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Size dependent and tunable elastic properties of hierarchical honeycombs with regular square and equilateral triangular cells
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Size dependent and tunable elastic properties of hierarchical honeycombs with regular square and equilateral triangular cells

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Simple closed-form results for all the independent elastic constants of macro-, micro- and nanosized first-order regular honeycombs …

Simple closed-form results for all the independent elastic constants of macro-, micro- and nanosized first-order regular honeycombs
with square and equilateral triangular cells and for the self-similar hierarchical honeycombs were obtained. It is found that, if the cell wall
thickness of the first-order honeycomb is at the micrometer scale, the elastic properties of a hierarchical honeycomb are size dependent,
owing to the strain gradient effects. Further, if the first-order cell wall thickness is at the nanometer scale, the elastic properties of a hierarchical
honeycomb are not only size dependent owing to the effects of surface elasticity and initial stresses, but are also tunable. In addition,
the cell size and volume of hierarchical nanostructured cellular materials can be varied, and hierarchical nanostructured cellular
materials could also possibly be controlled to collapse.

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  • 1. (This is a sample cover image for this issue. The actual cover is not yet available at this time.)This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
  • 2. Authors personal copy Available online at www.sciencedirect.com Acta Materialia 60 (2012) 4927–4939 www.elsevier.com/locate/actamat Size-dependent and tunable elastic properties of hierarchical honeycombs with regular square and equilateral triangular cells H.X. Zhu a,⇑, L.B. Yan a, R. Zhang a, X.M. Qiu b a School of Engineering, Cardiff University, Cardiff CF24 3AA, UK b School of Aerospace, Tsinghua University, Beijing 100084, People’s Republic of China Received 17 January 2012; received in revised form 9 May 2012; accepted 10 May 2012Abstract Simple closed-form results for all the independent elastic constants of macro-, micro- and nanosized first-order regular honeycombswith square and equilateral triangular cells and for the self-similar hierarchical honeycombs were obtained. It is found that, if the cell wallthickness of the first-order honeycomb is at the micrometer scale, the elastic properties of a hierarchical honeycomb are size dependent,owing to the strain gradient effects. Further, if the first-order cell wall thickness is at the nanometer scale, the elastic properties of a hier-archical honeycomb are not only size dependent owing to the effects of surface elasticity and initial stresses, but are also tunable. In addi-tion, the cell size and volume of hierarchical nanostructured cellular materials can be varied, and hierarchical nanostructured cellularmaterials could also possibly be controlled to collapse.Crown Copyright Ó 2012 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. All rights reserved.Keywords: Size effect; Honeycombs; Hierarchy; Elastic properties; Tunable properties1. Introduction the mechanical properties [8–14], and that, at the nanometer scale, both surface elasticity [8,15,16] and initial stresses In nature, living things evolve constantly to survive their [17,18] can greatly affect the mechanical properties of struc-changing environment. To support their own weight, to tural elements. Atomistic simulations [19] suggest that, forresist external loads and to enable different types of func- metallic structural elements with a size of a few nanometers,tions, their bodies should be structurally optimized and the strain gradient effect is irrelevant, and surface elasticitymechanically sufficiently strong. As a consequence, natural or surface energy dominates the mechanical properties. Theliving materials are usually made up of hierarchical cellular size-dependent bending rigidities have been obtained forstructures with basic building blocks at the micro- or microplates [11] and nanoplates [18]. The size-dependentnanoscale. transverse shear rigidities of micro- and nanoplates and The elastic properties of honeycombs at the macroscale the size-dependent elastic properties of the first-order hon-have been extensively studied and well documented [1–7]. eycombs with regular hexagonal cells were obtained inHowever, the results obtained for macrohoneycombs may Ref. [8]. Fan et al. [30] and Taylor et al. [31] have studiednot apply to their micro- and nanosized counterparts [8]. the effects of structural hierarchy on the elastic propertiesIt has been generally recognized that, at the micrometer of honeycombs. However, they did not study the size-scale, the strain gradient effect plays an important role in dependent effect or the tunable elastic properties for hierar- chical honeycombs. The aim of this paper is thus to obtain ⇑ Corresponding author. Tel.: +44 29 20874824. closed-form results for the size-dependent and tunable E-mail address: zhuh3@cf.ac.uk (H.X. Zhu). mechanical properties of regular self-similar hierarchical1359-6454/$36.00 Crown Copyright Ó 2012 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. All rights reserved.http://dx.doi.org/10.1016/j.actamat.2012.05.009
  • 3. Authors personal copy4928 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939honeycombs with square and equilateral triangular cells combined axial compression, transverse shear deformationwhose first-order cell wall thickness is at the micron and and plane-strain bending. Only a representative unit cellnanometer scales. shown in Fig. 1b is needed for mechanics analysis. By assuming that the bending stiffness of the cell walls is DB,2. Independent elastic constants of the first-order the shear stiffness is DS and the axial compression stiffnesshoneycombs is DC, it is easy to obtain all five independent elastic con- stants for a perfect regular honeycomb with equilateral tri- The first-order honeycombs are treated as materials angular cells and uniform cell walls.whose size is much larger than the individual cells and The in-plane Young’s modulus of a perfect regular hon-are assumed to have uniform cell walls of length L, width eycomb with equilateral triangular cells can be obtained asb and thickness h. The wall width b is assumed to be much (see Appendix A)larger than the thickness h. The focus of this paper is on 2DS Á D2 Á L2 þ 24DB Á D2 þ 24DB Á DS Á DC C Csmall elastic deformation and the elastic properties. E1 ¼ pffiffiffi 3 pffiffiffi pffiffiffiAlthough the cell wall can be of a metallic, biological or 3L Á DS Á DC þ 12 3L Á DB Á DC þ 4 3L Á DB Á DSpolymeric material, it is always assumed to be isotropic ð2Þand linear elastic, with Young’s modulus ES and Poisson The in-plane Poisson ratio is determined asratio vS, in the analysis that follows. As regular honey-combs with either square cells or equilateral triangular cells L2 Á DS Á DC þ 12DB Á DC À 12DB Á DS m12 ¼ ð3Þhave three orthogonal planes of symmetry, the maximum 3L2 Á DS Á DC þ 36DB Á DC þ 12DB Á DSpossible number of independent elastic constants is nine and the out-of-plane Young’s modulus is given by[4,8,20]. E3 ¼ f1 ES q ð4Þ2.1. Honeycombs with equilateral triangular cells where ES is the Young’s modulus of the solid material and f1 is a coefficient to be specified in the sections that follow, Fig. 1 shows a regular honeycomb with equilateral trian- the value of which depends on the type of honeycomb cellgular cells. It is easy to show that its in-plane mechanical and the size scale of the cell wall thickness.properties are isotropic. To fully determine the relationship The out-of-plane shear modulus can be obtained asbetween the applied state of effective stress and theresponding state of effective strain, only five independent G31 ¼ f2 GS q ð5Þelastic constants – E1, m12, E3, v31 and G31 – need be where GS is the shear modulus of the solid material and f2obtained. The detailed derivation of these independent is a coefficient to be specified.elastic constants is given in Appendix A. The out-of-plane Poisson ratio obviously remains the The relative density of the first-order honeycomb with same as that of the solid material at both the macro- andequilateral triangular cells is given by microscales (in this case, the strain gradient effect is pffiffiffi absent), thus 2 3hq¼ ð1Þ m31 ¼ mS ð6Þ L When the honeycomb is compressed by a uniform stress To simplify the analysis for honeycomb material within the x direction, there is no junction rotation because of cells at the nanometer scale, the surface is assumed to bethe symmetry, and the inclined cell walls undergo isotropic and to have the same Poisson ratio mS as that ofFig. 1. (a) A regular honeycomb with equilateral triangular cells of uniform cell wall thickness; (b) the loads applied to the cell walls of a unit cell when thehoneycomb is compressed in the x direction.
