Stimates for the elastic moduli of 2D aggregate of hexagonal - shape orthorhombic crystals with in - plane random crystalline orientations

 Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (2019), pp. 171 – 177
 DOI: https://doi.org/10.15625/0866-7136/13183
 ESTIMATES FOR THE ELASTIC MODULI
 OF 2D AGGREGATE OF HEXAGONAL-SHAPE
 ORTHORHOMBIC CRYSTALS WITH IN-PLANE RANDOM
 CRYSTALLINE ORIENTATIONS
 Vuong Thi My Hanh1,∗, Le Hoai Chau2, Vu Lam Dong1, Pham Duc Chinh1
 1Institute of Mechanics, VAST, Hanoi, Vietnam
 2Institute of Mechanics and Environmental Engineering, Hanoi, Vietnam
 ∗E-mail: vtmhanh@imech.vast.vn
 Received: 08 October 2018 / Published online: 13 June 2019
 Abstract. Numerical finite element simulations on the homogenization problem for large
 samples of particular 2D hexagonal-shape-geometry random orientation aggregates from
 the base crystals of orthorhombic symmetry have been performed. At sufficiently large
 random-aggregate samples, the scatter intervals of the macroscopic 2D bulk and shear
 elastic moduli converge toward the Voigt-Reuss-Hill bounds, and then our recently con-
 structed theoretical estimates, which have been specified for the aggregates.
 Keywords: effective elastic moduli; random cell polycrystal; 2D orthorhombic crystal.
 1. INTRODUCTION
 Polycrystalline materials often have irregular microstructures with random orien-
tations of constituent crystals. Hence the macroscopic (effective) elastic moduli of the
aggregates are expected to scatter over certain intervals depending on the sizes of the
representative volume element (RVE). We still do not know if the effective moduli should
converge to unique values as the size of RVE increases indefinitely or they should con-
verge to some finite (though small) scatter intervals. Another interesting question is how
the scatter intervals converge toward the known theoretical bounds constructed for the
moduli [1–10] as the sizes of RVE increase, and if the bounds could be improved signifi-
cantly.
 In this work, we use the numerical homogenization method to compute the effective
elastic moduli [11–13]. A homogenization method applies periodicity boundary condi-
tion and characteristic material force fields to obtain characteristic displacements within
the representative volume. The characteristic displacement is then used to compute the
effective moduli. We use the finite element method (see [14] for a review) to discretize
each and every crystal grain inside of the representative volume so that we can not only
 
c 2019 Vietnam Academy of Science and Technology
172 Vuong Thi My Hanh, Le Hoai Chau,Vu Lam Dong, Pham Duc Chinh
assign different properties for each grain but also solve for displacement. The period-
icity boundary condition in this method gives a better accuracy compared to uniform
boundary condition of physical tests or simple finite element analysis of the representa-
tive volume.
 For numerical simulations and comparisons with the bounds we choose 2D poly-
crystals [10, 15–17], which are relatively simpler than the more-complex 3D problems. In
the first stage, the hexagonal-shape-geometry of the aggregates has been chosen in [10].
In [10] the crystals of the simplest square symmetry have been considered, which showed
numerical scatter in the effective elastic shear modulus, while the effective elastic bulk
modulus is uniquely determinable. In this paper we consider more complex orthorhom-
bic crystals, which are expected to show the scatters in both elastic moduli.
 2. BOUNDS FOR 2D RANDOM POLYCRYSTALS
 The planar orthorhombic crystals have 4 independent elastic constants, which in
Voigt notations are C11, C22, C12, C33. The Voigt-Reuss-Hill bounds for the effective elastic
moduli of isotropic aggregates of the crystals have particular expressions
 e f f e f f
 KR ≤ K ≤ KV , µR ≤ µ ≤ µV , (1)
where
 1 1
 K = (C + 2C + C ) , µ = (C − 2C + C + 4C ) ,
 V 4 11 12 22 V 8 11 12 22 33
 − (2)
 C C − C2  C + C + 2C 1  1
 K = 11 22 12 , µ = 2 11 22 12 + .
