Continuous Relaxation Spectrum Research Articles

Modeling of the viscoelastic behavior of amorphous polymers by the differential and integration fractional method: the relaxation spectrum H( τ)

The dynamic mechanical behavior of poly(methyl methacrylate) (PMMA) from deep in the glass to the glass transition region has been studied by DMTA and analyzed by using a phenomenological fractional model in which the dynamic stress appears as a non-integer-order derivative of the strain. In order for the model to accurately represent the experimental data, three non-integer values for the derivative order are required. These values are related to two relaxation mechanisms. In the low temperature region (i.e. the β relaxation of PMMA), the derivative order is smaller and near 0.2, which indicates behavior close to the ideal elastic solid (glassy). For higher temperatures (between the β and the α relaxations), the derivative order is higher, indicating more viscoelastic behavior. In this work, modeling of the viscoelastic behavior of polymers using the fractional calculus approach is presented and the extended fractional solid (EFS) model is used to fit the experimental data of PMMA. In addition, the continuous relaxation spectrum H( τ) of PMMA is calculated from the model using the inverse Stieljes integral transformation. Finally, the effect of thermal treatment on the non-integer model parameters and on the distribution of relaxation times is obtained.

Non-linear viscoelasticity modeling of tomato paste products

This paper deals with the characterization and modeling of the non-linear rheological properties of tomato paste samples manufactured under different conditions (sieve pore size and breaking temperature). With this aim, flow measurements, dynamic linear viscoelastic and linear and non-linear stress relaxation tests have been carried out on the above-mentioned tomato paste samples. Different geometries with smooth and serrated surfaces have been used to optimize the viscous flow measurements and avoid wall-depletion phenomena. The viscous flow properties of tomato paste samples depend on water insoluble solids (WIS) content and particle size, which may be highly influenced by processing conditions. In general, viscosity follows a power-law relationship with tomato paste water insoluble content and particle diameter. A factorable non-linear viscoelasticity model, the Wagner integral equation, predicts the non-linear rheological response of these products under shear fairly well. The time-dependent part of this model is described by a continuous linear relaxation spectrum, calculated from regularization techniques. The use of the Soskey–Winter damping function provides the best predictions of the viscous flow curves.

Continuous Relaxation Spectrum for Concrete Creep and its Incorporation into Microplane Model M4

Efficient numerical finite-element analysis of creeping concrete structures requires the use of Kelvin or Maxwell chain models, which are most conveniently identified from a continuous retardation ...

On finite linear viscoelasticity of incompressible isotropic materials

The assumption of incompressibility as a kinematic constraint condition leads to consequences, which are of physical interest in view of a thermodynamically consistent material modelling. Some of these consequences are discussed within the concept of finite linear viscoelasticity. We present two natural possibilities to generalise the familiar Maxwell-model to finite strains; both tensor-valued differential equations are integrated to yield the present Cauchy stress as a functional of the relative Piola or Green strain history. Both types of Maxwell-models are related to a free energy functional in the sense that the dissipation inequality is satisfied. The stress and energy functionals are generalised to incorporate arbitrary kernel functions of relaxation; the only restriction for thermodynamic consistency is that the relaxation functions have a negative slope and a positive curvature. The linear combination of the two types of energy functionals can be understood to be a generalisation of the Mooney-Rivlin model to viscoelasticity. The concrete representation of relaxation functions, motivated from a finite series of Maxwell-elements in parallel, implies a Prony series, corresponding to a discrete relaxation spectrum. A compact notation to express a continuous relaxation spectrum is provided by the concept of a derivative of fractional order. In particular, differential equations of fractional order lead to relaxation functions coming very close to the relaxation behaviour of real materials. As an example, the evolution equation of a fractional Maxwell-model is solved, leading to a relaxation function of the Mittag-Leffler type. An essential advantage is that a rather small number of material constants is involved in this model, while the accuracy of its prediction is quite good. Physically nonlinear effects, such as process-dependent viscosity and relaxation behaviour can be incorporated into finite linear viscoelasticity, if the natural time is replaced with a material-dependent time scale. The rate of the material-dependent time then depends on further internal variables to account for the influence of the process history; this kind of proceeding is fully compatible with the entropy inequality. Temperature-dependence is included, starting from a multiplicative decomposition of the deformation gradient into mechanical and thermal parts. The resulting constitutive theory has a rather simple structure and offers a considerable degree of freedom to allow for physically important phenomena. The numerical simulations illustrate a small selection of these possibilities.

