Polymers

Research

15 pages, 3906 KiB

Open AccessArticle

Effect of Prestrain on Payne Effect and Hysteresis Loss of Carbon-Black-Filled Rubber Vulcanizates: Measurements and Modeling

by Boyuan Yin, Xinyue Jiao, Haibo Wen, Yan Li and Ming Li

Polymers 2024, 16(3), 436; https://doi.org/10.3390/polym16030436 - 4 Feb 2024

Viewed by 811

Abstract

The performance of a viscoelastic damper is governed by the mechanical properties of the viscoelastic material, which are sensitive to prestrain. Among viscoelastic materials, carbon black (CB)-filled rubber vulcanizate is commonly used in structural applications. In this paper, the prestrain-dependent Payne effect and [...] Read more.

The performance of a viscoelastic damper is governed by the mechanical properties of the viscoelastic material, which are sensitive to prestrain. Among viscoelastic materials, carbon black (CB)-filled rubber vulcanizate is commonly used in structural applications. In this paper, the prestrain-dependent Payne effect and hysteresis loss of CB-filled rubber vulcanizates are investigated through experimental and theoretical analysis. Based on the experimental results, the classic quantitative models proposed by Kraus, Huber–Vilgis, and Maier–Göritz are used to describe the Payne effect. The results show that the Maier–Göritz model is most suitable to describe the Payne effect, especially for the loss modulus. After calculating the area of the hysteresis loops, hysteresis loss curves at various dynamic strain amplitudes are parallel to each other. Through application of the time–strain superposition principle, the hysteresis loss at any arbitrary prestrain can be predicted. Thus, the aim of this paper is to provide guidance for researchers in choosing an accurate model for future investigations of the prestrain-dependent Payne effect. An accelerated characterization method is useful for the prediction of the hysteresis loss of rubber products using small amounts of experimental data, which can provide manufacturers with more attractive and lower cost opportunities for testing the mechanical properties of rubber products. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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35 pages, 8312 KiB

Open AccessArticle

How to Make the Stress Relaxation Experiment for Polymers More Informative

by Anna Stankiewicz and Sławomir Juściński

Polymers 2023, 15(23), 4605; https://doi.org/10.3390/polym15234605 - 2 Dec 2023

Cited by 1 | Viewed by 1224

Abstract

Different viscoelastic models and characteristics are commonly used to describe, analyze, compare and improve the mechanical properties of polymers. A time-dependent linear relaxation modulus next to frequency-domain storage and loss moduli are the basic rheological material functions of polymers. The exponential Maxwell model [...] Read more.

Different viscoelastic models and characteristics are commonly used to describe, analyze, compare and improve the mechanical properties of polymers. A time-dependent linear relaxation modulus next to frequency-domain storage and loss moduli are the basic rheological material functions of polymers. The exponential Maxwell model and the exponential stretched Kohlrausch–Williams–Watts model are, probably, the most known linear rheological models of polymers. There are different identification methods for such models, some of which are dedicated to specific models, while others are general in nature. However, the identification result, i.e., the best model, always depends on the specific experimental data on the basis of which it was determined. When the rheological stress relaxation test is performed, the data are composed of the sampling instants used in the test and on the measurements of the relaxation modulus of the real material. To build a relaxation modulus model that does not depend on sampling instants is a fundamental concern. The problem of weighted least-squares approximation of the real relaxation modulus is discussed when only the noise-corrupted time-measurements of the relaxation modulus are accessible for identification. A wide class of models, that are continuous, differentiable and Lipschitz with respect to parameters, is considered for the relaxation modulus approximation. The main results concern the models that are selected asymptotically as the number of measurements tends to infinity. It is shown that even when the true relaxation modulus description is completely unknown, the approximate optimal model parameters can be derived from the measurement data that are obtained for sampling instants that are selected randomly due to the appropriate randomization introduced whenever certain conditions regarding the adopted class of models are satisfied. It is shown that the most commonly used stress relaxation models, the Maxwell and Kohlrausch–Williams–Watts models, satisfy these conditions. Since the practical problems of the identification of relaxation modulus models are usually ill posed, Tikhonov regularization is applied to guarantee the stability of the regularized solutions. The approximate optimal model is a strongly consistent estimate of the regularized model that is optimal in the sense of the deterministic integral weighted square error. An identification algorithm leading to the best regularized model is presented. The stochastic-type convergence analysis is conducted for noise-corrupted relaxation modulus measurements, and the exponential convergence rate is proved. Numerical studies for different models of the relaxation modulus used in the polymer rheology are presented for the material described by a bimodal Gauss-like relaxation spectrum. Numerical studies have shown that if appropriate randomization is introduced in the selection of sampling instants, then optimal regularized models of the relaxation modulus being asymptotically independent of these time instants can be recovered from the stress relaxation experiment data. The robustness of the identification algorithm to measurement noises was demonstrated both by analytical and numerical analyses. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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32 pages, 7225 KiB

