicated that the nonaffine deformation of the micropore and connecting microchannels largely dictated the mechanical behavior of the porous elastomer.

Cyclic loading-unloading compressive test was conducted. Figure 9A shows representative stress-strain curves of solid elastomers with varied crosshead speeds. Like the cyclic tensile test, the cyclic compressive test did not show significant strain-rate dependency for the solid elastomer. However, the strain rate showed a more significant effect for the porous elastomer, as shown in Fig. 9B; with the increase of crosshead speed, the compressive stress at the preset strain and the compressive modulus both significantly increased. Figure 9C and D further compare the different cyclic mechanical behavior between the solid and porous elastomers. The two figures show cyclic stress-strain loops with three different preset strains. The inset figures show multiple cycles at a fixed preset strain of 0.3 mm/mm. As expected, for both solid and porous elastomers, the maximum stress increased with the increase of the preset strain. However, the porous elastomer showed relatively poorer repeatability during multiple cycles. This can be seen by comparing the inset figures in Fig. 9C and D. The different cycles for the solid elastomer fell into a single one but did not for the porous elastomer. Specifically for the porous elastomer, the first loading-unloading cycle had the largest hysteresis, and the permanent set increased with more testing cycles performed.

The experimental stress-strain cycles were also compared with those calculated using the affine deformation approximation. Figure 10 compared the experimental cycle with the calculated cycle at a preset strain of 0.3 mm/mm. The calculated curve significantly differed from the experimental curve of the porous elastomer, and the difference appeared to be even larger than in the tensile test.

Equation 1 was also used to determine the hysteresis quantitatively. The results are summarized in Table 4 and plotted in Fig. 11. The porous elastomer experienced much larger hysteresis than the solid one under all different testing conditions. The hysteresis of the porous elastomer was also found to increase with the increase of strain rate, whereas the corresponding one for the solid elastomer was almost invariant (see Fig. 11 A). The different mechanical behavior between the solid and porous elastomers during uniaxial compressive test further indicated that the microstructure existed in the porous elastomer played an important role. The deviation from affine calculations again supported the nonaffine deformation nature of such porous materials. Further experimental and theoretical studies on the role of the micropore and interconnecting microchannels are needed to quantitatively understand this effect.

Model Fit to Nonlinear Elastic Behavior

An attempt was made to determine a suitable model for fitting the nonlinear stress-strain relation of the porous elastomer. Well-established hyperelastic models including the Arruda-Boyce model, the Neo-Hookean model, the Mooney-Rivlin model, and the One-term Ogden model were tested. The constitutive equations of these models for uniaxial tensile test are provided as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [sigma] is the engineering stress, [C.sub.1] is a material constant, [lambda] is principle stretch, [beta] is the reciprocal of locking principle stretch square, [I.sub.1] is the first invariant of the left Cauchy-Green deformation tensor, and [[alpha].sub.i] is a set of constants ([[alpha].sub.1] = 1/2; [[alpha].sub.2] = 1/20; [[alpha].sub.3] = 11/1050; [[alpha].sub.4] = 19/7000; [[alpha].sub.5] = 519/673750).

Neo-Hookean Model : [sigma] = C([lambda] - 1/[[lambda].sup.2]) (3)

where [C.sub.1] is a material constant.

Mooney-Rivlin Model : [sigma] = (2[C.sub.1] + 2[C.sub.2]/[lambda])([lambda] - 1/[[lambda].sup.2] (4)

where [C.sub.1] and [C.sub.2] are material constants.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [C.sub.1] and [C.sub.2] are material constants.

The fitting plots of the aforementioned models to the stress-strain curves of solid and porous elastomers are shown in Fig. 12. The corresponding fitting statistics are summarized in Table 5. Among the four models used in this study, only the Arruda-Boyce model can satisfactorily fit the experimental data of both solid and porous elastomers, with coefficients of determination ([R.sup.2]) over 0.99.

The Arruda-Boyce Model was established based on an eight chain representation of the underlying macromolecular network structure of elastomer and non-Gaussian behavior of the individual chains [26-28], The excellent fitting shown above indicated that the deformation behavior of a porous structure containing micropores interconnected by microchannels may be analogous to that of elastic networks. A focused study on constitutive modeling of such a porous structure may, therefore, be interesting from fundamental point of view. In future study, it is worthy to carry out analytical modeling and finite element analysis to further understand the nature of this type of deformation.

