# A Novel Simple Approach to Material Parameters from Commonly Accessible Rheometer Data

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{2}and mean normalized error while fitting the obtained complex modulus data. Xu et al. [8] took this approach even further and used mathematical functions, derived from rheological models, in order to generate a master curve for the measured dynamic moduli. Mahiuddin et al. [9] displayed a completely different use for rheological modelling. He dedicated his research to the characterization of mechanical properties of fruits and vegetables in order to optimize transportation, storage and handling of harvested goods. Therefore, he compared the applicability of three elements generalized Maxwell model with a fractional power-law model, solved using Caputos’ fractional derivate. By using the fractional power-law model he could reduce the amount of fit parameters from seven to four, while still guaranteeing an adequate description of the measured stress–relaxation data. In the eyes of the authors, a reevaluation of the showcased data using the Burgers model, which uses four fitting parameters as well, should yield an adequate description as well. Shi et al. [10] and Meng et al. [11] used the three and four parameter Burgers model to predict stress relaxation properties of starch films and rice flour. An accurate description of the stress–strain relation and creep properties was achieved. Kundera et al. used a generalized Maxwell approach to describe the visco-elastic stress recovery behavior of O-rings, produced by an additive manufacturing approach [12]. A generalized Maxwell model, using five fitting parameters, was used to describe the stress–relaxation properties of the sealings. However, only the mechanical properties were correlated with the rheological model parameters. A direct correlation of sealing properties and model parameters did not take place. The here shown examples are representative for a wide field of rheological modelling. The majority of modelling approaches is based on the stress–relaxation test procedure [13]. While this is of major interest for the characterization of product properties, a lot of information affecting the production process is neglected. For nozzle and extrusion based processes, which play a major role in polymer processing and its correlated industrial fields, an analysis of frequency depended material properties is mandatory. However, research covering the classic rheological modelling of frequency dependent measurements is rarely conducted. Bhattacharyya et al. [14] derived a predictive function from an augmented Jeffreys model, where a friction element is added, for the dampening properties of shape memory polymers. He also derived such functions for various mechanical loading types, covering constant stress and stress rate as well as constant and periodic strain and constant strain rate. Sun et al. [15] used the Burgers model in a similar way to the presented approach. After a rheological characterization via frequency sweep, two fit functions for G’ and G″ were derived and fitted to the resulting data. However, they concluded that the Burgers model is not suitable for describing frequency dependent measurement data of dough. Our own experience shows that fit functions, as used in the publication, can lead to falsified conclusions due to the use of too many fit parameters. The data presented was subsequently fitted by a two parameter power-law function. However, shear rate or frequency dependent properties were not described by rheological model systems.

- (1)
- A novel, yet simple, concept is suggested to directly reveal the constitutive equation of a material from rheological measurements. In particular, the constitutive equation in the form of a differential equation is directly accessed, relating stress σ and deformation ε.
- (2)
- Various rheological model systems are benchmarked to reveal the simplest and statistically most significant ones. The presented approach is applicable to Dynamic Mechanical Analysis (DMA; typically used symbols are σ for stress and ε for deformation) and shear rheology (τ for stress and γ for deformation).
- (3)
- We present a material independent approach that is suitable for further in-depth interpretation of frequency dependent material properties, known to be of major importance for a variety of industrial applications. The presented concept additionally allows the analysis of specific model parameters and their respective influences on the targeted applications. Hypothetically, one can correlate key processing parameters such as shape fidelity for additive manufacturing or sealing properties for rubbers to the resulting model parameters. It should be noted that huge amounts of data are required in order to reveal correlations between model and process parameters. This paper, however, will focus on the approach of generating necessary data for possible future correlations.

