# Optimization of Functionally Graded Structural Members by Means of New Effective Properties Estimation Method

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation

#### 2.1. Basic Ideas

#### 2.2. Internal Energy

- be symmetric, positive definite and monotonic function of the volume fraction variable $\xi $;
- be reduced to the fourth-rank unit tensor in the absence of inclusions, $\xi =0$;
- transform the properties of the matrix material into the properties of the inclusion material when the volume fraction variable $\xi $ reaches unity.

#### 2.3. Effective Elastic Properties of Isotropic Composite Material

## 3. Comparison with Classical Averaging Schemes

#### 3.1. Concentration Factors

#### 3.2. Parametric Studies

#### 3.3. Validation

## 4. Optimal Distribution of Reinforcement in Circular Bar Subjected to Torsion

#### 4.1. Problem Formulation

_{2}+ Y

_{2}O

_{3}) will be examined. Table 6 contains the material data of aluminium and ceramic phase. The yield stress of the aluminium ${\tau}_{0}^{M}=95\mathrm{MPa}$ is assumed.

#### 4.2. Linear Distribution of Inclusions

#### 4.3. Nonlinear Distribution of Inclusions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Effective bulk and shear moduli versus volume fraction of inclusions, obtained with the use of the proposed method (TEE) and other approaches: V—Voigt, R—Reuss, MT—Mori-Tanaka, HS(U)—Hashin–Shtrikman Upper, HS(L)—Hashin–Shtrikman Lower.

**Figure 3.**Comparison of effective Young’s modulus obtained with the use of the proposed method (TEE) and other approaches: V—Voigt, R—Reuss, MT—Mori-Tanaka, HS(U)—Hashin–Shtrikman Upper, HS(L)—Hashin–Shtrikman Lower for carbon short-fibre reinforced polyacetal.

**Figure 6.**(

**a**) Changes of the inclusion volume fraction along the radius of a shaft; (

**b**) the external torque versus the unit angle of twist.

**Figure 8.**The distribution of shearing stress $\overline{\tau}=\tau /{\tau}_{0}^{M}$ along the radius of a shaft.

**Figure 9.**(

**a**) Changes of volume fraction $\xi $ along radius $\rho $ of the bar; (

**b**) torque versus unit angle of twist.

**Figure 10.**(

**a**) Unit angle of twist and (

**b**) dimensionless external torque versus square function coefficient $A$.

**Figure 11.**The distribution of shearing stress $\overline{\tau}=\tau /{\tau}_{0}^{M}$ along the radius of a shaft.

RVE: Representative volume element | ${N}_{ijkl}$: Inclusion-effect tensor |

V: Voigt’s estimate | ${\delta}_{ij}$: Kronecker’s symbol |

R: Reuss’ estimate | $K$: Bulk modulus |

HS (U): Hashin-Shtrikman upper | $\lambda ,\mu $: Lamé parameters |

HS (L): Hashin-Shtrikman lower | $\alpha ,\beta $: Concentration factors |

MT: Mori-Tanaka estimate | $R$: Radius of cross-section |

TEE: Estimate based on total energy equivalence | $\rho $: Distance from the cross-section centroid |

$\xi $: Volume fraction of inclusions in RVE | $L$: Length of bar |

$\psi $: Helmholtz’ free energy | $\theta $: Angle of twist |

${\sigma}_{ij}$: Cauchy stress tensor | $T$: Torque |

${\epsilon}_{ij}$: Strain tensor | $\tau $: Shearing stress |

${E}_{ijkl}$: Elasticity tensor |

Inclusion Effect Tensor | Effective Lamé Constants |
---|---|

$\begin{array}{ll}{N}_{ijkl}& =f\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right)\\ & +g{\delta}_{ij}{\delta}_{kl}\end{array}$ $f=\frac{1}{2}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +\frac{1}{2}$ $g=-\frac{1}{3}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-\sqrt{\frac{{K}^{I}}{{K}^{M}}}\right)\xi $ | $\begin{array}{l}\lambda ={\lambda}^{M}\left(2\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-\sqrt{\frac{{K}^{I}}{{K}^{M}}}\right)\left(\left(1-\frac{1}{2}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}+\sqrt{\frac{{K}^{I}}{{K}^{M}}}\right)\right){\xi}^{2}-\xi \right)+{\left(\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +1\right)}^{2}\right)\\ +\frac{4}{3}{\mu}^{M}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-\sqrt{\frac{{K}^{I}}{{K}^{M}}}\right)\left(\left(1-\frac{1}{2}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}+\sqrt{\frac{{K}^{I}}{{K}^{M}}}\right)\right){\xi}^{2}-\xi \right)\end{array}$ $\mu ={\left[\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +1\right]}^{2}{\mu}^{M}$ |

${N}_{ijkl}^{}=f\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right)$ $f=\frac{1}{2}\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +\frac{1}{2}$ $g=0$ | $\lambda ={\left[\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +1\right]}^{2}{\lambda}^{M}$ $\mu ={\left[\left(\sqrt{\frac{{\mu}^{I}}{{\mu}^{M}}}-1\right)\xi +1\right]}^{2}{\mu}^{M}$ |

**Table 3.**Concentration factors for chosen different averaging schemes: (in MT spherical shape of inclusions was assumed).

