Influence of Topology and Porosity on Size Effects in Stripes of Cellular Material with Honeycomb Structure under Shear, Tension and Bending
aa r X i v : . [ phy s i c s . c o m p - ph ] N ov Influence of Topology and Porosity on Size Effects inStripes of Cellular Material with HoneycombStructure under Shear, Tension and Bending
Rinh Dinh Pham, Geralf Hütter ∗ November 5, 2020
AbstractCellular solids are known to exhibit size effects, i. e. differences in the apparent effective elasticmoduli, when the specimen size becomes comparable to the cell size. The present contributionemploys direct numerical simulations (DNS) of the mesostructure to investigate the influences ofporosity, shape of pores, and thus material distribution along the struts, and orientation of loadingon the size effects and effective moduli of regular honeycomb structures. Beam models are comparedto continuum models for simple shear, uniaxial loading and pure bending of strips of finite width.It is found that the honeycomb structure exhibits a considerable anisotropy of the size effects andthat honeycomb structures with circular pores exhibit considerably stronger size effects than thosewith hexagonal pores (and thus straight struts). Positive (stiffening) size effects are observed undersimple shear and negative (softening) size effects under bending and uniaxial loading. The negativesize effects are interpreted in terms of the stress-gradient theory.
Keywords:
Cellular solids; Honeycomb structure; Topology; size effects; stress-gradient theory
1. Introduction
Cellular solids like foams and, more recently, architectured and additively manufactured meta-materialshave a high potential for lightweight constructions and smart devices. The assessment of such structuresrequires knowledge on the effective macroscopic behavior of these materials. In particular, a relationbetween effective macroscopic behavior and mesostructure is relevant as the latter can be influencedduring manufacturing, or even tailored to specific applications.It is well-known that the effective elastic moduli and the effective strength depend on mesoscopictopological properties of foams like connectivity of struts (bending/tension dominated) [6, 38, 47],distribution of material between struts and nodes [17, 27, 39, 40] as well as stochastic properties [27,28, 31, 34, 35, 41].Furthermore, considerable size effects have been observed in the behavior of cellular materials, bothin elastic and plastic regime, e. g. [5, 8, 12, 32, 36, 37, 44, 46]. This means that small specimens behavedifferently than would be anticipated from sufficiently large specimens using the classical Cauchy theoryof continuum mechanics. Size effects can be described by generalized continuum theories, starting fromGauthier and Jahsman [15] and Yang and Lakes [46] up to a large number of recent works [12, 19, 21,23, 25, 28, 29, 31–33, 36, 37, 44, 45]. Alternatively, direct numerical simulations (DNS) with discretelyresolved mesostructures of foams have been performed, firstly in order to investigate the reasons andmechanisms of size effects and, secondly, as benchmark for generalized continuum theories. Most DNS ∗ Technische Universität Bergakademie Freiberg, Institute of Mechanics and Fluid Dynamics, Lampadiusstr. 4, 09596Freiberg, Germany, [email protected] t d (a) hexagonal pore (prismatic struts) (b) circular pore Figure 1.: Hexagonal micro-structuresemployed beam lattices [10, 16, 19, 31, 34, 41, 47, 48], but also continuum models have been used[3, 23, 26, 28, 36, 45]. The quantitative comparison of DNS of tension tests and shearing tests in[23, 25] exhibited differences, even in normalized data for hexagonal structures between beam modelsand continuum models.The scope of the present contribution is to identify the reason for these differences. For this purpose,a systematic study of the effect of distribution of material in struts and alignment of struts in regularhexagonal microstructures on size effects and effective moduli in simple shearing, uniaxial tension andpure bending is performed.The present paper is structured as follows: Section 2 summarizes some basic geometric properties ofporous materials with hexagonal micro-structure and the employed finite element models, before thesubsequent sections 3–5 present models and results for simple shear, uniaxial loading and pure bending,respectively. Finally, section 6 closes with a summary.
