Elasticity of Random Multihase Materials: Percolation of the Stiffness Tensor The MIT Faculty has made this article oenly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Chen, Ying, and Christoher A. Schuh. Elasticity of Random Multihase Materials: Percolation of the Stiffness Tensor. Journal of Statistical Physics 62. (206): 232 24. htt://dx.doi.org/0.007/s0955-05-387-6 Sringer US Version Author's final manuscrit Accessed Wed Dec 9 2:4:42 EST 208 Citable Link htt://hdl.handle.net/72./0594 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms htt://creativecommons.org/licenses/by-nc-sa/4.0/
J Stat Phys (206) 62:232 24 DOI 0.007/s0955-05-387-6 Elasticity of Random Multihase Materials: Percolation of the Stiffness Tensor Ying Chen Christoher A. Schuh 2 Received: 5 May 205 / Acceted: 30 Setember 205 / Published online: 5 October 205 Sringer Science+Business Media New York 205 Abstract Toology and ercolation effects lay an imortant role in heterogeneous materials, but have rarely been studied for higher-order tensor roerties. We exlore the effective elastic roerties of random multihase materials using a combination of continuum comutational simulations and analytical theories. The effective shear and bulk moduli of a class of symmetric-cell random comosites with high hase contrasts are determined, and reveal shortcomings of classical homogenization theories in redicting elastic roerties of ercolating systems. The effective shear modulus exhibits tyical ercolation behavior, but with its ercolation threshold shifting with the contrast in hase bulk moduli. On the contrary, the effective bulk modulus does not exhibit intrinsic ercolation but does show an aarent or extrinsic ercolation transition due to cross effects between shear and bulk moduli. We also roose an emirical aroach for bridging ercolation and homogenization theories and redicting the effective shear and bulk moduli in a manner consistent with the simulations. Keywords Percolation Homogenization theory Elastic theory Effective elastic moduli Random multihase materials Comosite materials Predicting the effective elastic roerties of heterogeneous materials is a fundamental interdiscilinary roblem in hysics, materials, and mechanics due to the ervasive resence of random multihase materials in nature and engineering [,2]. Many comosite homogenization theories for effective elastic roerties have quantitatively incororated the effects of hase geometry and anisotroy, but largely overlook the role of disorder, toology, and er- B Ying Chen cheny20@ri.edu Deartment of Materials Science and Engineering, Rensselaer Polytechnic Institute, 0 8th Street, Troy, NY 280, USA 2 Deartment of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 0239, USA
Elasticity of Random Multihase Materials... 233 colation [3 5]. On the other hand, the ercolation community has exlored elastic rigidity ercolation [6 ] and couled diffusion-force ercolation [2,3] on random networks or lattices as well as the concet of continuum ercolation [4 20], which deals with toological ercolation effects in multihase solids. However, existing studies on ercolation are mostly limited to isotroic linear transort roerties where the second-order transort coefficient tensor for each hase reduces to a scalar [2 23]. Studies on continuum ercolation also tend to estimate effective roerties by maing the continuum to a discrete lattice [24 27], which is now known to roduce artifacts as comared to a true continuum resonse [28]. Elucidating ercolation effects in elasticity and incororating them in continuum homogenization theories could significantly advance mesoscale redictive caabilities that connect heterogeneous microstructures to macroscoic roerties. In this aer, we establish the effective elastic roerties of random multihase materials by continuum calculations that quantitatively cature toology and ercolation effects, and also demonstrate the deficiencies of classical homogenization theories when alied to ercolating systems. We further examine an aroach that bridges ercolation and homogenization theories and elucidates the observed disarate ercolation behavior in bulk and shear moduli. Our model system is a two-dimensional square domain containing 2766 cells from hexagonal tessellations. Each cell is randomly assigned a hase identity (either hase or hase 2) with a desired robability, and inter-hase boundaries are assumed to be erfect interfaces with continuity of field variables required across them. This