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Timoshenko beam model for chiral materials

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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.

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Abbreviations

b, h :

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(x, t), a(x, t):

Distributed transverse force and distributed longitudinal force, respectively

u(x, t), w(x, t):

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

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Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mechanics, Tianjin University, Tianjin, 300054, China

    T. Y. Ma, L. Yuan & J. S. Wang

  2. School of Engineering, Deakin University, Geelong, VIC, 3216, Australia

    Y. N. Wang

  3. Research School of Engineering, Australian National University, Canberra, ACT, 2601, Australia

    Q. H. Qin

Authors
  1. T. Y. Ma
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  2. Y. N. Wang
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  3. L. Yuan
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  4. J. S. Wang
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  5. Q. H. Qin
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Corresponding author

Correspondence to J. S. Wang.

Appendix

Appendix

Substituting Eqs. (40)–(42) into Eqs. (31), (32), (36), and (39) leads to the following four linear algebraic equations

$$\begin{aligned}&k(\mu +\alpha )\left( \frac{2}{3}mL^{3}-\frac{1}{2}mL^{4}-\frac{3}{2}aL^{4}\right. \nonumber \\&\left. \qquad +\frac{6}{5}aL^{5}\right) A-k(\mu -\alpha )n\left( \frac{1}{6}L^{4}-\frac{1}{10}L^{5}\right) A \nonumber \\&\qquad +b(k+1)(\chi +\nu )\left[ {n\left( \frac{1}{12}L^{4}-\frac{1}{20}L^{5}\right) } \right. \frac{2\chi +\kappa }{2\gamma +\beta }h \nonumber \\&\qquad +l(L^{2}-L^{3}-\frac{8L^{2}}{{\uppi }^{2}}+\frac{24L^{3}}{{\uppi }^{2}}+\frac{48L^{3}}{{\uppi }^{3}})\nonumber \\&\qquad \left. \hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] +F(L^{2}-L^{3})=0, \end{aligned}$$
(A1)
$$\begin{aligned}&-\frac{4}{3}L^{3}(2\mu +\lambda )In+(2\chi +\kappa )b\left\{ {\frac{h^{3}L^{3}}{9}n} \frac{2\chi +\kappa }{2\gamma +\beta } \right. \nonumber \\&\left. \qquad +l\frac{8L^{2}}{{\uppi }^{2}}\sqrt{\frac{\gamma +\eta }{2\gamma +\beta }}\left[ h\cosh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right. \right. \nonumber \\&\left. \left. \qquad -\sqrt{\frac{\gamma +\eta }{2\gamma +\beta }}\frac{4L}{{\uppi }}\sinh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] \right\} \nonumber \\&\qquad -k(\mu +\alpha )n\frac{8}{15}L^{5}A+k(\mu -\alpha )n\left( \frac{5}{6}mL^{4}-\frac{9}{10}aL^{5}\right) A \nonumber \\&\left. \qquad +b(k+1)(\chi -\nu )\left[ {n\frac{4}{15}L^{5}} \right. \frac{2\chi +\kappa }{2\gamma +\beta }h\right. \nonumber \\&\left. \qquad +l\frac{16L^{3}}{{\uppi }^{3}}\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] =0, \end{aligned}$$
(A2)
$$\begin{aligned}&k(\mu +\alpha )(2mL-3aL^{2})A-k(\mu -\alpha )L^{2}nA \nonumber \\&\qquad +\frac{1}{2}b(k+1)(\chi +\nu ) \left[ nL^{2}\frac{2\chi +\kappa }{2\gamma +\beta }h\right. \nonumber \\&\qquad \left. +2l\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] =-F, \end{aligned}$$
(A3)
$$\begin{aligned}&\frac{1}{2}(k+1)(\chi +\nu )\left[ \left( \frac{5}{6}mL^{4}-\frac{9}{10}aL^{5}\right) \frac{2\chi +\kappa }{2\gamma +\beta }\frac{h}{2}\right. \nonumber \\&\qquad \left. +\left( \frac{8mL^{2}}{{\uppi }^{2}}-\frac{24aL^{3}}{{\uppi }^{2}}\right. \right. \nonumber \\&\qquad \left. \left. +\frac{48aL^{3}}{{\uppi }^{3}}\right) \hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] \nonumber \\&\qquad -(k+1)(\chi -\nu )n\left[ \frac{4}{15}L^{5}\frac{2\chi +\kappa }{2\gamma +\beta }\frac{h}{2}\right. \nonumber \\&\qquad \left. +\frac{8L^{3}}{{\uppi }^{3}}\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] \nonumber \\&\qquad +(\gamma +\eta )\left\{ n\left[ \frac{4}{15}L^{5}\frac{2\chi +\kappa }{2\gamma +\beta }\frac{h}{2}\right. \right. \nonumber \\&\qquad \left. \left. +\frac{16L^{3}}{{\uppi }^{3}}\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] \frac{2\chi +\kappa }{2\gamma +\beta }\right. \nonumber \\&\qquad \left. +l\sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\left[ \frac{8L^{2}}{{\uppi }^{2}}\frac{2\chi +\kappa }{2\gamma +\beta }\frac{h}{2}\right. \right. \nonumber \\&\qquad \left. \left. +\frac{\pi }{4}\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right] \cosh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{4L}\right) \right\} =0.\nonumber \\ \end{aligned}$$
(A4)

