# Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Convergence Analysis of Viscoelastic Fractal Operators

#### 2.1. Stiffness Operator Method and Compliance Operator Method

#### 2.2. Algebraic Equations of Stiffness Operators for Fractal Cells

**Remark**

**1.**

#### 2.3. The Logical Foundation of the Equivalence Postulate

#### 2.4. Fractal Ladder

**Proof.**

**Remark**

**2.**

#### 2.5. Fractal Tree

**Proof.**

## 3. Viscoelastic Response Curves of Fractal Structures

#### 3.1. Correspondence between Operator Kernel Functions and Relaxation and Creep Functions

#### 3.2. Comparison of Mechanical Behavior between Several Classical Viscoelastic Models and Fractal Tree, Fractal Ladder Structures

## 4. Discretization of Continuous Viscoelastic Bodies into Fractal Topological Structures

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A1.**Schematic diagram of general Maxwell model. (

**a**) Fractal cell. (

**b**) General Maxwell model. (

**c**) Fractal element.

## Appendix C

## References

- Knauss, W.G.; Igor, E.; Lu, H. Mechanics of Polymers: Viscoelasticity; Springer: Boston, MA, USA, 2008. [Google Scholar] [CrossRef]
- Gargallo Ligia, R.D. Physicochemical Behavior and Supramolecular Organization of Polymers; Springer: Dordrecht, The Netherlands, 2009; pp. 43–162. [Google Scholar] [CrossRef]
- Yang, L.; Yang, L.; Lowe, R.L. A viscoelasticity model for polymers: Time, temperature, and hydrostatic pressure dependent Young’s modulus and Poisson’s ratio across transition temperatures and pressures. Mech. Mater.
**2021**, 157, 103839. [Google Scholar] [CrossRef] - Leblanc, J.L. A Multiparametric Approach of the Nonlinear Viscoelasticity of Rubber Materials. In Non-Linear Viscoelasticity of Rubber Composites and Nanocomposites: Influence of Filler Geometry and Size in Different Length Scales; Ponnamma, D., Thomas, S., Eds.; Springer International Publishing: Cham, Switzerland, 2014; pp. 273–300. [Google Scholar] [CrossRef]
- Oyen, M.L. Mechanical characterisation of hydrogel materials. Int. Mater. Rev.
**2014**, 59, 44–59. [Google Scholar] [CrossRef] - Xu, Q.; Engquist, B.; Solaimanian, M.; Yan, K. A new nonlinear viscoelastic model and mathematical solution of solids for improving prediction accuracy. Sci. Rep
**2022**, 10, 2202. [Google Scholar] [CrossRef] - Zhao, W.; Li, N.; Liu, L.; Leng, J.; Liu, Y. Mechanical behaviors and applications of shape memory polymer and its composites. Appl. Phys. Rev.
**2023**, 10, 011306. [Google Scholar] [CrossRef] - Christensen, R. Theory of Viscoelasticity, 2nd ed.; Academic Press: Cambridge, MA, USA, 1982; pp. 1–364. [Google Scholar] [CrossRef]
- Maxwell, J.C. On the dynamical theory of gases. Phil. Trans. R. Soc.
**1867**, 157, 49–88. [Google Scholar] [CrossRef] - Meyer, O.E. Theorie der elastischen Nachwirkung. Ann Phys.
**1874**, 227, 108–119. [Google Scholar] [CrossRef] - Boltzmann, L. Zur Theorie der elastischen Nachwirkung. Ann Phys.
**1878**, 241, 430–432. [Google Scholar] [CrossRef] - Volterra, V. Theory of Functionals and of Integral and Integro-Differential Equations; Dover: New York, NY, USA, 1959. [Google Scholar]
- Hu, K.X.; Zhu, K.Q. Mechanical analogies of fractional elements. Chin. Phys. Lett.
**2009**, 26, 108301. [Google Scholar] [CrossRef] - Schapery, R.A. Nonlinear viscoelastic solids. Int. J. Solids Struct.
**2000**, 37, 359–366. [Google Scholar] [CrossRef] - Bauwens, J.C. Two nearly equivalent aproaches for describing the non-linear creep behavior of glassy polymers. Colloid Polym. Sci.
**1992**, 270, 537–542. [Google Scholar] [CrossRef] - Schiessel, H.; Blumen, A. Mesoscopic Pictures of the Sol-Gel Transition Ladder Models and Fractal Networks. Macromolecules
**1995**, 28, 4013–4019. [Google Scholar] [CrossRef] - Schiessel, H.; Metzler, R.; Blumen, A.; Nonnenmacher, T.F. Generalized viscoelastic models Their fractional equations with solutions. J. Phys. A Math. Gen
**1995**, 28, 6567–6584. [Google Scholar] [CrossRef] - Heymans, N.; Bauwens, J.C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta
**1994**, 33, 210–219. [Google Scholar] [CrossRef] - Heymans, N. Hierarchical models for viscoelasticity: Dynamic behaviour in the linear range. Rheol. Acta
**1996**, 35, 508–519. [Google Scholar] [CrossRef] - Deseri, L.; Paola, M.D.; Zingales, M.; Pollaci, P. Power-law hereditariness of hierarchical fractal bones. Int. J. Numer. Meth. Biomed. Eng.
**2013**, 29, 1338–1360. [Google Scholar] [CrossRef] - Yin, Y.J.; Peng, G.; Yu, X.B. Algebraic equations and non-integer orders of fractal operators abstracted from biomechanics. Acta Mech. Sin.
**2022**, 38, 521488. [Google Scholar] [CrossRef] - Guo, J.Q.; Yin, Y.J.; Ren, G.X. Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons. Appl. Math. Mech.-Engl. Ed.
**2019**, 40, 1429–1448. [Google Scholar] [CrossRef] - Guo, J.Q.; Yin, Y.J.; Hu, X.L.; Ren, G.X. Self-similar network model for fractional-order neuronal spiking: Implications of dendritic spine functions. Nonlinear Dyn.
**2020**, 100, 921–935. [Google Scholar] [CrossRef] - Yin, Y.J.; Guo, J.Q.; Peng, G.; Yu, X.B.; Kong, Y.Y. Fractal Operators and Fractional Dynamics with 1/2 Order in Biological Systems. Fractal Fract.
**2022**, 6, 378. [Google Scholar] [CrossRef] - Peng, G.; Guo, J.Q.; Yin, Y.J. Self-similar functional circuit models of arteries and deterministic fractal operators: Theoretical revelation for biomimetic materials. Int. J. Mol. Sci.
**2021**, 22, 12897. [Google Scholar] [CrossRef] - Jian, Z.M.; Peng, G.; Li, D.A.; Yu, X.; Yin, Y. Correlation between Convolution Kernel Function and Error Function of Bone Fractal Operators. Fractal Fract.
**2023**, 7, 707. [Google Scholar] [CrossRef] - Yu, X.B.; Yin, Y.J. Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators. Fractal Fract.
**2023**, 7, 755. [Google Scholar] [CrossRef] - Anderson, W.; Duffin, R. Series and parallel addition of matrices. J. Math. Anal. Appl.
**1969**, 26, 576–594. [Google Scholar] [CrossRef] - Uchiyama, M. Operator means and matrix quadratic equations. Linear Algebra Appl.
**2021**, 609, 163–175. [Google Scholar] [CrossRef] - Mikusiński, J. Operational Calculus; Pergamon Press: Oxford, UK, 1959. [Google Scholar]
- Paola, M.D.; Zingales, M. Exact mechanical models of fractional hereditary materials. J. Rheol.
**2012**, 56, 983–1004. [Google Scholar] [CrossRef]

