Aggregate shape effect on the diffusivity of mortar: A 3D numerical investigation by random packing models of ellipsoidal particles and of convex polyhedral particles

Computers and Structures 144 (2014) 40–51

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Aggregate shape effect on the diffusivity of mortar: A 3D numerical investigation by random packing models of ellipsoidal particles and of convex polyhedral particles Lin Liu a, Dejian Shen a,⇑, Huisu Chen b, Wenxiang Xu c a b c

College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China Jiangsu Key Laboratory of Construction Materials, School of Materials Science and Engineering, Southeast University, Nanjing 211189, China Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China

a r t i c l e

i n f o

Article history: Received 11 April 2014 Accepted 29 July 2014

Keywords: Shape effect Diffusivity Particle packing Ellipsoidal particles Polyhedral particles

a b s t r a c t This paper investigates the influence of aggregate shape on the diffusivity of mortar in three dimensions by random packing models of ellipsoidal and convex polyhedral particles. From a digital mesostructure and by a lattice approach, the diffusivity of mortar can be predicted. Simulations are compared with experimental results from literature and are discussed for mono-sized and multi-sized particle packing with and without interfacial transition zones (ITZ). It was found that the shape of aggregate has a significant effect on the diffusivity of mortar, and for oblate ellipsoids, the shape effect is most pronounced comparing to prolate ellipsoids and polyhedrons. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Cement-based composite material, one of the most used materials in modern infrastructures, is a complex random multi-scale system of particle packing. At microscopic scale, fresh cement paste is composed of water and randomly packed irregular cement particles. At mesoscopic scale, the packing of aggregate particles is closely related to the mesostructures of concrete and mortar. In the last two decades, plenty of particle packing models have been developed to simulate the micro-/meso-structure of cement-based composite materials. According to the shape of particles, these models can generally be categorized as: packing models of spherical particles [1–10], packing models of 2D elliptical [11,12] or 3D ellipsoidal particles [13–17], packing models of 2D convex polygonal [18] or 3D convex polyhedral particles [19–22], packing models of arbitrary-shaped particles [23–31]. Among these packing models, the spherical particle based models have been developed first and take majority, e.g., the HYMOSTRUC model [1], the integrated spherical particle kinetics model [2], the SPACE model [3], the discrete element model (DEM) [4,5], the molecular dynamic model (MDM) [6,7], the particle suspension model [8], the HADES model [9] and the lic model [10], etc. Hereafter, ⇑ Corresponding author. Tel./fax: +86 25 83786183. E-mail addresses: [email protected] (L. Liu), [email protected] (D. Shen), [email protected] (H. Chen), [email protected] (W. Xu). http://dx.doi.org/10.1016/j.compstruc.2014.07.022 0045-7949/Ó 2014 Elsevier Ltd. All rights reserved.

considerable attention has been paid to the random packing of 2D elliptical and 3D ellipsoidal particles in cement-based composite materials. By employing 3D ellipsoidal particle packing systems, Xu and Chen [13–17] have revealed the particle shape effects on the micro-/meso-structure characteristics (e.g. the thickness and volume of ITZ, the volume fraction of solid phase, the pore size distribution, etc.). Furthermore, similar investigations about the particle shape effects have been addressed by employing 2D polygonal or 3D polyhedral particle packing systems [20–22]. By applying the Voronoi tessellation method, random close packing systems of complex-shaped particles (e.g., 2D convex polygons, spheropolygons, 3D convex polyhedra or spheropolyhedra) have been constructed [23–26]. As to arbitrary-shaped particles, based on a Fourier series (2D case) or a spherical harmonic series (3D case), the packing systems of arbitrary-shaped particles have been constructed in literature [27–31]. The particle packing models can provide us a fundamental understanding for the mechanical and transport properties of cement-based composite materials which are closely related to their corresponding micro-/meso- structures. Based on a particle packing, by coupling with a specific algorithm, the mechanical properties [32–36] and transport properties [37–45] of cementbased composite materials can be estimated. The particle packing effects on the mechanical properties including Young’s modulus, tensile strength, toughness, and fracture energy have been studied by some researchers in 2D [32–35] and attempted by Azéma et al.

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

in 3D [36]. Regarding to transport properties, by using random walk algorithm, the diffusivities of concrete and cement paste have been investigated by Bentz et al. [37] and Liu et al. [38] respectively. By converting the micro-/meso-structure of cement-based composite material into a transport network and using a lattice approach, Liu et al. [39] have revealed the diffusivity changes of cement paste with microcracks and Zheng et al. [40] have indicated the 2D elliptical aggregate shape effect on the diffusivity of concrete. By finite element method, Kamali-Bernard et al. [41,42] have computed the diffusivity of mortar from a digitized 3D mesostructure, and Li et al. [43] have obtained the diffusivity of concrete using 2D mesostructures with several aggregate shapes. Moreover, Zhang et al. [44] have studied the permeability of cement paste based on its 3D microstructure using lattice Boltzmann method, and Abyaneh et al. [45] have investigated the diffusivities of mortar and concrete based on their 3D mesostructures with ellipsoidal particles using finite difference method. The above models give us insights about the effects of various parameters (e.g. water-tocement ratio, degree of hydration, volume fraction of aggregate, shape effect of aggregate, diffusivity and volume fraction of interfacial transition zone (ITZ), etc.) on the diffusivity or permeability of cement-based composite materials. So far, most modeling works have focused on the packing systems of spherical particles [37–39,41,42,44]. Nevertheless, the particle shape may affect the properties of cement-based composite materials. Preliminary studies about the aggregate shape effect on the diffusivity of concrete have been carried out in two dimensions assuming elliptical aggregates [40] and convex polygons [43]. In three dimensions, Abyaneh et al. [45] have just attempted to investigate the aggregate shape effect on the diffusivity of mortar, but limited to spherical particle based systems. Following the present studies, this paper is going to investigate the influences of aggregate shape on the diffusivity of mortar by random packing models of ellipsoidal particles and of convex polyhedral particles. In order to investigate the aggregate shape effect, aggregate particles are modeled as ellipsoids with varying aspect ratios and as polyhedrons from tetrahedrons to icosahedrons. The diffusivity of mortar is predicted by employing a 3D lattice approach. The simulated diffusivities of mortars are compared with experimental results from literature to validate the numerical method. The effects of aggregate shape on the diffusivity of mortar are discussed under the following conditions: (1) for mono-sized particle packing, (2) for multi-sized particle packing of different particle volume fractions and (3) with and without ITZ zones. 2. Methodology The simulation process of obtaining the diffusivity of mortar consists of three steps: generation of mesostructure, digitalization of mesostructure and lattice modeling for diffusivity. First, a threedimensional mesostructure is generated where mortar is assumed as a composite of aggregate particles, bulk cement paste and ITZ. Random packing models of ellipsoidal particles and of convex polyhedral particles are employed to obtain the aggregate packing. The ITZ is assumed to be a shell around the aggregate particle with a certain thickness. Next, the mesostructure of mortar is converted into a corresponding mesh represented by voxels, and voxels are identified as the phase of bulk cement paste, ITZ or aggregate. Then, from the digitized mesostructure, 3D lattice modeling is carried out to obtain the diffusivity of mortar under steady-state regime. 2.1. Generation of mesostructure In this work, aggregate particles are modeled as a number of ellipsoids or polyhedrons with a given particle size distribution

