# Numerical modeling of systems based on SMA wires and pulleys: A finite element formulation within the Arbitrary Lagrangian–Eulerian framework

Guillaume Helbert, Jessy Simon

► To cite this version:

Guillaume Helbert, Jessy Simon. Numerical modeling of systems based on SMA wires and pulleys: A finite element formulation within the Arbitrary Lagrangian–Eulerian framework. International Journal of Non-Linear Mechanics, 2024, 165, pp.104800. 10.1016/j.ijnonlinmec.2024.104800. hal-05344457

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# Numerical modeling of systems based on SMA wires and pulleys: a finite element formulation within the Arbitrary Lagrangian-Eulerian framework

Guillaume Helbert¹, Jessy Simon¹

¹ ENI Brest, UMR CNRS 6027, IRDL, Technopôle Brest-Iroise, 29238 Brest, France

## Abstract

Thanks to their properties of superelasticity and shape memory effects, SMA wires are often considered for the design of damping devices or actuators coupled to pulleys. In order to dimension such devices for commercialization, it is necessary to develop reliable, robust and fast numerical tools. To this end, simplifying the management of the wire/pulley contact using an Arbitrary Lagrangian-Eulerian (ALE) formulation is an original and effective solution. In this work, the wires are modeled by three-dimensional truss thermo-mechanical finite elements associated with a superelastic law. For each node, the curvilinear abcissa is used as an additional degree of freedom in order to deal with material flow from one element to another. After presenting the ALE formalism and the necessary precautions (advection of variables), as well as the material behavior and thermo-mechanical contact laws at the pulleys, elementary validation tests are studied. Finally, two devices taken from the literature are modeled using the ALE formalism. The results obtained demonstrate the relevance and effectiveness of the approach adopted, as well as the importance of not neglecting friction at the pulleys and thermal effects on the material mechanical response.

**Keywords:** Shape Memory Alloys, Arbitrary Lagrangian-Eulerian Formulation, Finite Element Modeling, Pulleys, NiTi wires

## 1. Introduction

With new technologies has come a growing demand for microcontrollers, sensors and actuators. What's more, the reduction in the size of these devices as a result of technological improvements over the last few decades has led to phenomenal advances. Among the types of actuator commonly used, Shape Memory Alloys (SMA), and in particular Nickel-Titanium alloys, provide the best ratio of work per unit volume (>10 J/cm³ [1]). This is due to the remarkable properties of these materials, which enable high levels of displacement and force to be achieved for small devices. Indeed, the ability to memorize one shape at low temperature, then another at high temperature, and to switch almost reversibly from one to the other, is a remarkable opportunity for engineers. In particular, the assisted two-way memory effect makes it possible to achieve a high strain amplitude (up to around 8%) during a suitable temperature cycle and under mechanical loading, thanks to a diffusionless phase transition between austenite and martensite [2]. In addition, this phenomenon is associated with hysteretic behavior, reflecting a high capacity for

Email address: guillaume.helbert@enib.fr (Guillaume Helbert¹, Jessy Simon¹)

Dr J. Simon and Dr G. Helbert have the same contribution. They should be considered as the co-first author.
energy dissipation. Thus, under mechanical stress and above a temperature characteristic of the material ($A_f$ for "Austenite finish"), SMAs exhibit the superelastic property much sought-after in the design of structural damping devices [3]. According to [1], [4] and [5], who have carried out a wide-ranging survey of SMA-based actuators, the majority of these devices are based on the use of SMA in wire form, partly because of their higher fatigue strength and restituted force when compared with springs, for example. In order to ensure complex movements or to reduce the size of the devices, it is often necessary to redirect the wires by means of pulleys, which reduce friction on the wires to a minimum. The combination of SMA wires and pulleys has enabled the design of motors and servo-motors [6, 7], rotary actuators [8, 9, 10, 11, 12], rotating machines [13, 14, 15], robotic grippers, hands or arms [16, 17, 18]... Some of these works present analytical or numerical models based on strong assumptions concerning sources of heat generation or exchange affecting material temperature, or neglecting friction, which nevertheless causes energy dissipation. Yet the development of numerical tools is clearly identified as an opportunity to extend the commercialization of SMA-based devices, which remains modest in comparison with the number of patents filed on the subject [1].

A reliable, robust and fast numerical model for simulating the operation of such devices is an indispensable tool for the design phase, since it enables, among other things, the virtual evaluation of performance with a view to optimization. In this sense, the finite element method is a widely used approach for simulating complex thermo-mechanical systems. In the case of a system coupling SMA wires to pulleys, the major difficulty lies in modeling the interactions between these two components. Although powerful contact management algorithms have been developed, the number of elements required for the type of applications studied here is far too high to hope for real time simulations. Moreover, finer mesh discretization does not completely eliminate numerical oscillations due to the approximation of contact surfaces.

To overcome the problem of accurately and efficiently modeling the contact between SMA wires and pulleys using a Lagrangian description, the Arbitrary Lagrangian-Eulerian (ALE) method is adopted in this work. The original principle of this method is to exploit the advantages of the Lagrangian approach, in which the mesh is attached to the material it discretizes, and the Eulerian approach, in which the material moves relative to a fixed mesh. In particular, this method makes it possible to overcome problems of excessive deformation of elements in solid mechanics through material-independent mesh evolution [19, 20, 21, 22, 23], but also the management of free surfaces in fluid mechanics [24] and the treatment of fluid-structure interactions [25, 26]. A better description of contact surfaces in extrusion or rolling processes for example [27] is also a major asset of this approach. The ALE description was also used to simplify contact computations between yarns modelled as 1D finite element in biaxial woven fabrics [28, 29]. The special feature of the ALE method is therefore the ability of the material to flow through the moving mesh. This additional freedom, however, translates into a more complicated treatment of the resulting differential equation systems. Indeed, in the ALE framework, the total derivative $\frac{Df(x,t)}{Dt}$ of a quantity $f(x, t)$ is expressed as the sum of the time derivative in the reference frame attached to the mesh $\frac{\partial f(x,t)}{\partial t}|_m$ and a term related to the advection of this quantity in the mesh $c\frac{\partial f(x,t)}{\partial x}$:

$$ \frac{Df(x,t)}{Dt} = \frac{\partial f(x,t)}{\partial t}|_m + c \frac{\partial f(x,t)}{\partial x} \qquad (1) $$

With $c$ the advection velocity of the material in the mesh. The reader is referred to [24, 30] for a detailed description. There are two possible strategies for formulating an ALE thermo-mechanical problem. The first is to solve the coupled problem including conservation of mass, conservation of momentum and the heat equation with a suitable numerical stabilization method. The second,
more widely used in solid mechanics, solves the problem in two stages: a classical Lagrangian step followed by an Eulerian step consisting in advecting the variables while preserving the global solution obtained in the previous step.

In this work, the ALE formulation of the problem is based on numerical developments previously carried out to model cable (wire in our case) and pulley systems. In this approach, the cable is modeled by truss elements. A small pulley is then implicitly represented as a mesh node through which the material can flow. Figure 1 illustrates the interest of the ALE formulation in simplifying the mesh between a cable and a pulley. Capstan law for taking friction into account is implemented directly as an external force on the additional degree of freedom managing the relative motion between the material and the mesh. In the works cited above, the "ALE" term is often omitted, because the elastic behavior of the cable simplifies the problem. Indeed, the advection step is not necessary since there is no internal variable to be advected. Formulations of elastic cable-pulley systems based on an analytical approach [31, 32, 33], a prediction-correction method [34] or an implicit formulation with additional degrees of freedom at the pulleys [35] have been developed. In the work proposed here, the non-linear and thermo-mechanically coupled behavior of SMA wires requires a classical ALE formulation. Therefore, the approach proposed by [36, 29] using the curvilinear abscissa as an additional degree of freedom is taken up and reformulated in a thermo-mechanical framework with superelastic behavior to model SMA wires.

