The buckled beam is a fundamental building block used to produce morphing structures. The advancement of responsive materials has enabled these building blocks to actively transform structures between multiple states. Often the driving stimulus is applied to these structural elements in a longitudinally nonuniform fashion. However, most models assume uniform application of the stimulus or rely on numerical results, which obscure connections between the system parameters and the system response. In this work, we examine a thermally responsive, buckled beam and consider the temperature profile to be a half sine wave along its length. This assumption makes the analysis convenient and enables the derivation of an exact expression for the critical temperature rise leading to loss of stability under a range of conditions. In addition, a very simple expression is obtained for predicting the crossover from the symmetric snap-through mode to unsymmetrical transitions between buckled states. The results give insight into system design strategies and also lay the foundation for analyzing more complex systems.
ZhaoYHuaMYanY, et al. Stimuli-responsive polymers for soft robotics. Annu Rev Control Robot Auton Syst2022; 5(1): 515–545.
2.
ChiYLiYZhaoY, et al. Bistable and multistable actuators for soft robots: structures, materials, and functionalities. Adv Mater2022; 34(19): 2110384.
3.
HolmesD.Elasticity and stability of shape-shifting structures. Curr Opin Colloid In2019; 40: 118–137.
4.
VirginL.On the elastic snapping of structural elements. Int J Nonlin Mech2023; 149: 104329.
5.
TsudaTMochiyamaHFujimotoH. Quick stair-climbing using snap-through buckling of closed elastica. In: 2012 international symposium on micro-nanomechatronics and human science MHS, Nagoya, Japan, 4–7 November, 2012, pp. 368–373. New York: IEEE.
6.
SmithMGaoJSkandaniA, et al. Tuned photomechanical switching of laterally constrained arches. Smart Mater Struct2019; 28(7): 075009.
7.
CaoYDerakhshaniMFangY, et al. Bistable structures for advanced functional systems. Adv Funct Mater2021; 31(45): 2106231.
8.
LibrandiGTubaldiEBertoldiK.Snapping of hinged arches under displacement control: strength loss and nonreciprocity. Phys Rev E2020; 101(5): 053004.
9.
BonthronMTubaldiE.Dynamic behavior of bistable shallow arches: from intrawell to chaotic motion. J Appl Mech2024; 91(2): 021010.
10.
FungYKaplanA. Buckling of low arches or curved beams of small curvature. Technical Report
2840, National Advisory Committee for Aeronautics, 1952.
11.
SimitsesG.An introduction to the elastic stability of structures. Englewood Cliffs: Prentice Hall, 1976.
12.
BazantZCedolinL.Stability of structures: elastic, inelastic fracture, and damage theories. New York: Oxford University Press, 1991.
13.
PlautR.Snap-through of shallow elastic arches under end moments. J Appl Mech2009; 76(1): 014504.
14.
ChenJHungS.Snapping of an elastica under various loading mechanisms. Eur J Mech A-Solid2011; 30(4): 525–531.
15.
RadissonBKansoE.Elastic snap-through instabilities are governed by geometric symmetries. Phys Rev Lett2023; 130(23): 236102.
16.
CasalottiAD’AnnibaleFRosiG.Optimization of an architected composite with tailored graded properties. Z Angew Math Phys2024; 75(4): 126.
17.
HimaND’AnnibaleFDal CorsoF.Non-smooth dynamics of buckling based metainterfaces: rocking-like motion and bifurcations. Int J Mech Sci2023; 242: 108005.
18.
TurcoEBarchiesiE.Equilibrium paths of Hencky pantographic beams in a three-point bending problem. Math Mech Com Sys2019; 7(4): 287–310.
19.
BarchiesiEdell’IsolaFBersaniA, et al. Equilibria determination of elastic articulated duoskelion beams in 2d via a riks-type algorithm. Int J Nonlin Mech2021; 128: 103628.
20.
De AngeloMBarchiesiEGiorgioI, et al. Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: application to out-of-plane buckling. Arch Appl Mech2019; 89(7): 1333–1358.
21.
WangFWangTZhangX, et al. Wrinkling of twisted thin films. Int J of Solids Struct2023; 262–263: 112075.
22.
GiorgioIHarrisonPdell’IsolaF, et al. Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc R Soc A2018; 474: 20180063.
23.
BurgreenDRegalD.Higher mode buckling of bimetallic beam. J Eng Mech Div-ASCE1971; 97(4): 1045–1056.
24.
KianiYEslamiM.Thermal buckling analysis of functionally graded material beams. Int J Mech Mater Des2010; 6(3): 229–238.
25.