  • 4. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4929the bulk material. The out-of-plane Poisson ratio m31 of a The out-of-plane shear modulus can be obtained ashoneycomb with nanosized equilateral triangular cells is G31 ¼ f4 GS q ð12Þthus mS. Therefore, in the sections that follow, we only needto obtain the results of four independent elastic constants – where f4 is a coefficient to be determined.E1, m12, E3 and G13 – for perfect regular honeycombs with At the nanoscale, as the Poisson ratio of the surface isequilateral triangular cells at different size scales. assumed to be mS, the out-of-plane Poisson ratio remains the same as that of the solid material for different-order2.2. Honeycombs with square cells hierarchical honeycombs with regular square cells from macro- and micro- down to the nanoscale, The xy plane of perfectly regular honeycombs with m31 ¼ mS ð13Þsquare cells, as shown in Fig. 2a, is not isotropic. It is easyto verify this, and only six independent elastic constants In the sections that follow, we only need to obtain fourhave to be determined: E1, m12, G12, E3, v31 and G31. Their independent elastic constants – E1, E3, G12 and G31 – fordetailed derivation is given in Appendix B. the first-order honeycombs with regular square cells whose The relative density of the first-order honeycomb with cell wall thickness is uniform at different size scales.square cells and uniform cell wall thickness is given by 3. Elastic constants of macrosized first-order honeycombs 2hq¼ ð7Þ L For macrosized first-order honeycombs, the bending,The in-plane Young’s modulus is obtained as transverse shear and axial stretching/compression rigidities rx DC of the cell walls are given byE1 ¼ ¼ ð8Þ ex L ES bh3 DB ¼ Á ð14aÞThe in-plane Poisson ratio is determined as 1 À m2 12 S GS bh 1 DS ¼ ð14bÞm12 ¼ vS h=L ¼ mS q ð9Þ 1:2 2 DC ¼ ES bh ð14cÞwhich is the same as that given by Wang and McDowell[21] and applies to the first-order honeycomb with regular In Eq. (14b) a shear coefficient of 1.2 [22] is introduced forsquare cells at different size scales. The in-plane shear mod- the rectangular cross-section of the cell walls.ulus is obtained as 3.1. Honeycombs with regular equilateral triangular cells 6DB DSG12 ¼ ð10Þ DS L3 þ 12DB L Substituting Eqs. (1) and (14a–c) into Eqs. (2)–(5), theand the out-of-plane Young’s modulus is given as dimensionless results of all the four elastic constants canE3 ¼ f3 ES q ð11Þ be obtained as: q 2 q 2where f3 is a coefficient to be specified in the sections that E1 1 þ 5ð1ÀmS Þ þ 12ð1Àm2 Þfollow, the value of which depends on the size scale of E1 ¼ 1 ¼ S ð15Þ 3 ES q 1 þ 5ð1Àm Þ þ q2 2 q2 36ð1ÀmS Þthe cell wall thickness. SFig. 2. (a) A regular honeycomb with square cells of uniform cell wall thickness; (b) the loads applied to the cell walls of a unit cell when the honeycombundergoes in-plane shear deformation.
  • 5. Authors personal copy4930 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 1 3 þ 15ð1ÀmS Þ Á q2 À 36ð1Àm2 Þ Á q2 1 1m12 ¼ 1 1 S ð16Þ 1 þ 5ð1ÀmS Þ Á q2 þ 36ð1Àm2 Þ Á q2 S E3E3 ¼ ¼1 ð17Þ Es q G31 1G31 ¼ ¼ ð18Þ Gs q 2where the in-plane Young’s modulus is normalizedby 1 ES q; the out-of-plane Young’s modulus is normalized 3by ESq; the out-of-plane shear modulus is normalized byGSq; and the relative density q is given by Eq. (1). Theout-of-plane Poisson ratio is the same as that given inEq. (6). Wang and McDowell [21] obtained the in-plane(i.e. the xy plane) elastic properties for macrosized first-or- Fig. 4. Relationship between the in-plane Poisson ratio m12 and theder honeycombs with equilateral triangular cells. They relative density q of regular honeycombs with macrosized equilateral triangular cells.took cell wall bending as the sole deformation mechanism,and their result is slightly different from Eq. (15). For the first-order honeycombs with macrosized equilat-eral triangular cells, the relationship between the in-plane G12 1 G12 ¼ 1 ¼ 3 ð20Þdimensionless Young’s modulus E1 and the relative density 8ð1ÀvS Þ q3 GS 1 þ 5ð1ÀmS Þ q2q is shown in Fig. 3, and the relationship between the in- 1plane Poisson ratio m12 and the relative density q is pre- v12 ¼ vS q ð21Þ 2sented in Fig. 4. As can be seen from Figs. 3 and 4, both G31 1the in-plane dimensionless Young’s modulus E1 and the G31 ¼ ¼ ð22Þ Gs q 2Poisson ratio m12 vary so little with the honeycomb relativedensity q over the range from 0 to 0.35 that they can be The dimensionless in-plane shear modulus G12 , whichtreated as approximate constants: E1 % 1 and v12 % 0.33. has been normalized by 8ð1ÀvS Þ GS q3 , is plotted against the 1These are the same as the approximate results given by honeycomb relative density q, as shown in Fig. 5. TheWang and McDowell [21]. greater the honeycomb relative density, the smaller the dimensionless in-plane shear modulus. This is quite differ-3.2. Honeycombs with regular square cells ent from Wang and McDowell’s dimensionless result of 1 [21]. The dimensionless out-of-plane Young’s modulus Using Eqs. (8)–(12) and (14a)–(14c), the dimensionless (i.e. E3 ¼ E3 = ðqES ÞÞ is the same as that for equilateral tri-results of all the independent elastic constants of the first- angular honeycombs. Apart from G12 and v12 ¼ 1 vS q, all 2order honeycombs with regular square cells can be easily other dimensionless elastic constants (i.e.obtained as: E1 ¼ E2 ¼ E3 ¼ 1; G31 ¼ 1=2 and m31 = mS) are independent E1 of the honeycomb relative density. Note that E1 and E3 areE1 ¼ 1 ¼1 ð19Þ normalized by different factors. 2 qESFig. 3. Relationship between the in-plane dimensionless Young’s modulus Fig. 5. Relationship between the in-plane dimensionless shear modulusE1 and the relative density q of regular honeycombs with macrosized G12 and the relative density q of regular honeycombs with macrosizedequilateral triangular cells. square cells.