<figure><img src="image_2.png" /><figcaption>Figure 1: Comparison of cable-pulley meshes for a Lagrangian or ALE formulation: (a) Schematic of a cable-pulley mechanism, (b) Lagrangian mesh requiring fine discretization of both cable and pulley, ALE mesh for large (c) and small (d) pulleys</figcaption></figure>

Thus, the main objective of this work is to propose a reliable, robust and fast numerical model to simulate mechanical systems composed of SMA wires in contact with pulleys using truss-like elements with a thermo-mechanical formulation in an ALE framework. The model is designed to study phenomena relating to the intrinsic behavior of SMA wires in interaction with the pulleys involved in the operation of this type of device. The numerical model can thus be used to study the influence of parameters that are often difficult to control experimentally, particularly in the case of actuator control. In order to demonstrate the effectiveness of the proposed model, two application cases experimentally treated in the literature are modeled. The first is a damping device used in
civil engineering that exploits the superelasticity property of SMAs [37]. The second is a rotary actuator for medical applications using the assisted two-way shape memory effect [38]. The use of this model for other systems composed of SMA wires and pulleys may be the subject of further work.

This work is presented as follows. The next section is devoted to the presentation of the ALE formulation of a truss-like element in a thermo-mechanical framework with a superelastic behavior law to represent SMA wires. This is followed by elementary validation cases to validate the model implementation and familiarize the reader with the concept of the ALE formulation. Finally, the two validation cases are processed and the results analyzed to demonstrate the capabilities of the proposed numerical model.

## 2. Finite element numerical modeling

In this section, the thermo-mechanical ALE formulation of a truss finite element to model a SMA wire is described. The general formulation of the mechanical problem in the ALE framework is first given, and the terms relating to the thermo-mechanical behavior of SMAs will be explained. The advection scheme for internal variables is also described. Next, the thermal problem consisting of a Lagrangian step followed by the advection of nodal temperatures is formulated. Secondly, the behavior law used for SMA wires is presented. Finally, the modeling of interactions between SMA wires and pulleys is detailed.

### 2.1. ALE formulation

#### 2.1.1. Mechanical problem

The mechanical problem is solved by (explicitly) integrating the 3D mesh equation of motion and the 1D material flow equation through the mesh, since the SMA wires are modeled by truss elements. The differential systems are given by:

$$[M] \{\ddot{U}\} + \{F^{\text{int}}\} = \{F^{\text{ext}}\} \qquad [M_s] \{\ddot{U}_s\} + \{F_s^{\text{int}}\} = \{F_s^{\text{ext}}\} \qquad (2)$$

The vectors {$\ddot{U}$} and {$\ddot{U}_s$} are the acceleration vectors for displacement and material flow at the nodes. The vectors {$F^{\text{int}}$} and {$F_s^{\text{int}}$} are the vectors of associated internal forces, obtained after assembling the nodal forces calculated for each element {$f^{\text{int}}$} and {$f_s^{\text{int}}$}. Masses are assumed to be concentrated at the nodes, so that the [M] and [M_s] matrices can be considered diagonal. Additional forces due to inertia are neglected, given the load speeds involved. This is equivalent to neglecting the rate of change of the element mass, due to material flow.

For each truss element, the two nodes 1 and 2 are parametrized by their respective positions in space $\vec{x}_1$ and $\vec{x}_2$, and by their curvilinear abscissa $s_1$ and $s_2$:

$$\{q\} = \{\vec{x}_1, \vec{x}_2\}^T \qquad \{q_s\} = \{s_1, s_2\}^T \qquad (3)$$

We can thus deduce the length $l$ ($l_0$ in the initial state) of the element, as well as its length in the undeformed state, also known as the "material length" $\bar{L}$ ($\bar{L}_0$ in the initial state):

$$l = \|\vec{x}_2 - \vec{x}_1\| \qquad \bar{L} = s_2 - s_1 \qquad l_0 = \bar{L}_0 = \|\vec{x}_{02} - \vec{x}_{01}\| = s_{02} - s_{01} \qquad (4)$$

Note that unlike classical ALE formulations [24, 30] where mass conservation is described by a differential equation, the formulation using the curvilinear abscissa as an additional degree of freedom enforces it directly through an algebraic equation.
Shape functions are taken linear:

$$N_1(\xi) = \frac{1}{2}(1 - \xi) \qquad N_2(\xi) = \frac{1}{2}(1 + \xi) \qquad (5)$$

The parametric coordinate local to the element can be expressed from the associated curvilinear abscissa s and those corresponding to the two nodes s₁ and s₂:

$$\xi(s) = \frac{2s - s_1 - s_2}{s_2 - s_1} \qquad (6)$$

The shape functions depend on the curvilinear abscissas of the element nodes, which means they are time-dependent. In particular, this induces additional inertia terms, which are neglected here as mentioned earlier. The reader is referred to [36] for a detailed expression of these terms. The position $\vec{x}$ and curvilinear abscissa s can therefore be calculated for a point with coordinate $\xi$ in the element:

$$\vec{x} = [N_1 \mathbf{I}_{3 \times 3} \quad N_2 \mathbf{I}_{3 \times 3}] \{q\} \qquad s = [N_1 \quad N_2] \{q_s\} \qquad (7)$$

For convenience, we introduce displacement and material flow as follows:

$$\vec{u} = \vec{x} - \vec{x}_0 \qquad u_s = s - s_0 \qquad (8)$$

This translates to node i as:

$$\vec{u}_i = \vec{x}_i - \vec{x}_{0i} \qquad u_{si} = s_i - s_{0i} \qquad (9)$$

The Green-Lagrange strain in the uniaxial case is defined by:

$$\varepsilon = \frac{d\vec{x}}{ds} \cdot \vec{e}_1 - 1 = \frac{l - \bar{L}}{\bar{L}} \qquad (10)$$

With $\vec{e}_1$ the tangent unit vector of the element.

In order to determine the internal forces related to displacement and material flow, we need to derive the strain energy with respect to the vectors of the associated degrees of freedom:

$$\{f^{\text{int}}\} = \frac{\partial W}{\partial q} \qquad \{f_s^{\text{int}}\} = \frac{\partial W}{\partial q_s} \qquad (11)$$

Where:

$$W = \int_{\bar{L}} \frac{1}{2} A \sigma \varepsilon ds \qquad (12)$$

It comes,

$$\{f^{\text{int}}\} = A \int_{\bar{L}} [B]^T \sigma ds \qquad \{f_s^{\text{int}}\} = A \int_{\bar{L}} [B_s]^T \sigma ds \qquad (13)$$

Where, for an elastic material:

$$[B]^T = \frac{1}{\bar{L}} [-\vec{e}_1 \quad \vec{e}_1] \qquad [B_s]^T = \frac{l + \bar{L}}{2\bar{L}} [-1 \quad 1] \qquad (14)$$
With this formulation, a node can be Lagrangian, Eulerian or ALE. It is Lagrangian if the flowing degree of freedom is fixed ($u_{si} = 0$), Eulerian if its position is fixed ($\vec{u} = \vec{0}$) and ALE if none of them is force to be zero. Note, however, that external forces (e.g. friction with a pulley) must be applied to the ALE nodes, otherwise the system of equations is indeterminate. In a broader ALE framework, the mesh velocity is generally imposed [24, 30].

To illustrate the problem in ALE formulation, an elementary case is treated below. Three elastic truss elements are placed in series, as shown in Figure 2. The nodes at the ends are Lagrangian: node 1 is fixed, while node 4 is moved along $\vec{x}$ to tension and then monotonically release the assembly. Nodes 2 and 3 are simultaneously moved as shown in Figure 3-(a). Care must be taken to ensure that the size of the elements does not become too small for the time step considered, given the explicit nature of the integration.

