Interconversion of constitutive viscoelastic functions

Hajikarimi, Pouria; Nejad, Fereidoon Moghadas

doi:10.1016/b978-0-12-821210-3.00005-x

Cited by 3 publications

(9 citation statements)

References 16 publications

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“…T is an K-element vector of unknown coefficients of the models ( 6) and (7). The identification index (22) is rewritten in the compact form as…”

Section: Least-squares Regularized Identificationmentioning

confidence: 99%

“…The parameters of the optimal models in the example: optimal time-scale factors α opt , upper α min and lower α max bounds of the interval [α min , α max ] of suboptimal time-scale factors defined by inequality (66) for ε α = 0.1, regularization parameters λ GCV α opt , optimal identification indices Q Nopt α opt defined in (37), square norms: g K 2 of the vector of optimal model parameter g K (45) and H Kopt (τ) 2 of the optimal relaxation spectrum model H Kopt (τ) (46), relaxation spectrum approximation error ER1(K) defined in (64) and the distance between successive approximations H Kopt (τ) measured by ER2(K) (65). In Figure 5, the optimal models of the relaxation modulus G K t, α opt computed for g K (45) according to (7) are plotted, where the measurements G(t i ) of the 'real' modulus G(t) (62) are also marked. The optimal models G K t, α opt have been well fitted to the experimental data, as indicated by the mean-square model errors Q Nopt α opt /N, which for 8 ≤ K ≤ 12 vary in the range from 4.889•10 −9 Pa 2 to 5.0482•10 −9 Pa 2 .…”

Section: Optimal Modelsmentioning

confidence: 99%

“…The corresponding optimal regularization parameters are as follows: λ GCV (α min ) = 0.0167 and λ GCV (α max ) = 0.0094. For respective vectors of optimal model parameters, we have g 7b, the 'real' modulus G(t) (62) and the models: G K t, α opt (7) and (45) and G K (t, α min ), G K (t, α max ) computed for g λ GCV K (α min ), g λ GCV K (α max ) according to (7) are plotted. However, the relaxation moduli are almost identical (Figure 7b), the spectrum models differ (Figure 7a), which emphasizes the importance of the time-scale factor optimal selection.…”

Section: Optimal and Sub-optimal Time-scale Factorsmentioning

confidence: 99%

“…However, deeper insight into the complex behavior of viscoelastic materials is provided by the relaxation spectrum [2,4,5]. The relaxation spectrum is vital for constitutive models and for insight into the properties of a material, since, from the relaxation spectrum, other material functions used to describe rheological properties can be uniquely determined [4,6,7]. It is commonly used to describe, analyze, compare, and improve the mechanical properties of polymers [4,8,9], concrete [10], asphalt [11], rubber [12], wood [13], glass [14], dough [15], polymeric textile materials [16], geophysics [17] and biological materials [18].…”

Section: Introductionmentioning

confidence: 99%

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Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

Stankiewicz

2023

Materials

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The viscoelastic relaxation spectrum is vital for constitutive models and for insight into the mechanical properties of materials, since, from the relaxation spectrum, other material functions used to describe rheological properties can be uniquely determined. The spectrum is not directly accessible via measurement and must be recovered from relaxation stress or oscillatory shear data. This paper deals with the problem of the recovery of the relaxation time spectrum of linear viscoelastic material from discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation test. A two-level identification scheme is proposed. In the lower level, the regularized least-square identification combined with generalized cross-validation is used to find the optimal model with an arbitrary time-scale factor. Next, in the upper level, the optimal time-scale factor is determined to provide the best fit of the relaxation modulus to experiment data. The relaxation time spectrum is approximated by a finite series of power–exponential basis functions. The related model of the relaxation modulus is proved to be given by compact analytical formulas as the products of power of time and the modified Bessel functions of the second kind. The proposed approach merges the technique of an expansion of a function into a series of independent basis functions with the least-squares regularized identification and the optimal choice of the time-scale factor. Optimality conditions, approximation error, convergence, noise robustness and model smoothness are studied analytically. Applicability ranges are numerically examined. These studies have proved that using a developed model and algorithm, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and noise robust; small model errors are obtained for the optimal time-scale factors. The complete scheme of the hierarchical computations is outlined, which can be easily implemented in available computing environments.

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“…T is an K-element vector of unknown coefficients of the models ( 6) and (7). The identification index (22) is rewritten in the compact form as…”

Section: Least-squares Regularized Identificationmentioning

confidence: 99%

Section: Optimal Modelsmentioning

confidence: 99%

Section: Optimal and Sub-optimal Time-scale Factorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

Stankiewicz

2023

Materials

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“…Viscoelastic relaxation or retardation spectra are commonly used to describe, analyze, compare and improve the mechanical properties of polymers [ 1 , 2 , 3 , 4 , 5 ]. The spectra are vital for constitutive models and for the insight into the properties of a viscoelastic material, since, from the relaxation or retardation spectrum, other material functions used to describe rheological properties of various polymers can be uniquely determined [ 6 , 7 , 8 , 9 ]. However, the spectra are not directly accessible by measurement.…”

Section: Introductionmentioning

confidence: 99%

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

Stankiewicz

2023

Polymers

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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.

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Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements

Stankiewicz

2024

Polymers

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The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential–hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model’s best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model’s robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

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Interconversion of constitutive viscoelastic functions

Cited by 3 publications

References 16 publications

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

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

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements

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