(PSD). Platonic particles which are convex polyhedrons with faces composed of congruent convex regular polygons are employed in the generation of the mesostructure. There are five such particles: the tetrahedron, cube, octahedron, dodecahedron and icosahedron [20]. Illustration of Platonic particles is given in Fig. 1. An equivalent diameter Deq, defined as the diameter of a sphere having the same volume as that of a non-spherical particle [14–16], is introduced to connect the ellipsoidal PSD and the polyhedral PSD with the spherical PSD in cement-based composite materials [16,17,20]. For ellipsoids, Deq can be expressed as [17],

( Deq ¼

2cj2=3 2cj1=3

j<1 for ellipsoids jP1

ð1Þ

where Deq is the equivalent diameter of the ellipsoid or the polyhedron. c and j (i.e., if the ellipsoid shape is prolate, j = a/c; if the ellipsoid shape is oblate, j = c/a) are the semi-minor axis and aspect ratio of the ellipsoid, respectively. For Platonic particles, the relationship of the equivalent diameter Deq and the edge length a of each Platonic particle is given as [20],

Deq

8 pffiffi1=3 > 2 > a > > 2p > > > 6 1=3 > > a > > > p  < pffiffi 1=3 2 2 ¼ a p > > >   p ffiffi > 1=3 > 45þ21 5 > > a > 2p > > > > : 15þ5pffiffi5 1=3 ð 2p Þ a

for tetrahedron for cube for octahedron

ð2Þ

for dodecahedron for icosahedron

The particle size distribution of aggregate can be determined either experimentally in a conventional sieve analysis or generated from a theoretical gradation. In concrete, two particular distribution functions, the Fuller distribution and the equal volume fraction (EVF) distribution, are commonly used to represent the PSD of aggregates [46,47]. The Fuller and the EVF distribution functions, those stand for the lower and upper bound of aggregates respectively, can be given as [20],

F N ðDeq Þ ¼

eqq Dq eq  Dmin

ðDeqq max



Deqq min Þ



q ! 2:5 Fuller q ! 3:0 EVF

ð3Þ

where FN (Deq) is the cumulative number-based probability function of aggregates, Deq is the equivalent diameter of a non-spherical aggregate (mm), Dmaxeq and Dmineq are the maximum and minimum equivalent diameters (mm), respectively. According to Eq. (3), the particle number of ellipsoidal or polyhedral aggregates with various sizes can be obtained with a given volume fraction of aggregates Vf in concrete. Details of the determination of the number of ellipsoidal particles and polyhedral particles are described in literature [16,20]. In general, the geometric information of an ellipsoid can be described by the nine degrees of freedom, i.e., semi-major axis (a), semi-intermediate axis (b), semi-minor axis (c), center (x0, y0, z0) and three Euler angles (a, b, c). It is a so-called nine parameters method. In the field of computer simulation of ellipsoidal particle packing, a quadratic curve as described by Eq. (4) is preferred to express the geometrical model of an ellipsoid [14].

X T QX ¼ 0

ð4Þ

2 3 2 3 x A D F G 6y7 6D B E H7 7 6 7 where X ¼ 6 4 z 5 and Q ¼ 4 F E C I 5; A, B, . . ., and J are coeffi1 G H I J cients of a quadratic curve equation, which determine a unique ellipsoid in 3D space. As a consequence, coefficients of the geometric model must be associated with nine parameters for an ellipsoid.

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

(a) Tetrahedron

(b) Cube

(c) Octahedron (d) Dodecahedron (e) Icosahedron

Fig. 1. Illustration of Platonic particles [21].

The relationship between coefficients of the geometric model and nine parameters for an ellipsoid is described in Appendix A [14]. For Platonic particles, the set of vertices can be used as a useful geometrical representation of such a particle (as described in more detail in Appendix B [21]). Other important geometrical properties of the Platonic particle, such as its faces and edges, can be represented as certain subsets of the vertices. Take a tetrahedron as example, Fig. 2 illustrates its geometrical representation. To characterize the particle shape, sphericity is used to unify the shape of a polyhedral particle with the shape of an ellipsoidal particle. Sphericity s is defined as the ratio of surface area between spherical and non-spherical particles with the same volume [20]. For mono-shaped particle packing, its corresponding sphericity s is constant. Sphericity of mono-shaped Platonic and ellipsoidal particles utilized in this study is listed in Table 1. After the total number of non-spherical particles with various sizes is known, the mesostructure model of mortar or concrete can be constructed by random packing of such ellipsoidal or polyhedral particles in the bulk cement paste. The key issue of the random packing of aggregate particles is to ensure that each particle does not overlap with the others, while the neighboring interfacial layers around the aggregate particles may overlap. A contact detection algorithm of ellipsoids is presented and the accuracy and efficiency of the algorithm are tested by Xu et al. [13–17]. For polyhedral particles, the overlapping detection problem is solved by implementing separation axis algorithm [20–22]. To eliminate wall effects, periodic boundary conditions are applied. In this study, the mesostructures of mortar of ellipsoidal aggregates with different aspect ratios (e.g. j = 0.4, 0.6, 0.8, 1, 2, 3) and of Platonic particles (i.e., tetrahedron, cube, octahedron, dodecahedron and icosahedron) are generated, see Figs. 3 and 4. Previous simulation investigations from Abyaneh et al. [45] indicated that the sample size should be at least 2.5 times the largest aggregate size. In all the simulations hereinafter, this criterion will be satisfied.

