Nonlinear Buckling and Post-buckling of Shape Memory Alloy Shallow Arches

Document Type : Research Paper


1 Department of Mathematics, University of Patras, University Campus, Rio, GR-26504, Greece

2 Department of Civil, Geo and Environmental Engineering, Technische Universität München, Arcisstr. 21, Munich, D-80333, Germany

3 School of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens, GR-15773, Greece


In this work, the nonlinear buckling and post-buckling behavior of shallow arches made of Shape Memory Alloy (SMA) is investigated. Arches are susceptible to large deflections, due to their slenderness, especially when the external load exceeds the serviceability limit point. Beyond this, loss of stability may occur, the famous snap-through buckling. For this reason, curved beams can be used in passive vibration control devices for seismic response mitigation, and the geometrically nonlinear analysis is needed for the accurate prediction of their response. Thus, in this research effort, the assumptions of the Euler-Bernoulli beam theory are considered, and the Von Karman strain field is employed to account for large deflections. The formulation of the problem is displacement-based regarding the axial (tangential) and transverse (normal) displacements, while the two governing equations are coupled and nonlinear. In order to introduce the SMA constitutive law, the stress-strain experimental curves described in the literature are employed together with a fiber approach at specific control cross-sections along the beam. The numerical solution of the longitudinal problem is achieved using the Analog Equation Method (AEM), a Boundary Element Method (BEM) based technique, and the iterative procedure is based on a Newton-Raphson scheme by using a displacement control algorithm to trace the fully nonlinear equilibrium path and overcome the limit points. Several representative examples are studied, not only to validate the proposed model but also to investigate the nonlinear buckling and post-buckling of SMA shallow arches.


[1] Song, G., Ma, N, Li, H.N.: Applications of Shape Memory Alloys in Civil Structures, Engineering Structures, 28, 2006, 1266–1274.
[2] Vieta, N.V., Zakia, W., Umer, R., Analytical Model of Functionally Graded Material/Shape Memory Alloy Composite Cantilever Beam Under Bending, Composite Structures, 203, 2018, 764-776.
[3] Levitas, V.I., Roy, A.M., Preston, D.L., Multiple twinning and variant-variant transformations in martensite: Phase-field approach, Physical Review B, 88, 2013, 054113.
[4] Levitas, V.I., Roy, A.M., Multiphase phase field theory for temperature- and stress-induced phase transformations, Physical Review B, 91, 2015, 174109.
[5] Levitas, V.I., Roy, A.M., Multiphase phase field theory for temperature-induced phase transformations: Formulation and application to interfacial phases, Acta Materialia, 105, 2016, 244-257.
[6] Lagoudas, D.C. (ed), Shape Memory Alloys modeling and Engineering Applications, Springer, New York, 2008.
[7] Tanaka, K., A Thermomechanical Sketch of Shape Memory Effect: One-Dimensional Tensile Behavior, Res Mechanica, 18, 1986, 251-263.
[8] Liang, C., Rogers, C.A., One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials, Journal of Intelligent Material Systems and Structures, 8, 1997, 285-302.
[9] Auricchio, F., Sacco, E., A One-Dimensional Model for Superelastic Shape-Memory Alloys with Different Elastic Properties Between Austenite and Martensite, International Journal of Non-Linear Mechanics, 32(6), 1997, 1101-1114.
[10] Auricchio, F., Sacco, E., A Superelastic Shape-Memory-Alloy Beam Model, Journal of Intelligent Material Systems and Structures, 8, 1997, 489-501.
[11] Auricchio, F., Taylor, R.L., Lubliner, J., Shape Memory Alloys: Macromodelling and Numerical Simulations of the Superelastic Behavior, Computer Methods in Applied Mechanics and Engineering, 146, 1997, 281-312.
[12] Mori, T., Tanaka, K., Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions, Acta Metallurgica, 21, 1973, 571-574.
[13] Brinson, L.C., One-Dimensional Constitutive Behavior of Shape Memory Alloys: Thermo-Mechanical Derivation with Non-Constant Material Functions and Redefined Martensite Internal Variable, Journal of Intelligent Materials Systems and Structures, 4, 1993, 229-242.
[14] Souza, A.C., Mamiya, E.N., Zouain, N., Three-Dimensional Model for Solids Undergoing Stress-Induced Phase Transformations, European Journal od Mechanics-A/ Solids, 17, 1998, 789-806.