CONCLUSIONS

In this work, porous polysiloxane having micropores interconnected with microchannels was fabricated and its mechanical behavior was evaluated. The results from mechanical test, in tensile and compressive modes, and optical microscopy both indicated that the deformation process of such porous elastomer was highly nonaffine, meaning microscale deformation does not follow macroscale deformation. Particularly, the stress-strain curves of the porous elastomer largely deviated from the calculated ones from affine deformation. The microscopic observation revealed nonhomogeneous microscale deformation of the porous structure. The extent of deformation was different between the pore and connecting channels. The deformation was also different in channels with different orientation; channels with axis along the direction of the force deformed less, and those with perpendicular axis deformed more. Cyclic loading-unloading tests were also used to study the hysteresis. The hysteresis of the porous elastomer was considerably higher and more sensitive to strain rate change in comparison with the solid elastomer. A further effort was made to fit the tensile stress-strain data. Among different models tested, the Arruda-Boyce model satisfactorily fitted the stress-strain plots of both solid and porous elastomers.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Fred Cook and Dr. Meisha Shofner in Georgia Institute of Technology for the use of Rotap particle sorting machine and Instron testing machine. They would also like to show appreciations to Dr. Jack G. Zhou, Mr. Haibo Gong, and Mrs. Miao Yu in Drexel University for the help in porosity measurement with their MicroCT device.

REFERENCES

[1.] B. Amsden, Soft Matter, 3, 1335 (2007).

[2.] L.Q. Zhang, R. Shi, D.F. Chen, Q.Y. Liu, Y. Wu, X.C. Xu, and W. Tian, hit. J. Mol. Sci., 10, 4223 (2009).

[3.] E.S. Place, J.H. George, C.K. Williams, and M.M. Stevens, Chem. Soc. Rev., 38, 1139 (2009).

[4.] G. Vunjak-Novakovic, R. Maidhof, A. Marsano, and E.J. Lee, Biotechnol. Prog., 26, 565 (2010).

[5.] R.A. Hoshi, S. Behl, and G.A. Ameer, Adv. Mater., 21, 188 (2009).

[6.] J. Zhu, S. Wei, J. Ryu, and Z. Guo, J. Phys. Chem. C, 115, 132115 (2011).

[7.] J.H. Zhu, S.Y. Wei, R. Patil, D. Rutman, A.S. Kucknoor, A. Wang, and Z.H. Guo, Polymer, 52, 1954 (2011).

[8.] J.P. Santerre, S. Sharifpoor, C.A. Simmons, and R.S. Labow, Acta Biomater, 6, 4218 (2010).

[9.] T. Yoshii, A.E. Hafeman, J.S. Nyman, J.M. Esparza, K. Shinomiya, D.M. Spengler, G.R. Mundy, G.E. Gutierrez, and S.A. Guelcher, Tissue Eng. Part A, 16, 2369 (2010).

[10.] P.M. Crapo and Y. Wang, Biomaterials, 31, 1626 (2010).

[11.] Y. Hong, J.G. Zhou, and D. Yao, Adv. Polym. Technol., 31, 21330 (2013).

[12.] M.A. Islam, R.N. Stoicheva, and A. Dimov, J. Membr. Sci., 118, 9 (1996).

[13.] M. Danielsson, D.M. Parks, and M.C. Boyce, Mech. Mater., 36, 347 (2004).

[14.] O. Lopez-Pamies and P.P. Castaneda, J. Mech. Phys. Solids, 55, 1677 (2007).

[15.] O. Lopez-Pamies and P.P. Castaneda, J. Mech. Phys. Solids, 55, 1702 (2007).

[16.] B. Markert, Arch. Comput. Methods Eng., 15, 371 (2008).

[17.] O. Lopez-Pamies and M. Idiart, J. Elast., 95, 99 (2009).

[18.] J. Zhou and W.O. Soboyejo, Mater. Manuf. Processes, 19, 863 (2004).

[19.] R.M. Sullivan, L.J. Ghosn, and B.A. Lerch, Int. J. Solids Struct., 45, 1754 (2008).

[20.] A.D. Brydon, S.G. Bardenhagen, E.A. Miller, and G.T. Seidler, J. Mech. Phys. Solids, 53, 2638 (2005).

[21.] L. Leung, J. Perron, and H.E. Naguib, Polym. Polym. Compos., 15, 437 (2007).

[22.] F. Migneco, Y.C. Huang, R.K. Birla, and S.J. Hollister, Biomaterials, 30, 6479 (2009).

[23.] C.G. Jeong and S.J. Hollister, J. Biomed. Mater. Res. Part B, 93B, 141 (2010).

[24.] A.G. Mitsak, A.M. Dunn, and S.J. Hollister, J. Mech. Beliav. Biomed. Mater., 11, 3 (2012).

[25.] American Society for Testing and Materials (ASTM) Standard D412-06, "Standard Test Methods for Vulcanized Rubber and Thermoplastic Elastomers--Tension" (2008).

[26.] E.M. Arruda and M.C. Boyce, J. Mech. Phys. Solids, 41, 389 (1993).

[27.] A. Ali, M. Hosseini, and B.B. Sahari, Am. J. Eng. Appl. Sci., 3, 232 (2010).