Open AccessArticle

On Applicability of the Relaxation Spectrum of Fractional Maxwell Model to Description of Unimodal Relaxation Spectra of Polymers

by Anna Stankiewicz

Polymers 2023, 15(17), 3552; https://doi.org/10.3390/polym15173552 - 26 Aug 2023

Cited by 1 | Viewed by 854

Abstract

The relaxation time and frequency spectra are vital for constitutive models and for insight into the viscoelastic properties of polymers, since, from the spectra, other material functions used to describe rheological properties of various polymers can be uniquely determined. In recent decades the [...] Read more.

The relaxation time and frequency spectra are vital for constitutive models and for insight into the viscoelastic properties of polymers, since, from the spectra, other material functions used to describe rheological properties of various polymers can be uniquely determined. In recent decades the non-integer order differential equations have attracted interest in the description of time-dependent processes concerning relaxation phenomena. The fractional Maxwell model (FMM) is probably the most known rheological model of non-integer order. However, the FMM spectrum has not yet been studied and used to describe rheological materials. Therefore, the goal of the present paper was to study the applicability of the relaxation spectrum of FMM to the description of the relaxation spectra of polymers. Based on the known integral representation of the Mittag-Leffler two-parameter function, analytical formulas describing relaxation time and frequency spectra of FMM model were derived. Monotonicity of the spectra was analyzed and asymptotic properties were established. Relaxation frequency spectrum grows for large frequencies with a positive power law, while the relaxation time spectrum decays for large times with a negative power of time. Necessary and sufficient conditions for the existence of the local extrema of the relaxation spectra were derived in the form of two trigonometric inequalities. A simple procedure for checking the existence or absence of the spectra extrema was developed. Direct analytical formulas for the local extrema, minima, and maxima are given in terms of model fractional and viscoelastic parameters. The fractional model parameters, non-integer orders of the stress and strain derivatives of FMM uniquely determine the existence of the spectrum extrema. However, the viscoelastic parameters of the FMM, elastic modulus, and relaxation time affect the maxima and minima of the relaxation spectra and the values of their local peaks. The influence of model parameters on their local extrema was examined. Next, the applicability of the continuous–discrete spectrum of FMM to describe Baumgaertel, Schausberger and Winter (BSW) and unimodal Gauss-like relaxation spectra, commonly used to describe rheological properties of various polymers, was examined. Numerical experiments have shown that by respective choice of the FMM parameters, in particular by respective choice of the orders of fractional derivatives of the stress and strain, a good fit for the relaxation modulus experiment data was obtained for polymers characterized both by BSW and Gauss-like relaxation spectra. As a result, a good approximation of the real spectra was reached. Thus, the viscoelastic relaxation spectrum of FMM, due to the availability of the two extra degrees of freedom (non-integer orders of the stress and strain derivatives), provides deep insights into the complex behavior of polymers and can be applied for a wide class of polymers with unimodal relaxation spectra. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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42 pages, 19037 KiB