  • 6. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 49314. Elastic constants of microsized first-order honeycombs assumed to be absent. Therefore the out-of-plane dimen- sionless shear modulus is the same as that given by Eq. For uniform plates with thickness at the micrometer (18) for macrosized first-order honeycombs.scale, the size-dependent bending stiffness is given by Yang For the first-order honeycombs with regular microsizedet al. [12] as equilateral triangular cells and different values of l/h, the dimensionless in-plane Young’s modulus E1 , given by Eq. ES bh3 2DB ¼ Á Á ½1 þ 6ð1 À mS Þðl=hÞ Š ð23Þ (25), and the in-plane Poisson ratio v12, given by Eq. 1 À m2 12 S (26), are plotted against the relative density in Figs. 6and the transverse shear rigidity is given by Zhu [8] as and 7, respectively. The Poisson ratio, vS, of the solid mate- 2 2 rial is chosen as 0.3 in the calculation of the figures. As can GS bh ½1 þ 6ð1 À mS Þðl=hÞ Š be seen, the thinner the cell walls (i.e. the larger the value ofDS ¼ Á ð24Þ 1:2 1 þ 2:5ð1 þ mS Þðl=hÞ2 l/h), the larger the dimensionless in-plane Young’s modulusIn Eqs. (23) and (24), h is the thickness of the cell walls and E1 and the smaller the in-plane Poisson ratio v12.l is the material intrinsic length for the strain gradient ef- Eq. (1) can be rewritten as pffiffiffifect. If the first-order honeycomb is made of a metallic l 2 3l l qlmaterial, l is at the submicron to micron scale. The axial ¼ or ¼ pffiffiffi ð27Þ h qL L 2 3hstretching/compression stiffness of the cell walls with a uni-form thickness at the micrometer scale is the same as that where L is the length of the cell walls, which can be definedgiven in Eq. (14c) because the strain gradient effect is ab- as the size of the cells. Figs. 6 and 7 demonstrate that, forsent. In this section, we consider the elastic properties of the first-order honeycombs with equilateral triangular cellsthe first-order honeycombs made of uniform cell walls at the micrometer size scale and with a fixed relative densitywhose thickness h is at the micrometer or submicron scale. q, the smaller the cell size, the larger the dimensionless in- plane Young’s modulus and the smaller the in-plane Pois-4.1. Honeycombs with regular equilateral triangular cells son ratio. Substituting DB, DS and DC given in Eqs. (23), (24) and 4.2. Honeycombs with regular square cells(14c) into Eqs. (2) and (3), the in-plane Young’s modulus(which is normalized by 1 ES qÞ of the first-order honey- For the first-order honeycombs with microsized regular 3combs with microsized equilateral triangular cells can be square cells, the strain gradient effect is clearly absent whenobtained as it is uniaxially deformed in either the x, y or z direction. Therefore, its dimensionless elastic constants: E1 ; E3 ; G31 E1E1 ¼ 1 and the Poisson ratios: v12 and m31 are exactly the same Á ES q 3 as those of the first-order honeycombs with macrosized 2 2 1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 12ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2 1þ2:5ð1þvS Þ Þðl=hÞ 1 regular square cells. S S ¼ S Substituting Eqs. (23) and (24) into Eq. (10), the in- 1þ2:5ð1þvS Þ2 2 1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2 1 plane shear modulus can be obtained as S S Þðl=hÞ S ð25Þ G12 1 þ 6ð1 À vS Þðl=hÞ2 G12 ¼ ¼ ð28Þand the in-plane Poisson ratio is derived as 1 q3 GS 1 þ 3 Á 1þ2:5ð1þvS Þðl=hÞ2 q2 8ð1ÀvS Þ 5ð1ÀvS Þ 1þ6ð1Àv Þðl=hÞ2 S 1þ2:5ð1þvS Þ2 ðl=hÞ2 1 þ 15ð1Àv Þ½1þ6ð1Àv Þðl=hÞ2 Š q2 À 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2 1 1m12 ¼ 3 S S S which is normalized by G q3 . 8ð1ÀvS Þ S 2 2 1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2 1þ2:5ð1þvS Þ Þðl=hÞ 1 S S S ð26Þwhen a microsized first-order honeycomb undergoes uniax-ial compression or tension in the z direction (i.e. the out-of-plane direction), the strain gradient effect is absent. Thedimensionless out-of-plane Young’s modulus E3 in the zdirection is the same as that of the macrosized first-orderhoneycomb, and the out-of-plane Poisson ratio m31 is thesame as vs. As the size of a honeycomb material is much larger thanits cells, when the honeycomb material with microsized cellsundergoes a globally uniform out-of-plane shear deforma-tion in the zx or zy plane (see Fig. 1), the shear strain in each Fig. 6. Relationship between the in-plane dimensionless Young’s moduluscell wall is always in the cell wall plane and uniform in E1 and the relative density q of regular honeycombs with microsizedamplitude, and any strain gradient effect can thus be equilateral triangular cells.
  • 7. Authors personal copy4932 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 and surface elasticity or surface energy dominates the influ- ence on the mechanical properties. For uniform plates of thickness at the nanoscale, the combined effects of surface elasticity and initial stress on the bending stiffness is obtained as [18] ! Es bh3 6S 2s0 Db ¼ 1þ À vs ð1 þ vs Þ 1 À v2 12 s Es h Es h ! Es bh3 6ln vs ð1 þ vs Þ ¼ 1þ þ e0s ð29Þ 1 À v2 12 s h 1 À ms The transverse shear stiffness is derived as [8] h i2Fig. 7. Relationship between the in-plane Poisson ratio m12 and the GS bh 1 þ 6ln þ e0s mS ð1þmS Þ h 1ÀmSrelative density q of regular honeycombs with microsized equilateral DS ¼ Á À ln Á 2 ð30Þ 1:2 1þ 10ln þ 30triangular cells. h h and the axial stretching/compression stiffness is given by [15] Dc ¼ Es bhð1 þ 2ln =hÞ ð31Þ In Eqs. (29)–(31), S is the surface elasticity modulus, ln = S/Es is the material intrinsic length at the nanoscale and s0 is the initial surface stress, the amplitude of which can be varied by adjusting the applied electric potential [23–25]. When the initial surface stress s0 is present, the ini- tial elastic residual strains of the bulk material are 2s e0s ¼ À Es 0 ð1 À vs Þ in both the cell wall length and width h directions, and the initial elastic residual strain in the cell wall thickness direction is eh ¼ 4vsss0 ¼ À 1Àvs e0s . Although 0 E h 2vs the amplitude of e0scan be varied, the range of the recover- able elastic residual strain is limited by the yield strain of the bulk material. For single crystal nanomaterials or poly-Fig. 8. Relationship between the in-plane dimensionless shear modulus meric materials, the yield strain can be 10% or even larger.G12 and the relative density q of regular honeycombs with microsizedsquare cells. Atomistic simulation [26] has shown that, if the diameter of a gold wire is sufficiently small, it can automatically under- go plastic deformation solely owing to the presence of the The relationship between the in-plane dimensionless initial surface stresses. It is well known that the yieldShear modulus G12 and the relative density q of the first- strength, ry, of some conductive polymer materials ororder honeycombs with microsized regular square cells is