Timoshenko beam model for chiral materials

T.Y.M a·Y.N.Wang·L.Yuan·J.S.Wang·Q.H.Qin

1 Introduction

A wide range of natural and human-made materials such as deoxyribonucleic acid(DNA), flagellar filaments,wood,bone,carbon nanotubes,and chiral sculptured thin films have helical or twisted microstructure,e.g.,helical arrangements of protein lattices or atoms and twisted inclusions,at different length scales[1–14].These materials are essentially chiral in terms of their materials properties and can be regarded as chiral materials,due to the chiral nature of their microstructure.Such chiral symmetry of material properties is mainly manifested by coupling of different physical fields or deformations,e.g.,piezoelectricity and twist–stretch deformation,often leading to unusual optical or electrical properties and superior elastic properties[14–16].For example,chiral carbon nanotubes and DNA undergo distinct twist deformation when stretched because of their helical structure[10,16].Chiral lattice materials can refract elastic waves with negative refraction angle and even exhibit negative Poisson’s ratio,suggesting that chiral lattices could be used to realize and tailorelastic metamaterials[17–19].Thus,chiral materials hold great promise for use in a diversity of critical applications in certain important fields,for instance,smart sensors and actuators,force probes in biomedical engineering,structural elements for absorption of microwaves and elastic waves,and electrical and optical devices with novel functions,due to their unique properties.

Chiral effects in elasticity are not invariant with respect to inversion,i.e.,are noncentrosymmetric,and can be described by odd-rank rather than even-rank tensors[13–15].Thus,classical elasticity theory cannot capture the chiral behavior of such materials because the four-rank modulus tensor is invariant with respect to inversion.So far,two theories,i.e.,non centrosym metric micropolar elasticity theory[13,15]and strain gradient elasticity theory[20],have mainly be used to describe the mechanical behavior of chiral materials.In noncentrosymmetric micropolar elasticity theory,chiral materials are treated as hemitropic micropolar media or chiral Cosserat elastic solids[13].It is assumed that,beyond the classical displacement degrees of freedom,each material point has an extra microrotation.The gradient of microrotation is coupled with the stresses and couples tresses.Meanwhile,in the case of strain gradient theory,chirality is introduced through material parameters that are pseudo tensors directly linked to the three-dimensional deformation[20].

Motivated by the unique properties of chiral materials and their promising applications as structural and functional elements,the mechanical behavior of chiral materials has been intensively studied during the past decade[13–21].Utilizing the concept of hemitropic micropolar media,Sharma[14]solved the Eshelby inclusion problem for chiral solids and derived closed-form solutions for the size-dependent elastic fields of embedded spherical and circular cylindrical inclusions.On the basis of noncentrosymmetric micropolar elastic theory, the mechanical behavior of chiral lattice materials and chiral Cosserat elastic cylinders has also been investigated[18,22].Recently,Ie¸san[23]solved the deformation problem for homogeneous and isotropic chiral materials subjected to concentrated body forces using strain gradient elasticity theory.Most of these previous studies mainly focused on two-or three-dimensional chiral materials such as chiral lattice materials and chiral Cosserat cylinders.The mechanical behavior of slender and thin chiral rods or beams,including DNA,carbon nanotubes, flagellar filaments,and other quasi-one-dimensional chiral structures,has also attracted much research attention in recent years.Healey and Smith[24,25]established a hemitropic elastic rod model incorporating chirality and used it to predict the onset of DNA super coiling.Zhao et al.[26]investigated the bending and buckling behavior of pretwisted beams from the bionic point of view.Recently,Wanget al.[21]developed a refined Euler–Bernoulli beam model for chiral nanowires incorporating surface effects and material chirality to characterize their bending and buckling.It is known that the Euler–Bernoulli beam model neglects effects of transverse shear deformation and rotary inertia,holding only when the length of the beam is much greater than its thickness.If the length of chiral beam or rod structures is not sufficiently long,the Euler–Bernoulli model is inapplicable and the influence of transverse shear deformation and rotary inertia should be taken into account.Furthermore,chiral materials generally exhibit distinct shear–bending coupling deformation due to their microstructure chirality,in which transverse shear deformation cannot be simply ignored.

