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Abstract

A novel lumped-element modeling approach for magnetic shape memory alloys is presented. Building on concepts borrowed from rate-independent plasticity, the model describes the magnetic and magneto-mechanical behavior of a magnetic shape memory component subjected to a particular load case in a thermodynamically consistent way. The approach remedies the common issues of existing models regarding the representation of inner hysteresis loops and small-signal behavior. The model is parametrized in terms of a small number of parameters, which can be determined from single variant magnetic curves and mechanical first order reversal curves at constant magnetic input. The results indicate that the model predicts the magnetic and magneto-mechanical behavior with sufficient accuracy, which makes it appropriate for system design.

Keywords

Magnetic shape memory (MSM) actuators lumped-element model hysteresis model

1. Introduction

Single crystal magnetic shape memory alloys (MSMA), such as NiMnGa, exhibit comparatively large strains of typ. 6% in response to moderate external magnetic fields well below 1 MA/m due to twin boundary motion within their martensitic phase. For this reason, they are considered promising candidates for active elements in compact electromagnetic actuators, sensors, generators, and dampers. Since the first works in the middle of the 1990s (James and Wuttig, 1998; Ullakko et al., 1996), this class of active materials has gained a remarkable scientific attention, and a significant improvement with regard to alloy composition and magneto-mechanical properties (Wilson et al., 2007), fabrication technology (Kellis et al., 2012; Pagounis and Laufenberg, 2012) as well as operation temperature (Pagounis et al., 2014) has been attained. Despite these advances, the technology has not entered the market yet, although a multitude of demonstrators has been introduced for a variety of possible applications (Pagounis and Müllner, 2014; Pagounis and Schmidt, 2012; Schmidt and Quandt, 2010). One of the main technical reasons for this lack of commercialization appears to be the complexity of the design process of MSM-based actuator systems related to the nonlinear and hysteretic magneto-mechanical coupling of MSMA. Further complication arises from the fact that the performance of MSM actuators is a result of the mutual interaction of all contributing subsystems and mostly a tradeoff between contradictory design objectives. Taking these aspects into account, the need for a model-based design of MSM actuators is self-evident.

Different modeling approaches for the magneto-mechanical behavior of single crystal MSMA (mostly NiMnGa) have been published in the past. These may be roughly categorized into microscopic, macroscopic, state-space and lumped-element models (LEM).

A comprehensive overview of published constitutive models and how they build on each other can be found in Jafarzadeh and Kadkhodaei (2017), Kiang and Tong (2005), and LaMaster et al. (2015). Most of these microscopic and macroscopic models are based on continuum thermodynamics and on the minimization of free energy.

Microscopic models like Ahluwalia et al. (2006), James and Wuttig (1998), Mennerich et al. (2011), and Peng et al. (2016) consider the material behavior on the microscopic level with an explicit representation of the twinned structure. Usually, driving forces for twin boundary motion are computed as a function of the external loading, the material properties of the different domains, and their geometric arrangement. These models can enhance the basic understanding of coupling and allow for computing the load-dependent development of the twin structure evolution. However, due to the length scale to be resolved microscopic models are usually too expensive for actuator design with regard to computational cost.

The majority of published models describes the material state in terms of macroscopic quantities, which are based on a homogenization over the actual twin structure. The reader is referred to the papers mentioned above for an overview. LaMaster et al. (2015) note that a considerable part of the published models is purely two-dimensional, and many of them are restricted to load cases where magnetic and mechanical load are perpendicular to each other. Further, only one load is allowed to change in many models. These shortcomings have been overcome by several authors, for example, Bartel et al. (2021) and LaMaster et al. (2015). A difficulty is posed by the fact that twin boundaries typically extend along the entire sample width and, in addition, also the formation of broad twins has been reported (Aaltio et al., 2010; Marioni et al., 2004). As a consequence, the preconditions for homogenization implicit in many macroscopic models are not fulfilled. Moreover, the system response can be considered only in a very limited manner.

State-space system models are typically used for control applications, for example, for hysteresis compensation in position control (Binetti et al., 2015; Riccardi et al., 2012; Ruderman and Bertram, 2014). Usually, the entire electro-mechanical actuator behavior of an already existing MSM actuators is considered. Frequently, approaches are based on modified Preisach and Prandtl-Ishlinskii models. Although these models exhibit an exceptional prediction accuracy, they can hardly be used for system design since their prediction is limited to the particular (already existing) actuator used for parameter identification.

LEM consider the integral behavior of an MSM component subjected to a particular load case based on differential-algebraic equations without a spatial resolution of fields. Since the physical subsystems are explicitly represented, at least in a simplified manner, system design is readily possible. Although most of the models cited in the sequel are referred to as “one-dimensional constitutive models” with scalar input and output are formulated in terms of field quantities they consider only suitably defined effective values of the latter. Therefore, they are classified as LEM in this work.

Several authors adapted modeling approaches originally developed for thermally activated shape memory alloys (SMA) to MSMA (Bindl et al., 2012; Couch and Chopra, 2007; Guo et al., 2014). These approaches compute the martensitic volume fraction from approximation functions depending on load, direction, and current state. Similarly, the Tellinen hysteresis model, originally introduced to model the scalar ferromagnetic hysteresis, has, in a univariate (Suorsa et al., 2004) and a bivariate (Ziske et al., 2015) form, been adapted to MSM modeling. For the last two, only limiting hysteresis curves and surfaces, respectively, are required for their full parametrization, which is experimentally laborious for the bivariate version. Naturally, the large-signal behavior¹ is well represented in this class of models, especially if data are directly interpolated. In contrast, small-signal behavior (see Note 1) is poorly predicted to some extent, and, under certain conditions, an unphysical model behavior such as clockwise hysteresis drifting under cyclic loading occurs (Ziske et al., 2015). Also, thermodynamic consistency is not always ensured. Moreover, the magnetic behavior is not even considered in many models (Bindl et al., 2012; Couch and Chopra, 2007; Guo et al., 2014; Ziske et al., 2015). It has been attempted to overcome these issues with vector Preisach hysteresis models (Adly et al., 2006; Visone et al., 2010). However, implementation is complex, and parameterization is laborious since a considerable amount of first-order reversal curves (FORCs) in the full bivariate input space is required. LEMs building on continuum thermodynamics have been presented (Gauthier et al., 2007; Sarawate and Dapino, 2010). However, the agreement with practically relevant loading scenarios is only moderate and an appropriate minor loop prediction is not proven.

