A simplified analytical model to simulate constrained recovery mechanisms of SMA rings exhibiting two-way shape memory effect

Abstract

Shape memory alloys (SMAs) with shape recovery capabilities have been investigated recently at the European Organization for Nuclear Research (CERN) to develop ring-shaped pipe couplers for vacuum applications. SMA couplers exploit the One-Way and Two-Way Shape Memory Effect (OW- and TW-SME) to mount/dismount vacuum pipe, by temperature variations. A phenomenological analytical model based on simplified elastic-plastic axisymmetric theory has been developed and implemented in a commercial software to simulate biaxial constrained recovery mechanisms in thick-walled SMA rings with rectangular cross section. The model is particularly useful to predict the stress field in the SMA coupler as well as the contact pressure developed at the SMA ring/pipe interface during thermal mounting/dismounting operations, knowing their initial geometry and material properties. The predictions of the analytical model have been compared with experimental data and Finite Element (FE) simulations based on a user-defined material routine.

Keywords

Shape memory alloy analytical solution SMA rings constrained recovery pipe coupling

1. Introduction

Shape Memory Alloys (SMAs) arouse great interest from the scientific community and are nowadays employed increasingly in many fields of engineering (Borden, 1991; Jani et al., 2014), due to the Pseudoelastic Effect (PE), the One Way-Shape Memory Effect (OW-SME), and the Two-Way Shape Memory Effect (TW-SME) that give the material the ability to recover large forces and deformations (Duerig et al., 1990). These features are due to a reversible solid-state transition between two different microstructures: the body-centered cubit austenite (B2) and the monoclinic martensite (B19’) (Otsuka and Ren, 2005). In particular, the austenite is stable at high temperature and low stress, and the martensite is stable at high stress and low temperature. Consequently, the phase transformation can be induced by means of two different processes: Thermally Induced Martensite (TIM), between the so-called Transformation Temperatures (TTs), and Stress-Induced Martensite (SIM), between the critical Transformation Stresses (TSs).

Biocompatibility, high power-to-mass ratio, and relatively low cost mean that applications of SMAs span various fields of engineering, such as medicine, robotics, electronics, aerospace, and mechanics (Duerig et al., 1990; Jani et al., 2014; Lester et al., 2015; Mohd Jani et al., 2017; Niccoli et al., 2021). One of the most promising and successful applications includes thermomechanical couplings of tubes/pipes for aerospace, marine or industrial use (Kapgan and Melton, 1990; Malukhin et al., 2012; Proft and Duerig, 1990; Stalmans et al., 1997; Tabesh et al., 2012). The coupling is performed by thermally inducing the martensite to austenite transition of a SMA ring-shaped connector under constrained conditions, that is, the free recovery of a the pre-deformed SMA element (OW-SME) is prevented by to the presence of a mechanical obstacle (the pipes to be connected). This results in the generation of a radial force at the contact interface between the SMA coupler and the internal pipe. The disassembly, related to the austenite to martensite transition, may be induced by cooling thanks to the Two-Way Shape Memory Effect (TW-SME). Unlike OW-SME, TW-SME is not an inherent feature of SMAs, but can be obtained after specific thermomechanical treatments, the so-called training (Atli et al., 2013; Liu and Van Humbeeck, 1998; Meng et al., 2006; Perkins and Hodgson, 1990; Scherngell and Kneissl, 1998; Wang et al., 2003).

The possibility to remotely control the mounting/dismounting of a system by temperature changes has paved the way for the use of SMA connectors in the nuclear engineering domain, where the human presence must be limited as much as possible at sites dangerous to health (Besseghini et al., 1996; Kornfeldt et al., 1997; Nishikawa et al., 1986, 1989). In this regard, some recent studies have been carried out at CERN assessing the possibility to develop SMA-based couplers exhibiting two-way shape memory effect (TW-SME) to be employed in the Ultra-High-Vacuum (UHV) systems of the Large Hadron Collider (LHC) particle accelerator (Niccoli et al., 2017a, 2017b). Such couplers are able to realize leak-tight, radiation-resistant and demountable connections without the need of human presence in loco, since the clamping/unclamping mechanisms are remotely induced by controlling the temperature (Niccoli et al., 2019, 2022b). Preliminary analytical and numerical studies together with laboratory-scale experimental testing are the initial steps in the development of SMA connectors. The mechanical and functional behavior of such connectors has been extensively investigated numerically and experimentally (Niccoli et al., 2017a, 2017b, 2019) before their implementation within the vacuum systems at CERN (Niccoli et al., 2022b). Part of the research was devoted to the development of constitutive models for PE and SME to be implemented in commercial Finite Element (FE) software (Auricchio and Petrini, 2004; Auricchio et al., 2007; Lagoudas and Entchev, 2004; Souza et al., 1998). Due to the complexity of SMA behavior in constrained recovery applications, the material models already implemented in commercial FE codes are not suitable to properly simulate all stages of a coupling process. To this end, a new three-dimensional phenomenological model accounting for TW-SME and plasticity was recently developed and calibrated based on experimental measurements, with the aim of increasing the accuracy of numerical solutions (Scalet et al., 2019). However, the use of simple geometries for the connection system, such as axial symmetric bodies (pipes and ring), encourages an analytical approach of the problem, entailing a number of advantages with respect to a FEA. First, no special skills are required by the user to run the analytical code, as there is no need to model the geometry and process, set the contact conditions between coupler and pipes, or design the mesh. In addition, the computational cost in terms of solution time and calculator memory required for the analytical solution is estimated to be significantly lower than in the case of a FEA.

The modeling of TW-SME is found in several works, where general macroscopic/phenomenological models have been developed (Hakimi and Ashrafi, 2023; Lexcellent et al., 2000; Xu et al., 2021; Zhang et al., 1997). Most of them are based on energy criteria and, in particular, on the definition of Gibbs free energy (Xu et al., 2021) and Helmholtz free energy (Hakimi and Ashrafi, 2023; Lexcellent et al., 2000). Tensor-based equations allow their application to three-dimensional geometric configurations and different loading conditions. However, a high computational cost is required, which necessitates the use of numerical solutions (Xu et al., 2021). Alternatively, to reduce complexity, some approaches simplify the problem to the uniaxial case (Hakimi and Ashrafi, 2023; Zhang et al., 1997) or to stress-free conditions (Lexcellent et al., 2000).

In the literature, only a few works deal with the constitutive analytical modeling of SMA behavior under constrained recovery conditions. Research works are mainly focused on the uniaxial process (Sittner et al., 2000; Videnic et al., 2012; Vokoun et al., 2003) although most constrained recovery applications of SMA are based on multiaxial phenomena. Recent studies have explored the thermal recovery mechanisms of SMA rings (Piotrowski et al., 2012; Tabesh et al., 2018; Videnic et al., 2008) and the axisymmetric loading of thick-walled SMA cylinders in both elastic and transformation regimes (Liu and Du, 2014; Liu et al., 2013; Mirzaeifar et al., 2011; Radi, 2018). Specifically, these investigations include numerical simulations, semi-analytical analyses (Mirzaeifar et al., 2011), and analytical studies (Liu and Du, 2014; Liu et al., 2013; Radi, 2018; Tabesh et al., 2018; Videnic et al., 2008) of the behavior of thick-walled superelastic SMA cylinders subjected to internal pressure under plane stress or plane strain conditions, without accounting for plastic behavior. To this regard, only one study was found, dealing with the biaxial constrained recovery of a SMA ring, and, to this aim, a simple mathematical model based on the generalized plasticity was developed (Videnic et al., 2008). Such model is based on some simplifications. In particular, it does not account for any unrecoverable process (plastic strain accumulation). Moreover, it simulates only the thermal activation of the SMA during the heating phase (martensite to austenite transition), where the OW-SME is exploited. The disassembly process (austenite to martensite transition (TW-SME) is not modeled.

This paper presents a simplified analytical model to simulate the constrained recovery mechanisms of thick-walled SMA rings exhibiting TW-SME and plasticity. Both thermal mounting (martensite to austenite transition, M→A) and dismounting phases (austenite to martensite transition, A→M) are considered. The proposed model represents a continuation and completion of previous work (Niccoli, 2022a), where an analytical model was developed to simulate the martensitic ring pre-deformation (reorientation) mechanisms. It is based on modified elastic-plastic theories for axisymmetric bodies and calibrated by means of experimental data obtained from uniaxial thermo-mechanical measurements of commercial SMAs. The model is based on a semi-empirical approach, which results in a limited number of phenomenological parameters. The model can be regarded as a straightforward tool for predicting the constrained recovery behavior of SMA rings based on their material and geometric parameters. It serves as an initial step towards developing a robust and reliable design tool for SMA joints. The effectiveness of the analytical model was validated by evaluating different SMA ring materials and geometries. Its predictions in terms of mechanical stresses/contact pressure were compared with results from axisymmetric finite element (FE) analyses using specially developed material models. Experimental tests were also performed to validate the analytical results.

2. SMA-based pipe coupling: Mounting and dismounting features

As-manufactured SMA couplers are subjected to complex thermo-mechanical processes, the so-called training, to induce proper geometrical and functional properties in terms of TTs, one-way (OW) and two-way (TW) shape memory effects (SME). Ring pre-straining at a temperature lower than martensite finish TT ( $M_{f}$ ), represents a critical step of the training process, that is to induce stable detwinned martensitic variants. Modeling of the pre-strain phase is out of the scope of this work as it was developed in a previous study (Niccoli, 2022a). Subsequent thermal mounting and dismounting processes of pre-strained SMA rings are modeled in this investigation. Figure 1 schematically shows the basic mechanisms associated with both thermal mounting and dismounting of a SMA-based coupler, exhibiting both one-way and a (OW-SME) two-way (TW-SME) shape memory effects, that is in terms of contact pressure, p_i (upper graph) and circumferential strain (lower graph) as a function of the temperature under thermal cycle between TTs. $ε_{SMA}$ represents the circumferential strain at the SMA ring inner diameter while $ε_{PIPE}$ indicates the circumferential strain at the pipe outer diameter. Such curves refer to a rigid elastic pipe. The stress-free thermal recovery curve (ε–T ) of the SMA ring (no ring/pipe interaction) is also reported in the lower graph (dotted red curve) together with one way ( $ε_{OW}$ ) and two way ( $ε_{TW}$ ) shape recovery deformations.

These recovery mechanisms are due to a reversible transformation occurring from martensite to austenite (M-A) during heating, and its reversion (A-M) during cooling. OW-SME is always observed in a pre-deformed SMA; it is the main responsible of force generation in the coupler at the first thermal mounting process, due to the impeded deformation. On the contrary, TW-SME is achievable through specific thermo-mechanical treatments (training) that induce oriented internal stress fields. TW-SME allows the SMA ring dismounting.