ArnaudABoughalebJMonfrayS, et al. Thermo-mechanical efficiency of the bimetallic strip heat engine at the macro-scale and micro-scale. J Micromech Microeng2015; 25(10): 104003.
26.
XiYLyuQZhangN, et al. Thermal-induced snap-through buckling of simply-supported functionally graded beams. Appl Math Mech2020; 41(12): 1821–1832.
27.
RavindranSKroenerMShabanianA, et al. Analysis of a bimetallic micro heat engine for energy harvesting. Smart Mater Struct2014; 23(3): 035011.
WangXHoG.Design of untethered soft material micromachine for life-like locomotion. Mater Today2022; 53: 197–216.
30.
TimoshenkoS.Analysis of bi-metal thermostats. J Opt Soc Am1925; 11(3): 233–255.
31.
MichaelAYuKKwokC. Theoretical analysis of initially buckled thermally actuated snapping bimorph microbridge. In: Proc. SPIE 5276, device and process technologies for MEMS, microelectronics, and photonics III, Perth, NSW, Australia, 2 April, 2004, pp. 540–547. San Diego, CA: SPIE.
32.
OrthweinW.Thermal buckling of a bimetallic strip. J Vib Acoust Stress1984; 106(4): 538–542.
33.
PiYBradfordM.Nonlinear thermoelastic buckling of pin-ended shallow arches under temperature gradient. J Eng Mech2010; 136(8): 960–968.
34.
CaiJXuYFengJ, et al. In-plane elastic buckling of shallow parabolic arches under an external load and temperature changes. J Struct Eng2012; 138(11): 1300–1309.
35.
KeibolahiAKianiYEslamiM.Dynamic snap-through of shallow arches under thermal shock. Aerosp Sci Technol2018; 77: 545–554.
36.
RaoRYeZLvJ, et al. Nonlinear instability behavior and buckling of shallow arches under gradient thermo-mechanical loads. Front Mater2022; 9: 894260.
37.
SmithMShankarMBackmanR, et al. Designing light responsive bistable arches for rapid, remotely triggered actuation. In: Proc. SPIE 9058, behavior and mechanics of multifunctional materials and composites, San Diego, CA, 9 March, 2014, p. 90580F. San Diego, CA: SPIE.
38.
KornerKKuenstlerAHaywardR, et al. A nonlinear beam model of photomotile structures. Proc Natl Acad Sci USA2020; 117(18): 9762–9770.
39.
BurgreenDManittP.Thermal buckling of a bimetallic beam. J Eng Mech Div-ASCE1969; 95(2): 421–432.
40.
CabalARossD.Snap-through bilayer microbeam. In: Technical proceedings of the 2002 international conference on modeling and simulation of microsystems, San Juan, Puerto Rico, 22 April, 2002, pp. 230–233.
ShankarMSmithMTondigliaV, et al. Contactless, photoinitiated snap-through in azobenzene-functionalized polymers. Proc Natl Acad Sci USA2013; 110(47): 18792–18797.
43.
DaiCZhuQKhoruzhenkoO, et al. Reversible snapping of constrained anisotropic hydrogels upon light stimulations. Adv Sci2024; 11: 2402824.
44.
ByskovE.Elementary continuum mechanics for everyone: With applications to structural mechanics. Solid Mechanics and Its Applications, Dordrecht: Springer, 2013.
45.
PressWTeukloskySVetterlingW, et al. Numerical recipes in FORTRAN: The art of scientific computing. 2nd ed.Cambridge: Cambridge University Press, 1992.
46.
PostonTStewartI.Catastrophe theory and its applications. Mineola: Dover Publications, 1996.
47.
ThompsonJHuntG.A general theory of elastic stability. London: John Wiley, 1973.
48.
ThompsonJ.Stability predictions through a succession of folds. Philos T R Soc S-A1979; 292(1386): 1–23.
49.
ChenJHungS.Exact snapping loads of a buckled beam under a midpoint force. Appl Math Model2012; 36(4): 1776–1782.
50.
SeideP.Dynamics stability of laterally loaded buckled beams. J Eng Mech1984; 110(10): 1556–1569.
51.
GomezMMoultonDVellaD.Critical slowing down in purely elastic ‘snap-through’ instabilities. Nat Phys2017; 13(2): 142–145.
52.
HoffNBruceG.Dynamics analysis of the buckling of laterally loaded flat arches. J Math Phys1953; 32: 276–288.
53.
SaedMTorbatiAStarrC, et al. Thiol-acrylate main-chain liquid-crystalline elastomers with tunable thermomechanical properties and actuation strain. J Polym Sci Pol Phys2017; 55(2): 157–168.
54.
GiorgioIPlacidiL.A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’. Meccanica2024; 59(10): 1745–1756.