model system belongs to a class of materials widely known as symmetric-cell materials [4]. Each hase is described by a stiffness tensor C that relates strain ε to stress σ via Hooke s law: σ ij = C ijkl ε kl. Under the assumtion of isotroy, there are two indeendent elastic constants in C, e.g., the bulk modulus K and the shear modulus G. σ ij = K δ ij ε kk + 2Gε ij () where ε kk is the volumetric strain, ε ij is the deviatoric strain, and δ ij is the Kronecker delta. Under the lane strain condition that is alied to the system, the indices i, j, k can be or 2, and K differs from its three-dimensional counterart while G remains the same. We imose an orthogonal mixed boundary condition (where dislacements are alied on all sides of the secimen without friction), as the static (uniform traction) and kinematic (uniform dislacement) conditions usually lead to a lower and uer bound for the effective elastic roerties [29,30]. Finite element calculations are carried out using quadratic triangular elements and an adative re-meshing technique that yields finer meshes at high stress locations, to ensure a reliable and sufficiently resolved continuum solution for the stress and strain fields; the mesh size is therefore generally much smaller than the hase domain size. The effective lane strain bulk and shear moduli, K ef f and G ef f, are defined through σ ij =K ef f δ ij ε kk +2G ef f ε ij (2) where <> denotes an average over the entire samle. We start with a simle case where the two hases have equal bulk moduli (K = K 2 = K = 6.25 0 6 ) but different shear moduli (G =5 0 2, G 2 =5 0 6 ). The units for all moduli and stresses are the same (and are arbitrary). As shown in Fig. a, the effective shear modulus G ef f increases nonlinearly with the fraction of hase 2,. The simulation results lie between the widely-searated, rigorous Hashin lower and uer bounds for effective shear modulus of isotroic comosites (drawn as blue dashed lines) [3]. G ( ) = G + G 2 G + ( )(K +2G ) 2G (K +G ) (3)
234 Y. Chen, C. A. Schuh Fig. (Color online) a The effective shear modulus G e f f when the two hases have equal bulk moduli (K = K 2 = 6.25 06 ) but different shear moduli (G = 5 02 and G 2 = 5 06 ) as a function of the volume fraction of hase 2. The blue dashed lines are the Hashin lower and uer bounds for G e f f [Eqs. (3 4)]. The red dash-dot line shows the redictions from the Budiansky EMT equation [Eq. (5)]. The two green solid lines result from the GEM equation [Eq. (6)] using two sets of scaling exonents. b e The distribution of shear stress τ2 at different hase fractions, showing the develoment of ercolative stress aths with increasing hase fraction.
Elasticity of Random Multihase Materials... 235 G (+) = G 2 + G G 2 + (K 2+2G 2 ) 2G 2 (K 2 +G 2 ) Figure a also comares the simulation results with Budiansky effective medium theory (EMT) for G ef f [32], which was derived based on the assumtion that each article is embedded in an effective medium and can be described by the Eshelby solution [33]. G G ef f G 2 G ef f ( ) G + (β + )G ef f G 2 + (β = 0 (5) )G ef f where the strain roortionality arameter β = (6K + 0G ef f )/(5K + 5G ef f ) in three dimensions and (3K + 5G ef f )/(6K + 4G ef f ) in two dimensions for the case of K = K 2 = K. Predictions from Eq. (5) were lower than the simulation results when < 0.5 but overshoot at higher. Equation (5) imlicitly assumes a ercolation threshold at c = β (regardless of hase geometry and toology), which is deendent on hase bulk moduli and is aroximately 3/6 = 0.5 in the resent two dimensional case (as K > G 2 G and therefore K G ef f ). Equation (5) also tacitly assumes scaling exonents of unity around the ercolation threshold. These imlicit assumtions are made clear by comarison with the so-called generalized effective medium (GEM) equation [34,35], which is a semi-emirical mixing rule that conforms to the well-known ower-law scalings of ercolation theory near the threshold: ( ) G /s G /s ef f G /s + (c )G /s ef f + G /t 2 G /t ef f G /t 2 + (c )G /t ef f (4) = 0 (6) The GEM equation Eq. (6) incororates the ercolation threshold c in the same osition as β from Eq. (5). It also incororates the ercolation scaling exonents s and t, and imlicitly catures the couling between K and G ef f through c and ossibly s and t. In the resent elasticity roblem, the relevant ercolation threshold and scaling exonents are not known a riori, so we roceed by fitting the data as follows. Considering G ef f G G 2 as