Finite element modeling of transient ultrasonic waves in linear viscoelastic media.

Linear viscoelasticity offers a minimal framework within which to construct a causal model for wave propagation in absorptive media. Viscoelastic media are often described as media with 'fading memory,' that is, the present state of stress is dependent on the present strain and the complete time history of strain convolved with appropriate time-dependent shear and bulk stress relaxation moduli. An axisymmetric, displacement-based finite element method for modeling pulsed ultrasonic waves in linear, homogeneous, and isotropic (LHI) viscoelastic media is developed that does not require storage of the complete time history of displacement at every node. This is accomplished by modeling stress relaxation moduli as discrete or continuous spectra of decaying exponentials and relaxation times. Details of the construction and computation of the time-dependent stiffness matrix are presented. As an application of the finite element method, a finite number of exponentials (amplitudes and relaxation times) are employed to represent a typical model for a continuous relaxation spectrum. It is demonstrated that a small number of discrete exponentials are required to model ultrasonic wave propagation of a typical band-limited pulse in a model material accurately. Previous work has shown this model to be consistent with other analytic models for wave propagation in viscoelastic media [1].

Inflation of polymer melts into elliptic and circular cylinders

A thin sheet (membrane) of the polymeric material is clamped between a Teflon-coated thermostated plate and a thermostated aluminium cylinder. By applying thermostated air through the plate, the polymer membrane deforms into an elliptic or a circular cylinder. The position of the top of the inflating membrane is detected by fibreoptic sensors positioned in the cylinder. The pressure difference across the inflating membrane is measured as well. Measurements were performed on a polyisobutylene melt. As the deformation in this device is highly non-uniform, the response of the material is modelled by a finite element method (the 3D Lagrangian integral method). Here, the non-linear properties are modelled with a constitutive equation of the Factorised K–BKZ type, using a potential function F( 〈 ln u′〉 ), where 〈 ln u′〉 represents the potential function from the Doi–Edwards reptation theory. The linear viscoelastic properties of the material (the time dependence) have been obtained from oscillatory shear measurements and modelled with a continuous relaxation spectrum (BSW) as a memory function, using a leastsquare method. This allows the strain dependence in the constitutive equation to be estimated, based on numerical simulations and experimental measurements of the membrane inflation.

On the dynamic behaviour of polymers under finite strains: constitutive modelling and identification of parameters

In this paper we apply a model of finite viscoelasticity and propose an identification technique to represent the dynamic properties of polymers. The model is based on a multiplicative split of the deformation gradient into a thermal and a mechanical part, the latter being decomposed further into elastic and viscous parts. In order to formulate the constitutive equations we transfer the concept of discrete relaxation spectra to finite strains, specify the free energy as a function of elastic strain tensors and evaluate the dissipation principle of thermodynamics in the form of the Clausius–Duhem inequality. Then we investigate the dynamic moduli of a polyethylene melt under harmonic shear deformations and determine the material parameters. To this end we linearise the constitutive model and calculate the analytical solution of the evolution equations. In addition we formulate a second model which represents the experimental data on the basis of a fairly small number of fitting parameters. This so-called substitute model is based on the fractional calculus and corresponds to a continuous relaxation spectrum. In order to identify the material constants of the finite strain model we are looking for, we proceed as follows. We determine the parameters of the substitute model, calculate the so-called cumulative relaxation spectrum and approximate it by means of a series of step functions: the height of the steps corresponds to the stiffness parameters of the finite strain model and their locations to the inverse relaxation times.

Modeling of the Non-Linear Rheological Behavior of a Lubricating Grease at Low-Shear Rates

This paper deals with modeling the non-linear rheological behavior of lubricating greases at very low shear rates. With this aim, dynamic linear viscoelastic, non-linear stress relaxation, transient and steady-state shear flow, and transient first normal stress difference measurements have been carried out on a diurea-derivative lubricating grease. A factorable non-linear viscoelasticity model, the Wagner integral model, derived from the K-BKZ constitutive equation, was used in order to predict the non-linear rheological response of the above-mentioned lubricating grease under shear. The time-dependent part of the model was described by its linear relaxation spectrum, whilst two different damping functions (Wagner and Soskey-Winter’s damping functions) were analysed as the strain-dependent factor. The continuous linear relaxation spectrum was estimated, using regularization techniques, from the dynamic linear viscoelasticity functions. The damping function was calculated from non-linear stress relaxation tests. The constitutive model, with Soskey-Winter’s damping function, predicted the steady-state flow curve, the transient shear stress and the transient first normal stress differences of the lubricating grease studied fairly well. [S0742-4787(00)01603-9]