2. Hexagonal micro-structures
Cells of a periodic hexagonal micro-structure with respective dimensions are depicted in Figure 1. Forlater normalisation of results, the diameter d = q √ /πl ≈ . l of a circle of equivalent area isintroduced as characteristic dimension. The porosity c is defined for all configurations as the areafraction c = A pore /A cell of pores.In the present study, slender rectangular specimens L × H are simulated under different loadingconditions, whose length is assumed to be much larger than height L/H → ∞ . Consequently, only arepeatable cutout of the structure is required in the DNS as shown in Figure 2.It is well-known that regular hexagonal structures have isotropic effective in-plain elastic moduli. Fora hexagonal porous structure with prismatic struts, these effective moduli have been determined byAshby and Gibson [6] based on beam theory as E eff = 4 √ E (cid:18) tl (cid:19)
11 + α E ( t/l ) G eff = 1 √ E (cid:18) tl (cid:19)
11 + α G ( t/l ) . (1)2 a) Orientation 1 (b) Orientation 2 Figure 2.: Cutouts (dashed rectangle) of rectangular specimens for beam modelsTable 1.: Coefficients for effective moduli in Eqs. (1) [6] α E α G bending, axial and shear deformation . . ν . . ν bending and axial deformation / only bending The coefficients α E and α G depend on whether axial and shear deformations of the struts are consideredor neglected and are listed in Table 1. Therein, E and ν refer to Young’s modulus and Poisson ratio ofthe bulk material, respectively. However, higher-order effects may still be anistropic. That is why twodistinguished orientations are employed in the present study as shown in Figure 2.For the finite element simulations, the unit cells have been meshed by triangular linear plane stresselements using the values ν = 0 . as shown in Figure 3. After some mesh convergence studies havebeen performed in advance, it was decided to employ meshes with an element edge length of l/ to l/ depending on porosity c . Such a mesh resolution is considerably finer than shown in Figure 3(unrecognizable if depicted), but is necessary for the structures with high porosity to ensure that thereis a number of elements within each cross section of the struts. The structures with prismatic strutshave been simulated additionally with Timoshenko beam elements.Figure 3.: Coarse finite element meshes of unit cells (the employed meshes are finer)3 e r i o d i c p e r i o d i c p e r i o d i c p e r i o d i c Figure 4.: Simple shearing of an infinite layer
3. Simple Shearing
Firstly, the behavior of an infinite layer under simple shearing is simulated. Due to the periodicity alongthe layer, only a single slice with thickness of a single unit cell needs to be resolved in the FEM modelas depicted in Figure 4. The periodicity conditions at the left and right faces read u I ( x ) = u III ( x ) , u I ( x ) = u III ( x ) , ϕ I ( x ) = ϕ III ( x ) . (2)Thereby and in the following, the condition with respect to the rotational degree of freedom ϕ appliesonly to beam models. Note that the cut-outs of the beam models are shifted quarter a period in x -direction compared to those of continuum models since beams must not be located directly at lines ofsymmetry. The boundary conditions at top and bottom read u II ( x ) = u II ( x ) = u IV ( x ) = 0 , u IV ( x ) = γH, ϕ II ( x ) = ϕ IV ( x ) = 0 , (3)thus assuming that the rods must not rotate at top and bottom, neither in continuum model nor inbeam model.Figure 5 shows the deformation of the struts for differently shaped pores and both orientations ofthe structure relative to the direction of loading. It can be seen, in particular for orientation 2 atthe right-hand side of Figure 5, that the rods in the center exhibit an S-shaped deformation, whichis not possible at the top and bottom faces. These hindered deformations imply a higher stiffness ofthe surface layer, which leads to a stiffening size effect, being well-known for simple shear of cellularmaterials [23, 28, 41, 42].In order to quantify the size effect, the apparent shear modulus of each specimen is extracted fromthe simulations as G app = F / ( Lγ ) , where F denotes the sum of nodal forces at the top surface IV inhorizontal direction. Due to linearity of the problem, G app is independent of the applied magnitude ofloading γ .Figures 6 and 7 show the extracted apparent shear moduli G app for both shapes of pores and for beammodels, for both orientations and a number of porosities c , each being normalized with the respectiveeffective modulus G eff of each micro-structure. The effective shear modulus is defined as G eff = lim H/d →∞ G app (4)for each micro-structure, i. e. for each combination of porosity, orientation and shape of pores. Practi-cally, it is extracted