<figure><img src="image_3.png" /><figcaption>Figure 2: Three truss elements in series</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Imposed displacement at nodes ; (b) Material flow observed at nodes ; (c) Stress in the elements</figcaption></figure>

The material flow observed in Figure 3-(b) illustrates the transfer of matter from one element to another during the simulation. Thus, the curvilinear abscissa of each node naturally follows the associated position along $\vec{x}$. This ensures that the strain remains homogeneous along the assembly (the change in size $l$ of each element is compensated by the change in $\vec{L}$), as shown by the change in stress in each element, according to Figure 3-(c). Slight oscillations are nevertheless observed, in the absence of viscous damping.

### 2.1.2. Advection of internal variables

For a purely elastic material, the ALE formulation of the mechanical problem can be reduced to that described in the previous section. However, in the case of more complex materials, the internal variables used to describe its behavior require special treatment, since they are defined at material points that can move in the mesh. To take into account the effect of material flow
in the mesh, internal variables defined at element integration points must be advected between neighboring elements that exchange matter. The reader may refer to [39] for an in-depth study of the various possible strategies for carrying out this step. In the approach used here, an internal variable $\vartheta$ is advected from the following formula based on the finite volume method [39]:

$$V\vartheta = V^L\vartheta^L - \Delta t \vartheta_i^L \sum_i \int_{\Gamma_i} \vec{c}_i \cdot \vec{n}_i da \quad (15)$$

$V^L$ and $V$ are respectively the volumes of the element before and after advection, while $\vartheta^L$ and $\vartheta$ are the values of the internal variable before and after advection. $\Delta t$ is the time increment and $\vartheta_i^L$ is the value of the internal variable exchanged at the $\Gamma_i$ boundary of the element before advection. The integral term in the sum on the element $\Gamma_i$ boundaries corresponds to the exchange of matter with neighboring elements across these boundaries. Thus, $\vec{c}_i$ is the velocity of the matter in the mesh at node $i$ of the element and $\vec{n}_i$ is the outgoing normal at the same node. In the case of a linear truss element with a single integration point, the following simplifications apply:

$$ 
\begin{aligned}
V &= A_0 \bar{L} \\
\bar{L} &= \bar{L} + du_{s1} - du_{s2} \\
\vec{c}_i &= \frac{du_{si}}{\Delta t} \vec{e}_1
\end{aligned} 
\qquad 
\begin{aligned}
V^L &= A_0 \bar{L}^L \\
\bar{L}^L &= \bar{L} \\
\vec{n}_i &= (-1)^i \vec{e}_1
\end{aligned} 
$$

$$\Delta t \int_{\Gamma_i} \vec{c}_i \cdot \vec{n}_i da = (-1)^i A_0 du_{si}$$

with $du_{si}$ the last material flow increment at node i and $A_0$ the cross-section area of the truss. The reader may refer to [39] for elements of dimension greater than one or of greater degree of interpolation.

Thus, taking into account the previous simplifications, one obtains:

$$\vartheta = \frac{1}{\bar{L} + du_{s1} - du_{s2}} (\bar{L}\vartheta^L + du_{s1}\vartheta_1^L - du_{s2}\vartheta_2^L) \quad (16)$$

With $\vartheta_1^L$ and $\vartheta_2^L$ the internal variable values obtained before the advection step for the neighboring element connected to node 1 and the neighboring element connected to node 2, respectively. The values actually taken by $\vartheta_1^L$ and $\vartheta_2^L$ depend on the direction of material flow. Indeed, if the current element loses material to a neighbor, the value of the internal variable exchanged is that of the element itself:

$$\left\{ 
\begin{array}{ll}
\vartheta_1^L = \vartheta^L & \text{if } du_{s1} < 0 \\
\vartheta_2^L = \vartheta^L & \text{if } du_{s2} > 0
\end{array}
\right. 
\quad (17)$$

### 2.1.3. Thermal problem

The Lagrangian solution of the thermal problem consists in the integration (explicit [40]) of the following differential system:

$$[C] \{\dot{\theta}\} + \{\phi^{\text{int}}\} = \{\phi^{\text{ext}}\} \quad (18)$$
The vector {$\dot{\theta}$} contains the rates of temperature change at the nodes. {$\phi^{ext}$} and {$\phi^{int}$} are the external and internal flux vectors respectively, obtained after assembly. The internal flux vector {$\phi^{int}$} applied to the nodes associated with a element is given by:

$$ \{\phi^{int}\} = \left( A_0 \int_{\bar{L}} [N] \dot{r} ds + k \int_{\bar{L}} [B]^T [B] ds + h P_0 \int_{\bar{L}} [N]^T [N] ds \right) \cdot \{\theta_1, \theta_2\}^T \quad (19) $$

With, $A_0$ and $P_0$, the cross-section area and perimeter of the element respectively, $k$ and $h$ are the thermal diffusion and convection coefficients, $\dot{r}$ is the volume heat generation rate. $[N]$ and $[B]$ are the matrices constructed from the same shape functions as for the mechanical problem. Unlike a classical Lagrangian problem, the integration of the {$\phi^{int}$} terms is performed with respect to the varying curvilinear abcissa. In this way, the variation in the amount of matter in the elements is taken into account.

However, this formulation alone is not compatible with the motion of the mesh relative to the material. Rather than formulating the thermal problem in a purely ALE framework as in [41], a temperature advection step follows the time integration of the differential system (18) treated in the Lagrangian reference frame [41]. This step is described in the next section.

### 2.1.4. Temperature advection

The advection scheme adopted for internal variables defined for elements cannot be applied to nodal variables. One possible strategy, discussed in [39], would be to advect the mean temperature of the elements using the previously defined scheme and then calculate the new nodal temperatures from the advected mean temperatures. However, another approach used in [42] allows the advection of nodal variables to be carried out directly according to the following scheme:

$$ \theta = \theta^L + c \Delta t \nabla \theta^L \quad (20) $$

Where $c$ is the velocity of the material relative to the mesh. In the ALE formulation used here, the term $c\Delta t$ corresponds to the material flow increment $du_s$. This scheme therefore relies solely on calculating the gradient $\nabla\theta^L$ of the variable at the node under consideration, using the element located upstream of the material flow. Thus, if we consider two adjacent elements, the gradient at the common node $\nabla\theta_2^L$ is defined as a function of the sign of the flowing increment $du_{s2}$ as follows:

$$ \nabla\theta_2^L = \begin{cases} \frac{\theta_2^L - \theta_1^L}{\bar{L}_1} & \text{if } du_{s2} < 0 \\ \frac{\theta_3^L - \theta_2^L}{\bar{L}_2} & \text{if } du_{s2} > 0 \end{cases} \quad (21) $$

with $\theta_2^L$ the value at the current node, $\theta_1^L$ the value at the previous node and $\theta_3^L$ the value at the next node.

To make it easier to understand the sequence of the various integration steps involved in the ALE formulation, a flowchart is shown in Figure 4.

To illustrate the importance of temperature advection, the configuration shown in Figure 2 is repeated. However, node 4 is fixed and a uniform temperature gradient (to simplify interpretation of the curves) is imposed initially. Equilibrium is thus reached at the initial instant, and is maintained throughout the calculation, although the temperature is only imposed at nodes 1 and 4. Nodes 2 and 3 are ALE and the mesh is deformed as in the previous example.
<figure><img src="image_5.png" /><figcaption>Figure 4: Flowchart of the explicit time integration solver</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Imposed displacement at nodes; (b) Material flowing at nodes; (c) Temperature at nodes (dotted line in absence of advection)</figcaption></figure>

The change in node positions along $\vec{x}$ is identical to that of the curvilinear abscissas (see Figure 5-(a)-(b)). This is because the elements are not loaded in tension, which means that their respective lengths $l$ are identical to their undeformed lengths $\bar{L}$. According to Figure 5-(c), the temperature change at the nodes follows the same profile as that of the curvilinear abscissae, which
is expected in the case of a uniform temperature gradient. If nodal temperature advection is not taken into account, the temperature change is negligible, as shown by the dotted curves in Figure 5-(c). It would be necessary to impose a sufficiently slow node displacement (over a period much longer than the characteristic time of the thermal problem) to observe a correct instantaneous temperature at the nodes.