## 2. Materials and Methods

#### 2.1. Theory/Strategy

_{i}is sufficient, and can be chosen arbitrarily. However, this must be done for each applied angular frequency ${\mathsf{\omega}}_{\mathrm{j}}$. Thus, t

_{ij}corresponds to:

#### 2.2. Sample Preparation for Rheological Characterization

^{®}, J. Rettenmaier & Söhne GmbH & Co KG, Rosenberg, Germany) were used. The concentration is varied from 2% (w/v) to 8% (w/v). Sample preparation was performed according to [40]. The storage time at 37 °C was reduced to 16 h, since complete solubility had already been achieved. This holds true for sample concentrations up to 5%. Alginate solutions with higher polymer content were stored for 24 h before measurements took place in order to ensure homogenous solution properties. Constant stirring was performed with a magnetic stirrer at 150 rpm. However, stirring was not possible for concentrations higher than 4% due to the high viscosity of the solution. A calcium and magnesia free Dulbecco’s phosphate buffered saline solution DPBS (used as purchased; Sigma Aldrich, St. Louis, MO, USA) was used as solvent in order to prevent uncontrolled ionic crosslinking of the alginate solutions. For the fiber composite samples, PCL fiber fragments are added as dry filler in total weight percent simultaneously with the alginate powder. The stirring time of 16 h was also sufficient for a homogenous filler distribution.

#### 2.3. Shear Rheology

^{−1}to 100 s

^{−1}(revolutions per second). An optical evaluation of the gap after every measurement showed no negative influences, like drying or ejection of material during both measurements. The relation of Cox–Merz could be confirmed for the used alginate solutions [42].

#### 2.4. Printing of Alginate

## 3. Results and Discussion

#### 3.1. Rheological Model Evaluation of Alginate

^{2}, and the adjusted R

^{2}can be used.

#### 3.1.1. Single Parameter Model: Spring and Dashpot

#### 3.1.2. Two Component Model Systems: Kelvin–Voigt and Maxwell

#### 3.1.3. Four Parameter Model: The Burgers Model

#### 3.1.4. Resulting Model Parameters for the Burgers Model

_{2}parameter, which is given by the spring in the Kelvin–Voigt part of the Burgers model, becomes nearly 0 if no polymer is present. This is reasonable due to the assumption that the Kelvin–Voigt model is representative for regions with high polymer concentrations and therefore completely reversible deformation processes. A value slightly higher than 0 could be correlated to ionic interactions, which are induced by the salts that are dissolved in the DPBS puffer system. Additionally, the difference between spring constant values decreases with increasing alginate concentration, leading to a crossover in the region of 6.5 % alginate concentration.

#### 3.1.5. Correlation of Rheological Model Parameters and Key Processing Parameters

_{0}= Delay time to allow a finite strut diameter at t = 0; τ

_{s}= Characteristic time for spreading.

_{0}to allow a finite strut diameter at t = 0. Furthermore, a characteristic time τ

_{s}is used to quantify the spreading behavior. This characteristic time resembles the time point where the strut reaches 63.2% of the equilibrium strut diameter. Here a higher characteristic time resembles slower spreading and would therefore result in a better shape fidelity for the majority of commonly used shape fidelity assessments. The asymptotic strut diameter for infinite times can potentially be correlated to printing results as well. However, an artifact free determination of this value can be highly challenging due to several environmental effects, such as drying of hydrogel or inhomogeneity of the used printbed such as scratches, dirt particles, or a slight tilt angle. The resulting fitting parameters are shown in Table 6.

_{1}is proportional to the characteristic spreading time τ

_{s}of printed struts:

#### 3.2. Rheological Model Evaluation of Further Materials

#### 3.2.1. Analysis of Elastosil 7670

^{2}parameter. This parameter describes the quality of a fit function adjusted to the number of terms used. This is especially helpful for the identification of the most simple, yet suitable rheological model system. The values, calculated by Statistica, are given in the following Table 8.

^{2}greater than 0.99. For further analysis, a comparison of the resulting observed against predicted plots is carried out (see Figure 8). Due to the nearly fully elastic material properties of Elastosil 7670 one could already assume the Kelvin–Voigt or Zener model as suitable from a theoretical point of view. Both model systems are describing the measured data accurately.