ercolating (ercolation robability is one) and otherwise not ercolating (ercolation robability is zero), the average ercolation robability at a certain hase fraction can be evaluated by averaging over the ercolation robability of many simulations run at this value. The square root roerty roduct ( G G 2 ) is chosen for its secial significance in ercolation theory, being derived from hase-interchange or duality relations [,4]. We thus obtain the average ercolation robability () from our data and define c as the hase fraction corresonding to = 0.5 (see the inset in Fig. 3a showing as a function of ) [36]. c is found to be 0.503, which haens to be similar to the geometric connectivity or conductivity ercolation threshold of 0.5 for the resent hexagonal cell geometry (i.e., for site ercolation on its dual, triangular lattice) [36]. Subsequently, fitting Eq. (7) { (c ) G ef f () s for < c ( c ) t (7) for > c to simulation data on the range c 0.2 leads to s.38, t.26, both of which haen to be similar to s = t =.3 for conductivity roblems [37]. Predictions from Eq. (6) using the extracted c, s, andt values, lotted as the green solid line in Fig. a, are close to simulation data although slightly underestimate below c and overestimate above c. Fitting Eq. (7) to a much broader range of simulation data ( c 0.4) results in s.55 and t.55 (which are no longer ercolation scaling exonents, strictly seaking), and using them in Eq. (6) leads to a slightly better fit (green dashed line in Fig. a). Figure b e
236 Y.Chen,C.A.Schuh Fig. 2 (Color online) The effective bulk modulus K ef f of comosite materials comrised of two hases with the same shear moduli (G = G 2 =5 0 2 ) but different bulk moduli (K =6.25 0 2 and K 2 = 6.25 0 6 ) as a function of hase fraction. Theblack circles reresent simulation results, which are in excellent agreement with redictions of the Hashin Hill equation [Eq. (0)] drawn as the green dashed line. The ressure distribution at = 0.5 shown as theinset is consistent with Hill s theory that roosed iecewise ressure (or volumetric strain) when G = G 2. show that with increasing, ercolative shear stress aths emerge and grow, suggesting that shearing in materials is toology-deendent and that G ef f exhibits ercolation behavior. At low, there is some localized shear stress concentration. At high, ercolating aths form, involving regions under both ositive and negative high shear stresses because of deformation comatibility. We turn now to the oosite case where the two hases have the same shear moduli but different bulk moduli. The effective bulk modulus K ef f as a function of is lotted in Fig. 2. The Hashin bounds for K ef f [3]are K ( ) = K + K (+) = K 2 + K 2 K + K K 2 + (8) K +G (9) K 2 +G 2 When G = G 2 = G, the lower and uer bounds coincide (K ( ) = K (+) ),andk ef f can be calculated exactly from either equation regardless of the hase geometry [3,38 40]. K ef f = K + K 2 K + K +G = K 2 + K K 2 + K 2 +G Our simulation results agree with redictions from Eq. (0) in the entire range of as shown in Fig. 2. The curve shae resembles a lower bound, which has a ercolation threshold at unity and therefore never ercolates. K ef f does not exhibit intrinsic ercolation. This is a very suggestive result, imlying that volume change in materials (as modulated by the bulk modulus) is intrinsically indeendent of toology. Equation (0) being alicable to any hase geometry corroborates this hyothesis. (0)
Elasticity of Random Multihase Materials... 237 Hill [38,39] roosedthatasufficientconditionforeq. (0)tobetrueistohaveiecewise constant volumetric strain (or ressure) in each hase. If the volumetric strains are e and e 2 in the two hases resectively, and the ressures are P and P 2,then K ef f = f P + f 2 P 2 = f K e + f 2 K 2 e 2 () f e + f 2 e 2 f e + f 2 e 2 An equivalence between Eqs. (0) and() would require that e /e 2 =(K 2 + G)/(K + G) or P /P 2 =[K (K 2 + G)]/[K 2 (K + G)], which renders the dislacements and normal tractions continuous at hase boundaries and also satisfies the equilibrium and comatibility requirements. The ressure in our simulations is indeed iecewise constant (e.g., see the inset in Fig. 2), although the individual stress comonents are not. Also, although the shear moduli of the two hases are the same, the shear stress is not uniform in the material (not shown). Now we move on to the more general case where the two hases have different bulk and shear moduli. We first examine the effects of the hase bulk moduli