# Design of Particulate-Reinforced Composite Materials

^{*}

## Abstract

**:**

## 1. Introduction

_{H}, so that the methods, which are analogous to the ones discussed herein may also be successfully applied to the fluid flow problems.

## 2. The Effective Field Method

#### 2.1. Isolated Inclusions

#### 2.2. Chain of Inclusions

^{0}is described by Equation (1).

## 3. Numerical Homogenization Strategy

_{RVE}. Thus, it is obvious that the results, understood in the sense of the average property tensor, are directly dependent on the dimensions and form of the RVE. It is worth pointing out that, even for the 2D two-phase periodic composites, the RVE may be of an arbitrary form, not necessarily rectangular as it shown in Figure 3. Since in our numerical analysis we intend to give information on the variations of the property tensor components with respect to the volume fraction p, we define it in the following way (see Figure 4):

_{f}/r

_{p}and b/r

_{p}for both 2D and 3D cases. Let us note that, for the constant volume fraction p and the constant interparticle distance g

_{f}, the geometrical dimensions of the representative cell (i.e., b and h) are uniquely determined.

**E**and

**D**are replaced by their average values evaluated in the local cell (RVE).

**D**–

**E**characteristic curve for the inclusions. Commonly, the inclusion is modeled as isotropic material but for the unit cell the local property matrix can possess anisotropic properties as it is shown for example in Section 2.

_{y}. We have some more compact aggregates. The microscopic analysis on the structure of the two-phase composites revealed that there were aggregates forming rather than chains of spheres that can be approximated by ellipsoids, stripes or cylinders. Therefore, it is interesting to verify the correctness of the introduced FE model in cases when the external field is rotated with respect to coordinates defining RVE in order to consider the relationship between the orientation of inclusions and to explore symmetries in the constitutive relations in Equation (1) and their relevance to the homogenized composite medium. In the case of a generalized anisotropic structure for which principal axis of external fields and the unit cell do not coincide, the property tensor should satisfy the classical transformation rules of the second rank tensors. To interpret and investigate those effects, let us analyze the form of the property (permeability) matrix for the 3D ring structure shown in Figure 6. The geometry of unit cells is defined in the cylindrical coordinate system but the external field is directed along the line joining the center of the central spherical inclusion and the center of the ring curvature.

_{αβ}(α ≠ β) are not equal to zero, which means that it may be for instance the origin of clusters and of the inclusion aggregation at the beginning of magnetization. On the other hand, it may be easily verified that the terms of the property matrix satisfy the classical transformation rule:

^{0}.

## 4. Numerical Results

_{p}= 2000 and c

_{f}= 1. The analysis is conducted for 2D and 3D unit cells to test the capability and limitations of the proposed model. Similar to the previous case, the numerical model corresponds to the analysis of MR fluids.

#### 4.1. 2D Problems

_{zz}decreases as the interparticle distance increases and it is the highest for the highest volume fraction p = 0.5. The decrease of the effective permeability c

_{xx}is associated with the increase of the effective permeability c

_{zz}. Let us note that, for the constant volume fraction p, the variations of the ratio r

_{p}/h (or g

_{f}/r

_{p}) results in the change of the ratio b/r

_{p}(see Equation (9)). Using the single unit cell (Figure 4), one can observe that, for the external magnetic field having the non-zero component H

_{z}only, the chains of ferromagnetic particles are completely isolated since there is no interaction at the x direction. In fact, the experiments evidently demonstrate evidently that they form clusters of different shapes (see Bossis et al. [33]). In Figure 9, a relationship between the volume fraction and the effective permeability for arbitrary assumed value of ratio r

_{p}/h is depicted. As can be observed in the case of c

_{zz}, these relations can be considered as almost linear. In the case of c

_{xx}for small values of volume fraction, they are also linear. However, for the values of volume fraction greater then p > 0.35, the effective permeability values increases exponentially. For both parameters, c

_{zz}and c

_{xx}, the values are achieved for volume fraction p = 0.5.

#### 4.2. 3D Problems

_{xx}= c

_{yy}). The plots are drawn both for spherical (a = a

_{x}= a

_{y}= a

_{z}) and spheroid (a = a

_{x}= a

_{y}≠ a

_{z}) inclusions. For spherical inclusions, the distributions of the homogenized properties for the 3D case are similar to those for the 2D case (see Figure 8 and Figure 9). However, assuming the identical geometric ratios of the RVE, for the 3D case, the averaged values in the x direction are higher than those evaluated for the planar case, whereas the values in the z direction are almost identical. The shape of inclusions has a significant influence on the averaged values. For the increasing a

_{z}/h parameter, the effective property c

_{zz}decreases, and c

_{xx}increases since the x axis corresponds to the longer axis of spheroids. Figure 10 and Figure 11 show that the proposed model provides a transversely isotropic effective permeability, whereas the Maxwell–Garnett model gives an isotropic one. The Maxwell–Garnett model always yields the same estimates for any r