nanosized metallic materials can be 0.1E (E is the Young’splotted in Fig. 8. The thinner the cell walls, the larger the modulus) or larger [27]. Biener et al. [28] have experimen-dimensionless in-plane shear modulus. As the size of the tally found that, for nanoporous Au material, by control-cells can be defined as L, for microsized honeycombs with ling the chemical energy, the adsorbate-induced surfacea fixed relative density, the smaller the cell size, the larger stress s0 can reach 17–26 N mÀ1. If the diameter of the lig-the dimensionless in-plane shear modulus. aments is 5 nm, rx would be 20 GPa. As the bulk material 0 It is easy to verify that, when the strain gradient effect is discussed in this paper can be metallic, polymeric or biolog-absent (i.e. h/l tends to 0), all the elastic constants of the ical, without losing generality, the tunable range offirst-order honeycombs with either microsized equilateral 2s e0s ¼ À Es 0 ð1 À vs Þ is assumed to be from À0.1 to 0.1. In htriangular cells or square cells reduce to those of their mac- other words, the von Mises yield strength of the solid mate-rosized counterparts. rial is assumed to be 0.1ES/(1 À vS). If the actual yield strength of the honeycomb solid material is larger or smal-5. Elastic constants of nanosized first-order honeycombs ler than this, the results obtained for the nanosized first-or- der honeycombs can still be obtained by scaling those At the nanometer scale, both the surface elasticity [15] presented in this paper up or down.and the initial stresses [17,18] can greatly affect the mechan- When the effect of the initial surface stress s0 is absent,ical properties of structural elements. Atomistic simula- the initial dimensions of the nanosized cell walls aretions [19] show that, for metallic structural elements with assumed to be length L0, width b0 and thickness h0 for boththe size of a few nanometers, no strain gradient effect exists, regular triangular and square honeycombs. When the effect
  • 8. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4933of the initial surface stress is present, the dimensions of the first-order honeycombs with microsized cells, for nanosizedcell walls become first-order honeycombs with a fixed relative density q0, the smaller the cell size L0, the larger the normalized in-planeL ¼ L0 ð1 þ e0s Þ ð32aÞ Young’s modulus. If the surface elasticity modulus is nega-b ¼ b0 ð1 þ e0s Þ ð32bÞ tive (i.e. ln = S/ES < 0), the effects are reversed. For example,   2vs when ln/h is À0.2, the dimensionless Young’s modulush ¼ h0 ð1 þ eh Þ ¼ h0 1 À 0 e0s ð32cÞ 1 À vs becomes 0.6, which is smaller than that of the macrosized honeycombs (i.e. ln/h = 0). However, ln/h must be largerand thus the relative density of both the first-order square than À1/2, otherwise the elastic modulus E1 would be nega-and equilateral triangular honeycombs becomes tive and the honeycomb structure would deform automati-   cally [8] to a stable configuration. On the other hand, when hL0 2vs the effect of the surface elasticity modulus is absent and theq¼ q0 ¼ 1 À e0s q0 =ð1 þ eos Þ ð33Þ h0 L 1 À vs effect of the initial stress/strain is present, Eq. (34) reduces to 2s h iwhere e0s ¼ À Es 0 ð1 À vs Þ and 1 2 1 mS ð1þmS Þ pffiffiffi h 1þ mS ð1þmS Þ Á q þ 12ð1Àm2 Þ Á 1 þ eos Á 1Àm Á q2 5ð1ÀmS Þ½1þeos Á 1Àm Š S Sq0 ¼ 2 3h0 =L0 ð33aÞ E1 ¼ S h i 1þ h 1 i Á q2 þ 1 2 Á 1 þ eos Á mS ð1þmS Þ Á q2 36ð1Àm Þ 1ÀmSfor equilateral triangular honeycombs and 5ð1ÀmS Þ 1þeos Á mS ð1þmS Þ S 1ÀmSq0 ¼ 2h0 =L0 ð33bÞ 2vS 1 À 1ÀvS eosfor square honeycombs. Á 1 þ eos ð36Þ5.1. Honeycombs with regular equilateral triangular cells Eq. (36) can be approximated well by E1 % ð1 À 2vS eos = Substituting DB, DS and DC and the dimensions of the ð1 À vS ÞÞ=ð1 þ eos Þ, and the error is smaller than 1% ifnanosized cell walls given by Eqs. (29)–(33) into Eq. (2), q0 6 0.35. Thus the in-plane dimensionless Young’s modu-the in-plane dimensionless Young’s modulus of honey- lus of a nanosized first-order regular honeycomb with equi-combs with nanosized equilateral triangular cells can be lateral triangular cells can be varied by adjusting theobtained as amplitude of the initial surface stress (and hence the initial E1 strain eos), and the tunable range depends on the PoissonE1 ¼ 1 3 E S q0 ratio vS of the solid materials. When eos is varied from  à À Á n 2 1þ10ln þ30ðlh Þ ð1þ2ln Þ 2 h i 0.1 to À0.1, the tunable range of the dimensionless in-plane 1 þ 2ln þ 5ð1ÀmS Þ Á h h 1 ih q þ 1 2 Á 1 þ 6ln þ eos Á mS ð1þmS Þ Á q2 h 1þ6ln þeos Á h 12ð1Àm Þ mS ð1þmS Þ 1ÀmS h 1ÀmS S Young’s modulus will be from 0.909 to 1.111 if vS = 0, ¼ h i 0.831–1.206 if vS = 0.3, and 0.727–1.5 if vS = 0.5. mS ð1þmS Þ 1þ10ln þ30ðlh Þ n 2 1þ6ln þeos Á 1 þ 5ð1ÀmS Þ Á h 1 h i q2 þ 1 Á h 1ÀmS Á q2 The in-plane Poisson ratio can be obtained as mS ð1þmS Þ 36ð1Àm2 Þ ½1þ2lhn Š 1þ6ln þeos Á S h 1ÀmS h i 2vS m ð1þm Þ 1 À 1ÀvS eos 10ln ln 2 1þ6ln þeos Á S1Àm S h 1þ h þ30ð h Þ i q2 À 1 2 Á h Á 1 1 þ 15ð1ÀmS Þ Á S Á q2 1 þ eos 3 mS ð1þmS Þ 36ð1ÀmS Þ ð1þ2lhn Þ 6ln 1þ h þeos Á 1Àm ð34Þ S h i m12 ¼ mS ð1þmS Þwhere the in-plane Young’s modulus is normalized by n 2 1þ6ln þeos Á 1þ10ln þ30 lh ð Þ i 2 h 1ÀmS1 Á ES q0 and h and q0 are given by Eqs. (32c) and (33a), 1 þ 5ð1ÀmS Þ Á h 1 h q þ 1 36ð1Àm2 Þ Á Á q23 m ð1þm Þ 1þ6ln þeos Á S1Àm S S ð1þ2lh n Þ hrespectively. S When the effect of the initial surface stress s0 is absent ð37Þ(i.e. the initial elastic residual strain e0S of the bulk material When the effect of the initial surface stress s0 is absent, Eq.is 0) and the effect of the surface elasticity is present, h = h0 (37) reduces toand q = q0, and Eq. (34) reduces to 2 1þ10ln þ30ðlh Þ n ½1þ6lhn Š 2  à 1 1 þ 15ð1ÀmS Þ h Á q2 À 36ð1Àm2 Þ Á 1 Áq À Á ð1þ2lhn Þ 1þ10ln þ30ðlh Þ n 2 À Á 3 ½ 1þ6lnŠ ð1þ2lhn Þ 1 þ 2ln þ 5ð1ÀmS Þ Á 1 h Á q2 þ 12ð1Àm2 Þ Á 1 þ 6ln Á q2 1 m12 ¼ h S ð38Þ h ð1þ6lhn Þ h n 2E1 ¼ 2 S ð35Þ 1 1þ10ln þ30ðlh Þ h 1 ½1þ6ln Š 1 1 þ 5ð1ÀmS Þ Á 1þ10ln þ30ðlh Þ h n 1 Á q2 þ 36ð1Àm2 Þ Á ð1þ6lhn Þ Á q2 1þ Á q2 þ 36ð1Àm2 Þ Á 1þ2hn Á q2 ð1þ6lhn Þ S ð1þ2lhn Þ 5ð1ÀmS Þ ½1þ h Š 6ln S ð lh Þ Fig. 9 shows the effect of the surface elasticity on the rela- Fig. 10 shows that the effect of the surface elasticity ontionship between the normalized in-plane (i.e. xy plane) the in-plane Poisson ratio (given by Eq. (38)) of nanosizedYoung’s modulus (given by Eq. (35)) and the relative density first-order honeycombs depends upon the value of ln/h.q0 of the nanosized first-order honeycombs with regular The thinner the cell walls, the smaller the Poisson ratio.equilateral triangular cells. If the surface elasticity modulus On the other hand, the effect of the initial stress or strainis positive (i.e. ln = S/ES > 0), the thinner the cell walls, the on the Poisson ratio is so small that the result is notlarger the normalized Young’s modulus. Similarly to the presented.
