On the deformations of an electroactive solid subject to a uniform electric field in an orthogonal rheometer
Abstract
In this short paper, we extend the seminal study by Berker [4] of pseudoplanar flows that occur in an orthogonal rheometer, essentially two parallel disks rotating about non-coincident axes at the top and bottom, to the case of an electroactive elastic solid. We obtain the expression for the stress, which is a function of the deformation as well as the electric field, in an electroactive elastic solid using standard representation theorems. We show that in the case of elastic solids, pseudoplanar displacements can take place, with each layer z = constant rotating about a distinct center of rotation. We determine the nature of the locus of the centers of rotation, which can take on profiles that are distinctly different, based on the nature of the electric field, the applied pressure gradient and the rotation of the top and bottom plates.Keywords:
electroactive solid, orthogonal rheometer, pseudo-planar motion, rotation about non-coincident axesReferences
- T. von Karman, Über laminare und turbulente Reibung, ZAMM – Journal of Applied Mathematics and Mechanics, 1, 233–252, 1921.
- G.B. Batchelor, Note on a class of solutions of the Navier–Stokes equations representing rotationally symmetric flow, Quarterly Journal of Applied Mathematics, 4, 29–41, 1951.
- K. Stewartson, On the flow between two rotating co-axial disks, Proceedings of the Cambridge Philosophical Society, 49, 333–341, 1953.
- R. Berker, A new solution of the Navier–Stokes equation for the motion of a fluid contained between two parallel plates rotating about the same axis, Archiwum Mechaniki Stosowanej, 31, 2, 265–280, 1979.
- R. Berker, Integration des Equations du Movement d’un Fluide Visquex, Incompressible, Handbuch der Physik, Vol. VIII/2, Springer-Verlag, Berlin, 1963.
- B. Maxwell, R.P. Chartoff, Studies of a polymer melt in an orthogonal rheometer, Transactions of the Society of Rheology, 9, 1, 41–52, 1965.
- R.B. Bird, E.K. Harris, Jr., Analysis of steady state shearing and stress relaxation in the Maxwell orthogonal rheometer, American Institute of Chemical Engineers Journal, 14, 5, 758–761, 1968.
- L.L. Blyler, Jr., S.J. Kurtz, Analysis of the Maxwell orthogonal rheometer, Journal of Applied Polymer Science, 11, 1, 127–131, 1967.
- E.A. Kearsley, On the flow induced by a Maxwell–Chartoff rheometer, Journal of Research of the National Bureau of Standards, 74, C, 19–20, 1970.
- R.J. Gordon, W.R. Schowalter, On the relation between complex viscosity and steady state shearing in the Maxwell orthogonal rheometer, American Institute of Chemical Engineers Journal, 16, 2, 318–320, 1970.
- C. Goldstein, W.R. Schowalter, Further studies of fluid nonlinearity: The orthogonal rheometer and the oscillating sphere, Transactions of the Society of Rheology, 19, 1, 1–19, 1975.
- F. Ahrens, C. Goldstein, Viscous dissipation in the flow between eccentric rotating disks (orthogonal rheometer), Transactions of the Society of Rheology, 21, 2, 207–217, 1977.
- K.R. Rajagopal, The flow of a second order fluid between rotating parallel plates, Journal of Non-Newtonian Fluid Mechanics, 9, 185–190, 1981.
- K.R. Rajagopal, A.S. Wineman, Flow of a BKZ fluid in an orthogonal rheometer, Journal of Rhedology, 27, 5, 509–516, 1983.
- K.R. Rajagopal, M. Renardy, Y. Renardy, A.S. Wineman, Flow of viscoelastic fluids between plates rotating about distinct axes, Rheologica Acta, 25, 459–467, 1986.
- M. Bower, A.S. Wineman, K.R. Rajagopal, Flow of K-BKZ fluids between parallel plates rotating about distinct axes: Shear thinning and inertial effects, Journal of Non-Newtonian Fluid Mechanics, 22, 3, 289–307, 1987.
- L. Fusi, R. Tozzi, A. Farina, K.R. Rajagopal, Flow of a limited stress fluid between plates rotating about different axes, Acta Mechanica, 234, 12, 6691–6703, 2023.
- K. Yanamundra, C.C. Benjamin, K.R. Rajagopal, Flow of a colloidal solution in an orthogonal rheometer, Physics of Fluids, 36, 043334, 2024, https://doi.org/10.1063/5.0200595.
- K.R. Rajagopal, A.S. Gupta, Flow and stability of a second grade fluid between parallel plates, Archives of Mechanics, 33, 663–674, 1981.
- K.R. Rajagopal, Flow of viscoelastic fluids between rotating disks, Theoretical and Computational Fluid Dynamics, 4, 185–206, 1992.
- K.R. Rajagopal, On the flow of a simple fluid in an orthogonal rheometer, Archive for Rational Mechanics and Analysis, 79, 39–47, 1982.
- W. Noll, Motions with constant stretch history, Archive for Rational Mechanics and Analysis, 11, 97–105, 1962.
- R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4, 323–425, 1955.
- W. Noll, A mathematical theory of the mechanical behavior of continuous media, Archive for Rational Mechanics and Analysis, 2, 197–226, 1958.
- W. Noll, A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48, 1–50, 1972.
- K.R. Rajagopal, A.S. Wineman, New exact solutions in non-linear elasticity, International Journal of Engineering Science, 23, 217–234, 1985.
- M.M. Carroll, K.R. Rajagopal, Pseudo-planar elastic deformations and motions of non-linear elastic solids, Special Volume in honor of Professor R. Berker’s 75th Birthday, Bulletin of the Technical University of Istanbul, 27, 317–332, 1986.
- K.R. Rajagopal, A.S. Wineman, On a class of deformations of a material with nonconvex stored energy functions, Journal of Structural Mechanics, 12, 471–482, 1984.
- K.R. Rajagopal, M. Massoudi, A.S. Wineman, Flow of granular materials between rotating disks, Mechanics Research Communications, 21, 629–634, 1994.
- K.R. Rajagopal, G. Gupta, R.C. Yalamanchili, A rheometer for measuring the properties of granular materials, Particulate Science and Technology, 18, 39–55, 2000.
- K.R. Rajagopal, A.S. Wineman, A constitutive equation for nonlinear electroactive solids, Acta Mechanica, 135, 219–228, 1999.
- K.R. Rajagopal, M. Ruzicka, On the modeling of electrorheological materials, Mechanics Research Communications, 23, 401–407, 1996.
- K.R. Rajagopal, M. Ruzicka, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13, 1, 59–78, 2001.
- R. Bustamante, K.R. Rajagopal, On a new class of electroelastic bodies: Part I, Proceedings of the Royal Society, Series A, London, 469, 20120521, 2013.
- R. Bustamante, K.R. Rajagopal, On a new class of electroelastic bodies: Part 2, Proceedings of the Royal Society, Series A, London, 489, 20130106, 2013.
- K.R. Rajagopal, On implicit constitutive theories, Applications of Mathematics, 28, 4, 279–319, 2003.
- K.R. Rajagopal, Elasticity of elasticity, Zeitschrift für Angewandte Mathematik und Physik, 58, 309–317, 2007.
- K.R. Rajagopal, A.S. Wineman, Universal relations for electroactive solids undergoing shear and triaxial extension, International Journal of Non-Linear Mechanics, 169, 104954, 2025.
- K.R. Rajagopal, A.S. Wineman, New universal relations for nonlinear isotropic elastic materials, Journal of Elasticity, 17, 75–83, 1987.