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1.
F. R. Schwarzl 《Rheologica Acta》1969,8(1):6-17
Summary Numerical formulae are given for calculation of storage and loss compliance from the course of the creep compliance for linear
viscoelastic materials. These formulae involve values of the creep compliance at times which are equally spaced on a logarithmic
time scale. The ratio between succeeding times corresponds to a factor of two.
A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the
value of the damping, tanδ. The calculation of the storage compliance is easier with the lower damping values. This calculation involves the value of
the creep compliance at timet
0=1/ω, and that of its derivative with respect to the logarithm of time in a rather narrow region aroundt
0. In contrast the calculation of the loss compliance is more difficult with the lower damping values. This calculation involves
the value of the derivative of the creep compliance with respect to the logarithm of time in a broad interval aroundt
0.
Dedicated to Prof. Dr.J. Meixner, Aachen, on the occasion of his 60th birthday. 相似文献
Zusammenfassung Numerische Formeln werden angegeben, die die Berechnung der dynamischen Nachgiebigkeit aus der Kriechkurve erm?glichen. In diesen Formeln treten Werte der Kriechkurve auf, die zu logarithmisch ?quidistanten Zeitpunkten gemessen wurden. Das Verh?ltnis zweier aufeinanderfolgender Zeitpunkte entspricht stets einem Faktor 2. Für alle Formeln werden obere und untere Schranken für den relativen Fehler abgeleitet. Diese Schranken h?ngen vom Werte der D?mpfung (tanδ) ab, die bei der Kreisfrequenzω auftritt, für die die Berechnung erfolgt. Die Berechnung der Speicherkomponente der dynamischen Nachgiebigkeit ist desto leichter, je niedriger der Wert der D?mpfung ist. Zu dieser Berechnung ben?tigt man den Wert der Kriechfunktion zum Zeitpunktt 0=1/ω und deren logarithmische Zeitableitung in einem ziemlich engen Zeitintervall umt 0. Die Berechnung der Verlustkomponente der dynamischen Nachgiebigkeit ist desto leichter, je h?her der Wert der D?mpfung ist. Zu dieser Berechnung ben?tigt man den Wert der logarithmischen Zeitableitung der Kriechfunktion in einem breiten Zeitintervall umt 0.
Dedicated to Prof. Dr.J. Meixner, Aachen, on the occasion of his 60th birthday. 相似文献
2.
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The creep strains in linear viscoelastic materials under nonstationary loading of various types (incremental loading, complete unloading, and cyclic loading) are determined. Boltzmann–Volterra hereditary theory with fractional exponential kernel is used. Nonstationary loads are specified by Heaviside functions. The calculated results are validated by experimentally determining nonstationary creep strains of glass-reinforced plastic, plastic laminate, polymer concrete, duralumin, and nylon 相似文献
4.
B. Möginger 《Rheologica Acta》1993,32(4):370-379
Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress 0. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains
. The creep complicance C(t; 0) defined by the ratio
becomes, in principle, dependent on the initial stress 0. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength max and a nonlinearity parameter n
l
. The creep behavior of PBT and PC can be described in the range of long times up to initial stresses 0, being 75% for PBT and 60% for PC of the yield stress max with only two or one free fit parameter, respectively. 相似文献
5.
E. H. Lee 《Rheologica Acta》1961,1(4-6):426-430
Summary The relations between stress and strain prescribing linear viscoelastic behavior are discussed from the standpoint of application to problems of stress analysis. This consideration involves some important differences from the assessment of linear viscoelastic laws for the representation of material properties only.The use of the differential operator relation between stress and strain is usually most convenient, although the integral operators comprising the creep function or relaxation function can also sometimes be conveniently utilized. Examples of these are given, including the problem of indentation, in which the region of contact of a rigid spherical indentor under constant load can be expressed directly in terms of the creep function. When differential operator relations, which correspond to the usually considered viscoelastic models, are used, it is shown that a consistent application of the theory of delta functions and the associated symbolic differentiation permits initial values to be determined for the evaluation of the resulting differential equations. No mathematical difficulties arise if the loading functions are not smooth, as might be anticipated by the high order time derivatives which appear in the viscoelastic relations. An application of this theory is given which falls outside the scope of theLaplace transform type of analysis. The latter is often used to deal with this kind of problem, but has a restricted field of application.