from the simulation with the largest height H for each material. Consequently, thenormalized plots exhibit an asymptote G app /G eff → (dashed line in the figures). Firstly, all figures4igure 5.: Local deformation during simple shear with differently aligned microstructures and circularand hexagonal pores for two stacked unit cells ( c = 0 . )show an increase of the maximum deviation of G app /G eff from 1 with increasing porosity c for all typesof models and both orientations. This maximum size effect is obtained for a height of the shear layerof a single unit cell.Figure 6a shows that beam model overestimates the size effect for lower values of porosity c = 0 . and c = 0 . . But for higher porosities c ≥ . , beam model and the volume model with hexagonal pore (andthus prismatic struts) yield virtually identical results as expected. Furthermore, the normalized curvesdo hardly change anymore when going to even higher porosities c , but a saturation of the maximumattainable size effect is reached with respect to c . Figure 6a incorporates also the results of Tekoğluet al. [41] with stochastic beam lattices, which exhibit an even stronger size effect than the presentregular hexagonal lattices.Comparing the results for both orientations in Figure 6b, a strong difference is observed with respectto the maximum attainable size effect, i. e. the maximum value of G app /G eff . This means that the sizeeffect is anisotropic, although the classical elastic properties of hexagonal structures are isotropic. Theanisotropy of non-classical effects in hexagonal structures has been reported also in [36].Figure 7 shows the respective data for a hexagonal arrangement of circular pores, which can be sim-ulated by volume models only. The same tendencies are found than with hexagonal pores in Figure 6b.However, a quantitative comparison with Figure 6b shows that stronger size effects are obtained withhexagonal pores in orientation 1 than with circular pores and that the difference between both orienta-tions is weaker with circular pores. This behavior is plausible insofar as non-interacting circular poreswould be isotropic. This finding complies with the comparison of results for circular pores with resultsfrom literature for hexagonal pores and beam models in [23]. To investigate this effect further, modelshave been created, which posses a rounding r at the connecting point of the rods as shown in Figure 8a.The extracted apparent shear moduli in Figure 8b show a continuous transition from the hexagonalpores ( r = 0 ) to the results of circular pores with increasing r .Note that the maximum attainable size effect G app /G eff ≈ . with any of the present regular hexago-nal foam materials lies considerably below the maximum value G app /G eff ≈ . , which has been reportedby Tekoğlu and Onck [42] for Voronoi-tesselated stochastic lattices of beams for the same loading con-ditions. Apparently, there is a significant contribution of the stochastic distribution. Liebenstein et al.[31] compared stochastic and regular lattice under simple shear. However, they considered shear lay-ers of finite width, where the stiffening size effect from the clamped faces competes with a softeningsize effect from unconnected struts at the free lateral faces, making a distinction between both effectsimpossible. 5 .002.003.004.00 0 5 10 15 G a pp / G e ff H / dc =0.3 c =0.5 c =0.8 c =0.9 beam c =0.3 c =0.5 c =0.8 c =0.9 volume stoch. (T.&al.'11) (a) orienta (cid:1) on 1orienta (cid:0) on 2 H / d G a pp / G e ff c =0.3 c =0.5 c =0.8 c =0.9 c =0.3 c =0.5 c =0.8 c =0.9 (b) Figure 6.: Apparent shear modulus for simple shear with hexagonal pore: (a) Comparison of volumemodel with regular and stochastic beam models (latter results from Tekoğlu et al. [41]) and(b) comparison between both orientations H / d G a pp / G e ff (a) G a pp / G e ff H / d c =0.3 c =0.5 c =0.7 c =0.8 (b) Figure 7.: Apparent shear modulus for simple shear with continuum model with circular pore6 a) G a pp / G e ff H / d hexagonal r = l /20 r = l /10 r = l /5 r = l /2circular r (b) Figure 8.: Effect of rounding r of pore on size effect in simple shear ( c = 0 . , orientation 1)Figure 9 compares the effective shear moduli from the DNS to common analytical models. Firstly, it isfound that the results of the Timoshenko beam models coincide virtually with the analytical solution (1)of Gibson and Ashby incorporating shear deformations, thus verifying the present implementation.The present results for the continuum model with hexagonal pores, and thus prismatic struts, complyqualitatively and quantitatively with those of Lee et al. [30] obtained by a computational unit cellanalysis with periodic boundary conditions (not shown here). For high porosities c & . , anagreement of beam model and continuum model with hexagonal pores is observed as expected. Forlower porosities, c . . , the beam model underestimates the effective shear moduli of the more exactcontinuum models with hexagonal and circular pores. At same porosity, circular pores lead to highereffective modulus compared to hexagonal pores as predicted qualitatively by Warren and Kraynik [43].Though, the model of Warren and Kraynik overestimates the effective modulus in this regime, cf. Leeet al. [30]. The relative difference in the effective moduli between both pore geometries decreaseswith further decreasing c as the interaction between neighboring pores decreases. In this regime, theDNS results lie between the estimates of the Mori-Tanaka scheme and differential scheme (both from[18]). The latter provides a good approximation from high to medium porosities. Remarkably, itis outperformed by the Gibson-Ashby model (1) incorporating only bending and axial deformations(i.e. based on Euler-Bernoulli beam theory). Apparently, the additional compliance due to the sheardeformations of the struts is overcompensated by the additional stiffness due to material concentrationat the nodes of the structure. The Gibson-Ashby model with bending but no axial deformations (thirdentry in Table 1) overestimates the effective modulus considerably (not shown). The composite shellapproach of Hashin [20], adopted to the plane case in form of a circular pore in a circular unit cell asoutlined in the appendix A, with kinematic or static boundary conditions (“kBC”/“sBC”), correspondingto the Hashin-Shtrikman bounds for the present problem, enclose the aforementioned data, but are toofar away to be of practical relevance. The Taylor-Voigt estimate yields yields an even higher upperbound as expected.
4. Uniaxial Loading
As second load case, the uniaxial tension or compression of an infinitely long specimen is investigatedby the simulation of a periodically continuable section as shown in Figure 10, making additional use of7 aylor-VoigtGibson-Ashby shearcomposite shell kBCbeam la (cid:2) cehexagonal porecircular poreMori-Tanakacomposite shell sBC1.00.90.80.70.60.50.40.30.30.20.100.40.5 G e (cid:3) / G Porosity c Di (cid:4) eren (cid:5) al schemeDNS of hexagonal struct.Analy (cid:6) cal modelsTaylor-VoigtMori-Tanakacomposite shell kBCDScompositeshell sBCG-A sh. Gibson-Ashby axialG-A ax. Figure 9.: Effective shear moduli from simple shearing tests in comparison with analytical modelsthe mirror symmetry. The essential boundary and periodicity conditions at the four faces read u I ( x ) = 0 , u III ( x ) = Lε, u I ( x ) = u III ( x ) , ϕ I ( x ) = ϕ III ( x ) , u II ( x ) = 0 , ϕ II ( x ) = 0 . (5)All other degrees of freedom are left unconstrained. Therein, ε denotes the macroscopic longitudinalstrain. The apparent Young’s modulus is extracted from each simulation as E app = F / ( Hε ) , whereby F denotes the sum of all reaction forces at the right face III. The effective Young’s modulus is definedin analogy to the previous shearing case for each material as E eff = lim H/d →∞ E app .Respective deformed configurations for small specimens with two unit cells in height are shown inFigure 11. It can be seen for orientation 2 at the right-hand side that the struts at the free surface attop get warped, which is not possible for the struts at the line of symmetry at bottom. In contrast,no significant difference in the deformation modes of struts at top and bottom can be observed fororientation 1 at the left-hand side of Figure 11.Consequently, virtually no size effect can be observed for orientation 1 as shown in Figure 12. Incontrast, Figure 13 indicates a considerable softening size effect for orientation 2, i. e. smaller specimenshave a smaller apparent Young’s modulus than larger ones in accordance with [31, 41]. Again, themagnitude of size effect increases with increasing porosity c . Note that the normalized curves forhexagonal pores in Figure 13a converge for c & . , whereas no such convergence can be observed forthe circular pores in Figure 13b. Rather, the size effect increases with circular pores up to the maximumporosity c max ≈ . , whereby the sensitivity with respect to changes of c even increases in this regime.Remarkably, such a type of behavior had been predicted for circular pores in [25] by a homogenizationapproach within the stress-gradient theory (other shapes of pores were not considered).Comparing the magnitudes of size effects for simple shear and uniaxial loading in figures 6, 7 and 13shows that a notable size effect in shearing can be observed for specimens H/d . , whereas a notable The term “stress-gradient theory” is not used uniquely in literature, but different non-equivalent formulations have beenproposed under this name. Within the present paper, this name refers solely to the formulation of Forest and Sab [14]. x x B u u symmetry H / p e r i o d i c p e r i o d i c Figure 10.: Uniaxial tension of an infinite layerFigure 11.: Deformed configurations in uniaxial tension