The rheological and microstructural characterisation of the non-linear flow behaviour of concentrated oil-in-water emulsions

This paper reports experimental observations on the rheology and microstructure of concentrated oil-in-water emulsions stabilised by macromolecular and low-molecular-weight emulsifiers. From the four different samples that were tested, certain general trends were identified and the measured linear viscoelastic and non-linear viscoelastic properties were compared using a phenomenological factored integral constitutive equation with a continuous relaxation spectrum. The self-consistency of the model was quite good in two cases (vegetable protein and polyoxyethylene glycol non-ionic surfactant emulsifiers) with the general exception of low steady shear rate data where the model overpredicted experimental stress measurements. It is shown that the fitting of the experimental data is sensitive to a wide range of inter-relating factors. In addition, optical observations of the sheared materials consistently showed low shear rate surface slip and this observation was correlated with the rheology miss match. For some systems the optical microstructure studies reveal a range of behaviour for the emulsions dependent on emulsifier composition. Wall slip, microdomain movement, chaining and changes in droplet size distribution were all observed under different conditions and in some cases it has been possible to correlate these microstructure observations with the sample rheology.

Interpretation of shear material properties of vocal fold mucosal tissues with Fung’s quasilinear viscoelastic theory

Fung’s quasilinear viscoelastic theory (QLV) [Y. C. Fung, Biomechanics (Springer-Verlag, New York, 1993), pp. 277–292] has been successfully applied to model the time- and history-dependent properties of a variety of biological soft tissues in the passive state. In particular, the hysteresis or damping curves of many soft tissues remain relatively insensitive to the rate of deformation (strain rate or frequency) across several decades of variation. This insensitivity to frequency cannot readily be accounted for by traditional discrete models of linear viscoelasticity. Rather, Fung’s QLV provides a mathematically powerful framework in which a continuous stress relaxation spectrum can be used to model the relatively flat damping characteristics. The shear properties of human vocal fold mucosal tissues were measured with a rotational rheometer. Data on complex dynamic shear modulus, complex dynamic viscosity, and damping ratio as functions of frequency were obtained. It was found that vocal fold mucosal tissues show the shear-thinning behavior and have a relatively flat damping curve typical of biological soft tissues. Theoretical predictions based on the QLV matched well with the empirical data. The results suggest that the vocal fold mucosa can be modeled by the QLV under small-amplitude oscillation conditions. [Work supported by NIH Grant No. P60 DC00976.]

Effect of a filler on the long‐term mechanical behavior of an epoxy matrix/mineral particle composite

AbstractAn analysis is provided for the continuous relaxation and retardation spectra of epoxy matrix composites filled with different amounts with marble micro‐particles. The particle size distribution is characterized by the diameter sampling medium d50 = 13 μm and the sampling quantile d97 = 80 μm. The concentration of this filler was 10, 29 and 44.5 vol%. The creep compliance and the relaxation modulus were determined from experimental results. Alfrey's method is used to define the continuous spectra from these long‐term (103 days) creep and stress relaxation data. A polynomial approximation is applied to the analytical form of the results. The influence of the filler fraction on the spectra's distribution and on the equilibrium values of relaxation moduli and creep compliances is discussed. The application of discrete relaxation and retardation spectra and the mutual transition between them are demonstrated using a Laplace transform.

Non-Linear response of a long, discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced τE + (t, $$\dot \varepsilon$$ ) and $$\dot \varepsilon$$ − (t, τE) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of λ with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.

Fractional relaxation in anelastic solids

The ordinary relaxation phenomenon exhibiting a pure exponential decay is generalized by replacing the first time derivative by the α-fractional derivative (0 < α ⩽ 1) in the basic equation. Mathematical aspects are discussed with emphasis on the related continuous relaxation spectrum. From the physical point of view the thermoelastic coupling in anelastic solids is considered to take into account a temperature fractional relaxation due to diffusion. A viscoelastic model, formerly introduced by Caputo and Mainardi, is then recovered which generalizes the standard linear solid.

Constants underlying frequency changes in biological rhythmic movements.