In this paper,a formulation for the Timoshenko beam model with microstructure chirality is developed based on noncentrosymmetric micropolar elasticity theory.The governing equations and boundary conditions for the chiral beam are derived by means of the variational approach and Hamilton’sprinciple.This model is then employed to investigate the static bending and free vibration behavior of a chiral beam,and the effects of microstructure chirality on the mechanical behavior of the beam are examined.

2 Timoshenko beam model with chirality

In noncentrosymmetric micropolar elasticity theory,each material point of the chiral solid possesses microrotation degrees of freedom(DOF)besides the classical displacement DOF[13–15].Note that,unlike an elastic solid without chirality,the gradient of the microrotation is coupled with the deformation of a chiral solid,demonstrating the existence of an indirect relationship between the microrotation and displacement.The stored strain energyUfor a chiral Cosserat elastic solid undergoing infinitesimal deformation can be expressed in terms of the strain tensorεijand curvature tensorkijas follows[13]

whereσijis the stress tensor whilemijis the couple stress tensor.

The geometrical equations relating the strain and curvature tensors to the displacement vectoruiand microrotation vectorφiare[14]

whereeijkis the permutation tensor.

The constitutive equations for a chiral Cosserat elastic solid read[14]

whereμandλare the classical Lamé constants,α,γ,η,andβare the elastic constants introduced in micropolar theory,andχ,κ,andνare the elastic constants representing the material chirality.δklis the Kronecker delta.

The equilibrium equations for a chiral Cosserat elastic solid read

wherefjandmjare the body force and body couple,respectively.

Fig.1 Deformed chiral beam:scanning electron photomicrograph of tendril cell of towel gourd containing cellulose helix

As shown in Fig.1,we consider a chiral Cosserat elastic beam with rectangular cross-section of widthb,thicknessh,and lengthL.The beam elements contain chiral microstructures such as cellulose helices or stiff helical molecular chains,as shown in the scanning electron photomicrograph image.We refer to a Cartesian coordinate systemo−xyz,withxandzaxes along the length and thickness directions of the beam,respectively.The beam is subjected to a distributed transverse loadingq(x).By the assumptions of the Timoshenko beam,the displacement of a chiral beam can be expressed as

whereu1,u2,andu3are the three components of the displacement vector along thex,y,andzdirection,respectively.u(x,t)andw(x,t)are the displacement of the midsurface inxandzdirection,respectively.θ(x,t)is the angle of rotation of the normal to the midsurface of the beam.u(x,t),w(x,t),andθ(x,t)are functions ofxand timetonly.

The three components of the microrotation vector along thex,y,andzdirection are

whereφ1denotes the twist deformation of the beam along the central axis,andφ2represents the microrotation of the beam along theyaxis.

Substituting Eqs.(5)and(6)into Eq.(2)yields the following nonzero strain components

According to the constitutive relationships Eq.(3)and introducing the shear coefficient,the nonzero stresses and couple stress components can be obtained as follows

Substituting Eqs.(7)–(10)into Eq.(11)and performing the variational calculation,the first-order calculation of the strain energy in the time range[t1,t2]can be obtained as

whereis the second moment of area andAis the cross-sectional area of the beam.It is assumed that the initial and final configurations of the beam are specified with finite values,thus the virtual displacements att=t1andt=t2are zero.

The kinetic energy of the beam is given by

whereρis the mass density of the chiral material.

From Eqs.(5)and(13),the variation of the kinetic energy of the beam can be expressed as

Here,we assume that the mass density of the chiral materialρis constant,thusm0=ρA,m2=ρI.

The work done by applied external forces can be written as

whereq(x,t)anda(x,t)stand for the distributed transverse force and distributed longitudinal force,respectively.andare the axial force,lateral force,and moment applied at the two ends of the beam,respectively.