In early product design stages different principal variants of different underlying operating principles need to be evaluated regarding their desired system function. Usually, main dimensions and parameters are already determined, but a detailed design or layout does not yet exist. A model-based evaluation must consider this on an abstract level in consideration of all involved subsystems. An example might be the comparison of an electromagnetic and an MSM circuit breaker for a certain current range in several operating conditions. Modeling approaches based on generalized Kirchhoffian networks (LEM) are well suited to this purpose. Therefore, the latter approach is chosen below. In this context, it is aimed to use a minimum number of fitting parameters on the one hand while ensuring thermodynamic consistency and providing a sufficient prediction accuracy of large- and small-signal behavior on the other. The paper is structured as follows: Firstly, the modeling approach is described. Secondly, the necessary temporal discretization is discussed along with implementation aspects. Thirdly, the identification of parameters is exemplified. Finally, the model is verified by comparison with experimental data.

2. Model description

2.1. Chosen approach

Within this work, an approach based on rate-independent plasticity (Prandtl-Reuss) is chosen, which has been successfully extended to other coupled field problems in the past, especially to ferroelectric ceramics, see for example, Landis (2002). This procedure is transferred to the present problem in a dimensionally reduced form as a LEM. In particular, the magneto-mechanical terminal behavior of a specific sample (“component”) in a specific load case is considered, in which magnetic and mechanical loads are perpendicular to each other (Holz et al., 2012). Further, to emphasize the lumped-element character of the proposed model, we exclusively use component-related quantities (integral quantities, see below), which can also be measured directly (Ehle et al., 2020, 2021).

2.2. Considered load case

A cuboid, single-crystal MSM component with the volume

V = l_{0} d_{0} w,

(1)

where $l_{0}$ , $d_{0}$ and $w$ are component length, width, and depth in the fully compressed state, respectively, is situated in the air gap of a highly permeable, virtually closed magnetic core with pole dimensions $L > l_{\max}$ , $W > w$ and $D > ~ d_{0}$ (see Figure 1). Here, $l_{\max}$ is the component length in the fully elongated state. The core is excited by an arbitrary magnetomotive force driving an (integral) magnetic flux $Φ$ through air gap and MSM component. For the sake of simplicity, it is assumed that the magnetic core behaves magnetically reversible and remains unsaturated for the investigated excitation range. The pole faces are further assumed to be equipotential faces. Consequently, a magnetic voltage $V_{m}$ ² can be defined, dropping across air gap and MSM component.

Figure 1.

Schematic of the considered load case with relevant quantities and dimensions.

The MSM component is supported on one side of the mechanical loading axis, and an external force $F$ is applied on the other (with positive values of $F$ corresponding to tension). In response to the applied loads, a mechanically/magnetically induced reorientation (MIR) of twin variants occurs, and an elongation $Δ l$ is measurable along the mechanical loading axis. Since one of the sample faces is supported, a displacement of the mechanically free sample side

u = Δ l

(2)

can be defined, and the component length

l = l_{0} + u

(3)

changes accordingly. As discussed in more detail below, it is assumed that MIR preserves the volume and does not affect the depth $w$ of the sample. Therefore, the axial shape change affects the MSM component width according to

d = \frac{d_{0} l_{0}}{l_{0} + u} .

(4)

As part of this approximation, the area of the magnetically permeated component face also depends on the displacement:

A_{Φ} (u) = lw = (l_{0} + u) w .

(5)

Due to the non-negligible shape change of typ. 6% and more, the magnetic energy within the air gap is also displacement-dependent and contributes to the driving forces for MIR. For this reason, the magnetic behavior of the entire air gap will be considered, which is common practice in the modeling of conventional electromagnetic actuators. This unit comprising MSM component and air gap will be referred to as “MSM unit” below. As a positive side effect, Maxwell force contributions are automatically included in the considerations.

2.3. Additional assumptions

The following fundamental assumptions are additionally made for the model described below, based on the specific characteristics of the considered load-case, the aimed application, and experimental observations:

The elasticity of the component is neglected, so that the entire deformation of the component is ascribed to the motion of twin boundaries, since, for typically applied loads, the ratio between elastic and reorientation strain

\frac{ε^{el}}{ε^{r}} \approx \frac{σ}{E ε^{r}} = 2.5 ‰

(6)

is negligible. The latter result is calculated with an applied compressive stress of $σ = 3 MPa$ , an elastic modulus of $E = 20 GPa$ in single-variant state, and an effective maximum reorientation strain of $ε^{r} = 6 %$ . Furthermore, the corresponding elastic driving forces are negligible since the ratio between elastic and dissipative energy density ( $w^{el}$ and $w^{r}$ ) is about

\frac{w^{el}}{w^{r}} \approx \frac{σ^{2}}{2 E σ^{tw} ε^{r}} \approx 5 ‰

(7)

for the values assumed above and a twinning stress of $σ^{tw} = 0.7 MPa$ .

The time constants of practical magnetic circuits (defined by resistance and inductance) and mechanical loading mechanisms (defined by moving mass and reset spring stiffness) both typically lie in the single to double-digit millisecond range, as the evaluation of representative publications indicates (Faran et al., 2017, Henry et al., 2003, Minorowicz et al., 2016, Riccardi et al., 2014). This lowpass behavior limits both the step response and the large-signal magneto-mechanical frequency response to a few milliseconds and a few hundred hertz, respectively, see Aaltio and Ullakko (2000), Faran et al. (2017), Henry et al. (2003), and Tellinen et al. (2002). The practically attainable response times are thus at least two magnitudes larger than the reported intrinsic ones (Marioni et al., 2003, Saren et al., 2016). Thus, system dynamics will mainly determine the transient response of MSM actuators. Therefore, the lumped-element MSM component model is chosen to be rate independent.

The elongation of the component is changed thermodynamically irreversibly by MIR, while the change of magnetization is a thermodynamically reversible process in the absence of twin boundary motion. The latter assumption is based on the experimental observation of a negligible magnetic hysteresis in single variant states (Heczko et al., 2015).

Consistent with experimental results of tensile-compressive tests of different martensitic phases and twin boundary types (Söderberg et al., 2004, Straka et al., 2010, Suorsa, 2005), the twin boundary motion is associated with kinematic hardening behavior. That is, about twice the twinning stress needs to be overcome to reinduce reorientation upon load reversal.

The reorientation of martensitic variants is assumed to be volume-conserving.

Typically, investigated 5M NiMnGa single-crystal are cut along their {100} planes and undergo magneto-mechanical training in order to create a preferable twin structure, ideally consisting of only one type of twin boundaries and two types of martensitic variants (Heczko et al., 2009). Thus, it is assumed that the magnetic easy axes of the variants lie in the plane spanned by the magnetic and mechanical loading axes and are either parallel or perpendicular to the magnetic and mechanical loading axes.

Isothermal conditions are assumed throughout.

The intuitive viewpoint is adopted that the elongation of the MSM component is related linearly to the microstructural volume fraction of martensitic variants having their magnetic easy axis aligned with the magnetic loading axis.