#### 3.2.2. Analysis of TPU 1180A

#### 3.2.3. Analysis of PCL Filled Alginate

_{1}(the dashpot in serial alignment) can be observed. All other parameters are increasing by a factor 4–7 as well. Those observations can potentially be correlated to key process parameters, such as shape fidelity in biofabrication process or cell viability for biological experiments. Similar experimental results are currently investigated in the framework of the SFB TRR225 (http://trr225biofab.de/) and will surely yield new insights.

#### 3.3. Theoretical Comparison to Already Existing Methods for Model Parameter Determination

_{1}is accessible from the ${\mathsf{\epsilon}}_{0}$ deformation and initial stress measured immediately after the deformation step. Therefore, a typical relaxation experiment reveals only three out of four parameters of a Burgers model, where one parameter must be chosen arbitrarily. Similar problems also occur in creep recovery experiments [17]. Nevertheless, it must be pointed out that a huge variety of data from stress relaxation, creep and creep recovery experiments or even linear increasing loading and unloading, with varying loading/unloading speeds, can enable data sets where one can generate tables like Table 1 and can perform the same statistical analysis as suggested in this work. However, only by considering rheological measurements in oscillation mode and frequency sweeps a sufficient large and reliable variation of $\mathsf{\epsilon},\dot{\mathsf{\epsilon}}$, $\ddot{\mathsf{\epsilon}},$ $\dot{\mathsf{\sigma}},\text{}\ddot{\mathsf{\sigma}}$ $\mathrm{is}\text{}\mathrm{accessible}\text{}\mathrm{to}\text{}\mathrm{model}\text{}\mathsf{\sigma}$ to reveal the material model.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Calculated vs. measured shear stresses plot for a simple spring for describing the viscoelastic properties of 2% (

**a**), 5% (

**b**), and 8% (

**c**) alginate solutions.

**Figure 2.**Calculated vs. measured shear stresses plot for a single dashpot for describing the viscoelastic properties of 2% (

**a**), 5% (

**b**), and 8% (

**c**) alginate solutions.

**Figure 3.**Evaluation of the Kelvin–Voigt model as suitable model to describe the viscoelastic properties of 2% (

**a**), 5% (

**b**), and 8% (

**c**) alginate solutions.

**Figure 4.**Calculated against measured shear stress plot for the evaluated Maxwell model for 2% (

**a**), 5% (

**b**), and 8% (

**c**) alginate solutions.

**Figure 5.**Calculated against measured shear stress plot for the used burgers model system for 2% (

**a**), 5% (

**b**), and 8% (

**c**) alginate solutions.

**Figure 6.**Resulting E-moduli (

**a**) and Dashpot viscosities (

**b**) for the Burgers model. Solid lines represent the power law like fit according to [51], dashed lines are fitted with a fixed scaling exponent of n = m = 3.

**Figure 8.**Measured shear stress against calculated shear stress for Kelvin–Voigt (

**a**) and Zener m/k (

**b**).

$\mathsf{\epsilon}$ | $\dot{\mathsf{\epsilon}}$ | $\ddot{\mathsf{\epsilon}}$ | $\mathsf{\sigma}$ | $\dot{\mathsf{\sigma}}$ | $\ddot{\mathsf{\sigma}}$ | $\mathsf{\omega}$ |
---|---|---|---|---|---|---|

$\mathsf{\epsilon}\left({\mathrm{t}}_{11}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{11}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{11}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{11}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{11}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{11}\right)$ | ${\mathsf{\omega}}_{1}$ |

$\mathsf{\epsilon}\left({\mathrm{t}}_{21}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{21}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{21}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{21}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{21}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{21}\right)$ | ${\mathsf{\omega}}_{1}$ |

…. | ….. | ….. | ….. | ….. | ….. | …. |

$\mathsf{\epsilon}\left({\mathrm{t}}_{91}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{91}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{91}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{91}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{91}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{91}\right)$ | ${\mathsf{\omega}}_{1}$ |

$\mathsf{\epsilon}\left({\mathrm{t}}_{12}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{12}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{12}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{12}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{12}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{12}\right)$ | ${\mathsf{\omega}}_{2}$ |