Then,the variation of the work done by the external force is

Substituting Eqs.(12),(14),and(17)into Eq.(18)and considering thatδu,δw,andδφare arbitrary in 0＜x＜L,we obtain the following governing equations

According to Eq.(18),the boundary conditions of the Timoshenko chiral beam are given by

We have now derived the general Timoshenko beam model for a chiral material,including the governing equations(19)–(23)together with the boundary conditions in Eqs.(24)–(30).This model can effectively capture the deformation characteristics of a chiral material such as the twist–stretch.When the seven parameters related to chirality and micropolar materials are zero,the derived model degenerates to that for a linear elastic material.It should be mentioned that,in the degenerated beam model,the Poisson effect has been considered.If the Poisson effect is neglected,the degenerated beam model can further degenerate to the classical Timoshenko beam model.In general,it is difficult to solve these equations analytically,thus numerical solutions are sought.To investigate the mechanical behavior of chiral materials,the derived formulations are simplified for two cases,i.e.,the cantilever beam and the simply supported beam.

3 Case study

In the following,the formulations of the bending and free vibration problems of a cantilever beam and simply supported beam are derived based on a few simple assumptions.Here,it is assumed that the effect of the beam longitudinal inertia can be neglected compared with its transverse inertia effect,.Also,there is no distribute daxial force along the beam length,i.e.,a(x,t)=0.In addition,it is further assumed that chirality only exists inxaxis,i.e.,

3.1 Cantilever beam

We consider a cantilever beam of chiral material.The geometrical shape and size are the same as those in Fig.1.One end of the beam is fixed,while the other end is loaded by a forceF.

From Eq.(18),we have the following governing equations for the cantilever beam

Considering Eqs.(24)–(30),the boundary conditions for the cantilever beam can be written as

The governing equations and the boundary conditions Eqs.(31)–(39)can be solved approximately using the Galerkin method.It is assumed that the deflection and the angle of rotation satisfying the boundary conditions are in the form

Substituting Eqs.(40)–(42)into Eqs.(31),(32),(36),and(39)leads to four linear algebraic equations with four unknown parameters includingm,n,a,andl,which can be easily solved numerically.The four linear algebraic Eqs.(A1)–(A4)are given in “Appendix”.

In case of the free vibration problem of a cantilever beam,the governing Eqs.(31)–(33)and boundary conditions Eqs.(34)–(39)remain unchanged,except the loadingFbecomes zero.In addition,the deflectionw,rotation angleθ,and microrotation angleφ1are functions ofxand timetonly,i.e.,w=w(x,t),θ=θ(x,t),φ1=φ1(x,t).They have the following forms

where e andωare the base of the natural logarithm and the circular frequency of vibration,respectively.Following a procedure similar to that in Eqs.(A1)–(A4),the unknown parametersm,n,a,l,andωcan be easily obtained.

3.2 Sim ply supported beam

We con sider a simply supported beam of chiral material with the same geometrical shape and size as that in Fig.1.The beam is under constant distributed loadq(x).

The boundary conditions are:

Following a procedure similar to that in Eqs.(A1)–(A4),one canal so obtain the three linear algebraic equations Eqs.(A5)–(A7)in“Appendix”.The unknown parametersm,n,andlcan be easily solved.In case of the free vibration problem,the deflectionw,rotation angleθ,and microrotation angleφ1can be similarly solved.

4 Numerical results

To better understand the mechanical behavior of beams of chiral material,a chiral polymer lamella containing helical chains with finite stiffness is considered as a typical example in this section.The elastic properties of the chiral polymer lamella are taken to beλ=8.7 GPa andμ−α=2.2 GPa[21,27].Values of the elastic constantsα,β,γ,η,κ,χ,andνare not available in literature.However,in micropolar elasticity theory,αand(μ−α)are generally on the same order,and can be approximately considered as 2α=μ−α,i.e.,μ+α=4.4GPa.Other constants for chiral polymer lamellae can be approximately determined as 2γ+β=3×10−7N andγ+η=0.2×(2γ+β)=6×10−8N based on the twisting deformation of chiral polymer lamellae during grow th[27].The values above are proved to satisfy the thermodynamic restrictions on the elastic constants of chiral materials[13].According to the definition of chirality in material property,we haveis the chiral parameter indicating the degree of chirality in a material property with a range from 0 to 1.Ch=0 means that the material has no chirality,whileCh=1 means the material has the highest degree of chirality[1].The value ofChmainly depends on the geometrical shape and material properties of the chiral microstructure,the deformation state,and other factors such as material properties of the matrix.For most chiral materials in nature such as DNA,climbing tendrils,and chromatin fibers,the values ofChare usually smaller than 0.4 based on their twisting deformation;For example,the value ofChfor tendril of towel gourd is often smaller than 0.2.However,under large strain,the values ofChcan be up to 0.8 or even higher[2].For man-made chiral materials,values ofChcan be controlled by optimizing the microstructu redesign.The geometrical parameters of the beam are taken asL=20mm andb=0.5mm.