Below, the general framework used to describe the response of an MSM unit for the time interval $[t_{0}, t_{1}]$ is described, where $t_{0}$ and $t_{1}$ are the initial and the final time, respectively.

2.4. Description of the thermodynamic state

The thermodynamic state of the MSM unit is described in terms of the state variables $u$ and $Φ$ , both being considered as the primary unknowns of the problem. Effects of the loading history are captured through appropriate evolution equations for the elongation $u = u (t)$ (with $t$ being the time) as described below. The elongation is measured w.r.t. the hypothetical state of a single variant with its magnetically hard axis aligned with the magnetic loading axis. As the initial state need not be a single-variant state, $u (t_{0}) \neq 0$ in general. For the single-variant state with its magnetic easy axis aligned with the magnetic loading axis, the displacement is assumed to take the value $u = u^{\max}$ , so that $u$ can take values in between $0$ and $u^{\max}$ .

2.5. Helmholtz free energy and external work

The Helmholtz free energy $Ψ$ of the MSM unit is assumed to be a function of the thermodynamic state according to

Ψ (u, Φ) = Ψ^{mag} (u, Φ) + Ψ^{r} (u),

(8)

where $Ψ^{mag}$ is the magnetic contribution to the Helmholtz free energy and $Ψ^{r}$ will be used to describe the kinematic hardening behavior. In particular, the latter will be chosen such that states $u$ outside the range $(0, u^{\max})$ will be energetically unattainable. Moreover, the Helmholtz free energy will be assumed to be strongly convex. The influence of the environment on the behavior of the MSM unit is incorporated through the work done on the unit, the rate of which is assumed to be given by the power

P = F \overset{\cdot}{u} + V_{m} \overset{\cdot}{Φ} .

(9)

Here and henceforth, a superimposed dot indicates the rate of a quantity.

2.6. Second law of thermodynamics

For the assumed isothermal situation, the second law of thermodynamics demands that

P - \overset{\cdot}{Ψ} \geq 0 .

(10)

Using (8) and (9), this can be rewritten as

(F - \frac{\partial Ψ}{\partial u}) \overset{\cdot}{u} + (V_{m} - \frac{\partial Ψ}{\partial Φ}) \overset{\cdot}{Φ} \geq 0 .

(11)

Based on the assumptions that the change of magnetization is a thermodynamically reversible process, while the motion of twin boundaries is a thermodynamically irreversible process, this inequality is satisfied as follows:

1. The magnetic voltage is related to the Helmholtz free energy by

V_{m} = \frac{\partial Ψ}{\partial Φ} .

(12)

2. With regard to the mechanical part, a differentiable, convex “reorientation function”

$φ = φ (\hat{F})$ is introduced, with

\hat{F} = F - \frac{\partial Ψ}{\partial u}

(13)

being the thermodynamic driving force for twin boundary motion. The reorientation function is assumed to be minimal for $\hat{F} = 0$ , with $φ (0) < 0$ . Based on this reorientation function, the rate $\overset{\cdot}{u}$ is related to the corresponding driving force $\hat{F}$ as follows:

\overset{\cdot}{u} = {\begin{matrix} 0 & \begin{matrix} if \end{matrix} & φ < 0 \\ λ \frac{d φ}{d \hat{F}} & if & φ = 0 . \end{matrix}

(14)

Here, $λ$ is the reorientation multiplier and $\overset{\cdot}{φ}$ and $λ$ must satisfy the customary Karush-Kuhn-Tucker conditions

\begin{matrix} λ \geq 0 \\ \overset{\cdot}{φ} \leq 0 \\ λ \overset{\cdot}{φ} = 0 \end{matrix}

(15)

if $φ = 0 .$

2.7. Co-energy

Due to the specifics of the magnetic behavior, it later turns out to be advantageous to work with the co-energy defined through the Legendre transformation

\begin{matrix} Ψ' (u, V_{m}) = sup_{Φ} [V_{m} Φ - Ψ (u, Φ)] \\ = sup_{Φ} [V_{m} Φ - Ψ^{mag} (u, Φ)] - Ψ^{r} (u) \\ = Ψ'^{, mag} (u, V_{m}) - Ψ^{r} (u) \end{matrix}

(16)

instead of directly with the Helmholtz free energy $Ψ$ . Using $Ψ'$ , (12) and (13) are replaced by the equivalent relations

Φ = \frac{\partial Ψ'}{\partial V_{m}}

(17)

and

\hat{F} = F + \frac{\partial Ψ'}{\partial u} .

(18)

In particular, below the relation

\begin{matrix} Ψ'^{, mag} (u, V_{m}) = \frac{u}{u^{\max}} Ψ'^{, mag, e} (V_{m}) \\ + (1 - \frac{u}{u^{\max}}) Ψ'^{, mag, h} (V_{m}) \end{matrix}

(19)

will be used for the magnetic contribution of the co-energy, where it is assumed that both, $Ψ'^{, mag, e}$ and $Ψ'^{, mag, h},$ are strongly convex and possess a minimum at $V_{m} = 0$ . It is remarked that the relation (19) is based upon an empirical linear interpolation between the states associated with $u = 0$ and $u = u^{\max}$ .

2.8. Summary

Assuming that appropriate forms for $Ψ'$ and $φ$ are known, the response of the MSM unit in terms of the unknown functions $u (t)$ and $Φ (t)$ upon given, smooth histories $V_{m} (t)$ and $F (t)$ are described by the following combined set of equations:

\begin{matrix} 0 = Φ - \frac{\partial Ψ'}{\partial V_{m}} \\ \hat{F} = F + \frac{\partial Ψ'}{\partial u} \\ \overset{\cdot}{u} = {\begin{matrix} 0 & \begin{matrix} if \end{matrix} & φ < 0 \\ max {0, \frac{\frac{d φ}{d \hat{F}} (\overset{\cdot}{F} + \frac{\partial^{2} Ψ'}{\partial u \partial V_{m}} {\overset{\cdot}{V}}_{m})}{\frac{d φ}{d \hat{F}} \frac{\partial^{2} Ψ^{r}}{\partial u^{2}} \frac{d φ}{d \hat{F}}}} \frac{d φ}{d \hat{F}} & if & φ = 0 . \end{matrix} \end{matrix}

(20)

It is remarked that the last equation describing the evolution of $\overset{\cdot}{u}$ can be obtained by combining (14), (15), and (18). Furthermore, it has been utilized that

\frac{\partial^{2} Ψ'}{\partial u^{2}} = - \frac{\partial^{2} Ψ^{r}}{\partial u^{2}},

(21)

which follows from (16) and the specific form (19) assumed for the magnetic contribution of the co-energy. When using the set of equations (20), it is generally assumed that the prescribed force $F$ at the initial time $t_{0}$ is such that the initial state is admissible in that $φ \leq 0$ initially.