The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, U. S. Army under Contract No. DA-19-020-ORD-4550. 相似文献
Zusammenfassung Die vorliegende Arbeit behandelt die Beziehungen zwischen Spannung und Dehnung in linear viskoelastischen Körpern vom Standpunkt der Spannungsberechnung. Diese Problemstellung geht wesentlich über die Erwägungen, die beim Studium der materiellen Eigenschaften auftreten, hinaus.Die Benützung der Differentialoperatorbeziehung zwischen Spannung und Dehnung ist meist günstiger, obgleich die Integraloperatoren, die aus Kriechfunktionen oder Relaxationsfunktionen bestehen, manchmal auch zweckdienlich benützt werden können. Als Beispiel der Anwendung dieser Integraloperatoren wird das Stempelproblem gelöst, wobei die Ausdehnung des Kontaktbereichs des festen, kugelförmigen Stempels unter konstanter Last direkt durch die Kriechfunktion ausgedrückt wird.Bei der Benützung von Differentialoperator-beziehungen (die den bekannten viskoelastischen Modellen entsprechen) wird gezeigt, daß eine Anwendung der Theorie der Deltafunktionen die Bestimmung der Anfangsbedingungen für die Differentialgleichungen zuläßt. Auch wenn die Belastungsfunktionen nur stückweise glatt sind, treten keine Schwierigkeiten auf, trotz der höheren Ableitungen, die in den viskoelastischen Stoffbeziehungen auftreten. Ein zweites Beispiel zeigt eine Anwendung der Deltafunktionen auf ein Problem, das der Behandlung mit Hilfe derLaplace-Transformation nicht zugänglich ist.
The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, U. S. Army under Contract No. DA-19-020-ORD-4550. 相似文献
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7.
For viscoelastic materials, the energy release rate is a fairly important parameter for determining whether the crack extends, but so far its meaning for viscoelastic materials is not yet clear enough. In this paper, the thermodynamic mechanical theory is used to derive the local and global energy release rate of viscoelastic materials when constitutive equations are given. Moreover, the method of deriving the viscoelastic energy release rate is discussed. The relation between the energy release rate and internal energy, Helmholz free energy is described. The equivalence between the local and global energy release rates is also proved after some doubts in previous papers are dispelled, although the upshot is not original. Furthermore, the concrete forms of the energy release rate for anisotropic, orthotropic and isotropic viscoelastic media are presented. The forms of the local and global energy release rates in the failure zone are discussed and the difference between the equation presented here and the one given in previous papers is pointed out. 相似文献
8.
9.
10.
The complex effective moduli of viscoelastic materials can be experimentally measured by a number of techniques giving relaxation
data for materials at discrete frequencies. In this paper we present a method which allows one to find bounds on the moduli
of the material at the unmeasured frequencies. Our bounds can be very tight, and are, therefore, good approximations to the
data. Using our bounds, experimentalists can concentrate their efforts on gathering highly accurate data at only a few frequencies
and numerically generate data at other frequencies that may be of interest.
Received: 10 August 1999 Accepted: 21 December 2000 相似文献
11.
On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation 总被引:1,自引:0,他引:1
The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G ∞)/(G 0-G ∞) and the retardation function r(t) = (J ∞+t/η-J(t))/(J ∞-J 0) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)β), can r(t) be represented as exp (?(t/λ)µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η?1 is finite for a fluid and zero for a solid), G ∞ is the equilibrium modulus G e for a solid or zero for a fluid, J ∞ is 1/G e for a solid or the steady-state recoverable compliance for a fluid, G 0= 1/J 0 is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed. 相似文献
12.
Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical
step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal
variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based
on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in
which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied
to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according
to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by
Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated
fields. 相似文献
13.
14.
H. A. Waterman 《Rheologica Acta》1977,16(1):31-42
Summary Starting from the relations between complex dynamic moduli simple diagrams are deduced connecting the locus of complex Poisson's ratiov
* v +iv in the complexv
* plane with differences between various loss angles. From these diagrams the sequence of magnitudes of several loss angles appearing in linear viscoelastic theory is deduced. Although theoretically this sequence depends on the values ofv andv, it is found experimentally that for polymeric materials, due to the fact that the values ofv andv are constrained to limited ranges, general rules can be given. The sequences deduced are compared with experimental data. Finally some relations are used to illustrate the phase relationships between stress and deformations in an uniaxial stress experiment. From these relations a new method for measuring the loss angle in compression is suggested.
Part of this study was performed while the author was an employee of TNO, Delft.
