Acta Mechanica Sinica

, Volume 34, Issue 3, pp 549–560 | Cite as

Timoshenko beam model for chiral materials

  • T. Y. Ma
  • Y. N. Wang
  • L. Yuan
  • J. S. Wang
  • Q. H. Qin
Research Paper


Natural and artificial chiral materials such as deoxyribonucleic acid (DNA), chromatin fibers, flagellar filaments, chiral nanotubes, and chiral lattice materials widely exist. Due to the chirality of intricately helical or twisted microstructures, such materials hold great promise for use in diverse applications in smart sensors and actuators, force probes in biomedical engineering, structural elements for absorption of microwaves and elastic waves, etc. In this paper, a Timoshenko beam model for chiral materials is developed based on noncentrosymmetric micropolar elasticity theory. The governing equations and boundary conditions for a chiral beam problem are derived using the variational method and Hamilton’s principle. The static bending and free vibration problem of a chiral beam are investigated using the proposed model. It is found that chirality can significantly affect the mechanical behavior of beams, making materials more flexible compared with nonchiral counterparts, inducing coupled twisting deformation, relatively larger deflection, and lower natural frequency. This study is helpful not only for understanding the mechanical behavior of chiral materials such as DNA and chromatin fibers and characterizing their mechanical properties, but also for the design of hierarchically structured chiral materials.


Timoshenko beam model Chiral material Chirality Deflection Microrotation 

List of symbols


Width and thickness of beam, respectively


Base of natural logarithm


Body force


Shear coefficient

\(k_{ij} \)

Curvature tensor


Coefficients to be determined


Body couple


Couple stress tensor

q(xt), a(xt)

Distributed transverse force and distributed longitudinal force, respectively

u(xt), w(xt)

Displacement of midsurface in x and z direction, respectively

\(u_{1}, u_2 , u_3\)

Three components of displacement vector along x, y, and z direction, respectively

\(w_{(0)} \,, w_{(1)}\)

Deflection at midpoint of beam central axis of cantilever beam and simply supported beam, respectively


Cross-sectional area of beam


Chiral parameter indicating degree of chirality in material property


Second moment of area of cross-section of beam


Length of beam

\(\overline{M} , \overline{N} ,\overline{V} \)

Axial force, lateral force, and moment applied at the two ends of the beam, respectively


Stored strain energy

\(\alpha , \beta , \gamma , \eta \)

Elastic constants introduced in micropolar theory

\(\chi ,\kappa ,\nu \)

Elastic constants representing material chirality

\(\delta _{kl}\)

Kronecker delta

\(\varepsilon _{ij}\)

Strain tensor

\(\phi _i\)

Microrotation vector

\(\mu ,\lambda \)

Classical Lame’ constants

\(\theta (x,t)\)

Angle of rotation of normal to midsurface of beam

\(\rho \)

Mass density of chiral material

\(\sigma _{ij} \)

Stress tensor

\(\omega \)

Circular frequency of vibration



This study was supported by the National Natural Science Foundation of China (Grants 11472191, 11272230, and 11372100).


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • T. Y. Ma
    • 1
  • Y. N. Wang
    • 2
  • L. Yuan
    • 1
  • J. S. Wang
    • 1
  • Q. H. Qin
    • 3
  1. 1.Department of MechanicsTianjin UniversityTianjinChina
  2. 2.School of EngineeringDeakin UniversityGeelongAustralia
  3. 3.Research School of EngineeringAustralian National UniversityCanberraAustralia

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