Abstract
Background: Soft materials, including elastomers and gels, are pervasive in biological systems and technological applications. Despite the rapid developments of soft materials in the recent decades, it is still challenging to theoretically model and predict the large-deformation behaviors of soft structures.
Objective: The goal of this work is to give a general theoretical model to investigate the large deformation of a cantilevered soft beam under various loads. In particular, the applicability of the inextensibility assumption of the beam centerline is explored.
Methods: The governing equations of the soft beam system are derived according to the principle of minimum potential energy. In order to investigate the large deformation of the soft beam, the curvature of the beam centerline is exactly considered and the Yeoh model is utilized to account for the hyperelasticity of the soft beam. The derived ordinary differential equations are discretized by the Galerkin method and then solved by the iterative algorithm.
Results: Based on the proposed theoretical model, large bending deformations of the cantilevered soft beam are analyzed for various types of external loads, including uniformly distributed force, tipend concentrated force, and non-uniformly distributed force. Different values of the amplitude of the external loads are considered and fruitful deformed configurations are presented.
Conclusion: The proposed model is able to study the large deformation of the soft beam effectively. The inextensibility assumption of the beam centerline is applicable when the amplitude of the external load is relatively small. When the amplitude of the external load is sufficiently large, the extension of the centerline needs to be considered.
Keywords: Cantilevered, hyperelastic, inextensibility assumption, large deformation, soft beam, beam centerline.
Graphical Abstract
[1]
S. Kim, C. Laschi, and B. Trimmer, "Soft robotics: a bioinspired evolution in robotics", Trends Biotechnol., vol. 31, no. 5, pp. 287-294, 2013.
[http://dx.doi.org/10.1016/j.tibtech.2013.03.002] [PMID: 23582470]
[http://dx.doi.org/10.1016/j.tibtech.2013.03.002] [PMID: 23582470]
[2]
D. Rus, and M.T. Tolley, "Design, fabrication and control of soft robots", Nature, vol. 521, no. 7553, pp. 467-475, 2015.
[http://dx.doi.org/10.1038/nature14543] [PMID: 26017446]
[http://dx.doi.org/10.1038/nature14543] [PMID: 26017446]
[3]
B.D. Gates, "Materials science. Flexible electronics", Sci., vol. 323, no. 5921, pp. 1566-1567, 2009.
[http://dx.doi.org/10.1126/science.1171230] [PMID: 19299605]
[http://dx.doi.org/10.1126/science.1171230] [PMID: 19299605]
[4]
A. Nathan, A. Ahnood, M.T. Cole, S. Lee, Y. Suzuki, P. Hiralal, F. Bonaccorso, T. Hasan, L. Garcia-Gancedo, A. Dyadyusha, S. Haque, P. Andrew, S. Hofmann, J. Moultrie, D. Chu, A.J. Flewitt, A.C. Ferrari, M.J. Kelly, J. Robertson, G.A.J. Amaratunga, and W.I. Milne, "Flexible electronics: the next ubiquitous platform", Proc. IEEE, vol. 100, pp. 1486-1517, 2012.
[http://dx.doi.org/10.1109/JPROC.2012.2190168]
[http://dx.doi.org/10.1109/JPROC.2012.2190168]
[5]
Z.G. Yan, B.L. Wang, K.F. Wang, and C. Zhang, "A novel cellular substrate for flexible electronics with negative Poisson ratios under large stretching", Int. J. Mech. Sci., vol. 151, pp. 314-321, 2019.
[http://dx.doi.org/10.1016/j.ijmecsci.2018.11.026]
[http://dx.doi.org/10.1016/j.ijmecsci.2018.11.026]
[6]
M. Billinghurst, and T. Starner, "Wearable devices: new ways to manage information", Computer, vol. 32, pp. 57-64, 1999.
[http://dx.doi.org/10.1109/2.738305]
[http://dx.doi.org/10.1109/2.738305]
[7]
D. Son, J. Lee, S. Qiao, R. Ghaffari, J. Kim, J.E. Lee, C. Song, S.J. Kim, D.J. Lee, S.W. Jun, S. Yang, M. Park, J. Shin, K. Do, M. Lee, K. Kang, C.S. Hwang, N. Lu, T. Hyeon, and D.H. Kim, "Multifunctional wearable devices for diagnosis and therapy of movement disorders", Nat. Nanotechnol., vol. 9, no. 5, pp. 397-404, 2014.
[http://dx.doi.org/10.1038/nnano.2014.38] [PMID: 24681776]
[http://dx.doi.org/10.1038/nnano.2014.38] [PMID: 24681776]
[8]
S. Bose, S. Vahabzadeh, and A. Bandyopadhyay, "Bone tissue engineering using 3D printing", Mater. Today, vol. 16, pp. 496-504, 2013.
[http://dx.doi.org/10.1016/j.mattod.2013.11.017]
[http://dx.doi.org/10.1016/j.mattod.2013.11.017]
[9]
C. Schubert, M.C. van Langeveld, and L.A. Donoso, "Innovations in 3D printing: a 3D overview from optics to organs", Br. J. Ophthalmol., vol. 98, no. 2, pp. 159-161, 2014.
