Non-linear and hysteretical finite element formulation applied to magnetostrictive materials

Abstract

Giant magnetostrictive actuators are suitable for applications requiring large mechanical displacements under low magnetic fields; for instance Terfenol-D made out of rare earth-iron materials can produce important strains. But these actuators exhibit hysteretic non-linear behavior, making it very difficult to experimentally characterize them. Therefore, sophisticated numerical algorithms to develop computational tools are necessary. In this work, theoretical and numerical formulations within the finite element method are developed to simulate magnetostriction. Theoretically, within the framework of non-equilibrium thermodynamics, the hysteresis is introduced by the Debye-memory relaxation. Numerically, the main novelty is the time integration, coupled Newmark-$\beta$ (for mechanical) and convolution integrals (for magnetic constitutive equations); the non-linearity is solved with the standard Newton–Raphson algorithm. Constitutive non-linearities are incorporated with the Maxwell stress tensor, quadratically dependent on the magnetic field. The numerical code is validated using analytical and experimental solutions; several examples are presented to demonstrate the capabilities of the present formulation.

Appendix

For the FE implementation, the tangent stiffness $\varvec{\mathcal {K}}$ and mass $\varvec{\mathcal {M}}$ matrices may be explicitly expressed as:

$$\begin{aligned} \varvec{\mathcal {K}}^{uu}_{ab}&= \displaystyle \int _{\varOmega _e} {\varvec{\mathcal {B}}}^{sy \top }_{a}~\displaystyle \frac{\partial \varvec{\sigma }_{_{\!T}} ^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}} \ \text{ d }\varOmega _e,\nonumber \\ \varvec{\mathcal {K}}^{u\varphi }_{ab}&=\displaystyle \int _{\varOmega _e} {\varvec{\mathcal {B}}}^{sy \top }_{a}~\displaystyle \frac{\partial \varvec{\sigma }_{_{\!T}} ^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1} } \ \text{ d }\varOmega _e,\nonumber \\ \varvec{\mathcal {K}}^{\varphi u}_{ab}&= -\displaystyle \int _{\varOmega _e} {\varvec{\mathcal {B}}}^{\top }_{a}~\displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}} \ \text{ d }\varOmega _e,\nonumber \\ {\mathcal {K}}^{\varphi \varphi }_{ab}&= -\displaystyle \int _{\varOmega _e} {\varvec{\mathcal {B}}}^{\top }_{a}~\displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1} } \ \text{ d }\varOmega _e,\nonumber \\ \varvec{\mathcal {M}}^{uu}_{ab}&=\displaystyle \int _{\varOmega _e} \mathcal {N}_a~\rho _m~\varvec{I}~\mathcal {N}_b \ \text{ d }\varOmega _e. \end{aligned}$$

(46)

First, the mechanic and magnetic derivatives are calculated using (32) and (5) along with (10):

$$\begin{aligned} \displaystyle \frac{\partial \varvec{\sigma }_{_{\!T}} ^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1} }= & {} \mathbf {C}~{\varvec{\mathcal {B}}}^{sy}_{b} + \displaystyle \frac{\partial \varvec{\sigma }_{_{\!Mx}} ^{sy} \vert ^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1} },\nonumber \\ \displaystyle \frac{\partial \varvec{\sigma }_{_{\!T}} ^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}}= & {} (\varvec{e}^\varphi )^\top ~{\varvec{\mathcal {B}}}^{}_{b} + \displaystyle \frac{\partial \varvec{\sigma }_{_{\!Mx}} ^{sy} \vert ^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}} ,\nonumber \\ \displaystyle \frac{\partial \varvec{\sigma }_{_{\!Mx}} ^{sy} \vert ^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}}= & {} \displaystyle \frac{1}{2}~ \bigg [ \displaystyle \frac{\partial \varvec{B}_{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}}\varvec{H}_{n + 1}^\top + \varvec{H}_{n + 1}\left( \displaystyle \frac{\partial \varvec{B}_{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}}\right) ^\top \nonumber \\&-\displaystyle \frac{2}{\mu _0}\displaystyle \frac{\partial \varvec{B}_{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1}}\varvec{B}_{n + 1}^\top \varvec{I}\bigg ],\nonumber \\ \displaystyle \frac{\partial \varvec{\sigma }_{_{\!Mx}} ^{sy} \vert ^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}}= & {} \displaystyle \frac{1}{2}~ \bigg [\displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}}(\varvec{H}^{n + 1})^\top + \varvec{H}^{n + 1}\left( \displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}} \right) ^\top \nonumber \\&-\varvec{B}^{n + 1}~{\varvec{\mathcal {B}}}^{\top }_{b} -{\varvec{\mathcal {B}}}^{}_{b}~(\varvec{B}^{n + 1})^\top \nonumber \\&-\displaystyle \frac{2}{\mu _0}\displaystyle \frac{\partial \varvec{B}_{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}}\varvec{B}_{n + 1}^\top \varvec{I}\bigg ]. \end{aligned}$$

(47)

Second, the time-dependent derivatives for the magnetic induction are obtained from (39) to give:

$$\begin{aligned} \begin{array}{lllllllr} \displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varvec{u}}_b^{n + 1} } = \displaystyle \Big [1 - \phi (\varDelta \tau )\Big ]~\varvec{e}^\varphi ~{\varvec{\mathcal {B}}}^{sy}_{b},\\ \displaystyle \frac{\partial \varvec{B}^{n + 1}}{\partial \tilde{\varphi }_b^{n + 1}} = -\displaystyle \Big [1 - \phi (\varDelta \tau )\Big ]~\varvec{\mu }~{\varvec{\mathcal {B}}}^{}_{b} . \end{array} \end{aligned}$$

(48)

Finally, this numerical formulation is implemented in the FE research software FEAP [19].