The macroscopic elastic properties of discrete assemblies are fundamental characteristics of such systems. The contribution uses homogenization procedure based on equivalence of virtual work between the isotropic elastic continuum and the discrete system to develop analytical formulas for estimation of macroscopic elastic modulus and Poisson’s ratio. Such homogenization was recently used to derive formulas for discrete assemblies where (i) there is no vacant space between the discrete units, (ii) the orientation of contacts is uniformly distributed and (iii) the contact normals are parallel to contact vectors (directions connecting centers of discrete units). The third assumption is now removed, three dimensional systems with arbitrary relation between contact vectors and contact normals are studied here. It is shown that the limits of Poisson’s ratio of such system depends on the relation between contact normal and contact vector. The widest limits are however obtained when these two vectors are parallel. This means that arbitrary manipulations with discrete geometry cannot extend Poisson’s ratio of the system outside the known boundaries

Eliás, Jan

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Advanced Numerical MethodsElastic properties of isotropic discrete ystems with skew contact normalsJ. EliášXV International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS 2019E. On˜ate, D.R.J. Owen, D. Peric, M. Chiumenti & Eduardo de Souza Neto (Eds)ELASTIC PROPERTIES OF ISOTROPIC DISCRETESYSTEMS WITH SKEW CONTACT NORMALSJAN ELIA´Sˇ∗∗ Brno University of TechnologyFaculty of Civil EngineeringInstitute of Structural MechanicsVeverˇ´ı 331/95, Brno, 60200, Czech Republice-mail: jan.elias@vut.czKey words: Discrete model, homogenization, elasticity, Poisson’s ratio, normal vectorAbstract. The macroscopic elastic properties of discrete assemblies are fundamentalcharacteristics of such systems. The contribution uses homogenization procedure basedon equivalence of virtual work between the isotropic elastic continuum and the discretesystem to develop analytical formulas for estimation of macroscopic elastic modulus andPoisson’s ratio.Such homogenization was recently used to derive formulas for discrete assemblies where(i) there is no vacant space between the discrete units, (ii) the orientation of contacts isuniformly distributed and (iii) the contact normals are parallel to contact vectors (direc-tions connecting centers of discrete units). The third assumption is now removed, threedimensional systems with arbitrary relation between contact vectors and contact normalsare studied here.It is shown that the limits of Poisson’s ratio of such system depends on the relationbetween contact normal and contact vector. The widest limits are however obtained whenthese two vectors are parallel. This means that arbitrary manipulations with discretegeometry cannot extend Poisson’s ratio of the system outside the known boundaries.1 INTRODUCTIONDiscrete modeling allows to explain or predict complex behavior of heterogeneous,cohesive or granular materials. It represents material random heterogeneity and alsodirectly works with discrete and oriented nature of cracks. Its elastic behavior still possesopen challenges. Besides the minor issue of inevitable boundary layer [1], the majorproblem lies in inability to exhibit Poisson’s ratios greater than 1/4 for three dimensionalmodels. The usage of the discrete models is therefore limited to materials with relativelylow Poisson’s ratio.This paper is motivated by long