  • 9. Authors personal copy4934 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 Young’s modulus and shear modulus are proportional to (1 + 2ln/h). On the other hand, when the effect of surface elasticity is absent (or fixed) and the effect of the initial stress/strain is present, both the out-of-plane dimensionless Young’s modulus and shear modulus depend not only on the amplitude of the initial strain eos, but also on the Pois- son ratio vS of the solid material. 5.2. Honeycombs with regular square cells Substituting Eqs. (29)–(33) into Eq. (8), the in-plane dimensionless Young’s modulus of the first-order honey- combs with nanosized square cells can be obtained asFig. 9. Relationship between the in-plane dimensionless Young’s modulus    E1 1 2ln 2vSE1 and the relative density q of regular honeycombs with nanosized E1 ¼ ¼ 1þ 1À eos =ð1 þ eos Þ ð41Þequilateral triangular cells. Es q0 2 h 1 À vS which is normalized by ESq0. The in-plane dimensionless shear modulus of nanosized 1 first-order honeycombs, which is normalized by 8ð1ÀvS Þ q3 GS , 0 can be obtained as !3 G12 1 þ 6ln þ e0S mS ð1þmS Þ h ð1ÀmS Þ 2vS 1 À 1ÀvS eos G12 ¼ 1 ¼ n 2 Á q3 G 8ð1ÀvS Þ 0 S 3 1þ10ln þ30ðlh Þ 1 þ eos 1 þ 5ð1ÀvS Þ Á 6lnh mS ð1þmS Þ q2 1þ h þe0S ð1ÀmS Þ ð42Þ When the effect of the initial surface stress is absent (i.e. e0S is 0) and the effect of the surface elasticity is present, Eq. (42) reduces to G12 1 þ 6ln h G12 ¼ 1 ¼ 2 ð43ÞFig. 10. Size-dependent effect on the relationship between the in-plane G q3 8ð1ÀvS Þ S 0 3 1þ10ln þ30ðlh Þ h nPoisson ratio m12 and the relative density q of regular nanosized equilateral 1þ 5ð1ÀmS Þ Á 1þ6ln q2 htriangular honeycombs when the effect of the cell wall initial elasticresidual strain is absent. The relationship between the dimensionless in-plane shear modulus G12 and the relative density q of the first-order From the stretching stiffness of nanoplate given by Eq. honeycombs with nanosized square cells is plotted in(31), the normalized out-of-plane Young’s modulus of Fig. 11. If the surface elasticity modulus is positive (i.e.the first-order honeycombs with nanosized equilateral tri- ln = S/ES > 0), the thinner the cell walls, the larger theangular cells can be easily obtained as dimensionless shear modulus. If the relative density q of     nanosized honeycombs is fixed, the smaller the cell wall E3 2ln 2vSE3 ¼ ¼ 1þ Á 1À eos =ð1 þ eos Þ ð39Þ E s q0 h 1 À vSThe normalized out-of-plane shear modulus can be easilyderived as    G31 1 2ln 2vSG31 ¼ ¼ 1þ 1À eos =ð1 þ eos Þ ð40Þ Gs q0 2 h 1 À vSFor the nanosized first-order honeycombs with a fixed rel-ative density, the thinner the cell walls, the larger will be thenormalized Young’s modulus and the out-of-plane shearmodulus if ln = S/ES is positive. If the surface elasticitymodulus is negative, the trend of the effects is reversed.As the Poisson ratio of the surface is assumed to be thesame as that of the bulk material, the out-of-plane Poisson Fig. 11. Size-dependent effect on the relationship between the in-planeratio of a nanohoneycomb, m13, is thus equal to vS. When dimensionless shear modulus G12 and the relative density q of regularthe effect of the surface elasticity is present and the effect honeycombs with nanosized square cells when the effect of the cell wallof the initial stress/strain is absent, both the out-of-plane initial elastic residual strain is absent.
