J Mater Sci (2014) 49:5690–5702 DOI 10.1007/s10853-014-8288-y
The elastic and geometrical properties of micro- and nano-structured hierarchical random irregular honeycombs H. X. Zhu • H. C. Zhang • J. F. You • D. Kennedy Z. B. Wang • T. X. Fan • D. Zhang
•
Received: 7 February 2014 / Accepted: 28 April 2014 / Published online: 15 May 2014 Ó Springer Science+Business Media New York 2014
Abstract All the five independent elastic properties/ constants of micro- and nano-structured hierarchical and self-similar random irregular honeycombs with different degrees of cell regularity are obtained by analysis and computer simulation in this paper. Cell wall bending, stretching, and transverse shearing are the main deformation mechanisms of hierarchical honeycombs. The strain gradient effects at the micro-meter scale, and the surface elasticity and initial stress effects at the nano-meter scale are incorporated into all the deformation mechanisms in the analysis and finite element simulations. The results show that the elastic properties of hierarchical random irregular honeycombs strongly depend on the thickness of the firstorder cell walls if it is at the micro-meter scale, and that if the thickness of the first-order cell walls is at the nanometer scale, the elastic properties of hierarchical random irregular honeycombs are not only size-dependent, but are also tunable and controllable over large ranges. In addition, the geometrical properties of nano-structured hierarchical H. X. Zhu (&) H. C. Zhang D. Kennedy School of Engineering, Cardiff University, Cardiff CF24 3AA, UK e-mail:
[email protected] J. F. You National Key Laboratory of Combustion, Flow and ThermoStructure, The 41st Institute of the Forth Academy of CASC, Xi’an 710025, China Z. B. Wang CNM & IJRCNB Centers, Changchun University of Science and Technology, Changchun, China T. X. Fan D. Zhang State Key Laboratory of Composites, Department of Materials Science and Engineering, Shanghai Jiaotong University, Shanghai 200240, China
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random irregular honeycombs are also tunable and controllable.
Introduction The understanding of the mechanical properties of regular and random irregular macro-sized single-order (i.e. conventional) honeycombs [1–9] has well been established and documented. These available results, however, may not apply to their micro- and nano-sized counterparts [10] because of the effects of the surface elasticity [11–14] and the initial stress or strain [15, 16] at the nano-metre scale, and the strain gradient effect at the micro-metre scale [10, 14, 17–28]. Experimental evidence [11, 29–33] has shown that the initial surface stress can be controlled to vary by adjusting the amplitude of an applied electric potential. For example, although the initial surface stress of Au (111) is about 1.13 N/ m, Biener et al. [29] experimentally demonstrated that the adsorbate-induced initial surface stress of nano-porous Au material could be controlled to reach 17–26 N/m by adjusting the chemical energy. Their experimental results [11, 29–33] imply that for a single-order nano-sized honeycomb made of a solid material with Young’s modulus ES & 100 GPa and Poisson’s ratio vS = 0.3, the cell diameter of the honeycomb could be controlled to increase or to reduce about 10 % if the cell wall thickness is about 1.5 nm. This opens the possibility to change and to control the colour, wettability, electric capacitance, or natural frequency for nano-structured honeycomb/cellular materials. Experiments by Weissmuller et al. [31, 32] have demonstrated large recoverable deformation of nano-porous materials by adjusting the amplitude of the initial surface stress via controlling an applied electric potential. Moreover, there is a linear correlation between the surface stress and surface charge in anion adsorption on Au
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(111) [33]. These results suggest that nano-structured honeycomb/cellular materials could be used as the actuating materials or structures in practical applications. It may be difficult to produce sufficiently large honeycomb/cellular materials with all the cells at the nano-metre scale if their structure is of just a single order. This difficulty could be overcome if the honeycomb/cellular materials is nanostructured and hierarchical, i.e. having a number of structural levels. Lakes [34] and Taylor et al. [35] studied the mechanical properties of regular and conventional hierarchical cellular materials, but did not explore the size effects at the micro- or nano-metre scale and the possible tunable properties. The size-dependent and tunable elastic properties have been theoretically studied for first-order nano-sized regular hexagonal honeycombs [10] and for nano-structured hierarchical and self-similar honeycombs with square, triangular [36] and hexagonal [37] cells and open-cell foams with regular BCC cells [37]. Living natural materials, such as bones or plant stems, are usually cellular materials with random irregular cells and multilevel structural hierarchy. Their basic building blocks are usually at the micro- or nano-metre scale and their mechanical properties may be not only size-dependent, but also tunable and/or controllable. The degree of cell irregularity can thus greatly affect the mechanical properties of the nano-structured and hierarchical cellular materials, as has already been observed for their macro-sized single-order counterparts [7, 8]. The objective of this paper is to obtain effects of cell regularity on the size-dependent and tunable mechanical properties of micro- and nano-structured hierarchical and self-similar random irregular honeycombs, and further to briefly discuss the tunable geometrical properties.