The trained SMA ring is positioned onto the pipe at a temperature $T_{o}$ with an initial assembly clearance. The clearance corresponds to the gap between the initial internal radius of the trained SMA ring and the external radius of the pipe ( $Δ_{r 0}$ in Figure 1). The SMA ring pipe coupling process consists of the following subsequent operations: thermal mounting and thermal dismounting, which are described below.

Thermal mounting: the trained SMA ring is heated above its austenite finish temperature to activate OW-SME up to a maximum temperature $T_{\max}$ and then cooled down to the operative temperature ( $T_{o}$ ) corresponding to a certain contact pressure ( $p_{o}$ ). The mounting consists of four subsequent phases (phase 1 to 4) detailed at the end of the paragraph.

Thermal dismounting: it can be obtained by cooling the assembly from $T_{o}$ to a temperature to $T_{\min} < M_{f}$ , thanks to the TW-SME capabilities of the ring which induce its diameter expansion (see Figure 2). It consists of four phases (phase 5 to 8).

The subsequent phases of the mounting/dismounting process (phase 1 to 8) occur in specific thermal ranges of the complete heating/cooling cycle ( $T_{o} \to T_{\max} \to T_{\min}$ ) as described below.

- Phase (1) Thermoelastic stress-free phase ( $T_{o} < T < A_{s}$ ). No transformation mechanisms occur, and both SMA ring and the pipe expand, according to their linear thermal expansion coefficients (CTE), under stress-free conditions.

- Phase (2) Inelastic stress-free phase ( $A_{s} < T < T_{c}$ ). M-A transition (OW-SME) starts at $T = A_{s}$ (Austenite start temperature) and a stress free contraction continues until $T = T_{c}$ (ring-pipe contact), corresponding to full recovery of $ε_{free}$ ;

- Phase (3) Inelastic constrained phase $(T_{c} < T < T_{\max})$ ). The M-A transition in the SMA ring continues in constrained recovery condition until the rings reaches a certain temperature ${A_{f}}^{σ}$ , at which the martensitic variants are not anymore thermo-mechanically stable and the ring is fully austenitic. A radial pressure is developed at the contact interface which increases with increasing temperature. It results from the prevented OW-SME as well as from SMA/steel CTE mismatch. A maximum contact pressure ( $p_{\max}$ ) is reached at $T_{\max} > {A_{f}}^{σ}$ .

- Phase (4) Thermoelastic constrained phase ( $T_{\max} < T < T_{o})$ . The contact pressure decreases with respect to the temperature due to SMA/steel thermal expansion coefficient mismatch. Only thermo-elastic effects occur in this phase as $T_{o} > {M_{s}}^{σ}$ , where ${M_{s}}^{σ}$ is the martensite start temperature under stress. The operative contact pressure of the assembly ( $p_{o}$ ) is recorded at $T_{o}$ .

- Phase (5) Thermoelastic constrained phase ( ${M_{s}}^{σ} < T < T_{o}$ ). Initially, from $T_{o}$ to ${M_{s}}^{σ}$ (martensite start temperature under stress), both SMA and steel rings are subjected only to thermoelastic contraction.

- Phase (6) Inelastic constrained phase ( $T_{s} < T < {M_{s}}^{σ}$ ). At. $T = {M_{s}}^{σ}$ , the A-M phase transformation takes place and a further decrease in temperature causes significant stress relaxations resulting from SMA inelastic expansion (TW-SME). The contact between the rings is lost at $T = T_{s}$ (separation temperature).

- Phase (7) Inelastic stress-free phase ( $M_{f} < T < T_{s}$ ). the A-M phase transformation continues (SMA ring expansion) under free recovery conditions until $M_{f}$ (Martensite finish temperature) is reached.

- Phase (8) Thermoelastic stress-free phase ( $T_{\min} < T < M_{f}$ ). Both SMA ring and pipe contracts due thermal strains until $T_{\min}$ is reached, allowing the disassembly of the system thanks to the SMA/steel clearance.

3. SMA recovery stress: Modeling and assumptions

3.1. Basic features of SMA recovery stress

Recovery stresses are generated when the austenitic shape recovery of a SMA exhibiting SME is prevented while heating above A_s. The analysis of a uniaxial case is reported in Figure 2. A uniaxial case is analyzed in Figure 2. The generation of recovery stresses is due to a previous macroscopic deformation, up to a maximum value $ε_{\max}$ , at a temperature $T_{def}$ aimed to induce stable detwinned martensitic variants (Figure 2(a)). $ε_{unl}$ represents the residual inelastic deformation after unloading. If the SMA element is heated, free shape recovery occurs due to SME. In absence of mechanical obstacle (red curves) the entire one-way deformation $ε_{OW}$ can be recovered during the thermally induced martensite to austenite transformation. $ε_{irr}$ represents the irrecoverable part of deformation. If the material shows TW-SME capabilities, an elongation corresponding to $ε_{TW}$ can be induced by cooling below its martensitic TTs. Instead, in the presence of a mechanical rigid obstacle, free recovery stage occurs only up to a temperature $T_{c}$ (Figure 2(b)), namely the contact temperature. SMA partially transforms to austenite recovering a certain strain $ε_{free} = ε_{unl} - ε_{c}$ .

The recovery of the remaining deformation, the contact strain $ε_{ct} = ε_{c} - ε_{irr}$ , is impeded. Consequently, at temperature higher than $T_{c}$ , recovery stress is generated (Figure 2(c)). It increases by increasing temperature reaching a maximum value $σ_{r}^{\max}$ occurring in correspondence of $A_{f}^{σ}$ which corresponds to the temperature at which the SMA element, under stress, becomes fully austenitic, that is, without thermally induced or stress induced martensitic variants. This latter corresponds indicatively to the so-called martensite desist temperature $M_{d}$ . Thanks to the TW capability of the SMA, by subsequent cooling it is possible to induce the forward A-M transition starting from the martensite start temperature under stress $M_{s}^{σ}$ of the material associated with a significant relaxation until a zero-stress state at $T_{s}$ .

It is worth pointing out that for polycrystalline SMAs the slope of the measured σ–T curve is not constant as the evolution of recovery stress is non-linear. Moreover, the average slope, $β$ in Figure 2(c), is not a material parameter but depends to some extent on the martensitic pre-strain $ε_{\max}$ , the contact strain $ε_{ct}$ and in general on the thermomechanical history of the alloy (Šittner et al., 2018, 2019; Stalmans et al., 1995). However, $β$ is frequently estimated through linear fits and used as material characteristic parameters as, in specific conditions, was found close to the Clausius Clapeyron coefficient of the alloy.

In a very general case, the total strain of a material showing SME, $ε_{t}$ , can always be subdivided into a transformation component $ε_{tr}$ , an elastic component $ε_{el}$ , a plastic component $ε_{pl}$ and a thermal component $ε_{th}$ .

During constrained recovery, the total strain $ε_{t}$ is equal to the contact strain $ε_{ct}$ , so:

ε_{t} = ε_{ct} = ε_{tr} + ε_{el} + ε_{th} + ε_{pl}

(1)

Under the hypothesis of a mechanical obstacle infinitely rigid and with identical thermal expansion coefficient to the shape memory element the quantity $ε_{cth} = ε_{ct} - ε_{th}$ is independent of both temperature and stress, that is, $ε_{cth}$ is constant during constrained recovery:

ε_{cth} = ε_{ct} - ε_{th} = ε_{tr} + ε_{el} + ε_{pl} = const .

(2)

From equation (2) it follows that, during the constrained shape recovery mechanism, a certain amount of detwinned martensite variants transform to austenite when they are not anymore thermally stable and therefore the quantity $ε_{tr}$ decreases while $ε_{el} + ε_{pl}$ increases of the same extent.

Consequently, if $ε_{cth}$ is sufficiently high, plastic deformation occurs. The yield stress of the material, σ_y, is then an upper limit to σ_r. Recently, experimental evidence demonstrated that reversible twinned austenite formation (B2^T) is coupled with slips mechanisms (Šittner et al., 2018, 2019).

Conversely, if $ε_{cth}$ is within the elastic deformation range of the SMA sample, a recovery stress equal to $E_{A} \cdot ε_{cth}$ is generated, where $E_{A}$ is the Young’s modulus of the austenite. This represents a lower bound limit for σ_r.

All the considerations mentioned pertain to shape memory alloys (SMAs) whose recovery is limited by a perfectly rigid substrate that has the same thermal expansion coefficient as the memory material. In practical applications, however, the substrate experiences elastic or elastic-plastic deformations. During the constrained recovery process, once the temperature exceeds T_c, the thermo-mechanical behavior of the SMA is influenced by the substrate stiffness and the difference between the thermal expansion coefficients of the SMA and the substrate. Therefore, recovery stress depends on the specific recovery path and cannot be predicted immediately. Maximum recovery stress, $σ_{r}^{\max}$ , in NiTi-based alloys, ranges from about 300 to 900 MPa depending on the their microstructure (Cross et al., 1969; Friend, 1986; Jackson et al., 1972; Perkins, 2012; Šittner et al., 2018, 2019; Stalmans et al., 1995). Generally, $σ_{r}^{\max}$ , is a function of $ε_{\max}$ and $ε_{cth}$ (Cross et al., 1969) (cfr. Figure 2 and equation (2)). $ε_{OW}$ represents the upper bound value of $ε_{cth}$ .

3.2. Model assumptions

In general, the recovery stress for a given $ε_{cth}$ (cfr. equation (2)) at a given temperature (T) is an indication of the stability of the martensite variants at that temperature, so it is directly related to the stress required to induce the same amount of inelastic strain ( $ε_{cth}$ ) in the same material at that temperature (Perkins, 2012). Several experimental studies confirmed that the $σ_{r}$ (T, $ε_{cth}$ ) curve obtained by recovery stress tests (constrained thermal cycles between TTs for different $ε_{cth}$ ) can be approximated with a certain error (normally within 15%) by the stress-strain curve of the same material obtained from isothermal tensile tests σ(T, ε) (Duerig et al., 1990; Perkins, 2012). The typical transformation strains $ε_{OW}$ induced by martensite pre-deformation are lower than 7%. Consequently, a bi-linear stress-strain response is assumed for SMA constrained recovery modeling (see Figure 3(b)).