2.9. Model extension

The model described above is unable to accurately describe minor hysteresis loops. In order to remedy this issue, an extension similar to the Mróz model known from plasticity (Mróz, 1967) is proposed. In particular, the presence of $n = 1 \dots N$ “reorientation systems” is assumed, with each contributing equally to the total elongation according to

u = \sum_{n = 1}^{N} u^{(n)} .

(22)

In this equation $u^{(n)}$ is the contribution of the $n$ th reorientation system to the total elongation. It is assumed that the maximum contribution is identical for all reorientation systems, that is, $u^{(n)}$ is in the range between $0$ and $u^{\max} / N$ . It is noted that this approach is empiric and not directly linked to structural mechanisms. With regard to the Helmholtz free energy, the modified form

Ψ (u^{(1)}, \dots, u^{(N)}, Φ) = Ψ^{mag} (u, Φ) + \sum_{n = 1}^{N} Ψ^{r (n)} (u^{(n)})

(23)

is assumed, where $u$ is given by (22), and $Ψ^{r (n)}$ is the hardening related contribution associated with the $n$ th reorientation system. Since $Ψ^{r (n)}$ is only a function of $u^{(n)}$ , no crosshardening effects are considered. Similar to the formulation in the state variables $(u, Φ)$ the latter will be used to enforce the condition that $u^{(n)}$ stays in the range $(0, u^{\max} / N)$ . Moreover, the co-energy

\begin{matrix} Ψ' (u^{(1)}, \dots, u^{(N)}, V_{m}) = sup_{Φ} [V_{m} Φ - Ψ (u^{(1)}, \dots, u^{(n)}, Φ)] \\ = sup_{Φ} [V_{m} Φ - Ψ^{mag} (u, Φ)] - \sum_{n = 1}^{N} Ψ^{r (n)} (u^{(n)}) \\ = Ψ'^{, mag} (u, V_{m}) - \sum_{n = 1}^{N} Ψ^{r (n)} (u^{(n)}) \end{matrix}

(24)

is introduced again, with $Ψ'^{, mag}$ still given through (19). The same procedure as before is used to satisfy the second law of thermodynamics. In particular, $n = 1 \dots N$ reorientations function $φ^{(n)} ({\hat{F}}^{(n)})$ are introduced, with ${\hat{F}}^{(n)}$ being the thermodynamic driving force associated with the $n$ th reorientation system. Then, the governing set of equations becomes

\begin{matrix} 0 = Φ - \frac{\partial Ψ'}{\partial V_{m}} \\ {\hat{F}}^{(n)} = F + \frac{\partial Ψ'}{\partial u^{(n)}} \\ {\overset{\cdot}{u}}^{(n)} = \\ {\begin{matrix} 0 & \begin{matrix} if \end{matrix} & φ^{(n)} < 0 \\ max {0, \frac{\frac{d φ^{(n)}}{d {\hat{F}}^{(n)}} (\overset{\cdot}{F} + \frac{\partial^{2} Ψ'}{\partial u^{(n)} \partial V_{m}} {\overset{\cdot}{V}}_{m})}{\frac{d φ^{(n)}}{d {\hat{F}}^{(n)}} \frac{\partial^{2} Ψ^{r (n)}}{\partial {u^{(n)}}^{2}} \frac{d φ^{(n)}}{d {\hat{F}}^{(n)}}}} \frac{d φ^{(n)}}{d {\hat{F}}^{(n)}} & if & φ^{(n)} = 0 . \end{matrix} \end{matrix}

(25)

As before, it is assumed that the prescribed loading at time $t = t_{0}$ and the initial state are such that $φ^{(n)} \leq 0$ initially. It is also emphasized at this point that the decoupling of the different reorientation systems evident in (25) is caused by the particular choice for the magnetic co-energy and the neglect of crosshardening. This choice leads to a situation where the driving force ${\hat{F}}^{(n)}$ of the $n$ th reorientation system is only a function of $u^{(n)}$ and $V_{m}$ , but does not depend on the reorientation displacements of other reorientation systems. This aspect strongly simplifies the practical implementation of the model, while it is at the same time believed to give an appropriate description of the phenomena under consideration. It is finally noted that the extended formulation collapses to the original formulation if $n = 1$ . Hence, only the extended model will be discussed in the sequel.

3. Temporal discretization and implementation

As the set of equations (25) can usually not be solved analytically, temporal discretization is necessary. In this context, it is assumed that the loading in terms of $V_{m} (t)$ , $F (t)$ is prescribed and that approximate solutions for $u^{(n)} (t), Φ (t)$ are to be computed, corresponding to the typical load case to be modeled. For the latter purpose, the time interval $[t_{0}, t_{1}]$ is divided into discrete time steps. In particular, a time step between the discrete times $t^{(k - 1)}$ and $t^{(k)}$ is considered below. The loading at a discrete time $t^{(k)}$ is given by $V_{m} (t^{(k)})$ , $F (t^{(k)})$ , and the approximate solution for $u^{(n)} (t^{(k)})$ , $Φ (t^{(k)})$ will be denoted by $u^{(n) (k)}$ , $Φ^{(k)}$ .

The backward Euler method is used for temporal discretization. In particular, let

{\hat{F}}^{(n) (k)} (u^{(n)}) = {\hat{F}}^{(n)} [u^{(n)}, F (t^{(k)}), V_{m} (t^{(k)})]

(26)

and

φ^{(n) (k)} (u^{(n)}) = φ^{(n)} [{\hat{F}}^{(n) (k)} (u^{(n)})]

(27)

be the time-discrete driving forces for reorientation and time-discrete reorientation functions, respectively. Then, the equations

\begin{matrix} 0 = Φ^{(k)} - \frac{\partial Ψ'}{\partial V_{m}} [u^{(n) (k)}, V_{m} (t^{(k)})] \\ 0 = u^{(n) (k)} - u^{(n) (k - 1)} - Δ λ^{(n) (k)} \frac{\partial φ^{(n)}}{\partial {\hat{F}}^{(n)}} [{\hat{F}}^{(n) (k)} (u^{(n) (k)})] \\ 0 = {\begin{matrix} \begin{matrix} Δ λ^{(n) (k)} \end{matrix} & if & φ^{(n) (k)} (u^{(n) (k - 1)}) \leq 0 \\ φ^{(n) (k)} (u^{(n) (k)}) & if & φ^{(n) (k)} (u^{(n) (k - 1)}) > 0 \end{matrix} \\ 0 \leq Δ λ^{(n) (k)} \end{matrix}

(28)

are used to obtain updated values $u^{(n) (k)}$ , $Φ^{(k)}$ based on the previous solution $u^{(n) (k - 1)}$ and the loading, where $Δ λ^{(n) (k)}$ are time-discrete reorientation multipliers. The latter must be non-negative in order to ensure thermodynamic consistency.

