Poynting-Thomson Model Research Articles

The mechanical creep property of shale for different loads and temperatures

The analysis of the mechanical properties and the failure mechanism of shale with the effect of temperature are of great significance to the shale gas exploitation. In this paper, triaxial creep mechanical tests and AE monitoring method for shale under step loading were carried out at different temperatures and confining pressures. The classic Poynting-Thomson creep model was improved and the modified creep model considering temperature and viscoelastic-plastic deformation were built. The test results showed that the strength of the shale was related to the confining pressures and temperatures. The creep mechanism of shale was analyzed by the acoustic emission parameters. The peak frequency of acoustic emission signals reflected the difference of the rock fracture mode. The Poynting-Thomson model was modified on the theory of nonlinear rheology, which is more suitable for creep deformation of shale. The correlation coefficient of the modified model was higher than that of the classical model. The modified model considering temperature fit the shale creep curve well and could explain the creep characteristics under the influence of temperature and stress. The new model can be widely used in the research of shale creep deformation, such as the deep exploitation of shale gas, wellbore stability and other aspects of engineering problem of shale.

Effect of dynamic stiffness of fasteners on vibration and acoustic radiation of a ballastless track

A new model for a single wheel rolling over a metro ballastless track is developed. It is used to carry out the analysis on the effect of dynamic characteristics of fastener on the vibration and noise radiation of the wheel and the track in the vertical direction, under the excitation of the wheel/rail uneven surfaces in detail. In this analysis, a rail is modeled as a Timoshenko beam resting on discrete rubber booted short sleepers, and the sleepers are connected with the slab through linear springs and damping units, the slab is modeled by using the FE method, the fastener is characterized by using the Poynting-Thomson model which takes into account that the stiffness and dumping of the fasteners vary with vibration frequency in their service. The dynamic characteristics of the fastener include the variation of its stiffness and damping with frequency. The analysis considers that the fastener dynamic stiffness increases with the excitation frequency while its damping decreases. The vibration and acoustic radiation of the wheel/track is, to varying degree, affected by the dynamic property of the fastener. The vibration and acoustic radiation of the sleeper and the slab is greatly affected by the dynamic property of the fastener. But the effect of the total noise level of the wheel and the track by the dynamic property of the fastener is not so large because the wheel and rail noise is dominant in the whole analyzed system. These conclusions have certain reference values for the study of the vibration and noise reduction measures of the wheel and track coupled system using the fastening characteristic.

Fractional viscoelastic models with novel variable and constant order fractional derivative operators

This paper deals with the application of a novel variable-order and constant-order fractional derivative without singular kernel of Atangana-Koca type to describe the fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under the variable-order and constant-order fractional derivative without singular kernel. Our results show that the relaxation modulus and creep compliance exhibit viscoelastic behaviors producing temporal fractality at different scales. For each viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under novel variable-order and constant-order fractional derivative with no singular kernel.

Fractional viscoelastic models with non-singular kernels

New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin–Voigt model, fractional Zener model and fractional Poynting–Thomson model. For each fractional viscoelastic model, the stress relaxation modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models under different fractional derivative definitions is further compared. The results show that the fractional Zener model and fractional Poynting–Thomson model are equivalent in all conditions. Compared with the classical fractional derivative definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag–Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.

Creep and Quasi-Relaxation Examination of Artificially Aged Plasticized PVC

Abstract This paper introduces long-term tensile and quasi-relaxation tests of polyvinylchloride (PVC) fibers. During creep, the longitudinal and cross-sectional changes were measured of unaged and aged PVC fibers loaded with the same weight. In the quasi-relaxation section, where deformation was frozen, the stress changes were determined. During the examination of the creep, the Poynting–Thomson model described the phenomenon with sufficient accuracy (min R2 0.9628) and the calculated parameter values characterized the aging process well. In the relaxed phase, the Poynting–Thomson model was not adequate, and, therefore, the second-order time derivatives were also involved in the applied model. Aging significantly resulted in parameter changes in this section as well.

Heat capacities and volumetric changes in the glass transition range: a constitutive approach based on the standard linear solid

A novel approach to represent the glass transition is proposed. It is based on a physically motivated extension of the linear viscoelastic Poynting–Thomson model. In addition to a temperature-dependent damping element and two linear springs, two thermal strain elements are introduced. In order to take the process dependence of the specific heat into account and to model its characteristic behaviour below and above the glass transition, the Helmholtz free energy contains an additional contribution which depends on the temperature history and on the current temperature. The model describes the process-dependent volumetric and caloric behaviour of glass-forming materials, and defines a functional relationship between pressure, volumetric strain, and temperature. If a model for the isochoric part of the material behaviour is already available, for example a model of finite viscoelasticity, the caloric and volumetric behaviour can be represented with the current approach. The proposed model allows computing the isobaric and isochoric heat capacities in closed form. The difference \(c_\mathrm{p} -c_\mathrm{v} \) is process-dependent and tends towards the classical expression in the glassy and equilibrium ranges. Simulations and theoretical studies demonstrate the physical significance of the model.