Figure 3(a) schematically shows hysteretic recovery stress curve of a SMA exhibiting TWSME obtained by constrained heating/cooling reported on a typical SMA phase diagram including characteristic TTs (Otsuka and Ren, 2005) (Duerig et al., 1990). Figure 3(b) shows the isothermal bi-linear $σ - ε$ (or $σ_{r} - ε_{cth}$ ) curves of a SMA at three different temperatures ( $T_{1} < T_{2} < T_{3}$ ). The volume fraction of martensite at the end of the stress plateau, $ξ$ , goes from $ξ = 1$ (fully martensitic state at $T_{1}$ ) to $ξ = 0$ (fully austenitic state at $T_{3}$ ). $σ_{cr}^{S}$ represents the critical plateau stress to induce inelastic strain in the SMA. The latter may result from reoriented/stress induced martensite phenomena and/or austenite twins/slip formations depending on the temperature. For $σ < σ_{cr}^{S}$ (elastic region), the SMA is elastically deformed, according to the value of the equivalent Young’s modulus, which is defined as a function of the volume fraction of martensite E(ξ). For $σ \geq σ_{cr}^{S}$ (inelastic region), the SMA undergoes inelastic deformations and the stress-strain relation is described by a hardening parameter, H.

Figure 3(a) also shows the TTs of the trained SMA ( $A_{s}$ , $A_{f}$ , $M_{s}$ , $M_{f}$ ) that are assumed to vary linearly with the applied stress field according to Clausius Clapeyron equation. In constrained recovery applications, the stress field occurring in the SMA is not known a-priori. However, the austenite finish temperature under stress $A_{f}^{σ}$ and the martensite start temperature under stress $M_{s}^{σ}$ (cfr. Figure 3(a)) can be expressed as a function of the corresponding recovery stress calculated at the end of the martensite to austenite transformation, $σ (A_{f}^{σ})$ , and at the beginning of the austenite to martensite transition, $σ (M_{s}^{σ})$ , respectively:

A_{f}^{σ} = A_{f} + \frac{σ (A_{f}^{σ})}{β_{Af}}

(3)

M_{s}^{σ} = M_{s} + \frac{σ (M_{s}^{σ})}{β_{Ms}}

(4)

Where $A_{f}$ and $M_{s}$ are the austenite finish temperature and martensite start temperature in stress-free conditions and $β_{Af} and β_{Ms}$ the Clausius-Clapeyron coefficients for austenite finish and martensite start temperature, respectively. It is worth noting that when $T \geq A_{f}^{σ}$ , in case of a sufficiently rigid obstacle, austenite twins and plastic deformations may occur due to the high stress-temperature state. Consequently, $σ (A_{f}^{σ})$ corresponds indicatively to the yielding stress of the material: $σ (A_{f}^{σ}) = σ_{y}$ (cfr. Figure 3(a)); $A_{f}^{σ}$ can be immediately computed from equation (3). $M_{s}^{σ}$ instead, can be estimated iteratively from equation (4). To this purpose, $σ (M_{s}^{σ})$ can be expressed as a function of $M_{s}^{σ}$ and $σ (A_{f}^{σ})$ by subtracting from $σ (A_{f}^{σ})$ the thermoelastic stress variation occurring during the cooling from $A_{f}^{σ}$ to $M_{s}^{σ}$ due to the SMA/obstacle thermal expansion coefficient mismatch (cfr. Figure 3(a)).

TTs, $σ_{y}, β_{Af}, and β_{Ms}$ are experimentally evaluated by isobaric strain-temperature tests and isothermal stress-strain tests. Martensitic and austenitic yield stresses values exhibit a limited variation with temperature (see Section 6.1), therefore $σ_{y}$ is modeled as temperature independent.

The proposed model considers as internal variable the volumetric fraction of detwinned martensite. As experimentally demonstrated (Heller et al., 2019; Šittner et al., 2018, 2019), the volume fraction of martensite measured at the end of the stress plateau as a function of the temperature exhibits a sigmoidal trend between the transformation temperatures during both the inverse (M-A) and direct (A-M) transformations. Therefore, the evolution of the volume fraction of detwinned martensite in constrained recovery applications can be analytically described by the following sigmoidal kinetic laws:

ξ_{M \to A} = 1 - \frac{k N_{0}}{N_{0} + (k - N_{0}) e^{- r_{ξ} (T - (| A_{s} | + \frac{| A_{f}^{σ} | - | A_{s} |}{2}))} + q}

(5)

ξ_{A \to M} = 1 - \frac{k N_{0}}{N_{0} + (k - N_{0}) e^{- r_{ξ} (T + (| M_{s}^{σ} | + \frac{| M_{f} | - | M_{s}^{σ} |}{2}))} + q}

(6)

where $k, N_{0}, r_{ξ}$ , and $q$ are dimensionless tuning parameters, defining the amplitude ( $k = 1$ ), the inflection point temperature ( $N_{0} = 0.5$ corresponds to a curve with an inflection point temperature equal to the median temperature of the thermal range of interest), the maximum asymptotic intensity ( $r_{ξ} \leq 1$ ), and the slope ( $q = 0$ ) of the sigmoidal curve at the extremities of the temperature range.

The recovery strain $ε_{tr}$ is the recoverable deformation due to the OW-SME (contraction due to martensite to austenite transformation). In literature (Lubliner and Auricchio, 1996) the recovery strain is considered to be proportional to the volume fraction of martensite, as reported below:

ε_{tr} = ε_{OW} (ξ - ξ_{0})

(7)

Where $ξ_{0}$ is the volume fraction of martensite at the initial state of the considered transformation step while $ε_{OW}$ is the maximum recoverable strain available after the martensitic pre-strain (Figure 2). $| ε_{tr} |$ increases from zero to $ε_{OW}$ during M to A transformation (OW-SME) and decreases during A-M transformation (TW-SME). During M-A transformation ( $A_{s} - A_{f}^{σ})$ , $| ε_{tr} |$ describes both the evolution of the free recovery strain (when $| ε_{tr} | \leq | ε_{free} |)$ as well as the evolution of the constrained strain (when $| ε_{free} | < | ε_{tr} | \leq ε_{OW})$ . The latter can be computed as $| ε_{tr} | - | ε_{free} |$ , neglecting the thermoelastic effects.

For the sake of simplicity, it is also assumed that the SMA during constrained A to M transformation ( $M_{s}^{σ} - M_{f})$ elongates by the same amount of strain, $ε_{TW} = ε_{OW}$ , thanks to the TW-SME induced during the previous training/martensitic pre-deformation. Experimental evidence (Niccoli et al., 2017a, 2022b; Scalet et al., 2019) including specific tests carried out within this investigation (cfr. Figure 12) show that recovery stress of properly trained SMAs drops to zero by cooling down to $M_{f}$ even if two-way recoverable deformation obtained from stress-free thermal cycles is lower than the one way. This is due to the complex deformation mechanisms that occur during M-A constrained heating, which, as explained above, result in plastic and/or pseudoplastic strain generation. This plastic/pseudoplastic deformation increases the two-way shape memory effect (TW-SME) of the material. Additionally, the applied stress has a training effect on the SMA, leading to an increased volume fraction of martensite variants that are favorably oriented. This, in turn, enhances the material TW thermal recovery capabilities.

The Equivalent Young’s modulus $E$ (cfr. Figure 3(b)) is defined by the inverse rule of mixtures (Reuss formula), as follows:

E (ξ) = {(\frac{ξ}{E_{M}} + \frac{1 - ξ}{E_{A}})}^{- 1}

(8)

where $E_{A}$ and $E_{M}$ are the Young’s modulus of austenite and detwinned martensite, respectively.

$ε_{OW}$ , $E_{M}, E_{A}$ can be identified by uniaxial tests.

As mentioned in Section 2.1, experimental evidences show that the critical plateau stress $σ_{cr}^{s}$ (see Figure 3(b)) that define the evolution of stress in a constrained SMA exhibiting OW-SME increases non linearly with respect to the temperature. This is because recovery stress generation intrinsically involves complex deformation mechanisms at high temperatures and stresses (B19= > B2^T+slips) beyond the linear Clausius Clapeyron regime (Duerig et al., 2017; Heller et al., 2019; Šittner et al., 2019).

However, for the sake of simplicity, a linearization was performed and $σ_{cr}^{s}$ with respect to the temperatures during both martensite to austenite ( $A_{s} - A_{f}^{σ}$ ) and austenite to martensite ( $M_{s}^{σ} - M_{f}$ ) transition temperatures evolve according to the following laws:

σ_{=}^{s} σ_{cr}^{s} (A_{s}) + β_{M \to A} (T - A_{s})

(9)

σ_{cr A \to M}^{s} = σ_{cr}^{s} (M_{s}^{σ}) - β_{A \to M} (T - M_{s}^{σ})

(10)

During the constrained heating, $σ_{M \to A}^{s}$ increases from $σ_{cr}^{s} (A_{s})$ until it reaches the yield stress of the material at $T_{3} = A_{f}^{σ}$ ( $σ_{cr}^{S} = σ_{y}$ ) when twinning and plastic mechanism in the austenite phase occur and the martensite cannot be stress-induced anymore ( $ξ = 0$ ).

Similar considerations apply for the A-M transition occurring during the cooling stage starting from $M_{s}^{σ}$ . The critical stress relaxation is described by $β_{A \to M} .$ The two material parameters. $β_{M \to A}$ and $β_{A \to M}$ represent the average slope of the critical stress as a function of the temperature within the thermal range of interest that is, $A_{s} < T < A_{f}^{σ}$ and $M_{f} < T < M_{s}^{σ}$ , respectively. They can be experimentally evaluated by uniaxial isothermal stress-strain tests or isobaric strain temperature tests (crf. Section 6.1).

The tangent modulus $H$ (cfr. Figure 3(b)) is assumed to be constant during the entire process, as it does not show a significant variation with temperature (cfr. Section 6.1)).

Previous studies concerning stress free/stress applied thermal cycles on pre-deformed SMA rings and rods demonstrated that the evolution of circumferential strain/stress at the SMA ring inner diameter, $ε_{θ} (r_{i, SMA}), σ_{θ} (r_{i, SMA}),$ as a function of the temperature as well as the ring TTs can be correctly captured by $ε -$ T and $σ$ −T curves recorded from uniaxial thermo-mechanical tests of SMA samples subjected to the same martensitic pre-deformation $ε_{θ} {(r_{i, SMA})}_{\max} = ε_{\max}$ (cfr Figure 2) (Niccoli et al., 2017a; Scalet et al., 2019). In fact, equivalent stress and strain quantities at the inner radius of SMA rings with wall thickness to diameter ratio of engineering interest ( $h / D < 0.20$ ) can be approximated to circumferential ones. Consequently, it is possible to describe the ring TTs and the evolution of the martensite volume fraction in SMA rings by equations (3–6). Moreover, the circumferential recovery strain at the SMA ring inner diameter, $ε_{θ, tr} (r_{i, SMA})$ can be computed as the uniaxial strain $ε_{tr}$ (cfr. equation (7)).

