From material to structure: Quantifying the structure effect in magnetoelectric devices by coupling energy-based constitutive laws to finite elements

Modelling the anhysteretic magneto-elastic behaviour of a giant magnetostrictive material such as Terfenol-D is not straightforward. The discrete energy-averaged model (DEAM), developed and validated for Galfenol, is cubic and routes the large λ₁₁₁ constant into shear, so it predicts a near-zero longitudinal piezomagnetic coefficient for Terfenol-D unless it is re-parameterized ad hoc. An analytical energy-averaged model was proposed in 2022 for finite-element use, but its closed-form solution, built on the Dawson function, becomes singular under compressive stress and needs an empirically fitted validity limit. We show that this singularity is not physical: the partition integral is always finite, and a direct one-dimensional quadrature evaluates it, together with the magnetization, the magnetostriction and the full piezomagnetic tensor, including the shear coefficient d₁₅, exactly and in closed form for any field and stress. The same three-parameter law reproduces the longitudinal Galfenol response of the DEAM, while the shear channel of cubic materials keeps the DEAM necessary: the two laws are complementary by channel. The law is validated against measured Terfenol-D curves, where adding an anisotropy term to the energy balance improves the agreement. Embedding it at every integration point of an axisymmetric multiphysics finite-element model then quantifies the structure effect, the gap between the material curve and the device response, governed by two dimensionless numbers, χ_0\ N_d for the magnetic effect and k/E/h for the mechanical one, and illustrated on an annular Terfenol-D/PZT current sensor. A frequency study closes the loop: eddy-current screening multiplies the dynamic coefficient by a complex factor, the associated Joule losses heat the material, and the Bloch-law temperature dependence of the constitutive parameters derates the coefficients; thin laminated layers push this limit to the megahertz range.