belief of the author that Poisson’s ratio of discretesystems can be increased by changing model geometry. Models described in literature1305Jan Elia´sˇusually use contact faces between discrete units perpendicular to contact vectors [2, 3].This restriction is here abandoned allowing to construct model of completely arbitrarygeometry. Poisson’s ratio is then analyzed using strong assumptions about rotationsand translations in the model according to [4]. The paper unfortunately proves thatgeometrical changes lead only to shrinking of the interval of achievable Poisson’s ratios.The previously published results derived for 2D models [5] are extended here into 3D.The studied system is composed of ideally rigid bodies filling a domain continuouslywithout gaps or overlapping. It is assumed that the system is isotropic in statistical sense– there is no directional dependence. The rigid bodies interact at their borders, wherenormal and tangential displacement discontinuities ∆ results in normal and tangentialforces. Critical parameter governing the macroscopic Poisson’s ratio is the ratio betweentangential and normal contact stiffness denoted α hereinafter.Equations are derived from virtual work equality between the discrete system andBoltzmann continuum subjected to equal straining. The discrete system yields non-symmetric stress tensor on contrary to the Boltzmann continuum symmetric stress quan-tity. The virtual work equivalence is therefore accomplished with help of symmetrizationof the tensor of elastic constants from the discrete model. To simplify the notation, weintroduce operation transposition Tij on arbitrary tensorA of sufficient order by swappingindices i and j.ATij...i...j... = A...j...i... (1)2 FUNDAMENTAL GEOMETRIC RELATIONSEach if the rigid bodies have 6 degrees of freedom associated with translations androtations of some inner node, xa. The contact element between two nodes xa and xb hascontact area A, length l, unit normal vector n and contact vector t. The situation issketched in Fig. 1a in two dimensions.The vector n is here defined in Cartesian coordinate system by two angles, ξ and ζn = (cos ξ sin ζ, sin ξ sin ζ, cos ζ) (2)Figure 1: a) Contact between two rigid bodies, its normal and contact vector, area and volume; b)angles determining directions of normal and contact vector.2306Jan Elia´sˇwhere ζ is angle between z axis and normal and ξ is the rotation of n around the z axis- see Fig. 1. We assume that the system has no directional bias, therefore all normaldirections share the same probability of occurrence. Therefore, ξ must be uniform overinterval from 0 to 2pi and ζ has the following probability density functionfξ(ξ) =12pifor ξ ∈ (0, 2pi)0 otherwisefζ(ζ) =sin ζ2for ζ ∈ (0, pi)0 otherwise(3)The contact vector t is defined relatively to normal vector n by angles χ and θ -see Fig. 1b. To ensure isotropic, directionally unbiased 3D structure θ is required to beuniformly distributed over 0–2pi interval, probability distribution of fχ can be arbitrary.fθ(θ) =12pifor θ ∈ (0, 2pi)0 otherwise(4)For sake of simplicity, it will be assumed now that the maximum angle between n and tis γ ∈ (0, pi) and all directions within this range are equally probable.fχ(χ) =sinχ1− cos γ for θ ∈ (0, γ)0 otherwise(5)Rotation matrix is the second order tensor that provides the following relation betweenn and tt = R · n (6)One can imagine construction of n via taking the vector(0 0 1), rotate it along they axis by angle ζ and then along z axis by angle ξ (Fig. 1b). In the same way, theconstruction of t is done