## 2.2. Thermo-mechanical behavior of SMAs

In order to calculate the stress required to determine the internal forces given by equation (13), a constitutive law specific to the SMAs must be integrated for each element. For the truss elements presented above, a single integration point is required. First of all, the strain decomposition for the case of uniaxial loading is expressed simply:

$$ 
\begin{aligned}
\varepsilon &= \varepsilon^{el} + \varepsilon^{tr} + \varepsilon^{th} \\
&= \frac{\sigma}{E} + \gamma z + \alpha \theta
\end{aligned} 
\qquad (22) 
$$

With $\varepsilon^{el}$ the elastic strain (related to the stress $\sigma$), $\varepsilon^{tr}$ the martensitic transformation strain (related to the martensite volume fraction $z$) and $\varepsilon^{th}$ the thermo-elastic strain (related to the temperature increase of the element $\theta$).

The law is based on the use of yield functions $f_1$ and $f_2$, driving forward (austenite to martensite) and reverse phase transformations, respectively:

$$ 
\left\{ 
\begin{aligned}
f_1 &= |\sigma| & - (M(z) + \sigma^y(T)) &\le 0 \\
f_2 &= |\sigma - X(z)| & - (M(z) + \sigma^y(T)) &\le 0
\end{aligned}
\right. 
\qquad (23) 
$$

With $M$ an isotropic pseudo-hardening and $X$ a kinematic pseudo-hardening, linear functions of $z$. The transformation yield stress $\sigma^y$ is a function of temperature on the element [43]:

$$ \sigma^y(T) = \mu (T - T_{ref}) + \sigma^y(T_{ref}) \qquad (24) $$

With, $T = [N] \{T_1, T_2\}^T$, where $T_1 = \theta_1 + T_0$ and $T_2 = \theta_2 + T_0$ are the nodal temperatures. $T_0$ and $T_{ref}$ are the initial and reference temperatures, respectively, which can be taken to be identical.

Phase transformation kinetics are obtained from the normality law, which is simply expressed for uniaxial loading:

$$ 
\left\{ 
\begin{aligned}
\dot{\varepsilon}^{tr} &= \dot{\lambda}_1 \frac{\partial f_1}{\partial \sigma} & \text{if } \dot{z} > 0 \\
\dot{\varepsilon}^{tr} &= \dot{\lambda}_2 \frac{\partial f_2}{\partial \sigma} & \text{if } \dot{z} < 0
\end{aligned}
\right. 
\qquad (25) 
$$

Derivation of the free-energy potential gives an expression for the heat equation [44]. Thermal phenomena associated with phase transformations (intrinsic dissipation and thermo-mechanical couplings) as well as thermo-elastic couplings are combined in the term associated with the volume heat rate:

$$ 
\dot{r} = \underbrace{\gamma (\pi \dot{z} - \dot{H})}_{\text{intrinsic dissipation}} + \underbrace{\gamma \mu T \dot{z}}_{\text{thermo-mechanical couplings}} - \underbrace{E \alpha T \left(\frac{\dot{\sigma}}{E}\right)}_{\text{thermo-elastic couplings}} \qquad (26) 
$$

With $\pi$ the thermodynamic force associated with the internal variable $\varepsilon^{tr}$, while that associated with the stored energy $H$ is equal to $-\gamma$.
<table><caption>Table 1: Material parameters</caption><thead><tr><th>ρ</th><th>E</th><th>σ<sup>y</sup>(298K)</th><th>M<sub>max</sub></th><th>δ<sub>min</sub></th><th>δ<sub>max</sub></th><th>γ</th><th>α</th><th>μ</th><th>λ</th><th>C<sub>p</sub></th></tr></thead><tbody><tr><td>6500<br>(kg/m<sup>3</sup>)</td><td>50<br>(GPa)</td><td>450<br>(MPa)</td><td>100<br>(MPa)</td><td>250<br>(MPa)</td><td>200<br>(MPa)</td><td>0.05<br>(-)</td><td>14.10<sup>-6</sup><br>(K<sup>-1</sup>)</td><td>7.0<br>(MPa/K)</td><td>18<br>(W/m/K)</td><td>500<br>(J/K/kg)</td></tr></tbody></table>

In what follows, and unless otherwise stated, the material parameters used are those given in the table 1, obtained from [43]. To illustrate the model, two loading cases are shown in Figure 6. To reduce the number of parameters, it is assumed that the parameters do not depend on the phase. In addition, due to a lack of information on the materials considered in the following applications, the intermediate transformation from austenite to R-phase is not taken into account. Finally, in order not to make the behaviour law too complex, the phenomena of strain localisation and functional fatigue are not implemented, but may be the subject of future studies. To make a direct link with the material parameters given in Table 1, thermal effects are not considered here despite their importance, as we shall see later. The stress-strain curve in Figure 6-(a) shows the hysteretic behavior in superelasticity (6.5% imposed strain), while the stress-temperature curve in Figure 6-(c) shows the assisted two-way memory effect (thermal amplitude of 80 K under a stress of 405 MPa). The two loading paths are superimposed on the pseudo-phase diagram (see Figure 6-(b)).

<figure><img src="image_7.png" /><figcaption>Figure 6: (a) Curve $\varepsilon - \sigma$; (b) Phase pseudo-diagram; (c) Curve $T - \varepsilon$</figcaption></figure>

## 2.3. Pulley modeling

### 2.3.1. Friction law

The law of friction is given by the Capstan formula (Euler-Eytelwein). In the case of the ALE formulation, the direction of material flow given by the sign of $du_s$ indicates the direction of rotation of the pulley:

$$ \left\{ \begin{array}{ll} N_2 = e^{-f\beta}N_1 & \text{if } du_s > 0 \\ e^{-f\beta} < \frac{N_2}{N_1} < e^{+f\beta} & \text{if } du_s = 0 \\ N_1 = e^{-f\beta}N_2 & \text{if } du_s < 0 \end{array} \right. \qquad (27) $$
Where $N_1$ and $N_2$ are the tensions of the elements adjacent to the node under consideration. By convention, a positive $du_s$ increment corresponds to a material flow from element 1 to 2. $f$ is the coefficient of friction between the pulley and its axis and $\beta$ is the angle corresponding to the winding arc on the pulley. Depending on the size of the pulley, it can be discretized by a unique node or by one or several elements. The choice of the discretisation has an effect on the length of the wire passing through the pulley.

For validation purposes, three truss elements are placed in series with a $\pi$ rotation between the first two elements and a $\pi/2$ rotation between the last two. The pulleys are assumed to be of sufficiently small radius to model them by one node each (i.e. nodes 2 and 3 from Figure 7-(a)). They are associated with an identical coefficient of friction of $f=0.05$. The other two nodes are "Lagrangian": node 1 is fixed, while node 4 is subjected to a monotonically increasing displacement up to $\Delta u = 4\% 2.4$ m. According to Figure 7-(b), the material flow change is linear at nodes 2 and 3, and greater at the latter since it is associated with the element directly under tension. On unloading (sign inversion of $du$), both pulleys stick (material flow remains constant). The one corresponding to node 3 begins to turn in the opposite direction, followed by the one corresponding to node 2. Slip then decreases linearly. The results obtained are qualitatively in line with the reference case given in [45]. The ratios of tension $N_2/N_1$ and $N_3/N_2$ are given in Figure 7-(c) and are in good agreement with the expected value given by the dotted lines.