${\alpha}^{I}$ | ${\alpha}^{M}$ | ${\beta}^{I}$ | ${\beta}^{M}$ | |

V | 1 | 1 | 1 | 1 |

R | $\frac{{K}^{M}}{\xi {K}^{M}+(1-\xi ){K}^{I}}$ | $\frac{1-\xi {\alpha}^{I}}{1-\xi}$ | $\frac{{\mu}^{M}}{\xi {\mu}^{M}+(1-\xi ){\mu}^{I}}$ | $\frac{1-\xi {\beta}^{I}}{1-\xi}$ |

HS (U) | $\frac{{K}^{M}+\frac{4}{3}{\mu}^{\mathrm{max}}}{\xi {K}^{M}+(1-\xi ){K}^{I}+\frac{4}{3}{\mu}^{\mathrm{max}}}$ | $\frac{1-\xi {\alpha}^{I}}{1-\xi}$ | $\frac{{K}^{\mathrm{max}}(2{\mu}^{M}+3{\mu}^{\mathrm{max}})+\frac{4}{3}{\mu}^{\mathrm{max}}(3{\mu}^{M}+2{\mu}^{\mathrm{max}})}{{K}^{\mathrm{max}}(2{\mu}^{*}+3{\mu}^{\mathrm{max}})+\frac{4}{3}{\mu}^{\mathrm{max}}(3{\mu}^{*}+2{\mu}^{\mathrm{max}})}$ | $\frac{1-\xi {\beta}^{I}}{1-\xi}$ |

HS (L) | $\frac{{K}^{M}+\frac{4}{3}{\mu}^{\mathrm{min}}}{\xi {K}^{M}+(1-\xi ){K}^{I}+\frac{4}{3}{\mu}^{\mathrm{min}}}$ | $\frac{1-\xi {\alpha}^{I}}{1-\xi}$ | $\frac{{K}^{\mathrm{min}}(2{\mu}^{M}+3{\mu}^{\mathrm{min}})+\frac{4}{3}{\mu}^{\mathrm{min}}(3{\mu}^{M}+2{\mu}^{\mathrm{min}})}{{K}^{\mathrm{min}}(2{\mu}^{*}+3{\mu}^{\mathrm{min}})+\frac{4}{3}{\mu}^{\mathrm{min}}(3{\mu}^{*}+2{\mu}^{\mathrm{min}})}$ | $\frac{1-\xi {\beta}^{I}}{1-\xi}$ |

MT | $\frac{{K}^{M}+\frac{4}{3}{\mu}^{M}}{\xi {K}^{M}+(1-\xi ){K}^{I}+\frac{4}{3}{\mu}^{M}}$ | $\frac{1-\xi {\alpha}^{I}}{1-\xi}$ | $\frac{5}{2}\frac{{\mu}^{M}}{\xi ({\mu}^{M}-{\mu}^{I})+\frac{3}{2}{\mu}^{M}+{\mu}^{I}}$ | $\frac{1-\xi {\beta}^{I}}{1-\xi}$ |

TEE | $\frac{\xi \sqrt{{K}^{I}}+(1-\xi )\sqrt{{K}^{M}}}{\sqrt{{K}^{I}}}$ | $\frac{\xi \sqrt{{K}^{I}}+(1-\xi )\sqrt{{K}^{M}}}{\sqrt{{K}^{M}}}$ | $\frac{\xi \sqrt{{\mu}^{I}}+(1-\xi )\sqrt{{\mu}^{M}}}{\sqrt{{\mu}^{I}}}$ | $\frac{\xi \sqrt{{\mu}^{I}}+(1-\xi )\sqrt{{\mu}^{M}}}{\sqrt{{\mu}^{M}}}$ |

Material | Young Modulus E (GPa) | Poisson Ratio ν (-) | Lamé Constant λ (GPa) | Lamé Constant μ (GPa) |
---|---|---|---|---|

POM T-300 | 3.42 | 0.350 | 2.96 | 1.27 |

CF Fortafil F-3 | 227.00 ^{1} | 0.320 ^{2} | 152.86 | 85.98 |

Material | Young Modulus E (GPa) | Poisson Ratio ν (-) | Lamé Constant λ (GPa) | Lamé Constant μ (GPa) |
---|---|---|---|---|

PE | 1.30 ^{1} | 0.400 | 1.86 | 0.46 |

HAp | 15.95 ^{2} | 0.140 ^{2} | 2.72 | 7.00 |

Material | Lamé Constant λ (GPa) | Lamé Constant μ (GPa) |
---|---|---|

Al | 42.12 | 28.08 |

ZrO_{2} + Y_{2}O_{3} | 138.05 | 77.65 |

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**MDPI and ACS Style**

Wiśniewska, A.; Egner, H.
Optimization of Functionally Graded Structural Members by Means of New Effective Properties Estimation Method. *Materials* **2019**, *12*, 3139.
https://doi.org/10.3390/ma12193139

**AMA Style**

Wiśniewska A, Egner H.
Optimization of Functionally Graded Structural Members by Means of New Effective Properties Estimation Method. *Materials*. 2019; 12(19):3139.
https://doi.org/10.3390/ma12193139

**Chicago/Turabian Style**

Wiśniewska, Anna, and Halina Egner.
2019. "Optimization of Functionally Graded Structural Members by Means of New Effective Properties Estimation Method" *Materials* 12, no. 19: 3139.
https://doi.org/10.3390/ma12193139