[http://dx.doi.org/10.1136/bjophthalmol-2013-304446] [PMID: 24288392]
[http://dx.doi.org/10.1136/bjophthalmol-2013-304446] [PMID: 24288392]
[10]
G.A. Sydney, E.A. Matsumoto, and R.G. Nuzzo, "Mahadevan. L.; Lewis, J. A. Biomimetic 4D printing", Nat. Mater., vol. 15, pp. 413-418, 2016.
[http://dx.doi.org/10.1038/nmat4544]
[http://dx.doi.org/10.1038/nmat4544]
[11]
F. Momeni, X. Liu, and J. Ni, "A review of 4D printing", Mater. Des., vol. 122, pp. 42-79, 2017.
[http://dx.doi.org/10.1016/j.matdes.2017.02.068]
[http://dx.doi.org/10.1016/j.matdes.2017.02.068]
[12]
J.S. Katz, and J.A. Burdick, "Light-responsive biomaterials: development and applications", Macromol. Biosci., vol. 10, no. 4, pp. 339-348, 2010.
[http://dx.doi.org/10.1002/mabi.200900297] [PMID: 20014197]
[http://dx.doi.org/10.1002/mabi.200900297] [PMID: 20014197]
[13]
R. Yoshida, K. Sakai, T. Okano, Y. Sakurai, H.B. You, and W.K. Sung, "Surface-modulated skin layers of thermal responsive hydrogels as on-off switches: I. Drug release", J. Biomater. Sci. Polym. Ed., vol. 3, pp. 155-162, 1992.
[14]
D. Miyajima, K. Tashiro, F. Araoka, H. Takezoe, J. Kim, K. Kato, M. Takata, and T. Aida, "Liquid crystalline corannulene responsive to electric field", J. Am. Chem. Soc., vol. 131, no. 1, pp. 44-45, 2009.
[http://dx.doi.org/10.1021/ja808396b] [PMID: 19128171]
[http://dx.doi.org/10.1021/ja808396b] [PMID: 19128171]
[15]
Y. Kim, H. Yuk, R. Zhao, S.A. Chester, and X. Zhao, "Printing ferromagnetic domains for untethered fast-transforming soft materials", Nature, vol. 558, no. 7709, pp. 274-279, 2018.
[http://dx.doi.org/10.1038/s41586-018-0185-0] [PMID: 29899476]
[http://dx.doi.org/10.1038/s41586-018-0185-0] [PMID: 29899476]
[16]
M. Gong, P. Wan, D. Ma, M. Zhong, M. Liao, J. Ye, R. Shi, and L. Zhang, "Flexible breathable nanomesh electronic devices for on-demand therapy", In: Adv. Funct. Mater, vol. 29. 2019, p. 1902127.
[http://dx.doi.org/10.1002/adfm.201902127]
[http://dx.doi.org/10.1002/adfm.201902127]
[17]
T. Li, G. Li, Y. Liang, T. Cheng, J. Dai, X. Yang, B. Liu, Z. Zeng, Z. Huang, Y. Luo, T. Xie, and W. Yang, "Fast-moving soft electronic fish", Sci. Adv., vol. 3, no. 4, 2017.e1602045
[http://dx.doi.org/10.1126/sciadv.1602045] [PMID: 28435879]
[http://dx.doi.org/10.1126/sciadv.1602045] [PMID: 28435879]
[18]
D.C. Pamplona, H.I. Weber, and G.R. Sampaio, "Analytical, numerical and experimental analysis of continuous indentation of a flat hyperelastic circular membrane by a rigid cylindrical indenter", Int. J. Mech. Sci., vol. 87, pp. 18-25, 2014.
[http://dx.doi.org/10.1016/j.ijmecsci.2014.05.028]
[http://dx.doi.org/10.1016/j.ijmecsci.2014.05.028]
[19]
C. Jin, A. Davoodabadi, J. Li, Y. Wang, and T. Singler, "Spherical indentation of a freestanding circular membrane revisited: analytical solutions and experiments", J. Mech. Phys. Solids, vol. 100, pp. 85-102, 2017.
[http://dx.doi.org/10.1016/j.jmps.2017.01.005]
[http://dx.doi.org/10.1016/j.jmps.2017.01.005]
[20]
C. Zhang, L. Fan, and Y. Tan, "Sequential limit analysis for clamped circular membranes involving large deformation subjected to pressure load", Int. J. Mech. Sci., vol. 155, pp. 440-449, 2019.
[http://dx.doi.org/10.1016/j.ijmecsci.2019.03.011]
[http://dx.doi.org/10.1016/j.ijmecsci.2019.03.011]
[21]
Y. Anani, and G.H. Rahimi, "Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials", Int. J. Mech. Sci., vol. 108, pp. 122-128, 2016.
[http://dx.doi.org/10.1016/j.ijmecsci.2016.02.003]
[http://dx.doi.org/10.1016/j.ijmecsci.2016.02.003]
[22]
A. Almasi, M. Baghani, and A. Moallemi, "Thermomechanical analysis of hyperelastic thick-walled cylindrical pressure vessels, analytical solutions and FEM", Int. J. Mech. Sci., vol. 130, pp. 426-436, 2017.