[28.] M. Kroon, J. Elast., 102, 99 (2011).

Yifeng Hong and Donggang Yao

School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30318

Correspondence to: Donggang Yao; e-mail: yao@gatech.edu

Contract grant sponsor: NSF; contract grant number: CMMI-0927697.

DOI 10.1002/pen.23688

Published online in Wiley Online Library (wileyonlinelibrary.com).

There has been a growing interest in developing interconnected porous elastomers, primarily driven by many emerging applications in the biomedical field [1-7]. Currently, salt leaching is the most widely used method for creating porous polymers including porous elastomers [8-10]. However, the structures created typically lack in controllability over interconnectivity and pore morphology. Recently, the authors [11] developed a microsphere templating process for creating open-cell elastomer with well-defined porous structures. In this process, prepolymers and catalysts of elastomer were cast into a sintered wax microsphere template. The template was then sacrificially removed after curing of the elastomer to obtain the desired material. The porous elastomer fabricated by this process had a unique cocontinuous porous structure with micropores interconnected by microchannels. Such a structural feature is expected to be particularly beneficial to cell migration in soft tissue engineering. While this fabrication method has been discussed in detail, the mechanical behavior of such porous elastomer is little understood.

Because of the unique porous structure containing micropores interconnected by microchannels, the mechanical behavior of the resulting elastomer is expected to be different from that of normal close-cell [12-17] and open-cell [18-21] materials. Figure 1 offers a possible example to illustrate this point; the open-cell elastomer with interconnected and well-defined micropores experiences deformation caused by a tensile force applying in the lateral direction. Under affine deformation, the deformation of connecting microchannels is in proportion to the bulk material. For instance, the ratio of diameter [alpha]' over [alpha] is the same as the primary principal stretch. However, if the deformation is not affine, the size change of a microchannel is not necessary to be proportional to the bulk materials and can be affected by the channel orientation. This can lead to much larger increase in diameter for channels with axis perpendicular to the tensile direction (i.e., from [alpha] to [alpha]"). For the same reason, the diameter of a parallel channel can become inproportionally smaller (e.g., from [beta] to [beta]"). One would expect that the situation should become more complicated for analyzing and predicting the mechanical behavior of such elastomeric materials.

Some related work on the mechanical behavior of open-cell elastomers in finite strain deformation is surveyed here. Migneco et al. [22] investigated the nonlinear elastic behaviors of poly(glycerol-dodecanoate) porous scaffolds for tissue engineering and were able to fit their data using the Ogden Model. An exponential model was used by Jeong and Hollister [23] to analyze the compressive stress-strain relation of poly(1,8-octanediol-co-citrate) scaffolds with different porosity during biodegradation. Mitsak et al. [24] characterized mechanical properties of poly(glycerol sebacate) scaffolds under various curing conditions and fitted their nonlinear tensile stress-strain behavior by Neo-Hookean Model. However, structure-property relations of open-cell porous elastomer, in general, are not quite known.

In the current work, the mechanical behavior of the porous elastomer having micropores interconnected by microchannels was studied. Polysiloxane was chosen as the model material. The deformation characteristics and mechanical properties of the porous material under tension and compression were then studied. The microscale deformation was subsequently examined by microscopic observations. Cyclic loading and unloading tests were also performed to study the hysteresis of the material. Finally, an attempt was further made to fit the stress-strain curve using existing hyperelastic models.

EXPERIMENTAL

Materials

White jojoba wax microspheres, obtained from Natural Sourcing, LLC (Oxford, CT), were used to form porous templates. The microspheres with diameters of 425 to 500 pm were selected from the as-received materials by two sieves with mesh sizes 35 and 40 (W.S. Tyler Industrial Group, Mentor, OH). Polysiloxane elastomers were prepared from commercially available Sylgard[R] 184 silicone elastomer (Dow Coming Corporation, Midland, MI). Acetone, 99.5% purity, acquired from VWR International, LLC (Radnor, PA) was used to extract the wax templates.

Porous Elastomer Fabrication

White jojoba wax microspheres were sintered at 66[degrees]C (close to the wax melting temperature 70[degrees]C) for 5 h to form porous wax templates. The base and catalyst for the silicone elastomer were mixed at 10:1 weight ratio and sequentially cast into the wax template. After being degassed in a vacuum oven, the cast sample was cured at 40[degrees]C for 72 h. The wax template was then extracted in an acetone bath at 50[degrees]C with 300 rpm stirring for 48 h. Solid polysiloxane elastomers as control were prepared and washed in acetone under the same condition to remove unreacted residuals.

