Abstract
The aim of this article is to generalize previous works in order to provide a systematic method to derive the equilibrium equations and the constitutive ones for deformable semiconductors accounting for first-order strain, polarization, and magnetization gradients. This is done by use of the “principle of virtual power” subject to the objectivity requirement (i.e., translational and rotational invariances) to which we add the first and the second laws of thermodynamics associated with the conservation of energy and the entropy production. This leads to a generalized expression of the Clausius–Duhem inequality, from which constitutive equations are derived. The interactions of the electromagnetic fields with the deformable and the semiconducting continua appear naturally by generalized non-symmetric stress tensors and the body and surface forces of electromagnetic origin. A comparison with some previous works is made putting emphasis on flexoelectricity that will be dealt with in a future work. Finally, special attention is given to particular cases relative to dissipative phenomena associated with semiconducting properties. In order to be close to what is nowadays done by physicists, the SI units have replaced the Lorentz–Heaviside units often used in previous works.
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References
Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–642 (1968)
Tagantsev, A.K.: Theory of flexoelectric effect in crystals. Zhurnal Eksp. Teor. Fiz. 88, 2108–2122 (1985)
Tagantsev, A.K.: Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys. Rev. B. 34, 5883–5889 (1986)
Tolpygo, K.B.: Physical properties of the salt lattice constructed from deforming ions. J Exp Theor PhysUSSR. 20, 497 (1950)
Tolpygo, K.B.: Physical properties of a rock salt lattice made up of deformable ions. Ukr. J. Phys. 53, 93–102 (2008)
Kogan, S.M.: Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov. Phys.-Solid State 5, 2069–2070 (1964)
Meyer, R.B.: Piezoelectric Effects in Liquid Crystals. Phys. Rev. Lett. 22, 918–921 (1969)
Zubko, P., Catalan, G., Tagantsev, A.K.: Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013)
Yudin, P.V., Tagantsev, A.K.: Fundamentals of flexoelectricity in solids. Nanotechnology. 24, 432001 (2013)
Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: Théorie du second gradient. J Mécanique 12, 236–274 (1973)
Germain, P.: The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)
Collet, B., Maugin, G.A.: Sur l’électrodynamique des milieux continus avec interactions. C.R.A.S. (Compte Rendu à l’Académie des Sciences, Paris) t 279, 379–382 (1974)
Collet, B., Maugin, G.A.: Thermodynamique des milieux continus électromagnétiques avec interactions. C.R.A.S. (Compte Rendu à l’Académie des Sciences, Paris) t 279, 439–442 (1974)
Collet, B.: Sur une théorie des premier et second gradients des milieux continus électromagnétiques. PhD thesis. Univ. P. & M. Curie, Paris (1976)
Maugin, G.A.: A continuum theory of deformable ferromagnetic bodies. I. Field equations. J. Math. Phys. 17, 1727–1738 (1976)
Collet, B.: Higher order surface couplings in elastic ferromagnets. Int. J. Eng. Sci. 16, 349–364 (1978)
Maugin, G.: The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech. 35, 1–70 (1980)
Maugin, G.A., Pouget, J.: Electroacoustic equations for one-domain ferroelectric bodies. J. Acoust. Soc. Am. 68, 575–587 (1980)
Daher, N., Maugin, G.A.: Virtual power and thermodynamics for electromagnetic continua with interfaces. J. Math. Phys. 27, 3022–3035 (1986)
Daher, N., Maugin, G.A.: Deformable semiconductors with interfaces: Basic continuum equations. Int. J. Eng. Sci. 25, 1093–1129 (1987)
Maugin, G.A.: Continuum mechanics of electromagnetic solids. North-Holland, (1988)
Maugin, G.A.: The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Contin. Mech. Thermodyn. 25, 127–146 (2011)
Fousek, J., Cross, L.E., Litvin, D.B.: Possible piezoelectric composites based on the flexoelectric effect. Mater. Lett. 39, 287–291 (1999)
Cross, L.E.: Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients. J. Mater. Sci. 41, 53–63 (2006)
Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B. 77, 125424 (2008)
Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I: Foundations and Solid Media. (1990)
Maugin, G.A., Eringen, A.C.: On the equations of the electrodynamics of deformable bodies of finite extent. J. Mécanique. 16, 101–147 (1977)
Eringen, A.C.: Mechanics of Continua. Krieger Pub Co, Huntington, N.Y. (1980)
Kiréev, P.S., Medvedev, S.: La physique des semiconducteurs. (1975)
Wang, Z., Devel, M., Langlet, R., Dulmet, B.: Electrostatic deflections of cantilevered semiconducting single-walled carbon nanotubes. Phys. Rev. B. 75, 205414 (2007)
Wang, Z., Devel, M.: Electrostatic deflections of cantilevered metallic carbon nanotubes via charge-dipole model. Phys. Rev. B Condens. Matter Mater. Phys. 76, 195434 (2007)
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Lecoutre, G., Daher, N., Devel, M. et al. Principle of virtual power applied to deformable semiconductors with strain, polarization, and magnetization gradients. Acta Mech 228, 1681–1710 (2017). https://doi.org/10.1007/s00707-016-1787-y
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DOI: https://doi.org/10.1007/s00707-016-1787-y