500 600 70
0 80
0 901 001 101 201 301 401 50 E a pp / E e ff H / d c =0 3 c =0 5 c =0 8 c =0 9 Figure 12.: Apparent Young’s modulus for uniaxial loading from continuum models with orientation 19 ..... H / d E a pp / E e ff c =0.3 c =0.5 c =0.8 c =0.9 (a) ...... E a pp / E e ff H / d c = c =0.90 (b) Figure 13.: Apparent Young’s modulus for uniaxial loading from continuum models with orientation 2deviation between E app and E eff can be observed with specimens of at least twice that cross section H/d . . . . . And a saturation of the size effect with increasing porosity was observed in Figure 7even with circular pores.Figure 14 shows the effect of the rounding radius r of the pore (cmp. Figure 8a) on the size effectunder uniaxial loading. It is found that the magnitude of size effect increases with rounding radius.This the opposite trend than observed for simple shear in Figure 8. However, the effect of r is weakerunder uniaxial loading than for shearing.Figure 15 compares the predictions of the present DNS of regular hexagonal structures with stochasticDNS of Tekoğlu et al. [41], experimental data of Andrews et al. [5] and the aforementioned predictionsof Hütter et al. [25] obtained by means of a homogenization approach within the stress-gradient theory.Both, the present DNS with circular pores and the stochastic beam models of Tekoğlu et al. [41] agreewith the experimental data in principle. The data points from the experiments look as if the investigatedspecimens have not been large enough to reach the asymptotic effective modulus E eff . Comparing thedata of Tekoğlu et al. [41] with the present DNS shows that the stochastic structure exhibits a similarsize effect than the present data for regular structures with prismatic struts, in contrast to the previousshearing case. The stress-gradient theory gives a very good prediction for c = 0 . , but underestimatesthe strong increase of the size effect when going to c = 0 . . This discrepancy is surely related tothe simplified geometry of a circular pore in a circular volume element, which has been obtained forhomogenization within the stress-gradient theory.A plot of the ratio of effective Young’s modulus E eff and respective matrix value E versus the porosity c is qualitatively even almost quantitatively identical to the respective plot for the shear moduli inFigure 9 (as the effective Poisson ratio depends only very weakly on c ) and thus omitted here.
5. Pure Bending
The last considered loading case is pure bending of an infinitely long and thus ideally slender beam,implemented as sketched in Figure 16. The essential boundary and periodicity conditions at the left10 a pp / E e ff H / d
51 2 3 4 6 hexagonal r = l /20 r = l /10 r = l /5 r = l /2circular Figure 14.: Effect of rounding r of pore on size effect under uniaxial loading ( c = 0 . , orientation 2) ....... e xp. Alporasexp. Duocellstoch. (T.&al.'11) SG c =0.80SG c =0.90 E a pp / E e ff c = c =0.80 c =0.90 H / d Figure 15.: Size effect in apparent Young’s modulus of regular hexagonal structure (orientation 2 withcircular pore) in comparison to stochastic beam models of Tekoğlu et al. [41], experimentaldata of Andrews et al. [5] and predictions of stress-gradient theory (“SG”) from [25]11 B p e r i o d i c p e r i o d i c p e r i o d i c x x x Figure 16.: Pure bending of an infinitely long beamFigure 17.: Deformed configurations in pure bending ( c = 0 . )and right faces are formulated according to the classical beam theory as u I ( x ) = 0 , u I ( x = 0) = 0 , u III ( x ) = − κLx , u III ( x ) = u I ( x ) + 12 κL , ϕ III ( x ) = ϕ I ( x ) + κL, (6)whereby κ denotes the mean curvature of the neutral axis x = 0 . The top and bottom surfaces are freeand are thus left unconstrained (trivial natural boundary conditions). The bending moment is extractedfrom the simulations as M = P i x F ( i )1 from the horizontal reaction forces F ( i )1 of all nodes i at the rightface. Reaction moments as potential contributions to M for beam models turned out to be negligible.The bending stiffness is then defined as K = M/κ and normalized to the value K class = E eff H / ,which would be expected from the classical theory of bending. Thereby, values of the effective Young’smodulus E eff from uniaxial loading in Section 4 are used.The local deformations under bending are shown in Figure 17 for both orientations and both shapesof pores. It can be seen that the unit cells at the free surfaces at bottom and top are deformed similarlythan under tension, compare Figure 11.Figure 18 shows the normalized bending stiffnesses from the beam models. Firstly, it can be seenthat all curves approach K = K class asymptotically for H/d → ∞ as expected, thus verifying theemployed periodicity and boundary conditions (6). A negative size effect (smaller specimens are less12 (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) H / d K / K c(cid:17)(cid:18)(cid:19)(cid:20) c = c =0.5 c =0.8 c =0.9 (a) Orientation 1 ........ stoch. (T.&al.'11) K / K (cid:21)(cid:22)(cid:23)(cid:24)(cid:25) H / d c = (cid:26)(cid:27)(cid:28) c = c =0.8 c =0.9 (b) Orientation 2 Figure 18.: Bending stiffness with beam models for both models in comparison with results for stochasticbeam lattices from Tekoğlu et al. [41]stiff than expected) is observed for both orientations, in contrast to uniaxial loading. Furthermore, thenormalized curves converge towards an asymptotic one with increasing porosity c for both orientations.Though, the maximum deviation from the classical solution is considerably larger for orientation 2 thanfor orientation 1. Having an additional look at the abscissas in Figure 18a and Figure 18b shows, thata notable size effect is obtained till H/d . for orientation 1, but even till H/d . for orientation 2.Figure 18b incorporates additionally the results of Tekoğlu et al. [41] with stochastic beam lattices ofhigh porosity c > . . It is found that the size effect with stochastic lattice is even slightly larger thanthe large-porosity curve with regular lattices in orientation 2 as it was found already for the shearingcase.A negative size effect was observed already for orientation 1 in [45] and has been attributed to theunconnected struts at the free surface, which do not carry any load and thus form effectively a stress-freesurface layer of half a unit cell thickness. This is why the size effect for orientation 1 in Figure 18adepends only weakly on the porosity c . Figure 17 shows that the negative size effect for orientation2 arises rather from a stronger bending deformation of the struts close to the specimen surface. Thisdifferent mechanism explains the strong effect of c on the size effect in Figure 18b. Wheel et al. [45]observed even positive size effects for configurations where the specimen surface consists of continuousmatrix material. However, such a configuration can be found in the considered regular hexagonal micro-structures only for low porosities, but not for the present foam-like materials. Though, positive sizeeffects under bending have been also observed experimentally by Lakes et al. [4, 37, 46]. They cannotbe explained by the present highly idealized model.Figure 19 shows the extracted bending stiffnesses of the different models in normalized form. It canbe seen that the beam model complies to the results of the continuum model with hexagonal pores forsufficiently high porosities c & . . For a medium porosity c = 0 . , the models with hexagonal andcircular pores predict the same size effect. For the higher porosity c = 0 . , the predicted size effect withcircular pores is even stronger than the size effect with hexagonal pores. This is the opposite trend thanobserved for shearing.Figure 20 compares the size effects for hexagonal and circular pores in more detail and includinghigher porosities for orientation 2. Again, hexagonal pores lead to a saturation of the size effect for c & . (Figure 20a), whereas no such saturation is observed for circular pores in Figure 20b.Figure 20b incorporates additionally the respective prediction of the stress-gradient theory (Ap-13 / K (cid:29)(cid:30)(cid:31) ! H / " c = Figure 19.: Size effects in bending: Comparison of models (orientation 2) ...... H / d K / K ’()*+ (a) ...... K / K ,-/12 H / (b) Figure 20.: Size effects in bending in orientation 2 for (a) hexagonal and (b) circular pores in comparisonto predictions of stress-gradient theory (SG)14endix B) with higher-order constitutive parameters derived by homogenization with a composite-shellapproach in [25]. It can be observed that the stress gradient theory provides very good predictions formedium porosities . ≤ c ≤ . . For a high porosity c = 0 . the stress-gradient theory underestimatesthe size effect, presumably due to the employed composite shell approach which cannot reflect the poreinteractions adequately for large porosities. For a lower porosity c = 0 . , the stress-gradient theoryoverestimates the size effect. Note that the composite-shell approach yields isotropic behavior and canthus not reflect the anisotropy. Though, the stress-gradient theory itself requires a sixth-order tensorfor the higher-order relation which could reflect an anisotropy in principle [7].