  • 10. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4935 treated as materials whose size is much larger than the indi- vidual cells at the same hierarchy level. The relative density of the nth-order self-similar hierarchical honeycombs with either equilateral triangular cells or square cells can be eas- ily obtained as   2vS qn ¼ 1 À eos ðq0 Þn =ð1 þ eos Þ ð45Þ 1 À vS where q0 is given by Eq. (33a) for the first-order honey- combs with equilateral triangular cells and by Eq. (33b) for the first-order honeycombs with square cells. When the initial strain eos is 0, qn reduces to (q0)n. For the first-order honeycombs with regular equilateralFig. 12. Effect of cell wall initial elastic residual strain on the relationship triangular cells of wall thickness at the macro-, micro- orbetween the in-plane dimensionless shear modulus G12 and the relative nanoscale, the five independent elastic constants aredensity q of regular honeycombs with nanosized square cells when theeffect of the surface elasticity modulus is absent. obtained as given in Sections 3–5, depending upon the size scale of their cell wall thickness. For the nth-order self-sim-length or the cell size L, the larger the dimensionless elastic ilar hierarchical honeycombs with equilateral triangularmodulus. If the surface elasticity modulus is negative (i.e. cells, the five independent dimensionless elastic constantsln = S/ES < 0), the effects are reversed. For example, if can be obtained asln =h ¼ À0:1; G12 becomes 0.4, which is smaller than that q nÀ1 ðE1 Þnfor macrosized honeycombs (i.e. ln/h = 0). Eq. (43) implies ðE1 Þn ¼ 1 % 0 Á ðE1 Þ1 ð46Þthat ln/h must be larger than À1/6, otherwise the in-plane E q 3 S 0 3shear modulus G12 becomes negative and the honeycomb ðv12 Þn % 1=3 ð47Þstructure will automatically deform until the structural sta- ðE3 Þn nÀ1bility is regained [8]. ðE3 Þn ¼ ¼ ðq0 Þ Á ðE3 Þ1 ð48Þ Es q0 When the effect of the surface elasticity modulus is ðG31 Þn q0 nÀ1absent and only the effect of the initial surface stress is pres- ðG31 Þn ¼ ¼ Á ðG31 Þ1 ð49Þent, Eq. (42) reduces to Gs q0 2 !3 and (v31)n = vS, where n is the hierarchy level of the self- G12 1 þ e0S mS ð1þmS Þ 2vS 1 À 1ÀvS eosG12 ¼ 1 ¼ ð1ÀmS Þ similar hierarchical honeycombs. In Eqs. (46), (48) and 3 1 G q3 1 þ 5ð1ÀmS Þ Á 8ð1ÀvS Þ S 0 mS ð1þmS Þ q 2 1 þ eos (49), ðE1 Þ1 ; ðE3 Þ1 and ðG31 Þ1 are the dimensionless Young’s 1þe0S ð1ÀmS Þ moduli in the x and z directions and the dimensionless ð44Þ shear modulus in the xz plane of the first-order honey- Fig. 12 shows the effect of the initial residual elastic comb, respectively. It is easy to check that, when n P 2,strain e0S of the bulk material (or the initial stress) on the the errors of Eqs. (46) and (47) are smaller than 0.2% ifin-plane dimensionless shear modulus of the first-order q0 6 0.35.honeycombs with nanosized perfect regular square cells. For the nth-order self-similar hierarchical honeycombsWhen vS = 0.3 and the initial strain eos is varied from 0.1 with square cells, the six independent dimensionless elasticto À0.1, the dimensionless in-plane shear modulus G12 constants can be obtained ascan change from 0.6 to 1.5. q nÀ1 ðE1 Þn For the first-order nanosized honeycombs with regular ðE1 Þn ¼ 1 ¼ 0 Á ðE1 Þ1 ð50Þsquare cells, it is easy to obtain the same dimensionless qE 2 0 S 2out-of plane Young’s modulus E3 and shear modulus G31 ðE3 Þn ðE3 Þn ¼ ¼ ðq0 ÞnÀ1 Á ðE3 Þ1 ð51Þas those given by Eqs. (39) and (40) for the nanosized first- q0 ESorder honeycombs with regular equilateral triangular cells. ðG12 Þn q 3ðnÀ1Þ It is easy to verify that, for regular nanosized honey- ðG12 Þn ¼ 1 3 % 0 Á ðG12 Þ1 ð52Þ 8ð1ÀvS Þ ðq0 Þ GS 8combs with either square cells or equilateral triangular q nÀ1cells, all the elastic constants will reduce to those of their ðv12 Þn ¼ 0 Á ðv12 Þ1 ð53Þmacrosized counterparts when the effects of both the sur- 2face elasticity and the initial stress/strain are absent. ðG31 Þn q0 nÀ1 ðG31 Þn ¼ ¼ Á ðG31 Þ1 ð54Þ Gs q0 26. Size-dependent and tunable elastic properties ofhierarchical honeycombs and (m31)n = mS. In Eqs. (50)–(54), the elastic constants of the first-order honeycombs with regular square cells of wall The hierarchical honeycombs are assumed to be self- thickness at the macro-, micro- or nanoscale, ðE1 Þ1 ;similar [29]. At all different hierarchy levels, they are ðE3 Þ1 ; ðG12 Þ1 ; ðm12 Þ1 and ðG31 Þ1 , are given in Sections 3–5,
  • 11. Authors personal copy4936 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939depending upon the size scale of their cell wall thickness. cellular material collapse automatically [8]. In practicalTherefore, the elastic constants of the nth-order self-similar applications, one failing part could help protect others.hierarchical honeycombs are functions of those of the first-order honeycombs. It is easy to check that, when n P 2, the 7. Conclusionerror of Eq. (53) is smaller than 0.4% if q0 6 0.35. Fan et al.[30] and Taylor et al. [31] studied the effects of structural This paper presents the detailed derivation and closed-hierarchy on the elastic properties of honeycombs. How- form results of all the independent elastic constants ofever, they did not study the size-dependent effect or the tun- self-similar hierarchical honeycombs with either regularable elastic properties of hierarchical honeycombs. Zhu [8] square or equilateral triangular cells. The results imply thatstudied the size-dependent and tunable mechanical proper- many interesting geometrical, mechanical and physicalties for single-order regular hexagonal honeycombs, but properties and functions that do not exist in single-ordernot for hierarchical honeycombs. macrosized cellular materials become possible in their hier- For regular self-similar hierarchical honeycombs with archical nanostructured counterparts. If the cell wall thick-the thickness of their first-order cell walls at the nanometer ness of the first-order honeycomb is at the micrometerscale, the dimensionless elastic properties ðE1 Þn ; ðE3 Þn ; scale, the elastic properties of a hierarchical honeycomb   are size dependent owing to the strain gradient effects. If 2vSðG12 Þn and ðG31 Þn contain a common factor 1 À 1ÀvS eos = the cell wall thickness of the first-order honeycomb is atð1 þ eos Þ and can thus be varied over a large range by the nanometer scale, in addition to the size dependenceadjusting the amplitude of the initial strain eos. When the owing to the effects of surface elasticity, the elastic proper-initial strain is absent, the initial cell diameter, area and ties of a hierarchical honeycomb are tunable because of thevolume of an nth-order self-similar hierarchical honeycomb effects of the initial stresses/strains, the amplitudes of whichare assumed to be (L0)n, (A0)n and (V0)n respectively. When are controllable. More interestingly, if the cell wall/strutthe initial strain eos is present, the dimensionless cell diam- thickness of the first order cellular material is at the nano-eter, area and volume of an nth-order self-similar hierarchi- meter scale, the cell size, surface color, wettability, materialcal honeycomb become strength, stiffness, natural frequency and many other inter- esting physical properties of a hierarchical nanostructuredðLÞn =ðL0 Þn ¼ 1 þ eos ð55Þ cellular material could be varied over a large range. 