For a system of field equations of micropolar thermoelasticity we derive a propagation condition for thermoelastic disturbance in a form of monochromatic plane wave in deformation, micro rotation and temperature. The corresponding dispersion relation is given in an explicit form, together with dependence of characteristic coefficients on principal material constants forming the constitutive tensor of isothermal macro and micro elasticity, phenomenological heat conductivity and coupled macroscopic thermoelasticity. It is shown that due to the centro-symmetric nature of microelasticity and particular form of temperature coupling in a free energy function, the separation between the optical and acoustical branch of dispersion relation is inherent. For such systems dispersion relations due to the micropolar fields on one side and macroscopic thermoelastic fields on the other side are completely independent without any cross-coupling.
FoundationsWe consider a material body B in IR 3 defined on B × [0, ∞), so that the time t is within the interval t ∈ [0, ∞). A material point in B is determined by its position vector x relative to the fixed Cartesian coordinate system with the orthonormal basis {e 1 , e 2 , e 3 }, e i · e j = δ ij where δ ij is the Kronecker's delta and the summation convention over repeated indices is implied.A description of a thermoelastic system with microstructure in the linear approximation that corresponds to infinitesimal theory is given by a set of equations where deformation is determined by two independent fields, the displacement field u governed by momentum balance, and the field of micropolar rotation ϕ governed by an equation of rotational momentum balance. Leading equations together with expressions for the asymmetric strain tensor Υ ij , which is conjugate to the stress tensor σ ij , and the torsion curvature tensor κ ij conjugate of the couple stress tensor M ij , are given asVectors X i and Y i represent the body force and body moment as a counterpart to inertia force − ü i (x, t) and moment of inertia −Jφ i (x, t) with designating mass density and J density of microrotation as a new measure of inertia of motion due to microrotation. We assume that the free energy density depends on the fields Υ ij (x, t), κ ij (x, t) as well as on temperature difference θ(x, t) = T (x, t) − T 0 . Conditions Υ ij = 0, κ ij = 0 and T = T 0 define the natural state. In what follows we consider a homogeneous, isotropic and centro-symmetric micropolar continuum. In mathematical terms this means that the medium satisfies the principle of translational invariance, rotational invariance, and invariance with respect to space inversion. For such a continuum, the free energy density admits a representationin which µ and λ are the Lame's constants, c θ is specific heat per unit volume, γ = α θ (3λ + 2µ), α θ is the coefficient of linear thermal expansion while a, b, c and d are new elastic constants. With the help of Eq. (2), by employing the standard thermodynamics formalism, we obtain a coupled set of equati...