2.2. Digitalization of mesostructure In this section, the generated mesostructure of mortar with non-spherical aggregates is digitized into a voxel-presented mesostructure. The phase of each cubic voxel is determined based on a nine points controlling algorithm [29]. The nine points are the vertices and the center of the cubic voxel. If the majority of points in one voxel are aggregate phase, then the voxel is designated as aggregate. Accordingly, each voxel is designated as aggregate, bulk cement paste or ITZ. The key issue of digitalization is to judge whether the point of each cubic voxel is inside the aggregate particle or the ITZ zone. For a point P with a position (xp, yp, zp) in global coordinate system O–XYZ, the detecting of its phase in a packing system of ellipsoids as follows, (1) Transform the spatial coordinates of point P (xp, yp, zp) in global coordinate system O–XYZ into P (x*, y*, z*) in a local coordinate system O0 –X0 Y0 Z0 . The origin of the O0 –X0 Y0 Z0 is the center of the ellipsoid E (x0, y0, z0) and the x’, y’, z’ axis of the O0 –X0 Y0 Z0 is along the semi-major axis (a), the semiintermediate axis (b), the semi-minor axis (c), respectively. (2) If

ðx Þ2 a2

 2

 2

þ ðyb2Þ þ ðzc2Þ 6 1, point P is assumed to be inside

ellipsoid E, go to step (5).  2

ðx Þ (3) If we have, ðaþt

2 ITZ Þ

jast Þ2

ðy þ ðbþt

2 ITZ Þ

jast Þ2

ðz þ ðcþt

2 ITZ Þ

6 1, point P is assumed to

be in the ITZ zone around ellipsoid E. tITZ represents the thickness of the ITZ zone. (4) Replace ellipsoid E with another ellipsoid in the packing model, and repeat step (1) to step (3) until the relationship of point P with all the ellipsoids in the packing model is clear. (5) If P is inside any one ellipsoid, the phase of P is aggregate. If not, but P is inside the ITZ zone around any one ellipsoid, the phase of P is ITZ. Else, the phase of P is bulk cement paste. In the random packing models of polyhedrons, the geometrical information of a Platonic particle H is represented by the set of vertices P1 (x1, y1, z1), P2 (x2, y2, z2), . . ., Pk(xk, yk, zk). For a point P with a position (xp, yp, zp) in global coordinate system O–XYZ, the detecting of its phase in a packing system of polyhedrons as follows, (1) Surface F1 which consists of vertices P1 (x1, y1, z1), P2 (x2, y2, z2) and P3 (x3, y3, z3) on the polyhedron H, can be expressed as,

Aðx  x1 Þ þ Bðy  y1 Þ þ Cðz  z1 Þ ¼ 0

Fig. 2. Geometrical representation of a tetrahedron.

ð5Þ

where A ¼ n1 p2  n2 p1 , B ¼ p1 m2  p2 m1 , C ¼ m1 n2  m2 n1 and m1 ¼ x2  x1 , n1 ¼ y2  y1 , p1 ¼ z2  z1 , m2 ¼ x3  x1 , n2 ¼ y3  y1 , p2 ¼ z3  z1 .

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L. Liu et al. / Computers and Structures 144 (2014) 40–51 Table 1 Sphericities of mono-shaped Platonic and ellipsoidal particles. Ellipsoidal particles

j = 0.4

j = 0.6

j = 0.8

j=1

j=2

j=3

Sphericity Platonic particles Sphericity

0.850 Tetrahedron 0.671

0.951 Cube 0.806

0.991 Octahedron 0.846

1 Dodecahedron 0.910

0.929 Icosahedron 0.939

0.848 – –

(a) Oblate spheroidal, = 0.4

(b) Oblate spheroidal, = 0.6

(c) Oblate spheroidal, = 0.8

(d) Spheroidal, = 1

(e) Prolate spheroidal, = 2

(f) Prolate spheroidal, = 3

Fig. 3. Illustration of random packing of ellipsoidal particles (Fuller distribution, Dmaxeq = 2 mm, Dmineq = 0.2 mm,Vf = 0.3, container length = 6 mm).

(2) Substituting point P and other vertices on polyhedron H but not on surface S1 in Eq. (5), we have,

np ¼ Aðxp  x1 Þ þ Bðyp  y1 Þ þ Cðzp  z1 Þ n4 ¼ Aðx4  x1 Þ þ Bðy4  y1 Þ þ Cðz4  z1 Þ n5 ¼ Aðx5  x1 Þ þ Bðy5  y1 Þ þ Cðz5  z1 Þ  nk ¼ Aðxk  x1 Þ þ Bðyk  y1 Þ þ Cðzk  z1 Þ If np cannot satisfy with npn4 P 0, npn5 P 0, . . ., and npnk P 0, point P isn’t inside polyhedron H, go to step (4). Else, go to step (3). (3) Replace surface F1 with another surface Fi on polyhedron H, and repeat step (1) to step (2). If for all surfaces on polyhedron H, point P is satisfied with step (2), P is assumed as a point in polyhedron H. (4) Replace polyhedron H with another polyhedron, and repeat step (1) to step (3) until the relation of point P with all polyhedrons in the packing system is clear. If P is inside any one polyhedron, the phase of P is aggregate. Else, the phase of P is assumed as bulk cement paste. Similarly, the phase of point P in the same packing system of polyhedrons but with ITZ zones is detected. If the phase of P in polyhedron system without ITZ is bulk cement paste, and in the one with ITZ is aggregate, the phase of P then is updated as ITZ.

