# Generalized Fractional Derivative Anisotropic Viscoelastic Characterization

## Abstract

**:**

## 1. Introduction

## 2. Analysis

#### 2.1. Material Property Dependence on Temperature

Temperature | Modulus | Type of Constitutive Relations, Equation (2) | Type of Material |
---|---|---|---|

${T}_{0}$ | $E(t-t,\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{}^{\prime}\phantom{\rule{0.166667em}{0ex}}{T}_{0})$ | convolution | homogeneous |

${T}_{0}$ | $E(x,t-t,\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{}^{\prime}\phantom{\rule{0.166667em}{0ex}}{T}_{0})$ | convolution | non-homogeneous |

$T\left(x\right)$ | $E[x,t-t,\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{}^{\prime}\phantom{\rule{0.166667em}{0ex}}T\left(x\right)]$ | convolution | non-homogeneous |

$T(x,t)$ | $E[x,t,t,\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{}^{\prime}\phantom{\rule{0.166667em}{0ex}}T(x,{t}^{\prime})]$ | non-convolution | non-homogeneous |

#### 2.2. General Concepts—Isotropic Materials

**Figure 3.**Comparison of exact and approximate relaxation moduli Laplace transforms of $1/({p}^{\beta}+1/\tau )$ and $1/{(p+c)}^{\mu}$.

#### 2.3. Anisotropic Relations

- (1)
- different parametric values ${E}_{ijkln}$ and ${\eta}_{ijkln}$ with equal numbers N in all directions
- (2)
- same parameters ${E}_{ijkln}$ and ${\eta}_{ijkln}$ in all directions but with distinct ${N}_{ijkl}$, thus generating different numbers of GKM parameters in each direction
- (3)
- combinations of (1) and (2) above

## 3. Discussion

- (a)
- Performing “simple” experiments for which analytic solution can be formulated and evaluated
- (b)
- Curve fitting of actual creep and/or relaxation data by least square or other methods in order to determine modulus, creep function or compliance parameters
- (c)
- Inversion of FT or LT expressions for moduli, stresses and deformations

- exact procedures, such as table look up or analytical evaluation of the ω FT inversion integrals
- numerical procedures, such as FFT [41]

## 4. Conclusions

## Acknowledgement

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Hilton, H.H.
Generalized Fractional Derivative Anisotropic Viscoelastic Characterization. *Materials* **2012**, *5*, 169-191.
https://doi.org/10.3390/ma5010169

**AMA Style**

Hilton HH.
Generalized Fractional Derivative Anisotropic Viscoelastic Characterization. *Materials*. 2012; 5(1):169-191.
https://doi.org/10.3390/ma5010169

**Chicago/Turabian Style**

Hilton, Harry H.
2012. "Generalized Fractional Derivative Anisotropic Viscoelastic Characterization" *Materials* 5, no. 1: 169-191.
https://doi.org/10.3390/ma5010169