<figure><img src="image_8.png" /><figcaption>Figure 7: (a) Two-pulley test; (b) Material flow at nodes; (c) Tension ratio</figcaption></figure>

### 2.3.2. Contact heat exchange

The strong dependence of SMA behavior on temperature calls for careful management of thermal exchanges with the environment. In particular, contact heat transfer at the pulleys must be taken into account. As a first approach, a thermal conductance model is proposed:

$$ \{\varphi_c^{int}\} = \int_{\bar{L}} [N] h_c(T - T_0)ds \quad (28) $$

With $h_c$ the coefficient of thermal contact conductance per unit length depending on the materials involved, $T$ the temperature of the element and $T_0$ the temperature of the pulley, generally taken to be equal to the initial temperature when thermal equilibrium is achieved.
### 3. Validation and application cases

#### 3.1. *Elementary cases*

Before proceeding to the study of real application cases, the test presented in Figure 7 is extended to the case of superelastic behavior. In order to demonstrate the importance of thermal effects on the response of SMA wires, two extreme configurations will be highlighted, namely the isothermal case (which would correspond to quasi-static loading where thermal equilibrium is achieved at all times) and the adiabatic case (which would correspond to the case where thermal exchange is inoperative on the time scale considered).

For the isothermal case, Figure 8-(a) compares the behavior of the three elements of the assembly with the reference behavior when friction at the pulleys is assumed to be zero. The latter results in uniform behavior along the entire length of the wire (and linear material flow). In the presence of friction, heterogeneous behavior is observed: element 3 is fully transformed at maximum displacement, while element 1 is only slightly transformed. Figures 8-(b)-(c) clearly illustrate the effect of friction, since material flow remains lower overall than in the frictionless case, and the tension ratio is not unitary. It should be noted that the changes in material flow associated with nodes 2 and 3 is more complex than that shown in Figure 7-(b) for an elastic material: indeed, we can observe slope breaks associated with each change of slope (beginning or end of transformation) for each of the elements, to which we can add the sticking phases of the pulleys during unloading. Finally, we can also see that the duration of the pulley sticking phase is reduced in comparison with Figure 7-(c).

<figure><img src="image_9.png" /><figcaption>Figure 8: Isothermal configuration: (a) $\varepsilon - \sigma$ curves; (b) Material flow at nodes; (c) Tension ratio</figcaption></figure>

In the adiabatic case, the same heterogeneous behavior can be seen in Figure 9-(a). However, the slopes of the transformation plateaus are increased by thermo-mechanical couplings [44]. These result in a substantial temperature amplitude during loading, as shown in Figure 9-(c). The latter also illustrates thermo-elastic coupling (visible at the beginning and end of the curves by a temperature variation while the 3 elements are in the elastic domain), as well as intrinsic material dissipation (by a temperature increase measured at the end of the loading-unloading cycle). Temperature effects are directly characterized by an attenuation of the stiffness ratio between the elastic and transformation behavior ranges. Incidentally, material flow tends towards the values observed in the case of elastic behavior, and the duration of pulley sticking phases is thus slightly lengthened (see Figure 8-(c)).
<figure><img src="image_10.png" /><figcaption>Figure 9: Adiabatic configuration: (a) $\varepsilon - \sigma$ curves; (b) Material flow at nodes; (c) Temperature changes on the elements</figcaption></figure>

Taking into account thermal exchanges between the SMA wire and the environment will result in an intermediate response to those obtained for isothermal and adiabatic configurations, depending on the time scale over which the mechanical load operates. It is therefore essential to study the influence of thermal effects and friction in order to gain a better understanding of the operation of devices such as those presented hereafter.

## 3.2. Structure-isolating device

### 3.2.1. Description of the device

The first application deals with a system for isolating a civil engineering structure from its base to protect it against earthquakes. A system exploiting the superelasticity property of SMAs was presented and laboratory tested in [37] (see Figure 10-(a)), then tested under real conditions in [46]. An analytical model was also proposed in [47]. The system consists of one plate associated with the base while the other is associated with the structure. Two SMA wires join the plates together by passing through pulleys, while a roller attached to the top plate rolls with friction on the second. The former provides lateral stiffness, system centering and seismic vibration damping, while the latter ensures vertical load transfer and horizontal mobility.

Different configurations of the device have been studied in [37] and two of them will be discussed in what follows:

* "Cross" configuration with pulleys for wire/plate junctions (see Figure 10-(a) left).

* "Straight" configuration with pulleys for wire/plate junctions (see Figure 10-(a) right).

### 3.2.2. Description of the finite element model

The approach based on an ALE formulation makes perfect sense in this application, given the natural flow of the wires through the pulleys. Whatever the configuration, the wires are composed of 4 truss elements each (see Figure 10-(b)). The natural discretization consists in positioning the nodes at the wire/plate junctions as the dimensions of the pulleys are small compared to the length of the wires. All these nodes are described using an ALE formulation (material flow degrees of freedom are left free) and friction at the pulleys is modeled using the dedicated law. Due to the spatial symmetry of the problem considered, a single wire could be considered, but the size of the
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10: (a) Isolating devices shown in [37]; (b) Associated Finite Element Models</figcaption></figure>

model remains incomparably smaller than that of the problem treated in the classical Lagrangian formulation.

In order to validate the modeling approach, the tests carried out in [37] are reproduced numerically. A horizontal displacement $u_x$ was imposed along $\vec{x}$ at the nodes of the top plate at a constant speed of $\pm 1 \text{ mm/s}$, via a reference point (R.P.). Various increasing $\Delta u_x$ amplitudes are imposed (see Figure 11). Following the example of the tests carried out in [37], it was decided to decouple the effect of the rollers from that of the SMA wires, and to focus solely on the latter. Modeling the effect of the rollers could easily be taken into account by applying Coulomb friction forces to the R.P. The locking loop system of the wires was not taken into account in the simulation, for lack of precise information. However, the effect of wire flowing inherent in this system could be favorably treated by the ALE formulation presented in this work. Since this system also allows initial tensioning of the SMA wires, another strategy was considered: in a first step, a vertical displacement $u_z$ was imposed along $\vec{z}$ on the nodes of the upper plate via the R.P. until the correct distance between the two plates was reached (see Figure 11). This results in elongation of the wires. The level of displacement $\Delta u_z$ has been determined to achieve a particular value of relative wire elongation. It therefore depends on the configuration ("cross" or "straight") on which the length of the wires for a given device size will depend. The default pretension is $\epsilon_{pt}=0.1\%$, the friction coefficient $f$ is 0.025 and the convection coefficient $h$ is $25 \text{ W.m}^{-2}.\text{K}^{-1}$, considering that the device may be subjected to the effect of wind. Contact heat exchange at the wire/plate junctions is neglected, since the dimensions of the pulleys are very small compared with the length of the wires. For the same reason, the portion of wire located in the pulleys is small in relation to the length of the wire: the phase transformation potentially due to the bending of the wire in the pulleys is therefore neglected and the material is thus assumed to be initially deformation-free. The material parameters used are identical to those given in the table 1, except for the modulus of elasticity ($E=41 \text{ GPa}$), the transformation yield stress at room temperature ($\sigma^y=552 \text{ MPa}$) and the minimum and maximum hysteresis loop heights ($\delta_{min}=400 \text{ MPa}$ and $\delta_{max}=300 \text{ MPa}$), in
agreement with the data in [37].

### 3.2.3. Comparison with experimental results

Figure 11 shows the material flow at the nodes for both configurations in relation to the displacement of the top plate.

For the "straight" configuration, perfect symmetry is demonstrated between nodes 1 and 6 on the one hand, and 2 and 5 on the other. This is to be expected, since they play the same role: this is equivalent to changing the reference frame and considering the bottom plate as mobile. An interesting point is that the wire flow is totally reversible, since its value returns to that just after the pretensioning step when $u_x=0$. What's more, the material flow operates in an "alternative" way: its absolute value is never simultaneously minimal at all nodes. This phenomenon does not occur in the absence of friction. It would appear that this is due to the phenomenon of pulley sticking previously discussed and illustrated in Figure 7-(b).

In the "cross" configuration, this symmetry is lost in favor of alternating operation: material flow is identical (in absolute value) between nodes 1 and 6 on the one hand, and 2 and 5 on the other. In this configuration, material flow remains reversible, but takes place in a coordinated manner in all nodes, with or without friction. In contrast to the "straight" configuration, material flow remains quasi-proportional to the displacement of the moving plate: indeed, the maximum value is obtained for each node only unilaterally.