Fig.2 Variation of normalized deflection with length of cantilever beam for different thickness-to-length ratios

Fig.3 Variation of normalized deflection with length of simply supported beam for different thickness-to-length ratios

Fig.4 Effect of chirality on normalized deflection of cantilever beam

In Figs.2 and 3,the variation of the normalized deflection of the cantilever beam and simply supported beam along the beam length for different thickness-to-length ratios are plotted.Here,w(0)andw(1)are the deflection of the midpoint of the beam central axis of the cantilever beam and simply supported beam forh=1/5L,respectively.To verify the derived model,the results from both the Euler–Bernoulli beam model for a chiral material(EBBM-CM)[21]and the Timoshenko beam model for elasticmaterial(TBM-EM)are also shown in Figs.2 and 3.The results show that both the Timoshenko and Euler–Bernoulli beam for chiral material are more flexible than the Timoshenko beam of elastic material.The EBBM-CM and TBM-CM results do not differ greatly,and the deflection from EBBM-CM is even larger than that from TBM-CM.This is because the EBBM-CM is derived directly from the differential equations,which may not be self-consistent,while the TBM-CM derived here is self-consistent.Thus,the TBM-CM can degenerate to the TBM-EM rather than the EBBM-CM.Furthermore,the shear stress is incorporated in the EBBM-CM,being induced by the chiral properties of the material,and the nonzero microrotation inydirection is also considered,in contrast to the classical Euler–Bernoulli beam model for elastic materials.Figure 2 shows that,for the cantilever beam,the normalized deflection increases along the beam length,reaching its maximum at the end of the beam.Meanwhile,in Fig.3,the normalized deflection of the simply supported beam reaches its maximum at the midpoint of the beam length.In addition,Figs.2 and 3 show that the normalized deflection increases distinctly with decrease of the thickness-to-length ratio.These results are similar to those for the elastic beam without chirality.

Figures 4 and 5 display the effect of chirality on the normalized deflection of the cantilever beam and simply supported beam,respectively.During the computation,h=1mm.w(0)andw(1)are the deflection of the midpoint of the beam central axis of the cantilever beam and simply supported beam forCh=0,respectively.These two figures show that chirality can significantly affect the bending behavior of the beam.Ch=0 denotes that the beam is made of normal micropolar elastic material without chirality.ForCh=0,the beam has minimal deflection, compared with the chiral beam.It can be seen that the beam has relatively larger deflection for greater chirality parameter values,indicating that chirality can effectively decrease the bending stiffness of a beam;i.e.,make the beam more flexible.This is because we consider that chirality exists only along thexaxis,indicating that the axial line of the chiral microstructure such as helical fibers is parallel to thexaxis.Such arrangement of helical fibers increases the elastic properties in the axial direction,but decreases the elastic properties in transverse directions,e.g.,the bending stiffness.