The practical implementation is straightforward and has been done using the free mathematical software GNU Octave (Eaton et al., 2020). A classical return mapping algorithm was used, which is a standard method for classical plasticity (Simo and Hughes, 2006) and has as well been utilized for MSM constitutive models (Kiefer et al., 2012). In particular, each time step is started with a “trial step,” which involves evaluation of $φ^{(n) (k)} (u^{(n) (k - 1)})$ . For each $n$ with $φ^{(n) (k)} (u^{(n) (k - 1)}) \leq 0$ , $u^{(n) (k)}$ is set to $u^{(n) (k - 1)}$ and $Δ λ^{(n) (k)}$ to zero. Otherwise, $u^{(n) (k)}$ is obtained by solving the nonlinear equation $0 = φ^{(n) (k)} (u^{(n) (k)})$ , which is in practice done with the function fsolve of GNU Octave using $u^{(n) (k - 1)}$ as initial guess. Subsequently, the $Δ λ^{(n) (k)}$ are calculated, and if any $Δ λ^{(n) (k)} < 0$ are found, the solution is rejected and the time-step size is decreased. If, in contrast, all $Δ λ^{(n) (k)}$ are non-negative, $Φ^{(k)}$ is finally calculated from the first equation in (28), which completes the time-step.

4. Identification of unknown functions and parameters

For a given number of reorientation systems $N$ , the model still involves the unknown functions $Ψ'^{, mag, e}$ , $Ψ'^{, mag, h}$ , $Ψ^{r, (n)}$ , $φ^{(n)}$ and the parameter $u^{\max}$ . In the following, specific forms for the former functions will first be discussed. In this context, additional parameters are introduced. The identification of these parameters and of $u^{\max}$ is subsequently detailed in the context of a particular example.

4.1. Magnetic energy contribution

In order to obtain specific forms of $Ψ'^{, mag, e}$ and $Ψ'^{, mag, h}$ , the two special cases are considered that $u = u^{\max}$ and $u = 0$ . For these situations,

Φ = {\begin{matrix} Φ^{e} (V_{m}) = \frac{\partial {Ψ'}^{mag, e}}{\partial V_{m}} & if & u = u^{\max} \\ Φ^{h} (V_{m}) = \frac{\partial {Ψ'}^{mag, h}}{\partial V_{m}} & if & u = 0 . \end{matrix}

(29)

With (16), (17), and the specific form of $Ψ'^{, mag}$ in (19), the $Φ - V_{m}$ relation can be expressed as

Φ (u, V_{m}) = \frac{u}{u^{\max}} Φ^{e} (V_{m}) + (1 - \frac{u}{u^{\max}}) Φ^{h} (V_{m}) .

(30)

Starting from (29), ${Ψ'}^{mag, e}$ and ${Ψ'}^{mag, h}$ can in principle be directly determined by measuring $Φ - V_{m}$ curves for fixed $u = u^{\max}$ and $u = 0$ , respectively, followed by integration of the results. However, in order to simplify the procedure, the relations

\begin{matrix} Φ^{e} = G_{m}^{0} V_{m} + Φ_{sat}^{M} \tanh [\exp (\frac{V_{m}}{V_{m}^{sat, e}}) - 1] \\ Φ^{h} = G_{m}^{0} V_{m} + Φ_{sat}^{M} \tanh [\exp (\frac{V_{m}}{V_{m}^{sat, h}}) - 1] \end{matrix}

(31)

are assumed, with $Φ_{sat}^{M}$ , $V_{m}^{sat, e}$ and $V_{m}^{sat, h}$ being magnetic parameters to be fitted to match the experimental data, and

G_{m}^{0} = μ_{0} \frac{WL}{D}

(32)

the free space magnetic permeance of the entire air gap, where $μ_{0}$ is the vacuum permeability. It is noted that these relations have been determined by splitting the magnetic flux into a free space and a magnetization contribution. The latter is referred to as “magnetization related flux,” defined as

Φ^{M} (u, V_{m}) = Φ - G_{m}^{0} V_{m},

(33)

and is described for $Φ^{e}$ and $Φ^{h}$ by extended $\tanh$ functions, which have been found to give good fits to measured data. Using the identified magnetic relations, a magnetic driving force can be derived from the magnetic part of the co-energy:

F^{mag} = \frac{\partial {Ψ'}^{mag}}{\partial u} .

(34)

It is noted that the proposed functions in (31) appear to be not elementary integrable and consequently, $Ψ'^{, mag}$ cannot be expressed in closed form, which is, however, also not necessary for the numerical solution of the problem.

4.2. Hardening terms

In order to restrict the admissible reorientation displacements $u^{(n)}$ to the range $(0, u^{\max} / N)$ the “back force”

F^{B, (n)} (u^{(n)}) = - \frac{\partial Ψ^{r (n)}}{\partial u^{(n)}} .

(35)

must be such that $F^{B (n)} \to \infty$ as $u^{(n)} \to 0^{+}$ and $F^{B (n)} \to - \infty$ as $u^{(n)} \to (u^{\max} / N)^{-}$ . This requirement is satisfied by using the function

F^{B, (n)} = - F_{0}^{B} \tan [\frac{π}{2} (2 \frac{u^{(n)} N}{u^{\max}} - 1)],

(36)

where it is assumed that the parameter $F_{0}^{B}$ is the same for all reorientation systems. Based on this relation, $Ψ^{r (n)}$ can, in principle, be obtained by integration.

4.3. Reorientation functions

Since each reorientation function $φ^{(n)}$ depends only on the single scalar driving force ${\hat{F}}^{(n)}$ , the determination of the shape of a reorientation surface $φ^{(n)} = 0$ amounts to specifying the two boundaries of the range of admissible values for ${\hat{F}}^{(n)}$ (which is known as the “elastic domain” in classical plasticity). In this regard, it is assumed that both boundaries have the same distance from the origin ${\hat{F}}^{(n)} = 0$ . Then, the reorientation function may, without further loss of generality, be expressed as

φ^{(n)} = {(\frac{{\hat{F}}^{(n)}}{F^{tw, (n)}})}^{2} - 1,

(37)

where the (positive) “twining force” $F^{tw, (n)}$ of the $n$ th reorientation system describes the distance of the boundaries of the “elastic domain” from the origin ${\hat{F}}^{(n)} = 0$ . For the sake of simplicity, the twinning forces are assumed to be distributed according to

F^{tw, (n)} = F_{0}^{tw} + \frac{n}{N} F_{1}^{tw}

(38)

among the reorientation systems, with the new parameters $F_{0}^{tw}$ and $F_{1}^{tw}$ .