The material flow field is shown in Figure 12-(a)-(b). These images clearly illustrate the symmetrical nature of the "straight" configuration versus the alternating nature of the "cross" configuration discussed earlier, between nodes 1 and 6 on the one hand and 2 and 5 on the other.

The mechanical responses of the devices are shown in Figure 13-(a)-(b). They represent the force measured at the R.P. versus displacement. Curves associated with reference parameters are shown in black. The others will be discussed in the following paragraph. The curves presented are in agreement with the experimental results obtained from [37]. Indeed, there are two operating regimes:

* A regime associated with non-linear stiffness that results in hysteresis loops for the "cross" configuration (up to approx. 0.025 mm amplitude), while it does not show loops for the "straight configuration" and is of lesser importance (up to approx. 0.015 mm amplitude).

* A second regime that begins with a sudden decrease in stiffness. Only this regime exhibits hysteresis loops in the "straight" configuration.

In the "cross" configuration, the first regime (which should logically correspond to the elastic behavior of the material) predominates. This explains why the phenomenon of pulley sticking has little influence in the presence of friction (it is thus clearly shortened, in line with the remarks in section 3.1), unlike the "straight" configuration, where the second regime dominates (attributable to phase transformation).

In order to make the link with material-scale behavior, the stress-strain curves are shown in Figure 14, for the four portions of the wire. Whatever the configuration, there is almost perfect symmetry between the "long" portions (e.1 and e.3) on the one hand and the "short" portions (e.2 and e.4) on the other. It can be seen that the wires are much more stressed in the "straight" configuration. This is due to the shorter wire length and different orientation for the same device size, while the displacement of the moving plate is similar. As a result, stress values are higher for this same configuration, and the wires are expected to break more quickly due to fatigue. A direct link can be made between the hysteresis loops observable at the material scale and the loops
<figure><img src="image_13.png" /><figcaption>Figure 11: Imposed displacement along $\vec{x}$ and $\vec{z}$ and material flow in both configurations ($h=25 \text{ W.m}^{-2}.\text{K}^{-1}$, $f=0.025$ and $\varepsilon_{pt} = 0.1\%$)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: Material flow field from Paraview ($h=25 \text{ W.m}^{-2}.\text{K}^{-1}$, $f=0.025$ and $\varepsilon_{pt}=0.1\%$): (a) "cross" configuration; (b) "straight" configuration</figcaption></figure>

observable during the second operating regime of the device. Thus, it appears that the "cross" configuration causes loops to appear while the material remains elastic. This source of dissipation is therefore due to friction at the pulleys, as will be shown below. Thermal effects affect the material behavior, since the slopes associated with the plateaus and the height of the hysteresis loops are greater than expected, given the parameters entered in the model. This logically translates into
<figure><img src="image_15.png" /><figcaption>Figure 13: Mechanical response of the structure and influence of heat exchange ($f = 0.025$ and $\epsilon_{pt} = 0.1\%$): (a) "cross" device; (b) "straight" device</figcaption></figure>

higher stress values than would be predicted in a model based on the isothermal assumption, which is not insignificant in terms of wire fatigue life. A third regime is very slightly observable at the last loading level for the "straight" configuration, since there is a brief incursion into the elastic martensitic domain, which is very slightly apparent in the response of the structure for the reference parameters (see Figure 13-(b)).

<figure><img src="image_16.png" /><figcaption>Figure 14: Material response in the four sections of a wire ($h=25$ W.m$^{-2}$.K$^{-1}$, $f=0.025$ and $\epsilon_{pt}=0.1\%$): (a) "cross" configuration; (b) "straight" configuration</figcaption></figure>
### 3.2.4. Parametric analysis

The strength of the proposed model lies in its ability to obtain information that is difficult to access experimentally (stress, temperature, material flowing...), and also to take into account phenomena that the analytical model proposed in [47] does not incorporate. These include:

* effects of loading speeds in relation to wire heating/cooling,

* the influence of friction at wire/plate junctions,

* wire pretension level.

In particular, the natural frequency of the structure tested in [46] is of the order of 0.1 Hz. The loading is thus outside the quasi-static range for this type of material, given the inherent thermal effects. Thus, the material parameters of the model must correspond to the loading rate (i.e. the strain rate, which is a function of wire length and loading frequency) observed ad-hoc, unless thermal effects are explicitly taken into account as in the behavior law presented above. Moreover, we have little information on these three phenomena. It was therefore a natural choice to look at their influence, while taking care to retain realistic values for the associated parameters.

Figure 13 compares the influence of the convection coefficient (and therefore of heat exchange between the wires and the environment outside the device). This effect, which only affects the second operating regime of the device (linked to phase transformation), is much more pronounced for the "straight" configuration. Indeed, according to Figure 14, this configuration is associated with a phase transformation level almost four times higher. It is also detectable for the "cross" configuration, albeit to a lesser extent. Whatever the configuration, the isothermal case is associated with the lowest slope for the second regime, while slopes increase as exchanges decrease (i.e. $h$ decreases).

Figure 15 shows the influence of the friction coefficient at the pulleys, with $f$ varying from 0 to 0.1. It can be seen that both operating regimes are affected by increasing $f$, particularly the second. Moreover, the "cross" configuration shows greater sensitivity to this parameter. The explanation lies in the fact that this configuration involves greater wire flow at the pulleys, and therefore a more perceptible effect of friction on the device response. Thus, an increase in $f$ results in a widening of the loops and a more vertical unloading curve, approaching an hourglass shape. A value of $f$ between 0.05 and 0.1 seems to restore a response very similar to the experimental curve from [37]. More generally and whatever the configuration, an increase in $f$ implies an increase in the stiffness of the device for both operating regimes. For the "cross" configuration, it should be noted that a zero $f$ coefficient does not cause loops to appear when the wire portions are in the elastic domain, in contrast to cases where $f$ is non-zero. Friction at the wire/plate junctions is therefore a source of energy dissipation in addition to that intrinsic to the material.

Figure 16 shows the influence of initial wire pretension, with $\epsilon_{pt}$ varying from 0 to 0.4%. The first effect of increasing the pretension is to increase the initial stiffness of the device (tangent of the curve at the origin). The second, more predictable effect is to bring the wire closer to the phase transformation, since the second operating regime of the device is reached at lower displacements of the moving plate. The effect is therefore logically more pronounced for the "cross" configuration, since the first operating regime predominates. Although the increase in initial stiffness may be an advantage for the device in recentering the structure, the downside is that higher stress levels are reached. Thus, the maximum pretension level results in total martensitic transformation for the "straight" configuration for the maximal amplitude displacement.
<figure><img src="image_17.png" /><figcaption>Figure 15: Influence of the friction coefficient ($h=25 \ W.m^{-2}.K^{-1}$ and $\epsilon_{pt}=0.1\%$): (a) "cross" configuration; (b) "straight" configuration</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 16: Influence of pretension ($h=25 \ W.m^{-2}.K^{-1}$ and $f=0.025$): (a) "cross" configuration; (b) "straight" configuration</figcaption></figure>

Thus, the three parameters studied showed a real influence on device operation. Their influence is summarized in Table 2. In particular, fine-tuning the pretension can help optimize the device efficiency in terms of stiffness and energy dissipation, while ensuring a long fatigue life. The other two phenomena are less controllable. Nevertheless, their understanding is important for the design of the device. Based on the photographs of the device shown in Figure 10, the wire radii do not appear to be negligible compared with the curvature radii at the wire/plate junctions. An improvement to the model would therefore be to take into account the bending stiffness of the
wires in order to better predict stress levels and therefore the risk of wire breakage, particularly at these junctions.

