# Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Governing Equations

#### 2.2. Scale-Dependent Homogenization

#### 2.2.1. Hill–Mandel Macrohomogeneity Condition

- Uniform displacement (Dirichlet):$$\mathit{u}\left(\mathit{x}\right)={\u03f5}^{0}\xb7\mathit{x},\phantom{\rule{10.0pt}{0ex}}\forall \mathit{x}\in \partial {B}_{\delta}\left(\omega \right).$$
- Uniform traction (Neumann):$$\mathit{t}\left(\mathit{x}\right)={\sigma}^{0}\xb7\mathit{n},\phantom{\rule{10.0pt}{0ex}}\forall \mathit{x}\in \partial {B}_{\delta}\left(\omega \right).$$
- Uniform displacement-traction (mixed-orthogonal):$$[\mathit{u}\left(\mathit{x}\right)-{\u03f5}^{0}\xb7\mathit{x}]\xb7[\mathit{t}\left(\mathit{x}\right)-{\sigma}^{0}\xb7\mathit{n}]=0,\phantom{\rule{10.0pt}{0ex}}\forall \mathit{x}\in \partial {B}_{\delta}\left(\omega \right).$$

#### 2.2.2. Apparent and Effective Properties

#### 2.2.3. Scaling

#### 2.3. Random Fields

#### 2.4. Cellular Automata

## 3. Numerical Results and Discussion

#### 3.1. Cauchy Random Field Responses

#### 3.2. Dagum Random Field Responses

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A cell (i,j) and its von Neumann neighbours for cellular automaton discretization. (

**b**) neighbor forces and the external force acting on cell (i,j).

**Figure 2.**Apparent shear modulus responses corresponding to Cauchy random fields for various combinations of $(\alpha ,\beta ,\delta )$.

**Figure 3.**The scaling function $g({R}_{\alpha ,\beta}^{C},\delta )$ as a function of $\delta $ and the corresponding stretched exponential fit.

**Figure 4.**The scaling function $g({R}_{\alpha ,\beta}^{C},\delta )$ as a function of $\delta $ and the corresponding Cauchy function fit.

**Figure 5.**Apparent shear modulus responses corresponding to Dagum random fields for various combinations of $(\alpha ,\beta ,\delta )$.

**Figure 6.**The scaling function $g({R}_{\alpha ,\beta}^{D},\delta )$ as a function of $\delta $ and the corresponding stretched exponential fit.

**Figure 7.**The scaling function $g({R}_{\alpha ,\beta}^{D},\delta )$ as a function of $\delta $ and the corresponding Dagum function fit.

**Table 1.**Different combinations of $\alpha $ and $\beta $ in Cauchy and Dagum random fields. (A, B, C), (${A}_{C}$, ${B}_{C}$, ${C}_{C}$), and (${A}_{D}$, ${B}_{D}$, ${C}_{D}$) are the coefficients in the stretched exponential, the Cauchy function, and the Dagum function, respectively. $SSE$, $SS{E}_{C}$, and $SS{E}_{D}$ are the error sums of the squares corresponding to the stretched exponential and Cauchy and Dagum functions’ fits, respectively.

$\alpha $ | $\beta $ | D | H | A | B | C | $SSE$ | ${A}_{C}$ | ${B}_{C}$ | ${C}_{C}$ | $SS{E}_{C}$ |

1 | 1 | 2.5 | 0.5 | 0.24 | 0.34 | 0.005 | 2.5 × 10${}^{-6}$ | 0.46 | 0.34 | 2.3 | 2.1 × 10${}^{-6}$ |

1 | 1.8 | 2.5 | 0.1 | 0.24 | 0.46 | 0.003 | 2.2 × 10${}^{-5}$ | 0.11 | 4.60 | 0.16 | 1 × 10${}^{-6}$ |

1 | 0.2 | 2.5 | 0.9 | 0.50 | 0.05 | −0.090 | 2.7 × 10${}^{-5}$ | 0.12 | 0.53 | 0.35 | 3.9 × 10${}^{-5}$ |

