Novel C-Shaped Shape Memory Alloy Connectors for Vacuum Flanges: Modeling and Tests

Shape memory alloy (SMA)-based fasteners and connectors represent a class of successful SMA components that are increasingly used especially in marine or aerospace applications. The present paper aims to investigate novel C-Shaped SMA connectors for room-temperature vacuum pipes within particle accelerators. The proposed connectors exhibit the two-way shape memory effect (TW-SME), so they can generate significant axial recovery forces and they can be dismounted by temperature variations. Proper thermo-mechanical training procedures were performed to tune the mechanical and functional response of the connectors to make them suitable for the investigated application. The thermo-mechanical and functional response of the SMA fasteners in both free and constrained recovery conditions were assessed by strain gauge and extensometer tests as well as by finite element analyses based on a constitutive model, recently developed by the authors, accounting for TW-SME and plasticity. Comparison between experimental and numerical results validates the proposed model. Moreover, results give useful, important information in terms of SMA transformation temperatures, generated recovery forces as well as shape recovery capabilities.


Introduction
Shape Memory Alloys (SMAs) exhibit exceptional strain and force recovery capabilities owing to the Shape Memory Effect (SME) and Pseudoelastic Effect (PE) [1,2].These peculiar functional properties are linked to reversible phase transitions between two crystallographic phases: the austenite (body centered cubic, B2) and the martensite (monoclinic, B19') [2].Phase transformations can be induced mechanically (SIM, Stress-Induced Martensite) or by temperature changes (TIM, Thermally Induced Martensite) between the Transformation Temperatures (TTs).
The applications of NiTi-based SMA devices are continuously increasing in a variety of sectors, especially in the last two decades.The first large scale commercial solutions were medical devices for vascular surgery, orthopedics, and orthodontics [3].
SMA connectors, fasteners and clamps represent a class of successful SMA components used since decades especially for medical, marine or aerospace applications [1,3,6].This invited article is part of a special issue of Shape Memory and Superelasticity honoring Etienne Patoor for his contributions to the field of phase transforming materials and shape memory alloys.The special issue was organized by Dr. Fodil Meraghni, Ecole Nationale Supe ´rieure d'Arts et Me ´tiers (Arts et Me ´tiers Institute of Technology), and Dr. Dimitris Lagoudas, Texas A&M University.& F. Niccoli fabrizio.niccoli@cern.ch;fabrizio.niccoli@unical.it In the last years, SMA ring connectors found noticeable interest for interference couplings in civil engineering [9][10][11][12][13], in particular for seismic bracing and damping devices; studies are carried out regularly for further investigating their functional and structural behavior by numerical [12][13][14][15] and experimental [16,17] analyses to develop design methods for each specific application.
SMA clamps were developed mainly for medical applications, in particular bone surgery [18][19][20][21] and as fixator systems for suturing tissue in minimal access surgery [22].Such systems are normally heated to 40-50 °C after implantation to produce axial compressive forces that stabilize the bone fractures or tissue lesions.Clinical studies demonstrated the effectiveness of SMA clamp devices in orthopedics [18][19][20] although design methodologies and analytical or numerical studies [22] are very limited in the literature.
In these devices recovery forces are generated when the SMA element, pre-deformed in martensitic state, is heated above the austenite TTs (One-Way Shape Memory Effect, OW-SME) in a constrained state, i.e. with constant residual deformation (because of the presence of a mechanical obstacle).SMA fasteners and connectors can also exhibit the Two-Way Shape Memory Effect (TW-SME), so they can be detached by thermally inducing their martensitic configuration.In fact, they can change their shape reversibly by temperature variations between TTs, under zero mechanical load.TW-SME can be obtained by thermomechanical training processes that generate an internal stress field which fosters the nucleation of preferentially oriented martensitic variants [1,2,6].Certainly, the functional response of such NiTi-based devices, in terms of TTs, OW-SME, and TW-SME, must be tuned for each application as the stability as well as the reliability of such systems strongly depend on recovery strain/stress amplitude and evolution.
