# Design of Reinforcement in Nano- and Microcomposites

^{*}

## Abstract

**:**

## 1. Introduction

- Boundary conditions of the representative volume element,
- Form of the representative volume element,
- Shape of the reinforcement,
- Distribution of the reinforcement.

## 2. Preliminary Remarks

## 3. Boundary Conditions

- Linear displacement boundary condition (Dirichlet condition),
- Constant traction boundary condition (Neumann condition),
- Periodic boundary condition.

_{1}= 0, x

_{1}= a, x

_{2}= 0, x

_{2}= b, x

_{3}= 0 and x

_{3}= c—see Figure 2. The u

_{1}, u

_{2}and u

_{3}denote the displacements along x

_{1}, x

_{2}and x

_{3}directions, respectively, and e is the assumed small displacement in the elastic regime. The effective material properties were computed using the appropriate boundary conditions (Table 1) and Equations (1)–(3).

## 4. Form of the Representative Volume Element

_{i}—the inner diameter of CNT, and t—the thickness of CNT—see Table 3. Other characteristic parameters of the conducted FEA are presented in Table 3.

## 5. Optimization Problems with the Use of 2D Curves

- Set of nodes’ approximates curve Γ (see, e.g., Lee et al. [64]) created by discretization of finite elements, then the number of design variables is large and equal to the number of nodes on the edge Γ, and additionally increases in cases of description concentration of stresses; this method is used very rarely.
- Curve Γ is described by elementary analytic functions, in which case it is unambiguously defined by giving a finite fixed number (see Pedersen [65,66]), much smaller than in the previous case; however, there is always a problem with the introduction of the function and assessment, whether it is sufficient to solve a specific optimization problem; this method has unfortunately too little generality.
- Application of spline functions to the definition of curve Γ, where the spline functions are uniquely determined by specifying the coordinates of a finite number of key points P
_{i}(i = 0,1,2, ..., I); the number of basis points is equivalent to the number of design variables.

#### 5.1. Representation of the Curve Γ through Elementary Functions

^{IV}− λf = 0:

#### 5.2. Generation of Key Point Population

_{i}is located. In the adopted polar coordinate system, it is necessary to define the angular positions of the radii ${r}_{i}$. Below are some options for defining them depending on the imposed additional restrictions. It should be emphasized that, in this case, limitations are not boundary conditions imposed at the ends of curve Γ. They are taken into account directly in the methodology of its construction.

#### 5.2.1. No Constraints—Equidistant Key Points

#### 5.2.2. Generation of Convex Curves

_{0},...,P

_{4}) is shown in Figure 7b. Next, we check if the vectors form a convex polygon, investigating whether the vector associated with the P

_{i}point is above or below the line joining the points (P

_{i−1}, P

_{i+1}). If it is below the straight line, then we generate a new length so that the end of the vector is beyond the straight line. We start checking the condition of the convexity from the inside, i.e., by examining the position of the vector P

_{2}relative to the two extreme ones, i.e., P

_{0}and P

_{4}. We examine the positions of further points analogously, i.e., dividing subsequent intervals into halves. Of course, the most comfortable is to take an odd number of key points. After completing these operations, repeat the verification of the protuberances relative to the neighbors. In the case shown in Figure 7b, the point P

_{2}is above the line joining points P

_{0}and P

_{4}, but not with respect to the neighboring P

_{1}and P

_{3}. Thus, it is necessary to generate a new key point (by generating a new length of the vector) marked in Figure 7b as P

_{2}Currently, the set of key points forms a convex polygon, and we can construct a convex curve Γ. In computer graphics, the non-uniform rational basis spline (NURBS) mathematical model is commonly used for generating and representing curves and surfaces. In NURBS, control points are entered and the edition of the curve is realized based on the element’s control points for Bézier curves or based on spline modeling.

#### 5.2.3. Generation of a convex Curve with Constraints in the Form of a Constant Area Bounded by a Curve Γ

- (a)
- one of the angles on the edge of the curve is 0° and the other 90°,
- (b)
- one of the angles on the edge of the curve is not equal to 0° or 90°.

_{I}point on the OY axis) is determined by assuming initially that the searched curve is described as the so-called Ferguson curve, which in parametric form is expressed by the formula:

- Convexity of a polygon stretched on basis vectors,
- Boundary conditions at the ends of the curve.

_{i−1}P

_{i}is used here:

_{I}and repeat the operations of selecting the positions of the vectors starting from the correction of the angular positions of the base vectors. The construction of the curve that meets the condition of the convexity and constancy of the area is a set of geometrical observations based on the idea of rejecting solutions (positions) that do not meet the imposed conditions (unsuccessful trials). This method is simple for numerical algorithms and, in our opinion, has large generalisations in the sense of being easily adaptable to a range of similar problems.

