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
  • 114 Downloads

Abstract

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.

Keywords

Timoshenko beam model Chiral material Chirality Deflection Microrotation 

List of symbols

bh

Width and thickness of beam, respectively

e

Base of natural logarithm

\(f_i\)

Body force

k

Shear coefficient

\(k_{ij} \)

Curvature tensor

\(m,\,n,\,a,\,l\)

Coefficients to be determined

\(m_{j}\)

Body couple

\(m_{ij}\)

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

A

Cross-sectional area of beam

\(C_{\mathrm{h}}\)

Chiral parameter indicating degree of chirality in material property

I

Second moment of area of cross-section of beam

L

Length of beam

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

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

U

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

Notes

Acknowledgements

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

References

  1. 1.
    Wang, J.S., Wang, G., Feng, X.Q., et al.: Hierarchical chirality transfer in the growth of Towel Gourd tendrils. Sci. Rep. 3, 3102 (2013)CrossRefGoogle Scholar
  2. 2.
    Wang, J.S., Cui, Y.H., Shimada, T., et al.: Unusual winding of helices under tension. Appl. Phys. Lett. 105, 043702 (2014)CrossRefGoogle Scholar
  3. 3.
    Chen, Z., Majidi, C., Srolovitz, D., et al.: Tunable helical ribbons. Appl. Phys. Lett. 98, 011906 (2011)CrossRefGoogle Scholar
  4. 4.
    Yu, X.J., Zhang, L.N., Hu, N., et al.: Shape formation of helical ribbons induced by material anisotropy. Appl. Phys. Lett. 110, 091901 (2017)CrossRefGoogle Scholar
  5. 5.
    Ji, X.Y., Zhao, M.Q., Wei, F., et al.: Spontaneous formation of double helical structure due to interfacial adhesion. Appl. Phys. Lett. 100, 263104 (2012)CrossRefGoogle Scholar
  6. 6.
    Zhao, Z.L., Li, B., Feng, X.Q.: Handedness-dependent hyperelasticity of biological soft fibers with multilayered helical structures. Int. J. Nonlinear Mech. 81, 19–29 (2016)CrossRefGoogle Scholar
  7. 7.
    Okushima, T., Kuratsuji, H.: DNA as a one-dimensional chiral material: application to the structural transition between B form and Z form. Phys. Rev. E 84, 021926 (2011)CrossRefGoogle Scholar
  8. 8.
    Coombs, D., Huber, G., Kessler, J.O., et al.: Periodic chirality transformations propagating on bacterial flagella. Phys. Rev. Lett. 89, 118102 (2002)CrossRefGoogle Scholar
  9. 9.
    Wang, X.L., Sun, Q.P.: Mechanical model of the bistable bacterial flagellar filament. Acta Mech. Solida Sin. 24, 1–16 (2011)CrossRefGoogle Scholar
  10. 10.
    Chandraseker, K., Mukherjee, S., Paci, J.T., et al.: An atomistic-continuum Cosserat rod model of carbon nanotubes. J. Mech. Phys. Solids 57, 932–958 (2015)CrossRefGoogle Scholar
  11. 11.
    Robbie, K., Breet, M.J., Lakhtakia, A.: Chiral sculpted thin films. Nature 384, 616–616 (1996)CrossRefGoogle Scholar
  12. 12.
    Rong, Q.Q., Cui, Y.H., Shimada, T., et al.: Self-shaping of bioinspired chiral composites. Acta Mech. Sin. 30, 533–539 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lakes, R.S., Benedict, R.L.: Noncentrosymmetry in micropolar elasticity. Int. J. Eng. Sci. 20, 1161–1167 (1982)CrossRefMATHGoogle Scholar
  14. 14.
    Sharma, P.: Size-dependent elastic fields of embedded inclusions in isotropic chiral solids. Int. J. Solids Struct. 41, 6317–6333 (2004)CrossRefMATHGoogle Scholar
  15. 15.
    Lakes, R.: Elastic and viscoelastic behavior of chiral materials. Int. J. Mech. Sci. 43, 1579–1589 (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Upmanyu, M., Wang, H.L., Liang, H.Y., et al.: Strain-dependent twist-stretch elasticity in chiral filaments. J. R. Soc. Interface 5, 303–310 (2008)CrossRefGoogle Scholar
  17. 17.
    Tallarico, D., Movchan, N.V., Movchan, A.B., et al.: Tilted resonators in a triangular elastic lattice: chirality, bloch waves and negative refraction. J. Mech. Phys. Solids 103, 236–256 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, X.N., Huang, G.L., Ku, G.K.: Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices. J. Mech. Phys. Solids 60, 1907–1921 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Spadoni, A., Ruzzene, M.: Elasto-satic micropolar behavior of a chiral auxetic lattice. J. Mech. Phys. Solids 60, 156–171 (2012)CrossRefGoogle Scholar
  20. 20.
    Papanicolopulos, S.A.: Chirality in isotropic linear gradient elasticity. Int. J. Solids Struct. 48, 745–752 (2011)CrossRefMATHGoogle Scholar
  21. 21.
    Wang, J.S., Shimada, T., Wang, G.F., et al.: Effects of chirality and surface stresses on the bending and buckling of chiral nanowires. J. Phys. D: Appl. Phys. 47, 015302 (2014)CrossRefGoogle Scholar
  22. 22.
    Leşan, D.: Chiral effects in uniformly loaded rods. J. Mech. Phys. Solids 58, 1272–1285 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Leşan, D.: Fundamental solutions for chiral solids in gradient elasticity. Mech. Res. Commun. 61, 47–52 (2014)CrossRefGoogle Scholar
  24. 24.
    Healey, T.J.: Material symmetry and chirality in nonlinearly elastic rods. Math. Mech. Solids 7, 405–420 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Smith, M.L., Healey, T.J.: Predicting the onset of DNA supercoiling using a non-linear hemitropic elastic rod. Int. J. Nonlinear Mech. 43, 1020–1028 (2008)CrossRefGoogle Scholar
  26. 26.
    Zhao, Z.L., Zhao, H.P., Chang, Z., et al.: Analysis of bending and buckling of pre-twisted beams: a bioinspired study. Acta Mech. Sin. 30, 507–515 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ye, H.M., Wang, J.S., Tang, S., et al.: Surface stress effects on the bending direction and twisting chirality of lamellar crystals of chiral polymer. Macromolecules 43, 5762–5770 (2010)CrossRefGoogle Scholar
  28. 28.
    Zhao, Y.P.: Modern Continuum Mechanics. Science Press, Beijing (2016). (in Chinese)Google Scholar

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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