Geometrical model and treatment of finite element simulations Geometrical model of micro- and nano-structured hierarchical and self-similar random irregular Voronoi honeycombs The micro- and nano-structured random irregular hierarchical honeycombs are assumed to be self-similar at different hierarchy levels. Zhu et al. [7, 8] developed a computer code to construct representative unit periodic random irregular Voronoi honeycombs, in which the degree of the cell regularity a is defined as d a¼ ; d0
ð1Þ
where d is the minimum distance between the centres of any two neighbouring cells in a random irregular honeycomb and d0 is the distance between two neighbouring
cells in a perfect regular hexagonal honeycomb with the same number of complete cells, and given as [7, 8] sffiffiffiffiffiffiffiffiffiffi 2A0 pffiffiffi ð2Þ d0 ¼ N 3 In Eq. 2, A0 is the area of a representative unit volume element (RVE) model/honeycomb and N is the total number of complete cells within the unit RVE models. Figure 1a and b shows the RVEs of hierarchical and self-similar honeycombs with 300 complete cells and different degrees of regularity a = 0 and a = 0.7 at different hierarchy levels. It is assumed that for first-order random irregular honeycombs, all the cell walls have the same uniform initial thickness h0 and a unit initial width which is much larger than h0. When the effects of the initial surface stresses/ strains are absent, the initial relative density of the firstorder random irregular micro- or nano-sized honeycomb (i.e. the RVE model) is given as [5, 7, 8] ! M X q0 ¼ h0 l0i =L20 ; ð3Þ i¼1
where l0i is the cell wall initial lengths and L0 is the initial side length of the periodic unit honeycomb RVE models (as shown in Fig. 1) and M is the total number of cell walls. Size-dependent rigidities of first-order micro- or nanosized cell walls In mechanical analyses of cellular materials, the main deformation mechanisms [1, 2, 4–10] include cell wall/strut bending, transverse shearing and axial stretching or compression. When a first-order micro- or nano-sized honeycomb is subjected to in-plane deformation, plane strain bending of the cell walls is the dominant deformation mechanism [9, 10]. For micro-sized honeycombs, if the cell wall thickness is between submicron and micron size, strain gradient [17–28] may affect the bending and transverse shear rigidities. For a flat and wide (compared to the thickness) plate with a uniform thickness at the micro-metre scale, the size-dependent bending, transverse shear and axial stretching/compression rigidities have been obtained and given as [10] Db ¼
ES h 3 ½1 þ 6ð1 vS Þðlm =hÞ2 ; 12ð1 v2S Þ
ð4Þ
Ds ¼
hEs ½1 þ 6ð1 þ vS Þðlm =hÞ2 2 2:4ð1 þ vS Þ 1 þ 2:5ð1 þ vS Þ2 ðlm =hÞ2
ð5Þ
Dc ¼ ES h;
ð6Þ
where ES and vS are the Young’s modulus and Poisson’s ratio of the solid material, h is the cell wall thickness and lm is the length scale parameter for strain gradient effect (i.e. a
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Fig. 1 Unit periodic RVE models of the micro- or nanostructured hierarchical and selfsimilar random irregular honeycombs with different degrees of regularity. a a = 0.0; b a = 0.7
material constant which can be experimentally measured). For metal materials, lm is usually between submicron and micron size. In Eqs. 4 and 5, the cell wall width is assumed to be a unity and much larger than the thickness h. It is easy to check that when the plate thickness h is much larger than lm (i.e. lm/h approaches 0), the bending and transverse shear rigidities given in Eqs. 4 and 5 reduce to those of conventional mechanics. The axial stretching/compression stiffness always matches the conventional results because there is no strain gradient effect when a cell wall undergoes uniaxial tension or compression. For nano-sized honeycombs, as the cell walls are very thin and the surface to volume ratio is large, both the surface elasticity and the initial stress or strain can greatly affect the bending, transverse shearing and axial stretching/compression rigidities. For simplicity in the analyses and simulations that are to follow, both the surface and the bulk materials of the first-order cell walls are assumed to be isotropic and to have the same Poisson’s ratio vS. When a surface stress s0 is present, the amplitudes of the initial stresses in the bulk material in the length and the width directions of the first-order cell walls are equal [38] and are obtained as rL0 = rW 0 = -2s0/h, where h is the current thickness of the cell walls. At the nano-metre scale, the yield strength of the first-order cell wall material, ry, could reach 0.1ES or even a larger value [29, 39]. For recoverable
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elastic deformation, the initial von Mises stress should not exceed the yield strength ry of the bulk material, thus 2s0 re ¼ jrL0 j ¼ jrW 0 j ¼ j h j ry ¼ 0:1ES . The amplitudes of the initial elastic residual strains in the length and width directions of the first-order cell walls are equal, and are related to the initial stress in the bulk material by [38] eL0 ¼ eW 0 ¼
rL0 ð1 vS Þ ES
ð7Þ
The initial elastic residual strain in the thickness direction of the first-order cell walls is obtained as et0 ¼
Dh 4vS s0 2vS L ¼ r h0 ES h ES 0
ð8Þ
When the effects of the initial stresses are present, the bending, transverse shearing and axial stretching/compression rigidities of a nano-sized first-order cell wall have been obtained in Refs. [17], [10] and [12], respectively, as ES bh3 ln mS ð1 þ vS Þ L Db ¼ e ð9Þ 1þ6 þ 1 vS 0 h 12ð1 v2S Þ 1þv L 2 6l Gs bh ½1 þ hn þ vS 1vSS e0 Ds ¼ 1:2 1 þ 10lh n þ 30ðln =hÞ2
ð10Þ
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Dc ¼ Es bhð1 þ 2ln =hÞ;
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ð11Þ
where the width of the cell walls, b, is initially unity and is assumed to be much larger than the thickness h. In Eqs. 9– 11, ln = S/ES is the intrinsic length of the material at the nano-metre scale and S is the surface elasticity modulus. It should be noted that the bulk material from which the hierarchical honeycombs are made could be any type of metallic, or polymeric or biological materials. The material nano-scale intrinsic length ln is typically in the range from 0.01 to 1 nm. When the effects of the cell wall initial stresses are present, the current width, thickness and length of the first-order cell walls can be obtained as L b ¼ b0 ð1 þ eW 0 Þ ¼ 1 þ e0 2vS L h ¼ h0 ð1 þ et0 Þ ¼ h0 1 r0 ES
