Abstract
Even for a moderate actuation, a large electric voltage requirement hinders the application of electro-active polymers (EAPs) in many areas. Hence, among other mechanisms, the actuation enhancement in EAPs is performed via inclusions of high-dielectric-permittivity fillers in the matrix material in the uncured stage. Moreover, to obtain an optimum advantage from the high-dielectric-permittivity fillers, an electric field can be applied during the curing process which helps the particles to align in a preferred direction. To be specific, recent experimental evidences show that these particles form a dispersed anisotropy rather than a perfect transverse anisotropic structure. The polymer curing process is a complex (visco-) elastic phenomenon where a liquid polymer gradually transforms into a solid macromolecular structure due to cross-linking of the initial solution of short polymer chains. This phase transition comes along with an increase in the material stiffness and a volume shrinkage. In this paper we present a phenomenologically inspired large strain framework for simulating the curing process of particle-filled electro-active polymers with a dispersion-type anisotropy that can work under the influence of an electro-mechanically coupled load. The application of the proposed approach is demonstrated with some numerical examples. These examples illustrate that the model can predict common features in particle-filled dispersed electro-active polymers undergoing curing processes in the presence of an electro-mechanically coupled load.
Similar content being viewed by others
References
Adolf, D.B., Martin, J.E., Chambers, R.S., Burchett, S.N., Guess, T.N.: Stresses during thermoset cure. J. Mater. Res. 13, 530–550 (1998)
Alastrue, V., Martinez, M., Doblare, M., Menzel, M.: Anisotropic micro-sphere-based finite elasticity applied to blood vessel modeling. Int. J. Mech. Phys. Solids 57, 178–203 (2009)
Ask, A., Menzel, A., Ristinma, M.: Phenomenological modeling of viscous electrostrictive polymers. Int. J. Nonlinear Mech. 47(2), 156–165 (2012)
Bazant, Z.P., Oh, B.H.: Efficient numerical integration on the surface of a sphere. Z. Angew. Math. Mech. 66, 37–49 (1986)
Büschel, A., Klinkel, S., Wagner, W.: Dielectric elastomers—numerical modeling of nonlinear visco-elasticity. Int. J. Numer. Methods Eng. 93, 834–856 (2013)
Bustamante, B.: Transversely isotropic nonlinear electro-active elastomers. Acta Mech. 206(3–4), 237–259 (2009)
Carpi, F., Rossi, D.D.: Improvement of electromechanical actuating performances of a silicone dielectric elastomer by dispersion of titanium dioxide powder. IEEE Trans. Dielectr. Electr. Insul. 12, 835–843 (2005)
Carpi, F., Gallone, G., Galantini, F., Rossi, D.D.: Silicone-poly(hexylthiophene) blends as elastomers with enhanced electromechanical transduction properties. Adv. Funct. Mater. 18, 235–241 (2008)
Cortes, D.H., Lake, S.P., Kadlowec, J.A., Soslowsky, L.J., Elliot, D.M.: Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches. Biomech. Model Mechanobiol. 9, 651–658 (2010)
Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity. Acta Mech. 174(3), 167–183 (2005)
Dorfmann, L., Ogden, R.W.: Nonlinear electroelasticity: material properties, continuum theory and applications. Proc. R. Soc. A 473, 20170311 (2017)
Diaconu, I., Dorohoi, D.O., Ciobanu, C.: Eletromechanical response of polyurethane films with different thickness. Roman. J. Phys. 53(1–2), 91–97 (2008)
Dang, Z.M., Yuan, J.K., Zha, J.W., Zhou, T., Li, S.T., Hu, G.H.: Fundamentals, processes and applications of high-permittivity polymer-matrix composites. Prog. Mater. Sci. 57, 660–723 (2012)
Dal, H., Zopf, C., Kaliske, M.: Micro-sphere based viscoplastic constitutive model for uncured green rubber. Int. J. Solids Struct. 132–133, 201–217 (2018)
Dal, H., Kaliske, M.: A micro-continuum-mechanical material model for failure of rubber-like materials: application to ageing-induced fracturing. Int. J. Mech. Phys. Solids 57(8), 1340–1356 (2009)
Dal, H., Cansiz, B., Miehe, C.: A three-scale compressible microsphere model for hyperelastic materials. Int. J. Numer. Methods Eng. 116, 412–433 (2018)
Ehret, A.E., Itskov, M., Schmid, H.: Numerical integration on the sphere and its effect on the material symmetry of constitutive equations—a comparative study. Int. J. Numer. Methods Eng. 81, 189–206 (2010)
Fliege, J., Maier, U.: The distribution of points on the sphere and corresponding cubature formulae. IMA J. Numer. Anal. 19(2), 317–334 (1999)