[15] Auricchio, F., Petrini, L., A Three-Dimensional Model Describing Stress-Temperature Induced Solid Phase Transformations. Part I: Solution Algorithm and Boundary Value Problems, International Journal for Numerical Methods in Engineering, 61, 2004, 807-836.
[16] Ghomshei, M.M., Tabandeh, N., Ghazavi, A., Gordaninejad, F., Nonlinear Transient Response of a Thick Composite Beam with Shape Memory Alloy Layers, Composites Part B: Engineering, 36, 2005, 9-24.
[17] Khalili, S.M.R., Botshekanan Dehkordi, M., Carrera, E., Shariyat, M., Non-Linear Dynamic Analysis of a Sandwich Beam with Pseudoelastic SMA Hybrid Composite Faces Based on Higher Order Finite Element Theory, Composite Structures, 96, 2013, 243-255.
[18] Ostadrahimi, A., Arghavani, J., Poorasadion, S., An Analytical Study on the Bending of Prismatic SMA Beams, Smart Materials and Structures, 24, 2015, 125035.
[19] Razavilar, R., Fathi, A., Dardel, M., Arghavani Hadi, J., Dynamic Analysis of a Shape Memory Alloy Beam with Pseudoelastic Behavior, Journal of Intelligent Material Systems and Structures, 29(9), 2018, 1835-1849.
[20] Fahimi, P., Eskandari, A.H., Baghani, M., Taheri, A., A Semi-Analytical Solution for Bending Response of SMA Composite Beams Considering SMA Asymmetric Behavior, Composites Part B, 163, 2019, 622-633.
[21] Chung, J. H., Heo, J.S., Lee, J.J., Implementation Strategy for the Dual Transformation Region in the Brinson SMA Constitutive Model, Smart Materials and Structures, 16, 2006, N1-N5.
[22] DeCastro, J.A., Melcher, K.J., Noebe, R.D., Gaydosh, D.J., Development of a Numerical Model for High-Temperature Shape Memory Alloys, Smart Materials and Structures, 16, 2007, 2080-2090.
[23] Khandelwal, A., Buravalla, V.R., A Correction to the Brinson's Evolution Kinetics for Shape Memory Alloys, Journal of Intelligent Materials Systems and Structures, 19(1), 2008, 43-46.
[24] Poorasadion, S., Arghavani, J., Naghdabadi, R., Sohrabpour, S., An Improvement on the Brinson Model for Shape Memory Alloys with Application to Two-Dimensional Beam Element. Journal of Intelligent Materials Systems and Structures, 25(15), 2013, 1905-1920.
[25] Auricchio, F., Reali, A., Stefanelli, U., A Macroscopic 1D Model for Shape Memory Alloys Including Asymmetric Behaviors and Transformation-Dependent Elastic Properties, Computer Methods in Applied Mechanics and Engineering, 198, 2009, 1631-1637.
[26] Mirzaeifar, R., DesRoches, R., Yavari, A., Gal, K., On Super-Elastic Bending of Shape Memory Alloy Beam, International Journal of Solids and Structures, 50(10), 2013, 1664-1680.
[27] Zaki, W., Moumni, Z., A Three-Dimensional Model of the Thermomechanical Behavior of Shape Memory Alloys, Journal of the Mechanics and Physics of Solids, 55, 2007, 2455-2490.
[28] Zaki, W., Moumni, Z., Morin, C., Modeling Tensile-Compressive Asymmetry for Superelastic Shape Memory Alloys, Mechanics of Advanced Materials and Structures, 18(7), 2011, 559-564.
[29] Viet, N.K., Zaki, W., Umer, R., Analytical Model for a Superelastic Timoshenko Shape Memory Alloy Beam Subjected to a Loading-Unloading Cycle, Journal of Intelligent Material Systems and Structures, 29(20), 2018, 3902-3922.
[30] Rejzner, J., Lexcellent, C., Raniecki, B., Pseudoelastic Behavior of Shape Memory Alloy Beams under Pure Bending: Experiment and Modelling, International Journal of Mechanical Sciences, 44, 2002, 665–86.
[31] Watkins, R.T., Reedlunn, B., Daly, S., Shaw, J.A., Uniaxial, Pure Bending, and Column Buckling Experiments on Superelastic Niti Rods and Tubes, International Journal of Solids and Structures, 146, 2018, 1–28.
