Linear constitutive equations of a thermopiezomagnetic medium involving mechanical, electrical, magnetic, and thermal elds are presented with the aid of a thermodynamic potential. A thermopiezomagnetic medium can be formed by bonding together a piezoelectric and magnetostrictive composite. Two energy functionals are de ned. It is shown via Hamilton's principle that these functionals yield the equations of motion for the mechanical eld, Maxwell's equilibrium equations for the electrical and magnetic elds, and the generalized heat equation for the thermal eld. Finite element equations for the thermopiezomagnetic media are obtained by using the linear constitutive equations in Hamilton's principle together with the nite element approximations. The nite element equations are utilized on an example two-layer smart structure, which consists of a piezoceramic (barium titanate) layer at the bottom and a magnetoceramic (cobalt ferrite) layer at the top. An electrostatic eld applied to the piezoceramic layer causes strain in the structure. This strain then produces magnetic eld in the magnetoceramic layer.
NomenclatureA = vector of magnetic potential B = vector of magnetic ux density b = matrix of electromagnetic coef cients c = matrix of elastic stiffness coef cients D = vector of electrical displacement E = vector of electrical eld intensity e = matrix of piezoelectric coef cients G = thermodynamic potential H = vector of magnetic eld intensity h = vector of external (applied) heat ux I = area moment of inertia about the neutral axis J= vector of volume current density K = matrix of heat conduction coef cients = matrix of piezomagnetic stress coef cients n = vector of surface normal P = vector of pyroelectric coef cients P b = vector of body forces P c = vector of concentrated forces P s = vector of surface forces q = vector of heat ux r = vector of thermomagnetic coef cients S = strain vector T = stress vector u = displacement vector W = heat source density ® = entropy constitutive coef cient " = matrix of dielectric coef cientś = entropy density 2 = absolute temperature 2 0 = reference temperature µ = small temperature variatioņ = vector of thermal stress coef cients ¹ = matrix of permeability coef cients ½ = mass density
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