4.4. Parameter identification

Besides $N$ and the dimensions of the air gap, the model involves the parameters $u^{\max}$ , $Φ_{sat}^{M}$ , $V_{m}^{sat, e}$ , $V_{m}^{sat, h}$ , $F_{0}^{tw}$ , $F_{1}^{tw}$ , and $F_{0}^{B}$ . These are in the following identified for a single-crystal NiMnGa sample (ETO Magnetic) situated in the magnetic circuit in a dedicated test setup for the simultaneous magneto-mechanical characterization, published recently by the authors (Ehle et al., 2020, 2021).

Regarding the identification of the magnetic parameters, that is, $Φ_{sat}^{M}$ , $V_{m}^{sat, e}$ , and $V_{m}^{sat, h}$ , the simplest way appears to be the direct measurement of $Φ^{e} (V_{m})$ and $Φ^{h} (V_{m})$ in the fully elongated ( $u \approx u^{\max}$ ) and a the fully compressed ( $u \approx 0)$ state, respectively. For large laboratory-scale electromagnet cores with $L > l$ and $W > w$ , as used here, this is challenging since $Φ$ has comparatively high free-space contributions. As a result, the change of $Φ$ due to $Φ^{M}$ can be obscured by drift and noise. Therefore, another procedure is proposed. Based on the measured magnetic flux through the MSM component, the total flux is derived by means of a simple lumped-element model of the magnetic core. The latter is described in detail in “Appendix A: Model-Based Magnetic Curve Identification” and is, despite the error in saturation of $Φ^{M}$ (approx. 10%), used in the following. The experimental curves for $Φ^{e} (V_{m})$ and $Φ^{h} (V_{m})$ determined in this way were used to identify the parameters $Φ_{sat}^{M}$ , $V_{m}^{sat, e}$ , and $V_{m}^{sat, h}$ from (31). With the identified parameters (see Table 1), a good agreement is obtained for sample states $0 < u < u^{\max}$ , apart from the prediction of the initial slope, as visible from Figure 2. It is remarked in this context that, here and in the following, $Φ^{M} (u, V_{m}) = Φ - G_{m}^{0} V_{m}$ is shown in the figures in order to eliminate the large gap-related contribution of the flux, which would otherwise be the dominant contribution in the plots.

Table 1.

Identified model parameters for the investigated NiMnGa-sample from ETO Magnetic.

Parameter	Value	Significance
$Φ_{sat}^{M}$	$29 μ Wb$	saturation magnetization related flux
$V_{m}^{sat, e}$	$260 A$	saturation magnetic voltage for $u = 0$
$V_{m}^{sat, h}$	$1500 A$	saturation magnetic voltage for $u = u^{\max}$
$F_{0}^{tw}$	$2.5 N$	twinning force distribution parameters
$F_{1}^{tw}$	$2.5 N$	twinning force distribution parameters
$F_{0}^{B}$	$0.5 N$	back force constant
$u^{\max}$	$0.89 mm$	max. displacement of the sample
$N$	$20$	number of reorientation systems
$l_{0} \times w_{0} \times d_{0}$	$14.18 mm \times 3.04 mm \times 2.01 mm$	MSM component dimensions
$L \times W \times D$	$35 mm \times 12 mm \times 2.3 mm$	air gap dimensions

Figure 2.

Comparison between measured (solid) and predicted (dash-dotted) magnetic curves $Φ^{M} (u, V_{m})$ at different constant displacement levels. Intermediate curves ( $0 \leq u \leq u^{\max}$ ) are predicted according to (30) with parameters from Table 1.

Regarding the remaining mechanical model parameters, mechanical first order reversal curves (FORC) were used. Advantageously, this would be done at zero magnetic voltage (where $F^{mag} = 0$ ) to identify the mechanical parameters without requiring knowledge of the magnetic properties. However, in most experimental setups, as in the one used here, only compressive forces can be applied to the sample. Consequently, the parameters must be identified using experimental data for $V_{m} > 0$ in order to shift the measured mechanical curves into the compressive load range. The latter was done here using $V_{m} > V_{m}^{sat, h}$ , thereby ensuring a state-independent and constant magnetic driving force. The result of the iterative parameter identification is shown in Figure 3. While $F_{0}^{B}$ mainly influences the curvature of the outer loops, $F_{0}^{tw}$ and $F_{1}^{tw}$ determine hysteresis width and onset of “flow” w.r.t. the hysteresis width. The identified model parameters and dimensions are summarized in Table 1.

Figure 3.

Model fit (dash-dotted) to the measured force-displacement curve (solid) at magnetic saturation. Parameters are listed in Table 1.

5. Experimental Validation

In this section, a selection of model predictions is compared to experimental results, obtained in the magneto-mechanical test bench described in Ehle et al. (2020). The measured magnetization-related magnetic flux through the MSM component was full-scale adjusted from a reference measurement with a Nickel sample and corrected for drift if necessary. In most of the following figures, measured data were slightly smoothed in order to facilitate distinction between individual curves. Displacement data have further been adjusted for elastic contributions, arising mostly from the compliance of test rig and load cell. In order to compare results from a defined initial state, the experimentally performed mechanical resetting cycle with a high compressive mechanical load prior to each experiment were considered in all simulations. After the resetting cycle, the displacement $u$ is slightly larger than zero in the simulations due to the action of backforces. In contrast, the state of the real sample after resetting cannot readily be determined as this would require the measurement of the volume fractions of the martensitic variants. However, the deviation from a single variant state should be small, so that the experimental displacement has been set to zero after the resetting cycle.

5.1. Constant force loading experiments

In this load case, the component is magnetically loaded and subsequently unloaded by means of a sinusoidal magnetic voltage while the applied compressive loading force is held constant. The resulting curves, also referred to as “butterfly curves,” indicate the attainable displacement at a certain preload depending on the magnitude of the magnetic input. Figure 4 compares six different large-signal loops over the practically relevant loading range. For clarity, all curves are shown for positive $V_{m}$ only and rising and falling hysteresis branches are plotted in separate diagrams. Deviations occur mostly in terms of curvature (the experimental curves appear magnetically more sheared, see Note 3) and incomplete elongation and resetting, in particular for low preloading forces. In contrast, maximum displacement, hysteresis corner points and hysteresis width are reasonably predicted.

Figure 4.

Comparison between measured (solid) and simulated (dash-dotted) magneto-mechanical (a and c) and magnetic (b and d) large-signal behavior at constant force for six different mechanical preloads. For the sake of clarity, rising and falling branches are plotted separately.