Characterization

An Olympus BX51 optical microscope installed with an Olympus UC30 digital camera (Olympus Corporation of the Americas, Center Valley, PA) was used to observe the morphology of sintered wax templates under the reflective mode. Morphology of elastomer porous structure was captured by a Hitachi SEM S800 scanning electron microscopy (Hitachi High-Technologies Corporation, Tokyo, Japan). Sliced samples were gold-sputtered for 45 s before SEM measurement.

The porosity of fabricated porous elastomers was measured by a micro computed tomography (Micro-CT) system (model: 1172; Bruker Corporation, Billerica, MA). Binary images generated from two dimensional (2D) scanning images were used to reconstruct 3D models. The porosity was obtained by calculating void volume fraction in the 3D model.

To observe the in situ deformation of the pore structure during tension, a thin porous elastomer sheet was installed on a customized device with two ends fixed at mobile clamps. The sample was deformed with tension under the Olympus BX51 optical microscope. Selected pores were dyed black by a marker pen and their shape and position during deformation were tracked.

An Instron 5667 universal testing machine (Instron Corporation, Norwood, MA) with a 100 N load cell was used to conduct tensile and compressive tests. For tensile tests, both porous and solid elastomer sheets, 2.0 mm thick, were cut into dumbbell testing samples by an ASTM D412-C cutting die [25], For compressive tests, the elastomer sheets were cut into cubic testing samples with an approximate dimension of 6.0 X 6.0 X 6.0 mm. Before compressive testing, an approximately 0.01 N preload was applied to ensure a complete contact between the sample and the surface of mounting cylinder on the machine. Both tensile and compressive tests were operated at room temperature in the air. Each representative stress-strain curve was obtained by averaging the results from at least 10 tests.

RESULTS AND DISCUSSION

Porous Elastomer Fabricated

Jojoba wax microspheres were sintered at 66[degrees]C for 5 h to prepare porous templates. The sintering condition was determined based on the results from the previous work [11]. The morphology of wax templates obtained from optical microscopy is shown in Fig. 2A. Sintering necks were formed between wax microspheres. These necks later served as the templates for the formation of microchannels interconnecting neighboring micropores. This is confirmed by the SEM image of the elastomer porous structure as shown in Fig. 2B. Microchannels with gradually enlarged outlets were found to connect adjacent micropores, indicating the formation of a cocontinuous porous structure. Additionally, the pores were round with a diameter between 400 and 500 pm, reflecting the size and shape of jojoba wax microspheres in the template. The overall structure observed is quite similar to the schematic one shown in Fig. 1A.

Tensile Mechanical Behaviors

Uniaxial tensile test was used to study the influence of the porous structure on the mechanical strengths and extensibility of the porous elastomer. The solid elastomer was studied first as control. Figure 3A shows representative stress-strain curves of solid elastomers under five different crosshead speeds. The standard derivative curves were found to be very close to the mean curve. As all the testing samples had the same dimension, the crosshead speed was considered to be proportional to the engineering strain rate. Therefore, the uniaxial tensile properties of solid polysiloxane in this crosshead speed range can be considered to be independent of strain rate. For this reason, the tensile test of the porous elastomer was performed at a constant crosshead speed of 30 mm/min. The uniaxial tensile stress-strain curves of the porous are shown in Fig. 3B. For comparison, the stress-strain curve for the solid elastomer is also included. Both solid and porous elastomer demonstrated a nonlinear stress-strain relation. Table 1 summarizes the mechanical properties of the two types of elastomers under uniaxial tension. The ultimate tensile strength, the tensile modulus, and the ultimate tensile strain of the porous elastomer were 0.34 [+ or -] 0.01 MPa, 0.46 [+ or -] 0.02 MPa, and 55.8 [+ or -] 0.9%, respectively, whereas, for the solid elastomer, these values were 6.64 [+ or -] 0.16 MPa, 1.96 [+ or -] 0.04 MPa, and 160.8 [+ or -] 2.8%. One can notice that the mechanical strengths and extensibility of the porous elastomer were considerably lower than those of the solid elastomer. This can be attributed to the existence of porous structure.

The modulus and strength of the porous elastomer were also estimated using an additive rule based on the affine deformation assumption. This was achieved by multiplying the modulus or strength of the solid elastomer by the solid volume fraction (equal to one minus porosity). The tensile modulus of the porous elastomer with 56% porosity was calculated to be 0.86 MPa. Likewise, the strength was calculated to be 0.43 MPa. One can notice that the calculated values are considerably larger than the experimental ones, indicating the porous elastomer did not follow the affine deformation. The actual difference between the calculated and experimental stress-strain curves are presented in Fig. 3B. The calculated one largely deviated from the experimental one, and this deviation started from a very small strain that was less than 0.5% (see inset graph of Fig. 3B).