l ¼ l0 ð1 þ eL0 Þ
ð12Þ ð13Þ ð14Þ
The bending, axial stretching/compression and transverse shearing rigidities can all be controlled to vary [10, 12, 16, 38] over large ranges because the cell wall width, thickness and length can be controlled to reduce or increase by about 10 % at the nano-metre scale by application of an electric potential. Treatment of the equivalent deformation rigidities of first-order micro- or nano-sized cell walls in finite element simulations Micro- and nano-sized irregular honeycombs were simulated using the ANSYS finite element software [40]. Each of the cell walls was partitioned into a number of BEAM3 elements. This type of beam element has two nodes and takes account of the bending, stretching and transverse shearing deformation mechanisms. For a first-order microor nano-sized 2D Voronoi honeycomb with an initial relative density q0, the initial cell wall thickness h0 can be obtained from Eq. 3. In almost all commercial finite element software, such ABAQUS or ANSYS, there is no type of element which can directly incorporate the size-dependent effects in the finite element simulations. We thus use an equivalent solid beam with thickness he, Young’s modulus Ee and Poisson’s ratio ve for the beam elements in the finite element simulations. These equivalent values are obtained from the following three simultaneous equations: Ee h3e ¼ Db 12 G e he Ee h e ¼ ¼ Ds 1:2 2:4ð1 þ ve Þ Ee he ¼ Dc ;
ð15Þ ð16Þ ð17Þ
where Db, Ds and Dc are given in Eqs. 4–6 for micro-sized first-order honeycombs, and in Eqs. 9–11 for nano-sized first-order honeycombs. As Db, Ds and Dc are size-dependent and may also be tunable, the effects of the strain gradient, or the surface elasticity and initial stress/strain on the mechanical properties of the first-order micro- or nanosized random irregular honeycombs can be incorporated into the finite element simulations using the equivalent values of he, Ee and ve obtained from Eqs. 15–17. The Poisson’s ratio vS of the solid material of the first-order cell walls is always assumed to be 0.4 in the simulations of this paper. It is noted that the value vS of the solid material has very little effect on the dimensionless elastic properties of the macro- and micro-structured hierarchical honeycombs. It is well known that if the sized effects are absent (i.e. lm/h = 0 in Eqs. 4–6 or ln/h = 0 in Eqs. 9–11), the obtained dimensionless Young’s modulus of a Voronoi honeycomb would be a function of the relative density q0 and the degree of regularity a, and be independent of the actual cell wall thickness h. When constructing each individual Voronoi honeycomb model with a given relative density, the actual cell wall thickness has been obtained, which is different for each individual models. For each model, we use the same Young’s modulus ES and Poisson’s ratio vS for the solid material, but a different cell wall thickness h (which is obtained when constructing the geometrical model). The strain gradient effects or the surface effects are incorporated into the simulations by choosing different values of lm/h or ln/h in Eqs. 3–6 or 9– 11. Therefore, we can obtain the correct equivalent values of he, Ee and ve for each of the individual model from Eqs. 15–17 and 3–6 or 9–11. As there is no identical cell wall thickness h for different random Voronoi honeycomb models (even with the same degree of regularity or the same relative density), we always have different values of he, Ee and ve for each different models. To validate the use of the equivalent values of he, Ee and ve obtained from Eqs. 15–17, the exact theoretical deflection and the computed deflection were compared for a single horizontal micro- or nano-sized plate/beam cantilever structure when a concentrated transverse load was applied to the free end. Although the equivalent value of ve obtained from Eqs. 15–17 could be negative or larger than 0.5 in some cases, which is physically meaningless for conventional solid materials, nevertheless the numerical results for the mechanical responses obtained from the finite element simulations were exactly the same as those of the theoretical predictions. This suggests that the adoption of the equivalent values of he, Ee and ve can properly incorporate the size effects in finite element simulations and hence correctly predict the mechanical behaviours of micro- or nano-sized first-order random irregular honeycombs.
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Boundary conditions in finite element simulations
ðKÞ1 ¼
As the constructed random irregular Voronoi honeycomb models shown in Fig. 1 are periodic [7], periodic boundary conditions [5, 7] can easily be imposed in the finite element simulations. The periodic boundary conditions are given by ¼ uright uright ; i0 j0 left ¼ vright vright ; vleft i vj i0 j0 top top ubottom ; ui uj ¼ ubottom i0 j0 top top bottom bottom vi vj ¼ vi0 vj0 ; ¼ hright ; hleft i i0 top bottom hi ¼ hi 0 : uleft i
uleft j
ð18Þ
Mechanical properties of first-order micro- and nanosized random irregular honeycombs The first-order micro- or nano-sized random irregular honeycombs are assumed to be made of a solid material with Young’s modulus ES, shear modulus GS and Poisson’s ratio vS. The elastic properties of random irregular honeycombs are obtained by finite element simulations from periodic models with 300 complete cells [7, 8]. It has been found that although the obtained Young’s moduli could be quite different for different random irregular honeycomb models, they are almost identical in the x and y directions for each of the same models [7]. Each data point in the figures in this paper represents the mean value of 20 different random irregular honeycomb models with the same degree of regularity and the same set of other parameters (e.g. degree of regularity or cell wall thickness). It is noted that the Poisson’s ratio of the solid material of the firstorder honeycombs is assumed to be 0.4 throughout this paper. For both micro- and nano-sized first-order random irregular honeycombs, the in-plane Young’s modulus is normalised by 1.5ESq30/(1 - v2S), i.e. ðE1 Þ1 ð1 v2S Þ 1:5ES q30
ð19Þ
The out-of-plane Young’s modulus is normalised by ESq0, i.e. ðE3 Þ1 ¼
ðE3 Þ1 ES q0
The bulk modulus is normalised by ESq0, i.e.
