Finite element formulation for implicit magnetostrictive constitutive relations

Abstract

Magnetostrictive materials that couple mechanical and magnetic domains have been widely explored for use in sensors and actuators. These materials often exhibit a nonlinear material response under applied magnetic fields, which limits the use of linear constitutive models. Furthermore, the nonlinear constitutive relations tend to be implicit in nature. Hence, a finite element scheme that can handle the implicit relationship between the mechanical (stresses and strains) and magnetic (magnetic flux density and magnetic field) quantities is proposed in order to arrive at solutions to boundary value problems. In the proposed scheme, while the physical requirements of equilibrium and strain–displacement relation are satisfied point-wise, the constitutive relations hold in a weak integral sense. A fully coupled magnetostrictive plane stress rectangular element is developed based on the proposed scheme and its efficacy in arriving at solutions to coupled field boundary value problems is illustrated by subjecting the element to standard loading conditions.

Appendix

1.1 Appendix A: Shape function matrix for magnetic scalar potential $\psi$

The Lagrange shape functions assumed for the scalar potential, displacements and geometry are

$$\begin{aligned} N_1&= \frac{(1-s)(1-t)}{4}&N_2&= \frac{(1+s)(1-t)}{4} \end{aligned}$$

(34a)

$$\begin{aligned} N_3&= \frac{(1+s)(1+t)}{4}&N_4&= \frac{(1-s)(1+t)}{4} \end{aligned}$$

(34b)

The shape function matrix for the scalar potential and the [B] matrix for the magnetic field are thus given by

$$\begin{aligned} {[}N^{\psi }]&= \begin{bmatrix} N_1&N_2&N_3&N_4 \end{bmatrix} \end{aligned}$$

(35a)

$$\begin{aligned} {[}B^{\psi }]&= -[J]^{-T}\begin{bmatrix} N_{1,s} &{} N_{2,s} &{} N_{3,s} &{} N_{4,s} \\ N_{1,t} &{} N_{2,t} &{} N_{3,t} &{} N_{4,t} \end{bmatrix} \end{aligned}$$

(35b)

1.2 Appendix B: Shape function matrix for magnetic vector potential A

The shape functions for a non-conformal element are given as

$$\begin{aligned} N_1&= -\frac{(s - 1)(t - 1)(s^2 + s + t^2 + t - 2)}{8} \end{aligned}$$

(36a)

$$\begin{aligned} N_2&= -\frac{(s - 1)^2(s + 1)(t - 1)}{8} \end{aligned}$$

(36b)

$$\begin{aligned} N_3&= -\frac{(s - 1)(t - 1)^2(t + 1)}{8} \end{aligned}$$

(36c)

$$\begin{aligned} N_4&= \frac{(s + 1)(t - 1)(s^2 - s + t^2 + t - 2)}{8} \end{aligned}$$

(36d)

$$\begin{aligned} N_5&= -\frac{(s - 1)(s + 1)^2(t - 1)}{8} \end{aligned}$$

(36e)

$$\begin{aligned} N_6&= \frac{(s + 1)(t - 1)^2(t + 1)}{8} \end{aligned}$$

(36f)

$$\begin{aligned} N_7&= \frac{(s + 1)(t + 1)(- s^2 + s - t^2 + t + 2)}{8} \end{aligned}$$

(36g)

$$\begin{aligned} N_8&= \frac{(s - 1)(s + 1)^2(t + 1)}{8} \end{aligned}$$

(36h)

$$\begin{aligned} N_9&= \frac{(s + 1)(t - 1)(t + 1)^2}{8} \end{aligned}$$

(36i)

$$\begin{aligned} N_{10}&= \frac{(s - 1)(t + 1)(s^2 + s + t^2 - t - 2)}{8} \end{aligned}$$

(36j)

$$\begin{aligned} N_{11}&= \frac{(s - 1)^2(s + 1)(t + 1)}{8} \end{aligned}$$

(36k)

$$\begin{aligned} N_{12}&= -\frac{(s - 1)(t - 1)(t + 1)^2}{8} \end{aligned}$$

(36l)

Thus, the [B] matrix for the magnetic flux is thus given by

$$\begin{aligned} {[}B^A] = \begin{bmatrix} \frac{\partial s}{\partial y} &{} \frac{\partial t}{\partial y} \\ -\frac{\partial s}{\partial x} &{} \frac{\partial t}{\partial x} \end{bmatrix}\begin{bmatrix} N_{1,s} &{} N_{2,s} &{} N_{3,s} &{} \dots &{} N_{12,s} \\ N_{1,t} &{} N_{2,t} &{} N_{3,t} &{} \dots &{} N_{12,t} \end{bmatrix} \begin{bmatrix} \mathbf {T_1} &{} \mathbf {0} &{} \mathbf {0} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {T_1} &{} \mathbf {0} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {0} &{} \mathbf {T_1} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {0} &{} \mathbf {0} &{} \mathbf {T_1} \end{bmatrix} \end{aligned}$$

(37)

where $\mathbf {T_1}=\begin{bmatrix} 1 &{} 0 &{} 0 \\ 0 &{} J_{11} &{} J_{21} \\ 0 &{} J_{12} &{} J_{22} \end{bmatrix}$ and $\mathbf {0}$ is a null matrix of size 3 $\times$ 3.