Optical microscopy was used to study in situ morphological changes of the porous elastomer during tensile test. The results are shown in Fig. 4. The tensile force was applied along the lateral direction of the figure. The micrograph in Fig. 4A shows the morphology at the onset of the test at zero strain. The micropore at the center of the micrograph was tracked and highlighted by the white circle. The micropore was round with microchannels connecting to neighboring micropores. Three microchannels are shown in the figure, one with axis nearly aligned along the tensile direction and the other two aligned in the perpendicular direction. From Fig. 4A-E, it can be observed that, as the strain increased, the micropore was elongated along the tension direction and became elliptic. One can also notice that the microchannels with axes perpendicular to the force direction were widened but the third one did not undergo obvious changes. The different deformation modes observed for the micropore and the microchannels seem to conform the schematic shown in Fig. 1. For example, the micropore was elongated by 30% from Fig. 4A-C and by another 25% from Fig. 4C-E, whereas, for the two perpendicular microchannels, these two increments were approximately 66 and 45%. These inhomogeneous changes in the porous structure indicated a nonaffine deformation nature of the porous elastomer. Additionally, the results in Fig. 4 revealed high resilience of the porous elastomer; the recovered structure (Fig. 4F) was almost identical to undeformed one (Fig. 4A).

Cyclic loading-unloading tensile test was performed to study the hysteresis of the porous elastomer under tension. Representative stress-strain curves of the solid elastomer with different crosshead speeds are shown in Fig. 5A. The prestrain was fixed at 0.3 mm/mm. Only small deviation was observed between the standard derivative curve and the mean curve, indicating the hysteresis was insensitive to the change of strain rate. The results in Fig. 5B illustrate the cyclic loading-unloading behavior of the porous elastomer with varied crosshead speeds. Slight strain rate hardening was observed. However, the strain rate seemed to have no influence over the permanent sets after unloading. The effects of prestrain on the hysteresis of the solid and porous elastomers are shown in Fig. 5C and D, respectively. As expected, with increase of preset strain, the maximum tensile stress and the permanent set both increased. The loading-unloading curves were found to be repeatable. Examples of multiple cycles are shown in the inset graphs in Fig. 5C and D. For comparison purposes, the loading-unloading curves for both solid and porous elastomers are plotted in the same figure, Fig. 6. The calculated stress-strain curve from the affine deformation approximation is also shown in the figure. A large deviation was observed between the calculated curve and experimental curve of the porous elastomer. This agreed with the uniaxial tension data previously discussed.

For more quantitative investigation, the hysteresis, H, was represented as the normalized strain energy density difference between the loading and unloading processes:

H = [integral]([[sigma].sub.1] - [[sigma].sub.u])d[epsilon]/[integral][[sigma].sub.1]d[epsilon] (1)

where [[sigma].sub.1] and [[sigma].sub.u] denote loading and unloading stresses, and e is tensile strain. Effectively, the integral on the numerator represents the area enclosed by the loading and unloading curves, and the integral on the denominator stands for the area underneath the loading curve. Table 2 summarizes the results. Generally, the hysteresis of the porous elastomer was considerably higher than that of the solid elastomer. The hysteresis data are also plotted in Fig. 7. From Fig. 7A, it can be seen that the hysteresis of the porous elastomer was quite sensitive to the change of the crosshead speed, whereas the hysteresis of the solid elastomer was almost independent of the crosshead speed. Figure 7B shows that the hysteresis of the porous elastomer was considerably larger than that of the solid elastomer at all preset strains used in the experiment.

Compressive Mechanical Behaviors

Uniaxial compressive test was used to study the compressive mechanical behaviors of the porous elastomer. Figure 8A shows the representative stress-strain curves of the solid elastomer with varied crosshead speeds. Similar to the tensile testing results, the compressive stress-strain curves were nearly independent of strain rate. A fixed crosshead speed (3 mm/min) was, therefore, chosen for determining the stress-strain curves. Figure 8B compares the compressive stress-strain curves of the solid and porous elastomers. Table 3 summarizes the mechanical properties obtained from compressive test. As expected, the compressive properties of the porous elastomer were much lower than those of the solid one. The compressive stress at a preset strain of 0.6 mm/mm and the compressive modulus of the porous elastomer were 0.45 [+ or -] 0.02 and 0.39 [+ or -] 0.01 MPa, respectively, whereas for solid elastomer, these values were 9.46 [+ or -] 0.28 and 1.79 [+ or -] 0.01 MPa. The calculated compressive stress and modulus from the affine deformation approximation were 4.16 and 0.79 MPa, respectively. Comparing these calculated values with the experimental results, one can find that the deformation process under uniaxial compression for the porous elastomer was not affine either. For comparison, the experimental and calculated stress-strain curves were plotted in a single chart, as shown in Fig. 8B. The calculated curve considerably deviated from the experimental one. From the inset figure in Fig. 8B, it can further be seen that the large difference between the two curves existed even at very small strain where linear behavior was expected. Overall, the compressive test showed similar nonaffine behavior as the tensile test. Both indicated that the nonaffine deformation of the micropore and connecting microchannels largely dictated the mechanical behavior of the porous elastomer.

