Application of smoothing spline approximations in problems on identification of creep parameters

Golub, V. P.; Pogrebnyak, A. D.; Romanenko, I.

doi:10.1007/bf02700657

Cited by 9 publications

(10 citation statements)

References 3 publications

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“…Table 1 summarizes the coefficients A j , B j , C j , and D j of the smoothing splines for the fibers under consideration. The procedure of smoothing is outlined in [5].…”

Section: Unified Isochronous Deformationmentioning

confidence: 99%

“…Figure 2 shows initial creep curves for the FM 3001 fibers at θ = 23°C (à) and for the FÌ 10001 fibers at θ = 25°C (b). The circles stand for the experimental data, and the solid lines for smoothed spline approximations [5]. The creep curves have been plotted for stresses σ k = 3.28 ( ), 4.14 ( ), 8.27 ( ), 12.41 ( ), 16.55 ( ) MPa in the case of FM 3001 and for stresses σ k = 3.20 ( ), 4.96 ( ), 6.78 ( ), 9.27 ( ), 12.41 ( ) MPa in the case of FM 10001.…”

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confidence: 99%

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Nonlinear Creep of Viscoelastic Organic Fibers under Tension

2005

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The nonlinearity of the creep of nylon fibers is justified based on the similarity of a set of isochronous creep curves, which also includes the instantaneous deformation curve. Nonlinear hereditary constitutive equations of creep are derived. The real values of the influence function are determined as the basic rheological characteristic of the material. The applicability of Boltzmann's, Abel's, Rzhanitsyn's, and Rabotnov's kernels is estimated quantitatively. The choice of an Abel-type kernel is justified. The calculated and experimental data are in satisfactory agreement for a loading duration of up to 1,000 hours and an order of magnitude change in the stresses Introduction. The creep of metals at high temperatures and most polymeric and composite materials occurs in the nonlinear domain of deformation [1,7,10,17]. In this domain, experimental compliance curves do not agree and there is no a unified creep function. Also, depending on the type of material and loading conditions, either the initial creep curves or the isochronous creep curves are similar.There are many publications devoted to nonlinear hereditary models. Some of these models, methods for determining their material constants, and their results were discussed in [8, 10-16, 19, 21, 22, 24-27, 30, 31]. The best justification was given to Rabotnov's constitutive equation of nonlinear viscoelasticity [16,17] based on the similarity conditions for isochronous creep curves.The similarity condition for isochronous creep curves is used in [29] to formulate the concept of a unified isochronous deformation curve. This curve also includes the instantaneous tension curve as an isochrone at time zero. The same concept underlies a generalized rheological model describing both linear and nonlinear creep. This model was used in [3,4] to analyze the linear creep of viscoelastic fibers. In the present paper, we employ this model to analyze the nonlinear creep strains of viscoelastic nylon fibers.1. Problem Formulation. Subject of Inquiry. Let us analyze the nonlinear creep of viscoelastic organic fibers that are long subjected to a tensile force. The external load does not vary during creep, and the tensile stress σ(t) is defined by

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“…Table 1 summarizes the coefficients A j , B j , C j , and D j of the smoothing splines for the fibers under consideration. The procedure of smoothing is outlined in [5].…”

Section: Unified Isochronous Deformationmentioning

confidence: 99%

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Nonlinear Creep of Viscoelastic Organic Fibers under Tension

2005

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“…The symbols stand for the strains obtained from the creep curves for the following durations of loading: t j = 77 ( ), 200 ( ), 1000 ( ), 4000 ( ), 8800 ( ) h and t j = 10 ( ), 50 ( ), 200 ( ), 1000 ( ), 3000 ( ), 9400 ( ) h, respectively. The thin solid lines represent creep isochrones approximated by smoothing cubic splines [3]. The bold line is the spline approximation of the tensile stress-strain curve, which is, in essence, an isochrone at t j = 0.…”

Section: Linearity Of Viscoelastic Properties From Creep Isochronesmentioning

confidence: 99%

“…The symbols stand for normalized experimental data obtained from primary creep curves based on relation (2.3) for stresses s k = 331 ( ), 1000 ( ), 1660 ( ), 2330 ( ) MPa and s k = 300 ( ), 450 ( ), 600 ( ), 800 ( ), 1200 ( ) MPa, respectively. The thin solid lines represent creep functions approximated by smoothing cubic splines[3] used to calculate the sample mean J t j ( ). The dashed lines show the maximum deviation d max = ±5% from J t j ( ).…”

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Nonlinear creep of unidirectional fibrous composites tensioned along the reinforcement