 R + − R 2
 C11 C22 2C12 C11C22 − C12 C33
 The recently constructed general bounds for the effective moduli of the random
polycrystals can be well approximated by the simpler bounds for circular cell polycrys-
tals [10, 15–17], which are specified for the random orthorhombic crystal aggregate
 KL ≤ Ke f f ≤ KU , µL ≤ µe f f ≤ µU , (3)
where
 L U
 K = PK(µR, µ∗R), K = PK(µV, µ∗V ),
 KRµR KV µV
 µ∗R = , µ∗V = , (4)
 KR + 2µR KV + 2µV
 L U
 µ = Pµ(µR, µ∗R), µ = Pµ(µV, µ∗V ),
 2
 (C11 + µ0 + µ∗)(C22 + µ0 + µ∗) − (C12 + µ0 − µ∗)
PK(µ0, µ∗) = − µ0 ,
 C11 + C22 − 2C12 + 4µ∗
 h C + C + 2C + 4µ 1 i−1
 ( ) = 11 22 12 0 + −
Pµ µ0, µ∗ 2 2 µ∗ ,
 (C11 + µ0 + µ∗)(C22 + µ0 + µ∗) − (C12 + µ0 − µ∗) C33 + µ∗
 K0µ0
µ∗ = .
 K0 + 2µ0
 (5)
 Numerical results of the estimates (1)–(5) for some orthorhombic crystals taken from
[18] are presented in Tabs.1,2.
 Estimates for the elastic moduli of 2D hexagonal-shape orthorhombic crystals with in-plane random crystalline . . . 173
 Table 1. The elastic constants (C11, C22, C12, C33) of some orthorhombic crystals
 from [18], all in GPa
 C11 C22 C12 C33
 S(1) 2.05 4.83 1.59 0.43
 S(3) 2.40 4.83 1.71 0.87
 U(1) 19.86 26.71 10.76 12.44
 TiO2 27.30 48.40 14.90 12.50
 Cu 16.84 16.84 12.14 7.54
 In(2) 4.45 4.44 4.05 0.65
 Table 2. The estimates (1)–(5), all in GPa
 U L U L
 KV K K KR µV µ µ µR
 S(1) 2.52 2.1612 2.1365 1.9900 0.6770 0.5735 0.5686 0.542
 S(3) 1.78 1.7680 1.7678 1.7600 0.6040 0.5762 0.5757 0.560
 U(1) 17.00 16.7490 16.7400 16.6000 9.3500 8.5767 8.5544 8.180
 TiO2 26.40 24.8080 24.7670 24.0000 12.0000 11.5821 11.5744 11.400
 Cu 14.50 14.4900 14.4900 14.4900 4.9400 4.0909 4.0279 3.580
 In(2) 4.25 4.2475 4.2475 4.2475 0.4240 0.3579 0.3494 0.303
 The self-consistent approximation is the solution KSCA = K and µSCA = µ of the
equations
 K = PK(µ, µ∗) , µ = Pµ(µ, µ∗) . (6)
 3. A 2D RANDOM MODEL AND SIMULATIONS
 The homogenization method assumes that the medium contains a large number of
repeating patterns (unit cells) stacked against each other. Only one unit cell need to be
analysed to compute the effective moduli. In this case, our unit cell, again, contains a
large number of grains. Each grain is assigned with a uniform crystal property, but the
property is random across the grains. In this study, the grain takes the hexagonal shape.
We use the finite element method to solve the homogenization numerically, and in this
case, each hexagonal grain is discretized using 64 quadrilateral finite elements.