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ð21Þ
and the out-of-plane shear modulus is normalised by GSq0, i.e. ðG31 Þ1 ¼
ðG31 Þ1 GS q0
ð22Þ
Size-dependent elastic properties of micro-sized first-order random irregular honeycombs
where i and j are nodes on the left or top edge of the random irregular periodic honeycomb, while i0 and j0 are the corresponding nodes on the right or bottom edge.
ðE1 Þ1 ¼
ðKÞ1 ES q0
ð20Þ
For the first-order random irregular micro-sized honeycombs with degrees of regularity a = 0.0 and 0.7, the size-dependent relationships between the dimensionless in-plane Young’s modulus and the honeycomb relative density are shown in Fig. 2a and b. As can be seen, the thinner the cell walls, the larger the dimensionless Young’s modulus; and the larger the relative density, the smaller the dimensionless Young’s modulus. If the relative density remains unchanged, Eq. 3 and Fig. 2a and b suggest that the smaller the cell size, the larger would be the dimensionless Young’s modulus. If the cell wall thickness is much larger than the material length parameter lm of the strain gradient effect (i.e. lm/h0 = 0), the results in Fig. 2a and b reduce to those of the macro-sized honeycombs as given in Ref. [7]. Figure 3 shows the effects of cell regularity on the dimensionless Young’s modulus of irregular honeycombs with lm/h0 = 0.5 and constant initial relative densities q0 = 0.01 and q0 = 0.2, each data point represents the mean value of 20 similar models and the error bar shows the standard deviation. As can be seen from Fig. 3, when the honeycomb relative density is small (e.g. q0 = 0.01), the dimensionless Young’s modulus generally reduces with the increase of the cell regularity; in contrast, when the honeycomb relative density is large (e.g. q0 = 0.2), the dimensionless Young’s modulus generally increases with the increase of the cell regularity. The trend change takes place when q0 & 0.15 for random irregular honeycombs with lm/h0 = 0.5, and occurs due to the increasing honeycomb relative density, as shown in Fig. 3. This observation is consistent with the results given in Fig. 7 of Ref. [7] for the macro-sized counterparts (i.e. lm/h0 = 0), where the dimensionless Young’s modulus reduces with the increasing cell regularity when q0 = 0.01, increases with the increment of the cell regularity when q0 = 0.25 or larger, and the trend change occurs when q0 & 0.22. It can thus logically be conjectured that the thinner the cell walls (i.e. the larger the value of lm/h0), the smaller is the relative density at which the trend change takes place. Figure 4 shows the relationships between the dimensionless bulk modulus and (a) the honeycomb relative density, (b) the degree of regularity. For first-order random irregular honeycombs with cell wall thickness at the micro-
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lm/h=0.0
5
lm/h=0.2 4
lm/h=0.5 lm/h=1.0
3
2
1
Non-dimensional Young's Modulus
Non-dimensional Young's Modulus
(a) 6
ρ0 =0.01 ρ0 =0.20
2.5
2.0
1.5
1.0 0 0.00
0.0 0.05
0.10
0.15
0.20
0.25
0.2
0.4
0.30
0.6
0.8
1.0
Degree of Regularity
Relative Density
lm/h=0.0
5
lm/h=0.2 4
Fig. 3 Effects of cell regularity on the dimensionless Young’s modulus of micro-sized random irregular honeycombs with lm/ h = 0.5
(a) 0.243 Non-dimensional Bulk Modulus
Non-dimensional Young's Modulus
(b) 6
lm/h=0.5 lm/h=1.0
3
2
1
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Relative Density
metre scale, generally, the thinner the cell walls, the larger is the dimensionless bulk modulus because at the micrometre scale, thinner cell walls have a larger dimensionless bending stiffness. As can be seen from Fig. 4a, the dimensionless bulk modulus increases very slightly with an increase of the relative density and with a reduction of the cell wall thickness (i.e. an increase of lm/h0); however, this change of the dimensionless bulk modulus is so small that it could be simply neglected. On the other hand, cell regularity has a much larger effect on the dimensionless bulk modulus, i.e. the larger the degree of the cell regularity, the larger is the dimensionless bulk modulus as shown in Fig. 4b. For perfectly regular micro-sized honeycombs, the degree of regularity is a = 1.0 and the dimensionless bulk modulus is 0.25, which is the same as that of their macrosized counterparts [7] because the strain gradient effect is completely absent in this case.