Recently, research studies undertaken at CERN (European Organization for Nuclear Research) investigated for the first time the use of active SMA leak-tight pipe connectors for room-temperature vacuum systems of particle accelerators [6][7][8][23][24][25][26]. Their compactness, bi-metallic joining capabilities, and possible remote mounting/disassembly by temperature variations (exploiting the TW-SME) are some of the advantages which make them particularly interesting for applications in the high energy physics domain.Cylindrical SMA connectors can be installed onto vacuum pipe extremities, and they are smaller and lighter than conventional pipe coupling systems such as the standard bolted (CF) and quick-connect ConflatÒ (QC) flanges.
CF and QC flanges are widely utilized in vacuum systems, especially in particle accelerators.Very large stainless-steel conical chain clamps (2-4 bolts) are normally used to apply axial forces to the QC flanges.This is necessary to deform metal gaskets placed at the tubes interface to create leak tight vacuum joints.The use of such large, chained flanges could result in critical operational issues and significant design limitations, especially for restricted access/radioactive areas where there are tight space occupancy constraints.Furthermore, possible lubricant contaminations in radiation environments could occur in case of the use of these traditional joining devices.
In this paper, novel C-Shaped SMA connectors were studied as an alternative smart connection system for vacuum pipes.Several C-Shaped SMA elements can be used to realize active bolt-free chains to be installed onto the flanges.They can be thermally activated to generate significant axial recovery forces.Moreover, they can be easily dismounted by cooling to subzero temperatures exploiting their TW behavior.Such SMA chains would be much smaller and lighter than the traditional QC stainlesssteel collars and can be controlled remotely by temperature changes.Furthermore, owing to their intrinsic modular nature, they represent a very flexible solution, compatible with a variety of commercial flanges of different sizes.Another important advantage with respect to the conventional connection systems would be the absence of lubrication between the chains and the flanges thus creating reliable clean joint (of importance for ultra-high vacuum application) and avoiding frequent friction-related mounting issues.Finally, they would also represent a smart solution for bi-metallic connections.
A commercial NiTi alloy (50.8Ni-49.2Tiat.%) was selected to realize such connectors.This alloy exhibits suitable functional/mechanical properties for applications in vacuum systems which need to be leak-tight at room temperature.As confirmed by previous investigations [23,24,26], NiTi ring connectors made of the same alloy ensure a proper thermal mounting by heating them above 150 °C as well as an easy dismounting phase by cooling them below -50 °C with no risk of premature relaxations.A dogbone-like simplified geometry was chosen for a preliminary design phase.Based on previous studies [23,24,26], proper thermo-mechanical training procedures were performed to tune the mechanical and functional response of the fasteners to make them suitable for room temperature vacuum system applications.Proper alloy selection and training is essential to avoid premature thermal activation and dismounting.The thermo-mechanical and functional response of the trained SMA fasteners in both free (without mechanical obstacle) and constrained (with mechanical obstacle) recovery conditions were assessed by strain gauge (SG) and extensometer tests as well as by finite element (FE) analyses.The latter were carried out using a commercial software and a constitutive model recently developed by the authors [23] and validated within this work.The choice of model reported in [23] is dictated by the fact that it is able to describe the behavior of severely pre-strained NiTi-based SMAs exhibiting TW-SME and plasticity, compared to other state-of-the-art models.In fact, SMA modeling approaches generally concern the description of transformation-induced plasticity due to cyclic thermo-mechanical transformation (see, e.g., [27][28][29][30][31][32]) or plastic strains due to slip mechanisms at sufficiently high stresses (see, e.g., [33][34][35]).However, to the authors' knowledge, the analysis of the effect of plastic deformation, occurring during severe martensitic pre-strain and possible subsequent constrained thermal cycles, on the OW-SME, TW-SME, and TTs is considered only in [23].

