A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials

Yélomè Judicaël Fernando Kpomahou; Koffi Judicaël Agbélélé; Arnaud Edouard Yamadjako; Bachir Koladé Adélakoun Ambelohoun

doi:doi:10.11648/j.ijmsa.20231201.11

A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials

Yélomè Judicaël Fernando Kpomahou, Koffi Judicaël Agbélélé, Arnaud Edouard Yamadjako, Bachir Koladé Adélakoun Ambelohoun

Published in International Journal of Materials Science and Applications (Volume 12, Issue 1)

Received: 21 November 2022 Accepted: 13 December 2022 Published: 4 March 2023

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Abstract

Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.

Published in	International Journal of Materials Science and Applications (Volume 12, Issue 1)
DOI	10.11648/j.ijmsa.20231201.11
Page(s)	1-7
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Nonlinear Rheological Model, Integer and Non-Integer Orders Nonlinearities, Stress Relaxation, Viscoelastic Materials

References

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[2]	Lockett, F. J.: Nonlinear Viscoelastic Solids, Academic Press, London. New York (1972).
[3]	Chakespari, A. G., Rajabipour, A., Mobli, H.: Anisotropic Relaxation and Creep Properties of Apple, Adv J Food SciTechnol, 2 (4), 200-205 (2010).
[4]	O'Connell, G. D., Jacobs, N. T., Sen, S., Vresilovic, E. J., Elliott, D. M.: Axial creep loading and unloaded recovery of the human intervertebral disc and the effect of degeneration, J. Mech. Behav. Biomed. Mater, 4 (7), 933-942 (2011).
[5]	Figueroa, J. D. C., Hernanderz, Z. J. E., Pena, R. J.: Stress relaxation and Creep Recorvery Test Performed on Wheat Kernels Versus Doughs: Influence of Glutenins on Rheological and Quality Properties, Cereal Foods World, 48 (3), 139-143 (2013).
[6]	Sopakayang, R., De Vita, R.: A mathematical model for creep, relaxation and strain stiffening in parallel-fibered collagenous tissues, Med. Eng. Phys., 33, 1056–1063 (2011).
[7]	Fung, Y. C.: Biomechanics Properties of Living Tissues, New York Springer-Verlag (1993).
[8]	Findley, W. N., Lai. J. S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover Publications, Inc, York (1976).
[9]	Hadjikov, L., Kirilova, M., Stoytchev, S., Pashkouleva, D.: Visco-elastic mechanical behaviour of human abdominal fascia, Ser. Biomech., 1 (1), 39-45 (2007).
[10]	Kim, J. Tay, B. K., Stylopoulos, N., Rattner, D. W., and Srinivasan, M. A.: Characterization of Intra-abdominal Tissues from in vivo Animal Experiments for Surgical Simulation, Med Image Comput Comput Assist Interv., 2878, 206-213 (2003).
[11]	Corr, D. T., Starr, M. J., Vanderby, R. Jr., Best, T. M.: A Nonlinear Generalized Maxwell Fluid Model for Viscoelastic Materials,” J. Appl. Mech., 68, 787-790 (2001).
[12]	Launay, B.: A Simplified Nonlinear Model for Describing the Viscoelastic Properties of Wheat Flour Doughs at High Shear Strain, Cereal Chem, 67 (1), 25-31 (1990).
[13]	Monsia, M. D.: A Hyperlogistic-TypeModel for the Prediction of Time-Dependent Nonlinear Behavior of Viscoelastic Materials, Int. J. Mech. Eng., Serials Publications, New Delhi, India, (2011a).
[14]	Monsia, M. D.: A Nonlinear Generalized Standard Solid Model for Viscoelastic Materials, Int. J. Mech. Eng., 4 (1), pp. 11-15 (2011b).
[15]	Monsia, M. D.: A Modified Voigt Model for Nonlinear Viscoelastic Materials,” Int. J. Mech. Eng., 4 (1), 17-20 (2011c).
[16]	Monsia, M. D.: A Simplified Nonlinear Generalized Maxwell Model for Predicting the Time Dependent Behavior of Viscoelastic Materials, World J. Mech., 1, 158-167 (2011d).
[17]	Liu Z., Yeung, K.: The Preconditioning and Stress Relaxation of Skin Tissue, Int. J. Pharm. Biomed. Res., 2 (1), 22-28 (2008).
[18]	Myhan, R., Markowski, M., Daszkiewicz, T., Zapotoczny, P., and Sadowski, P.: Non-linear stress relaxation model as a tool for evaluating the viscoelastic properties of meat products, J. Food Eng., 146, 107–115 (2015).
[19]	Lin, Che-Yu, 2020, Alternative form of Standard Linear Solid model for characterizing stress relaxation and creep: Including a novel parameter for quantifying the ratio of fluids to solids of a viscoelastic solid, Front. Mater., 7, 1-11 (2020).