<table><caption>Table 2: Qualitative influence of the parameters studied on the response of the isolator device</caption><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td colspan="2">"cross"</td><td>stiffness (1<sup>st</sup> regime)</td><td>stiffness (2<sup>nd</sup> regime)</td><td>dissipation</td></tr><tr><td>h ↑</td><td></td><td>0</td><td>+</td><td>+</td></tr><tr><td>f ↑</td><td></td><td>++</td><td>++</td><td>++</td></tr><tr><td>ε<sub>pt</sub> ↑</td><td></td><td>++</td><td>0</td><td>+</td></tr><tr><td colspan="2">"straight"</td><td>stiffness (1<sup>st</sup> regime)</td><td>stiffness (2<sup>nd</sup> regime)</td><td>dissipation</td></tr><tr><td>h ↑</td><td></td><td>0</td><td>++</td><td>++</td></tr><tr><td>f ↑</td><td></td><td>+</td><td>+</td><td>+</td></tr><tr><td>ε<sub>pt</sub> ↑</td><td></td><td>+</td><td>0</td><td>+</td></tr></tbody></table>

### 3.3. Rotating actuator for medical applications

#### 3.3.1. Description of the device

The second application involves an actuator device with small dimensions (25.0 × 9.0 × 3.5 mm³), low weight (0.52 g) and low electrical power consumption, making it very attractive for biomedical applications. Operation is based on the use of two SMA wires, both attached to a main pulley (see Figure 17-(a)). The wires are heated independently by the Joule effect: the contraction of one wire relative to the other, due to a transformation from martensite to austenite, causes the pulley to rotate and transmit torque. Alternating activation of the wires results in periodic, symmetrical operation. Initial pretension of the wires enables operation within the operating range of the material in question. Disymmetrical pretension allows adjustment of the average angle. The system also makes it possible to quickly reach a target angle and maintain position. Two additional pulleys reduce the dimensions of the device without reducing the size of the wires, which are dimensioned to ensure a certain angular amplitude of the pulley as a function of its radius.

#### 3.3.2. Description of the finite element model

Unlike the previous application, little information is available in the associated article [38]. Dimensions have therefore been estimated from the overall dimensions of the device. Given the presence of three pulleys in the device, the ALE formulation is again entirely appropriate. The model is shown in Figure 17-(b). Only the nodes at the ends of the two wires are Lagrangian, since the matter does not flow through them. A vertical displacement $u_z$ is initially imposed on nodes 1 and 16 to ensure initial pretensioning of the wires (see Figure 18). Nodes 8 and 9 are kinematically linked via a reference point (R.P.), which was used as a reference for measuring the angle of rotation of the main pulley. The remaining 12 nodes associated with the pulleys are fixed in the reference frame. However, the associated material flow degrees of freedom are not fixed. In order to maintain an efficient model, the pulleys have been discretized in a "coarse" way: two nodes (or one element) for each mini-pulley and 5 nodes (or 4 elements) for the main pulley for each wire. The angles and approximate lengths of the wires wound on the pulleys must be respected for the respective treatment of friction and heat exchange by contact of the wires with the device. Once pretension has been achieved, the position of nodes 1 and 16 is maintained, while an alternating heat flux is imposed on the two wires by means of a power per unit length dissipated by the Joule effect (see Figure 18). In the absence of a PID corrector, unlike the experimental
<figure><img src="image_19.png" /><figcaption>Figure 17: (a) Actuator shown in [38] ; (b) Associated Finite Element Model</figcaption></figure>

device, a simplified square-wave signal was imposed on the current flowing through each of the two
wires. The very first square-wave is associated with a lower amplitude (like the signals imposed
in [38]) to take account of the fact that the second wire has not yet been activated, thus ensuring
symmetrical operation. Finally, it should be noted that for purely practical display reasons, the
two wires are not placed in the same plane, and nodes 8 and 9 linked by the R.P. are offset to
compensate for the loss of wire length on the pulley due to discretization, and to ensure an element
length (and therefore time step) that is not too small for large rotations of the main pulley.

The wire diameter is taken to be 76 µm to be able to refer to the curves presented in [38]. The radii of the main and secondary pulleys are 1.95 mm and 1 mm respectively. By default, the pretension corresponds to a relative elongation of the two wires of 2.4% and the friction coefficient is taken to be 0.01. Finally, the thermal contact exchange coefficient is taken to be $h_c=25 \text{ W.m}^{-1}.\text{K}^{-1}$ by default, while the natural convection coefficient is taken to be $h=10 \text{ W.m}^{-2}.\text{K}^{-1}$. In the absence of precise information concerning the material, the parameters used are those given in table 1 corresponding to standard values.

3.3.3. *Comparison with experimental results*

Figure 18 shows the material flow at the nodes associated with the two secondary pulleys on
the one hand, and at the nodes associated with the main pulley on the other.

A quasi-symmetrical alternation between the two wires is systematically observed, suggesting
an angular signal centered at zero. Material flow values are highest at nodes 2, 3 and 4 for the first
wire, and 13, 14 and 15 for the second. Part of the explanation lies in the fact that these nodes
are associated with the largest elements, whose elongation would naturally be highest assuming
homogeneous strain along the wires. Material flow at the nodes becomes weaker the closer you get
to the ends connected to the main pulley.

Figure 19 shows the temperature changes of several elements of the device. This is a valuable
piece of information, not documented in [38], and of prime importance since it is temperature that
<figure><img src="image_20.png" /><figcaption>Figure 18: Imposed displacement and current and material flow ($h_c=25 \text{ W.m}^{-1}.\text{K}^{-1}$, $f=0.01$ and $\varepsilon_{pt}=2.4\%$)</figcaption></figure>

drives wire contraction. The alternation observed for material flow is confirmed with temperature. The temperature amplitude is highest in elements 1, 3, 12 and 14, which are the portions not in contact with the pulleys and therefore not directly subject to heat exchange with them. These elements, which are therefore the largest, are also the ones that will logically undergo the greatest levels of strain, reinforcing the previous comment on material flow at the associated nodes. Figure 20-(a), which illustrates the material flow field along the wires that constitute the device, clearly shows that the maxima are associated with nodes 2, 3, 14 and 15.

Figure 20-(b) shows the changes in transformation strain of the longest elements of each of the two wires (1, 3, 12 and 14) as a function of temperature. The curves illustrate a relatively different hysteretic behavior from that shown as an example in Figure 6-(c). Indeed, unlike the case of imposed loading in the latter, the case studied here refers to an imposed displacement at the end of the wires and not to a stress. As a result, stress varies throughout the heating cycles. Nevertheless, there are similarities between the paired elements, again due to the alternating operation of the system.

Finally, the response of the device is given in Figure 21, through the evolution of the rotation angle $\alpha$ of the main pulley and the associated torque $\tau$. The latter is deduced from the forces measured at nodes 1 and 16, as in the formula used in [38] ($\tau = r_P(F_1 - F_{16}) = r_P\Delta F$, where $r_P$ is the radius of the main pulley). The reference curves (in black) give orders of magnitude
<figure><img src="image_21.png" /><figcaption>Figure 19: Temperature changes along the two wires ($h_c=25 \text{ W.m}^{-1}.\text{K}^{-1}$, $f=0.01$ and $\varepsilon_{pt}=2.4\%$)</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Figure 20: (a) Material flow field from Paraview ($h_c=25 \text{ W.m}^{-1}.\text{K}^{-1}$, $f=0.01$ and $\varepsilon_{pt}=2.4\%$); (b) Material response in the 4 longest elements of the device</figcaption></figure>

similar to those obtained experimentally with an amplitude $2\alpha_{max}$ close to 20° and a torque whose values are between 1 and 2 N.mm. The form of the angular signal remains close to a square-wave signal, due to the imposed flux, and a sinusoidal signal seems difficult to achieve without a PID corrector. A torque peak can be observed at the start of each half-period, which appears to be directly associated with the temperature peak (see Figure 19). A significant rise in temperature on one wire ensures an almost total reverse transformation to austenite, which translates into a
significant increase in stress, given the elongation imposed on this wire (although part of this is
provided by the second wire, which cools down). The consequence is reflected in the torque. This
effect must be avoided by the control system, both to ensure correct operation of the device and
the durability of the wires, and therefore of the device itself.