lm /h=0.2
0.242
lm /h=0.5 lm /h=1.0
0.241
0.240
0.239 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Relative Density
(b) 0.26 Non-dimensional Bulk Modulus
Fig. 2 Size-dependent effect on the relationship between the dimensionless Young’s modulus and the relative density of micro-sized first-order random irregular honeycombs with degrees of regularity. a a = 0.0; b a = 0.7
lm /h=0.0
ρ0 =0.01 0.24
ρ0 =0.20
0.22
0.20
0.18 0.0
0.2
0.4
0.6
0.8
1.0
Degree of Regularity Fig. 4 Relationships of the dimensionless bulk modulus of microsized first-order random irregular honeycombs against a relative density when a = 0.7; b degree of regularity
Figure 5 shows the size-dependent relationship between Poisson’s ratio and the relative density for micro-sized first-order random irregular honeycombs with degree of
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1.1
Non-dimensional Young's Modulus
(a) 10
Poisson's Ratio
1.0 0.9 0.8
lm/h=0.0
0.7
lm/h=0.2
0.6
lm/h=0.5 0.5
lm/h=1.0
0.4 0.00
0.05
0.10
0.15
0.20
0.25
ln/h=0.0 8
ln/h=0.5
6
ln/h=1.0 4
2
0 0.00
0.30
ln/h=0.2
0.05
0.10
Relative Density
regularity a = 0.7. Generally, the thinner the cell walls or the larger the honeycomb relative density, the smaller is Poisson’s ratio; and the degree of cell regularity has very little effect on Poisson’s ratio. When the relative density tends to zero, Poisson’s ratio always approaches unity. As it has already been demonstrated in Ref. [7] that random irregular honeycombs are isotropic in-plane, the sizedependent dimensionless in-plane shear modulus of the first-order micro-sized honeycombs can thus be obtained from the in-plane dimensionless Young’s modulus and Poisson’s ratio, and so the results for the in-plane shear modulus are not presented separately. As there is no strain gradient effect in the out-of-plane deformation, the out-of-plane dimensionless Young’s modulus of first-order micro-sized random irregular honeycombs is obviously unity, the out-of-plane Poisson’s ratio v31 = vS, and the out-of-plane dimensionless shear modulus are always approximately 0.5. Thus, all the outof-plane elastic properties of micro-sized first-order random irregular honeycombs are exactly the same as those of their macro-sized counterparts. Size-dependent and tunable elastic properties of nanosized first-order random irregular honeycombs When the cell wall thickness is at the nano-metre scale, both the surface elasticity [12] and the initial stresses/ strains [10, 15, 16] can affect the elastic properties of firstorder random irregular honeycombs. When the effects of the initial stresses/strains are absent, the size-dependent relationships between the dimensionless Young’s modulus and the relative density are given in Fig. 6a and b for firstorder nano-sized random irregular honeycombs with degrees of regularity a = 0.0 and a = 0.7. As can been
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0.20
0.25
0.30
(b) 10 Non-dimensional Young's Modulus
Fig. 5 Size-dependent effect on the relationship between Poisson’s ratio and the relative density of micro-sized first-order random irregular honeycombs with a = 0.7
0.15
Relative Density
ln/h=0.0 8
ln/h=0.2 ln/h=0.5
6
ln/h=1.0 4
2
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Relative Density Fig. 6 Size-dependent effect on the relationship between the dimensionless Young’s modulus and the relative density of nano-sized random irregular honeycombs when the effect of the cell wall initial stress/strain is absent. a a = 0.0; b a = 0.7
seen, the thinner the cell walls, the larger is the dimensionless Young’s modulus; and the larger the relative density, the smaller the dimensionless Young’s modulus. If the relative density of the first-order nano-sized irregular honeycombs remains the same, Eq. 3 and Fig. 6a and b suggest that the smaller the cell size, the larger would be the dimensionless Young’s modulus. These results are consistent with previous findings for nano-sized first-order honeycombs with perfectly regular hexagonal cells (i.e. a = 1.0) [10]. When the cell wall thickness h0 is much larger than the material intrinsic length ln at the nano-metre scale (i.e. ln/h0 = 0), the results reduce to those of the macro-sized counterparts given in Ref. [7]. When the effects of the surface elasticity are absent (i.e. ln/h0 = 0), the effects of the initial stresses/strains on the relationships between the dimensionless Young’s modulus and the relative density are shown in Fig. 7a and b for the first-order nano-sized random irregular honeycombs with
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Non-dimensional Young's Modulus
ε L0= -0.06 ε L0= -0.03 ε L0= 0 ε L0= 0.03 ε L0= 0.06
2.0 1.6 1.2 0.8 0.4 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Relative Density Non-dimensional Young's Modulus
ρ 0 =0.01 ρ 0 =0.20
5
4
3
2 0.0
0.2
0.4
0.6
0.8
1.0
Degree of Regularity
(b) 2.4
ε L0 = -0.06 ε L0 = -0.03
2.0
Non-dimensional Young's Modulus
6
(a) 2.4
Fig. 8 Relative density effect on the relationship between the dimensionless Young’s modulus and the cell regularity of nano-sized random irregular honeycombs with ln/h = 0.5 when the cell wall initial stress effect is absent
ε L0 = 0 1.6
ε L0 = 0.03 ε L0 = 0.06
1.2 0.8 0.4 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Relative Density Fig. 7 Effects of the cell wall initial stress/strain on the relationship between the dimensionless Young’s modulus and the relative density of nano-sized random irregular honeycombs when the surface elasticity effect is absent. a a = 0.0; b a = 0.7