**Figure 1.**Some typical viscoelastic models. (

**a**) The Maxwell model. (

**b**) The Kelvin–Voigt model. (

**c**) The General Kelvin–Voigt (GKV) model.

**Figure 2.**Schematic of the force–electricity analogy. (

**a**) The stiffness interaction among mechanically series-connected elements is similar to the behavior of electrical resistors in a parallel configuration; (

**b**) conversely, the stiffness of mechanically parallel-connected elements corresponds to electrical resistors arranged in series.

**Figure 4.**Schematic diagram of fractal structure cells. (

**a**) Fractal ladder structure. (

**b**) Fractal tree structure.

**Figure 5.**Schematic diagram of fractal ladder structures from level 1 and level 2 to infinite level. (

**a**–

**c**) The first, the second and the infinite level of fractal ladder. (

**d**) Fractal ladder cell.

**Figure 6.**Schematic diagram of fractal tree structures from level 1 and level 2 to infinite level. (

**a**–

**c**) The first, the second and the infinite level of fractal tree. (

**d**) Fractal tree cell.

**Figure 7.**Creep response functions to step stress for structures at various levels. $\tau =\eta /E$ denotes the characteristic time. The red dashed line represents the response at the characteristic time $\tau =1$.

**Figure 8.**Quasi-static responses of classical viscoelastic models and fractional viscoelastic models. (

**a**–

**e**) Schematic diagrams and operator expressions of classical linear viscoelastic models and fractal viscoelastic models. (

**f**–

**j**) Creep function curves of the structures, with black line segments indicating applied step stress, loaded at time $t/\tau =1$ and unloaded at time $t/\tau =5$. The horizontal axis represents dimensionless time, while the vertical axis represents the dimensionless creep function. (

**k**–

**o**) Relaxation function curves, with black line segments indicating applied step strain, loaded at time $t/\tau =1$. The vertical axis is dimensionless stress. (

**p**–

**t**) Creep curves under multiple loading cycles, with black line segments indicating applied step stress, loaded at time $t/\tau =1$.

**Figure 9.**Schematic diagram of the discretization process from a continuous model to a physical fractal model. (

**a**) Elastic body subjected to distributed viscous constraints. (

**b**) Discretization of continuous structure. (

**c**) Schematic diagram of force transmission paths. (

**d**) Renormalization to fractal ladder structure. (

**e**) Equivalent relationship between fractal ladder cells and elements.

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**MDPI and ACS Style**

Yu, X.; Yin, Y.
Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory. *Fractal Fract.* **2024**, *8*, 200.
https://doi.org/10.3390/fractalfract8040200

**AMA Style**

Yu X, Yin Y.
Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory. *Fractal and Fractional*. 2024; 8(4):200.
https://doi.org/10.3390/fractalfract8040200

**Chicago/Turabian Style**

Yu, Xiaobin, and Yajun Yin.
2024. "Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory" *Fractal and Fractional* 8, no. 4: 200.
https://doi.org/10.3390/fractalfract8040200