{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T06:16:03Z","timestamp":1789193763333,"version":"build-2803163510"},"reference-count":63,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,12,23]],"date-time":"2018-12-23T00:00:00Z","timestamp":1545523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper proposes a new force-displacement model for superelastic shape memory alloy (SMA) springs under complex loading and unloading. For the SMA wires used to make superelastic springs, a new multilinear constitutive model based on a modification of the 1D Motahari model is developed. In the modified model, the stress-strain relation curves are changed to fit the experimental results. Furthermore, the established force-displacement relationship of the springs considers the impact of not only the torque but also the moment on the cross sections of the SMA wires. Afterwards, a series of tension tests are performed on four NiTi helical spring specimens under various loading conditions. From the numerical simulations and experimental results, it is shown that, compared with the force-displacement curves for the SMA springs simulated by the Motahari model, those simulated by the proposed model can better approximate the experimental results. The new model inherits the advantage of simple computation of the multilinear constitutive model and can predict the force-displacement relation for superelastic SMA springs very well. Furthermore, due to the self-sensing properties of the SMA springs, the new model is very significant for establishing a new strategy for measuring the displacements or forces of SMA springs under complex loading and unloading.<\/jats:p>","DOI":"10.3390\/s19010050","type":"journal-article","created":{"date-parts":[[2018,12,24]],"date-time":"2018-12-24T10:37:49Z","timestamp":1545647869000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Numerical Simulation and Experimental Study of a Simplified Force-Displacement Relationship in Superelastic SMA Helical Springs"],"prefix":"10.3390","volume":"19","author":[{"given":"Bin","family":"Huang","sequence":"first","affiliation":[{"name":"School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongwang","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Song","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1243\/09544100JAERO211","article-title":"Aerospace applications of shape memory alloys","volume":"221","author":"Hartl","year":"2007","journal-title":"Proc. Inst. Mech. Eng. Part G\u2013J Aerosp. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1017\/S026357470800458X","article-title":"A shape memory alloy-based tendon-driven actuation system for biomimetic artificial fingers, part I: Design and evaluation","volume":"27","author":"Bundhoo","year":"2009","journal-title":"Robotica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1016\/j.matdes.2013.11.084","article-title":"A review of shape memory alloy research, applications and opportunities","volume":"56","author":"Jani","year":"2014","journal-title":"Mater. Des."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.12989\/sss.2011.7.1.001","article-title":"Design and Control of a Proof-of-Concept Active Jet Engine Intake Using Shape Memory Alloy Actuators","volume":"7","author":"Song","year":"2004","journal-title":"Smart. Struct. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"045012","DOI":"10.1088\/0964-1726\/25\/4\/045012","article-title":"Shape memory alloy actuated accumulator for ultra-deepwater oil and gas exploration","volume":"25","author":"Patil","year":"2016","journal-title":"Smart Mater. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1088\/0964-1726\/11\/2\/305","article-title":"An overview of vibration and seismic applications of NiTi shape memory alloy","volume":"11","author":"Saadat","year":"2002","journal-title":"Smart Mater. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1088\/0964-1726\/16\/1\/023","article-title":"A one-dimensional strain-rate-dependent constitutive model for superelastic shape memory alloys","volume":"16","author":"Ren","year":"2007","journal-title":"Smart Mater. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1088\/0964-1726\/16\/4\/017","article-title":"Phenomenological modeling of the cyclic behavior of superelastic shape memory alloys","volume":"16","author":"Ren","year":"2007","journal-title":"Smart Mater. Struct."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2202","DOI":"10.1088\/0964-1726\/16\/6\/023","article-title":"Investigation of vibration mitigation of stay cables incorporated with superelastic shape memory alloy dampers","volume":"16","author":"Liu","year":"2007","journal-title":"Smart Mater. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1266","DOI":"10.1016\/j.engstruct.2005.12.010","article-title":"Applications of shape memory alloys in civil structures","volume":"28","author":"Song","year":"2006","journal-title":"Eng. Struct."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1177\/1045389X11411220","article-title":"Seismic Response Control Using Shape Memory Alloys: A Review","volume":"22","author":"Ozbulut","year":"2011","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/S0141-0296(98)00097-2","article-title":"Base isolation system with shape memory alloy device for elevated highway bridges","volume":"22","author":"Wilde","year":"2000","journal-title":"Eng. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125026","DOI":"10.1088\/0964-1726\/25\/12\/125026","article-title":"Experimental investigations of building structure with a superelastic shape memory alloy friction damper subject to seismic loads","volume":"25","author":"Qian","year":"2016","journal-title":"Smart Mater. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1002\/eqe.2777","article-title":"Shake table test and numerical study of self-centering steel frame with SMA braces","volume":"46","author":"Qiu","year":"2017","journal-title":"Earthq. Eng. Struct. Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"e2180","DOI":"10.1002\/stc.2180","article-title":"Experimental investigations on seismic control of cable-stayed bridges using shape memory alloy self-centering dampers","volume":"25","author":"Zhou","year":"2018","journal-title":"Struct. Control Health Monit."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"075029","DOI":"10.1088\/1361-665X\/aac28f","article-title":"An innovative re-centering SMA-lead damper and its application to steel frame structures","volume":"27","author":"Li","year":"2018","journal-title":"Smart Mater. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1088\/0964-1726\/15\/2\/010","article-title":"Health monitoring and rehabilitation of a concrete structure using intelligent materials","volume":"15","author":"Song","year":"2006","journal-title":"Smart Mater. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"025024","DOI":"10.1088\/0964-1726\/24\/2\/025024","article-title":"Energy-dissipating and self-repairing SMA-ECC composite material system","volume":"24","author":"Li","year":"2015","journal-title":"Smart Mater. Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1007\/s11665-009-9433-7","article-title":"Shape Memory Alloy Tension\/Compression Device for Seismic Retrofit of Buildings","volume":"18","author":"Speicher","year":"2009","journal-title":"J. Mater. Eng. Perform."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1080\/13632469.2011.562406","article-title":"Innovative superelastic isolation device","volume":"15","author":"Attanasi","year":"2011","journal-title":"J. Earthqu. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"095016","DOI":"10.1088\/0964-1726\/22\/9\/095016","article-title":"An improved tuned mass damper (SMA-TMD) assisted by a shape memory alloy spring","volume":"22","author":"Mishra","year":"2013","journal-title":"Smart Mater. Struct."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"065009","DOI":"10.1088\/0964-1726\/23\/6\/065009","article-title":"Passive base isolation with superelastic nitinol SMA helical springs","volume":"23","author":"Huang","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s11803-016-0351-z","article-title":"Experimental and numerical study on hysteretic performance of SMA spring-friction bearings","volume":"15","author":"Zhuang","year":"2016","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.measurement.2015.02.058","article-title":"Bidirectional angular control of an integrated sensor\/actuator shape memory alloy based system","volume":"69","author":"Ruth","year":"2015","journal-title":"Measurement."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"055019","DOI":"10.1088\/0964-1726\/19\/5\/055019","article-title":"Modeling of the electrical resistance of shape memory alloy wires","volume":"19","author":"Cui","year":"2010","journal-title":"Smart Mater. Struct."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"025001","DOI":"10.1088\/0964-1726\/22\/2\/025001","article-title":"Sensorless displacement estimation of a shape memory alloy coil spring actuator using inductance","volume":"22","author":"Kim","year":"2012","journal-title":"Smart Mater. Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.12989\/sss.2007.3.2.189","article-title":"Position estimation and control of SMA actuators based on electrical resistance measurement","volume":"3","author":"Song","year":"2007","journal-title":"Smart. Struct. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7682","DOI":"10.3390\/s120607682","article-title":"An accurately controlled antagonistic shape memory alloy actuator with selfsensing","volume":"12","author":"Wang","year":"2012","journal-title":"Sensors"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"112","DOI":"10.3390\/s100100112","article-title":"Tracking Control of Shape-Memory-Alloy Actuators Based on Self-Sensing Feedback and Inverse Hysteresis Compensation","volume":"10","author":"Liu","year":"2010","journal-title":"Sensors"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"12958","DOI":"10.3390\/s131012958","article-title":"Electrical Resistivity-Based Study of Self-Sensing Properties for Shape Memory Alloy-Actuated Artificial Muscle","volume":"13","author":"Zhang","year":"2013","journal-title":"Sensors"},{"key":"ref_31","first-page":"83","article-title":"Deformation of a shape memory alloy helical spring:analysis based on stress-strain-temperature relation","volume":"34","author":"Tobushi","year":"1991","journal-title":"JSME Int. J. Ser. I"},{"key":"ref_32","first-page":"251","article-title":"A Thermomechanical sketch of shape memory effect:one-dimensional tensile behavior","volume":"18","author":"Tanaka","year":"1986","journal-title":"Res. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1177\/1045389X9700800404","article-title":"Design of shape memory alloy springs with applications in vibration control","volume":"8","author":"Liang","year":"1997","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1177\/1045389X9000100205","article-title":"One-dimensional thermomechanical constitutive relations for shape memory materials","volume":"1","author":"Liang","year":"1990","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.1016\/j.compstruc.2004.03.025","article-title":"Finite element analysis of superelastic, large deformation behavior of shape memory alloy helical springs","volume":"82","author":"Toi","year":"2004","journal-title":"Comput. Struct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1177\/1045389X9300400213","article-title":"One-dimensional constitutive behavior of shape memory alloys:thermomechanical derivation with non-constant material functions and redefined martensite internal variable","volume":"4","author":"Brinson","year":"1993","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"025008","DOI":"10.1088\/0964-1726\/19\/2\/025008","article-title":"Experimental and numerical investigations of shape memory alloy helical springs","volume":"19","author":"Aguiar","year":"2010","journal-title":"Smart Mater. Struct."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3439","DOI":"10.1016\/j.ijsolstr.2004.11.006","article-title":"A constitutive model for shape memory alloys considering tensile\u2013compressive asymmetry and plasticity","volume":"42","author":"Paiva","year":"2005","journal-title":"Int. J. Solids Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1016\/j.ijsolstr.2010.10.026","article-title":"A combined analytical, numerical, and experimental study of shape-memory-alloy helical springs","volume":"48","author":"Mirzaeifar","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"035002","DOI":"10.1088\/0964-1726\/18\/3\/035002","article-title":"Nonlinear finite element formulation for analyzing shape memory alloy cylindrical panels","volume":"18","author":"Mirzaeifar","year":"2009","journal-title":"Smart Mater. Struct."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.mechmat.2010.06.003","article-title":"Exact solutions for pure torsion of shape memory alloy circular bars","volume":"42","author":"Mirzaeifar","year":"2010","journal-title":"Mech. Mater."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1051\/meca\/2014040","article-title":"Experimental and finite element analysis of superelastic, behaviour of shape memory alloy for damping applications","volume":"15","author":"Thiebaud","year":"2014","journal-title":"Mech. Ind."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.mechmat.2011.04.003","article-title":"Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation","volume":"43","author":"Chemisky","year":"2011","journal-title":"Mech. Mater."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"085018","DOI":"10.1088\/0964-1726\/23\/8\/085018","article-title":"Nonlinear dynamics of a pseudoelastic shape memory alloy system-theory and experiment","volume":"23","author":"Enemark","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"035012","DOI":"10.1088\/0964-1726\/24\/3\/035012","article-title":"Nonlinear geometric influence on the mechanical behavior of shape memory alloy helical springs","volume":"24","author":"Savi","year":"2015","journal-title":"Smart Mater. Struct."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/S0045-7825(96)01232-7","article-title":"Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behavior","volume":"146","author":"Auricchio","year":"1997","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"183","DOI":"10.12989\/sss.2015.16.1.183","article-title":"Simulation of superelastic SMA helical springs","volume":"16","author":"Mehrabi","year":"2015","journal-title":"Smart. Struct. Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2666","DOI":"10.1016\/j.ijsolstr.2014.03.039","article-title":"A thermodynamically-consistent microplane model for shape memory alloys","volume":"51","author":"Mehrabi","year":"2014","journal-title":"Int. J. Solids Struct."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1016\/j.engstruct.2006.06.007","article-title":"Multilinear one-dimensional shape memory material model for use in structural engineering applications","volume":"29","author":"Motahari","year":"2007","journal-title":"Eng. Struct."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.ijplas.2011.10.009","article-title":"Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys","volume":"32\u201333","author":"Lagoudas","year":"2012","journal-title":"Int. J. Plast."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1177\/1045389X9600700112","article-title":"Simplifications and Comparisons of Shape Memory Alloy Constitutive Models","volume":"7","author":"Brinson","year":"1996","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Lagoudas, D.C., Mayes, J.J., and Khan, M.M. (2001, January 5\u20138). Simplified shape memory alloy (SMA) material model for vibration isolation. Proceedings of the Smart Structures and Materials 2001 Conference, Newport Beach, CA, USA.","DOI":"10.1117\/12.436514"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"3649","DOI":"10.1016\/S1359-6454(97)00490-4","article-title":"Phase diagram based description of the hysteresis behavior of shape memory alloys","volume":"46","author":"Bekker","year":"1998","journal-title":"Acta Mater."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"085032","DOI":"10.1088\/0964-1726\/20\/8\/085032","article-title":"Combining thermodynamic principles with Preisach models for superelastic shape memory alloy wires","volume":"20","author":"Doraiswamy","year":"2011","journal-title":"Smart Mater. Struct."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/j.msea.2007.08.028","article-title":"A coupled thermomechanical model for shape memory alloys-From single crystal to polycrystal","volume":"481","author":"Heintze","year":"2008","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"075028","DOI":"10.1088\/0964-1726\/21\/7\/075028","article-title":"Suppressing nonlinear resonances in an impact oscillator using SMAs","volume":"21","author":"Sitnikova","year":"2012","journal-title":"Smart Mater. Struct."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/S0921-5093(96)10510-4","article-title":"Shear and tensile thermomechanical behavior of near equiatomic NiTi alloy","volume":"222","author":"Manach","year":"1997","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1271","DOI":"10.12989\/sss.2015.15.5.1271","article-title":"Theoretical analysis of superelastic SMA helical structures subjected to axial and torsional loads","volume":"15","author":"Zhou","year":"2015","journal-title":"Smart. Struct. Syst."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2721","DOI":"10.1177\/1045389X16635845","article-title":"Modelling, characterisation and uncertainties of stabilised pseudoelastic shape memory alloy helical springs","volume":"27","author":"Enemark","year":"2016","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_60","unstructured":"Wahl, A.M. (1944). Mechanical Springs, Penton Publishing Co.. [1st ed.]."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2591","DOI":"10.1007\/s11665-014-0906-y","article-title":"Simulations of Mechanical Response of Superelastic NiTi Helical Spring and its Relation to Fatigue Resistance","volume":"23","author":"Frost","year":"2014","journal-title":"J. Mater. Eng. Perform."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"075021","DOI":"10.1088\/0964-1726\/23\/7\/075021","article-title":"Constitutive modeling of tension-torsion coupling and tensioncompression asymmetry in NiTi shape memory alloys","volume":"23","author":"Mehrabi","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_63","unstructured":"Song, Y. (2017). Experimental and Numerical Study of Vibration Control of Frame Structure with SMAS-TMD System. [Master\u2019s Thesis, Wuhan University of Technology]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/1\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:35:47Z","timestamp":1760196947000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/1\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,23]]},"references-count":63,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1]]}},"alternative-id":["s19010050"],"URL":"https:\/\/doi.org\/10.3390\/s19010050","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,23]]}}}