Cyclic loading-unloading compressive test was conducted. Figure 9A shows representative stress-strain curves of solid elastomers with varied crosshead speeds. Like the cyclic tensile test, the cyclic compressive test did not show significant strain-rate dependency for the solid elastomer. However, the strain rate showed a more significant effect for the porous elastomer, as shown in Fig. 9B; with the increase of crosshead speed, the compressive stress at the preset strain and the compressive modulus both significantly increased. Figure 9C and D further compare the different cyclic mechanical behavior between the solid and porous elastomers. The two figures show cyclic stress-strain loops with three different preset strains. The inset figures show multiple cycles at a fixed preset strain of 0.3 mm/mm. As expected, for both solid and porous elastomers, the maximum stress increased with the increase of the preset strain. However, the porous elastomer showed relatively poorer repeatability during multiple cycles. This can be seen by comparing the inset figures in Fig. 9C and D. The different cycles for the solid elastomer fell into a single one but did not for the porous elastomer. Specifically for the porous elastomer, the first loading-unloading cycle had the largest hysteresis, and the permanent set increased with more testing cycles performed.

The experimental stress-strain cycles were also compared with those calculated using the affine deformation approximation. Figure 10 compared the experimental cycle with the calculated cycle at a preset strain of 0.3 mm/mm. The calculated curve significantly differed from the experimental curve of the porous elastomer, and the difference appeared to be even larger than in the tensile test.

Equation 1 was also used to determine the hysteresis quantitatively. The results are summarized in Table 4 and plotted in Fig. 11. The porous elastomer experienced much larger hysteresis than the solid one under all different testing conditions. The hysteresis of the porous elastomer was also found to increase with the increase of strain rate, whereas the corresponding one for the solid elastomer was almost invariant (see Fig. 11 A). The different mechanical behavior between the solid and porous elastomers during uniaxial compressive test further indicated that the microstructure existed in the porous elastomer played an important role. The deviation from affine calculations again supported the nonaffine deformation nature of such porous materials. Further experimental and theoretical studies on the role of the micropore and interconnecting microchannels are needed to quantitatively understand this effect.

Model Fit to Nonlinear Elastic Behavior

An attempt was made to determine a suitable model for fitting the nonlinear stress-strain relation of the porous elastomer. Well-established hyperelastic models including the Arruda-Boyce model, the Neo-Hookean model, the Mooney-Rivlin model, and the One-term Ogden model were tested. The constitutive equations of these models for uniaxial tensile test are provided as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [sigma] is the engineering stress, [C.sub.1] is a material constant, [lambda] is principle stretch, [beta] is the reciprocal of locking principle stretch square, [I.sub.1] is the first invariant of the left Cauchy-Green deformation tensor, and [[alpha].sub.i] is a set of constants ([[alpha].sub.1] = 1/2; [[alpha].sub.2] = 1/20; [[alpha].sub.3] = 11/1050; [[alpha].sub.4] = 19/7000; [[alpha].sub.5] = 519/673750).

Neo-Hookean Model : [sigma] = C([lambda] - 1/[[lambda].sup.2]) (3)

where [C.sub.1] is a material constant.

Mooney-Rivlin Model : [sigma] = (2[C.sub.1] + 2[C.sub.2]/[lambda])([lambda] - 1/[[lambda].sup.2] (4)

where [C.sub.1] and [C.sub.2] are material constants.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [C.sub.1] and [C.sub.2] are material constants.

The fitting plots of the aforementioned models to the stress-strain curves of solid and porous elastomers are shown in Fig. 12. The corresponding fitting statistics are summarized in Table 5. Among the four models used in this study, only the Arruda-Boyce model can satisfactorily fit the experimental data of both solid and porous elastomers, with coefficients of determination ([R.sup.2]) over 0.99.

The Arruda-Boyce Model was established based on an eight chain representation of the underlying macromolecular network structure of elastomer and non-Gaussian behavior of the individual chains [26-28], The excellent fitting shown above indicated that the deformation behavior of a porous structure containing micropores interconnected by microchannels may be analogous to that of elastic networks. A focused study on constitutive modeling of such a porous structure may, therefore, be interesting from fundamental point of view. In future study, it is worthy to carry out analytical modeling and finite element analysis to further understand the nature of this type of deformation.