by four successive rotations along axes y, z, y and z by anglesχ, θ, ζ and ξ, respectively.n = Rz(ξ) ·Ry(ζ) ·(0 0 1)t = Rz(ξ) ·Ry(ζ) ·Rz(θ) ·Ry(χ) ·(0 0 1)(7)The rotation matrix from Eq. (6) is thereforeR(ξ, ζ, χ, θ) = Rz(ξ) ·Ry(ζ) ·Rz(θ) ·Ry(χ) ·RTy (ζ) ·RTz (ξ) (8)The cosine of angle χ between n and t readscosχ = n · t = n ·R · n = R : (n⊗ n) = R :N (9)where the second order tensor N is according to Kuhl et al. [4] defined as N = n ⊗ n.Since no gaps or overlaps exist between the rigid bodies, the volume of the domain issummation over volume of individual mechanical elementsV =∑eVe =∑ecosχeAele3=∑eRe :NeAele3(10)Note that the volume of individual element is negative whenever χ > pi/2.3307Jan Elia´sˇ3 TENSOR OF EQUIVALENT ELASTIC CONSTANTSLet us strain the discrete system in macroscopic sense by constant strain tensor ε. Ac-cording to [4], it is assumed that all the rotations are zeros and differences in translationsare dictated by differences in positionϕ = 0 ub − ua = ε · (xb − xa) (11)The displacement jump on contact between cells a and b reads∆ = ub − ua = lε · t (12)where l and t are length and contact vector belonging to element connecting bodies a andb. The normal and shear strain and stress directly followeN =n ·∆l= n · ε · t eT = ∆l− eNn = ε · t− (n · ε · t)n (13)sN = E0eN sT = E0αeT (14)where E0 is the normal stiffness coefficient and α is the tangential/normal stiffness ratioconsidered constant in the whole domain.The virtual work done by single element readsδW = Al (sNδeN + sT · δeT ) (15)and summation of individual contributions provides the total virtual work in the discretesystem.Let us now define two additional tensors: the fourth order tensor I vol and the thirdorder tensor T .I vol =1⊗ 13(16)T = 3n · (I vol)T13 − n⊗ n⊗ n (17)where 1 is the identity matrix of size 3. Note that T is different from definition in [4, 1]because the symmetry implied by equality t = n is no longer present. The transpositionT13 means that dimensions 1 and 3 are swapped. Eq. (13) can be rewritten aseN =(N ·RT ) : ε eT = (T ·RT ) : ε (18)using transposition T of the second order tensor swapping its two dimensions T = T12.The virtual work of single element expressed in Eq. (15) can be rewritten as wellδW = Al (sNδeN + sT · δeT )= AlE0([(N ·RT ) : ε] [(N ·RT ) : δε]+ α [(T ·RT ) : ε] · [(T ·RT ) : δε])= AlE0[ε :(R ·N ⊗N ·RT )T12 : δε+ αε : (R · T T13 · T ·RT ) : δε]= AlE0ε :((R ·N ⊗N ·RT )T12 + αR · T T13 · T ·RT) : δε (19)= AlE0ε : (N + αT ) : δε4308Jan Elia´sˇwhereN =(R ·N ⊗N ·RT )T12 T = R · T T13 · T ·RT (20)The total virtual work of readsδW dis =∑eδWe =∑eAeleE0ε : (N e + αT e) : δε (21)The discrete system is now related to equally strained elastic isotropic BoltzmannBoltzmann continuum occupying the same domain of volume V . Stress in the continuumis provided by constitutive equation σ =D : ε, where D is fourth order tensor of elasticconstants. The virtual work of the continuum isδW con = V σ : δε = V ε :D : δε (22)The equivalence of the discrete and continuous system implies equality of virtual worksδW dis = δW con (23)Substituting Eqs. (21) and (22) into Eq. (23), expression for tensor of elastic constants isderivedD =〈1V∑eAeleE0 (N e + αT e)〉SYM(24)The symmetrization is needed because the tensors N and T do not posses the symme-tries required for Boltzmann continuum, which are the major symmetry (derived fromequivalence of mixed derivatives of elastic potential) and the minor symmetry (derivedfrom symmetry of stress and strain tensors). The symmetric part can be easily obtainedusing transposition T34.