With 16 figures and 1 table 相似文献
Zusammenfassung Mit Hilfe der Beziehungen zwischen komplexen dynamischen Moduln wurden einfache Diagramme konstruiert, in denen der geometrische Ort des komplexen Poisson-Verhältnissesv * v +iv in der komplexenv *-Ebene mit Differenzen zwischen verschiedenen Verlustwinkeln verbunden werden. Aus diesen Diagrammen kann auf die Reihenfolge der Größe mehrerer in der linearen viskoelastischen Theorie auftretender Verlustwinkel gefolgert werden. Obwohl diese Reihenfolge theoretisch von den Werten vonv undv abhängig ist, wird experimentell gefunden, daß für polymere Substanzen allgemeine Regeln gegeben werden können. Dies findet seine Ursache in der Tatsache, daß die Werte vonv undv auf begrenzte Gebiete beschränkt sind. Zum Schluß werden für einen einachsigen Spannungsversuch die Phasenbeziehungen zwischen den Spannungen und den Verformungen illustriert. Mit Hilfe dieser Phasenbeziehungen wird eine neue Methode für die Messung des Verlustwinkels in Kompression vorgeschlagen.
Part of this study was performed while the author was an employee of TNO, Delft.
With 16 figures and 1 table 相似文献
15.
《International Journal of Solids and Structures》2007,44(3-4):1291-1303
In this paper, interconversion between linear viscoelastic material functions is studied emphasizing materials with relatively fast rate of relaxation. The aim of this paper is to study the whole material function determination process from a linear viscoelastic experiment to interconversion by taking into account non-ideal loading and noisiness of the data in such an experiment. No assumptions are made concerning the form of the relaxation modulus or the creep compliance. Interconversion is carried out by evaluating numerically the convolution integral. Three different yet similar approaches are studied. In numerical interconversion, the resulting matrix equation is ill-posed. Due to this, Tikhonov regularization is applied to solve the related matrix system. Numerical simulations indicate that reliable results can be obtained with proposed numerical procedures. 相似文献
16.
Creep and stress relaxation are known to be interrelated in linearly viscoelastic materials by an exact analytical expression. In this article, analytical interrelations are derived for nonlinearly viscoelastic materials which obey a single integral nonlinear superposition constitutive equation. The kernel is not assumed to be separable as a product of strain and time dependent parts. Superposition is fully taken into account within the single integral formulation used. Specific formulations based on power law time dependence and truncated expansions are developed. These are appropriate for weak stress and strain dependence. The interrelated constitutive formulation is applied to ligaments, in which stiffness increases with strain, stress relaxation proceeds faster than creep, and rate of creep is a function of stress and rate of relaxation is a function of strain. An interrelation was also constructed for a commercial die-cast aluminum alloy currently used in small engine applications. 相似文献
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18.
There are several models for the determination of molecular weight distributions (MWDs) of linear, entangled, polymer melts via rheometry. Typically, however, models require a priori knowledge of the critical molecular weight, the plateau modulus, and parameters relating relaxation time and molecular weight (e.g., k and in =kM). Also, in an effort to obtain the most general MWD or to describe certain polymer relaxation mechanisms, models often rely on the inversion of integral equations via regularization. Here, the inversion of integral equations is avoided by using a simple double-reptation model and assuming that the MWD can be described by an analytic function. Moreover, by taking advantage of dimensionless variables and explicit analytic relations, we have developed an unambiguous and virtually parameter-free methodology for the determination of MWDs via rheometry. Unimodal MWDs have been determined using only a priori knowledge of the exponent and dynamic moduli data. In addition, the uncertainty in rheological MWD determinations has been quantified, and it is shown that the reliability of the predictions is greater for the high-molecular-weight portion of the distribution. 相似文献
19.
20.
A method is presented whereby the molecular weight distribution of a molten polymer can be determined from measured values of the storage and loss moduli (the response of the system to small amplitude oscillatory shear flow). The Curtiss-Bird theory for the constitutive behavior of polymer melts is used. This results in an integral equation relating the moduli to the molecular weight distribution. The method that we describe here uses a regularization scheme with quadratic programming to extract the MWD from the components of the moduli (or, equivalently, from the components of the complex viscosity). We verify the applicability of the scheme by considering concocted data for given molecular weight distributions. The scheme is also used to predict the molecular weight distributions of polystyrene from complex viscosity data that have been presented in the literature. Variations between the predicted and measured molecular weight distributions can be attributed to the inadequacy of the underlying molecular theory at large frequencies. The predicted average molecular weights obtained using the truncated dynamic data are in reasonable agreement with the experimental values. 相似文献