[32] Shang, Z., Wang, Z., Nonlinear Tension-Bending Deformation of a Shape Memory Alloy Rod, Smart Materials and Structures, 21, 2012, 115004.
[33] Atanackovic, T., Achenbach, M. Moment curvature relations for a pseudoelastic beam, Continuum Mechanics and Thermodynamics, 1, 1989, 73–80.
[34] Mirzaeifara, R., DesRochesb, R., Yavarib A., Gall, K., A Closed-form Solution for Superelastic Shape Memory Alloy Beams Subjected to Bending, Proc. SPIE 8342, Behavior and Mechanics of Multifunctional Materials and Composites, 83421O, 2012.
[35] Sepiani, H., Ebrahimi, F., Karimipour, H., A mathematical model for smart functionally graded beam integrated with shape memory alloy actuators, Journal of Mechanical Science and Technology, 23, 2009, 3179–3190.
[36] Bingfei, L, Dui, G, Yang, S., On the transformation behavior of functionally graded SMA composites subjected to thermal loading, European Journal of Mechanics - A/Solids, 40, 2013, 139-147.
[37] Marfia, S., Sacco, E., Reddy, J.N., Superelastic and shape memory effects in laminated shape-memory-alloy beams, AIAA Journal, 41(1), 2003, 100-109.
[38] Viet, N.V., Zaki, W., Umer, R., Bending models for superelastic shape memory alloy laminated composite cantilever beams with elastic core layer, Composites: Part B, 147, 2018, 86–103.
[39] Bodaghi, M., Damanpack, A.R., Aghdam, M.M., Shakeri, M., Active shape/stress control of shape memory alloy laminated beams, Composites: Part B, 56, 2014, 889-899.
[40] Bayat, Y., Ekhteraei Toussi, H., Analytical layerwise solution of nonlinear thermal instability of SMA hybrid composite beam under nonuniform temperature condition, Mechanics of Advanced Materials and Structures, 2019, 1-14 (in press).
[41] Babaee, A., Sadighi, M., Nikbakht, A., Alimirzaei, S., Generalized differential quadrature nonlinear buckling analysis of smart SMA/FG laminated beam resting on nonlinear elastic medium under thermal loading, Journal of Thermal Stresses, 41(5), 2018, 583-607.
[42] Akbaş, S.D., Post-Buckling Analysis of Axially Functionally Graded Three-Dimensional Beams, International Journal of Applied Mechanics, 7(3), 2015, 1550047.
[43] Torki, M.E., Reddy, J.N., Buckling of Functionally Graded Beams with Partially Delaminated Piezoelectric Layers, International Journal of Structural Stability and Dynamics, 16(3), 2016, 1450104.
[44] Anish, Chaubey, A., Kumar, A., Kwiatkowski, B., Barnat-Hunek, D., Widomski, M.K., Bi-Axial Buckling of Laminated Composite Plates Including Cutout and Additional Mass, Materials, 12(11), 2019, 1750.
[45] Kaveh, A., Dadras, A., Malek, N.G., Buckling load of laminated composite plates using three variants of the biogeography-based optimization algorithm, Acta Mechanica, 229(4), 2018, 1551-1566.
[46] Onkar, A.K., Nonlinear buckling analysis of damaged laminated composite plates, Journal of Composite Materials, 53(22), 2019, 3111–3126.
[47] Ou, X., Zhang, X., Zhang, R., Yao, X., Han, Q., Weak form quadrature element analysis on nonlinear bifurcation and postbuckling of cylindrical composite laminates, Composite Structures, 188, 2018, 266-277.
[48] Zhou, Z., Ni, Y., Tong, Z., Zhu, S., Sun, J., Xu, X., Accurate nonlinear buckling analysis of functionally graded porous graphene platelet reinforced composite cylindrical shells, International Journal of Mechanical Sciences, 151, 2019, 537-550.
[49] Quan, T.Q., Cuong, N.H., Duc, N.D., Nonlinear buckling and post-buckling of eccentrically oblique stiffened sandwich functionally graded double curved shallow shells, Aerospace Science and Technology, 90, 2019, 169-180.
[50] Khoa, N.D., Thiem, H.T., Duc, N.D., Nonlinear buckling and postbuckling of imperfect piezoelectric S-FGM circular cylindrical shells with metal–ceramic–metal layers in thermal environment using Reddy's third-order shear deformation shell theory, Mechanics of Advanced Materials and Structures, 26(3), 2019, 248-259.