## 6. Shape of the Reinforcement

#### 6.1. 2D Homogenization Problem

#### 6.2. Isoperimetric Problem—Verification of the Accuracy of Numerical Solutions

^{2}/2. Due to the symmetry of the problem concerning the y-axis, we only look for half of the curve. Finally, we formulate the optimization problem in the following way:

#### 6.3. Shape of Fibre Bundles

## 7. Distribution of the Reinforcement

_{t}and p

_{b}denote the corresponding properties of the top and bottom faces of the plate, respectively, and V(z) is a function describing the distribution of reinforcement density in the thickness direction. For FGM, the function V(z) is defined as:

_{l}and ρ

_{t}are the mass densities of the leading edge and trailing edge, respectively; w denotes the width of the panel. Second density grading in the length direction is as follows:

_{r}and ρ

_{t}are the mass densities of the root edge and tip edge, respectively; L denotes the length of the panel. Next definition of density diagonal grading across the xy plane (in two directions) is as follows:

_{u}and ρ

_{l}are the mass densities of upper and lower bounds, respectively. A model for characterizing the mechanical properties of functionally graded materials (FGMs) with the regular polygonal cross-section is also developed by introducing the power-law rule—see Ref. [41]. However, modelling of their mechanical behaviour still leads to many problems, particularly in the description of 1D or 2D structures, such as beams, plates or shells. The broader discussion of those problems and solutions can be found in Muc et al. [76,77,78,79,80].

_{f}—the strict formulation is given by Banichuk et al. [81]. The optimisation problem deals with the topology optimisation since we are looking for the optimal material distribution. The results are presented in Figure 15 and show a good correlation between numerical and analytical studies. Upper values (red lines) in Figure 15 were calculated for the lower value of the assumed beam deflection.

## 8. Conclusions

- The method of description (representation) of the 2D curve related to the number of design variables included in the optimization process was presented.
- There was a good agreement of theoretical and numerical solutions, both in the case of using genetic algorithms (GA) as well as a modified evolutionary strategy (MEA) for the shape optimization of the single fibre.
- In shape optimization of fibre bundles, a circle was assumed as the original shape. In the numerical computations, the algorithm of modified evolutionary strategy (MEA) was used. The shape of the fibre bundle depends on the conditions limiting the form of the curve Γ understood in the sense of its convexity or not.
- The composite beam reinforced by short fibres, loaded at the centre and having a variable density of fibres along the length, was considered, being an example of FGM. The optimization problem deals with topology optimization since we were looking for the optimal material distribution. The results showed a good correlation between numerical and analytical studies.
- Some examples of numerical homogenization of various 2D and 3D RVE have been studied to present the influence of boundary conditions, form of RVE, shape and distribution of the reinforcement on the effective material properties.
- The comparison of numerical homogenization and micromechanical models showed that micromechanical models are appropriate in the case of the simple shapes of reinforcement.
- However, for complicated shapes of the reinforcement, the application of the numerical homogenization is crucial because the determination of characteristic relations such as d/l ratio is problematic.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 2.**A quarter of representative volume element (RVE) having a square array of fibres with characteristic dimensions.

**Figure 3.**Schematic diagram of unidirectional fibres distribution in a matrix and selected representative volume elements: (

**a**) square; (

**b**) hexagonal array, respectively.

**Figure 5.**Description of 2D optimization problems with the use of 2D curves: (

**a**) shape optimization—the shape of the middle surface is described by the curve $\Gamma $; (

**b**) shape optimization—the curve describes boundaries of structures $\Gamma ={\Gamma}_{1}{\displaystyle \cup}{\Gamma}_{2}{\displaystyle \cup}{\Gamma}_{3}{\displaystyle \cup}{\Gamma}_{4}$; (

**c**) dimensional optimization—structure thickness distribution is described by the curve Γ; (

**d**) material optimization—a rectangle is made of three different composite materials (the curves ${\Gamma}_{1}$ and ${\Gamma}_{2}$ characterize the division of the area); (

**e**) optimization of the shape of the mid-surface ${\Gamma}_{2}$ and dimensional optimization (the curve ${\Gamma}_{1}$ describes the distribution of the thickness).

**Figure 6.**(

**a**) parameterization of the curve Γ through eigenfunctions in regions of stress concentration; (

**b**) superellipse shapes for different values of parameter n.

**Figure 7.**(

**a**) generation of key points in the absence of constraints (five basis points); (

**b**) construction of convex curves (five basis points).

**Figure 9.**2D RVE: (

**a**) a structure with a uniform distribution of short fibre; (

**b**) a bilayered structure; (

**c**) a structure with a circular reinforcement; and (

**d**) a structure with a triangular shape of reinforcement.

**Figure 10.**Generation of nine elements of the initial population with the solution of the Equation (14).

**Figure 11.**Optimal curves—GA and MEA (black color) and obtained using the optimization procedure (red color).

**Figure 13.**Planar representative volume element—initial (circular) and optimal shapes of fibre bundles.

**Figure 14.**Bending of a simply supported beam having a variable fibre volume fraction distribution along the length.

**Figure 15.**Comparison of fibre volume fraction distribution along the beam length: analytical (a dotted line) Banichuk et al. [81] and numerical (a continuous line) results.