6. Summary and conclusions
The present contribution employed a plane, regular hexagonal arrangement of pores as minimum modelsto investigate size effects of porous and cellular materials under simple shear, uniaxial loading and purebending. Considering theoretically infinitely long specimens excludes potential coupling with effects oflateral surfaces. From a numerical point of view, this approach allows to resolve many more unit cellsover the cross section than in previous studies from literature with specimens of finite length. Firstly,it can be said that the present results comply qualitatively with the known trends from literaturefor (regular and disturbed) honeycomb structures, that positive size effects occur under simply shear[8, 26, 28, 31, 41] and negative size under uniaxial loading or bending [31, 41]. This facts contradicts theprediction of Abdoul-Anziz and Seppecher [1] that honeycomb lattices with low relative density shouldbehave purely classically. In the present contribution, models with circularly and hexagonally-shapedpores have been investigated and compared. The latter have prismatic struts, which have been modeledeither as beams or as plane stress continua. Furthermore, two orientations of the lattice relative tothe axes of loading have been investigated. It was found that the classical effective elastic moduli areindependent of the orientation as expected and that beam models are suitable for porosities largerthan about 70%. But the size effects depend considerably on the alignment between lattice orientationand direction of loading. This means that the size effects are anisotropic, as can be anticipated fromsymmetry considerations in generalized continuum theories [7]. In addition, the negative size effectsunder tension and bending, which originate from free surfaces, depend strongly on porosity and shape ofthe pores and thus on the material distribution along the struts. These size effects have been interpretedsuccessfully by the stress-gradient theory, for which the bending solution was derived. Under simpleshear, a positive (stiffening) size effect was observed which shows a significant anisotropy for hexagonalpores and thus prismatic struts. The magnitudes of size effects in simple shearing are smaller inmagnitude than for bending and uniaxial loading. In particular, it turned out that the rule of thumbthat classical homogenization would be applicable when the ratio of scale separation
H/d is larger than10 [2] is not strict enough for bending and uniaxial loading at high porosities. A comparison of thepresent results from regular beam lattices with respective date from stochastic lattices from literature[41] showed that the size effects may be even stronger for the stochastic lattices.With its systematic sensitivity study, the present contribution provides a comprehensive data basefor the benchmark of generalized continuum theories. Size effects in honeycomb structures have beenmodeled macroscopically by (first) strain-gradient [8, 9, 36], Cosserat [9, 11, 16, 23, 31, 41], (firstorder) micromorphic [28] or (first) stress-gradient theories [25]. First strain gradient and micromorphictheories, including the special case of Cosserat and couple stress theory, predict only positive size effectswhereas the first stress gradient theory predicts only softening size effects. Thus, a second gradienttheory or a second order micromorphic theory is necessary to describe the appearance of both type ofeffects for a single material under different loading conditions [13]. Future studies need to investigatethe size effects for more realistic 3D models, potentially with a view on the rapidly evolving field ofarchitectured materials. 15 eferences [1] Abdoul-Anziz, H., Seppecher, P., 2018. Strain gradient and generalized continua obtained by ho-mogenizing frame lattices. Mathematics and mechanics of complex systems 6 (3), 213–250.[2] Ameen, M. M., Peerlings, R. H. J., Geers, M. G. D., 2018. A quantitative assessment of the scaleseparation limits of classical and higher-order asymptotic homogenization. Eur. J. Mech. A-Solid.71, 89–100.[3] Ameen, M. M., Rokoš, O., Peerlings, R. H. J., Geers, M. G. D., 2018. Size effects in nonlinearperiodic materials exhibiting reversible pattern transformations. Mech. Mater. 124, 55–70.[4] Anderson, W. B., Lakes, R. S., 1994. 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A. Composite Shell Homogenization for the Plane Case