Open AccessArticle

On Recovery of a Non-Negative Relaxation Spectrum Model from the Stress Relaxation Test Data

by Anna Stankiewicz, Monika Bojanowska and Paweł Drozd

Polymers 2023, 15(16), 3464; https://doi.org/10.3390/polym15163464 - 18 Aug 2023

Cited by 1 | Viewed by 816

Abstract

The relaxation spectra, from which other material functions used to describe mechanical properties of materials can be uniquely determined, are important for modeling the rheological properties of polymers used in chemistry, food technology, medicine, cosmetics, and many other industries. The spectrum, being not [...] Read more.

The relaxation spectra, from which other material functions used to describe mechanical properties of materials can be uniquely determined, are important for modeling the rheological properties of polymers used in chemistry, food technology, medicine, cosmetics, and many other industries. The spectrum, being not directly accessible by measurement, is recovered from relaxation stress or oscillatory shear data. Only a few models and identification methods take into account the non-negativity of the real spectra. In this paper, the problem of recovery of non-negative definite relaxation spectra from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test is considered. A new hierarchical identification scheme is developed, being applicable both for relaxation time and frequency spectra. Finite-dimensional parametric classes of models are assumed for the relaxation spectra, described by a finite series of power-exponential and square-exponential basis functions. The related models of relaxation modulus are given by compact analytical formula, described by the products of power of time and the modified Bessel functions of the second kind for the time spectrum, and by recurrence formulas based on products of power of time and complementary error functions for frequency spectrum. The basis functions are non-negative. In result, the identification task was reduced to a finite-dimensional linear-quadratic problem with non-negative unknown model parameters. To stabilize the solution, an additional smoothing constraint is introduced. Dual approach was used to solve the stated optimal identification task resulting in the hierarchical two-stage identification scheme. In the first stage, dual problem is solved in two levels and the vector of non-negative model parameters is computed to provide the best fit of the relaxation modulus to experiment data. Next, in second stage, the optimal non-negative spectrum model is determined. A complete scheme of the hierarchical computations is outlined; it can be easily implemented in available computing environments. The model smoothness is analytically studied, and the applicability ranges are numerically examined. The numerical studies have proved that using developed models and algorithm, it is possible to determine non-negative definite unimodal and bimodal relaxation spectra for a wide class of polymers. However, the examples also demonstrated that if the basis functions are non-negative and the model is properly selected for a given type of the real spectrum (unimodal, multimodal), the optimal model determined without non-negativity constraint can be non-negative in the dominant range of its arguments, especially in the wide neighborhood of the spectrum peaks. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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35 pages, 4215 KiB

Open AccessArticle

A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions

by Anna Stankiewicz

Polymers 2023, 15(4), 958; https://doi.org/10.3390/polym15040958 - 15 Feb 2023

Cited by 4 | Viewed by 1224

Abstract

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of [...] Read more.

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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21 pages, 539 KiB

Open AccessArticle

Lifetime Predictions for High-Density Polyethylene under Creep: Experiments and Modeling

by A. D. Drozdov, R. Høj Jermiin and J. de Claville Christiansen

Polymers 2023, 15(2), 334; https://doi.org/10.3390/polym15020334 - 9 Jan 2023

Cited by 3 | Viewed by 2318

Abstract

Observations are reported in uniaxial tensile tests with various strain rates, tensile relaxation tests with various strains, and tensile creep tests with various stresses on high-density polyethylene (HDPE) at room temperature. Constitutive equations are developed for the viscoelastoplastic response of semicrystalline polymers. The [...] Read more.