2ðAÞn =ðA0 Þn ¼ ð1 þ eos Þ ð56Þ Acknowledgementand This work is supported by the EC project PIRSES-GA- 3 2009-247644.ðV Þn =ðV 0 Þn ¼ ð1 þ eos Þ ð57Þ Appendix A. Derivation of the elastic properties of regularIf eos can be controlled to change from À0.1 to 0.1, the honeycombs with equilateral triangular cellsdimensionless cell diameter, area and volume of an nth-or-der self-similar hierarchical honeycomb would vary over A.1. In-plane Young’s modulus and Poisson ratioranges from 0.9 to 1.1, 0.81 to 1.21 and 0.729 to 1.331,respectively. This can be of very important applications. When a first-order regular honeycomb with equilateralFor example, by adjusting the cell size, a hierarchical cellu- triangular cells is uniaxially compressed in the x direction,lar material could possibly be controlled to change its color as shown in Fig. 1a, there is no junction rotation because ofor wettability. Micro- or nanosized porous materials are the symmetry of the structure and the applied load. Only aoften used to select/separate materials with a specific parti- representative unit cell structure, as shown in Fig. 1b, iscle size in medical industry. The results given in Eqs. (55)– thus needed for the analysis. Node A is assumed to have(57) suggest that the size of the selected/separated particles no displacement and rotation. The horizontal load appliedis tunable and controllable. Although it is difficult to alter at B is assumed to be P1 and the horizontal load applied atthe cell size and the mechanical properties, such as the stiff- C is P2. Obviously, there is no load in the y direction at Bness, the buckling force and the natural frequency, of a sin- and C.gle-order macro- or microsized porous material, the results The inclined cell wall AB undergoes bending, transverseobtained in this section suggest that it is possible to realize shear and axial compression. Its deformation in the x direc-those for its hierarchical counterpart with the first-order tion is given bycell wall/strut thickness at the nanometer scale. Dx ¼ Dx þ Dx þ Dx b s c ðA1Þ In addition, Eqs. (42) and (52) imply the possibility that,if ln/h is very close to À1/6 (but ln/h should be slightly lar- where Dx ; Dx and Dx are the deformations in the x direction b s cger than À1/6, the structure would not be stable), control- owing to cell wall bending, transverse shear and cell wallling eos to a negative value (say À0.1) could result in a axial compression, respectively. By assuming that the bend-negative ðG12 Þ, and hence could make the hierarchical ing stiffness of the cell walls is DB, the shear stiffness is DS,
  • 12. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4937and the axial compression stiffness is DC, Eq. (A1) can be The deformation of the inclined cell wall in the y directionrewritten as is  À Á3  P 1 Á sin 60 Á L 2  P 1 Á sin 60 Á L 2 Dy ¼ Dy þ Dy þ Dy b s c ðA10ÞDx ¼ cos 30 þ 3DB DS Eq. (A10) can be rewritten as   P 1 Á cos 60 Á L 2  À Á3 ÀÁ Â cos 30 þ sin 30 P 1 Á sin 60 Á L2  P 1 Á sin 60 Á L 2 DC Dy ¼ sin 30 þ 3DB DS P 1 L3 3P 1 L P 1 L ÀÁ ¼ þ þ ðA2Þ  P 1 Á cos 60 Á L 32DB 8DS 8DC Â sin 30 À 2 cos 30 DCwhere L is the length of the cell walls. pffiffiffi pffiffiffi pffiffiffi From Fig. 1b, the dimension of the representative unit 3P 1 Á L3 3P 1 Á L 3P 1 Á L ¼ þ À ðA11Þcell in the x direction is 96DB 8DS 8DC L  L From Fig. 1b, the dimension of the representative unit celllx ¼ Á cos 60 ¼ ðA3Þ in the y direction is 2 4 pffiffiffiThe compressive strain of the inclined element AB in the x L  3L ly ¼ Á sin 60 ¼ ðA12Þdirection is therefore 2 4 Dx P 1 L2 3P 1 P1 The expansion strain in the y direction can be obtained asex ¼ ¼ þ þ ðA4Þ lx 8DB 2DS 2DC Dy P 1 L2 P1 P1 ey ¼ ¼ þ À ðA13ÞThe stress component of the honeycomb in x direction due ly 24DB 2DS 2DCto force P1 is thus The Poisson ratio is therefore P1 2P 1  rx1 ¼ ¼ pffiffiffi ðA5Þ ey  L2 DS DC þ 12DB DC À 12DB DS v12 ¼   ¼ 2  L Á sin 60 Á b 3L e  3L D D þ 36D D þ 12D D ðA14Þ x S C B C B Swhere the width b of the honeycomb is much larger thanthe cell wall thickness h and is assumed to be 1 for As the honeycomb structure is isotropic, the in-plane shearsimplicity. modulus G12 can be obtained from Ex and v12, and given as For the horizontal element AC, the deformation com- Ex G12 ¼patibility condition requires that the compressive strain 2ð1 þ v12 Þof AC due to force P2 should be the same as that of the pffiffiffi 2 pffiffiffi pffiffiffi 3L Á DS Á DC þ 12 3DB Á DC þ 12 3DB Á DSinclined element AB in the x direction. Thus, ¼   4L3 Á DS þ 48L Á DB P 1 L2 3P 1 P1 ðA15ÞP 2 ¼ D C Á ex ¼ D C Á þ þ ðA6Þ 8DB 2DS 2DCThe stress component of the honeycomb in x direction due A.2. Out-of-plane shear modulus and Young’s modulusto force P2 is derived as P2 2P 2 To derive the out-of-plane shear modulus, the size of therx2 ¼ ¼ pffiffiffi honeycomb material is assumed to be much larger than the L Á sin 60 Á b 3L pffiffiffi honeycomb cells and thus much larger than the cell wall P 1 Á L Á DC 3P 1 Á DC P1 length L. When the honeycomb material is subjected to ¼ pffiffiffi þ þ pffiffiffi ðA7Þ 4 3DB DS Á L 3L an out-of-plane pure shear stress syz (which is in the yzThe total compressive stress in the x direction is thus ob- plane), the shear load T on the cell wall of the representa-tained as tive unit cell is shown in Fig. A1. pffiffiffi pffiffiffi The equilibrium in the y direction requires P 1 LDC 3P 1 DC 3P 1rx ¼ rx1 þ rx2 ¼ pffiffiffi þ þ ðA8Þ Q ¼ T Á sin 600 ðA16Þ 4 3DB DS L L andThe Young’s modulus of the honeycomb is therefore pffiffi pffiffi pffiffiffi P 1 ÁLÁDC 3P 1 ÁDC DS ÁDC ÁL2 þ12DB ÁDC þ12DB ÁDS pffiffi 3 2 rx 4pffiffiDB þ DS ÁL þ L 3P 1 3 4 3DB ÁDS ÁL Q¼ sÁyz L ðA17ÞEx ¼ ¼ P ÁL2 3P ¼ D ÁD ÁL2 þ12D ÁD þ4D ÁD 4 ex 1 þ 1 þ P1 S C B C B S 8DB 2DS 2DC 8DB ÁDS ÁDC where syzpis the effective out-of-plane shear stress acting on ffiffi 2DS Á D2 Á L2 þ 24DB Á D2 þ 24DB Á DS Á DC C C the area 43 L2 of the representative unit cell of the honey- ¼ pffiffiffi 3 pffiffiffi pffiffiffi ðA9Þ 3L Á DS Á DC þ 12 3L Á DB Á DC þ 4 3L Á DB Á DS comb material, as shown in Fig. A1.