By sequentially detecting the phase of points in global coordinate system and the nine points controlling algorithm, the digitized mesostructure can be obtained. Fig. 5 shows the digitized mesostructure of mortar with ellipsoidal aggregates and tetrahedral aggregates, respectively.

2.3. 3D lattice modeling for diffusivity From the voxel-presented mesostructure obtained above, the diffusivity of mortar is simulated by 3D lattice modeling. 3D lattice modelling can be employed to assess the internal deterioration [48–50], mechanical properties [51] and diffusivities [39,40] of material in question.In the 3D lattice modeling of the diffusivity of mortar, the first step is to convert the voxel-presented mesostructure of mortar into a lattice network consisting of diffusive lattice elements. In the voxel-presented mesostructure, the diffusivity of each voxel is equal to the diffusion coefficient of its composing phase. Two diffusive phases (i.e., bulk cement paste and ITZ) exist in mortar, while aggregate phase is assumed to be non-diffusive. A diffusive lattice element can be formed by connecting any two adjacent diffusive voxels. Fig. 6 illustrates how to construct a lattice network which consists of diffusive lattice elements. For the diffusive lattice element i  j relating voxel i and voxel j, its diffusion coefficient Dij is,

Dij ¼

1 Di

2 þ D1j

ð6Þ

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

(a) Tetrahedral

(d) Dodecahedral

(b) Cubic

(c) Octahedral

(e) Icosahedral

Fig. 4. Illustration of random packing of Platonic particles (Fuller distribution, Dmaxeq = 2 mm, Dmineq = 0.2 mm,Vf = 0.3, container length = 6 mm).

Fig. 5. Illustration of digitized mesostructures of mortar with ellipsoidal aggregates and tetrahedral aggregates (Fuller distribution, Dmaxeq = 2 mm, Dmineq = 0.2 mm, Vf = 0.3, container length = 6 mm, thickness of ITZ = 50 lm).

qij ¼ Dij

Fig. 6. Determination of the diffusive lattice elements [39].

where Di and Dj is the diffusion coefficient of voxel i and voxel j, respectively. The flow through the element i  j is satisfied with Fick’s first law,

dcij dx

ð7Þ

where qij is the flow density through element ij. cij is the concentration gradient from voxel i to voxel j. x is the center distance from voxel i to voxel j. According to experimental results measured by Delagrave et al. [52], the tritiated water diffusion coefficient of the w/c = 0.45 cement paste Dp is 9.8  1012 m2/s. In this study, this value is assigned to paste voxels. A value of DITZ = 6.4Dp is utilized for ITZ voxels, which corresponds to a homogeneous value associated to a given ITZ thickness and has been previously identified by the authors [41,42]. From the digitized mesostructure of mortar in Fig. 6, the corresponding lattice network is illustrated in Fig. 7, where the diffusive lattice elements are colored according to their diffusion coefficient Di-j in units of m2/s.

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

(a) Oblate spheroidal, κ = 0.6

(b) Tetrahedral

Fig. 8. Concentration distributions in mortar with ellipsoidal aggregates and tetrahedral aggregates at a steady-state diffusion regime.

regime. The water concentration, in units of mol/m3, is represented by colors from cool to warm. After the steady-state diffusion is achieved, the diffusion coefficient of mortar, Dm, can be obtained by,

(a) Oblate spheroida, κ = 0.6l

Dm ¼

(b) Tetrahedral Fig. 7. Illustration of lattice networks of mortar with ellipsoidal aggregates and tetrahedral aggregates (Fuller distribution, Dmaxeq = 2 mm, Dmineq = 0.2 mm,Vf = 0.3, container length = 6 mm, thickness of ITZ = 50 lm).

According to Eq. (7), the matrix equation for a diffusive lattice element can be written by,





Dij Aij 1 1 l 1 1



ci cj

"

¼

qij qji

# ð8Þ

where Aij and l represents the area and length of the element i  j, respectively. The whole matrix equation for the lattice network is assembled one element by one element. After the lattice network is constructed and whole matrix equation for the lattice network is assembled, the next work is to impose concentration boundary conditions. Concentrations ct and c0 are assigned to the inlet surface and the outlet surface, respectively. The inlet and outlet surfaces are assumed as two parallel surfaces of the cubic mesostructure. In this study, 1.0 mol/mm3 for ct and 0 mol/mm3 for c0 are used. No flux occurs through other surfaces. The concentration distribution in the lattice network can be obtained by solving the whole matrix equation. There are several methods to solve the whole matrix equation, for instance, finite difference method (FDM) and conjugate gradient method (CGM). The FDM can give the concentration changing with time but time consuming. The CGM cannot give concentration changing with time, but can calculate the concentration distribution at steadystate with a high efficiency [39]. CGM is utilized in this study. Fig. 8 shows concentration distributions in mortars with ellipsoidal aggregates and tetrahedral aggregates at a steady-state diffusion

Q L A ct  c0

ð9Þ

where Q is flux through the outlet surface, mol mm2 s1; L is the length of the mesostructure; A is the cross-sectional area of the mesostructure, mm2; ct and c0 is the concentration of the diffusive species on the inlet and outlet surface, mol mm3. Simulations in literature [45,53] indicated that the predicted transport property increases when smaller voxels are used, because the aggregate shape and the connectivity between the ITZ voxels are better represented. In order to capture the minimum aggregates and the ITZ zones, a resolution of 10 lm/voxel is utilized in this study. Since the lattice approach is applied to a digitized mesostructure, it serves as a platform for applying real images of microstructure as inputs in the future, and other transport properties such as pressure-induced water flow and ion’s diffusion. 3. Comparison with experimental results A number of experiments have investigated the influence of aggregate on the diffusivity of mortar, which could be utilized to validate the simulation results at different aggregate volume fractions. Five sets of experimental results [52,54–57], given in Fig. 9, generally exhibit the same trend that the diffusivity ratios of mortar decrease with the increasing aggregate volume fraction. While the data from literature [58] indicate that the diffusivity ratios of

Fig. 9. Comparison of diffusivity ratios Dm/Dp of mortar by experiments from literature [52,54–58] and by simulations.