It is worth noting that, the modeling of the complex microstructural deformation mechanisms occurring during recovery stress generation is out of the scope of this work which aims at describing the SMA constrained recovery behavior in a simple phenomenological fashion.

4. SMA ring constrained recovery modeling

The set of equations, boundary conditions and load steps governing both M-A and A-M transitions in SMA rings exhibiting TW-SME are defined in the following paragraphs.

4.1. Equilibrium and compatibility equations

The ring-pipe system can be bi-dimensionally modeled, due to the axisymmetric conditions, as depicted in Figure 4. In particular, Figure 4(a) shows the initial and final state of the process, where a gap $(Δ r_{0}$ ) exists between the pipe and the SMA ring. Figure 4(b) represents all other intermediate steps, from contact to separation, where a radial pressure p_i exists at the contact interface. r_c corresponds to the SMA ring/pipe contact radius while $\bar{r}$ is the critical radius which correspond to elastic/inelastic transition, assuming a bi-linear stress-strain relationship (cfr. Figure 3(b)).

The problem can be studied in polar coordinates (r, θ). Stress and strain (σ, ε) in the SMA ring are function of the radius r. For the sake of brevity, the calculation procedure that leads to the final equilibrium and compatibility equations is not given in detail but can be found entirely in (Niccoli et al., 2021).

The equilibrium equation, in the absence of body forces and in terms of radial and hoop stresses (σ_r, σ_θ), is given by:

\frac{d σ_{r}}{dr} + \frac{σ_{r} - σ_{θ}}{r} = 0

(11)

The compatibility equations, in terms of radial and hoop strain ( $ε_{r}$ , $ε_{θ}$ ) for the elastic and inelastic regions, respectively, are reported below:

\frac{d ε_{θ}^{el}}{dr} - \frac{ε_{r}^{el} - ε_{θ}^{el}}{r} = 0

(12)

\frac{d ε_{θ}^{in}}{dr} - \frac{ε_{r}^{in} - ε_{θ}^{in}}{r} = - \frac{1}{E} (\frac{d σ_{r}}{dr} + \frac{d σ_{θ}}{dr})

(13)

Where $ε_{r}^{el}, ε_{θ}^{el}$ , and $ε_{r}^{in}$ and $ε_{θ}^{in}$ are the elastic and inelastic components of strain, respectively. Total strains are therefore $ε_{r} = ε_{r}^{el} + ε_{r}^{in}$ and $ε_{θ} = ε_{θ}^{el} + ε_{θ}^{in}$ .

Equations (9) and (10) can be also expressed in terms of stress to have a final equations system that can be solved directly. For the elastic region, equation (9) is combined with the Hooke’s law. Under plane stress conditions, the compatibility is rewritten as:

\frac{d σ_{θ}}{dr} - \frac{σ_{r} - σ_{θ}}{r} = 0

(14)

For the inelastic region, is assumed that beyond the elastic limit the deformations follow the plastic flow and they can be calculated, according to the total strain theory, as follows:

ε_{ij}^{in} = \frac{3}{2} \frac{ε_{e}^{in}}{σ_{e}} S_{ij}

(15)

where $σ_{e}$ is the equivalent von Mises stress, and $S_{ij}$ are the deviatoric stress components. The evolution of the equivalent inelastic strain $ε_{e}^{in}$ as a function of the equivalent stress $σ_{e}$ is obtained from the uniaxial stress-strain curve. Thus, the inelastic compatibility equation expressed in terms of stress is obtained by combining equation (10) with equation (12), as shown below:

\begin{matrix} \frac{1}{E (ξ)} (\frac{d σ_{θ}}{dr} + \frac{d σ_{r}}{dr}) + \frac{1}{2 r σ_{e}} [3 (σ_{θ} - σ_{r}) + r (2 \frac{d σ_{θ}}{dr} - \frac{d σ_{r}}{dr})] \\ (σ_{cr}^{s} - σ_{e}) (\frac{1}{E (ξ)} - \frac{1}{H}) - \frac{1}{4 σ_{e}^{3}} \\ [(2 σ_{θ} - σ_{r}) (σ_{θ} (2 \frac{d σ_{θ}}{dr} - \frac{d σ_{r}}{dr}) - σ_{r} (\frac{d σ_{θ}}{dr} - 2 \frac{d σ_{r}}{dr}))] \\ (\frac{σ_{cr}^{s}}{E (ξ)} - \frac{σ_{cr}^{s}}{H}) = 0 \end{matrix}

(16)

4.2. Load steps and boundary conditions

During phase transformations under constrained-recovery conditions, the prevented circumferential strain at the SMA ring inner diameter, increases by increasing temperature (M-A transition) and decreases to zero by decreasing temperature (A-M transition).

Three different loading cases occur in the SMA ring exhibiting bilinear stress-strain evolution (cfr. Section 6.1), and can be summarized in the following three steps:

- Load step #1: Elastic strain at both $r_{i, SMA}$ and $r_{e, SMA}$ ;

- Load step #2: Inelastic strain at $r_{i, SMA}$ and elastic strain at $r_{e, SMA}$ ;

- Load step #3: Inelastic strain at both $r_{i, SMA}$ and $r_{e, SMA}$ .

The SMA ring goes through all load steps, in the order defined by the direction of the transformation: from step #1 to #3 in the case of M-A (heating), from step #3 to #1 in the case of A-M (cooling).

The ring is subjected to an elastic stress state if $σ_{e} (r_{i, SMA}) < σ_{cr}^{s}$ , where $σ_{e} (r_{i, SMA})$ is the equivalent Von Mises stress at the inner radius. When $σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s}$ , the stress evolution starts to follow the plastic-like rule and inelastic deformations occur in at the inner radius of the ring. As long as $σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s}$ and $σ_{e} (r_{e, SMA}) < σ_{cr}^{s}$ two regions coexist in the ring: inelastic at the inner radius and elastic at the outer radius (load step #2). For $σ_{e} (r_{e, SMA}) \geq σ_{cr}^{s}$ , the entire ring is in the inelastic region (load step #3), until the one-way strain at the inner diameter is fully prevented and the phase transformation is completed. It is worth noting that the pipe to be connected is not modeled as an infinitely rigid body, since it undergoes elastic deformations during the ring contraction. Therefore, the elastic deformation of the pipe is considered at the load step #1 of the M-A phase transformation as its contribution to the contact pressure is not negligible. At load steps #2 and #3 the calculus of the contact pressure is carried out by neglecting the pipe thermo-elastic strain in the computation of the prevented circumferential strain, as it has been shown experimentally to be at least an order of magnitude lower than the transformation strain of the SMA.

The initial stress-strain state of the A-M transformation $(T = M_{s}^{σ})$ , is assumed to be equal to that corresponding to the end point of the previous M-A transformation $(T = A_{f}^{σ})$ , minus the thermoelastic contribution occurring between $A_{f}^{σ}$ and $M_{s}^{σ}$ (see Section 5). The prevented circumferential strain at the inner diameter, decreases by decreasing temperature from its maximum value to zero according to equation (7). Initially, the ring is fully deformed inelastically (step #3). The temperature decrease causes gradual recovery of strain due to TW-SME and a subsequent contact relaxation in the system. For $σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s}$ and $σ_{e} (r_{e, SMA}) < σ_{cr}^{s}$ , the ring is in the mixed loading case (step #2), while it undergoes complete elastic deformations when $σ_{e} (r_{i, SMA}) < σ_{cr}^{s}$ (step #1).

The critical temperatures that define the three load steps (#1, #2, #3) in both phase transformations are not known a-priori, as they depend on geometry, mechanical properties and loading conditions of the SMA ring and need to be computed during process simulation.

Table 1 reports the system of equations and boundary conditions for the three load steps. External boundary conditions are applied in terms of pressure at the inner radius, where $σ_{r} (r_{i, SMA}) = p_{i}$ , and at the outer radius, where $σ_{r} (r_{e, SMA}) = 0$ . For the mixed load step (#2), additional conditions (internal stress equilibrium) are required in correspondence of the critical radius $\bar{r}$ (cfr Figure 4(b)).

Table 1.

System of equation and boundary conditions defining the load steps that occur during phase transformations.

Load step #	Condition	System of equations	Boundary conditions
1	$σ_{e} (r_{i, SMA}) < σ_{cr}^{s}$	${\begin{matrix} \begin{matrix} E_{el} (Eq . 11) \\ C_{el} (Eq . 14) \end{matrix} \end{matrix}$	${\begin{matrix} \begin{matrix} \begin{matrix} σ_{r}^{el} (r_{i, SMA}) = - p_{i} \\ σ_{r}^{el} (r_{e, SMA}) = 0 \end{matrix} \end{matrix} \end{matrix}$
2	${\begin{matrix} σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s} \\ σ_{e} (r_{e, SMA}) < σ_{cr}^{s} \end{matrix}$	${\begin{matrix} \begin{matrix} E_{el} (Eq . 11) \\ C_{el} (Eq . 14) \end{matrix} \\ E_{in} (Eq . 11) \\ C_{in} (Eq . 16) \end{matrix}$	${\begin{matrix} \begin{matrix} \begin{matrix} σ_{r}^{in} (r_{i, SMA}) = - p_{i} \\ σ_{r}^{el} (r_{e, SMA}) = 0 \end{matrix} \end{matrix} \\ σ_{r}^{el} (\bar{r}) = σ_{r}^{in} (\bar{r}) \\ σ_{θ}^{el} (\bar{r}) = σ_{θ}^{in} (\bar{r}) \end{matrix}$
3	$σ_{e} (r_{e, SMA}) \geq σ_{cr}^{s}$	${\begin{matrix} E_{in} (Eq . 11) \\ C_{in} (Eq . 16) \end{matrix}$	${\begin{matrix} \begin{matrix} \begin{matrix} σ_{r}^{in} (r_{i, SMA}) = - p_{i} \\ σ_{r}^{in} (r_{e, SMA}) = 0 \end{matrix} \end{matrix} \end{matrix}$

5. Analytical modeling of thermal mounting and dismounting

The model is developed under the assumption of linear elastic behavior of the pipe and small deformations of the SMA ring/pipe assembly. The mounting (M to A) and dismounting (A to M) phases are analyzed below. Several subphases are identified for each phase.

Two basic conditions occur depending on the geometric configuration of the system: stress free and stress applied (constrained).