contrast, K 2 /K, on both G ef f and K ef f by gradually decreasing K while fixing K 2, G, and G 2 (G 2 /G =0 4 ). Shown in Fig. 3 are the redicted G ef f in (a) and K ef f in (b) for K 2 /K =, 0, 0 2,0 3, and 0 4.InFig.3a, G ef f decreases with decreasing K, subtly and nonlinearly; G ef f values for K 2 /K =0 2 and 0 3 (blue and green data oints) do not always lie in between those for K 2 /K =and0 4, but rather coincide with the data for K 2 /K =atlow and with the data for K 2 /K =0 4 at high. The values of the ercolation threshold c are evaluated from the ercolation robability as shown in the bottom right inset, and are assembled in the to left inset in Fig. 3a, together with the scaling exonents s and t from fitting G ef f to Eq. (7) on the range c 0.2. With increasing K 2 /K, c increases from 0.503 to 0.558. s and t values for all K 2 /K contrasts are within the range of.3 ± 0. (excet for s =.48at K 2 /K =0 4 ), which resents a similarity with the universality class of conductivity roblems. Furthermore, the effect of K 2 /K on G ef f resembles that of short-range correlations [34], which cause a small shift in the ercolation threshold but do not alter scaling. We further run a second set of simulations where G 2 /G =. 0 6 is much higher and observe the same trend in the increase in c as K 2 /K is increased from 30 to 3 0 5 (data not shown). With an extremely high K 2 /K (relative to G 2 /G ), however, the effect of hase bulk moduli may become very significant and scaling behavior may change. In Fig. 3b, with decreasing K (increasing K 2 /K ), K ef f curves shift downward in the low range while reserving the overall curve shae. These curves for K ef f all show ercolation features, although more evidentathigherk 2 /K.AsK ef f in the case of equal shear moduli (G = G 2 ) shown in Fig. 2 does not ercolate, we suggest that the aarently extrinsic ercolation in K ef f seen in Fig. 3b originates from the intrinsic ercolation in G ef f shown in Fig. 3a in the resence of shear moduli contrast (G 2 /G =0 4 ). Figure 4 shows the effect of the hase shear moduli contrast, G 2 /G,onG ef f in (a) and K ef f in (b), obtained by gradually increasing G 2 (i.e., increasing G 2 /G from to 0 4 ) while fixing K, K 2 and G.InFig.4a, with increasing G 2, G ef f increases in a highly nonlinear fashion, by a larger extent at higher, but all curves have a shae characteristic of ercolation. In Fig. 4b, we observe an increase in K ef f as G 2 is increased, esecially at high. Interestingly, there is a drastic change in the curve shae as G 2 increases, from having a single curvature to incurring an inflection in curvature. In articular, the red data oints for the case of G 2 /G =0 4 show a shae clearly indicating a ercolation transition. We again suggest that such transition in the behavior of K ef f is extrinsic, and is triggered by the increasing dominance of ercolation in G ef f at high G 2 /G as shown in Fig. 4a. In light of the above observations, we roose that the Hashin Hill equation [Eq. (0)], written for the case of two hases having the same shear modulus, may also be rofitably
238 Y.Chen,C.A.Schuh Fig. 3 (Color online) The effective shear modulus G ef f (a) and effective bulk modulus K ef f (b) offive families of comosite materials with different K, while other hase moduli K 2, G and G 2 are all fixed (G 2 /G is held at a high contrast of 0 4 ).InFig.3a, the ercolation threshold c and the scaling exonents s and t extracted from G ef f data are assembled in the to left inset,andthebottom right inset shows how c is determined from the ercolation robability.theblack solid lines in Fig. 3b are redictions of K ef f from the modified Hashin Hill equation [Eq. (2)]. alied to comosites containing hases of different shear moduli, if we relace the common shear modulus by an effective shear modulus. In other words, we hyothesize that Eq. (0) is still alicable when G = G 2, rovided that G in the equation is relaced by G ef f. K ef f = K + K 2 K + = K 2 + (2) K +G ef f K K 2 + K 2 +G ef f For the data sets in Figs. 3 and 4, G ef f was already obtained from simulations as shown in Figs. 3a and4a. Using these G ef f results, Eq. (2) can redict K ef f values, which are