Two exemplary cases of small-signal behavior prediction are further discussed in the following. It is remarked that, due to the steepness of the hysteresis, even slight deviations of the model prediction from the measured curves lead to a different state at load reversal of $V_{m}$ . In order to enable a better inner hysteresis comparison, the load reversal points of the model input were therefore slightly adjusted. Figure 5(a) and (b) shows a set of inner hysteresis loops at constant force, where a unipolar sinusoidal magnetic voltage with increasing magnitude is applied while the mechanical loading force is kept constant. As a representative of the class of LEM building on scaled large-signal loops, the univariate Tellinen model, taking the hysteresis $u (V_{m}, F = const .)$ into account, is considered for comparison. The form used in the present paper is described in more detail in “Appendix B: Used Tellinen Model Formulation.” The inner hysteresis curves are reasonably predicted by the presented model for the entire range of states, whereas the Tellinen model shows a reversal point-dependent error. In particular, this concerns the displacement level-dependent width of the constant region of the inner hysteresis before $u$ and $Φ^{M}$ drop. Especially for load reversal points at small displacement biases, this width is predicted too narrow. This behavior is also observable in similar models as Guo et al. (2014). In contrast to other LEM as Couch and Chopra (2007), inner hysteresis curves appear not “angular” in the present model as consequence of both the back force contributions and the different twinning force levels of the individual reorientation systems.

Figure 5.

Comparison between the measured (solid) and simulated (dash-dotted) behavior at constant force loading: magneto-mechanical inner hysteresis loops and corresponding magnetic response (a and b) and small signal behavior (c and d).

The prediction of the small-signal behavior, where $V_{m}$ was cyclically varied (three cycles) with a small magnitude around different bias operating points of $V_{m}$ , is depicted in Figure 5(c) and (d). While the measured behavior is well predicted by the present model, the limits of the Tellinen model are clearly exposed by this load case.

5.2. Constant magnetic loading experiments

In this loading case, a mechanical compression test with subsequent unloading is performed at constant $V_{m}$ . From the resulting curves, the necessary work to compress the component (loading) and the maximum mechanical output work (unloading) at a certain magnetic input can be derived. In Figure 6, mechanical large-signal loops for five different constant magnetic voltages are shown. Besides slight deviations in curvature and hysteresis width, reasonable agreement is again observed.

Figure 6.

Comparison between the measured (solid) and simulated (dash-dotted) force-displacement curves at constant magnetic voltage: rising branches (a) and falling branches (b). Measured force data were slightly smoothed for clarity.

5.3. Combined magneto-mechanical loading

The prediction of the response to a combined magneto-mechanical loading cycle is depicted in Figure 7. Here, a bipolar sinusoidal magnetic voltage of increasing magnitude was applied to the MSM unit while the compressive loading force was increased linearly with time, as visible from Figure 7(a). As before, good agreement between model and experiment is achieved, although some quantitative deviations in the displacement response are observed.

Figure 7.

Comparison between the measured and simulated magneto-mechanical and magnetic behavior at combined loading: input signals w.r.t. time (a), magneto-mechanical response (b and d), and magnetic (c and e) response. For a better monochrome curve distinction, each of the eight half-wave cycles is enumerated and indicated in (b–e).

6. Discussion

With the chosen, simple forms of $Ψ'$ and $F^{tw, (n)}$ , the model predicts the magneto-mechanic behavior of the investigated MSM component in the used test setup qualitatively well. Experimentally observed effects like incomplete resetting for lower preloads, decreased displacement with increasing mechanical loading, or the load and the operation point dependent shape of hysteresis loops are also visible in the model response. The same applies for the corresponding magnetic response. In contrast to models building on interpolated large-signal loops (see Section 1), no operation point-dependent effects occur in the prediction of inner hysteresis loops (see Section 5.1.).

Quantitatively, magnetic and magneto-mechanical large-signal loops are predicted with reasonable accuracy for practically relevant loads. For certain loading ranges, the agreement is only moderate, for example, for butterfly curves or compressive tests. The deliberate choice of simple constitutive functions currently limits the adaptability to the behavior of the MSM unit. It is believed that more sophisticated forms for $Ψ^{', mag}$ , $Ψ^{r, (n)}$ , and $F^{tw, (n)}$ can improve the prediction quality, although this assumption needs to be validated. In particular, a different form for $Ψ'^{, mag}$ is necessary to improve the agreement to the experimentally determined magnetic curves, as visible from Figure 2. In this context, the approach of decoupling of the reorientation systems must be abandoned, leading to more effort for parameter identification and higher computational cost.

The model in its current form cannot account for certain effects observable experimentally. This concerns in particular the difference in hardening behavior between the two single variant states, that is, the different shape of the hysteresis as $u$ approaches $0$ or $u^{\max}$ , respectively. This effect is observable from Figure 3, as well as in results of other authors, for example, in Pagounis and Laufenberg (2012). It is currently not fully clear whether this effect is due to certain experimental boundary conditions or material behavior. Comparison with measurements of the supplier suggest the former. If this is the case, this substantiates even more the approach of a component-related characterization and modeling. Another effect is the appearance of different slopes of the $F - u$ curves in compression tests for $V_{m} > 0$ and $V_{m} = 0$ , visible from Figure 6 and again in results of other authors (Likhachev, 2013; Suorsa and Pagounis, 2004). It is believed that this effect is not predicted by the model due to the chosen form of $Ψ'^{, mag}$ , which is linear in $u$ .

Inherent to the chosen approach of considering the magnetic behavior of the MSM unit, the model validity is limited to the particular magnetic setup used for parameter identification. For other setups fulfilling the conditions $L > l_{\max}$ , $W > w$ and the same $D$ , that is, where stray field contributions do not differ significantly, the method described in “Appendix A: Model-Based Magnetic Curve Identification” may give a reasonable approximation. For entirely different setups, for example, for custom magnetic cores, a separate magnetic characterization or a finite element-based approach is necessary.

Despite the mentioned limitations and losing the claim of generality, the presented LEM can support system design. Its main strength is the ability to predict both magnetic and magneto-mechanical large-signal and small-signal behavior correctly, while maintaining thermodynamic consistency. Due to the strong influence of experimental conditions like friction on shape and curvature of the coupling curves, a thermodynamically correct model behavior appears to be more valuable than an exact prediction of large-signal loops, for which other models might be suited better. With its capabilities, the presented model is not only suited for the model-based investigation of actuators, but also for other applications like self-sensing actuators or generators where a correct minor loop prediction is mandatory and where the retroaction of the MSM component on the system level is function-determining. Moreover, proper efficiency and performance evaluations are possible, considering the entire system from electrical excitation to driven mechanics.