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

In our simulation, the particle size distribution is satisfied with Fuller’s distribution function, where a size interval of 0.1 mm is adopted. Fig. 10 shows several plateaus as the size increases. This is because no particle is generated at such sieve size by simulation. The simulation is generally in agreement with the sand grading in experiments. Assuming all aggregate particles are spherical, simulations were carried out at the ITZ thicknesses of 0 lm and 25 lm. Fig. 11 compares our numerical simulations with the experimental results, which are plotted against aggregate volume fraction. As expected, the diffusivity ratio Dm/Dp decreases with an increase in aggregate volume fraction. The simulations also show that Dm/Dp increases with an increase in ITZ thickness. Fig. 10. Comparison of sand grading in mortars by experiment [55] and by simulation.

1

Dm/Dp

0.8 0.6 0.4 Wong 2009 Simulation, k=1 t=0 Simulation, k=1 t=25µm

0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

4. Results and discussion In this section, by applying the computational methods presented in Section 2, the simulation dependency of mortar diffusivity on aggregate shape is investigated. Polyhedral particles including tetrahedral, cubic, octahedral, dodecahedral and icosahedral particles and ellipsoidal particles including oblate ellipsoidal (j = 0.4, 0.6 and 0.8) and prolate ellipsoidal (j = 2 and 3) particles are involved. The aggregate shape effect on the simulated diffusivity of mortar are presented in Figs. 12 and 13, and discussed under the following conditions: (1) for mono-sized particle packing with a 0.3 particle volume fraction; (2) for multi-sized particle packing with 0.3, 0.4 and 0.5 particle volume fractions and (3) with and without ITZ zones .

Vf Fig. 11. Comparison of diffusivity ratios by experiment [55] and by simulation.

mortar increase with the increasing aggregate volume fraction when a fine sand grading was utilized. The experimental results also show a certain variability attributed to different test methods (under steady-state regime [52,54–56] or non-steady-state regime [57,58]), different diffusion species (chlorides [54,56–58], oxygen [55], tritiated water [52]), different sand grading [54,58], different w/c, different hydration ages and so on. Simulations from ellipsoidal particle packing system with different aspect ratios are compared with these experimental results first. With a Fuller distribution and Deq in the range of 0.2–2 mm, the simulated results may show different trends with different aspect ratio and ITZ thickness. The simulated upper curve is obtained from spherical particle packing systems with a 50 lm ITZ thickness. Two lower simulated curves are obtained from oblate spheroidal particle with j = 0.4 and from tetrahedral particle packing systems without ITZ. The simulated upper curve shows an increase in the diffusivity ratio Dm/Dp with the increasing aggregate volume fraction, exhibiting the same trend as the experimental results from Caré [58] where a fine sand grading was utilized. The lower and middle curves show the same trend as other experimental results [52,54–57]. The diffusivity of mortar decreases with the increasing aggregate volume fraction. Except for the aggregate shape effects, this may be attributed to the change in ITZ content and ITZ connectivity in mortars no matter by simulation and by experiment. The influences of ITZ on the diffusivity ratio will be discussed in the next section. The predicted diffusivities are further compared with the experimental data from Wong et al. [55]. In the experiments, mortars were made of CEM I at 0.30 w/c ratio and Thames valley sand (<5 mm) complying with medium grading. The accumulative passing fraction of sand at different sieve sizes is given in Fig. 10. The samples were sealed cured for 3 days and then preconditioned by drying at gradually increasing temperature up to 50 °C to constant weight over a period of 90 days.

4.1. Aggregate shape effect for mono-sized particle packing The aggregate shape effect for mono-sized particle packing with a 0.3 particle volume fraction is illustrated in Fig. 12. At tITZ = 0, the predicted diffusivity ratio increases with the increasing sphericity s for both particle packing of convex polyhedral particles and

Fig. 12. Shape effect on the diffusivity ratio Dm/Dp for mono-sized particle packing of Deq = 1 mm and Deq = 2 mm with a 0.3 particle volume fraction.

L. Liu et al. / Computers and Structures 144 (2014) 40–51

47

Fig. 13. Shape effect on the diffusivity ratio Dm/Dp for multi-sized particle packing with 0.3, 0.4 and 0.5 particle volume fractions.

ellipsoidal particles. And at s = 1, the maximum diffusivity ratio is achieved by simulation. By comparing the oblate to prolate ellipsoids packing, it is found that the influence of sphericity on diffusivity ratios for oblate ellipsoidal is more pronounced. At the same s, the predicted diffusivity ratio by prolate ellipsoidal particle packing is higher than by the oblate one. By comparing simulations under two different equivalent diameters (Deq = 1 and 2 lm), it can also be found that at the same aggregate volume fraction, a higher diffusivity of mortar can be obtained by using coarse aggregate at tITZ = 0.

For ellipsoidal particle packing, the shape effect at tITZ = 25 lm illustrates the same trend comparing to tITZ = 0. However, for polyhedral particle packing, the shape effect at tITZ = 25 lm shows

4.2. Aggregate shape effect for multi-sized particle packing The shape effects for multi-sized particle packing of different particle volume fractions are illustrated in Fig. 13. For polyhedral particle packing, at tITZ = 0, the predicted diffusivity ratio increases with the increasing sphericity s. For oblate ellipsoidal particle packing, the predicted diffusivity ratio also indicates the same trend. However, for prolate ellipsoidal particle packing, the influence of sphericity is not significant. 4.3. Aggregate shape effect in the presence of ITZ zones The thickness of the ITZ around aggregates is typically about 25–50 lm, depending on the size of cement particles and bleeding effects [45]. In order to investigate the shape effects in the presence of ITZ zone, the simulated diffusivity ratios at tITZ = 25 lm for mono-sized packing and multi-sized packing are shown in Figs. 13 and 14.

Fig. 14. Changes of volume and connectivity of ITZ with sphericity for polyhedral particle packing and ellipsoidal particle packing.