Material Selection
A Ni-rich NiTi alloy (50.8Ni-49.2Ti% at.) was used in this investigation.It is the same alloy used for NiTi-based ring couplers for vacuum connections [6,23,24,26] which exhibit a similar operative thermo-mechanical cycle with respect to the C-Shaped fasteners under investigation.After proper thermo-mechanical training, such alloy ensures a proper thermal stability range and operating recovery force for room-temperature vacuum applications [6,23,24,26].
The connectors in martensitic state need to be placed onto the flanges at room temperature and mounted by heating above austenite TTs (contraction related to OW-SME).Owing to thermal hysteresis, the fastener keeps the contracted austenitic shape at room temperature after cooling and exerts a proper axial recovery force onto the flanges to create a sealed joint.The dismounting would be obtained owing to TW-SME (elongation), by cooling the fasteners below martensite TTs (subzero temperatures).The selected NiTi alloy, after a proper training based on severe mechanical deformations, fulfills such operating requirements in terms of TTs.Austenite TTs are sufficiently high to limit the risk of premature thermal contraction before for room-temperature.Martensite TTs ensure a fully austenitic structure (after thermal activation) in a large thermal range (20-200 °C) without significant transformation-induced relaxations for temperatures higher than 10 °C.
The DSC thermogram of the fully annealed alloy (T = 900 °C for 45 min) is reported in Fig. 1.The significant heat flow variations, identified by the tangent method, corresponds to the TTs of the alloy.The selected NiTi is austenitic at room temperature, in fact figure shows a martensite finish temperature, M f = -36 °C, a martensite start temperature M s = -22 °C, an austenite start temperature, As = -8 °C and austenite finish temperature, Af = 12 °C.An increase of the austenite TTs can be obtained, at least for the first heating stage, by a severe predeformation in martensitic condition.Systematic studies of such mechanism were carried out by proper thermo-mechanical tests [23,24,26] schematically depicted in Fig. 2. The latter shows the stress-strain response of the alloy (black curve) during the martensitic mechanical pre-strain as well as the temperature-strain evolution of the subsequent stress-free thermal cycle (red curve).Low temperature mechanical pre-strain was carried out in fully martensitic conditions (T \ M f ), by monotonic strain controlled loading up to a certain strain, e tot , followed by a complete unloading.Thermal cycles in stress-free condition were performed after martensitic pre-strain to identify the TTs, the one-way (OW) and two-way (TW) shape memory strain (respectively, e OW and e TW ) as well as the permanent strain e p .The first heating from T \ M f to T [ A f 1st causes the reverse martensite-to-austenite (M-A) transition, showing e OW and e P , accumulated during the pre-strain.
The austenite TTs of the pre-deformed material at the first cycle, A s 1st and A f 1st , can be identified.Subsequent cooling stage causes the austenite-to-martensite (A-M) transformation and allows to estimate the TW shape memory strain, e TW , and the martensitic TTs.The effects of mechanical pre-strain on both shape memory properties and on TTs were systematically analyzed in [23].In particular, a marked increase of e OW is observed up to a maximum value of about 8.4% at e tot = 14.4%, followed by a decrease with further increasing the pre-strain.A similar non monotonic trend is observed for e TW .This is a direct consequence of the slips occurring in the martensitic matrix at such large mechanical pre-strain, as confirmed by the marked increase of the permanent strain e P with increasing e tot .As an example, a martensitic pre-deformation up to e tot = 28.8%gives e OW = 5%, e TW = 2.4%, e p = 18%, A s 1st = 48 °C, A f 1st = 90 °C [23].Considering the actual application of the fasteners in vacuum systems, our attention will be focused on the alloy response in the first free/constrained thermo-mechanical cycle after martensitic pre-deformation.The analysis and modeling of the cyclic thermo-mechanical behavior of the fasteners is not of interest of this study.Consequently, from now on, A s , A f , M s , and M f refers only to the first thermal cycle after pre-deformation.
It is worth noting that the selected alloy, as confirmed by previous investigations [23,24,26], ensures a proper thermal mounting by heating above Af, at about 150 °C, without the risk of a premature activation as well as a suitable dismounting phase, obtained by cooling the fasteners below M f = -50 °C.Such NiTi alloy fulfills the operating requirements for room-temperature vacuum applications which include a large thermal stability range (15-200 °C) with a fully austenitic structure (after thermal activation) without significant transformation-induced relaxations for temperatures higher than 15 °C (cfr.Fig. 10a).