 The basis for numerical homozenization lies in the assumption that when an object
undergo uniform macroscopic deformation, the microscopic deformation in each unit cell
is not uniform. However, the boundary between two adjacent unit cells have to remain
compativble, i.e. there should not be any cracks in the material. The deformation within
a unit cell follows standart solid mechanics principals and constitutive model. These
assumptions are sufficient to solve for the microscopic unit cell deformation using any
numerical method. In this case the standard displacement-based finite element method is
used. In addition, the plane-strain formulation is used for this two-dimensional problem
and don’t anticipate that the conclusion is different if we use plane-stress.
174 Vuong Thi My Hanh, Le Hoai Chau,Vu Lam Dong, Pham Duc Chinh
 To arrive at the effective moduli, we first apply the periodicity condition [13] and 3
load cases of material forces (in 3D, it would requires 6 load cases) to the unit cell. The
material forces are calculated according to the following formula
 (ij) (kl)
 f = −divYC[e ] . (7)
 After solving the unit cell for characteristic displacements, the homogenized elastic-
ity tensor is calculated according to the following formula.
 e f f 1 Z
 C = (e(ij) + ∇ χ(ij))C(y)(e(kl) + ∇ χ(kl))dy , (8)
 ijkl | Y | y y
 Y
where | Y | is the unit cell size (the area of the unit cell in this 2D setting); e(ij) is the unit
 e f f
test strain; Cijkl is the component of the homogenized elasticity tensor (to be calculated);
C(y) is the given fourth order local elasticity tensor that varies according to the location
in the unit cell; χij is the so-called characteristic displacement corresponding to the unit
test strain e(ij).
 4. NUMERIAL RESULTS AND COMPARISONS
 The effective moduli of of some polycrystalline aggregates taken from [18] and re-
ported in Tab.1 with random crystal orientations are calculated using numerical homog-
enization. To represent the material microstructure, we assume periodic unit cells each
of which contains a large number of hexagonal grains with a random crystal orientation.
Each grain is described by 8 × 8 rectangular finite elements.
 (a) (b) (c)
Fig. 1. Unit cell finite element meshes with 4 × 4, 8 × 8, 16 × 16 hexagonal grains per unit cell (a)–
(c); Note that colors are only used to distinguish the grains and do not correspond to orientation.
 Grains with the same colors can have different orientation
 Five levels of grain refinements are considered, corresponding to roughly 4 × 4, 8 ×
8, 16 × 16, 32 × 32, and 64 × 64 hexagonal grains per unit cell (see Fig.1 for illustrations
of thee finite element meshes of the unit cell at the first 3 grain sizes). The grains are
assigned with random crystal orientations. The randomness in the angles between the
 Estimates for the elastic moduli of 2D hexagonal-shape orthorhombic crystals with in-plane random crystalline . . . 175
crystal axes and the Cartesian coordinates is uniformly distributed between 0 and 180
degrees. For each grain size, a number of random samples are generated and analysed,
each of which is represented by a marker point on the plot, c.f. the following figures.