0.2 | 1 | 2.9 | 0.5 | 0.25 | 0.60 | 0.002 | 3.7 × 10${}^{-6}$ | 25.36 | 0.22 | 8.11 | 7.5 × 10${}^{-6}$ |

1.8 | 1 | 2.1 | 0.5 | 0.25 | 0.27 | 0.001 | 9.4 × 10${}^{-7}$ | 0.17 | 0.69 | 0.82 | 2.7 × 10${}^{-5}$ |

0.2 | 0.8 | 2.9 | 0.6 | 0.24 | 0.58 | 0.003 | 2.2 × 10${}^{-5}$ | 14.88 | 0.23 | 7.33 | 1.2 × 10${}^{-5}$ |

0.6 | 0.8 | 2.7 | 0.6 | 0.23 | 0.37 | 0.008 | 2.8 × 10${}^{-5}$ | 1.11 | 0.23 | 3.59 | 5.6 × 10${}^{-6}$ |

0.6 | 1.4 | 2.7 | 0.3 | 0.24 | 0.48 | 0.004 | 2.4 × 10${}^{-5}$ | 1.90 | 0.29 | 4.36 | 1.4 × 10${}^{-5}$ |

(a) Cauchy random fields | |||||||||||

$\alpha $ | $\beta $ | D | H | A | B | C | $SSE$ | ${A}_{D}$ | ${B}_{D}$ | ${C}_{D}$ | $SS{E}_{D}$ |

0.2 | 0.4 | 2.9 | 0.8 | 0.22 | 0.45 | 0.012 | 4.9 × 10${}^{-5}$ | 2.35 | 0.58 | 0.056 | 8.5 × 10${}^{-5}$ |

0.2 | 0.6 | 2.9 | 0.7 | 0.23 | 0.51 | 0.007 | 3.7 × 10${}^{-5}$ | 2.43 | 0.75 | 0.055 | 6.5 × 10${}^{-6}$ |

0.2 | 0.8 | 2.9 | 0.6 | 0.24 | 0.52 | 0.004 | 3.3 × 10${}^{-5}$ | 0.76 | 0.90 | 0.187 | 2.3 × 10${}^{-5}$ |

0.4 | 0.8 | 2.8 | 0.6 | 0.24 | 0.42 | 0.006 | 3.1 × 10${}^{-5}$ | 0.54 | 0.75 | 0.273 | 8.7 × 10${}^{-6}$ |

0.6 | 0.8 | 2.7 | 0.6 | 0.23 | 0.36 | 0.006 | 2.1 × 10${}^{-5}$ | 0.66 | 0.60 | 0.215 | 4.5 × 10${}^{-6}$ |

1 | 1 | 2.5 | 0.5 | 0.24 | 0.32 | 0.004 | 3.3 × 10${}^{-6}$ | 0.22 | 0.64 | 0.821 | 4.9 × 10${}^{-6}$ |

1 | 1.8 | 2.5 | 0.1 | 0.25 | 0.42 | 0.002 | 6.4 × 10${}^{-6}$ | 0.15 | 0.95 | 1.33 | 5.8 × 10${}^{-8}$ |

0.6 | 1.4 | 2.7 | 0.3 | 0.24 | 0.44 | 0.003 | 2.6 × 10${}^{-5}$ | 1.18 | 0.81 | 0.119 | 6.3 × 10${}^{-6}$ |

(b) Dagum random fields |

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**MDPI and ACS Style**

Jetti, Y.S.; Ostoja-Starzewski, M.
Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects. *Fractal Fract.* **2021**, *5*, 255.
https://doi.org/10.3390/fractalfract5040255

**AMA Style**

Jetti YS, Ostoja-Starzewski M.
Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects. *Fractal and Fractional*. 2021; 5(4):255.
https://doi.org/10.3390/fractalfract5040255

**Chicago/Turabian Style**

Jetti, Yaswanth Sai, and Martin Ostoja-Starzewski.
2021. "Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects" *Fractal and Fractional* 5, no. 4: 255.
https://doi.org/10.3390/fractalfract5040255