2007

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Creep strains in unidirectional organic-fiber-and organic-glass-fiber-reinforced plastics subject to tension in the reinforcement direction are predicted. Based on homogeneity condition and statistical criteria, it is shown that the creep of the composites is essentially nonlinear. To predict creep strains, use is made of a nonlinear creep model based on a modified Rabotnov's similarity hypothesis for isochrones and of material constants determined from tests on specimens made of a composite as a whole and specimens made of its separate components, the mixing rule applied in the latter case. The calculated results and the experimental data are in satisfactory agreement Introduction. Many unidirectional fibrous composites are engineering materials in which a viscoelastic matrix is reinforced with various organic and inorganic fibers. These composites are widely used in aeronautical and space engineering, shipbuilding, oil refinary, and oil and gas pipelines [1,8,14,21,23]. Fibrous composites as reinforcement of concrete structures have recently aroused considerable interest [26,32].Under long-term loads, viscoelastic fibrous composites creep even at temperatures far below the glass-transition temperature. This effect necessitates using creep strain, creep rate, relaxation time, creep limits, and long-term strength as the main deformation and strength characteristics of composites. One of the important problems in the mechanics of viscoelastic composites is to predict their viscoelastic properties from the properties and volume fractions of its components.A great many theoretical and experimental studies are concerned with various aspects of creep in unidirectional fibrous composites. Some of the results from such studies are reviewed in [8,9,14,15,23,30]. It should be noted that the overwhelming majority of studies deal with the linear range of long-term viscoelastic deformation. All solutions found are based on the Boltzmann-Volterra theory of linear viscoelasticity [11,31]. When used to predict creep strains of composites from the properties of their components, the linear theory is usually supplemented with the mixing rule [8,14,15,23].Theoretical and experimental studies of creep of various polymer composites show that the linear theory of viscoelasticity can be directly applied to a certain class of problems. In the majority of composites, however, the range of linearity is relatively short, and satisfactory results can only be obtained with not too high stresses and short-term loads. It should be taken into account that a portion of the creep strain may be irreversible.There are several approaches to describe the nonlinear viscoelastic deformation of composites. The general nonlinear equation of viscoelasticity was derived in [33]. It is represented by an infinite series of multiple integrals; therefore, it is not practical to identify numerous kernels and to determine their parameters experimentally. Similar difficulties arise in using the nonlinear models proposed in [4,6,7,10,13,22].The approach based on ...

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“…Functions whose values are tabulated primary or transformed test data used to determine the parameters of the hereditary kernels are analytically represented by smoothing cubic splines [4]:…”

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A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

2011

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The analytical and experimental procedures of the method for determining the parameters of the hereditary kernels in the modified Rabotnov's nonlinear viscoelastic model are outlined. The method is based on the similarity of the isochronous creep curves to the instantaneous-deformation curve. The parameters of the kernels are determined by fitting the discrete values of the kernels that are found by differentiating the average similarity function. The discrete values of the kernels in the domain of singularity are calculated using weight functions Keywords: nonlinear viscoelasticity, parameters of hereditary kernel, creep rate, stress relaxation rate, similarity function, weight functionIntroduction. Selection of hereditary kernels and reliable determination of theirs resolvents and parameters constitute one of the basic problems in the hereditary theory of viscoelasticity. This problem is, as a rule, solved by seeking creep and relaxation kernels that characterize the mechanical properties of viscoelastic materials and appear in the constitutive equations relating stresses, strains, and time.There are various analytical interpretations of the creep and relaxation kernels and methods for determining their parameters [6,7,9]. The chief difficulties that arise in selecting kernels and methods to determine their parameters are due to the fact that the creep and relaxation rates tend to infinity as t ® 0 and asymptotically decrease as t ® ¥. Moreover, the hereditary kernels must be analytically invertible so that the parameters of one kernel can be used to find the values of another kernel.The quasistatic mechanical properties of materials in the linear theory of viscoelasticity can be quite accurately described in terms of creep and relaxation functions. These functions are set up using the data of constant-stress creep tests or constant-strain relaxation tests. A general approach, logical procedures, and their experimental and numerical implementation needed to determine mechanical characteristics in the linear theory of viscoelasticity are detailed in [6,7,9,12].The methods for determining mechanical characteristics in the nonlinear theory of viscoelasticity are more complicated and less justified than in the linear theory. This is primarily because the nonlinear theory uses several variations of constitutive equations and, respectively, several methods for determining viscoelastic characteristics.The most general approach to determining the mechanical characteristics of nonlinear viscoelastic materials uses the Volterra-Frechet multiple integral representation [7]. However, the great number and multidimensional nature of the hereditary kernels that enter into the multiple integral expressions, especially for high-order terms, considerably complicate the determination of the kernel parameters. To overcome these difficulties, simplified variations of the general nonlinear constitutive equation were proposed and numerical and experimental procedures for determining the necessary viscoelastic characteristics were dev...

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Application of smoothing spline approximations in problems on identification of creep parameters

Cited by 9 publications

References 3 publications

Nonlinear Creep of Viscoelastic Organic Fibers under Tension

Nonlinear Creep of Viscoelastic Organic Fibers under Tension

Nonlinear creep of unidirectional fibrous composites tensioned along the reinforcement

A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

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