When an animal increases or decreases the frequency of its limb motions, how should the transformation in timing be characterized? It has been hypothesized that the transformation is adiabatic, even though the biological conditions are nonconservative and non-rate-limited (Kugler and Turvey 1987). An adiabatic transformation requires that the rhythmic system's action (energy/frequency) and entropy production remain time-invariant throughout the transformation. The non-conservative adiabatic hypothesis was evaluated through an experiment on human rhythmic hand movements. On each trial, a subject began at a prescribed frequency and then, over a 30 s interval, increased (or decreased) the frequency continuously at will. For each subject, on each increasing and decreasing trial, cycle kinetic energy was a linear function of cycle frequency with a negative energy intercept. By the adiabatic hypothesis, the slope of the function defines the constant action and the intercept defines the constant dissipation - changes in cycle frequency incur no changes in energy dissipated per cycle. Slopes and intercepts were correlated suggesting a common basis for the two constants, and the variety of cycle amplitude-cycle duration relations were in agreement with the nonmonotonic, nonlinear space-time function predicted by the hypothesis. The possibilities of addressing aspects of the data through (a) muscle modeled as a continuum of Kelvin bodies with a continuous relaxation spectrum, and (b) various classes of autonomous differential equations, were discussed. Most importantly, the discussion focused on the puzzling independence of energy cost and speed exhibited by locomoting animals differing in morphology, physiology, size, and taxa. It was suggested that the independence may reflect a very general principle - adiabatic transformability of biological movement systems.

The constitutive behaviour of passive heart muscle tissue: A quasi-linear viscoelastic formulation

A quasi-linear viscoelastic law with a continuous relaxation spectrum describing triaxial constitutive behaviour of heart muscle tissue is presented. The elastic respose of the viscoelastic law is anisotropic, while the relaxation behaviour is assumed isotropic. The law is designed for a biphasic description (fluid-solid) of the myocardial tissue. Biaxial and uniaxial stress-strain curves from the literature are used to evaluate the parameters of the model. The non-linear elastic response, the difference between fibre and cross-fibre stiffness, the phenomenon of stress relaxation, the stiffening of the stress-strain relationship with increasing strain rate and the weak frequency dependency of the dissipated energy during cyclic loading are fairly well described by the proposed law. However, it is found that the model produces realistic values for the dissipated energy during cyclic loading only when relaxation parameter values are chosen which result in an overestimation of the stress relaxation data by more than 100%. This finding may indicate non-quasi-linearity of viscoelasticity of passive heart muscle tissue.

Torsional Shock Waves in a Viscoelastic Rod

The propagation of torsional shock waves in a thin circular viscoelastic rod is investigated theoretically. An analysis is carried out based on the approximate equations previously derived. Two typical viscoelastic models are considered, which possess, respectively, the discrete and continuous relaxation spectrum. One is the usual Maxwell- Voigt model and the other is a new model whose relaxation function is given by a power law with weak singularity. The structures of steady shock profiles are presented and compared for both types. Finally a brief discussion is included on the simplified evolution equations for a far field transient behavior.

Direct analysis of continuous relaxation spectra

The dynamic response of a complex biological or chemical system to a perturbation must often be described by an intergral over an effectively continuous relaxation spectrum. Because of its well known instability to experimental error, the direct estimation of the spectrum is generally considered unfeasible. However, we show that good estimates can be obtained by constraining the spectrum to be the smoothest one that is consistent with the data. Also constraining the spectrum to be non-negative, if there is a priori knowledge of this, can further increase its accuracy. The method is completely automatic in that no initial estimates or assumptions about the functional form of the spectrum are necessary. Therefore models can be tested more rigorously and objectively since the functional form that they predict for the spectrum need not be assumed at the outset of the analysis as with parameter-fitting procedures. The method is illustrated on simulated data on the photodissociation of CO from heme proteins at low temperatures. The non-uniqueness of the solutions is discussed.

Continuous relaxation spectrum of dipoles in VC:Vac copolymer films

Continuous relaxation spectrum of dipoles in VC:Vac copolymer films

THE TIME-DEPENDENT MECHANICAL PROPERTIES OF SKIN

The mechanical properties of the skin were investigated by applying a torsional deformation to a circumscribed area. Results of the tests indicate that the skin is nonlinear and time dependent. The dynamic response shows that the phase angle is insensitive to frequency below 1 Hz; the peak torque amplitude increases slowly above 0.004 Hz. A self-consistent description is presented utilizing continuous relaxation spectra. The technique describes the stress relaxation data and predicts the form of the dynamic response.

Velocity dispersion due to anelasticity; implications for seismology and mantle composition

The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann after-effect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the ‘standard linear solid’, and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms. The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.