March  2019, 8(1): 149-162. doi: 10.3934/eect.2019009

Nonlinear waves in thermoelastic dielectrics

DIBRIS, University of Genoa, 16145 Genoa, Italy

The research leading to this paper has been developed under the auspices of INDAM, Italy

Received  December 2017 Revised  February 2018 Published  March 2019 Early access  January 2019

This paper is addressed to the analysis of wave propagation in electroelastic materials. First the balance equations are reviewed and the entropy inequality is established. Next the constitutive equations are considered for a deformable and heat-conducting dielectric. To allow for discontinuity wave propagation, an appropriate objective rate equation of the heat flux is considered. The thermodynamic consistency of the whole set of constitutive equations is established. Next the nonlinear evolution equations so determined are tested in relation to wave propagation properties. Waves are investigated in the form of weak discontinuities and the whole system of equations for the jumps is obtained. As a particular simple case the propagation into an unperturbed region is examined. Both the classical electromagnetic waves and the thermal waves are found to occur. In both cases the mechanical term is found to be induced by the electrical or the thermal wave discontinuity.

Citation: Angelo Morro. Nonlinear waves in thermoelastic dielectrics. Evolution Equations and Control Theory, 2019, 8 (1) : 149-162. doi: 10.3934/eect.2019009
References:
[1]

R. Bustamante, A. Dorfmann and R. W. Ogden, On electric body forces and Maxwell stresses in nonlinearly electroelastic solids, Int. J. Engng Sci., 47 (2009), 1131-1141. doi: 10.1016/j.ijengsci.2008.10.010.

[2]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.

[3]

L. Dorfmann and R. W. Ogden, Nonlinear electroelasticity, Acta Mech., 174 (2005), 167-183.  doi: 10.1007/s00707-004-0202-2.

[4]

L. Dorfmann and R. W. Ogden, Electroelastic waves in a finitely deformed electroactive material, IMA J. Appl. Math., 75 (2010), 603-636.  doi: 10.1093/imamat/hxq022.

[5]

L. Dorfmann and R. W. Ogden, Nonlinear electroelasticity: Material properties, continuum theory and applications, Proc. R. Soc. A, 473 (2017), 20170311, 34 pp. doi: 10.1098/rspa.2017.0311.

[6]

A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I: Foundations and Solid Media, Springer, New York 1990.

[7] M. E. GurtinE. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2011.  doi: 10.1017/CBO9780511762956.
[8]

D. Griffiths, Electromagnetic theory, American Journal of Physics, 69 (2001), 829. doi: 10.1119/1.1371014.

[9]

A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comp. Modelling, 52 (2010), 1869-1876.  doi: 10.1016/j.mcm.2010.07.021.

[10]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105.  doi: 10.1007/s10659-010-9292-3.

[11]

A. Morro, Thermodynamic consistency of objective rate equations, Mech. Res. Comm., 84 (2017), 72-76.  doi: 10.1016/j.mechrescom.2017.06.008.

[12]

Y.-S. Pao and K. Hutter, Electrodynamics for moving elastic solids and viscous fluids, Proc. IEEE, 63 (1975), 1011-1021.  doi: 10.1109/PROC.1975.9878.

[13]

B. Straughan, Heat Waves, Springer, Berlin, 2011. doi: 10.1007/978-1-4614-0493-4.

[14]

H. F. Tiersten, On the nonlinear equations of thermoelectroelasticity, Int. J. Engng Sci., 9 (1971), 587-604.  doi: 10.1016/0020-7225(71)90062-0.

[15]

R. A. Toupin, A dynamical theory of elastic dielectrics, Int. J. Engng. Sci., 1 (1963), 101-126.  doi: 10.1016/0020-7225(63)90027-2.

[16]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, in "Encyclopedia of Physics", Ⅲ/3, Springer, Berlin, 1965.

[17]

C. Truesdell and R. Toupin, The Classical Field Theories, in "Encyclopedia of Physics", Ⅲ/1, Springer, Berlin, 1960.

[18]

M. W. Zemansky, Heat and Thermodynamics, McGraw-Hill, New York, 1968.

show all references

References:
[1]

R. Bustamante, A. Dorfmann and R. W. Ogden, On electric body forces and Maxwell stresses in nonlinearly electroelastic solids, Int. J. Engng Sci., 47 (2009), 1131-1141. doi: 10.1016/j.ijengsci.2008.10.010.

[2]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.

[3]

L. Dorfmann and R. W. Ogden, Nonlinear electroelasticity, Acta Mech., 174 (2005), 167-183.  doi: 10.1007/s00707-004-0202-2.

[4]

L. Dorfmann and R. W. Ogden, Electroelastic waves in a finitely deformed electroactive material, IMA J. Appl. Math., 75 (2010), 603-636.  doi: 10.1093/imamat/hxq022.

[5]

L. Dorfmann and R. W. Ogden, Nonlinear electroelasticity: Material properties, continuum theory and applications, Proc. R. Soc. A, 473 (2017), 20170311, 34 pp. doi: 10.1098/rspa.2017.0311.

[6]

A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I: Foundations and Solid Media, Springer, New York 1990.

[7] M. E. GurtinE. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2011.  doi: 10.1017/CBO9780511762956.
[8]

D. Griffiths, Electromagnetic theory, American Journal of Physics, 69 (2001), 829. doi: 10.1119/1.1371014.

[9]

A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comp. Modelling, 52 (2010), 1869-1876.  doi: 10.1016/j.mcm.2010.07.021.

[10]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105.  doi: 10.1007/s10659-010-9292-3.

[11]

A. Morro, Thermodynamic consistency of objective rate equations, Mech. Res. Comm., 84 (2017), 72-76.  doi: 10.1016/j.mechrescom.2017.06.008.

[12]

Y.-S. Pao and K. Hutter, Electrodynamics for moving elastic solids and viscous fluids, Proc. IEEE, 63 (1975), 1011-1021.  doi: 10.1109/PROC.1975.9878.

[13]

B. Straughan, Heat Waves, Springer, Berlin, 2011. doi: 10.1007/978-1-4614-0493-4.

[14]

H. F. Tiersten, On the nonlinear equations of thermoelectroelasticity, Int. J. Engng Sci., 9 (1971), 587-604.  doi: 10.1016/0020-7225(71)90062-0.

[15]

R. A. Toupin, A dynamical theory of elastic dielectrics, Int. J. Engng. Sci., 1 (1963), 101-126.  doi: 10.1016/0020-7225(63)90027-2.

[16]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, in "Encyclopedia of Physics", Ⅲ/3, Springer, Berlin, 1965.

[17]

C. Truesdell and R. Toupin, The Classical Field Theories, in "Encyclopedia of Physics", Ⅲ/1, Springer, Berlin, 1960.

[18]

M. W. Zemansky, Heat and Thermodynamics, McGraw-Hill, New York, 1968.

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