_{p}/h because it is insensitive to the microstructure. Thus, the Maxwell–Garnett model cannot be used for these composites because, even when the overall volume fraction is very small, the distance between particles of the same chain is so small that particle interactions cannot be disregarded. For the low volume fractions, the Maxwell–Garnett gives very good estimations in the x directions only (Figure 11). For spherical inclusions, as can be observed in Figure 10 and Figure 11, the effective field model gives much better approximations of the effective values evaluated with the use of the FE model than the Maxwell–Garnett model. However, the effectiveness of the effective field model decreases for the high particle interactions (a

_{z}/h < 0.1) and for the high volume fractions (p > 0.25). It is obvious that theoretical estimations, i.e., with the use of the Maxwell–Garnett model and the effective field model have limited applications in comparison with the FEM model since the first corresponds to the random (quasi-isotropic) structure of reinforced particles, and the second to the chain-like structure, i.e., in two directions the dimensions of the RVE tend to infinity.

## 5. Optimal Design

_{yy}and c

_{xx}reach their optimal values (see Figure 12 and Figure 13). However, the optimal values of the effective properties are strongly dependent on the values of the geometrical ratios r

_{p}/h and the volume fractions p. Let us note that the maximal value of the term c

_{zz}is much higher than those plotted previously in Section 4, and the values of c

_{zz}are much lower. Therefore, it seems to be reasonable to conclude that the optimal rectangular form of the particles can prevent their aggregation in ellipsoids or cylinders instead of linear chains and in this sense, the theoretical effective field model may be applicable in the estimations of the effective properties. It is worth mentioning that the obtained optimal designs completely resemble those obtained by Guest and Prevost [36] for fluid transportation problem (the Darcy law). They concluded that the Schwartz P minimal surface is believed to be the maximum permeability structure in the 3D case. However, the authors of the cited paper assumed in advance the isotropic properties of the permeability matrix.

## 6. Concluding Remarks

**B**–

**H**). From these models, it is found that the averaged property tensor components are strongly dependent on the dimensionless interparticle distance and the volume fraction.

## Author Contributions

## Conflicts of Interest

## Appendix A. Mathematical Formulation and Derivation of the Effective (Homogenized) Material Properties

_{α}(x) is the gradient of a scalar function φ(x) called the potential of the field:

^{0}containing a set of inclusions with the property tensor C, the system of differential Equations (A1) and (A2) may be reduced to integral equations for the fields E(x) inside the inclusions, i.e.,

_{0}is the external field in the medium without the inclusion (C

^{1}(x) = 0) by the action of the same sources of the field. The kernel K(x) is determined in the classical manner:

^{0}. The Green function satisfies the equation:

- -
- With the use of the potential $\mathsf{\phi}\left(\mathrm{x}\right)$, the solution of Equation (A1) can be decomposed as follows: $\mathsf{\phi}\left(\mathrm{x}\right)={\mathsf{\phi}}^{0}\left(\mathrm{x}\right)+{\mathsf{\phi}}^{1}\left(\mathrm{x}\right)$, where ${\mathsf{\phi}}^{0}\left(\mathrm{x}\right)$ is the potential in the medium without the inclusion, and ${\mathsf{\phi}}^{1}\left(\mathrm{x}\right)$ is the perturbation of the potential due to the presence of the inclusion that tends to zero when $\left|\mathrm{x}\right|\to \infty $.
- -
- The potential ${\mathsf{\phi}}^{0}\left(\mathrm{x}\right)$ satisfies the relation analogous to Equation (A1), i.e., ${\nabla}_{\mathsf{\alpha}}{\mathrm{C}}_{\mathsf{\alpha}\mathsf{\beta}}^{0}{\nabla}_{\mathsf{\beta}}{\mathsf{\phi}}^{0}(\mathrm{x})$ = −q(x).

^{0}by the action of the field ${\mathrm{E}}_{\mathsf{\beta}}^{\ast}(\mathrm{x})$, and, for ellipsoidal inclusions, it takes the following form:

_{1}, a

_{2}, and a

_{3}are the semiaxes of an ellipsoid, and ${\mathrm{e}}_{\mathsf{\alpha}}^{\mathrm{n}}$ are the unit vectors of the ellipsoid principal axes, the orientation of which is given by the normal m. For a spheroid inclusion with the semiaxes a

_{1}= a

_{2}= a, a

_{3}, the tensor A

^{0}takes the following form:

^{0}and using the definition in Equation (A1) of the flux tensor D(x) (for the inclusion and the medium) after a set of transformation of the result, it is possible to derive the average of the flux tensor, i.e.,

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**Figure 1.**Fields in a homogeneous medium with inclusions: (

**a**) particles with interfaces; and (

**b**) two phase composites.

**Figure 5.**The local distributions of magnetic fields for the boundary conditions in Equation (8)—spheroids a

_{x}/a

_{y}= 2, p = 0.3, r

_{p}/h = 0.485, c

_{p}= 2000, c

_{0}= 1: (

**a**) distribution of the magnetic scalar potential ϕ; (

**b**) distribution of the magnetic field intensity; and (

**c**) distribution of the magnetic flux density.

**Figure 6.**The system of 5 ferromagnetic particles—the external field is applied at the x, y or z direction.

**Figure 7.**Boundary conditions and the local (transformed off-axis) values of the permeability matrix.

**Figure 8.**Variations of the effective permeability at two perpendicular directions with particle radius.

**Figure 9.**Variations of the effective permeability at two perpendicular directions with volume fractions.

**Figure 10.**Variations of the permeability coefficients with interparticle vertical distance r

_{p}/h (transversely-isotropic body).