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

Table 2 Diffusivity increment/decrement percentage and variations of increment percentage with sphericity for oblate ellipsoidal particle packing and polyhedral particle packing with a 0.3 particle volume fraction. Mono-sized, Deq = 1 mm

Mono-sized, Deq = 2 mm

Multi-sized, 0.22 mm

tITZ = 0

tITZ = 25 lm

tITZ = 0

tITZ = 25 lm

tITZ = 0

tITZ = 25 lm

Dmðj¼0:4Þ Dmðj¼1Þ Dmðj¼1Þ

0.11

0.09

0.06

0.08

0.09

0.08

DmðtetraÞ Dmðj¼1Þ Dmðj¼1Þ

0.09

0.01

0.07

0.02

0.09

0.11

Dmðj¼0:4Þ Dmðj¼1Þ Dmðj¼1Þ Ds

0.72

0.63

0.42

0.51

0.59

0.54

DmðtetraÞ Dmðj¼1Þ Dmðj¼1Þ Ds

0.28

0.04

0.23

0.05

0.27

0.34

an opposite trend comparing to tITZ = 0. The diffusivity ratio decreases but not increases with the increasing sphericity, see Fig. 13. It indicates that the influence of ITZ on diffusivity is more pronounced to polyhedral particle packing than to ellipsoidal particle packing. This may be attributed to the changes of volume and connectivity of ITZ for different-shaped particle packing, see Fig. 14. Fig. 14 shows that the volume of ITZ decreases with the increasing sphericity for polyhedral particle packing, while no significant changes for ellipsoidal particle packing. The connectivity of ITZ decreases with the increasing sphericity both for polyhedral particle packing and for ellipsoidal particle packing. For polyhedral particle packing, with an increasing sphericity, the decrease of volume and connectivity of ITZ reduces the diffusivity of mortar. Comparing to the diffusivity of mortar with spherical particles, the maximum increment/decrement percentage for ellipsoidal particles and polyhedral particles, i.e., (Dm(j=0.4)  Dm(j=1))/Dm(j=1) and (Dm(tetra)  Dm(j=1))/Dm(j=1) are listed in Table 2. The variations of increment percentage with sphericity, i.e., (Dm(j=0.4)  Dm(j=1))/ Dm(j=1)Ds and (Dm(tetra)  Dm(j=1))/Dm(j=1)Ds are also given. It can be found that the shape effect on diffusivity for oblate ellipsoidal particles is more pronounced than for polyhedral particles.

(3) By comparing the oblate ellipsoids to polyhedrons, it is found that the influence of sphericity on diffusivity for oblate ellipsoids is larger than that for polyhedrons.

Acknowledgements The financial supports of National Natural Science Foundation of China via Grant No. 51308187 and Natural Science Foundation of Jiangsu Province via Grant No. BK20130837 are greatly acknowledged. The China Postdoctoral Science Foundation (No. 2013M531266), Jiangsu Postdoctoral Science Foundation (No. 1202022C), the foundation from State Key Laboratory of High Performance Civil Engineering Materials (No. 2012CEM005) are also greatly acknowledged. Appendix A. The relationship between coefficients of the geometric model and nine parameters for an ellipsoid [14] A matrix W is defined

0

B W ¼ @ e1 f1

5. Conclusions In order to investigate the influence of aggregate shape on the diffusivity of mortar in three dimensions, random packing models of ellipsoidal particles and of convex polyhedral particles are first utilized. Polyhedral aggregates including tetrahedral, cubic, octahedral, dodecahedral and icosahedral particles and ellipsoidal aggregates including oblate ellipsoidal (j = 0.4, 0.6 and 0.8) and prolate ellipsoidal (j = 2 and 3) particles are involved. A shape factor, sphericity, is introduced to unify the shapes of polyhedral particles and ellipsoidal particles. By converting the packing systems of polyhedral and ellipsoidal particles into voxel-presented mesostructures of mortars and by employing a 3D lattice approach, the diffusivities of mortars are simulated. The simulated diffusivities of mortars are compared with experimental results from literature to validate the numerical method. The effect of aggregate shape on the diffusivity of mortar is presented and discussed. Main conclusions that can be drawn from the modeling studies in this paper as follows, (1) For ellipsoidal particle packing, with and without ITZ zones, the simulated diffusivity of mortar show an increase with the increasing sphericity. The shape effect is more pronounced for oblate ellipsoids compared to prolate ellipsoids. (2) For polyhedral particle packing, simulations of diffusivity with sphericity show opposite trends with and without ITZ. The simulated diffusivity of mortar show an increase with the increasing sphericity at tITZ = 0, while a decrease at tITZ = 25 lm.

d1

d4

1

d2

d3

e2

e3

C e4 A

f2

f3

f4

ðA:1Þ

where coefficients of the matrix W can be represented as

d1 ¼

1 cos b cos c a

ðA:2Þ

d2 ¼

1 cos b sin c a

ðA:3Þ

d3 ¼

1 sin b a

ðA:4Þ

d4 ¼ x0 d1  y0 d2  z0 d3

ðA:5Þ

e1 ¼

1 ð cos a sin c  sin a sin b cos cÞ b

ðA:6Þ

e2 ¼

1 ðcos a cos c  sin a sin b sin cÞ b

ðA:7Þ

e3 ¼

1 sin a cos b b

ðA:8Þ

e4 ¼ x0 e1  y0 e2  z0 e3

ðA:9Þ

f1 ¼

1 ðsin a sin c  cos a sin b cos cÞ c

ðA:10Þ

f2 ¼

1 ð sin a cos c  cos a sin b sin cÞ c

ðA:11Þ

49

L. Liu et al. / Computers and Structures 144 (2014) 40–51

f3 ¼

1 cos a cos b c

ðA:12Þ

f 4 ¼ x0 f 1  y0 f 2  z0 f 3

ðA:13Þ

Hence, the relationship between the coefficient matrix Q and nine parameters can be written as