CONCLUSIONS

In this work, porous polysiloxane having micropores interconnected with microchannels was fabricated and its mechanical behavior was evaluated. The results from mechanical test, in tensile and compressive modes, and optical microscopy both indicated that the deformation process of such porous elastomer was highly nonaffine, meaning microscale deformation does not follow macroscale deformation. Particularly, the stress-strain curves of the porous elastomer largely deviated from the calculated ones from affine deformation. The microscopic observation revealed nonhomogeneous microscale deformation of the porous structure. The extent of deformation was different between the pore and connecting channels. The deformation was also different in channels with different orientation; channels with axis along the direction of the force deformed less, and those with perpendicular axis deformed more. Cyclic loading-unloading tests were also used to study the hysteresis. The hysteresis of the porous elastomer was considerably higher and more sensitive to strain rate change in comparison with the solid elastomer. A further effort was made to fit the tensile stress-strain data. Among different models tested, the Arruda-Boyce model satisfactorily fitted the stress-strain plots of both solid and porous elastomers.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Fred Cook and Dr. Meisha Shofner in Georgia Institute of Technology for the use of Rotap particle sorting machine and Instron testing machine. They would also like to show appreciations to Dr. Jack G. Zhou, Mr. Haibo Gong, and Mrs. Miao Yu in Drexel University for the help in porosity measurement with their MicroCT device.

REFERENCES

[1.] B. Amsden, Soft Matter, 3, 1335 (2007).

[2.] L.Q. Zhang, R. Shi, D.F. Chen, Q.Y. Liu, Y. Wu, X.C. Xu, and W. Tian, hit. J. Mol. Sci., 10, 4223 (2009).

[3.] E.S. Place, J.H. George, C.K. Williams, and M.M. Stevens, Chem. Soc. Rev., 38, 1139 (2009).

[4.] G. Vunjak-Novakovic, R. Maidhof, A. Marsano, and E.J. Lee, Biotechnol. Prog., 26, 565 (2010).

[5.] R.A. Hoshi, S. Behl, and G.A. Ameer, Adv. Mater., 21, 188 (2009).

[6.] J. Zhu, S. Wei, J. Ryu, and Z. Guo, J. Phys. Chem. C, 115, 132115 (2011).

[7.] J.H. Zhu, S.Y. Wei, R. Patil, D. Rutman, A.S. Kucknoor, A. Wang, and Z.H. Guo, Polymer, 52, 1954 (2011).

[8.] J.P. Santerre, S. Sharifpoor, C.A. Simmons, and R.S. Labow, Acta Biomater, 6, 4218 (2010).

[9.] T. Yoshii, A.E. Hafeman, J.S. Nyman, J.M. Esparza, K. Shinomiya, D.M. Spengler, G.R. Mundy, G.E. Gutierrez, and S.A. Guelcher, Tissue Eng. Part A, 16, 2369 (2010).

[10.] P.M. Crapo and Y. Wang, Biomaterials, 31, 1626 (2010).

[11.] Y. Hong, J.G. Zhou, and D. Yao, Adv. Polym. Technol., 31, 21330 (2013).

[12.] M.A. Islam, R.N. Stoicheva, and A. Dimov, J. Membr. Sci., 118, 9 (1996).

[13.] M. Danielsson, D.M. Parks, and M.C. Boyce, Mech. Mater., 36, 347 (2004).

[14.] O. Lopez-Pamies and P.P. Castaneda, J. Mech. Phys. Solids, 55, 1677 (2007).

[15.] O. Lopez-Pamies and P.P. Castaneda, J. Mech. Phys. Solids, 55, 1702 (2007).

[16.] B. Markert, Arch. Comput. Methods Eng., 15, 371 (2008).

[17.] O. Lopez-Pamies and M. Idiart, J. Elast., 95, 99 (2009).

[18.] J. Zhou and W.O. Soboyejo, Mater. Manuf. Processes, 19, 863 (2004).

[19.] R.M. Sullivan, L.J. Ghosn, and B.A. Lerch, Int. J. Solids Struct., 45, 1754 (2008).

[20.] A.D. Brydon, S.G. Bardenhagen, E.A. Miller, and G.T. Seidler, J. Mech. Phys. Solids, 53, 2638 (2005).

[21.] L. Leung, J. Perron, and H.E. Naguib, Polym. Polym. Compos., 15, 437 (2007).

[22.] F. Migneco, Y.C. Huang, R.K. Birla, and S.J. Hollister, Biomaterials, 30, 6479 (2009).

[23.] C.G. Jeong and S.J. Hollister, J. Biomed. Mater. Res. Part B, 93B, 141 (2010).

[24.] A.G. Mitsak, A.M. Dunn, and S.J. Hollister, J. Mech. Beliav. Biomed. Mater., 11, 3 (2012).

[25.] American Society for Testing and Materials (ASTM) Standard D412-06, "Standard Test Methods for Vulcanized Rubber and Thermoplastic Elastomers--Tension" (2008).

[26.] E.M. Arruda and M.C. Boyce, J. Mech. Phys. Solids, 41, 389 (1993).

