Quasi-Harmonic Solutions for Transversely Isotropic Magneto-Electro-Thermo-Elasticity: A Symbolic Mathematics Approach

The behaviour of magneto-electro-thermo-elastic materials is governed by a set of six differential equations in which the piezoelectric, the piezomagnetic, the magnetoelectric and thermal effects are coupled. Under the steady-state condition, this system of equations is homogeneous and can be rewritten as a set of uncoupled modified Laplace equations expressed as a function of the roots of a characteristic polynomial associated with the original set of governing equations. Differently from previous proposals, the presented approach employs the entire kernel of the adjoint differential operator so as to preserve completeness. Finally, due to the large number of constitutive parameters involved in the uncoupling process, two Mathematica scripts that compute the coefficients of the characteristic polynomial and the components of the adjoint differential operator are described in full detail.