$\mathsf{\epsilon}\left({\mathrm{t}}_{22}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{22}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{22}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{22}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{22}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{22}\right)$ | ${\mathsf{\omega}}_{2}$ |

…. | ….. | ….. | ….. | ….. | ….. | ….. |

$\mathsf{\epsilon}\left({\mathrm{t}}_{92}\right)$ | $\dot{\mathsf{\epsilon}}\left({\mathrm{t}}_{92}\right)$ | $\ddot{\mathsf{\epsilon}}\left({\mathrm{t}}_{92}\right)$ | $\mathsf{\sigma}\left({\mathrm{t}}_{92}\right)$ | $\dot{\mathsf{\sigma}}\left({\mathrm{t}}_{92}\right)$ | $\ddot{\mathsf{\sigma}}\left({\mathrm{t}}_{92}\right)$ | ${\mathsf{\omega}}_{2}$ |

………. | ………. | ………. | ………. | ………. | ………. | ………. |

………. | ………. | ………. | ………. | ………. | ………. | ………. |

………. | ………. | ………. | ………. | ………. | ………. | ………. |

………. | ………. | ………. | ………. | ………. | ………. | ${\mathsf{\omega}}_{\mathrm{max}}$ |

**Table 2.**Overview of rheological models with constitutive equations of first order differential equation type, having 2 or 3 main effects Ki.

Model Name | Scheme | $\mathbf{\sigma}-\mathbf{Form}\text{}\mathbf{of}\text{}\mathbf{the}\text{}\mathbf{Differential}\text{}\mathbf{Equation}$ | Linear Equation |
---|---|---|---|

Maxwell | $\mathsf{\sigma}=\mathsf{\eta}\dot{\mathsf{\epsilon}}-\frac{\mathsf{\eta}}{\mathrm{E}}\dot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\text{}\mathrm{K}}_{1}\dot{\mathsf{\epsilon}}+{\text{}\mathrm{K}}_{2}\dot{\mathsf{\sigma}}$ | |

Kelvin-Voigt | $\mathsf{\sigma}=\mathrm{E}\mathsf{\epsilon}+\mathsf{\eta}\dot{\mathsf{\epsilon}}$ | $\mathsf{\sigma}={\mathrm{K}}_{1}\mathsf{\epsilon}+{\text{}\mathrm{K}}_{2}\dot{\mathsf{\epsilon}}$ | |

Zener K | $\mathsf{\sigma}=\frac{{\mathrm{E}}_{1}{\mathrm{E}}_{2}}{{\mathrm{E}}_{1}+{\mathrm{E}}_{2}}\mathsf{\epsilon}+\frac{{\mathrm{E}}_{2}\mathsf{\eta}}{{\mathrm{E}}_{1}+{\mathrm{E}}_{2}}\dot{\mathsf{\epsilon}}-\frac{\mathsf{\eta}}{{\mathrm{E}}_{1}+{\mathrm{E}}_{2}}\dot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\text{}\mathrm{K}}_{1}\mathsf{\epsilon}+{\text{}\mathrm{K}}_{2}\dot{\mathsf{\epsilon}}+{\mathrm{K}}_{3}\dot{\mathsf{\sigma}}$ | |

Zener M | $\mathsf{\sigma}={\mathrm{E}}_{2}\mathsf{\epsilon}+\frac{\mathsf{\eta}\left({\mathrm{E}}_{1}+{\mathrm{E}}_{2}\right)}{{\mathrm{E}}_{1}}\dot{\mathsf{\epsilon}}-\frac{\mathsf{\eta}}{{\mathrm{E}}_{1}}\dot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\text{}\mathrm{K}}_{1}\mathsf{\epsilon}+{\text{}\mathrm{K}}_{2}\dot{\mathsf{\epsilon}}+{\mathrm{K}}_{3}\dot{\mathsf{\sigma}}$ |

**Table 3.**Overview of rheological models with constitutive equations of second order differential equation type, having 3 or 4 main effects Ki.