Gallone, G., Carpi, F., Rossi, D.D., Levita, G., Marchetti, A.: Dielectric constant enhancement in a silicone elastomer filled with lead magnesium niobate-leads titanate. Mater. Sci. Eng. C 27, 110–1162 (2007)
Gillen, K.T.: Effect of cross-links which occur during continuous chemical stress-relaxation. Macromolecules 21, 442–446 (1988)
Hossain, M., Possart, G., Steinmann, P.: A small-strain model to simulate the curing of thermosets. Comput. Mech. 43, 769–779 (2009a)
Hossain, M., Possart, G., Steinmann, P.: A finite strain framework for the simulation of polymer curing. Part I: elasticity. Comput. Mech. 44(5), 621–630 (2009b)
Hossain, M., Possart, G., Steinmann, P.: A finite strain framework for the simulation of polymer curing. Part II: viscoelasticity and shrinkage. Comput. Mech. 46(3), 363–375 (2010)
Hossain, M., Steinmann, P.: Degree of cure-dependent modelling for polymer curing processes at small-strain. Part I: consistent reformulation. Comput. Mech. 53(4), 777–787 (2014)
Hossain, M., Steinmann, P.: Continuum physics of materials with time-dependent properties: reviewing the case of polymer curing. Adv. Appl. Mech. 48, 141–259 (2015)
Hossain, M., Saxena, P., Steinmann, P.: Modelling the mechanical aspects of the curing process of magneto-sensitive elastomeric materials. Int. J. Solids Struct. 58, 257–269 (2015)
Hossain, M., Saxena, P., Steinmann, P.: Modelling the curing process in magneto-sensitive materials: rate-dependence and shrinkage. Int. J. Nonlinear Mech. 74, 108–121 (2015)
Hossain, M., Chatzigeorgiou, G., Meraghni, F., Steinmann, P.: A multi-scale approach to model the curing process in magneto-sensitive polymeric materials. Int. J. Solids Struct. 69–70, 34–44 (2015)
Hossain, M., Vu, D.K., Steinmann, P.: Experimental study and numerical modelling of VHB 4910 polymer. Comput. Mater. Sci. 59, 65–74 (2012)
Hossain, M., Vu, D.K., Steinmann, P.: A comprehensive characterization of the electro-mechanically coupled properties of VHB 4910 polymer. Arch. Appl. Mech. 85(4), 523–537 (2014)
Hossain, M., Steinmann, P.: Modelling electro-active polymers with a dispersion-type anisotropy. Smart Mater. Struct. 27(2), 1–17 (2018)
Heinrich, C., Aldridge, M., Wineman, A.S., Kieffer, J., Waas, A.M., Shahwan, K.W.: The role of curing stresses in subsequent response, damage and failure of textile polymer composites. J. Mech. Phys. Solids 61, 1241–1264 (2013)
Horgan, C.O., Saccomandi, G.: Constitutive models for compressible nonlinearly elastic materials with limiting chain extensibility. J. Elast. 77, 123–138 (2004)
Itskov, M.: On the accuracy of numerical integration over the unit sphere applied to full network models. Comput. Mech. 57(5), 859–865 (2016)
Itskov, M., Khiem, V.N., Waluyo, S.: Electroelasticity of dielectric elastomers based on molecular chain statistics. Math. Mech. Solids (2018). https://doi.org/10.1177/1081286518755846
Johlitz, M., Steeb, H., Diebels, S., Chatzouridou, A., Batal, J., Possart, W.: Experimental and theoretical investigation of nonlinear viscoelastic polyurethane systems. J. Mater. Sci. 42, 9894–9904 (2007)
Koh, S.J.A., Keplinger, C., Li, T., Bauer, S., Suo, Z.: Dielectric elastomer generators: how much energy can be converted? IEEE/ASME Trans Mechatron 16(1), 33–41 (2011)
Kashani, M.R., Javadi, S., Gharavi, N.: Dielectric properties of silicone rubber-titanium dioxide composites prepared by dielectrophoretic assembly of filler particles. Smart Mater. Struct. 19, 1–7 (2010)
Kussmaul, B., Risse, S., Kofod, G., Wache, R., Wegener, M., McCarthy, D.N., Krueger, H., Gerhard, R.: Enhancement of dielectric permittivity and electromechanical response in silicone elastomers: molecular grafting of organic dipoles to the macromolecular network. Adv. Funct. Mater. 21, 4589–4594 (2011)
Keip, M.A., Steinmann, P., Schröder, J.: Two-scale computational homogenization of electro-elasticity at finite strains. Comput. Methods Appl. Mech. Eng. 278, 62–79 (2014)
Kiasat, M.: Curing Shrinkage and Residual Stresses in Viscoelastic Thermosetting Resins and Composites. TU Delft, Delft (2000). PhD Thesis
Landgraf, R., Scherzer, R., Rudolph, M., Ihlemann, J.: Modelling and simulation of adhesive curing processes in bonded piezo metal composites. Comput. Mech. 54(2), 547–565 (2014)
Landgraf, R.: Modeling and simulation of the curing of polymeric materials. Ph.D. dissertation, TU Chemnitz, Germany (2015)
Lion, A., Höfer, P.: On the phenomenological representation of curing phenomena in continuum mechanics. Arch. Mech. 59, 59–89 (2007)
