Meccanica

, 44:47 | Cite as

Uniqueness theorem and Hamilton’s principle in linear micropolar thermopiezoelectric/piezomagnetic continuum with two relaxation times

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Abstract

The constitutive laws are obtained for the linear micropolar thermopiezoelectric/piezomagnetic continuum with two relaxation times. A positive decreasing dissipative pyroelectric/pyromagnetic function is introduced and a uniqueness theorem is proved. The equation of motions and the boundary condition are obtained from a Hamilton’s type variational principle. Extended Hamilton’s principle is given.

Keywords

Thermopiezoelectric/piezomagnetic continuum Micropolar theory Generalized thermoelasticity Uniqueness theorem Pyroelectric/pyromagnetic dissipative function Hamilton’s principle Continuum mechanics 

Nomenclature

Bi

Components of magnetic field strength

bi

Coefficients characterizing the lack of a centre of symmetry

c

Speed of light in vacuum

CE

Specific heat at constant strains constant electric and magnetic fields

Cijkl,Dijkl

Elastic moduli

Di

Components of electric displacement vector

Ei

Components of electric field vector

eij

Components of strain tensor

ei=−Θ,i

Thermal intensity

Fi

Mass force

\(\mathcal{G}\)

Pyroelectric/pyromagnetic dissipative function

gijk

Piezoelectric moduli

Hi

Magnetic field intensity

hijk

Piezomagnetic moduli

Jij

Micro-inertia tensor

K

The kinetic energy

kij

Thermal conductivity tensor

L

The Lagrangian function

Mi

Mass couple vector

mij

Components of couple stress tensor

ni

The outer unit vector normal to the surface

Q

Intensity of applied heat source per unit volume

qi

Components of heat flux vector

ri

Components of rotation vector

S

Entropy per unit volume

si

Components of Poynting vector

T

Absolute temperature

t

Time

T0

Reference temperature chosen so that \(\frac{|T-T_{0}|}{T_{0}}\ll 1\)

U

Enthalpy per unit volume

ui

Components of displacement vector

β

\(=\frac{\rho C_{E}}{T_{0}}\)

βij

Thermal moduli

γi

Pyromagnetic moduli

δij

Kronecker’s delta

ε0

Electric constant (permittivity of vacuum)

εij

Components of micro-strain tensor

ε

Internal energy per unit volume

ε1

Internal energy per unit mass

εij

Dielectric moduli (the permittivity tensor).

η

Entropy per unit mass

Θ

=T−T 0

λi

Pyroelectric moduli

μ0

Magnetic constant (permeability of vacuum)

μij

Permeability tensor

νij

Magneto-electric moduli

ρ

Density

σij

Components of stress tensor

τ,ν

Relaxation times

Φ

Free energy density

φ

Electric potential

Ψ

Generalized free energy density

ψ

Magnetic potential

ωi

Components of micro-rotation vector

ωi,j

=ω ij Microcurvature tensor

εijk

Permutation tensor

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematical and Physical SciencesNizwa UniversityNizwaOman

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