Preliminary Design Considerations
The geometry of the C-Shaped SMA connectors considered in this investigation is reported in Fig. 3.It is a simplified geometry resulting from preliminary finite element (FE) analyses and analytical/empirical considerations.
Dimensions are compatible with commercial QC flanges (DN100 or larger).A dogbone-like shape (one symmetry plane only) was chosen for the C-Shaped SMA fasteners with the aim of simplify the the ''martensitic pre-deformation phase (see Training), limiting the bending effects on the horizontal part.A cross section of 5 9 8 mm was selected for each SMA element to generate a significant recovery force.The latter is directly related to the maximum recovery stress generated by the selected NiTi (about 280 MPa at room temperature [23]).The recovery force of a single NiTi fastener (after activation) was experimentally evaluated (cfr.''Free and Constrained Recovery Measurements'' and Fig. 10a) and corresponds to about 6 kN at room temperature.The minimum axial force per unit arc length exerted by conventional bolts or chains in commercial Conflat Ò flanges is about 350 N/mm, leading to a total axial force of about 110kN for DN100 flanges.This latter causes plastic deformations in the copper gasket placed between the flanges, generating throttling effects on the leak paths thus creating leak-tight joints.The total axial force required for DN100 joints divided by the force generated by a single NiTi fastener gives the minimum number of C-Shaped connectors considered for the active SMA chain.However, it is worth pointing out that the detailed design of the chain and its connecting elements is out of the scope of this study and will be carried out in future works.

Thermo-Mechanical Tests of C-Shaped SMA Couplers
Training Thermo-mechanical training of the SMA connectors (Fig. 3) was performed aiming at providing a suitable OWand TW-SME as well as appropriate TTs and thermal stability range.This process is based on tensile tests in martensitic condition (T \ M f ).The latter were performed by a universal testing machine (Instron 1276, load capacity 1MN) equipped with an environmental chamber (-150 °C/ ?300 °C).The cross-head velocity was set to 1 mm/min.The sample temperature, measured by k-thermocouples was about -65 °C.
An axial displacement of 4 mm was imposed to provide a suitable mechanical pre-strain (cfr.Fig. 7) to cope with the strict functional requirements for room-temperature vacuum applications.The latter include TW transformation strains higher than 1% and a specific TT range (A s- [ 30 °C, M s \ -20 °C) [23,26] (cfr.Fig. 8).Martensitic SMA connectors after unloading were about 1.9 mm longer owing to accumulation of both plastic and pseudoplastic (transformation) strains (cfr.Fig. 7).Fig. 4 reports a comparison between the undeformed and trained geometry.

Free and Constrained Recovery Measurements
The strain and force recovery capabilities of the trained SMA connectors were assessed by thermal recovery tests in free and constrained conditions.Free recovery experiments were aimed at evaluating the thermally induced deformations of the connectors without preventing any shape change.They were carried out by performing thermal cycles between TTs (-100 °C/ ? 100 °C); recovery deformation in terms of relative displacement (point A-point B in Fig. 3) was measured by an electrical extensometer installed onto the sample.
Constrained recovery tests were carried out to estimate the maximum recovery force generated by the SMA element when its OW deformation is prevented by a mechanical obstacle, simulating the actual operative condition (SMA/flange interaction).These tests were performed by activating the SMA connector against a 316LN steel bar, installed with minimum clearance (0.02 mm) and instrumented by a uniaxial electrical strain gauge (Fig. 5).Strain signals were used to estimate the evolution of the axial contact force, generated by the SMA thermal activation, as a function of the temperature, according to the theory of elasticity.K-type thermocouples were used as temperature sensors.