**Figure 11.**Variations of the permeability coefficients with volume fraction p (transversely-isotropic body).

**Figure 12.**Variations of the effective properties with the particle shape—the constant volume fraction p: (

**a**) the x direction; and (

**b**) the y direction.

**Figure 13.**Variations of the effective properties with the particle shape—the constant interparticle distance r

_{p}/h: (

**a**) the x direction; and (

**b**) the y direction.

Magnetostatics | Electrostatics | Heat Flow | Diffusion | Porous Media | |
---|---|---|---|---|---|

Definition of scalar potentials ϕ | |||||

ϕ | Scalar potential ϕ (A) | Electric potential V (Volt) | Temperature T (°K) | Substance concentration γ (mol/m^{3}) | Hydrostatic pressure p (Pa) |

Definition of vector fields in the vacuum ${\mathrm{E}}_{\mathsf{\alpha}}={\nabla}_{\mathsf{\alpha}}\mathsf{\phi}$, $\mathrm{r}\mathrm{o}{\mathrm{t}}_{\mathsf{\alpha}\mathsf{\beta}}{\mathrm{E}}_{\mathsf{\beta}}=0$ | |||||

E_{α} | Intensity of magnetic field H_{α} (A/m) | Intensity of electric field E_{α} (V/m) | Not used | Not used | Not used |

Definition of material properties C_{αrβ} | |||||

C_{αβ} | Magnetic permeability coefficients μ_{0}μ_{αβ} (Tm/A) | Electric permittivity coefficients ε_{0}ε_{αβ} [(A s)/(V m)] | Heat conduction coefficients k_{αβ} [W/(°K m)] | Diffusion coefficients D_{αβ} (m^{2}/s) | Proportionality coefficients K_{αβ}/η [m^{2}/(Pa s)] * |

Definitions of vector fields in a particular medium D_{α} | |||||

D_{α} | Magnetic flux density B_{α} (T) | Electric flux D_{α} (C/m^{2}) | Not used | Not used | Not used |

Constative relations ${\mathrm{D}}_{\mathsf{\alpha}}={\mathrm{C}}_{\mathsf{\alpha}\mathsf{\beta}}{\mathrm{E}}_{\mathsf{\beta}}$ | |||||

Physical relations ${\nabla}_{\mathsf{\alpha}}{\mathrm{D}}_{\mathsf{\alpha}}=0$ | |||||

Gauss’s Law for magnetism ${\nabla}_{\mathsf{\alpha}}\left({\mathsf{\mu}}_{0}{\mathsf{\mu}}_{\mathsf{\alpha}\mathsf{\beta}}{\mathrm{H}}_{\mathsf{\beta}}\right)=0$ | Gauss’s Law ${\nabla}_{\mathsf{\alpha}}\left({\mathsf{\epsilon}}_{0}{\mathsf{\epsilon}}_{\mathsf{\alpha}\mathsf{\beta}}{\mathrm{E}}_{\mathsf{\beta}}\right)=\mathsf{\rho}$ | Fourier’s Law ${\nabla}_{\mathsf{\alpha}}\left({\mathrm{k}}_{\mathsf{\alpha}\mathsf{\beta}}{\nabla}_{\mathsf{\beta}}\mathrm{T}\right)+\mathrm{Q}=0$ | Fick’s Law ${\nabla}_{\mathsf{\alpha}}\left({\mathrm{D}}_{\mathsf{\alpha}\mathsf{\beta}}{\nabla}_{\mathsf{\beta}}\mathsf{\gamma}\right)=0$ | Darcy’s Law ${\nabla}_{\mathsf{\alpha}}\left[\left({\mathrm{K}}_{\mathsf{\alpha}\mathsf{\beta}}/\mathsf{\eta}\right){\nabla}_{\mathsf{\beta}}\mathrm{p}\right]=0$ |

_{0}, magnetic permeability of vacuum, μ

_{0}= 4π × 10

^{−7}(Tm/A); ε

_{0}, electric permittivity of vacuum, ε

_{0}= 8.854187817 × 10

^{−12}[(A s)/(V m)]; * η, dynamic viscosity of fluid (Pas).

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Muc, A.; Barski, M.
Design of Particulate-Reinforced Composite Materials. *Materials* **2018**, *11*, 234.
https://doi.org/10.3390/ma11020234

**AMA Style**

Muc A, Barski M.
Design of Particulate-Reinforced Composite Materials. *Materials*. 2018; 11(2):234.
https://doi.org/10.3390/ma11020234

**Chicago/Turabian Style**

Muc, Aleksander, and Marek Barski.
2018. "Design of Particulate-Reinforced Composite Materials" *Materials* 11, no. 2: 234.
https://doi.org/10.3390/ma11020234