Liu, B., Shaw, M.T.: Electrorheology of filled silicone elastomers. J. Rheol. 45, 641–657 (2011)
Mehnert, M., Hossain, M., Steinmann, P.: On nonlinear thermo-electro-elasticity. Proc. R. Soc. A 472(2190), 20160170 (2016)
Monk, P.: Finite Element Methods for Maxwell Equations. Oxford University Press, Oxford (2003)
Molberg, M., Crespy, D., Rupper, P., Nesch, F., Manson, J.A.E., Loewe, C., Opris, D.M.: High breakdown field dielectric elastomer actuators using encapsulated polyaniline as high dielectric constant filler. Adv. Funct. Mater. 20, 3280–3291 (2010)
Miehe, C., Göktepe, S., Lulei, F.: A micro–macro approach to rubber-like materials: part I, the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617–2660 (2004)
Mahnken, R.: Thermodynamic consistent modeling of polymer curing coupled to viscoelasticity at large strains. Int. J. Solids Struct. 50(13), 2003–2021 (2013)
Nateghi, A., Dal, H., Keip, M.A., Miehe, C.: An affine microsphere approach to modeling strain-induced crystallization in rubbery polymers. Contin. Mech. Thermodyn. 30(3), 485–507 (2018)
Opris, D.M., Molberg, M., Walder, C., Ko, Y.S., Fischer, B., Nuesch, F.A.: New silicone composites for dielectric elastomer actuator applications in competition with acrylic foil. Adv. Funct. Mater. 21, 3531–3539 (2011)
Oliva-Aviles, A.I., Aviles, F., Sosa, V.: Electrical and piezoresistive properties of multi-walled carbon nanotube/polymer composite films aligned by an electric field. Carbon 49, 2989–2997 (2011)
Pandolfi, A., Vasta, M.: Fiber distributed hyper elastic modelling of biological tissues. Mech. Mater. 44, 151–162 (2012)
Brochu, P., Pei, Q.: Advances in dielectric elastomers for actuators and artificial muscles. Macromol. Rapid Commun. 31, 10–36 (2010)
Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998)
Risse, S., Kussmaul, B., Krueger, H., Kofod, G.: Synergistic improvement of actuation properties with compatibilized high permittivity filler. Adv. Funct. Mater. 22, 3958–3962 (2012)
Risse, S., Kussmaul, B., Krueger, H., Kofod, G.: A versatile method for enhancement of electromechanical sensitivity of silicone elastomers. RSC Adv. 2, 9029–9035 (2012)
Romasanta, L.J., Lopez-Manchado, M.A., Verdejo, R.: Increasing the performance of dielectric elastomer actuators: a review from the materials perspective. Prog. Polym. Res. 51, 188–211 (2014)
Steinmann, P., Hossain, M., Possart, G.: Hyperelastic models for rubber-like materials: consistent tangent operators and suitability of Treloar’s data. Arch. Appl. Mech. 82(9), 1183–1217 (2012)
Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Continuum Physics, vol. 1, pp. 239–353. Academic, New York (1971)
Saxena, P., Vu, D.K., Steinmann, P.: On rate-dependent dissipation effects in electro-elasticity. Int. J. Nonlinear Mech. 62, 1–11 (2014)
Saxena, P., Pelteret, J.-P., Steinmann, P.: Modelling of iron-filled magneto-active polymers with a dispersed chain-like microstructure. Eur. J. Mech. A Solids 50, 132–151 (2015)
Skacel, P., Bursa, J.: Comparison of constitutive models of arterial layers with distributed collagen fibre orientations. Acta Bioeng. Biomech. 16(3), 47–58 (2014)
Tomer, V., Randall, C.A.: High field dielectric properties of anisotropic polymer-ceramic composites. J Appl Phys 104, 074106/1–074106/7 (2008)
Thylander, S.: Microsphere-based modeling of electro-active polymers. Ph.D. dissertation, Lund University, Sweden (2016)
Vogel, F.: On the modeling and computation of electro- and magneto-active polymers. Ph.D. dissertation, Friedrich-Alexander-University Erlangen-Nuremberg, Germany (2014)
Vu, D.K., Steinmann, P.: Numerical modeling of non-linear electroelasticity. Int. J. Numer. Methods Eng. 70, 685–704 (2007)
Wissler, M., Mazza, E.: Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators. Sens. Actuators A 134, 494–504 (2007)
Yang, Ta-I, Kofinas, P.: Dielectric properties of polymer nanoparticle composites. Polymer 48, 791–798 (2009)
Womersley, R.S.: Interpolation and cubature on the sphere—UNSW Sydney. https://web.maths.unsw.edu.au/~rsw/Sphere/. Accessed 11 June 2017
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Michael Johlitz, Lucien Laiarinandrasana, Yann Marco.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hossain, M. Modelling the curing process in particle-filled electro-active polymers with a dispersion anisotropy. Continuum Mech. Thermodyn. 32, 351–367 (2020). https://doi.org/10.1007/s00161-019-00747-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-019-00747-5