Loading | Constants | Boundary Displacements | |||||
---|---|---|---|---|---|---|---|

x_{1} Direction | x_{2} Direction | x_{3} Direction | |||||

x_{1} = 0 | x_{1} = a | x_{2} = 0 | x_{2} = b | x_{3} = 0 | x_{3} = c | ||

Axial normal | ${E}_{A}^{*}$ and ${\nu}_{A}^{*}$ | u_{1} = 0 | u_{1} = e | u_{2} = 0 | u_{3} = 0 | ||

Transverse normal | ${E}_{T}^{*}$ and ${\nu}_{T}^{*}$ | u_{1} = 0 | u_{2} = 0 | u_{2} = e | u_{3} = 0 | ||

Axial shear | ${G}_{A}^{*}$ | u_{2} = 0u _{3} = 0 | u_{2} = eu _{3} = 0 | u_{1} = 0u _{3} = 0 | u_{1} = 0u _{3} = 0 | u_{3} = 0 | u_{3} = 0 |

Material Constant | Carbon Nanotube | Matrix |
---|---|---|

Young’s modulus (GPa) | 1000 | 3.2 |

Poisson’s ratio | 0.3 | 0.3 |

Parameter | Value | |

Outer diameter of CNT (nm) | 10 | |

Thickness of CNT (nm) | 0.3 | |

Length of CNT (nm) | 100 | |

Volume fraction of CNTs | 2.75% | |

Parameter | Square Array | Hexagonal Array |

Dimensions of RVE: 2a × 2b × 2c (nm) | 100 × 20 × 20 | 100 × 20 × 40 |

Total number of FE for the whole RVE | 30,880 | 36,160 |

Element type | C3D8R—8-node linear hexahedrons with reduced integration |

Model | ${\mathit{E}}_{\mathit{A}}^{*}$ (GPa) | ${\mathit{E}}_{\mathit{T}}^{*}$ (GPa) | ${\mathit{\nu}}_{\mathit{A}}^{*}$ | ${\mathit{\nu}}_{\mathit{T}}^{*}$ | ${\mathit{G}}_{\mathit{A}}^{*}$ (GPa) | ${\mathit{G}}_{\mathit{T}}^{*}$ (GPa) |
---|---|---|---|---|---|---|

FEM–square array | 30.62 | 3.90 | 0.3 | 0.6 | 4.29 | 1.22 |

FEM–hexagonal array | 30.60 | 4.15 | 0.3 | 0.56 | 4.16 | 1.32 |

Vanin model | 30.61 | 3.62 | 0.3 | 0.41 | 1.3 | 1.28 |

Material Constant | Reinforcement | Matrix |
---|---|---|

Young’s modulus (GPa) | 70.4 | 3.2 |

Poisson’s ratio | 0.22 | 0.35 |

**Table 6.**Effective Young’s modulus of microcomposites (material data in Table 5).

2D RVEs | Effective Young’s Modulus (GPa) | ||
---|---|---|---|

FEM | Micromechanical Models | Rule of Mixture (Equation (A2)) | |

bilayered structure | 9.301 | 9.119—(Equation (A3)) | 24.704 |

align short fibre | 8.26 | 3.798—Cox (Equation (A5)) 8.777—H-T (Equation (A4)) | 24.704 |

triangular shape of the reinforcement | 4.588 | 6.987—Cox (Equation (A5)) 7.280—H-T (Equation (A4)) | 9.92 |

circular inclusion | 5.635 | - | 24.704 |

Notation | Theoretical Solution | The Type of Algorithm | |
---|---|---|---|

Genetic Algorithm | Modified Evolutionary Strategy | ||

The length of the arc—$\tilde{L}$ | 7.06858 | 7.06113 Error: 0.11% | 7.06344 Error: 0.073% |

Area—Area | 15.90431 | 15.7987 Error: 0.66% | 15.83521 Error: 0.43% |

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**MDPI and ACS Style**

Chwał, M.; Muc, A.
Design of Reinforcement in Nano- and Microcomposites. *Materials* **2019**, *12*, 1474.
https://doi.org/10.3390/ma12091474

**AMA Style**

Chwał M, Muc A.
Design of Reinforcement in Nano- and Microcomposites. *Materials*. 2019; 12(9):1474.
https://doi.org/10.3390/ma12091474

**Chicago/Turabian Style**

Chwał, Małgorzata, and Aleksander Muc.
2019. "Design of Reinforcement in Nano- and Microcomposites" *Materials* 12, no. 9: 1474.
https://doi.org/10.3390/ma12091474