The transient response of multiferroic composites

The behavior of cracked multiferroic composites: Fully coupled thermo-electro-magneto-elastic analysis

“…In Patron and Kudryavtesev (1988) and Kuang (2013), the thermo-piezoelectric equations have been derived. By incorporating the magnetic effects, the constitutive equations that govern the behavior of thermo-electro-magneto-elastic material have been established (Aboudi, 2018). These equations express the stress tensor σ ij , electric displacement D i , magnetic induction B i , entropy η and heat flux Q i in terms of the strain tensor ϵ ij , electric field E i , magnetic field H i , and temperature deviation θ from a reference temperature T R as follows…”

“…In the present fully coupled formulation, the mechanical deformation and temperature are governed by the equations of equilibrium and the coupled energy equation, whereas the electric and magnetic fields are governed by the steady-state Maxwell equations, in conjunction with the quasi-electrostatic and quasi-magnetostatic approximations. As to the fully coupled constitutive equations, they are derived by employing the energy balance, the Clausius–Duhem inequality, and a first-order expansion of the Gibbs potential, from which the expressions for the stresses, electric displacements, magnetic inductions, and entropy can be established (see Aboudi, 2018).…”