Finite Element Simulations
Numerical simulations were performed using the commercial finite element (FE) software Abaqus/CAE 2017 (Simulia, Providence, RI) to predict the thermo-mechanical cycles described in ''Thermo-Mechanical Tests of C-Shaped SMA Coupler'' as well as to validate a constitute model recently developed by the authors [23] on an applicative case-study.
The geometries of both the SMA C-shape connector and the 316LN stainless steel element are based on the information reported in ''Preliminary Design Considerations and Thermo-Mechanical Tests of C-Shaped SMA Couplers'' (Figs. 3 and 5).Only one-half of the components is modeled, exploiting symmetric conditions and thus applying standard symmetric boundary conditions.Specifically, the SMA C-shape connector geometry is meshed using 57,072 nodes and 51,750 eight-node linear hexahedral elements with full integration (C3D8) available in the element library of the software Abaqus.The 316LN stainless steel element mesh is defined by 183,528 nodes and 173,425 eight-node linear hexahedral elements with full integration (C3D8).
The thermo-mechanical behavior of the SMA is captured using the three-dimensional macroscopic constitutive model proposed by the authors [23], able to describe the TW-SME induced by severe pre-deformation and the related austenitic TT shift, in addition to the OW-SME, superelasticity, and plasticity.A brief description of the model is reported in Appendix A. Model parameters are listed in Table 1 and are taken from [23] since the same SMA alloy is used in both works.
The mechanical response of the 316LN stainless steel is modeled as linear elastic with Young's modulus E = 205 GPa, Poisson's ratio m = 0.3, and thermal expansion coefficient a = 1.65E-05 1/°C.
A quasi-static analysis is performed by imposing a uniform temperature field history on the couplings as well as appropriate boundary conditions for simulating the thermo-mechanical tests described in ''Thermo-Mechanical Tests of C-Shaped SMA Couplers''.Particularly, to simulate SMA martensitic pre-deformation (''Training''), an axial displacement (x-direction) of 2 mm is imposed to the free surface of the SMA connector at -65 °C (see Fig. 6).After loading, the connector is then unloaded.Subsequently, from this deformed (trained) shape, two analyses are performed to simulate the free and constrained recovery, without and with steel bar, respectively, as described in ''Free and Constrained Recovery Measurements''.The imposed thermal cycle consists in a heating stage up to 200 °C and subsequent cooling to -100 °C.
In constrained recovery analyses, a surface-to-surface contact is activated between the steel and SMA components.In particular, an isotropic penalty friction formulation was adopted with coefficient equal to 0.7 and a hard contact pressure-overclosure relationship for the normal component, enforced by augmented Lagrange method.

Results and Discussion
Figure 7a and b reports respectively the plastic strain, ke p k, and transformation strain, ke tr k, distributions in the SMA element after martensitic pre-strain and unloading.Results reveal a significant plastic deformation of the fastener.In fact, the pre-strain induces mainly tensile deformations in the component with maximum permanent strain of about 20% (see Fig. 7a) near the filet radius.In addition, Fig. 7b shows that the maximum allowable transformation strain (ke tr k = 5%) is reached in most of the fastener volume.
As explained in ''Material Selection'', TTs, OW-SME, and TW-SME are affected by the plastic strain occurring during severe martensitic pre-strain and the subsequent constrained thermal cycles (high recovery stress generation).Plastic deformations are essential to tune the functional properties of the SMA device in terms of TTs and transformation strains.In the following paragraphs, the thermo-mechanical behavior of the pre-deformed fastener is analyzed in both free and constrained recovery conditions simulating the absence and the presence of a mechanical obstacle (steel flange), respectively.