<figure><img src="image_23.png" /><figcaption>Figure 21: Response of the device and influence of heat exchange ($f = 0.01$ and $\varepsilon_{pt} = 2.4\%$)</figcaption></figure>

3.3.4. *Parametric analysis*

Parameters of the same nature as for the previous application are studied here to show their
effect on the operation of the device. Indeed, the analytical developments presented in [38] do not
take into account heat exchange and friction at the pulleys, as well as wire pretension.

Figure 21 compares the reference case with configurations in which heat exchanges linked to
contact between wires and pulleys are increased (2h<sub>c</sub> and 4h<sub>c</sub>) or reduced (0.2h<sub>c</sub>). The role of these
exchanges, which by definition tend to bring the temperature of the material towards the temper-
ature of the environment in which it is immersed, is of two kinds. During the heating phase, the
higher the coefficient h<sub>c</sub>, the greater the resistance to temperature rise. During the cooling phase
(when the Joule effect no longer operates), the higher the h<sub>c</sub> value, the greater their contribution
to temperature drop. Thus, the ability to achieve the widest possible temperature range over a
given heating/cooling period for a given power dissipation is based on contact exchanges that are
neither too high nor too low. The results confirm this intuition, since the maximum value of the h<sub>c</sub>
coefficient prevents the wire from reaching a temperature sufficient to ensure reverse phase trans-
formation from the very first heating period. The minimum value, on the other hand, allows an
effective temperature increase, while cooling is difficult to achieve. As a result, the angle increases
in line with expectations (i.e. broadly in the form of a square wave), before deteriorating as the
first wire cools. The intermediate values (h<sub>c</sub> and 2h<sub>c</sub>) provide more acceptable angular signals.
The two points that differentiate them are: the angular amplitude, which is in favor of the refer-
ence curve, and the fact that the heating associated with the first square wave is not sufficient to
achieve a functional temperature amplitude for a value of 2h<sub>c</sub>. On the other hand, the torque does
not peak at the beginning of the heating periods in the latter case, unlike the curves associated
with lower exchanges.

Figure 22 illustrates the effect of pretension on actuator rotation. Thus, the displacement imposed on nodes 1 and 16 is such that the relative elongation of the wires corresponds to values from 1.2 to 2.8%. This does not provide any information on the strain of each portion of wire, since
the law of friction of the pulleys imposes a different tension on them. Only a zero *f* coefficient would allow us to know the level of pretension along the wire, which is not the case here. The results show that the higher the pretension, the greater the angular amplitude. As heat exchange is identical here, temperature amplitudes in the elements are more or less the same. However, the tension drop in elements 3 and 12 due to friction means that the phase transformation and associated strain cannot be exploited identically. Moreover, the minimum pretension level (1.2 %) is too low to ensure cyclic transformation deformation in elements 1 and 14, which are unaffected by pulley friction. Pretension only influences torque at its maximum, since relaxation causes the curves to converge towards the same value (with the exception of the curve associated with minimum pretension).

<figure><img src="image_24.png" /><figcaption>Figure 22: Influence of pretension ($h_c$=25 W.m⁻¹.K⁻¹ and $f$=0.01)</figcaption></figure>

Figure 23 illustrates the effect of the friction coefficient on the pulleys. As discussed earlier, the influence of pretension and friction are closely linked. An increase in the *f* coefficient results in a greater drop in the initial strain of successive elements, especially as we move away from nodes 1 and 16. Thus, the transformation strains of elements 3 and 12 are not sufficiently exploited, resulting in a lower angular amplitude. Friction, on the other hand, plays an important role in the torque value at the end of relaxation. The absence of friction in the pulleys prevent the differential between the forces measured at nodes 1 and 16. Increasing the *f* coefficient increases torque, while reducing angular amplitude in counterpart. In both Figure 22 and Figure 23, the parameters studied have very little influence on the height of the torque peaks, unlike those related to contact heat exchange, which seems to confirm the thermal origin of these peaks as stated above. As with previous parameters, the search for a compromise will ensure that the device functions correctly, since the angular amplitude of the pulse is useless if no torque is transmitted.

The importance of working within the operating range of the device according to the material parameters of the two wires has been demonstrated. The influence of the parameters studied (summarized in Table 3) is therefore crucial, since they act directly or indirectly on the stress and temperature in each portion of the wire, which are the two global variables driving SMA operation (as illustrated by the pseudo-diagram shown in Figure 6-(b)). Such a numerical model cannot replace the work carried out by the PID corrector, which plays its role very satisfactorily, as shown by the experimental results presented in [38]. On the other hand, it can provide valuable assistance in dimensioning the device by estimating the role of the parameters mentioned above. Indeed, control assumes that the operating range of the material can be reached, given the boundary
<figure><img src="image_25.png" /><figcaption>Figure 23: Influence of friction ($h_c=25 \text{ W.m}^{-1}.\text{K}^{-1}$ and $\varepsilon_{pt} = 2.4\%$)</figcaption></figure>

conditions of the problem. The developed tool also enables different control strategies to be tested when designing such a device.

<table><caption>Table 3: Qualitative influence of the parameters studied on the response of the rotational actuator</caption><thead><tr><th></th><th>angular amplitude</th><th>torque peak decrease</th><th>average torque increase</th></tr></thead><tbody><tr><th>$h_c$ ↑</th><td>+ then –</td><td>+</td><td>–</td></tr><tr><th>$f$ ↑</th><td>–</td><td>++</td><td>++</td></tr><tr><th>$\varepsilon_{pt}$ ↑</th><td>++</td><td>–</td><td>0</td></tr></tbody></table>

## 4. Conclusion

In conclusion, this article has demonstrated the clear potential of the ALE formulation for better understanding, dimensioning and optimizing actuator devices and, more generally, systems coupling pulleys and SMA wires. In particular, the contributions of this work are as follows:

* A finite element truss in ALE formulation dedicated to strongly coupled non-linear thermo-mechanical problems and in particular to shape memory materials has been developed.

* Numerical integration of this type of element has been validated on elementary examples, enabling a better understanding of the concepts of the ALE formulation.

* Two application cases from the literature, based on different SMA characteristic properties and application domains at opposite ends of the spectrum, have been studied. Dedicated models demonstrating robustness and speed of execution were developed. The results obtained were directly compared with experimental results. In this way, the relevance of the ALE approach was established both qualitatively and quantitatively. In addition, the strength of the models was illustrated by the possibility of tracking the evolution of data that is in practice inaccessible experimentally. Finally, parametric analyses showed the influence of heat exchange, pulley friction and SMA wire pretension.

The pursuits of this work are numerous and were partly raised in the application cases studied:
* Taking bending stiffness into account will enable better prediction of stress levels with a view to preventing failure by developing a finite beam element within the ALE formulation framework.

* Taking into account thermal martensite to deal with the one-way memory effect, as well as the intermediate R-phase in the behavior law, will extend the possibilities offered by this approach. Care must be taken to advect the associated internal variables within the ALE formulation.

* Taking into account heat dissipation due to friction at the pulleys will enrich the prediction of heat exchange and therefore of system response.

* Taking account of functional fatigue phenomena (by improving the behaviour law) and strain localisation by combining the ALE formulation with alternative numerical methods (X-FEM, phase field) will make it possible to optimise the performance of such devices.

* By integrating a closed-loop control system, complex actuator responses can be imposed. This will make it possible to optimize the control law and associated parameters.

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Highlights

Numerical modeling of systems based on SMA wires and pulleys: a finite element formulation within the Arbitrary Lagrangian-Eulerian framework

Guillaume Helbert<fnref n="1" />, Jessy Simon<fnref n="1" /> <fnref n="1" />

* A thermo-mechanical finite element truss in ALE formulation has been developed.

* Two real devices based on SMA wires and pulleys were successfully modelled.

* A parametric analysis of the models revealed ways of optimising the devices.

<footnote n="1">¹Dr J. Simon and Dr G. Helbert have the same contribution. They should be considered as the co-first author.</footnote>