Objective decomposition of the stress tensor in granular flows

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1 Objective decomposition o the stress tensor in granular lows D. Gao Ames Laboratory, Ames, Iowa 50010, USA S. Subramaniam* and R. O. Fox Iowa State University and Ames Laboratory, Ames, Iowa 50010, USA D. K. Homan Ames Laboratory and Iowa State University, Ames, Iowa 50010, USA Received 21 April 2004; published 16 February 2005 A model or the stress tensor in granular lows Volson, Tsimring, and Aranson, Phys. Rev. Lett. 90, is correctly generalized to an objective orm that is independent o the coordinate system. The objective representation correctly models the isotropic and anisotropic parts o the stress tensor, whereas the original model or stress tensor components is dependent on the coordinate system. This general objective orm o the model also relaxes the assumption in the original model that the principal axes o the granular stress tensor be coaxial with that o the luid stress tensor. This generalization expands the applicability o the model to a wider class o granular lows. The objective representation is also useul in analyzing other models based on additive decomposition o the stress tensor in granular lows. DOI: /PhysRevE PACS number s : n, b I. INTRODUCTION Granular lows can exhibit dierent types o material characteristics and constitutive behavior, depending on the volume raction and the magnitude o applied shear rate relative to relevant particle time scales. Many recent attempts 1 3 to characterize the stress in granular lows involve an additive decomposition o the granular stress. In a recent continuum theory proposed by Aranson and Tsimring AT 1, the stress tensor ij in a granular low is decomposed into ij, a luid part, and s ij a solid part. Models are then proposed or the luid and solid parts. Similar decompositions into impulsive and enduring parts 2, or kinetic and rictional parts 3, are common in the granular low literature. It is important to note that although the term luid or luidlike is used in the granular low literature, this is potentially misleading since it may imply that the systems under consideration are granular mixtures solid particles with interstitial luid. The ocus o this work and those cited in Res. 1,2 is on the decomposition o the particulate solid stress into solidlike and luidlike parts. I the ambient luid is present, as in the case o granular mixtures, then an additional stress associated with the luid will appear in the models see Re. 3 or example. With this clariication the quotation marks on luid and solid are dropped. In the AT model 4,5, the luid and solid parts o the stress tensor are modeled in terms o the granular stress tensor ij. It is then assumed that the principal axes o all the stress tensors are coaxial. The coeicients in the AT model which are ratios o stress components are then determined by matching individual components o the luid stress tensor *Corresponding author. Electronic address: shankar@iastate.edu to data obtained rom molecular dynamics MD simulations o zero-gravity Couette low 4,5. Models or the luid or solid stress tensor which require speciication o coeicients that depend on ratios o individual stress components are obviously coordinate-system dependent, and do not guarantee the important requirement that the stress tensor be objective 6. In Euclidean space the objectivity requirement is that the tensor components in dierent coordinate systems satisy a transormation rule see or example Malvern 15. Thereore models such as the AT model are not general, but are restricted to the coordinate system and low conigurations in which they are speciied. Speciically, it is unclear how to generalize the AT model, which is ormulated or two-dimensional 2D Couette low, 1 to a coordinate-system-independent orm so that it may be applied to granular lows in more complex 3D geometries. Here we show how the AT model may be generalized to 3D, while satisying the objectivity requirement. I the luid and solid parts o the stress tensor are modeled in terms o the granular stress tensor ij, the general objective orm o the decomposition can obtained by using representation theorems 7,8. The assumption o coaxial principal axes or all the stress tensors is shown to be unnecessary, and an objective model which does not require this assumption is derived. The special case o coaxial principal axes is subsumed in the general objective orm. This objective model is 1 More precisely this is a 2C low, where by 2C we mean that the componentiality 13 o the stress tensor is 2, i.e., the stress tensor has only two nonzero singular values and the component index range is i, j=1,2. A 2D simulation can at best yield a 2C stress tensor, whereas in 3D the stress tensor can be either 3C or 2C. For simplicity o exposition we assume the stress tensor is nonsingular in the rest o this paper, in which case componentiality and dimensionality are the same /2005/71 2 / /$ The American Physical Society