Observations are reported in uniaxial tensile tests with various strain rates, tensile relaxation tests with various strains, and tensile creep tests with various stresses on high-density polyethylene (HDPE) at room temperature. Constitutive equations are developed for the viscoelastoplastic response of semicrystalline polymers. The model involves seven material parameters. Four of them are found by fitting observations in relaxation tests, while the others are determined by matching experimental creep curves. The predictive ability of the model is confirmed by comparing observations in independent short- and medium-term creep tests (with the duration up to several days) with the results of numerical analysis. The governing relations are applied to evaluate the lifetime of HDPE under creep conditions. An advantage of the proposed approach is that it predicts the stress-time-to-failure diagrams with account for the creep endurance limit. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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18 pages, 7201 KiB

Open AccessArticle

Dynamical Behaviors of a Translating Liquid Crystal Elastomer Fiber in a Linear Temperature Field

by Lin Zhou, Wangyang Yu and Kai Li

Polymers 2022, 14(15), 3185; https://doi.org/10.3390/polym14153185 - 4 Aug 2022

Viewed by 1414

Abstract

Liquid crystal elastomer (LCE) fiber with a fixed end in an inhomogeneous temperature field is capable of self-oscillating because of coupling between heat transfer and deformation, and the dynamics of a translating LCE fiber in an inhomogeneous temperature field are worth investigating to [...] Read more.

Liquid crystal elastomer (LCE) fiber with a fixed end in an inhomogeneous temperature field is capable of self-oscillating because of coupling between heat transfer and deformation, and the dynamics of a translating LCE fiber in an inhomogeneous temperature field are worth investigating to widen its applications. In this paper, we propose a theoretic constitutive model and the asymptotic relationship of a LCE fiber translating in a linear temperature field and investigate the dynamical behaviors of a corresponding fiber-mass system. In the three cases of the frame at rest, uniform, and accelerating translation, the fiber-mass system can still self-oscillate, which is determined by the combination of the heat-transfer characteristic time, the temperature gradient, and the thermal expansion coefficient. The self-oscillation is maintained by the energy input from the ambient linear temperature field to compensate for damping dissipation. Meanwhile, the amplitude and frequency of the self-oscillation are not affected by the translating frame for the three cases. Compared with the cases of the frame at rest, the translating frame can change the equilibrium position of the self-oscillation. The results are expected to provide some useful recommendations for the design and motion control in the fields of micro-robots, energy harvesters, and clinical surgical scenarios. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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12 pages, 3994 KiB

Open AccessArticle

Strain Rate-Dependent Hyperbolic Constitutive Model for Tensile Behavior of PE100 Pipe Material

by Yan Li, Wenbo Luo, Maodong Li, Bo Yang and Xiu Liu

Polymers 2022, 14(7), 1357; https://doi.org/10.3390/polym14071357 - 27 Mar 2022

Cited by 6 | Viewed by 3042

Abstract

It is not conservative to directly use the strength tested under the laboratory loading rates to evaluate the long-term creep strength of polymers. A suitable strain rate-dependent constitutive model is crucial for accurately predicting the long-term strength and mechanical behavior of polymer pressure [...] Read more.

It is not conservative to directly use the strength tested under the laboratory loading rates to evaluate the long-term creep strength of polymers. A suitable strain rate-dependent constitutive model is crucial for accurately predicting the long-term strength and mechanical behavior of polymer pressure pipes. In this study, the Kondner hyperbolic constitutive model is considered the base model in deriving the rate-dependent constitutive model for PE100 pipe material, and the yield stress and initial tangent modulus are the two rate-dependent parameters of the model. Uniaxial tension tests are carried out under five specified strain rates ranging from 10⁻⁵ s⁻¹ to 5 × 10⁻² s⁻¹ to obtain these two parameters. It is demonstrated that the strain rate dependence of the yield stress and the initial tangent modulus can be described by either a power or a logarithm law. The predictions from the two models are in good agreement with the experiments. In contrast, the power-law rate-dependent Kondner model is more suitable for describing the rate-dependent tensile behavior of PE100 pipe material than the logarithm-law rate-dependent Kondner model, especially for the cases of very low strain rates which relate to the polymer pressure pipe applications. Full article

(This article belongs to the Special Issue Time-Dependent Mechanical Behavior of Polymers and Polymer Composites)

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