, Volume 53, Issue 15, pp 3759–3777 | Cite as

A shearable and thickness stretchable finite strain beam model for soft structures

  • Liwen He
  • Jia Lou
  • Youheng Dong
  • Sritawat Kitipornchai
  • Jie YangEmail author


Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.


Finite strain Soft materials Hyperelastic Bending-to-stretching transition Shearable 



The work described in this paper was fully supported by the Australian Research Council Grant under Discovery Project scheme (DP160101978). The authors are very grateful for these financial supports. Dr. He and Dr. Lou are also grateful for the support from National Natural Science Foundation of China (Grant Nos. 11602118 and 11602117) and also sponsored by K.C. Wong Magna Fund in Ningbo University.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanics and Engineering ScienceNingbo UniversityNingboPeople’s Republic of China
  2. 2.School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China
  3. 3.School of Civil EngineeringUniversity of QueenslandBrisbaneAustralia
  4. 4.School of EngineeringRMIT UniversityBundooraAustralia

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