Free recovery Capabilities of C-Shaped Connectors
Figure 8 illustrates the numerical and experimental deformation-temperature response of a trained C-Shaped element obtained from a stress-free thermal cycle between TTs.The OW and TW shape recovery capabilities of the Fig. 7 Finite element results of the SMA fastener after martensitic pre-strain and unloading.Contour plot of the a plastic strain tensor norm (left), ke p k, and b transformation strain tensor norm (right), ke tr k fastener were studied in terms of relative displacement between point A and point B (cfr.Fig. 3).A OW displacement d OW = 1.50 mm and a TW displacement d TW- = 0.62 mm are obtained from experimental measurements confirming the effectiveness of the training stage.Fig. 8 shows also good agreement between simulations and experiments with errors on OW and TW deformations lowers than 13%.The figure also shows the TTs of the SMA fastener at the first thermal cycle (M s , M f , A s , and A f ).The austenitic TT increase is correctly captured by the simulations.These results highlight the good predictive capabilities of the adopted constitutive model [23].Mismatch between the TT values is mainly due their intrinsic experimental variability.In fact, they can vary in a range of about ± 8 °C within the same material batch.

Constrained Recovery Capabilities of C-Shaped Connectors
The constrained recovery capabilities of the fastener (SMA/steel coupling) were analyzed with the aim of measuring the evolution of the recovery force of the device induced by thermal cycling between TTs.The entire residual deformation after martensitic pre-strain is impeded by the steel element to generate the maximum recovery force.Figure 9a shows the normal stress distribution in the x-direction in the fastener at the maximum temperature (T = 200 °C) obtained from FE simulations, while Fig. 9b reports the plastic strain distribution at the same temperature.The internal stress field reveals significant tensile and bending effects occurring in the SMA connector.Normal stress (absolute value) is higher in the upper internal side of the horizontal element as its neutral axis is not barycentric but shifted towards the external side of the component.This is also evident from Fig. 9b which shows a significant accumulation of plastic strain in the upper inner volume (point C) due to the higher tensile stresses.
Figure 10a shows the experimental axial recovery force, F rec , (obtained from strain gauge tests) as a function of temperature, while Fig. 10b reports the corresponding numerical data from FE simulations.F rec sharply increases from the beginning of the SMA/steel contact (occurring approximately at the austenite start temperature A s ) up to the austenite finish temperature (under stress), A f r .A similar trend was obtained from constrained uniaxial SMA samples [23].
In Fig. 10 A f r corresponds to the temperature at which the constrained SMA connector becomes fully austenitic.The force increase from A f r to T max = 200 °C and its subsequent decrease from 200 °C to martensite start temperature under stress (M s r ) is mainly caused by thermal stresses related to the 316LN/NiTi thermal expansion coefficient mismatch.The force drop below M s r is due to the TW-SME.The fastener is dismounted (F rec = 0) when the martensite finish temperature (M f ) is reached.The simulations were found to be relatively accurate in comparison to experiments with errors lower than 16%.It is worth noting that NiTi clamps material show an error on the TTs of about ± 7 °C owing to metallurgy and manufacturing processes.This directly affects the value of the maximum recovery force generated when installed (± 15% variation within the same material batch).The good agreement between experimental (Fig. 10a) and numerical results (Fig. 10b) confirms the capability of the model in capturing the evolution of the recovery forces generated by such SMA devices by temperature variations.The mismatch between experiments and simulations in terms of force-temperature slope at the cooling stage (from about 200 C to 35 C) is attributed to the austenitic Young's modulus variation with temperature [36] which is not modeled and to the presence of possible residual martensitic variants as well as austenite twins accumulated during the heating stage [37,38].
Figure 10b shows also the evolution of plastic strains (point C in Fig. 9) as a function of temperature during the constrained recovery test.It is worth highlighting that they accumulate during both the isothermal pre-deformation phase and the subsequent constrained thermal cycle.In the latter stage, stress-induced martensitic variants transform to austenite involving slip deformations [23,37,38].