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Yélomè Judicaël Fernando Kpomahou, Koffi Judicaël Agbélélé, Arnaud Edouard Yamadjako, Bachir Koladé Adélakoun Ambelohoun. (2023). A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. International Journal of Materials Science and Applications, 12(1), 1-7. https://doi.org/10.11648/j.ijmsa.20231201.11

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ACS Style

Yélomè Judicaël Fernando Kpomahou; Koffi Judicaël Agbélélé; Arnaud Edouard Yamadjako; Bachir Koladé Adélakoun Ambelohoun. A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. Int. J. Mater. Sci. Appl. 2023, 12(1), 1-7. doi: 10.11648/j.ijmsa.20231201.11

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AMA Style

Yélomè Judicaël Fernando Kpomahou, Koffi Judicaël Agbélélé, Arnaud Edouard Yamadjako, Bachir Koladé Adélakoun Ambelohoun. A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. Int J Mater Sci Appl. 2023;12(1):1-7. doi: 10.11648/j.ijmsa.20231201.11

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@article{10.11648/j.ijmsa.20231201.11,
  author = {Yélomè Judicaël Fernando Kpomahou and Koffi Judicaël Agbélélé and Arnaud Edouard Yamadjako and Bachir Koladé Adélakoun Ambelohoun},
  title = {A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials},
  journal = {International Journal of Materials Science and Applications},
  volume = {12},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ijmsa.20231201.11},
  url = {https://doi.org/10.11648/j.ijmsa.20231201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmsa.20231201.11},
  abstract = {Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.},
 year = {2023}
}

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TY  - JOUR
T1  - A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials
AU  - Yélomè Judicaël Fernando Kpomahou
AU  - Koffi Judicaël Agbélélé
AU  - Arnaud Edouard Yamadjako
AU  - Bachir Koladé Adélakoun Ambelohoun
Y1  - 2023/03/04
PY  - 2023
N1  - https://doi.org/10.11648/j.ijmsa.20231201.11
DO  - 10.11648/j.ijmsa.20231201.11
T2  - International Journal of Materials Science and Applications
JF  - International Journal of Materials Science and Applications
JO  - International Journal of Materials Science and Applications
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2327-2643
UR  - https://doi.org/10.11648/j.ijmsa.20231201.11
AB  - Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.
VL  - 12
IS  - 1
ER  -

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Author Information

Yélomè Judicaël Fernando Kpomahou
Koffi Judicaël Agbélélé
Arnaud Edouard Yamadjako

Department of Physics, University of Abomey-Calavi (UAC), Abomey-Calavi, Benin
Bachir Koladé Adélakoun Ambelohoun

Laboratory of Energetics and Applied Mechanics (LEMA), University of Abomey-Calavi (UAC), Abomey-Calavi, Benin

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APA Style

Yélomè Judicaël Fernando Kpomahou, Koffi Judicaël Agbélélé, Arnaud Edouard Yamadjako, Bachir Koladé Adélakoun Ambelohoun. (2023). A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. International Journal of Materials Science and Applications, 12(1), 1-7. https://doi.org/10.11648/j.ijmsa.20231201.11

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ACS Style

Yélomè Judicaël Fernando Kpomahou; Koffi Judicaël Agbélélé; Arnaud Edouard Yamadjako; Bachir Koladé Adélakoun Ambelohoun. A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. Int. J. Mater. Sci. Appl. 2023, 12(1), 1-7. doi: 10.11648/j.ijmsa.20231201.11

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AMA Style

Yélomè Judicaël Fernando Kpomahou, Koffi Judicaël Agbélélé, Arnaud Edouard Yamadjako, Bachir Koladé Adélakoun Ambelohoun. A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials. Int J Mater Sci Appl. 2023;12(1):1-7. doi: 10.11648/j.ijmsa.20231201.11

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@article{10.11648/j.ijmsa.20231201.11,
  author = {Yélomè Judicaël Fernando Kpomahou and Koffi Judicaël Agbélélé and Arnaud Edouard Yamadjako and Bachir Koladé Adélakoun Ambelohoun},
  title = {A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials},
  journal = {International Journal of Materials Science and Applications},
  volume = {12},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ijmsa.20231201.11},
  url = {https://doi.org/10.11648/j.ijmsa.20231201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmsa.20231201.11},
  abstract = {Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - A Rheological Model with Integer and Non-Integer Orders Nonlinearities for Predicting Stress Relaxation Behavior in Viscoelastic Materials
AU  - Yélomè Judicaël Fernando Kpomahou
AU  - Koffi Judicaël Agbélélé
AU  - Arnaud Edouard Yamadjako
AU  - Bachir Koladé Adélakoun Ambelohoun
Y1  - 2023/03/04
PY  - 2023
N1  - https://doi.org/10.11648/j.ijmsa.20231201.11
DO  - 10.11648/j.ijmsa.20231201.11
T2  - International Journal of Materials Science and Applications
JF  - International Journal of Materials Science and Applications
JO  - International Journal of Materials Science and Applications
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2327-2643
UR  - https://doi.org/10.11648/j.ijmsa.20231201.11
AB  - Viscoelastic materials are widely used as devices for vibration control in modern engineering applications. They exhibit both viscous and elastic characteristic when undergoing deformation. They are mainly characterized by three time-dependent mechanical properties such as creep, stress relaxation and hysteresis. Among them, stress relaxation is one of the most important features in the characterization of viscoelastic materials. This phenomenon is defined as a time-dependent decrease in stress under a constant strain. Due to the inherent nonlinearity shown by the material response over a certain range of strain when viscoelastic materials are subjected to external loads, nonlinear rheological models are needed to better describe the experimental data. In this study, a singlenonlinear differential constitutive equation is derived froma nonlinear rheological model composed of a generalized nonlinear Maxwell fluid model in parallel with a nonlinear spring obeying a power law for the prediction of the stress relaxation behavior in viscoelastic materials. Under a constant strain-history, the time-dependent stress is analytically derived in the cases where the positive power law exponent, α ˂ 1 and α ˃1. The Trust Region Method available in MATLAB Optimization Toolbox is used to identify the material parameters. Significant correlations are found between the experimental relaxation data taken from literature and exact analytical predictions. The obtained results show that the developed rheological model with integer and non-integer orders nonlinearities accurately describes the experimental relaxation data of some viscoelastic materials.
VL  - 12
IS  - 1
ER  -

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