In case of stress-free conditions (no SMA/pipe interaction), an initial radial gap exists between the pipe and the SMA coupler ( $r_{i, SMA} - r_{e, PIPE}$ ). The final radial gap induced by a certain temperature variation ( $T_{fin} - T_{in})$ can be determined according to the total deformations, as written below:

Δ r = r_{i, SMA} (1 + ε_{SMA}) - r_{e, PIPE} (1 + ε_{PIPE})

(17)

where $ε_{SMA}$ and $ε_{PIPE}$ are the circumferential strain components at the inner radius of the SMA ring and at the outer radius of the elastic pipe. $ε_{SMA}$ includes a thermal and a transformation strain component: $ε_{SMA} = ε_{th, SMA} + ε_{tr}$ , while $ε_{PIPE}$ only a thermal component ( $ε_{PIPE} = ε_{th, PIPE}$ ).

The transformation component of SMA strain, $ε_{tr}$ , is calculated by equation (7) by knowing the maximum recoverable strain $ε_{OW}$ obtained from stress-free thermal cycles on pre-deformed SMA rings/rods (cfr. Figures 2 and 8).

The general thermal strain $ε_{th}$ is defined as:

ε_{th} = α (T_{fin} - T_{in})

(18)

where $α$ denotes the thermal expansion coefficient (CTE) of the material, and $T_{in}$ and $T_{fin}$ are the initial and final temperature of the considered phase, respectively.

In the case of stress-applied (constrained) conditions the SMA/pipe radial gap is zero, the prevented circumferential strains of the SMA ring at the inner diameter increases by increasing temperature and the evolution of the contact pressure $p_{i}$ can be obtained iteratively, as explained in Sections 4 and 5.3. At the generic temperature $T$ the total radial contact pressure $p_{i}$ is given by the sum of a thermoelastic, $p_{th}$ , and a transformation, $p_{tr},$ contribution, as shown below:

p_{i} (T) = p_{tr} (T) + p_{th} (T)

(19)

The thermoelastic component of pressure $p_{th}$ is determined according to the geometry and elastic properties of coupler and pipe, as written below:

\begin{matrix} Δ r_{c} = r_{c} p_{th} \\ [\frac{1}{E {(ξ)}_{SMA}} (\frac{r_{e, SMA}^{2} + {r_{c}}^{2}}{r_{e, SMA}^{2} - {r_{c}}^{2}} + ν_{SMA}) - \frac{1}{E_{PIPE}} (- \frac{{r_{c}}^{2} + r_{i, PIPE}^{2}}{{r_{c}}^{2} - r_{i, PIPE}^{2}} + ν_{PIPE})] \end{matrix}

(20)

Where $r_{c}$ is the SMA/pipe contact radius and $Δ r_{c}$ is the interference that would result due to the mismatch between the SMA/pipe strain occurring within a certain temperature variation (e.g. CTE mismatch). $Δ r_{c}$ can be calculated as follows:

Δ r_{c} = | r_{c} (ε_{PIPE} - ε_{SMA}) |

(21)

It is worth noting that equations (20) and (21) are valid only under thermoelastic behavior of both SMA and pipe.

The computational procedure is described in detail in Section 5.3, where a schematic representation of the calculation algorithm is provided by the flowcharts in Figures 5 –7.

5.1. Mounting

In agreement with Figure 1, the SMA ring/pipe mounting stage consists of the following steps:

- Phase 1: $T_{o} \to A_{s}$ , thermoelastic stress-free phase

- Phase 2: $A_{s} \to T_{c}$ , inelastic stress-free phase

- Phase 3: $T_{c} \to T_{\max}$ , inelastic stress-applied (constrained) phase

- Phase 4: $T_{\max} \to T_{o}$ , thermoelastic stress-applied phase

For the sake of simplicity, the starting and ending point of each phase will be denoted by subscripts N − 1 and N, respectively, where N is the number of the phase under consideration.

Phase 1: $T_{o} \to A_{s}$

In this phase, the temperature is always lower than the austenite start temperature $A_{s}$ . Therefore, no phase transformation mechanisms occur and both rings are subjected only to linear thermoelastic strain, calculated by applying equation (18) with the following boundary conditions: $T_{in} = T_{o}$ and $T_{fin} = A_{s}$ . The initial gap between the SMA and the pipe is $Δ r_{0} = r_{i, SMA, 0} - r_{e, PIPE, 0}$ . The thermal deformations establish a new gap between the rings that is determined as follows (equation (17)):

Δ r_{1} = r_{i, SMA, 0} (1 + ε_{th, SMA, 1}) - r_{e, PIPE, 0} (1 + ε_{th, PIPE, 1})

(22)

Phase 2: $A_{s} \to T_{c}$

At $T = A_{s}$ , the M-A phase transformation starts to take place and the SMA ring deforms based on both thermal (expansion) and recovery (contraction) strains. The combination of the two effects leads to the contraction of the ring since the recovery strain is at least an order of magnitude higher than the thermal strain (Figure 1). The pipe continues to expand, as it is only subjected to thermal deformation. The two rings come into contact at $T = T_{2} = T_{c}$ .

The thermal deformations of both rings are calculated by applying equation (18) by imposing the proper boundary conditions.

The contact condition ( $Δ r_{2} = 0)$ is given by:

Δ r_{1} = | r_{i, SMA, 1} ε_{SMA, 2} | + | r_{e, PIPE, 1} ε_{PIPE, 2} |

(23)

where $ε_{SMA, 2} = ε_{th, SMA, 2} + ε_{tr, 2}$ and $ε_{PIPE, 2} = ε_{th, PIPE, 2}$ .

The system of equations (5), (7), and (23), together with the proper boundary conditions (initial martensite volume fraction $ξ_{1} = 1)$ , allow to calculate the volume fraction of martensite when the contact occurs, $ξ = ξ_{2}$ , as well as the contact temperature $T_{c}$ and the maximum free recovery circumferential strain at the internal radius of the SMA ring, $ε_{tr, 2} = ε_{free}$ .

At the end of this phase, the two rings are in contact with zero contact pressure at the interface. The contact radius $r_{i, SMA, 2 =} r_{e, PIPE, 2} = r_{c, 2}$ can be calculated as follows:

r_{c, 2} = r_{e, PIPE, 1} (1 + ε_{PIPE, 2})

(24)

Phase 3: $T_{c} \to T_{\max}$

After the contact, the temperature increase induces the thermal expansion of the pipe, while the SMA ring would continue to contract due to the recovery properties. A radial contact pressure, $p_{i}$ occurs due to the impeded deformation $| ε_{tr} | - | ε_{free} |$ and can be computed iteratively as explained in Sections 4.2 and 5.3. The prevented deformation can be determined by equations (5) and (7); the system of equilibrium, compatibility and boundary conditions equations reported in Table 1 is solved in order to obtain the stress and strain state for each calculation step. At $T = A_{f}^{σ}$ , the MA transformation is completed $(ξ = ξ_{3} = 0)$ and the SMA is in a fully austenitic structure. The maximum contact pressure $p_{3} = p_{\max}$ is then generated at $T = T_{\max}$ . If $T_{\max}$ is slightly higher than $A_{f}^{σ}$ (typical operating condition) the hypothesis of constant $σ_{y}$ and H as a function of temperature are consistent. The impeded strain increment due to SMA/pipe thermal strain mismatch is at least two orders of magnitude lower than the total inelastic strain accumulated in the SMA. The stress/contact pressure variation occurring from $A_{f}^{σ}$ and $T_{\max}$ is therefore negligible. The latter assumption is confirmed experimentally (see Figures 13 and 14).

Phase 4: $T_{\max} \to T_{o}$

From $T_{\max} to T_{o}$ , only thermoelastic deformations (equation (18)) are considered, as it is assumed $M_{s}^{σ} < T_{o}$ (Section 2). The decrease in temperature results in a linear variation of the contact pressure due to the SMA/pipe thermal expansion coefficient mismatch ( $α_{PIPE} > α_{SMA}$ ). The thermoelastic contact pressure component $p_{th, 4}$ is obtained from equations (18), (20), and (21) by considering $ε_{SMA} = ε_{th, SMA}$ , $T_{fin} = T_{4} = T_{o}$ (operative temperature) and $T_{in} = T_{3} = T_{\max}$ . The total contact pressure (operative pressure) at $T_{o}$ is reported below:

p_{4} = p_{o} = p_{\max} - p_{th, 4}

(25)

5.2. Dismounting

The dismounting consists of the following steps (cfr. Figure 1):

- Phase 5: $T_{o} \to M_{s}^{σ}$ : thermoelastic constrained phase

- Phase 6: $M_{s}^{σ} \to T_{S}$ : inelastic constrained phase

- Phase 7: $T_{S} \to M_{f}$ : inelastic stress-free phase

- Phase 8: $M_{f} \to T_{\min}$ : thermoelastic stress-free phase

Phase 5: $T_{o} \to M_{s}^{σ}$

In the initial phase of the thermal dismounting, only thermoelastic deformations occur. Both the ring and pipe contract, and the decrease in temperature results in a linear decrease of the contact pressure according to equations (18), (20), and (21).

Phase 6: $M_{s}^{σ} \to T_{s} .$

At $T = M_{s}^{σ}$ , the AM transition starts to take place ( $ξ > 0)$ , and this causes the expansion of the SMA ring due to the TW-SME. At the same time, thermal strains also occur and the pipe continues to contract with decreasing temperature. This opposite behavior leads to the separation of the two rings at $T = T_{s}$ . The prevented transformation strain is calculated by applying equations (6) and (7), with $ξ_{0} = 0$ . The contact pressure is obtained by solving the system of equilibrium, compatibility, and boundary conditions according to the proper load step (Table 1). The separation condition $(p_{6} = 0)$ allows the identification of the dismounting temperature (T_s). The contact/separation radius $(r_{s})$ is determined as follows:

r_{c, 6} = r_{s} = r_{e, PIPE, 0} (1 + α_{PIPE} (T_{s} - T_{0}))

(26)

Phase 7: $T_{s} \to M_{f}$

For $T < T_{s}$ , the two rings are not in contact, the pipe and the SMA ring are unloaded and deform in free recovery conditions. The SMA ring is subjected to both transformation (expansion) and thermal (contraction) strain. The recovery strain in the SMA is determined by equations (6) and (7). The pipe is subjected only to thermoelastic deformations (equation (18)), therefore it contracts, allowing the formation of a new gap between the two rings $Δ r_{7}$ (equation (17)). At the end of this phase, the A→M transformation is completed.

Phase 8: $M_{f} \to T_{\min}$

In this phase, both rings contract because of thermal effects, according to equation (18). Since $α_{PIPE} > α_{SMA}$ a radial gap $Δ r_{8}$ (equation (17)) between the rings still exists at the end of the phase, which allows the complete disassembly of the system.

5.3. Solution procedure

The analytical model herein proposed is implemented in Wolfram Mathematica^®. The flowcharts in Figures 5 and 6 show the computational procedure for mounting and dismounting phases, respectively.