[27.] A. Ali, M. Hosseini, and B.B. Sahari, Am. J. Eng. Appl. Sci., 3, 232 (2010).

[28.] M. Kroon, J. Elast., 102, 99 (2011).

Yifeng Hong and Donggang Yao

School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30318

Correspondence to: Donggang Yao; e-mail: yao@gatech.edu

Contract grant sponsor: NSF; contract grant number: CMMI-0927697.

DOI 10.1002/pen.23688

Published online in Wiley Online Library (wileyonlinelibrary.com).

TABLE 1. Summary of solid and porous elastomer mechanical proper- ties under uniaxial tensile test. Ultimate Tensile Tensile Sample Strength (MPa) Modulus (MPa) Solid 6.64 [+ or -] 0.16 1.96 [+ or -] 0.04 Porous 0.34 [+ or -] 0.01 0.46 [+ or -] 0.02 Ultimate Tensile Sample Strain (%) Solid 160.8 [+ or -] 2.8 Porous 55.8 [+ or -] 0.9 TABLE 2. Summary of hysteresis of porous and solid elastomers under cyclic tensile tests with different crosshead speeds and pre-set strains. Crosshead 10 mm/min 30 mm/min Speeds Solid 0.072 [+ or -] 0.001 0.072 [+ or -] 0.001 Porous 0.075 [+ or -] 0.003 0.082 [+ or -] 0.002 Pre-set 0.1 mm/mm 0.3 mm/mm Strains Solid 0.065 [+ or -] 0.003 0.072 [+ or -] 0.001 Porous 0.071 [+ or -] 0.003 0.082 [+ or -] 0.002 Crosshead 90 mm/min Speeds Solid 0.072 [+ or -] 0.001 Porous 0.097 [+ or -] 0.003 Pre-set 0.5 mm/mm Strains Solid 0.078 [+ or -] 0.002 Porous 0.088 [+ or -] 0.001 TABLE 3. Summary of solid and porous elastomer mechanical proper- ties under uniaxial compressive test with a pre-set compressive strain of 0.6 mm/mm. Compressive Stress at Compressive Sample Preset Strain (MPa) Modulus (MPa) Solid 9.46 [+ or -] 0.28 1.79 [+ or -] 0.10 Porous 0.45 [+ or -] 0.02 0.39 [+ or -] 0.01 TABLE 4. Summary of hysteresis of porous and solid elastomers under cyclic compressive tests with different crosshead speeds and pre-set strains. Crosshead 1 mm/min 3 mm/min Speeds Solid 0.062 [+ or -] 0.003 0.062 [+ or -] 0.003 Porous 0.101 [+ or -] 0.005 0.117 [+ or -] 0.004 Pre-set 0.1 mm/mm 0.3 mm/mm Strains Solid 0.055 [+ or -] 0.002 0.062 [+ or -] 0.003 Porous 0.090 [+ or -] 0.006 0.117 [+ or -] 0.004 Crosshead 9 mm/min Speeds Solid 0.062 [+ or -] 0.003 Porous 0.138 [+ or -] 0.003 Pre-set 0.5 mm/mm Strains Solid 0.099 [+ or -] 0.005 Porous 0.123 [+ or -] 0.004 TABLE 5. Fitting Sstatistics of Arruda-Boyce model, Neo-Hookean model, Mooney-Rivlin model, and One-term Ogden model. Arruda-Boyce [C.sub.1] Model Solid 4.69 [+ or -] 0.39 Porous 0.059 [+ or -] 0.001 Neo-Hookean [C.sub.1] Model Solid 1.86 [+ or -] 0.10 Porous 0.24 [+ or -] 0.01 Mooney-Rivlin [C.sub.1] Model Solid 2.43 [+ or -] 0.10 Porous 0.32 [+ or -] 0.01 One-term Ogden [C.sub.1] Model Solid 0.17 [+ or -] 0.01 Porous 0.028 [+ or -] 0.002 Arruda-Boyce [beta] [R.sub.2] Model Solid 0.21 [+ or -] 0.01 0.99648 Porous 1.07 [+ or -] 0.01 0.99997 [R.sub.2] Neo-Hookean Model Solid 0.75927 Porous 0.92612 Mooney-Rivlin [C.sub.2] [R.sub.2] Model Solid -3.14 [+ or -] 0.20 0.96638 Porous -0.28 [+ or -] 0.01 0.99390 One-term Ogden [C.sub.2] [R.sub.2] Model Solid 4.86 [+ or -] 0.05 0.99640 Porous -13.57 [+ or -] 0.38 0.96625

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Author: | Hong, Yifeng; Yao, Donggang |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Geographic Code: | 1U5GA |

Date: | Jul 1, 2014 |

Words: | 4593 |

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