Fig.5 Effect of chirality on normalized deflection of simply supported beam

Fig.6 Variation of normalized rotation angle of cantilever beam along beam length for different chirality parameters

Fig.7 Variation of normalized rotation angle of simply supported beam along beam length for different chirality parameters

Fig.8 Variation of natural frequency of cantilever beam with length to-thickness ratio for different chirality parameters

Figures 6 and 7 show the variation of the normalized rotation angle of the cantilever beam and the simply supported beam with the length-to-thickness ratio for different chirality parameters.Here,θ(0)is the rotation angle at the midpoint of the beam central axis of the cantilever beam forCh=0,andθ(1)is the rotation angle at the end of the simply supported beam forCh=0.The curves in Figs.6 and 7 almost overlap eachother,respectively,illustrating that chirality has no obvious effects on the rotation angle of the beam.This is perhaps because chirality inxaxis can only induce axial tension–twisting coupling deformation,having no obvious effects on transverse shear deformation.Similar to the case of an elastic beam without chirality,the increase of the rotation angle of the chiral beam is accompanied by an increase of the deflection along the beam length for the cantilever beam.In the case of the simply supported beam,the rotation angle reaches its peak at the two ends,becoming zero at the midpoint of the beam length because there is no curvature of the beam.

In Figs.8 and 9,the variation of the natural frequency of the cantilever beam and simply supported beam along the length is plotted.As shown in Figs.8 and 9,the natural frequency of the cantilever beam and simply supported beam decreases sharply for relatively smaller length-to-thickness ratio at the beginning,then decreases much more slowly for large length-to-thickness ratios.Furthermore,the chirality of the material has distinct effects on the free vibration behavior of the cantilever beam and simply supported beam.The greater the chirality parameter is,the lower the natural frequency of the beam is.This also occurs because chiral microstructure makes materials much more flexible than nonchiral counterparts,resulting in relatively lower natural frequency.

Fig.9 Variation of natural frequency of simply supported beam with length-to-thickness ratio for different chirality parameters

Fig.10 Variation of microrotation angle of cantilever beam along length for different chirality parameters

Fig.11 Variation of microrotation angle of simply supported beam along length for different chirality parameters

Twisting–stretch coupling deformation is the main difference between chiral materials and nonchiral counterparts.Figures 10 and 11 display the twisting deformation during the bending of a cantilever beam and simply supported beam for different chirality parameters.Here,φ(0)is the microrotation angle at the midpoint of the beam central axis of the cantilever beam forCh=0.2,andφ(1)is the microrotation angle at the end of the simply supported beam forCh=0.2.

Fig.12 Distribution of normalized microrotation angle of cantilever beam elements in cross-section along z direction

The variation of the microrotation angle with the length-tothickness ratio is very similar to that of the rotation angle in Figs.6 and 7.However,in contrast to the rotation angle curves in Figs.6 and 7,chirality has distinct effects on the variation of the microrotation,which depends strongly on the curvature of the beam.In the computation,we plotted the variation of the microrotation angle of beam elements at the top surface,i.e.,z=h/2,along the beam length.For the bended beam,the beam elements at the upper part(i.e.,z＞0)undergo tensile deformation,coupled with twisting induced by the chirality.Here,we may as well assume that positive values of the chirality parameterChcorrespond to right-handed twist,while negative values ofChcorrespond to left-handed twist.We consider herein the case whereChonly takes positive values.Thus,at the bottom of the beam cross-section withz＜0,beam elements undergo compressive strain and twist in left-handed direction.

To further study the effects of chirality on the bending behavior of a beam,we present the variation of the normalized microrotation angle of the cantilever beam elements in cross-section alongzdirection in Fig.12.Here,the crosssection of the cantilever beam atx=h/4 is considered,andφ(2)is the microrotation angle of the structural element at the top surface of this cross-section forCh=0.2.A long the thickness direction,the microrotation angle of beam elements varies linearly due to the linear distribution of tensile strain along the thickness direction. The chirality also affects significantly the microrotation angle distribution.For material with small value of chirality parameter,the microrotation angle usually has relatively smaller values.Forz=0,the beam elements at the center line undergo zero strain and thus have no microrotation.For other cross-sections of the cantilever beam and the simply supported beam,the variation of the microrotation angle under different chirality parameters is expected to be similar.It should be noted that the beam under goeszero twist due to the reverse directions of the microrotation of structural elements in the upper and lower parts of the beam.