Elasticity of Random Multihase Materials... 239 Fig. 4 (Color online) The effective shear modulus G ef f (a) and effective bulk modulus K ef f (b) offive families of comosite materials with different G 2, while other hase moduli K, K 2 and G are all fixed (K 2 /K is held at a high contrast of 0 4 ).Theblack solid lines in (b) are redictions of K ef f from the modified Hashin Hill equation [Eq. (2)]. lotted as black solid lines in Figs. 3band4b. The redicted K ef f is in excellent agreement with K ef f obtained from simulations for all curves regardless of whether individual hase bulk or shear moduli are being changed, which is remarkable considering the random hase distribution, high hase contrast, and ercolation effects in G ef f resent in our model system. This discovery thus rovides an aroximation (rather than a rigorously-derived relation) to redict K ef f once G ef f is known or can be obtained by other means, by modifying the Hashin Hill equation to incororate G ef f. Given how wide the rigorous Hashin bounds on K ef f are, the ability to accurately redict K ef f for arbitrary values of G of the hases should
240 Y.Chen,C.A.Schuh offer some ractical value, and also affirms the utility of introducing ercolation concets into the field of continuum elasticity. The results shown in Figs. 3b and 4b also suort our rior conjecture that ercolation in K ef f is extrinsic and arises from the intrinsic ercolation in G ef f. Analytically, G ef f may be determined from the GEM equation Eq. (6). Although Eq. (6) does not exlicitly incororate the effect of hase bulk moduli (or K ef f ) on G ef f, it incororates ercolation threshold c and scaling exonents s and t that change with hase bulk moduli. The combination of Eqs. (6)and(2) can thus be used to simultaneously redict G ef f and K ef f for random multihase materials with a high degree of accuracy. In summary, while most existing ercolation studies deal with scalar roerties, we have exlored ercolation of a tyical higher-rank tensor that cannot be reduced to a scalar even for isotroic materials. Assessing the effective bulk and shear moduli, K ef f and G ef f,of random multihase materials by a variety of continuum calculations, we showed that G ef f exhibits tyical, intrinsic, ercolation behavior, with scaling exonents similar to those seen in conductivity ercolation roblems. The ercolation threshold for G ef f,however,shifts slightly with the contrast in hase bulk moduli. On the contrary, our results suggest that K ef f does not exhibit intrinsic ercolation. Rather, the cross effects between bulk and shear moduli give rise to what we refer to as extrinsic ercolation in K ef f, originating from the ercolation in G ef f. This observation allowed us to roose that by incororating G ef f in the Hashin Hill equation, one can redict K ef f even for systems with different hase shear moduli. Acknowledgments Y. Chen acknowledges the suort from RPI start-u fund. C.A.S. acknowledges the suort of the U.S. National Science Foundation, under contract CMMI-332789. Both authors enjoyed the suort of the U.S. National Science Foundation under contract DMR-0346848 in the early stages of this work. References. Milton, G.W.: The Theory of Comosites. Cambridge University Press, Cambridge (2002) 2. Sahimi, M.: Heterogeneous Materials : Linear Transort and Otical Proerties. Sringer, New York (2003) 3. Kim, I.C., Torquato, S.: Effective conductivity of susensions of hard sheres by Brownian motion simulation. J. Al. Phys. 69, 2280 2289 (99) 4. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscoic Proerties. Sringer, New York (2002) 5. Choy, T.C.: Effective Medium Theory. Clarendon Press, Oxford (999) 6. Kantor, Y., Webman, I.: Elastic roerties of random ercolating systems. Phys. Rev. Lett. 52,89 894 (984) 7. Moukarzel, C., Duxbury, P.M.: Stressed backbone and elasticity of random central-force systems. Phys. Rev. Lett 75, 4055 4058 (995) 8. Latva-Kokko, M., Timonen, J.: Rigidity of random networks of stiff fibers in the low-density limit. Phys. Rev. E. 64, 0667 (200) 9. Zhou, Z.C., Joos, B., Lai, P.Y.: Elasticity of randomly diluted central force networks under tension. Phys. Rev. E. 68, 0550 (2003) 0. Tighe, B.P., Socolar, J.E.S., Schaeffer, D.G., Mitchener, W.G., Huber, M.L.: Force distributions in a triangular lattice of rigid bars. Phys. Rev. E 72, 03306 (2005). Jacobs, D.J., Thore, M.F.: Generic rigidity ercolation: the ebble game. Phys. Rev. Lett. 75, 405 4054 (995) 2. Chen, Y., Schuh, C.A.: Percolation of diffusional cree: a new universality class. Phys. Rev. Lett. 98, 03570 (2007) 3. Chen, Y., Schuh, C.A.: Coble cree in heterogeneous materials: the role of grain boundary engineering. Phys.Rev.B76, 064 (2007)
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