 2.6 2.7
 2.5 V-R bounds
 2.6
 V-R bounds Bounds (3)
 2.4 Bounds (3)
 SC value
 SC value
 2.5
 2.3 FEM FEM
 (GPa)
 (GPa)
 eff
 eff
 2.2 K
 K 2.4
 2.1
 2.3
 1.9 2.2
 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
 Number of hecxagonal per unit cell side S(1) Number of hecxagonal per unit cell side S(3)
 (a) (b)
 26.4 17.1
 26.2 V-R bounds
 26
 Bounds (3) 17
 25.8 V-R bounds
 SC value
 25.6 Bounds (3)
 FEM 16.9
 25.4 SC value
 25.2
 (GPa) FEM
 (GPa) 16.8
 eff 25
 eff
 K
 24.8 K
 24.6 16.7
 24.4
 24.2 16.6
 24
 23.8 16.5
 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
 Number of hecxagonal per unit cell side TiO
 2 Number of hecxagonal per unit cell side U(1)
 (c) (d)
Fig. 2. The scatter of the effective 2D bulk moduli (in GPa) of S(1), S(3), TiO2 and U(1) aggregate
with hexagonal grain configuration and random crystal orientations versus the unit cell size (in
the number of grains). There are five clusters of points corresponding to the five meshes in Fig.1
(because of close clustering of the scatter points, the horizontal positions of points are slightly
shifted for plotting clarity). Finite element simulation results (FEM) are compared with Voigt-
Reuss-Hill (V-R) bound from Eq. (1), our theoretical bounds (3), and self-consistent (SC) value
 from (6)
 The finite element simulation results on the effective moduli compared with the
bounds (1)–(5) are reported in Figs.2–3. Maximal capacity of our computer has been
used to generate the results. A computer, with operating system windows 10 Pro 64-
bit, hardware includes processor Intel (R) Core (TM) i5-7400 CPU 3.00GHz(4CPUs) 3.00
GHz, memory RAM 8Gb DDR3, solid state drive 120Gb SSD, has been used. Computing
time for each figure is about 18 hours. One can see that the scatters of the simulation
results reduce as the sizes of RVEs increase. The numerical simulation results converge
toward Voigt-Reuss-Hill bounds (1), and then to the estimates (3), for both the macro-
scopic 2D bulk and shear moduli.
176 Vuong Thi My Hanh, Le Hoai Chau,Vu Lam Dong, Pham Duc Chinh
 5 0.45
 4.9
 4.8 V-R bounds
 4.7 Bounds (3)
 V-R bounds
 4.6 SC value
 Bounds (3)
 4.5 FEM 0.4
 4.4 SC value
 4.3 FEM
 (GPa) 4.2 (GPa)
 eff eff
 µ 4.1 µ
 4
 3.9
 3.8
 3.7
 3.6
 3.5 0.3
 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
 Number of hecxagonal per unit cell side Cu Number of hecxagonal per unit cell side In(2)
 (a) (b)
 0.68 0.92
 0.66 0.91
 V-R bounds
 0.9 V-R bounds
 0.64 Bounds (3)
 SC value Bounds (3)
 0.89
 0.62 FEM SC value
 0.88 FEM
 (GPa)
 (GPa)
 eff
 0.6 eff
 µ
 µ 0.87
 0.58
 0.86
 0.56 0.85
 0.54 0.84
 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
 Number of hecxagonal per unit cell side S(1) Number of hecxagonal per unit cell side S(3)
 (c) (d)
Fig. 3. The scatter of the effective shear moduli (in GPa) of Cu, In(2), S(1) and S(3) aggregate with
hexagonal grain configuration and random crystal orientations versus the unit cell size (in the
number of grains). There are five clusters of points corresponding to the five meshes in Fig.1
(because of close clustering of the scatter points, the horizontal positions of points are slightly
shifted for plotting clarity). Finite element simulation results (FEM) are compared with Voigt-
Reuss-Hill (V-R) bound from Eq. (1), our theoretical bounds (3), and self-consistent (SC) value
 from (6)
 5. CONCLUSION
 At the maximal RVE size of 64 × 64 grains, the numerical results on both the effec-
tive bulk and shear moduli of certain hexagonal-shape-crystal aggregates with random
crystalline orientations converge to our theoretical bounds on the moduli. Some other
crystalline aggregates would require larger size RVEs to achieve that convergence. Still,
this is a good bench mark for further studies on more complex random Voronoi poly-
crystals, which are more realistic and might require stronger mathematical methods and
computer resources, as both the grains’ shapes and crystalline orientations are random.
That is the subject of our further studies.
 ACKNOWLEDGMENT
 This work is funded by National Foundation for Science and Technology Develop-
ment of Vietnam (NAFOSTED) under grant number 107.02-2014.33.
 Estimates for the elastic moduli of 2D hexagonal-shape orthorhombic crystals with in-plane random crystalline . . . 177
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