In the case of the simply supported beam, following a procedure similar to that in Eqs. (A1)–(A4) results in the following equations

$$\begin{aligned}&-\frac{1}{2}k(\mu +\alpha )m\frac{{\uppi }^{2}}{L}A+\frac{1}{2}k(\mu -\alpha )n{\uppi }A \nonumber \\&\quad -\frac{1}{4}b(k+1)(\chi +\nu ){\uppi }\nonumber \\&\quad \times \left[ {n\frac{2\chi +\kappa }{2\gamma +\beta }h+2l\hbox {sinh}\left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{2L}\right) } \right] \nonumber \\&\quad +\frac{2ql}{{\uppi }}=0,\end{aligned}$$
(A5)
$$\begin{aligned}&b(2\chi +\kappa )\frac{{\uppi }^{2}}{L^{2}}\left\{ \frac{h^{3}}{12} \frac{2\chi +\kappa }{2\gamma +\beta }n\right. \nonumber \\&\left. \quad +\sqrt{\frac{\gamma +\eta }{2\gamma +\beta }}\frac{L}{{\uppi }}\left[ h \cosh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{2L}\right) \right. \right. \nonumber \\&\left. \left. \quad -2\sqrt{\frac{\gamma +\eta }{2\gamma +\beta }}\frac{L}{{\uppi }}\sinh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{2L}\right) \right] l \right\} \nonumber \\&\quad +k(\mu -\alpha )m\frac{\uppi }{{L }}A-k(\mu +\alpha )nA \nonumber \\&\quad -(2\mu +\lambda )In\frac{{\uppi }^{2}}{L^{2}}+\frac{1}{2}b(k+1)(\chi -\nu )\left[ n\frac{2\chi +\kappa }{2\gamma +\beta }h\right. \nonumber \\&\left. \quad +2l\sinh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{2L}\right) \right] =0, \end{aligned}$$
(A6)
$$\begin{aligned}&\frac{1}{2}(k+1)(\chi +\nu )m\frac{{\uppi }}{L}-\frac{1}{2}(k+1)(\chi -\nu )n\nonumber \\&\quad +(\gamma +\eta )\left[ n\frac{2\chi +\kappa }{2\gamma +\beta }\right. \nonumber \\&\left. \quad +l\sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }}{L}\cosh \left( \sqrt{\frac{2\gamma +\beta }{\gamma +\eta }}\frac{{\uppi }h}{2L}\right) \right] =0. \end{aligned}$$
(A7)

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Ma, T.Y., Wang, Y.N., Yuan, L. et al. Timoshenko beam model for chiral materials. Acta Mech. Sin. 34, 549–560 (2018). https://doi.org/10.1007/s10409-017-0735-y

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  • DOI: https://doi.org/10.1007/s10409-017-0735-y

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