The input are the initial material and geometrical properties of the two rings (Figure 5). Also, a counter i = 0 is set to be used for the step-by-step calculation of the contact pressure in the non-linear inelastic phases (#3 and #6 in Section 5.2). The step size in terms of temperature variation is defined by the value of ΔT, arbitrarily chosen by the user. The lower ΔT, the higher the accuracy of the solution and the computational time.

Phase 1 ( $T_{o} \to A_{s}$ ) consists in the calculation of the radial gap $Δ r_{1}$ between the rings at the beginning of the M to A transition ( $T = A_{s}$ ). In phase 2 ( $A_{s} \to T_{c}$ ), the value of $T_{c}$ is determined by solving the system of equations involving both transformation and thermoelastic effects. In phase 3 $(T_{c} \to T_{\max})$ the contact pressure is determined step-by-step as its evolution is not linear. More info about this phase are reported at the end of the section (cfr. Figure 7). The maximum pressure $p_{\max}$ is calculated at the maximum temperature of the process. In phase 4 ( $T_{\max} \to T_{o}$ ) The operating pressure p_o at the operating temperature T_o is determined simply by decreasing the maximum pressure by the thermoelastic pressure occurring in the considered temperature range (equation (20)).

The stress-strain distributions at the end of the mounting phase represents the input of the dismounting phase (Figure 6). In particular, the contact radius and the contact pressure at $T_{o}$ , which are $r_{c, 4} = r_{c, o}$ and $p_{o}$ , respectively, are given as input for the calculation of the stress-strain evolution upon final cooling.

In phase 5 $(T_{o} \to M_{s}^{σ})$ , first the martensite start temperature under stress-applied conditions ( $M_{s}^{σ}$ ) is determined. This calculation requires an iterative procedure, so an initial value of $M_{s}^{σ}$ is estimated. The iterations terminate when the difference between the equivalent stress at the inner radius corresponding to a contact pressure $p_{i} = p_{0} - p_{th}$ (equation (27)) and the equivalent stress calculated by equation (4) is lower than the tolerance value $δ σ$ . The equivalent stress at the inner radius is calculated as follows:

σ_{e} (r_{i, SMA}) = \sqrt{σ_{r} {(r_{i, SMA})}^{2} + σ_{θ} {(r_{i, SMA})}^{2} - σ_{r} (r_{i, SMA}) σ_{θ} (r_{i, SMA})}

(27)

Once $M_{s}^{σ}$ is determined, the linear contact pressure evolution is obtained by considering only the thermoelastic deformations of both the SMA ring and the pipe (equation (20)).

In phase 6 $(M_{s}^{σ} \to T_{s})$ , the contact pressure is calculated by a step-by-step procedure, as its evolution is non-linear (cfr. Figure 7). At $T_{s}$ the two rings are in contact with zero pressure, so phase 7 $(T_{s} \to M_{f})$ only involves the calculation of the SMA and pipe deformations due to transformation and thermoelastic phenomena in order to estimate the gap between the rings at the end of the A to M transition, $Δ_{r, 7}$ . The radial gap at the end of the process, $Δ_{r, 8}$ , is the determined in the last phase $(M_{f} \to T_{\min})$ by thermoelastic strain calculation.

The flowchart in Figure 7 shows the step-by-step iterative process to solve equilibrium, compatibility and boundary conditions equations (defined in Table 1) in the inelastic constrained phases (#3,6 in Section 5.2). The boundary conditions are expressed in terms of radial pressure; all systems of equations are solved numerically.

An initial value of temperature, contact pressure and critical radius (interface radius between elastic and inelastic phase) must be given as input. Such values are: $T = T_{c}$ , $p_{i} = 0$ and $\bar{r} = r_{i, SMA}$ for the mounting phase (#3), and $T = M_{s}^{σ}$ , $p_{i} = p_{i} (M_{s}^{σ})$ and $\bar{r} = r_{e, SMA}$ for the dismounting phase (#6). Moreover, three counters $j, k$ , and $h$ are defined and set to zero at the beginning of each phase. The counter $j$ is used to increment the temperature of a quantity equal to $j Δ T$ . Two other parameters, $Δ p$ and $Δ r$ , related to $k$ and $h$ , respectively, define the step size used for the iterative calculation of the contact pressure and the critical radius (pressure increment = $k Δ p$ ; critical radius increment = $h Δ r$ ).

Different equations and boundary conditions are used depending on the load step. This latter is determined according to the stress-strain state in the system. First, the impeded deformation $| ε_{tr} | - | ε_{free} |$ and the equivalent stress $σ_{e} (r_{i, SMA})$ are calculated at the internal radius of the SMA ring for each temperature value.

If $σ_{e} (r_{i, SMA}) < σ_{cr}^{s}$ the load steps #1 occurs. Load step #2 occurs if $σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s}$ and the critical radius $\bar{r}$ is lower than the outer radius of the SMA ring ( $r_{e, SMA}$ ) while load step #3 occurs when $σ_{e} (r_{i, SMA}) \geq σ_{cr}^{s}$ and $\bar{r} > r_{e, SMA}$ . For the non-mixed steps #1 and #3 the stress/strain values in the entire ring are within a single region, fully elastic or fully inelastic, respectively. In load step #1 the contact the pressure is directly calculated by using equations (20) and (21) while in load step #2 and #3 the contact pressure is calculated iteratively. The calculus of $p_{i}$ stops when the difference between the circumferential deformation at the SMA inner radius $(ε_{θ} (r_{i, SMA}) = ε_{SMA})$ corresponding to $p_{i}$ and the impeded strain $(| ε_{tr} | - | ε_{free} |)$ is lower than the tolerance value $δ ε$ .

In load step #2, the calculation of the stress/strain state in the ring involves equilibrium and compatibility equations of two regions, elastic and inelastic. Boundary conditions are defined at inner, outer and transformation radii of the ring. The iterative calculation of $\bar{r}$ continues until the difference between the equivalent stress at the critical radius $σ_{e} (\bar{r})$ and the critical start stress $σ_{cr}^{S}$ is lower than a tolerance value $δ σ$ .

All relative tolerance values used for the iterative calculations are set to be less than 0.05%.

6. Materials and experiments

6.1. Uniaxial tests and model calibration

Experimental tests were performed on Ni_50.8Ti_49.2 and Ni_46.4Ti_45.0Nb_8.6 dog-bones.

Specimen were heat treated and pre-strained $(ε_{\max} > 7 %)$ to induce the shape memory properties in the material through alignment of the martensitic variants. The effects of mechanical pre-strain, $ε_{\max}$ , on one way deformation, $ε_{OW}$ , two-way deformation, $ε_{TW}$ , and on TTs were systematically analyzed in (Niccoli et al., 2017a). In particular, both $ε_{OW}$ , and $ε_{TW}$ as a function of $ε_{tot}$ show a non-monotonic trend. Moreover, austenitic TTs can be increased for the first heating cycle. This is particularly evident for ternary NiTiNb SMAs (Niccoli et al., 2017b).

A total of about 30 samples were used to carry out the characterization tests, two sample for each test. Errors below 5% were found between samples. Average values were reported in the following graphs.

Isobaric strain-temperature tests $(σ = 0, 100, 200, 300 MPa)$ were carried out on NiTi specimens in order to identify the evolution of the austenitic and martensitic TTs as a function of the applied stress and the Clausius Clapeyron coefficients.

Figure 8 shows the curve true strain vs temperature $(ε - T)$ obtained from the first stress-free ( $σ = 0$ ) heating/cooling cycle (=3·10⁻²°C s⁻¹) between TTs. K-type thermocouples and high temperature extensometers were used. The figure shows the one way, $ε_{OW}$ , and two-way strain, $ε_{TW}$ , obtained by heating above $A_{f}$ and cooling below $M_{f}$ , respectively.

A comparable stress-free behavior was observed between rods and rings, despite the presence of residual stress in the rings after pre-straining. The TTs and OW/TW strain values were found to be nearly identical, along with a similar reduction in austenite transformation temperatures following the initial thermal activation (Niccoli et al., 2017b). This is an important finding, as the material data from SMA rods were used to model the behavior of SMA rings. Results in terms of TTs as a function of the applied stress for the rest of isobaric tests on NiTi (σ = 100, 200, 300 MPa) are reported in Figure 10. Instead, a complete characterization of NiTiNb samples is reported in (Niccoli et al., 2017a).

Figure 9(a) illustrates the isothermal true stress-true strain $(σ - ε)$ response of NiTi alloy (=5·10⁻⁴ s⁻¹) at different temperatures: T = −80, 40, 85,110, 150, 200°C.

At T = −80°C the material has a completely martensitic structure; At T = 150°C the material is assumed to be completely austenitic. The figure also shows the yielding stress of the material in both phases as well as the austenite and martensite Young’s moduli (E_{A, EM}) used for simulations.

Figure 9(b) shows the bilinear stress-strain response implemented in the analytical model. The critical plateau stress, $σ_{cr}^{s}$ , and the hardening coefficient, H, are computed by applying the OLS (Ordinary Least Squares) method.

Results from both isothermal and isobaric tests on NiTi samples are reported in Figure 10 that shows the stress versus temperature phase diagram of the material. $σ_{cr}^{s}$ increases by increasing temperature from M_s to $A_{f}^{σ}$ . However, it clearly deviates from the linear Clausius Clapeyron regime (described by β_Ms) at temperatures higher than T = 40°C due to irreversible transformation mechanisms and austenite twins formation (β_M-A ≠ β_Ms ≈ β_A-M). Figure 10 also shows the yielding stress, $σ_{y}$ , obtained from isothermal tests at low and high temperatures exhibiting a very limited variation with temperatures.

A list of the SMA (NiTi and NiTiNb) and steel (316LN and NiCrMo) material parameters used as input for the proposed analytical model is reported in Tables 2 and 3, respectively.

Table 2.

SMA material parameters used for analytical model calibration.

Parameter	NiTi	NiTiNb	Unit
E _M	18,735	15,000	MPa
E _A	70,000	64,000	MPa
H	1000	1000	MPa
ν	0.3	0.3	/
$ε_{OW}$	0.05	0.056	mm/mm
σ_y	482	600	MPa
α _A	$0.7 \times * 10^{- 5}$	$1.1 * \times 10^{- 5}$	°C⁻¹
α _M	$0.7 * \times 10^{- 5}$	$1.1 * \times 10^{- 5}$	°C⁻¹
β_Ms	6	5	MPa °C⁻¹
β_Af	6	5.5	MPa °C⁻¹
M _f	−66	−143	°C
M _s	−50	−105	°C
A _s	41	38	°C
A _f	50	75	°C
β_M−A	1.74	0.21	MPa °C⁻¹
β_A−M	3.46	2	MPa °C⁻¹

Table 3.