〈•〉SYM = •+ •T342(25)Thanks to assumed statistical independence between normal and contact vector andelemental area and length, the summation in Eq. (24) can be broken into the followingexpressionD =E0V〈E [N ] + αE [T ]〉SYM∑eAele (26)where E [•(x)] is the mean value of function • dependent on vector x with distributionfunction fX(x)E [•(x)] =∞∫−∞· · ·∞∫−∞•(x)fX(x) dx (27)Substituting V from Eq. (10) and utilizing the statistical independence again, oneobtainsD =3E0E[R :N ]〈E [N ] + αE [T ]〉SYM (28)5309Jan Elia´sˇ4 EVALUATION OF EXPECTATIONSThe integration of mean values from Eq. (28) is tedious. It was analytically donein [5] in two dimensions over two independent variable. For three dimensions it has tobe performed over four independent angles and the integration cannot be separated sincerotation matrixR depends on all four angles. The calculation was performed by computerwith a help of Python library for symbolic mathematics SymPy [6]. The following threeintegrations were delivered.E [R :N ] =γ∫−γ2pi∫0pi∫02pi∫0R :N12pisin ζ212pisinχ1− cos γ dξ dζ dθ dχ = cos2(g2)(29)E [N ] =γ∫−γ2pi∫0pi∫02pi∫0N12pisin ζ212pisinχ1− cos γ dξ dζ dθ dχ ==13(I vol)T23+2 cos γ + cos 2γ + 120(I vol +(I vol)T24 − 23(I vol)T23)(30)E [T ] =γ∫−γ2pi∫0pi∫02pi∫0T12pisin ζ212pisinχ1− cos γ dξ dζ dθ dχ ==23(I vol)T24 − 2 cos γ + cos 2γ + 120(I vol +(I vol)T23 − 23(I vol)T24)(31)Only the symmetric parts of these expectations are needed (Eq. 25). The symmetricparts of involved tensors are〈(I vol)T23〉SYM=〈(I vol)T24〉SYM=I3(32)where the fourth order tensor I = Iijkl = (δikδjl + δilδjk)/2 with δij ≡ 1 being theKronecker delta is employed. The symmetric part of expectations reads〈E [N ]〉SYM = 2 cos γ + cos 2γ + 21180I +2 cos γ + cos 2γ + 120I vol (33)〈E [T ]〉SYM = 39− 2 cos γ − cos 2γ180I − 2 cos γ + cos 2γ + 120I vol (34)5 MACROSCOPIC ELASTIC CHARACTERISTICSThe mechanical behavior of linearly elastic isotropic solid is determined by two in-dependent constants (here we choose elastic modulus E and Poisson’s ratio ν) definingtensor of elastic constantsD =E1 + νI +3Eν(1 + ν)(1− 2ν)Ivol (35)6310Jan Elia´sˇ0 0.5 1 1.5 2 2.5 3α0.51.01.52.0E/E0[-]γ → 0γ = pi8γ = pi4γ = pi2 or piγ = 3pi40 0.5 1 1.5 2 2.5 3α−0.20.00.2ν[-]γ → 0γ = pi8γ = pi4γ = pi2 or piγ = 3pi40 pi/4 pi/2 3pi/4 piγα = 0α = 1/3α = 2/3α = 1α = 3/2Figure 2: Macroscopic elastic characteristics according to Eqs. (38) and (37).Equation (28) along with symmetrized expectations (29), (33) and (34) providesD = E0[(1− α)(2 cos γ + cos(2γ)− 39) + 6030(cos γ + 1)I +3(1− α)5cos γI vol](36)Equality between respective scalar multipliers of tensors I vol and I in Eqs. (36)and (35) provides relations between macroscopic parameters E and ν and mesoscopicparameters E0, α and γ.ν =3(1− α)(cos γ + cos2(γ))(1− α)(7 cos γ + 7 cos2 γ − 20) + 30 (37)E = E02 [(1− α)(cos γ + cos2 γ − 20) + 30] [(1− α)(cos γ + cos2 γ − 2) + 3](1− α)(7 cos γ + 7 cos2 γ − 20) + 30 (38)These equations are plotted in Fig. 2 for range α ∈ (0, 3).Calculation limit for γ → 0 must yield relations for discrete system with n = t.limγ→0ν =1− α4 + αlimγ→0E = E02 + 3α4 + α(39)Indeed, the calculation of limits provides correct expressions derived in e.g. [1] underassumption of perpendicularity of contact vector and contact face. They are also identicalto those from microplane theory [7].By differentiate the expression with respect to γ and search for stationary point, themaximum and minimum possible values of Poisson’s ratio can be found. The stationarypoints are γ = 0, pi and arccos(−0.5) (≈ 2.09440). Plotting the Poisson’s ratio