Model Name | Scheme | $\mathbf{\sigma}-\mathbf{Form}\text{}\mathbf{of}\text{}\mathbf{the}\text{}\mathbf{Differential}\text{}\mathbf{Equation}$ | Linear Equation |
---|---|---|---|

Lethersich | $\mathsf{\sigma}={\mathsf{\eta}}_{2}\dot{\mathsf{\epsilon}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{\mathrm{E}}\ddot{\mathsf{\epsilon}}-\frac{{\mathsf{\eta}}_{1}+{\mathsf{\eta}}_{2}}{\mathrm{E}}\dot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\mathrm{K}}_{1}\dot{\mathsf{\epsilon}}+{\mathrm{K}}_{2}\ddot{\mathsf{\epsilon}}+{\mathrm{K}}_{3}\dot{\mathsf{\sigma}}$ | |

Jeffreys | $\mathsf{\sigma}=\left({\mathsf{\eta}}_{1}+{\mathsf{\eta}}_{2}\right)\dot{\mathsf{\epsilon}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{\mathrm{E}}\ddot{\mathsf{\epsilon}}-\frac{{\mathsf{\eta}}_{1}}{\mathrm{E}}\dot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\mathrm{K}}_{1}\dot{\mathsf{\epsilon}}+{\mathrm{K}}_{2}\ddot{\mathsf{\epsilon}}+{\mathrm{K}}_{3}\dot{\mathsf{\sigma}}$ | |

Burgers | $\mathsf{\sigma}={\mathsf{\eta}}_{1}\dot{\mathsf{\epsilon}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{{\mathrm{E}}_{2}}\ddot{\mathsf{\epsilon}}-\left(\frac{{\mathrm{E}}_{1}{\mathsf{\eta}}_{1}+{\mathrm{E}}_{1}{\mathsf{\eta}}_{2}+{\mathrm{E}}_{2}{\mathsf{\eta}}_{1}}{{\mathrm{E}}_{1}{\mathrm{E}}_{2}}\right)\dot{\mathsf{\sigma}}-\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{{\mathrm{E}}_{1}{\mathrm{E}}_{2}}\ddot{\mathsf{\sigma}}$ | $\mathsf{\sigma}={\mathrm{K}}_{1}\dot{\mathsf{\epsilon}}+{\mathrm{K}}_{2}\ddot{\mathsf{\epsilon}}+{\mathrm{K}}_{3}\dot{\mathsf{\sigma}}+{\mathrm{K}}_{4}\ddot{\mathsf{\sigma}}$ |

Parameter in Burgers Model | E_{0}/η_{0} (Pa/Pa∙s) | $\tilde{\mathit{c}}$ (g/l) | Exponent m/n |
---|---|---|---|

E_{1}(c) | 1.18 ± 0.21 | 0.23 (fixed) | 2.37 ± 0.06 |

E_{2}(c) | 0.06 ± 0.01 | 0.11 (fixed) | 2.71 ± 0.03 |

η_{1}(c) | 0.001 (fixed) | 0.23 ± 0.02 | 3.49 ± 0.08 |

η_{2}(c) | 0.001 (fixed) | 0.11 ± 0.01 | 2.70 ± 0.04 |

**Table 5.**Resulting fit parameters for fixed scaling exponents, according to Equations (15) and (16).

Parameter in Burgers Model | E_{0}/η_{0} (Pa/Pa∙s) | $\tilde{\mathit{c}}$ (g/l) | Exponent m/n |
---|---|---|---|

E_{1}(c) | 22.91 ± 5.67 | 1.45 ± 0.17 | 3 |

E_{2}(c) | 2.62 ± 0.71 | 0.63 ± 0.07 | 3 |

η_{1}(c) | 0.001 (fixed) | 0.14 ± 0.01 | 3 |

η_{2}(c) | 0.001 (fixed) | 0.16 ± 0.01 | 3 |

**Table 6.**Resulting fit parameters for the used asymptotic exponential fit function from Equation (17).

Alginate Concentration | A (mm) | t_{0} (s) | τ_{s} (s) |
---|---|---|---|

3% | 3.90 ± 0.002 | 10.65 ± 0.08 | 10.49 ± 0.05 |

4% | 3.24 ± 0.001 | 15.61 ± 0.05 | 13.92 ± 0.04 |

**Table 7.**Resulting characteristic times for 3% and 4% Alginate solutions concerning spreading and Burgers model.

Char. Time (s) | 3% Alginate | 4% Alginate | τ(4%)/τ(3%) | |
---|---|---|---|---|