2 GAO et al. easily extended to 3D. While the AT model in 2D speciies three model coeicients, it is shown that proper application o the requirement o objectivity results in ewer coeicients: only two coeicients can be independently speciied or this tensor decomposition. The objective model also shows that the stress components can only be matched in a least-squares sense 9 11, regardless o the assumption o coaxial principal axes or all stress tensors. The objective orm o the granular stress tensor decomposition is essential to compute granular lows using constitutive relations in continuum twoluid ormulations such as MFIX 12. The physical signiicance o using the objective orm which allows or noncoaxial stress tensors is that it extends the applicability o the model to granular lows with nonspherical grains, or other sources o noncoaxiality. It is convenient to express the granular stress ij, a second-order tensor that is assumed to be symmetric, 2 in isotropic and deviatoric parts: ij = 1 3 ii ij + ij = 0 ij + b ij, where ij is the symmetric deviatoric stress deined as ij = ij 1 3 ii ij, and b ij is the normalized, symmetric, traceless, anisotropy tensor deined as 4 5 II. DECOMPOSITION OF GRANULAR STRESS b ij = 1 0 ij = 1 0 ij ij. 6 The basic idea in AT s additive stress decomposition is to express the stress tensor in a granular low as the sum o a luid stress tensor and a solid stress tensor: ij = ij + s ij. The goal is to propose models or each o the solid and luid parts in terms o an order parameter and the granular stress tensor ij, and thereby come up with a model or stress in the granular low. The order parameter is deined by Volson et al. 4 as a mesoscopic space-time average raction o solidlike contacts between the particles in the granular system. A contact is considered solidlike i it is in a stuck state and its duration is longer than a typical collision time. Additional details o the calculation o the order parameter rom MD are given in 4. Aranson and Tsimring 1 express the luid stress in terms o the granular stress, the general orm o such a model being = M, where M is an isotropic tensor unction in the sense o Smith and Smith 7. Isotropic tensor unctions satisy the invariance property o Eq. 3 when subjected to unitary transormations. The remaining stress that is obtained by subtracting the luid stress rom total stress is denoted the solid stress. III. OBJECTIVE FORM The objectivity requirement is that i is an isotropic tensor unction M o the tensor as in Eq. 2, and Q is an arbitrary unitary transormation o the coordinate axes, such that * = Q Q T is the luid stress tensor in the transormed coordinate system, then QM Q T = M Q Q T Here 0 = ii /3 is the scale o the stress. 3 Einstein notation is used so summation is implied over repeated indices. An objective orm 4 in which the luid or solid stress tensor may by expressed as a unction o the granular stress tensor is ij = 0 ij + b ij + b 2 ij 1 3 b2 ll ij 7 where,, are undetermined scalar coeicients that are unctions o the invariants o b ij and the order parameter. Note that b ij has zero trace by deinition, so only its second and third invariants may be nonzero. The components o the second-order tensor b 2 are deined as b 2 ij = b ik b kj, and b 2 ll is a scalar that is deined as b 2 ll = b lk b kl. 9 I the solid stress tensor is also represented in a similar orm, then the requirement that the luid and solid stresses sum to the granular stress requires that the solid stress model expression be 2 The stress tensor in granular lows is assumed to be symmetric 14. However, in particle dynamics simulations which incorporate angular momentum transer between particles, this assumption needs to be veriied. Malvern 15 states that a symmetric stress tensor is implied by the moment o momentum principle or a collection o particles interacting through equal, opposite, and collinear orces, but the symmetry property is lost when even equal and opposite couples are included. Nevertheless, the objectivity requirements that we impose here can be extended to the general nonsymmetric stress tensor by decomposing it into symmetric and skewsymmetric parts. 3 In 2D the scale o the stress is deined as 0 = ii /2, and appropriate modiications are needed or the deinition o the deviatoric and anisotropy tensors. 4 This simpler version o the more general orm proposed by Pope 16 ollows rom Eq

3 OBJECTIVE DECOMPOSITION OF THE STRESS s ij = 0 1 ij + 1 b ij b 2 ij 1 3 b2 ll ij. 10 Clearly one can exactly match three components o the luid stress tensor model or the solid stress model, but not both to data rom MD simulations by speciying the three model coeicients,,. In a 3D granular low there are six independent nonzero components o the luid stress tensor. Thereore one can speciy the three model coeicients,, to match the six components rom simulation data only in a least-squares sense In the 2D case one can show that the characteristic equation or the stress tensor is a quadratic instead o a cubic or the 3D case, and there are only two invariants instead o three or the 3D case : the sum and product o the two principal values o the stress tensor. The Cayley-Hamilton theorem in the 2D case shows that b 2 instead o b 3 in the 3D case itsel can be expressed as a linear combination o b, and thereore the term in is redundant and can be dropped. Then there are only two coeicients and. Noting that in 2D there are three independent nonzero components o the luid stress tensor, again the two coeicients must be determined by