degrees of regularity a = 0.0 and a = 0.7. At the nanometre scale, the amplitude of the recoverable initial elastic strain in the cell wall length direction can be controlled to vary over a range from -0.1 to 0.1 by application of an electric potential [10, 32, 33, 36–38]. Figure 7a and b demonstrate that the dimensionless Young’s modulus of the nano-sized first-order random irregular honeycombs can be controlled either to increase about 60 % or to reduce about 35 % when the initial strain in cell wall length direction is controlled to vary from -6 to 6 %. In agreement with the previous results [10, 36–38], whether the effects of the surface elasticity are present (i.e. ln/h0 = 0) or absent (i.e. ln/h0 = 0), the percentage of the tunable range of the dimensionless Young’s modulus remains almost unchanged if the applied initial strain in the cell wall length direction is controlled to vary over the same range. If the values of the initial strain applied in the cell wall length direction are different from those given in Fig. 7a and b, the dimensionless Young’s modulus of the
nano-sized first-order random irregular honeycombs can be obtained by scaling up or scaling down the results. When the effects of the initial stresses/strains are absent, the relationships between the dimensionless Young’s modulus and the cell regularity are shown in Fig. 8 for nano-sized random irregular honeycombs with ln/h = 0.5 and different values of initial relative density, i.e. q0 = 0.01 and q0 = 0.2. The standard deviation is about 10 % of the dimensionless Young’s modulus or smaller. As for the case of micro-sized random irregular honeycombs, the dimensionless Young’s modulus reduces with an increase in the degree of the cell regularity when the honeycomb relative density is small, e.g. q0 = 0.01, and increases with an increase in the cell regularity when the honeycomb relative density is large, e.g. q0 = 0.20. For nano-sized random irregular honeycombs with ln/h = 0.5, the trend change occurs when the relative density is about q0 = 0.15. The larger the value of ln/h, the smaller is the relative density q0 at which the trend change occurs. Figure 9a shows the effects of the cell wall thickness (or the surface elasticity) on the relationship between the dimensionless bulk modulus and the relative density of the nano-sized first-order random irregular honeycombs with a = 0.7. As can be seen, it is the cell wall thickness rather than the relative density which affects the dimensionless bulk modulus, and cell wall stretching/compression is the dominant deformation mechanism when a honeycomb is under a hydrostatic stress. Figure 9b shows that when amplitude of the initial strain in the cell wall direction is controlled to vary from -6 to 6 %, the dimensionless bulk modulus of nano-sized random irregular honeycombs with a = 0.7 can be controlled to change from 15 % increment to 14 % reduction, while the effect of the initial relative
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J Mater Sci (2014) 49:5690–5702 0.55
ln/h=0.0
1.2
Non-dimensional Bulk Modulus
Non-dimensional Bulk Modulus
(a) 1.4
ln/h=0.2
1.0
ln/h=0.5
0.8
ln/h=1.0
0.6 0.4 0.2 0.0 0.00
Non-dimensional Bulk Modulus
0.40
0.32
0.10
0.15
0.20
0.25
0.30
0.40
ε ε ε ε ε
0.05
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 10 Relative density effect on the relationship between the dimensionless bulk modulus and the regularity of nano-sized random irregular honeycombs with ln/h = 0.5 when the cell wall initial stress effect is absent
L 0 = -0.06 L 0 = -0.03 L 0 =0 L 0 = 0.03 L 0 = 0.06
0.10
0.15
0.20
0.25
0.30
Relative Density Fig. 9 Relationship between the dimensionless bulk modulus and relative density of nano-sized random irregular honeycombs with a = 0.7. a Surface elasticity effect; b initial stress/strain effect
density on the dimensionless bulk modulus is almost negligible. The dimensionless bulk moduli of nano-sized random irregular honeycombs with a = 0.7, shown in Fig. 9a and b, can well be approximated by 2vS L L 0:24ð1 þ 2lhn Þð1 1v e Þ=ð1 þ e Þ. 0 S 0 Figure 10 shows the relationship between the dimensionless bulk modulus and the degree of regularity of nanosized random irregular honeycombs with ln/h = 0.5. The small standard deviation value indicates that the bulk moduli obtained from each of the 20 similar models is very close. As can be seen, the greater the degree of the cell regularity, the larger the dimensionless bulk modulus. This is because the greater the cell regularity, the more dominant a role the cell wall stretching/compression mechanism plays when a honeycomb is under a hydrostatic stress. The results of Fig. 10 agree well with those for micro-sized and the macro-sized first-order random irregular honeycombs, indicating that the honeycomb relative density does not affect this relationship. Comparing the results in Fig. 10 to
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Degree of Regularity
0.24
0.16 0.00
ρ0 =0.20
0.50
0.35
0.05
Relative Density
(b) 0.48
ρ0 =0.01