Spreading | τ_{s} | 10.49 ± 0.52 | 13.93 ± 0.41 | 1.33 ± 0.01 |

Maxwell-Part | τ_{1} | 0.016 ± 0.001 | 0.021 ± 0.001 | 1.29 ± 0.137 |

Kelvin–Voigt-Part | τ_{2} | 0.017 ± 0.001 | 0.018 ± 0.001 | 1.06 ± 0.153 |

Rheological Model System | Adjusted R^{2} |
---|---|

Maxwell | 0.286 |

Kelvin–Voigt | 0.992 |

Zener m/k | 0.996 |

Lethersich/Jeffreys | 0.326 |

Burgers | 0.457 |

Rheological Model System | E_{1} | η | E_{2} |
---|---|---|---|

(Pa) | (Pa∙s) | (Pa) | |

Kelvin–Voigt | 32,685.11 | 54.18 | / |

Zener m | 6926.5 | 138.2 | 31,340.72 |

**Table 10.**Applicability of different model systems for the thermoplastic poly-urethane (TPU) 1180A melt.

Rheological Model System | Adjusted R^{2} |
---|---|

Maxwell | 0.997 |

Kelvin–Voigt | 0.960 |

Zener m/k | 0.997 |

Lethersich/Jeffreys | 0.999 |

Burgers | 0.999 |

Model Parameter in Burgers Model | Elastollan 1180A |
---|---|

E_{1} (Pa) | 203,959.30 ± 5765.51 |

E_{2} (Pa) | 116,297.58 ± 7.06 |

η_{1} (Pa∙s) | 403.75 ± 0.48 |

η_{2} (Pa∙s) | 1662.26 ± 27.00 |

Rheological Model System | Adjusted R^{2} |
---|---|

Maxwell | 0.828 |

Kelvin–Voigt | 0.639 |

Zener m/k | 0.924 |

Lethersich/Jeffreys | 0.859 |

Burgers | 0.952 |

**Table 13.**Influence of 10% PCL fiber fragments on the Burgers model parameters of 3% (w/v) alginate.

Model Parameter | Pure 3% Alginate | 3% Alginate + 10 wt % PCL | Percentage Increase |
---|---|---|---|

E_{1} (Pa) | 697.69 ± 35.38 | 2916.71 ± 215.47 | 418% |

E_{2} (Pa) | 511.27 ± 22.54 | 2611.20 ± 7.06 | 511% |

η_{1} (Pa∙s) | 11.07 ± 0.12 | 139.74 ± 4.23 | 1262% |

η_{2} (Pa∙s) | 8.38 ± 0.47 | 54.50 ± 3.26 | 650% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schrüfer, S.; Sonnleitner, D.; Lang, G.; Schubert, D.W.
A Novel Simple Approach to Material Parameters from Commonly Accessible Rheometer Data. *Polymers* **2020**, *12*, 1276.
https://doi.org/10.3390/polym12061276

**AMA Style**

Schrüfer S, Sonnleitner D, Lang G, Schubert DW.
A Novel Simple Approach to Material Parameters from Commonly Accessible Rheometer Data. *Polymers*. 2020; 12(6):1276.
https://doi.org/10.3390/polym12061276

**Chicago/Turabian Style**

Schrüfer, S., D. Sonnleitner, G. Lang, and D. W. Schubert.
2020. "A Novel Simple Approach to Material Parameters from Commonly Accessible Rheometer Data" *Polymers* 12, no. 6: 1276.
https://doi.org/10.3390/polym12061276