7. Conclusions

A novel, thermodynamically consistent approach for the lumped-element modeling of MSM components has been presented. By comparison with experimental data, a qualitatively good agreement was observed, while the quantitative prediction is sufficient for rough actuator design. From the results, it is concluded that the presented approach is able to account for the underlying coupling effects in MSM-based actuator systems. An asset of the model is that it is parameterized with a few comprehensible model parameters and identified with only one set of mechanical FORC and magnetic curves of the single variant states.

Future work may focus on the following aspects. An appropriate way to parameterize the model for arbitrary magnetic cores needs to be identified. Currently, due to the deliberate choice of component-related quantities, this is not envisaged. However, this might be feasible in principle if geometrically similar setups are considered. A Finite Element Analysis based parameter identification, which is the state of the art for parameterizing lumped-element models of electromagnetic actuators, might be considered. In this regard, a data conversion to account for arbitrary sample sizes would also be desirable. In addition, the origin of the experimental effects described in Section 6 should be clarified.

Footnotes

Appendix A: Model-based magnetic curve identification

In order to improve resolution and robustness in the measurement of $Φ$ , it is proposed to derive the latter quantity from a measurement of the magnetic flux thorough the MSM component $Φ^{MSM}$ , measured with two pick-up coils in proximity of the sample surface, as recently proposed by the authors (Ehle et al., 2020, 2021). This is done with a simple magnetic equivalent circuit model, schematically depicted in Figure A1. In this model, the air gap is divided into two “flux tubes,” namely MSM component including the residual air gap between MSM component and pole faces (1), and remaining gap between the poles (2).

The magnetic permeance of (2) can approximately be computed from its geometry using

(A.1)

G_{m}^{gap} (u) = μ_{0} \frac{WL - A_{Φ}}{D} .

By applying Gauss’s law for magnetism to the present case, $Φ$ can be determined from the measured magnetic curves of the MSM component $Φ^{MSM} (u, V_{m})$ as

(A.2)

Φ = Φ^{MSM} + G_{m}^{gap} V_{m} .

A full-scale adjustment of $Φ^{MSM}$ was performed after a calibration with a Nickel sample of similar shape and dimension. The results of this model-based approach were then compared with those obtained from a direct measurement of $Φ$ using two pick-up coils of similar size as the pole faces and can be found in Figure A2. Although the equivalent network approach strongly simplifies the problem under consideration, a good agreement was attained besides a deviation of the saturation magnetic flux of about 10%. The error could be reproduced with a 3D magnetostatic finite element analysis using Nickel as sample material and is due to the assumption of uniform fields in the air gap, inherent in the used equivalent circuit approach. This error leads to an overestimation of the magnetic driving force of similar magnitude. In the context of this work, where primarily the modeling approach is of interest, this is acceptable. Substantially more accurate results may be obtained based on finite element simulations, a discussion of which is however beyond the scope of the present contribution. Furthermore, the advantage of much lower noise and lower drift outbalances the disadvantage. It is further noted that, with the equivalent network approach described above, the actual size of the core (apart from $D$ ) does not influence the model behavior, since it is evident from (33) and (A.2) that

(A.3)

\frac{\partial Φ}{\partial u} = \frac{\partial Φ^{MSM}}{\partial u} - μ_{0} \frac{w}{D} V_{m} .

In a superordinate system model, however, it is mandatory to consider the core size properly in order to account for inductance and back electromotive force correctly.

Appendix B: Used Tellinen model formulation

For the exemplary comparison of minor loop prediction shown in Figure 5, the univariate Tellinen hysteresis model (Tellinen, 1998) has been considered. Originally introduced to model the scalar magnetic hysteresis, it was modified here to predict the rate-independent magneto-mechanical hysteresis for the load case $u (V_{m}, F = const .)$ described in Section 5.1. The displacement increment is computed according to

(A.4)

d u = {\begin{matrix} k_{↑} d u_{↑} & for & d V_{m} > 0 \\ k_{↓} d u_{↓} & for & d V_{m} < 0 \\ 0 & else . \end{matrix}

The scaling factors

(A.5)

k_{↑} (u, V_{m}) = \frac{u_{↓} (V_{m}) - u}{u_{↓} (V_{m}) - u_{↑} (V_{m})} and k_{↓} = 1 - k_{↑}

are computed from the two so-called limiting hysteresis curves $u_{↑}$ (lower, rising hysteresis branch) and $u_{↓}$ (upper, falling hysteresis branch).

The model was implemented in GNU Octave. For a given initial state $u_{0}$ , the displacement increment $Δ u^{(k)}$ at each time step $t^{(k)}$ is computed by transforming (A.4) and (A.5) into their time-discrete forms. Depending on the sign of $Δ V_{m}^{(k)} = V_{m}^{(k)} - V_{m}^{(k - 1)}$ , (A.5) is evaluated for $u^{(k - 1)}$ and $V_{m}^{(k)}$ . Subsequently, $Δ u^{(k)}$ is computed with (A.4) where $Δ u_{↑, ↓}^{(k)} = u_{↑, ↓} (V_{m}^{(k)}) - u_{↑, ↓} (V_{m}^{(k - 1)})$ are the time-discrete displacement increments of rising and falling hysteresis branches, respectively. The displacement at each time step can then be computed according to $u^{(k)} = u^{(k - 1)} +$ $Δ u^{(k)}$ and is used as initial value for the subsequent time step. The required limiting hysteresis curves $u_{↓} (V_{m})$ and $u_{↑} (V_{m})$ were identified with interpolated and smoothed large-signal loops from Figure 4. The magnetic response of the Tellinen model has been computed according to (30), as for the presented model.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work has been supported by “Deutsche Forschungsgemeinschaft” (DFG) under Grant No. NE 1836/1-1.

ORCID iD

Fabian Ehle

Notes

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A thermodynamically consistent lumped-element model for magnetic shape memory components

Abstract

Keywords

1. Introduction

2. Model description

2.1. Chosen approach

2.2. Considered load case

2.3. Additional assumptions

2.4. Description of the thermodynamic state

2.5. Helmholtz free energy and external work

2.6. Second law of thermodynamics

2.7. Co-energy

2.8. Summary

2.9. Model extension

3. Temporal discretization and implementation

4. Identification of unknown functions and parameters

4.1. Magnetic energy contribution

4.2. Hardening terms

4.3. Reorientation functions

4.4. Parameter identification

5. Experimental Validation

5.1. Constant force loading experiments

5.2. Constant magnetic loading experiments

5.3. Combined magneto-mechanical loading

6. Discussion

7. Conclusions

Footnotes

Appendix A: Model-based magnetic curve identification

Appendix B: Used Tellinen model formulation

Declaration of conflicting interests

Funding

ORCID iD

Notes

References