withrespect to the limit angle γ (Fig. 2 on the right hand side) shows that the maximumrange of ν is obtained for γ = 0, i.e. when the contact vector equals the normal vector.This is the classic solution stated in Eq. (39). Increasing γ towards pi/2 shrinks theinterval of achievable Poisson’s ratios to zero. The interval opens again beyond pi/2 withopposite signs; its width maximizes at γ = arccos(−0.5). The limiting values at thispoint are obtained at α = 0 and α → ∞ as (−0.091, 0.034). The maximum values ofPoisson’s ratio are achieved for n = t, any deviation of the model geometry from thisrelation causes narrowing of the Poisson’s ratio limits.7311Jan Elia´sˇ6 CONCLUSIONS• It has been proven that under assumption (5) one cannot increase the Poisson’sratio limits beyond what is provided by model with n = t in Eq. (39).• The formulas are derived from strong and unrealistic assumption about rotationsand displacements in the model (Eq. 11). Behavior of the real model will be lessrigid, however the overall effect on macroscopic elastic constants should be qualita-tively the same.• The same conclusions were found for 2D models under the same assumptions in [5].• The theory will be extended to arbitrary distribution of χ and verified by comparisonwith real behavior of discrete systems.ACKNOWLEDGEMENTFinancial support provided by the Czech Science Foundation under project No. GA19-12197S is gratefully acknowledged.REFERENCES[1] Jan Elia´sˇ. Boundary layer effect on behavior of discrete models. Materials, 10:157,2017. ISSN 1996-1944. doi:10.3390/ma10020157.[2] Peter Grassl and John E. Bolander. Three-dimensional network model for coupling offracture and mass transport in quasi-brittle geomaterials. Materials, 9(9):782, 2016.ISSN 1996-1944. doi:10.3390/ma9090782.[3] John E. Bolander and Shigehiko Saito. Fracture analyses using spring networks withrandom geometry. Engineering Fracture Mechanics, 61(5-6):569–591, 1998. ISSN0013-7944. doi:10.1016/S0013-7944(98)00069-1.[4] Ellen Kuhl, Gian Antonio D’Addetta, Hans J. Herrmann, and EkkehardRamm. A comparison of discrete granular material models with continuous mi-croplane formulations. Granular Matter, 2(3):113–121, 2000. ISSN 1434-5021.doi:10.1007/s100350050003.[5] Jan Elia´sˇ. On macroscopic elastic properties of isotropic discrete systems: effect oftessellation geometry. In Proceedings of the Tenth International Conference on Frac-ture Mechanics of Concrete and Concrete Structures - FraMCoS-X held in Bayonne,France, 2019.[6] Aaron Meurer, Christopher P. Smith, Mateusz Paprocki, Ondrˇej Cˇert´ık, Sergey B.Kirpichev, Matthew Rocklin, AMiT Kumar, Sergiu Ivanov, Jason K. Moore, Sar-taj Singh, Thilina Rathnayake, Sean Vig, Brian E. Granger, Richard P. Muller,Francesco Bonazzi, Harsh Gupta, Shivam Vats, Fredrik Johansson, Fabian Pedregosa,Matthew J. Curry, Andy R. Terrel, Sˇteˇpa´n Roucˇka, Ashutosh Saboo, Isuru Fernando,8312Jan Elia´sˇSumith Kulal, Robert Cimrman, and Anthony Scopatz. Sympy: symbolic computingin Python. PeerJ Computer Science, 3:e103, 2017. ISSN 2376-5992. doi:10.7717/peerj-cs.103.[7] Ignacio Carol and Zdeneˇk P. Bazˇant. Damage and plasticity in microplane theory.International Journal of Solids and Structures, 34(29):3807–3835, 1997. ISSN 0020-7683. doi:10.1016/S0020-7683(96)00238-7.9313

Elastic properties of isotropic discrete systems with skew contact normals

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Elastic properties of isotropic discrete systems with skew contact normals

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