Steel material parameters used for analytical model calibration.

Parameter	316LN	NiCrMo	Unit
E	205,000	210,000	MPa
ν	0.3	0.3	/
α	$1.7 \times * 10^{- 5}$	$1.2 \times * 10^{- 5}$	°C⁻¹

NiTiNb and steel parameters are taken from (Niccoli et al., 2017b).

Further experimental tests were performed to validate the analytical model in a uniaxial case described in Section 3.2 before developing a two-dimensional axisymmetric formulation.

The constrained recovery capabilities of NiTi samples were analyzed by holding the residual strain during thermal cycles. Dog-bone shaped Ti–50.8 at.% samples (gauge length = 31 mm, cross section) similar to those used for isobaric test (cfr. Figure 8) were pre-deformed $(ε_{\max} > 7 %)$ and subsequently installed against a C-shaped steel frame instrumented with electrical SG as reported schematically in Figure 11.

Installation was obtained by means of circular pins and holes realized onto the SMA and steel samples. Steel frames and pins were designed in order to exhibit elastic behavior during the thermal activation. The steel structures were instrumented with uniaxial SG (XC11-HBM) as reported in Figure 11. The test was carried out with the aim of measuring the maximum recoverable force of the alloy, as well as to analyze the shift in the TTs under applied stress conditions. In this testing condition, the whole residual strain was prevented, in order to obtain the maximum recovery force and, consequently, the maximum increase of the TTs. The thermal cycle consisted in a heating stage to 200°C and subsequent cooling down to −65°C.

Thermal loading/unloading cycles were performed at a constant rate of 1.5°C/min in order to keep quasi-static conditions. Both temperature and strain data acquisition were performed at a frequency of 2 Hz. A quarter-bridge configuration was adopted for strain measurements. As well known, this configuration gives an apparent strain during temperature change, namely thermal output. This apparent strain signal was preliminary measured by stress-free thermal tests on instrumented steel parts and then was subtracted from the strain signals obtained from the SMA-steel coupling. The gauge factor variation with temperature was also taken into account. The evolution of the SMA recovery stress with temperature (σ_rec vs T) was estimated from strain measurements, by using the theory of elasticity considering the steel frame base as a beam subjected to bending and compression:

σ_{rec} = \frac{F_{rec}}{A_{SMA}} = \frac{ε}{A_{SMA}} \cdot \frac{E}{\frac{6 x}{b c^{2}} + \frac{1}{bc}}

(28)

Where:

F _rec = SMA recovery force

x = distance between F_rec and the neutral axis of the cross section of the steel frame base = 21 mm

$ε$ = normal longitudinal strain at strain gauge location

A _SMA = SMA sample cross section = 4.5 mm²

E = steel Young’s modulus = 205 GPa

Figure 12 shows a comparison between the predicted and measured strain-temperature (stress-free thermal cycles) and stress-temperature curves (constrained recovery tests).

Figure 12(a) shows a good agreement between the experimental curve from stress free test and the analytical solution especially in the heating stage where the OW-SME occurs. A discrepancy is evident in the subsequent cooling stage as the model assumes $ε_{TW} = ε_{OW}$ , so the simulated TW strain is higher than the experimental one (see consideration in Section 3.2).

The comparison between experimental and predicted stress in Figure 12(b) demonstrates that the model is able to properly simulate the development of recovery forces. The stress evolution with temperature is captured correctly in both heating and cooling phases. It is worth pointing out that $A_{f}^{σ}$ in Figure 12 as explained in Section 3.1 corresponds to the temperature at which the SMA sample, under stress, becomes fully austenitic. A decrease of the recovery stress is recorded by subsequent cooling from the maximum temperature mainly because of the austenite Young’s modulus variation with temperature and the mismatch of the thermal expansion coefficient (CTE) between the steel and the SMA. The higher stress drop starting at 25°C corresponds to the beginning of the austenite to martensite transition (M_s^σ) which leads to a stress-free state with further cooling at about M_f.

The latter result confirms that recovery stress of properly trained SMAs drops to zero by cooling down to M_f even if two-way recoverable deformation obtained from stress-free thermal cycles is lower than the one way (see Figure 12(a)). As explained in Section 3.2 this is attributed to the complex deformation mechanisms occurring during M-A constrained heating which increases the TW-SME of the material as well as the applied stress itself that has a training effect increasing the volume fraction of favorably oriented martensite variants.

6.2. Mechanical expansions of SMA couplers

Isothermal mechanical expansions of martensitic SMA rings at −80°C were conducted using a tapered punch mounted in a servo-hydraulic testing machine (Instron 1276, with a load capacity of 1 MN) equipped with a climatic chamber capable of temperatures ranging from −170°C to +300°C. Careful control of process parameters, including cross-head speed and ring temperature, was essential as they significantly impact the geometric and dimensional tolerance of the ring couplers. The test was performed in displacement control mode at a rate of approximately 10 mm/min. A K-type thermocouple, affixed directly to the sample, was used to measure the ring temperature. The maximum punch diameter was selected to ensure an adequate mechanical pre-strain for training the rings.

6.3. Constrained recovery tests of SMA rings

Constrained recovery tests were performed on SMA/steel ring assembly systems. Both NiTi and NiTiNb connectors were studied in order to validate the analytical model with experimental data. The tests were conducted in an Instron E10000 machine equipped with a climate chamber with a temperature range of −170°C/+300°C. Direct measurements of temperature and strains were carried out using k-type thermocouples and biaxial strain gages mounted on the inner surface of the steel rings. Axial and circumferential strain signal were recorded. An insulating glue was brushed inside the ring in the area corresponding to the terminals of the strain gauges to avoid undesired contact with the ring itself. Both stainless steel 316LN and NiCrMo3 steel were used for elastic ring manufacturing. The inner and outer diameter were, ID_St = 24 mm and OD_St = 44.3 mm, respectively. The NiTi and NiTiNb coupler dimensions were ID_NiTi = 45 mm/OD_NiTi = 51 mm and ID_NiTiNb =45 mm/OD_NiTiNb = 56 mm, respectively. Therefore, an initial gap of about ΔD = 0.7 mm is estimated between the inner and outer rings at the beginning of the process.

p = - \frac{E}{2 (1 + ν)} \frac{{OD}_{st}^{2} - {ID}_{st}^{2}}{{OD}_{st}^{2}} (ε_{θ} - ε_{z})

(29)

Where E and $ν$ are the steel Young’s modulus and Poisson coefficient, respectively. Two SMA/steel couplings per tests were used and average values were reported in the results section. An error below 5% was found between the strain data of different samples.

7. Finite element modeling

Finite element simulations were carried out in a commercial software (ANSYS^®) to analyze the constrained recovery behavior of the NiTi rings mounted on elastic steel rings. Their thermo-mechanical behavior is described by a recently developed 3D SMA model (Scalet et al., 2019) which was implemented in ANSYS as a User Programmable Feature (UPF). A user-defined material subroutine (USERMAT) was written in Fortran to be linked to the FE software. Such model for SMA material is able to simulate both OW- and TW-SME effects allowing the modeling of the entire assembly/disassembly processes. The model was calibrated with respect to the experimental data obtained from dog-bone samples, as explained in Section 6.1. The input parameters to the USERMAT are reported in Table 4.

Table 4.

NiTi mechanical and functional properties used as an input to the USERMAT in the FE simulations.

Parameter	Value	Unit
Martensite (twinned) Young’s modulus, E_i,M	30,000	MPa
Martensite (detwinned) Young’s modulus, E_f,M (=E_M)	18,735	MPa
Austenite Young’s modulus, E_iA (=E_A)	70,000	MPa
Transformation hardening, h_tr (=H)	1000	MPa
Plastic hardening, h_p	735	MPa
Transformation strain, ε_OW	0.05	mm/mm
Elastic domain radius (at low T), R_XH	326.6	MPa
Elastic domain radius (at high T), R_XL	90	MPa
Inelastic domain radius, R_Y (=σ_y)	482	MPa
Transformation and plastic interaction, B	1200	MPa
Thermal expansion coefficient, α (=α_M,α_A)	$0.7 \times * 10^{- 5}$	°C⁻¹
Clausius Clapeyron coefficient, β (=β_Ms)	6	MPa/°C

For the sake of clarity, the symbols in parentheses represent the equivalent parameters used in the proposed analytical model.

A comparison between the measured and simulated isothermal martensitic uniaxial loading-unloading cycle is reported in Figure 13.

The system was modeled in a 3-D space to allow the use of the USERMAT, based on a 3-D formulation (Scalet et al., 2019). A circular ring sector (2° angle) 10 mm long was considered by exploiting both the circumferential and planar symmetries. The mesh consists in about 13,000 quadratic elements (type SOLID187) of different sizes, since a refinement was made in correspondence of the contact surfaces, as shown in Figure 14, in order to increase the accuracy of the solution.

The pre-deformation of the NiTi rings was simulated under displacement control. A radial displacement was applied to the inner surface of the ring to obtain a maximum deformation at the inner diameter higher than 7%. An initial radial gap of $Δ_{r 0}$ = 0.35 mm was considered between the pre-deformed NiTi ring and the 316LN ring, representing the initial mounting conditions.

Thermal mounting and disassembly of the system were simulated by applying a temperature cycle starting from the martensitic conditions (T = −80°C), consisting in a heating stage up to T_max = 200°C and a cooling stage to T_min = −150°C.

The contact between the two rings was simulated as a frictional contact using the Augmented Lagrange method, which ensures minimum penetration and zero gap between the two contact surfaces. The friction coefficient was set to 0.3, which is a typical value for the NiTi-steel contact pairs.

8. Results and discussions

This section provides the results obtained from the proposed analytical model in comparison with experimental data and FE simulations. Results obtained from NiTi/316LN and NiTiNb/NiCrMo couplings are here presented and discussed. Different configurations, in terms of geometry and materials, were analyzed as reported in Table 5.

Table 5.

SMA ring and pipe geometry.

Case study #	SMA material	Pipe material	Initial SMA ring radius, $r_{i, SMA, 0}$ (mm)	SMA ring thickness, $t_{SMA}$ (mm)	Pipe thickness, $t_{PIPE}$ (mm)	Initial radial gap, $Δ r_{0}$ (mm)
1	NiTi	316LN	22.5	5	10.15	0.7
2	NiTi	316LN	22.5	3	13	0.075
3	NiTi	316LN	22.5	5	13	0.075
4	NiTi	316LN	22.5	8	13	0.075
5	NiTiNb	NiCrMo	22.75	8.2	13	0.075
6	NiTiNb	NiCrMo	22.75	8.2	13	0.425
7	NiTiNb	NiCrMo	22.75	8.2	13	0.8

8.1. NiTi/316LN ring coupling

The contact pressure evolution as a function of the temperature obtained by the analytical computation, numerical simulation and experimental test are reported in Figure 15. Results refers to configuration #1 in Table 5.

