Nonlinear Constitutive Relations for Magnetostrictive Materials with Applications to 1-D Problems

Abstract

In this paper, we present a nonlinear constitutive relation for magnetostrictive materials that includes nonlinear coupling effects arising between temperature/preload and magnetic field strengths. The relations are derived from thermodynamic principles using Gibbs free energy expanded in a Taylor series with only the pertinent constants included as determined from experimental evidence present in existing literature. By assuming that the magnetostrictive material is operated in a biased magnetic field and perturbing this field with a small value, relations between the nonlinear material constants and linear coefficients present in existing literature are derived. The accuracy of the nonlinear constitutive relation is evaluated by comparing experimental results obtained on a Terfenol-D rod operating under both magnetic field and stress biases with theoretical values. Results indicate that the model adequately predicts the nonlinear strain/field relations in specific regimes. The nonlinear constitutive relations are also integrated into a 1-dimensional nonlinear finite element model for studying structural components. The coupled magneto elasticity problem is derived using a weighted residual method along with a Newton-Raphson iteration technique. Results of the analytical model indicate that linear approaches are inappropriate for modeling the response of this material in a structure.

A general one-dimension nonlinear magneto-elastic coupled constitutive model for magnetostrictive materials

“…Thus, the theoretical researches of nonlinear magnetostrictive constitutive already have attracted many researchers' interest. For instance, the early standard square model (SS model) proposed by Carman and Mitrovic, 16 the hyperbolic tangent model (HT model) and the model based on density of domain switching (DDS model) proposed by Wan et al, 17,18 all of them can accurately simulate the magnetostrictive strain under low magnetic fields, but can not simulate the saturation phenomenon of magnetostrictive strain in the region of high magnetic field. Then Zheng et al 19 divided the elastic strain produced by pre-stress into nonlinear part associated with the domain rotation and linear part independent of domain rotation, and built Zheng-Liu model for Terfenol-D rod which first achieves an accurate prediction of magnetostrictive strain under different pre-stresses in the region of low, moderate and high field.…”

A Constitutive Model for Magnetostrictive Materials ‐ Theory and Finite Element Implementation

“…They show strong nonlinearities, which have to be considered in the development of constitutive relations. The published phenomenological material models commonly apply the mathematically and non thermodynamically motivated Preisach model [2] or they are based on a higher order thermodynamic potential [3], which leads to many material parameters. The presented constitutive model is thermodynamically consistent and employs only a few material parameters.…”

High-Field Characterization of Piezoelectric and Magnetostrictive Actuators

“…These onedimensional nonlinear models employing the physics of ferromagnetism have been successfully developed and tested against experimental data at low frequencies (Calkins et al, 1997). Another approach to writing nonlinear constitutive equations for the magnetostrictive material is to retain more than the first terms from the series expansion of the magneto-elastic Gibbs free energy (Carman and Mitrovic, 1995), using a parabolic approximation for the strain dependence on the magnetic field. Through finite element solving techniques, this approach is able to capture the magnetostrictive behavior from zero to approximately 75% of the saturation-driving fields for a large range of stresses, less the magnetostriction saturation occurring at higher fields.…”

“…Through finite element solving techniques, this approach is able to capture the magnetostrictive behavior from zero to approximately 75% of the saturation-driving fields for a large range of stresses, less the magnetostriction saturation occurring at higher fields. Carman and Mitrovic (1995) showed that this nonlinear model needs only a constant coupling coefficient at the expense of a strong variation of the compliance coefficient. This numerically efficient approach was implemented for magnetically unbiased magnetostrictive actuator operation (Pratt et al, 1999).…”