matching the three stress components in a leastsquares sense. The accuracy o the luid stress model or a given set o data is deined in terms o the p-norm 11 o the error matrix, which is deined as the dierence between the modeled luid stress tensor and the data. Here we use p=1 and deine the matrix error measure as : = model data 1 / data This measure o modeling error is useul because it applies to all the models considered in this study. Another way to measure the errors is speciic to the objective model and results in the norm o an error vector. For the 2D case the objective orm o stress tensor representation in Eq. 7 has three equations and two unknowns, which requires solving = In matrix notation this least-squares problem is Kx = y with K the coeicient matrix, and x the unknown vector o model coeicients. The error in the objective model can also be quantiied by calculating the vector norm o the relative error in the least-squares solution: = Kx y 2 / y Using the data,, and rom Figs. 6, 7, and 8 reported in Volson et al. 4 we obtain the coeicients or the objective model. The values o the coeicients and in the objective model are shown as unctions o the order parameter in Fig. 1. As expected the coeicients approach 1 as the order parameter approaches zero, corresponding to FIG. 1. Model coeicients as unctions o order parameter : and or the general objective orm, and CPA and CPA or models OCPA0 and OCPA or the equivalent objective orm assuming coaxial principal axes. the granular low reaching the ully luidized state. It is also to be expected that the coeicients will approach zero as the order parameter approaches 1. In this case it would be preerable to solve or the model coeicients o the solid stress tensor, 1 and 1. The errors incurred in terms o the matrix norm and the vector norm are depicted in Fig. 2. Over the entire range o the order parameter it is gratiying to note that the errors o the objective model are less than 10%. As approaches zero the granular low becomes more luidized, and the error drops rapidly i.e., the solid stress is negligible. Thereore, or the granular Couette low the objective orm Eqs. 7 and 10 accurately decomposes the stress into luid and solid parts with the ollowing expressions that it the variation o model coeicients with order parameter: = 1 1.8, 14 = These expressions ensure the correct limiting behavior o the model at =0 and 1. IV. COAXIAL PRINCIPAL AXES CASE In the AT model it is assumed that the principal axes o all three stress tensors,, and s are coaxial. The coeicients in the AT model are then determined by matching FIG. 2. Error in modeled luid stress characterized by matrix one-norm or all models: general objective and equivalent objective under coaxial principal axes assumption OCPA0 and OCPA

4 GAO et al. individual components xx, xy, yy o the luid stress tensor to data obtained rom molecular dynamics simulations o a canonical granular low 1,4. The objective model can be investigated under the assumption o coaxial principal axes. I the objective model is written in principal axes coordinates, then the deviatoric tensor is diagonal and given by ij = i ij 0 ij. 16 The normalized symmetric traceless anisotropy tensor b ij is also diagonal and is given by b ij = i 1 ij In the 2D case there are only two coeicients and. Note that in principal coordinates we have 1 = , 2 = By eliminating and rom the above equations we obtain the equivalent objective speciication o the coaxial principal axes CPA model to be CPA = , CPA = A technical detail arising rom nonzero angle between the principal axes o the luid and total granular stress data results in two slightly dierent ways in which the CPA model coeicients can be calculated. In both approaches 1 and 2 in Eqs. 18 and 19 are the singular values o the total granular stress matrix. In the irst approach OCPA we use the singular values o the luid stress matrix or 1, 2, even though the principal directions o the luid and total stress tensors are not identical. In the second approach OCPA0, the luid stress tensor is transormed into the principal coordinates o the total stress tensor, and the diagonal components o the transormed matrix are taken to be 1, 2. The values o CPA and CPA as a unction o the order parameter are shown in Fig. 1. It is ound that CPA is practically identical to or the objective model without the coaxial principal axes assumption. There are some dierences between CPA and or the objective model without the coaxial principal axes assumption. The modeling error measure is shown or both models in Fig. 2, and it is ound that the error incurred in the various models is comparable. V. VALIDITY OF COAXIAL PRINCIPAL AXES ASSUMPTION Denoting the principal axes o as u 1,u 2, and similarly the principal axes o as u 1,u 2 11, one can calculate the FIG. 3. Angle between principal axes o the total granular stress and the luid stress. angle between the principal axes o and as = arccos u 1 u 1 = arccos u 2 u The angles or the data obtained rom Volson et al. 4 are shown in Fig. 3. It is ound that the maximum angle between the two principal axis systems is about 8, so the principal axes o the stress tensors are almost collinear. As decreases, the angle increases to the maximum, and then drops rapidly when approaches