The effective moduli (in-plane bulk modulus and shear modulus) of materials with circular pores (orinclusions) have been determined in [23] for kinematic boundary conditions and plane stresses as K eff = E
12 1 − c − ν + c (1 + ν ) (7) G eff = G (cid:0) − (1 + ν )(1 − c ) (cid:1) (1 − c )(3 − ν )(1 + ν + (3 − ν ) c + (1 + ν ) c ) + (1 + ν ) c (3 − c + 4 c ) (8)The effective Young’s modulus follows as E eff = 4 G eff K eff / ( G eff + K eff ) . The corresponding effectiveshear modulus under statically uniform boundary conditions was determined in [24] as G eff = G (1 − c ) (1 + ν )1 + ν + (1 − ν ) c + (7 + 3 ν ) c + (3 − ν ) c . (9)The in-plane bulk modulus K eff in eq. (7) is the same for both types of boundary conditions. B. Bending Solution in Stress-Gradient Theory
The stress-gradient theory of Forest and Sab [14] introduces the field of “micro-displacements” Φ ijk as work-conjugate quantity to the gradient R ijk := ∂σ ij /∂x k of the Cauchy stress σ ij . The lattersatisfies the conventional balances ∂σ ij /∂x i = 0 and σ ij = σ ji . Energetic considerations imply that thekinematic relation ε ij = 12 (cid:18) ∂u i ∂x j + ∂u j ∂x i (cid:19) + ∂ Φ ijk ∂x k (10)18or the strain tensor ε ij must not involve only the symmetric part of the gradient of the conventionaldisplacement vector u i , but additionally the divergence of micro-displacement tensor Φ ijk . The linear-elastic non-classical constitutive relations for an isotropic material read for the plane case in a Voigt-type notation (cid:18) Φ (cid:19) = ˆ B · (cid:18) R R (cid:19) , (cid:18) Φ (cid:19) = ˆ B · (cid:18) R R (cid:19) . (11)Further details like symmetries of R ijk and Φ ijk can be found in [14]. The symmetric higher-ordercompliance matrix ˆ B was identified for a porous material by homogenization in [25].The stress-gradient solution for the pure bending problem in Figure 21 is obtained by the ansatz that
M Mx x x Figure 21.: Pure bendingthe only non-vanishing stress component σ = σ ( x ) depends only on the distance x to the neutralaxis. Hooke’s law for plane stresses thus yields ε = σ ( x ) /E eff and ε = − νσ ( x ) /E eff . Insertingthe only non-vanishing component of the stress gradient R = σ ′ ( x ) to the higher-order constitutivelaw (11) yields Φ = B σ ′ ( x ) and Φ = 1 / B σ ′ ( x ) . These micro-displacements and strainscan be inserted into the kinematic relation (10) to obtain the ODEs σ − E eff B σ ′′ = ∂u ∂x , νσ + 13 E eff B σ ′′ = ∂u ∂x . (12)It is known from micromorphic theories [15, 22] that the Bernoulli hypothesis remains valid and non-classical terms affect only the lateral contraction u lat . Thus, we choose ∂u /∂x = κx and ∂u /∂x = u ′ lat ( x ) with curvature κ =const (compare (6)). The first ODE from (12) thus yields σ = E eff κ " x − H sinh( x ℓ )2 sinh( H ℓ ) . (13)Therein, ℓ = √ E eff B abbreviates an internal characteristic length scale (identical to uniaxial loadingin [25]). The constants of integration in (13) have been determined by the trivial boundary condition σ ( ± H/
2) = 0 . A plot of the stress distribution in Figure 22a exhibits similarities with the torsionsolution in [29]. The second equation (12) would now allow to determine the lateral contraction u lat ( x ) . Finally, the bending stiffness K is obtained from the total strain-energy in each cross section H/ Z − H/
12 [ σ ij ε ij + R ijk Φ ijk ] d x = 12 H/ Z − H/ (cid:20) E eff σ + B ( σ ′ ) (cid:21) d x = 12 Kκ (14)as K = E eff H " − ℓ H H ℓ tanh (cid:0) H ℓ (cid:1) − ! . (15)Obviously, the term in the square bracket reflects the size effect. Its plot in Figure 22b shows thatthe stress-gradient theory predicts a softening size effect as known from uniaxial loading and torsion[25, 29]. The linear-elastic constitutive relation between R ijk and Φ kmp is formed in general by a tensor of sixth-order tensor,which may be anisotropic for materials with hexagonal structure [7]. . . . . . . . . σ E e ff κ H x /HH/ℓ = 5 H/ℓ = 10
H/ℓ = 20 lass. (a) . . . .
81 0 50 100 150 200 K / K c l a ss H/ℓ (b)(b)