  • 13. Authors personal copy4938 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 Fig. A1. The out-of-plane shear load on the area of a representative cell unit. In this part, we derive the out-of-plane shear modulus The dimensionless out-of-plane Young’s modulus of nano-G13 for the first-order honeycombs with nanosized equilat- sized first-order honeycombs with equilateral triangulareral triangular cells. Eqs. (A16) and (A17) lead to cells can be easily obtained as 1 E3 2lnT ¼ syz Á L2 ðA18Þ E3 ¼ ¼1þ ðA23Þ 2 Es q hAs the shear stress or strain is uniform in the cell walls (be- which is normalized by ESq. Similarly, when the surface ef-cause the honeycomb material is assumed to be much lar- fect is absent, it reduces to that of honeycombs with macro-ger than the cell size L), the shear strain ciyz in the or microsized cells.inclined cell wall shown on the right side of Fig. A1 is re-lated to the shear force T by Appendix B. Derivation of the elastic properties of regular 0  honeycombs with square cells i SLcyz ¼ T GS hL þ 2 Á 2ð1 þ mS Þ B.1. In-plane Young’s modulus and Poisson ratio syz L ¼ ðA19Þ 2GS hð1 þ 2ln =hÞ When a regular first-order honeycomb with square cellswhere S is the surface elasticity modulus, the surface Pois- is uniaxially compressed in the x direction, there is no junc-son ratio is assumed to be vS (the same as that of the solid tion rotation because of the symmetry of the structure and Smaterial), 2ð1þvS Þ is the shear modulus of the surface, ln = S/ the applied load. There is no stress or strain in vertical cellES is the material intrinsic length at nanometer scale, and h walls. By taking account the expansion of the horizontalis the thickness of the nanosized cell walls. cell walls, the in-plane Poisson ratio of the first-order hon- The shear strain of the inclined cell wall is related to the eycomb can be obtained as v12 ¼ v21 ¼ L vS ¼ 1 qvS . The in- h 2out-of plane shear strain (i.e. the shear strain in the yz plane dimensionless Young’s modulus can be easily E Dplane) of the first-order honeycomb by obtained as E1 ¼ 1E 1q ¼ ESCh, where q ¼ 2h is the honeycomb L 2 S relative density. ciyz Lsyzcyz ¼ ¼ pffiffiffi ðA20Þ To derive the in-plane shear modulus, a pure shear stress 0 cos 30 3GS hð1 þ 2ln =hÞ sxy is applied to the honeycomb shown in Fig. 2a. Only aAs the relative density of the first-order honeycombs with representative unit cell structure (shown in Fig. 2b) is pffiffiffi h needed for the analysis. The force equilibrium conditionequilateral triangular cells is q ¼ 2 3 L, the out-of-planeshear modulus of the nanosized honeycomb is thus requires syz 1 P ¼ sxy L ðA24ÞG13 ¼ Gyz ¼ =ðGS qÞ ¼ ð1 þ 2ln =hÞ ðA21Þ cyz 2 If junction A shown in Fig. 2b has no rotation, the vertical displacement of the middle point D of the horizontal cellwhich is normalized by Gsq. As can be seen in Eq. (A21), the wall owing to bending and transverse shear should bedimensionless out-of-plane shear modulus of honeycombs À Á3with nanosized cells is size dependent. When the surface ef- P L PLfect is absent (i.e. h ) ln), Eq. (A21) reduces to the dimen- DAD ¼ DB þ DS ¼ 2 þ 2 ðA25Þ 3DB DSsionless out-of-plane shear modulus for a honeycombswith macro- or microsized equilateral triangular cells. Obviously, under pure in-plane shear deformation, the The stretching stiffness of a nanoplate is given by [15] middle points of the horizontal cell walls of the deformed square honeycomb always remain in the same straight line,DC ¼ ES bhð1 þ 2ln =hÞ ðA22Þ which is assumed to be in the horizontal position in order
  • 14. Authors personal copy H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4939to simplify the analysis. Thus, the deformed cell wall AD [9] Toupin RA. Arch. Ration. Mech. Anal. 1962;11:385–414.has a clockwise rotation of hA, which is given as [10] Nix WD, Gao H. J. Mech. Phys. Solid 1998;46:411–25. [11] Gao H, Huang Y, Nix WD, Hutchinson JW. J. Mech. Phys. Solid DAD PL2 P 1999;47:1239–63.hA ¼ ¼ þ ðA26Þ [12] Yang F, Chong ACM, Lam DCC, Tong P. Int. J. Solid Struct. L=2 12DB DS 2002;39:2731–43.Consequently, owing to cell wall bending, transverse shear [13] Lam DCC, Yang F, Chong ACM, Wang J, Tong P. J. Mech. Phys. Solids 2003;51:1477–508.and junction rotation, the displacement of point C in the [14] Zhu HX, Karihaloo BL. Int. J. Plast. 2008;24:991–1007.horizontal direction is obtained as [15] Miller RE, Shenoy VB. Nanotechnology 2000;11:139–47. PL3 PL [16] Wang J, Duan HL, Huang ZP, Karihaloo BL. Proc. Roy. Soc.jDC j ¼ 2hA ðL=2Þ ¼ þ ðA27Þ 2006;A462:1355–63. 12DB DS [17] Zhu HX. Nanotechnology 2008;19:405703.The shear strain c is therefore [18] Zhu HX, Wang J, Karihaloo BL. J. Mech. Mater. Struct. 2009;4:589–604. jDC j sxy L3 2sxy L [19] Sun ZH, Wang XX, Soh AK, Wu HA, Wang Y. Comput. Mater. Sci.cxy ¼ ¼ þ ðA28Þ 2007;40:108–13. L=2 6DB DS [20] Kim B, Christensen RM. Int. J. Mech. Sci. 2000;42:657–76.The in-plane shear modulus of a regular square honey- [21] Wang AJ, McDowell DL. J. Eng. Mater. Technol. 2004;126:137–combs is thus 156. [22] Gere JM, Timoshenko SP. Mechanics of Materials. 3rd ed. Lon- sxy 6DB DSG12 ¼ ¼ ðA29Þ don: Chapman & Hall; 1995. cxy DS L3 þ 12DB L [23] Haiss W, Nichols RJ, Sass JK, Charle KP. J. Electroanal. Chem. 1998;452:199–202. [24] Weissmuller J, Viswanath RN, Kramer D, Zimmer P, Wurschum R,References Gleiter H. Science 2003;300:312–5. [25] Kramer D, Viswanath RN, Weissmuller J. Nano Lett. 2004;4:793–[1] Warren WE, Kraynik AM. Mech. Mater. 1987;6:27–37. 796.[2] Papka SD, Kyriakides S. J. Mech. Phys. Solids 1994;42:1499–532. [26] Diao D, Gall K, Duan ML, Zimmerman JA. Acta Mater.[3] Masters IG, Evans KE. Compos. Struct. 1996;35:403–22. 2006;54:643–53.[4] Gibson LJ, Ashby MF. Cellular Solids: Structures and Properties. [27] Lin ZC, Huang JC. Nanotechnology 2004;15:1509–18. 2nd ed. Cambridge: Cambridge University Press; 1997. [28] Biener J, Wittstock A, Zepeda-Ruiz LA, Biener MM, Zielasek V,[5] Zhu HX, Hobdell JR, Windle AH. J. Mech. Phys. Solids Kramer D, et al. Nat. Mater. 2009;8:47–51. 2001;49:857–70. [29] Lakes R. Nature 1993;361:511–5.[6] Zhu HX, Thorpe SM, Windle AH. Int. J. Solid Struct. [30] Fan HL, Jin FN, Fang DN. Compos. Sci. Technol. 2008;68: 2006;43:1061–78. 3380–7.[7] Wang AJ, McDowell DL. J. Eng. Mater. Technol. 2004;126:137–56. [31] Taylor CM, Smith CW, Miller W, Evans KE. Int. J. Solid Struct.[8] Zhu HX. J. Mech. Phys. Solids 2010;58:696–709. 2011;48:1330–9.