Conclusions
In this work, novel C-Shaped SMA connectors were proposed as an alternative smart and flexible connection system for vacuum pipes.C-Shaped SMA elements can be Fig. 8 Experimental and numerical results for a free thermal recovery of a C-Shaped fastener.Horizontal relative displacement-temperature (d-T) curve (points A-B in Fig. 3) obtained for first stress-free thermal activation between the TTs after martensitic pre-strain used to realize active bolt-free chains to be installed onto the flanges.To this aim, such systems exploits the two-way shape memory effect (TW-SME) to exert proper recovery forces onto the flanges as well as for detaching purposes.Accordingly, experimental and finite element (FE) investigations were performed to assess the performance of these components under various thermo-mechanical conditions.A commercial NiTi alloy (50.8Ni-49.2Tiat.%) was selected.A proper training procedure was adopted to tune the functional response of the connectors for the investigated application.The thermo-mechanical behavior of the SMA fasteners in free and constrained recovery conditions was assessed by strain gauge and extensometer tests, revealing the efficacy of the constitutive model in correctly describing the behavior of the analyzed components, exhibiting TW-SME and plasticity.Smart chains consisting of a number of C-Shaped SMA elements would be much smaller and lighter than the traditional stainless-steel collars used for vacuum flanges.Moreover, they can operate with no lubricant limiting potential contamination issues especially for ultra high vacuum applications.Such novel devices can be mounted/ dismounted remotely by temperature changes, representing a good solution also for bi-metallic connections.Research activities aimed at the thermo-mechanical and functional analyses of SMA-based chains are ongoing and they will be included in a future paper.

Appendix A
This section briefly reviews the principal features of the three-dimensional phenomenological constitutive model for SMAs investigated in the present paper.For a more detailed presentation, the reader is referred to ref. [23].
The model is able to describe the TW-SME induced by severe pre-deformation and the related austenitic TT shift, in addition to the OW-SME, superelasticity, and plasticity.Accordingly, the model introduces the transformation strain e tr and the plastic strain e p as tensorial internal variables, in addition to the total strain e and the absolute temperature T as control variables.Specifically, e tr describes the inelastic strain associated to TIM or SIM transformations and allows to take into account in an approximated form the martensite reorientation process; e p describes the plastic deformation during specific thermomechanical conditions.
After adopting an additive decomposition of the total strain into an elastic, thermal, and inelastic (i.e., transformation and plastic) contribution, the Helmholtz free energy function is defined as follows: h p ke p k 2 À Be tr : e p þ I e L e tr ð Þ ðA:1Þ where h and e are the volumetric and deviatoric strain, respectively; K and G = G(e tr ) are the bulk and shear modulus, respectively; s M ¼ bhT À T Ã i, where b [ 0 is a parameter, T* is the transformation temperature, and ‹Á› is the positive part function; h tr and h p define, respectively, phase transformation and plastic hardening; B is a positive parameter that controls the interaction between phase transformation and plastic effects; I e L e tr ð Þ is the indicator function included to satisfyke tr k e L .
Following classical theory, the stress-strain relationship as well as the thermodynamic forces associated to e tr and e p can be derived by proper differentiation of the free energy function W.
To describe phase transformation/martensite reorientation, and plasticity evolution, two classical Mises-type limit functions are adopted, completed by associative evolution equations for e tr and e p and Kuhn-Tucker and consistency conditions.
The solution algorithm, as proposed in [23], is simple and robust as it is based on the Fischer-Burmeister complementarity function and makes the model suitable for implementation within FE codes, allowing the simulation of complex SMA devices.
Funding Open access funding provided by Universita `della Calabria within the CRUI-CARE Agreement.
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Fig. 1
Fig. 1 Differential Scanning Calorimetry thermogram of the selected NiTi alloy

Fig. 3 Fig. 5
Fig. 3 Left: schematic depiction of a chain of C-Shaped SMA connectors for steel vacuum flanges.Right: geometry of single C-Shaped SMA-connector

Fig. 9
Fig. 9 FE results for a constrained thermal recovery of a C-Shaped fastener.a Normal stress distribution (x-direction), S11 (left).b plastic strain tensor norm ke p k at 200 °C (right)

Fig. 10
Fig. 10 Experimental (left) and numerical (right) results for a constrained thermal recovery of a C-Shaped fastener.a Experimental recovery force as a function of temperature from strain gauge measurements (left).b Numerical recovery force and permanent strain (point C in Fig. 9) as a function of temperature from FE simulations (right)

Table 1
SMA model parameters adopted in the numerical simulations