[51] Cho, H. K., Optimization of laminated composite cylindrical shells to maximize resistance to buckling and failure when subjected to axial and torsional loads, International Journal of Precision Engineering and Manufacturing, 19(1), 2018, 85-95.
[52] Tsiatas, G.C., Siokas, A.G., Sapountzakis, E.J., A Layered Boundary Element Nonlinear Analysis of Beams. Frontiers in Built Environment: Computational Methods in Structural Engineering, 4(52), 2018.
[53] Tsiatas, G.C., Babouskos, N.G., Linear and Geometrically Nonlinear Analysis of Non-Uniform Shallow Arches under a Central Concentrated Force, International Journal of Non-Linear Mechanics, 92, 2017, 92-101.
[54] Liang, C., Rogers, C. A., Design of Shape Memory Alloy Springs with Applications in Vibration Control, Journal of Intelligent Material Systems and Structures, 8(4), 1997, 314-322.
[55] McCormick, J., Tyber, J., DesRoches, R., Gall, K., Maier, H.J., Structural Engineering with Niti. Part II: Mechanical Behaviour and Scaling, Journal of Engineering Mechanics, 133(9), 2007, 1019-1029.
[56] Katsikadelis, J.T., The Boundary Element Method for Engineers and Scientists, Academic Press, Elsevier, Oxford, UK, 2016.
[57] Sanders, J.L., Nonlinear Theories of Thin Shells. Quarterly Applied Mathematics, 21, 1963, 21-36.
[58] Timoshenko, S., Woinowsky-Krieger, S., Theory of plates and shells, McGraw-Hill, 1959.
[59] Reddy, J.N., Mechanics of Laminated Composite Plates and Shells. Theory and Analysis, CRC Press, Florida, USA, 2003.
[60] Reddy, J.N., Mahaffey, P., Generalized beam theories accounting for von Kármán nonlinear strains with application to buckling, Journal of Coupled Systems and Multiscale Dynamics, 1(1), 2013, 120-134.
[61] Sapountzakis, E.J., Mokos, V.G., Shear deformation effect in nonlinear analysis of spatial beams, Engineering Structures, 30, 2008, 653-663.
[62] Liu, N., Jeffers, A.E., Isogeometric analysis of laminated composite and functionally graded sandwich plates based on a layerwise displacement theory, Composite Structures, 176, 2017, 143-153.
[63] Liu, N., Jeffers, A.E., Adaptive isogeometric analysis in structural frames using a layer-based discretization to model spread of plasticity, Computers & Structures, 196, 2018, 1-11.
[64] Liu, N., Jeffers, A.E., A geometrically exact isogeometric Kirchhoff plate: Feature-preserving automatic meshing and C1 rational triangular Bézier spline discretizations, International Journal for Numerical Methods in Engineering, 115, 2018, 395-409.
[65] Powell, M.J.D., A Fortran subroutine for solving systems of Nonlinear algebraic equations, Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Gordon and Breach, 115-161, 1970.
[66] Batoz, J.L., Dhatt, G., Incremental displacement algorithms for nonlinear problems. International Journal for Numerical Methods in Engineering, 14(8), 1979, 1262-1267.
[67] Liu, N., Plucinsky, P., Jeffers, A.E., Combining Load-Controlled and Displacement-Controlled Algorithms to Model Thermal-Mechanical Snap-Through Instabilities in Structures, Journal of Engineering Mechanics, 143, 2017, 04017051.
[68] Liu, N., Jeffers, A.E., Feature-preserving rational Bézier triangles for isogeometric analysis of higher-order gradient damage models, Computer Methods in Applied Mechanics and Engineering, 357, 2019, 112585.
[69] Tsiatas, G.C., Charalampakis, A.E., Optimizing the natural frequencies of axially functionally graded beams and arches, Composite Structures, 160, 2017, 256-266.
[70] Charalampakis, A.E., Tsiatas, G.C., A Simple Rate-Independent Uniaxial Shape Memory Alloy (SMA) Model, Frontiers in Built Environment: Computational Methods in Structural Engineering, 4(46), 2018.
[71] Zhang, Y., Zhu, S., A shape memory alloy-based reusable hysteretic damper for seismic hazard mitigation, Smart Materials and Structures, 16, 2007, 1603-1613.
[72] Pi, Y.-L., Bradford, M.A., Uy, B., In-plane stability of arches, International Journal of Solids and Structures, 39(1), 2002, 105-125.