2

3

A

D

F

G

6D 6 Q ¼6 4F

B

E

E

C

H7 7 7 I5

G

H

I

J

2

2

ðA:14Þ

2

2

ðA:15Þ

A ¼ d1 þ e21 þ f 1

The six square faces Fa of the cube are given by

B ¼ d2 þ e22 þ f 2 2

C ¼ d3 þ e23 þ f 3

ðA:16Þ

D ¼ d1 d2 þ e1 e2 þ f 1 f 2

ðA:17Þ

E ¼ d2 d3 þ e2 e3 þ f 2 f 3

ðA:18Þ

F ¼ d1 d3 þ e1 e3 þ f 1 f 3

ðA:19Þ

G ¼ d1 d4 þ e1 e4 þ f 1 f 4

ðA:20Þ

H ¼ d2 d4 þ e2 e4 þ f 2 f 4

ðA:21Þ

I ¼ d3 d4 þ e3 e4 þ f 3 f 4

ðA:22Þ

2

ðA:23Þ

Appendix B. Vertices, edges, and faces of the Platonic particles [21] Here, the size parameter of the Platonic particles is defined as the radius (r) of the circumscribed sphere. The origin of the coordinate system O0 XYZ is set as the centroid of the Platonic particles, and the z axis of the coordinate system is considered to pass through one of vertices. Thus, the vertices (Pv), edges (Se), and faces (Fa) of the Platonic particles can be obtained. For a tetrahedron, there are four vertices (v = 1, . . ., 4), six edges (e = 1, . . ., 6), and four triangular faces (a = 1, . . ., 4). The four vertices Pv of the tetrahedron are written as

P3 ¼

pffiffiffi 2  r; 3

!T pffiffiffi pffiffiffi 2 6 1 r; r;  r ; 3 3 3 !T !T pffiffiffi pffiffiffi 1 2 2 1 6  r;  r ; P4 ¼ r; 0;  r 3 3 3 3

P2 ¼

F1 ¼ fP1 ; P2 ; P8 ; P3 g; F2 ¼ fP1 ; P3 ; P6 ; P4 g; F3 ¼ fP1 ; P4 ;P7 ; P2 g; F4 ¼ fP5 ; P7 ; P4 ; P6 g; F5 ¼ fP5 ; P8 ; P2 ; P7 g; F6 ¼ fP5 ; P6 ;P3 ; P8 g

P1 ¼ ð0; 0; rÞT P4 ¼

F2 ¼ fP1 ; P3 ; P4 g;

F3 ¼ fP1 ; P4 ; P5 g;

F5 ¼ fP6 ; P3 ; P2 g; F8 ¼ fP6 ; P2 ; P5 g

F6 ¼ fP6 ; P4 ; P3 g;

P4 ¼ P6 ¼

ðB:1Þ P8 ¼

S2 ¼ ½P1 ; P3 ;

S3 ¼ ½P1 ; P4 ;

P13 ¼

S5 ¼ ½P3 ; P4 ;

S6 ¼ ½P4 ; P2 

ðB:2Þ

ðB:3Þ

pffiffiffi pffiffiffi !T pffiffiffi !T 2r r 3r 5r 5r ; 0; ; P3 ¼ ; ; ; 3 3 3 3 3 pffiffiffi pffiffiffi !T pffiffiffi pffiffiffi !T 5r 3r r r  3r 5r ; ; ; P5 ¼ ; ; ; 3 3 3 3 3 3 pffiffiffi pffiffiffiffiffiffi pffiffiffi !T pffiffiffi pffiffiffiffiffiffi pffiffiffi !T 3 5 3þ 5 15 þ 3 r 15  3 r r; r; ; P7 ¼  r; r; ; 3 3 6 6 6 6 pffiffiffiffiffiffi pffiffiffi !T pffiffiffiffiffiffi pffiffiffi !T pffiffiffi pffiffiffi 3þ 5 3 5 15  3 r 15 þ 3 r  r;  r; ; P9 ¼ r;  r; ; 3 3 6 6 6 6 ! ! pffiffiffi pffiffiffi pffiffiffi T T 5r  3r r 2r  5r ; ; ; P11 ¼ ð0; 0; rÞT ; P12 ¼ ; 0; ; 3 3 3 3 3 pffiffiffi pffiffiffi !T pffiffiffi pffiffiffi !T r  3r  5r r 3r  5r ; ; ; P14 ¼ ; ; ; 3 3 3 3 3 3 !T !T pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi  5r  3r r 3 5 15 þ 3 r ; ; ; P16 ¼  r;  r; ; 3 3 3 6 3 6 !T !T pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi 3þ 5 3þ 5 15  3 r 15  3 r r;  r; ; P18 ¼ r; r; ; 6 6 3 3 6 6 !T !T pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi 3 5  5r 3r r 15 þ 3 r  r; r; ; P20 ¼ ; ; ðB:8Þ 3 6 6 3 3 3

P1 ¼ ð0; 0; rÞT ; P2 ¼

S4 ¼ ½P2 ; P3 ;

F4 ¼ fP2 ; P3 ; P4 g

ðB:7Þ

where the connection of neighboring vertices comprises one edge of the particle. A dodecahedron has 20 vertices (v = 1, . . ., 20), 30 edges (e = 1, . . ., 30), and 12 pentagonal faces (a = 1, . . ., 12). The 20 vertices Pv of the dodecahedron are given by

S1 ¼ ½P1 ; P2 ;

F2 ¼ fP1 ; P2 ; P4 g;

ðB:6Þ

F4 ¼ fP1 ; P5 ; P2 g; F7 ¼ fP6 ; P5 ; P4 g;

P10 ¼

F3 ¼ fP1 ; P4 ; P3 g;

!T pffiffiffi pffiffiffi 2r 2r ; ; 0 ; P5 ¼ 2 2 !T pffiffiffi pffiffiffi  2r  2r ; ;0 2 2

P3 ¼

F1 ¼ fP1 ; P2 ; P3 g;

where the superscript T denotes the transpose of a vector. According to the four vertices, the six edges Se of the tetrahedron are given by

F1 ¼ fP1 ; P3 ; P2 g;

!T pffiffiffi pffiffiffi  2r 2r ; ;0 ; 2 2 !T pffiffiffi pffiffiffi  2r 2r ; ;0 ; 2 2