those in Fig. 6 of paper [7], it is easily deduced that the effect of the surface elasticity on the dimensionless bulk modulus of nano-sized honeycombs can be well described 2vS L by a factor ð1 þ 2lhn Þð1 1v e Þ=ð1 þ eL0 Þ because cell wall S 0 stretching/compression is the dominant deformation mechanism. When the effects of the initial stress/strain are absent, the effects of the cell wall thickness (i.e. ln/h or the surface elasticity effects) on the relationship between the in-plane Poisson’s ratio and the relative density of nano-sized honeycombs with a = 0.7 are shown in Fig. 11a. As can be seen, the in-plane Poisson’s ratio tends to unity as the honeycomb relative density tends to zero, reduces approximately linearly with an increase of the relative density, and becomes approximately 0.78 (for the case ln/ h = 0) when q0 = 0.30. Generally, the thinner the cell wall (i.e. the larger the ln/h value), the smaller is the inplane Poisson’s ratio. However, the effect of the surface elasticity (or the cell wall thickness) on the in-plane Poisson’s ratio is very small. Figure 11b shows that the relationship between the in-plane Poisson’s ratio and the relative density of nano-sized honeycombs (with a = 0.7) can be controlled to vary by adjusting the amplitude of the initial strain. The controllable range of the Poisson’s ratio is approximately in proportion to the range of the tunable initial strain applied in the cell wall direction and to the honeycomb relative density. For nano-sized first-order random irregular honeycombs, the normalised out-of-plane Young’s modulus can be easily obtained as ðE3 Þ1 2ln 2vS L ðE3 Þ1 ¼ ¼ 1þ e0 = 1 þ eL0 1 ES q0 h 1 vS ð23Þ
J Mater Sci (2014) 49:5690–5702
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Tunable geometrical properties of nano-sized firstorder random irregular honeycombs
(a) 1.1
Poisson's Ratio
1.0
For nano-sized first-order random irregular honeycombs, when the effects of the initial stress/strain are absent, the initial cell diameter, cross-sectional area and volume are assumed to be D0, A0 and V0, respectively. When the effects of the initial stress/stain are present, those geometrical parameters become
0.9
ln /h=0.0
0.8
ln /h=0.2 ln /h=0.5
0.7
ln /h=1.0 0.6 0.00
0.05
0.10
0.15
0.20
0.25
0.30
ðDÞ1 ¼
ðDÞ1 ¼ 1 þ eL0 D0
ð26Þ
ðAÞ1 ¼
ðAÞ1 ¼ ð1 þ eL0 Þ2 A0
ð27Þ
ðVÞ1 ¼
ðVÞ1 ¼ ð1 þ eL0 Þ3 ; V0
ð28Þ
Relative Density
(b) 1.05 1.00
where the initial strain eL0 in the cell wall length/width direction can be controlled to vary over the range -0.1 to 0.1 by application of an electric potential. Therefore, the geometrical properties of the nano-sized first-order random irregular honeycombs are tunable and controllable.
Poisson's Ratio
0.95 0.90 0.85
ε L0 = -0.06 ε L0 = -0.03 ε L0 = 0
0.80
ε L0 = 0.03
0.75
ε L0 = 0.06
0.70 0.00
0.05
Mechanical properties of micro- and nano-structured hierarchical and self-similar random irregular honeycombs
0.10
0.15
0.20
0.25
0.30
Relative Density Fig. 11 Relationship between the in-plane Poisson’s ratio and the relative density of nano-sized random irregular honeycombs with a = 0.7. a Surface elasticity effect; b initial stress/strain effect
The dimensionless bulk modulus and the out-of-plane shear modulus can well be approximated by ðKÞ1 2ln 2vS L ðKÞ1 ¼ ¼ KðaÞ 1 þ e0 = 1 þ eL0 1 ES q0 h 1 vS ð24Þ ðG31 Þ1 1 2ln 2vS L 1þ ðG31 Þ1 ¼ ¼ e0 = 1 þ eL0 1 2 Gs q0 h 1 vS
ð25Þ and the out-of-plane Poisson’s ratio is (v31)1 = vS, i.e. the same as that of the solid material. In the above equations, 2s0 eL0 ¼ hE ð1 vS Þ and K(a) give the relationship between S the dimensionless bulk modulus of the macro-sized firstorder random irregular honeycombs and the cell regularity, which is a function of the regularity a, obtained in Ref. [7] and is almost the same as the relationship given in Fig. 4b. The effect of the relative density q0 on K(a) is very small and thus negligible.
The micro- and nano-structured hierarchical random irregular honeycombs are assumed to be geometrically self-similar [34–37], as shown in Fig. 1a and b. At each of the different hierarchy levels, the honeycomb is assumed to be a material whose size is much larger than the individual cells at that hierarchy level. The relative density of an nth order hierarchical and self-similar random irregular honeycomb is tunable and given as 2vS L qn ¼ 1 e0 ðq0 Þn =ð1 þ eL0 Þ; ð29Þ 1 vS where q0 is given in Eq. 3 for the first-order random irregular honeycomb. When the initial strain eL0 is 0, qn reduces to (q0)n. For nth order micro- or nano-structured hierarchical and self-similar random irregular honeycombs, the five independent dimensionless elastic constants can be obtained as ðE1 Þn ¼
ðE1 Þn ð1 v2S Þ ½f ða; q0 Þn1 ðE1 Þ1 1:5ES q0
ðv12 Þn vðq0 Þ; ðE3 Þn ¼
ð30Þ
n2
ð31Þ
ðE3 Þn ¼ ðq0 Þn1 ðE3 Þ1 ES q0
ð32Þ
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5700
ðG31 Þn q ð 0 Þn1 ðG31 Þ1 Gs q0 2
and (v31)n = vS, where n is the hierarchy level of the micro- or nano-structured hierarchical and self-similar honeycombs. In Eqs. 30, 32 and 33, ðE1 Þ1 ; ðE3 Þ1 , and ðG31 Þ1 are the dimensionless in-plane and out-of-plane Young’s moduli, and the dimensionless out-of-plane shear modulus of the first-order micro- or nano-sized random irregular honeycomb, respectively. Their results have been obtained and given in ‘‘Size-dependent elastic properties of micro-sized first-order random irregular honeycombs’’ and ‘‘Size-dependent and tunable elastic properties of nanosized first-order random irregular honeycombs’’ sections. Hierarchical materials may have a number of structural levels. For hierarchical and self-similar random irregular honeycombs with n C 2 and fixed values q0 = 0.2 or q0 = 0.3, the numerical results of the function f(a, q0) = (E1)n/[(E1)n-1q30] in Eq. 30 