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- $$\beta $$ β (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.

中文翻译：

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应用于磁致伸缩材料的非线性和滞后有限元公式

巨磁致伸缩执行器适用于在低磁场下需要大机械位移的应用；例如由稀土铁材料制成的 Terfenol-D 可以产生重要的菌株。但是这些致动器表现出滞后非线性行为，因此很难通过实验表征它们。因此，需要复杂的数值算法来开发计算工具。在这项工作中，开发了有限元方法中的理论和数值公式来模拟磁致伸缩。理论上，在非平衡热力学的框架内，滞后是由德拜记忆弛豫引入的。在数值上，主要的新颖之处在于时间积分，耦合 Newmark- $$\beta $$ β（对于机械）和卷积积分（对于磁本构方程）；非线性是用标准的 Newton-Raphson 算法解决的。本构非线性与麦克斯韦应力张量相结合，二次依赖于磁场。使用解析和实验解决方案验证数值代码；提供了几个例子来证明本配方的能力。