[http://dx.doi.org/10.1016/j.ijmecsci.2017.06.033]
[http://dx.doi.org/10.1016/j.ijmecsci.2017.06.033]
[23]
R.M. Soares, P.F.T. Amaral, F.M.A. Silva, and P.B. Gonçalves, "Non-linear breathing motions and instabilities of a pressure-loaded spherical hyperelastic membrane", Nonlinear Dyn., vol. 99, pp. 351-372, 2020.
[http://dx.doi.org/10.1007/s11071-019-04855-4]
[http://dx.doi.org/10.1007/s11071-019-04855-4]
[24]
S.P. Jung, T.W. Park, and W.S. Chung, "Dynamic analysis of rubber-like material using absolute nodal coordinate formulation based on the non-linear constitutive law", Nonlinear Dyn., vol. 63, pp. 149-157, 2011.
[http://dx.doi.org/10.1007/s11071-010-9792-5]
[http://dx.doi.org/10.1007/s11071-010-9792-5]
[25]
Q. Xu, and J. Liu, "An improved dynamic model for a silicone material beam with large deformation", Lixue Xuebao, vol. 34, pp. 744-753, 2018.
[http://dx.doi.org/10.1007/s10409-018-0759-y]
[http://dx.doi.org/10.1007/s10409-018-0759-y]
[26]
E. Reissner, "On one-dimensional finite-strain beam theory: the plane problem", Z. Angew. Math. Phys., vol. 23, pp. 795-804, 1972.
[http://dx.doi.org/10.1007/BF01602645]
[http://dx.doi.org/10.1007/BF01602645]
[27]
J.C. Simo, "A finite strain beam formulation. The three-dimensional dynamic problem. Part I", Comput. Methods Appl. Math., vol. 49, pp. 55-70, 1985.
[28]
M.M. Attard, "Finite strain-beam theory", Int. J. Solids Struct., vol. 40, pp. 4563-4584, 2003.
[http://dx.doi.org/10.1016/S0020-7683(03)00216-6]
[http://dx.doi.org/10.1016/S0020-7683(03)00216-6]
[29]
L. He, J. Lou, Y. Dong, S. Kitipornchai, and J. Yang, "Variational modeling of plane-strain hyperelastic thin beams with thickness-stretching effect", Acta Mech., vol. 229, pp. 4845-4861, 2018.
[http://dx.doi.org/10.1007/s00707-018-2258-4]
[http://dx.doi.org/10.1007/s00707-018-2258-4]
[30]
M. Amabili, P. Balasubramanian, I.D. Breslavsky, G. Ferrari, R. Garziera, and K. Riabova, "Experimental and numerical study on vibrations and static deflection of a thin hyperelastic plate", J. Sound Vibrat., vol. 385, pp. 81-92, 2016.
[http://dx.doi.org/10.1016/j.jsv.2016.09.015]
[http://dx.doi.org/10.1016/j.jsv.2016.09.015]
[31]
P. Balasubramanian, G. Ferrari, M. Amabili, and Z.J.G.N. Del Prado, "“Experimental and theoretical study on large amplitude vibrations of clamped rubber plates”, Int. J. Non-Lin", Mech., vol. 94, pp. 36-45, 2017.
[32]
M. Amabili, I.D. Breslavsky, and J.N. Reddy, "Nonlinear higher-order shell theory for incompressible biological hyperelastic materials", Comput. Methods Appl. Math., vol. 346, pp. 841-861, 2019.
[33]
G. Marckmann, and E. Verron, "Comparison of hyperelastic models for rubber-like materials", Rubber Chem. Technol., vol. 79, pp. 835-858, 2006.
[http://dx.doi.org/10.5254/1.3547969]
[http://dx.doi.org/10.5254/1.3547969]
[34]
C. Semler, G.X. Li, and M.P. Païdoussis, "The non-linear equations of motion of pipes conveying fluid", J. Sound Vibrat., vol. 169, pp. 577-599, 1994.
[http://dx.doi.org/10.1006/jsvi.1994.1035]
[http://dx.doi.org/10.1006/jsvi.1994.1035]
[35]
O.H. Yeoh, "Characterization of elastic properties of carbon-black-filled rubber vulcanizates", Rubber Chem. Technol., vol. 63, pp. 792-805, 1990.
[http://dx.doi.org/10.5254/1.3538289]
[http://dx.doi.org/10.5254/1.3538289]
[36]
W. Chen, and L. Wang, "Theoretical modeling and exact solution for extreme bending deformation of hard-magnetic soft beams", In: ASME J. Appl. Mech., vol. 87. 2020.041002.
[http://dx.doi.org/10.1115/1.4045716]
[http://dx.doi.org/10.1115/1.4045716]
[37]
W. Chen, Z. Yan, and L. Wang, "Complex transformations of hard-magnetic soft beams by designing residual magnetic flux density", Soft Matter, vol. 16, pp. 6379-6388, 2020. https://pubs.rsc.org/en/content/articlelanding/2020/sm/c9sm02529d
Article Metrics
21
1