; P2 ¼ ð0; 0; rÞT ;

The eight triangular faces Fa of the octahedron are given by



where square brackets indicate the subtraction of two vectors Moreover, on the basis of Pv, the four triangular faces Fa of the tetrahedron can be expressed as

ðB:5Þ

where the connection of neighboring vertices consists of one edge of the particle. For an octahedron, there are six vertices (v = 1, . . ., 6), 12 edges (e = 1, . . ., 12), and eight triangular faces (a = 1, . . ., 8). The six vertices Pv of the octahedron are written as

P6 ¼

2

J ¼ d4 þ e24 þ f 4  1

P1 ¼ ð0; 0; rÞT ;

!T pffiffiffi pffiffiffi pffiffiffi !T 2 2r r  2r 6r r P1 ¼ ð0; 0; rÞ ; P2 ¼ ;0; ; P3 ¼ ; ; ; 3 3 3 3 3 ! !T pffiffiffi pffiffiffi pffiffiffi T  2r  6r r 2 2r r T ; ; ; P5 ¼ ð0;0; rÞ ; P6 ¼ ;0; ; P4 ¼ 3 3 3 3 3 !T !T pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2r  6r r 2r 6r r ; ; ; P8 ¼ ; ; ðB:4Þ P7 ¼ 3 3 3 3 3 3 T

where

2

where braces represent the assembly of vectors. It is worth noting that these vertex vectors comprising each face are arranged in counterclockwise order. A cube has eight vertices (v = 1, . . ., 8), 12 edges (e = 1, . . ., 12), and six square faces (a = 1, . . ., 6). The eight vertices Pv of the cube are denoted as

P15 ¼ P17 ¼ P19 ¼

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L. Liu et al. / Computers and Structures 144 (2014) 40–51

The 12 pentagonal faces Fa of the dodecahedron are given by

F1 ¼ fP1 ; P2 ; P5 ; P6 ; P3 g;

F2 ¼ fP1 ; P3 ; P7 ; P8 ; P4 g;

F3 ¼ fP1 ; P4 ; P9 ; P10 ; P2 g;

F4 ¼ fP2 ; P10 ; P17 ; P18 ; P5 g;

F5 ¼ fP3 ; P6 ; P19 ; P20 ; P7 g;

F6 ¼ fP4 ; P8 ; P15 ; P16 ; P9 g

F7 ¼ fP11 ; P13 ; P16 ; P15 ; P12 g; F8 ¼ fP11 ; P14 ; P18 ; P17 ; P13 g; F9 ¼ fP11 ; P12 ; P20 ; P19 ; P14 ; g F10 ¼ fP12 ; P15 ; P8 ; P7 ; P20 g; F11 ¼ fP13 ; P17 ; P10 ; P9 ; P16 g;

F12 ¼ fP14 ; P19 ; P6 ; P5 ; P18 g;

ðB:9Þ

where the connection of neighboring vertices comprises one edge of the particle. An icosahedron has 12 vertices (v = 1, . . ., 12), 30 edges (e = 1, . . ., 30), and 20 triangular faces (a = 1, . . ., 20). The 12 vertices Pv of the dodecahedron can be written as 0pffiffiffi 1T sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi

T 2r r 51 5þ1 r A @ pffiffiffi r; pffiffiffi r; pffiffiffi ; P1 ¼ ð0;0;rÞ ; P2 ¼ pffiffiffi ;0; pffiffiffi ; P3 ¼ 5 5 5 2 5 2 5 0 pffiffiffi 1T 0 pffiffiffi 1T sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi 5þ1 51 r A 5þ1 51 r A pffiffiffi r; pffiffiffi ; P5 ¼ @ pffiffiffi r;  pffiffiffi r; pffiffiffi ; P4 ¼ @ pffiffiffi r; 5 5 2 5 2 5 2 5 2 5 0pffiffiffi 1T sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi

T 2r r 51 5þ1 r A pffiffiffi r; pffiffiffi ; P7 ¼ ð0; 0; rÞT ; P8 ¼  pffiffiffi ;0; pffiffiffi ; P6 ¼ @ pffiffiffi r; 5 5 5 2 5 2 5 0 pffiffiffi 1T 0pffiffiffi 1T sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi 51 5 þ 1 r A 5þ1 5  1 r A pffiffiffi r; pffiffiffi ; P10 ¼ @ pffiffiffi r; pffiffiffi r; pffiffiffi ; P9 ¼ @ pffiffiffi r; 5 5 2 5 2 5 2 5 2 5 0pffiffiffi 1T 0 pffiffiffi 1T sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5þ1 5  1 r A 51 5 þ 1 r A pffiffiffi r; pffiffiffi ; P12 ¼ @ pffiffiffi r; pffiffiffi r; pffiffiffi P11 ¼ @ pffiffiffi r; 5 5 2 5 2 5 2 5 2 5 T

ðB:10Þ

The 20 triangular faces Fa of the icosahedron can be expressed as

F1 ¼ fP1 ;P2 ;P3 g; F2 ¼ fP1 ;P3 ;P4 g; F3 ¼ fP1 ;P4 ;P5 g; F4 ¼ fP1 ;P5 ;P6 g; F5 ¼ fP1 ;P6 ;P2 g; F6 ¼ fP2 ;P11 ; P3 g; F7 ¼ fP3 ;P12 ; P4 g; F8 ¼ fP4 ; P8 ; P5 g; F9 ¼ fP5 ; P9 ; P6 g; F10 ¼ fP6 ; P10 ; P2 g; F11 ¼ fP7 ;P9 ;P8 g; F12 ¼ fP7 ;P10 ; P9 g; F13 ¼ fP7 ; P11 ; P10 g; F14 ¼ fP7 ;P12 ; P11 g; F15 ¼ fP7 ;P8 ;P12 g; F16 ¼ fP8 ; P9 ; P5 g; F17 ¼ fP9 ; P10 ;P6 g; F18 ¼ fP10 ; P11 ;P2 g; F19 ¼ fP11 ; P12 ;P3 g; F20 ¼ fP12 ; P8 ; P4 g

ðB:11Þ

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