As mentioned above,we mainly consider the case in which the central line of the chiral microstructure,such as helical fibers,is parallel toxaxis;i.e.,chirality only exists inxaxis.This work could easily be extended to other cases,e.g.,where the central line of the chiral microstructure has a tilt angle withxaxis ino−xzplane and chirality exists inyorzaxis direction.

5 Conclusions

Using the variational method,we derived the governing equations and boundary conditions for a Timoshenko beam of chiral material based on noncentrosymmetric micropolar elasticity theory and Hamilton’s principle.The proposed Timoshenko beam model accounts for microstructure chirality and the effects of transverse shear deformation,rotary inertia,and microrotation that be come significant when dealing with beams with insufficiently large aspect ratio.Linear static bending and free vibration problems of cantilever and simply supported beams were solved approximately using the Galerkin method,and corresponding numerical results are presented.These results show that chirality can significantly affect the mechanical behavior of a chiral beam.Chirality can make a beam more flexible compared with that without chirality,resulting in relatively larger defection and lower natural frequency.Chirality also induces distinct twisting deformation of beam elements during the bending process.These results will be useful for characterizing mechanical properties of chiral materials such as DNA,flagellar filaments,and chromatin fibers and for designing hierarchically structured chiral materials,and could also be extended to investigate the tensile behavior of single molecules of DNA[28].

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

Appendix

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

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

1.Wang,J.S.,Wang,G.,Feng,X.Q.,et al.:Hierarchical chirality transfer in the grow th of Towel Gourd tendrils.Sci.Rep.3,3102(2013)

2.Wang,J.S.,Cui,Y.H.,Shimada,T.,et al.:Unusual winding of helices under tension.Appl.Phys.Lett.105,043702(2014)

3.Chen,Z.,Majidi,C.,Srolovitz,D.,et al.:Tunable helical ribbons.Appl.Phys.Lett.98,011906(2011)

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)

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)

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)

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)

8.Coombs,D.,Huber,G.,Kessler,J.O.,et al.:Periodic chirality transformations propagating on bacterial flagella.Phys.Rev.Lett.89,118102(2002)

9.Wang,X.L.,Sun,Q.P.:Mechanical model of the bistable bacterial flagellar filament.Acta Mech.Solida Sin.24,1–16(2011)

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)

11.Robbie,K.,Breet,M.J.,Lakhtakia,A.:Chiral sculpted thin films.Nature 384,616–616(1996)

12.Rong,Q.Q.,Cui,Y.H.,Shimada,T.,et al.:Self-shaping of bioinspired chiral composites.Acta Mech.Sin.30,533–539(2014)

13.Lakes,R.S.,Benedict,R.L.:Noncentrosymmetry in micropolar elasticity.Int.J.Eng.Sci.20,1161–1167(1982)

14.Sharma,P.:Size-dependent elastic fields of embedded inclusions in isotropic chiral solids.Int.J.Solids Struct.41,6317–6333(2004)

15.Lakes,R.:Elastic and viscoelastic behavior of chiral materials.Int.J.Mech.Sci.43,1579–1589(2001)

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)

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)

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)

19.Spadoni,A.,Ruzzene,M.:Elasto-satic micropolar behavior of a chiral auxetic lattice.J.Mech.Phys.Solids 60,156–171(2012)

20.Papanicolopulos,S.A.:Chirality in isotropic linear gradient elasticity.Int.J.Solids Struct.48,745–752(2011)

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)

22.Le¸san,D.:Chiral effects in uniformly loaded rods.J.Mech.Phys.Solids 58,1272–1285(2010)

23.Le¸san,D.:Fundamental solutions for chiral solids in gradient elasticity.Mech.Res.Commun.61,47–52(2014)

24.Healey,T.J.:Materialsymmetry and chirality innonlinearly elastic rods.Math.Mech.Solids 7,405–420(2002)

25.Smith,M.L.,Healey,T.J.:Predicting the onset of DNA super coiling using a non-linear hemitropic elastic rod.Int.J.Nonlinear Mech.43,1020–1028(2008)

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)

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)

28.Zhao,Y.P.:Modern Continuum Mechanics.Science Press,Beijing(2016).(in Chinese)