zero. Whether this nearcollinearity o the stress tensors is a universal characteristic o granular low in this regime is questionable. Certainly coaxial principal axes are not expected i the grains are anisotropic e.g., ellipsoids. The small angle between the principal axes explains why the orthotropic model incurs errors that are o the same magnitude as the objective model or this low. We also see rom Figs. 2 and 3 that the modeling errors ollow the same trend as the angle between the principal axes. The small angle between the principal axes also explains why the model coeicients with the coaxial principal axes assumption CPA, CPA are very close to the general objective model. One may expect larger dierences in other granular lows, although these comparisons are diicult to make because only the objective model is truly generalizable and independent o the coordinate system. VI. OBJECTIVE MODEL PERFORMANCE IN THICK GRANULAR LAYER Volson et al. 4 also report MD simulations or a thick granular layer under nonzero gravity driven by a moving upper plate, a granular system that is dierent rom the zerogravity Couette low with which the model coeicients were calibrated. The objective model is tested in this thick granular system, and these results o the model predictions or the granular layer driven by a moving upper plate under nonzero gravity 4 case P10V5 are shown in Fig 4. The agreement o the objective model s predicted stresses to the simulation data is remarkably good. VII. CONCLUSIONS An objective generalization o the stress model based on the order parameter 4,5 has been developed see Eqs. 7,

5 OBJECTIVE DECOMPOSITION OF THE STRESS FIG. 4. Comparison o objective model predictions o luid stress components with MD data or a thick granular system driven by a moving upper plate under nonzero gravity. 14, and 15. The objective model does not assume that the principal axes o the luid and total granular stress are coaxial. The objective model has ewer model coeicients than the original model and thereore the luid stress is matched only in a least-squares sense. Model coeicients and error measures are compared or both the general objective model and the equivalent objective model under coaxial principal axes assumption. It is ound that the error is comparable or both models and is below 10% or all values o the order parameter in the Couette low coniguration or which data rom MD are available. The angle between principal axes o and is computed and it was ound to increase sharply at very small 0.1, luidlike regime, reach a maximum o about 10 or the granular Couette low, and then decrease again at larger solidlike regime. The modeling errors ollow the same trend. This nonlinear variation with order parameter tells us that the stress in granular matter does not become luidlike in direct proportion to the order parameter. Further study is needed to understand i the discontinuity in the angle that is observed around =0.05 is indicative o a sort o phase transition, or not. The objective model is used to predict the stresses in a dierent granular system rom the one using which the model coeicients were calibrated thin Couette low with zero gravity. In this thick granular low driven by a moving upper plate under nonzero gravity, the objective model predictions are in excellent agreement with the simulation data, even though the objective model has ewer coeicients than the previous model. However, a more rigorous test o the objective model would require data rom MD o a ully 3D granular low or a range o order parameter and stress tensor anisotropy values. ACKNOWLEDGMENTS The authors would like to thank Dmitri Volson and Lev Tsimring or providing them the data rom their MD simulations. This manuscript has been authored at Iowa State University o Science and Technology under Contract No. W-7405-ENG-82 with the U.S. Department o Energy. S.S. would like to thank Sankaran Sundaresan and Gary Grest or useul comments on a drat o this article. 1 I. S. Aranson and L. S. Tsimring, Phys. Rev. E 65, M. Y. Louge, Phys. Rev. E 67, A. Srivastava and S. Sundaresan, Powder Technol. 129, D. Volson, L. S. Tsimring, and I. S. Aranson, Phys. Rev. E 68, D. Volson, L. S. Tsimring, and I. S. Aranson, Phys. Rev. Lett. 90, C. Truesdell, A First Course in Rational Continuum Mechanics Academic Press, New York, 1977, Chaps. 3 and 4. 7 M. M. Smith and G. F. Smith, Int. J. Eng. Sci. 19, S. Pennisi and M. Trovato, Int. J. Eng. Sci. 25, C. L. Lawson and R. J. Hanson, Solving Least Squares Problems Prentice-Hall, Englewood Clis, NJ, G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods or Mathematical Computations Prentice-Hall, Englewood Clis, NJ, 1977, Chap G. E. Golub and C. F. Van Loan, Matrix Computations Johns Hopkins University Press, Baltimore, M. Syamlal, W. Rogers, and T. J. O Brien, Technical Report DOE/METC-94/1004, U. S. Department o Energy, Morgantown Energy Technology Center, Morgantown, West Virginia, 1993 unpublished. 13 W. C. Reynolds and S. C. Kassinos, Proc. R. Soc. London, Ser. A 451, L. E. Silbert, D. Ertas, G. S. Grest, T. C. Halsey, D. Levine, and S. J. Plimpton, Phys. Rev. E 64, L. E. Malvern, An Introduction to the Mechanics o a Continuous Medium Prentice-Hall, Englewood Clis, NJ, S. B. Pope, Phys. Fluids 6,