are obtained by computer simulation, and are plotted against the honeycomb regularity a, as shown in Fig. 12. It is found that the value of f(a, q0) depends mainly on q0 and is not sensitive to the values of En-1 and (v12)n-1, and thus remains almost unchanged for hierarchy levels n C 2 if q0 is a constant. As can be seen from Fig. 12, when q0 is 0.2, the value of f(a, q0) is consistently larger than unity, and when q0 is 0.3, the value of f(a, q0) is slightly smaller than but very close to unity. The smaller the value q0 and the lower the degree of the cell regularity a, the larger is the value of f(a, q0). Therefore, it can be concluded that, for micro- or nano-structured hierarchical and self-similar honeycombs with the same value of q0 and the same first-order cell wall thickness (i.e. ðE1 Þ1 remains the same), the larger the hierarchy level n, the larger is the dimensionless in-plane Young’s modulus ðE1 Þn if q0 B 0.28; or the smaller is the dimensionless in-plane Young’s modulus ðE1 Þn if q0 [ 0.3. In other words, if the overall relative density qn = qn0 remains unchanged for micro- or nano-structured hierarchical and self-similar honeycombs and first-order cell wall thickness remains the same (i.e. ðE1 Þ1 remains the same), the dimensionless in-plane Young’s modulus ðE1 Þn of a hierarchical random irregular honeycomb given Eq. 30 can be designed to be larger or smaller than that of its singleorder counterpart, depending on the values q0 and a. For micro- or nano-structured hierarchical and selfsimilar regular hexagonal honeycombs (a = 1.0), when q0 = 0.2 and q0 = 0.3, the results given in Eqs. 30–33 are consistent with those of Ref. [37]. For hierarchical and selfsimilar honeycombs with n C 2, the value of the in-plane Poisson’s ratio (v12)n is approximately the same as that given by the solid curve in Fig. 11a for ln/h = 0, which
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1.6
ð33Þ
ρ0 =0.20 ρ0 =0.30 1.4
f (α , ρ0 )
ðG31 Þn ¼
J Mater Sci (2014) 49:5690–5702
1.2
1.0
0.8 0.0
0.2
0.4
0.6
0.8
1.0
Degree of Regularity Fig. 12 Relationship between the value of f(a, q0) = (E1)n/ [(E1)n-1q30] and cell regularity of micro- or nano-structured hierarchical and self-similar random irregular honeycombs with n C 2 and different relative densities q0 = 0.2 and q0 = 0.3
depends mainly on the value of q0 and is almost independent of the cell regularity a and the material Young’s modulus. Therefore, the Poisson’s ratio of micro- or nanostructured hierarchical honeycombs with n C 2 is in general almost not tunable. Equation 32 indicates that the dimensionless out-of-plane Young’s modulus ðE3 Þn of micro- or nano-structured hierarchical and self-similar honeycombs depends only on the value of the overall relative density qn and the thickness of the first-order cell walls. In contrast, Eq. 33 shows clearly that for micro- or nano-structured hierarchical and self-similar honeycombs with the same overall relative density qn, the larger the number of structural hierarchy level n, the smaller is the dimensionless out-of-plane shear modulus ðG31 Þn .
Tunable geometrical properties of nano-structured hierarchical and self-similar random irregular honeycombs For nth order nano-structured hierarchical and self-similar random irregular honeycombs, when the effects of the initial stress/strain are absent, the initial cell diameter, cross-sectional area and volume are assumed to be (D0)n, (A0)n and (V0)n, respectively. When the effects of the initial stress/stain are present, those geometrical parameters vary and are normalised to ðDÞn ¼
ðDÞn ¼ 1 þ eL0 ðD0 Þn
ð34Þ
ðAÞn ¼
ðAÞn ¼ ð1 þ eL0 Þ2 ðA0 Þn
ð35Þ
J Mater Sci (2014) 49:5690–5702
ðVÞn ¼
ðVÞn ¼ ð1 þ eL0 Þ3 ; ðV0 Þn
5701
ð36Þ
where eL0 is the initial strain of the first-order cell walls in their length/width direction, which can be controlled to vary over -0.1 to 0.1 by application of an electric potential. Therefore, the geometrical properties of the nano-sized nth order hierarchical and self-similar random irregular honeycombs are tunable and controllable. These are the same as those of perfect regular honeycombs and independent of the degree of the cell regularity.
Conclusion All the five independent elastic constants: E1, v12, E3, G31 and v31 = vS are obtained for the micro- and nano-structured firstorder and the hierarchical and self-similar random irregular honeycombs. It is found that if the thickness of the first-order cell walls is at the micro-metre scale, the mechanical properties of random irregular hierarchical honeycombs are sizedependent due to the strain gradient effects; and that if the thickness of the first-order cell walls is at the nano-metre scale, the mechanical properties of the random irregular hierarchical honeycombs are not only size-dependent due to the surface elasticity effects but are also tunable and controllable due to the effects of the initial stresses/strains. In addition, for nanostructured hierarchical random irregular honeycombs, the cell diameter, cross-section area and volume can all be controlled to vary over large ranges by application of an applied electric potential. The results obtained in this paper could serve as a guide in the design of many different advanced functional materials with tunable and controllable properties in strength/ stiffness, colour, wettability, selectivity, natural frequency, or electric capacitance.
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