A significant increase in contact pressure (experimental curve) is observed as the temperature rises from approximately Tc ≈ 60 °C (cfr. Fig. 1) to $A_{f}^{σ} \approx$ 130 °C. At $A_{f}^{σ}$ , most of the ring volume transitions to a fully austenitic state, meaning that neither thermally induced nor stress-induced martensitic variants are present. Upon cooling from the maximum temperature to the martensite start temperature under stress ( $M_{s}^{σ} \approx$ 0 °C), the pressure exhibits an almost linear decrease.

This behavior can be attributed to several factors: the mismatch in thermal expansion coefficients between the two materials, the temperature-dependent variation of the SMA’s Young’s modulus (Liu & Xiang, 1998), and the potential presence of residual martensitic variants (Fan et al., 2020). The model considers only thermoelastic effects within this temperature range ( $A_{f}^{σ} - M_{s}^{σ}$ ). A pronounced pressure drop occurs upon further cooling, corresponding to the thermally induced austenite-to-martensite transformation. These distinct stages of the process are also captured by both the finite element (FE) and analytical models.

The graph shows a very good agreement between the three curves, proving the accuracy of the analytical model.

Contact pressure predictions obtained from the analytical and FE analyses show similar trends and values. P–T curves obtained from FE results are smoother than the analytical one because of the intrinsically different formulation which does not consider a bilinear evolution of the critical stress, $σ_{cr}^{s}$ which increases by increasing temperature from M_s to $A_{f}^{σ}$ by clearly deviating from the Clausius Clapeyron regime (Figure 10). Moreover, the analytical model includes a sharp load step transition occurring during phase transformations as described in Section 4.2. Maximum differences between the FE-analytical predictions of about 20% were obtained at the maximum temperature and during subsequent cooling to $M_{s}^{σ}$ . This is mainly due to the higher capability of the analytical model to estimate the $A_{f}^{σ}$ temperature. The contact pressure plateau occurring during heating beyond $A_{f}^{σ}$ was also better simulated by the analytical model. In fact, as explained in Section 5 (phase 3), if $T_{\max}$ is slightly higher than $A_{f}^{σ}$ (typical operating condition), the impeded strain increment due to SMA/pipe thermal strain mismatch is at least two orders of magnitude lower than the total inelastic strain accumulated in the SMA. The stress/contact pressure variation occurring from $A_{f}^{σ}$ and $T_{\max}$ is therefore negligible, as confirmed experimentally.

A percentage error between the analytical prediction compared to the numerical or experimental data can be defined as reported below:

% e = | \frac{{p_{i}}_{, AN} - {p_{i}}_{, EXP / FEM}}{p_{i},_{EXP / FEM}} * 100 |

(30)

An absolute error of about 18% was calculated between analytical and numerical models at $T_{o} = 25^{°} C$ while an error lower than 18% is recorded in comparison with the experimental curve. Figure 13 also shows the accuracy of the model in predicting the SMA transformation temperatures in biaxial constrained recovery systems.

Figure 16 shows the analytical prediction and FEM simulation solution of the radial stress distribution as a function of the normalized radius of the SMA ring at $T_{o} = 25^{°} C$ . As expected, the radial stress at the inner radius of the SMA ring corresponds to the contact pressure and decreases by increasing radius up to the external surface, where the radial stress is zero. The mismatch in trend and values between the two solutions is mainly attributed to the residual stress distribution occurring after the preformation phase, which is not taken into account in the proposed analytical model.

8.2. NiTiNb/ NiCrMo ring coupling

Figure 17 shows contact pressure versus temperature curves obtained from the analytical modeling and experimental tests of a NiTiNb/NiCrMo ring coupling (configuration #5 in Table 5). Again, the graph shows a very good agreement between analytical and experimental data with a maximum error of about 17% at $T_{o} = 25^{°} C$ . For this coupling, the change of slope of the experimental curve during the cooling stage corresponding to the two different regions (thermo-elastic and inelastic) is more evident than the NiTi/316LN coupling thanks to the higher NiTiNb hysteresis.

A wider thermal stability range ( $A_{f}^{σ}$ − $M_{s}^{σ}$ ) upon cooling is measured experimentally (200/−55°C) and predicted by model calculations (200/−25°C). The pressure-temperature slope range is perfectly captured by the model accounting only for thermoelastic effects in this thermal range ( $A_{f}^{σ}$ − $M_{s}^{σ}$ ).

8.3. Effect of SMA ring thickness

The effect of the SMA rings thickness on the contact pressure was investigated by analyzing three different geometrical configurations of NiTi—316LN ring coupling system (case studies #2, 3, 4 in Table 5). The results of analytical calculations, FE simulations and experimental measurements at $T_{o} = 25^{°} C$ are reported in Table 6 with respective relative errors. Experimental data refers to (Niccoli et al., 2022b). Results show that the contact pressure as well as the analytical/experimental error increases with increasing SMA ring thickness. A maximum error of 20.4% between the analytical prediction and the experiments was estimated (configuration #4 in Table 4).

Table 6.

Contact pressure at the operating temperature (T_o = 25°C) obtained by analytical calculation, FE simulation and experimental measurement for NiTi/316LN coupling system at different values of SMA ring thickness.

Case study # (Table 4)	SMA ring thickness, $t_{SMA}$ (mm)	Analytical contact pressure, $p_{o, AN}$ (MPa)	FEM contact pressure, $p_{o, FEM}$ (MPa)	Experimental contact pressure, $p_{o, EXP}$ (MPa)	Error analytical-FEM, $E_{AN - FEM}$ (%)	Error analytical-experimental, $E_{AN - EXP}$ (%)
2	3	38.3	65.6	36	41.6	1.64
3	5	58.2	108.6	53	46.4	1.73
4	8	81.9	159.6	68	48.7	20.4

8.4. Effect of the initial gap

The effect of the initial gap between the SMA ring and the pipe on the contact pressure was also evaluated by analyzing three different geometries of the NiTiNb–NiCrMo coupling system (#5, 6, 7 in Table 5). The results of analytical calculations, FE simulations and experimental measurements are reported in Table 7 with the respective errors. FEM and experimental results were taken from (Niccoli et al., 2017a). Results show that the maximum contact pressure slightly decreases with increasing initial radial gap. However, a maximum decrease of about 18% was recorded by the analytical model by increasing the radial gap from 0.075 to 0.8 mm. This is confirmed by experimental results showing a pressure drop of about 16%. Results shows that the error of the analytical prediction increases by increasing the initial clearance. However, a maximum error of 17% (analytical-experimental comparison) was recorded for case study #7, where the effect of the maximum gap was investigated.

Table 7.

Contact pressure at the operating temperature (T_o = 25°C) obtained by analytical calculation, FE simulation and experimental measurement for NiTiNb/NiCrMo coupling system at different values of initial radial gap.

Case study #	Initial radial gap, $Δ r_{0}$ (mm)	Analytical contact pressure, $p_{o, AN}$ (MPa)	FEM contact pressure, $p_{o, FEM}$ (MPa)	Experimental contact pressure, $p_{o, EXP}$ (MPa)	Error analytical-FEM, $E_{AN - FEM}$ (%)	Error analytical-experimental, $E_{AN - EXP}$ (%)
5	0.075	104	84	89	23.8	17
6	0.425	95	82	84	15.9	10
7	0.8	85	72	75	18.1	13

9. Conclusion

An analytical model, based on the elastic-plastic theory of axisymmetric geometries, was developed to simulate the constrained recovery mechanisms of shape memory alloy (SMA) rings exhibiting the two-way shape memory effect (TW-SME). The model was used to calculate the pressure evolution at the contact interface between the SMA ring and the internal pipe, as well as the stress and strain distribution in the SMA ring, during the thermal activation cycle. The constrained recovery process was discretized into a number of subphases for each of the two main phases, mounting and dismounting, involving martensite to austenite and austenite to martensite phase transformations, respectively. Different loading steps were computed in the solution of each phase, involving fully elastic, mixed elastic-inelastic, and fully inelastic SMA ring deformations. Two coupling systems were studied, such as NiTi/316LN and NiTiNb/NiCrMo, with different SMA ring thicknesses and initial SMA ring-steel pipe gaps. Systematic comparison of the model predictions, in terms of contact pressure evolution as a function of temperature, with finite element analyses based on a user-defined material model and experimental measurements showed very good agreement. Results also revealed that the contact pressure increases by increasing the thickness of the SMA ring and reducing the initial gap between the SMA ring and the internal pipe. The proposed model provides a very powerful tool for the design of pre-strained SMA rings with one-way and two-way recovery capabilities.

Footnotes

Appendix

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 received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Carmine Maletta

Fabrizio Niccoli

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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A simplified analytical model to simulate constrained recovery mechanisms of SMA rings exhibiting two-way shape memory effect

Abstract

Keywords

1. Introduction

2. SMA-based pipe coupling: Mounting and dismounting features

3. SMA recovery stress: Modeling and assumptions

3.1. Basic features of SMA recovery stress

3.2. Model assumptions

4. SMA ring constrained recovery modeling

4.1. Equilibrium and compatibility equations

4.2. Load steps and boundary conditions

5. Analytical modeling of thermal mounting and dismounting

5.1. Mounting

Phase 1: T o → A s

Phase 2: A s → T c

Phase 3: T c → T max

Phase 4: T max → T o

5.2. Dismounting

Phase 5: T o → M s σ

Phase 6: M s σ → T s .

Phase 7: T s → M f

Phase 8: M f → T min

5.3. Solution procedure

6. Materials and experiments

6.1. Uniaxial tests and model calibration

6.2. Mechanical expansions of SMA couplers

6.3. Constrained recovery tests of SMA rings

7. Finite element modeling

8. Results and discussions

8.1. NiTi/316LN ring coupling

8.2. NiTiNb/ NiCrMo ring coupling

8.3. Effect of SMA ring thickness

8.4. Effect of the initial gap

9. Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

References

Phase 1: $T_{o} \to A_{s}$

Phase 2: $A_{s} \to T_{c}$

Phase 3: $T_{c} \to T_{\max}$

Phase 4: $T_{\max} \to T_{o}$

Phase 5: $T_{o} \to M_{s}^{σ}$

Phase 6: $M_{s}^{σ} \to T_{s} .$

Phase 7: $T_{s} \to M_{f}$

Phase 8: $M_{f} \to T_{\min}$