Creep constitutive models for viscoelastic materials based on fractional derivatives

To describe the time-dependent creep behavior of viscoelastic material, fractional constitutive relation models which are represented by the fractional element networks are studied. Three sets of creep experimental data for polymer and rock are employed to demonstrate the effectiveness of these fractional derivative models. The corresponding constrained problem of nonlinear optimization is solved with an interior-point algorithm to obtain best fitting parameters of these fractional derivative models. The comparison results of measured values and calculated values versus time are displayed through graphics. The results demonstrate that the fractional Poynting–Thomson model is optimal in simulating the creep behavior of viscoelastic materials. And it also shows that the interior-point method is effective in the inverse problem to estimate parameters of fractional viscoelastic models.

Influence of large strain preloads on the viscoelastic response of rubber-like materials under small oscillations

The viscoelastic response predicted by linearized internal variables models in the case of small oscillations superimposed on a large static preload is investigated, comparing simple forms of the Zener and the Poynting–Thomson models. It is shown that both of them predict a preload dependency of the equilibrium linearized stress, but only the latter take into account such a dependency on the out-of-equilibrium part. Yet, the Zener model is much more frequently linearized as the Poynting–Thomson model. The formulation of each model for finite deformations is quickly reminded, before linearizing them around a large static preload. Finally, a comparison of the influence of preload on each model is proposed for uniaxial extension, before discussing which kind of model has to be chosen regarding theoretical and practical aspects.

介电松弛的基本分数模型及其在频率域上的松弛特征

The dynamic process represented by the fractional calculus is in nature dissipative, and the fractional constitutive equations have been successfully applied in describing the viscoelastic phenomena. In this paper, the fractional element for the dielectric relaxation, the cap-resistor, is introduced, and the basic fractional models of dielectric relaxation are constructed. The constitutive equations of the fractional models are derived, and their complex permittivities are presented. Analysis on the relaxation characteristics shows that these fractional models can offer relaxation processes with diverse frequency dependence, and some classical models such as Cole-Cole equation can be regarded as a special case of the fractional model. In addition, the fractional Poynting-Thomson model is used to simulate the dielectric relaxation behavior of glycerol, the fitting results show the model can delineate the dielectric constant and the dielectric loss very well.

Elastic and anelastic properties of α- and b-quartz single crystals

The anisotropic complex Young’s modulus of synthetic quartz single crystals is investigated, using a symmetric three-point bending experimental setup in a Dynamic Mechanical Analysis (DMA) testing system. The storage modulus and the dissipation modulus are examined at ambient pressure between room temperature and 620°C, across the α - β phase transition (573°C). The moduli are determined from measured stress, strain and phase lag data of oriented single crystals at five to ten distinct frequencies between 0.5 and 20 Hz. Quartz shows a strong frequency and temperature dependence of its complex mechanical properties, especially in the vicinity of the α - β phase transition. Around 573°C, the storage modulus steadily increases with frequency, while the dissipation modulus reaches a maximum between 1 and 2 Hz. The observed elastic and anelastic behaviour can be well described by the Poynting-Thomson model, yielding low- and high-frequency limits of the Young’s modulus. This accounts for differences among published data obtained with different experimental techniques, where inconsistent Young’s moduli are reported for similar temperature conditions, especially close to the α - β phase transition. Softening of the Young’s modulus of α -quartz with increasing temperature is accompanied by a decrease of the dynamic viscosity over several orders of magnitude and can be described by two Arrhenian processes. Using published shear moduli and the derived dynamic viscosity, mean relaxation times are derived, which decline from > 1700 s to c -axis and from > 700 s to 0.5 s perpendicular to the c -axis of the crystal, when increasing the temperature from 59°C to 571°C. The ratio of storage to dissipation modulus corresponds to the seismic quality factor Q , a measure of the attenuation of seismic waves. It exhibits a strong frequency and temperature dependence and reaches a minimum at the α - β phase transition. Longitudinal sound wave velocities show significant dispersion of >6 % parallel and >14 % perpendicular to the c -axis of the crystal at 571°C, between 1 and 20 Hz. The characteristics of both attenuation and dispersion of seismic waves close to the α – β phase transition could be used as an in situ temperature probe for quartz-rich crustal rocks.

Theoretical model and solution for the rheological problem of anchor-grouting a soft rock tunnel

The paper presents an analytical method for analysis of the soft rock tunnel, based on a model dividing soft rock tunnels into a region of anchor-grouting and a region of non-anchor-grouting surrounding rock. The Poynting–Thomson model and the Kelvin–Hooke model are applied to the region of non-anchor-grouting surrounding rock and the region of anchor-grouting, respectively. Stress expressions in the region of non-anchor-grouting surrounding rock and the region of anchor-grouting are obtained. Expanding the expression of displacements in the region of anchor-grouting into a Maclaurin series, and utilizing the cumulative displacement curve of the surrounding rock through observation, a theoretical model is set up. This model and its solution for the rheology problem of anchor-grouting a soft tunnel have been proved to be effective in practical engineering; according to the Mohr–Coulomb yield condition, a safe criterion for an anchor-grouting soft rock of a tunnel can be found.

Modelling of a thermo-viscoelastic coupling for large deformations through finite element analysis

In this article we present the mathematical model leading to the equations of the thermoviscoelastic coupling for large deformations. We have chosen a Poynting-Thomson model to describe the rheological behavior of the material. In this model the source term generated by viscous frictions is independent of temperature. The two-dimensional heat transfer equation in Lagrangian form is solved via a finite element method for a two-layer elastomer-metal test piece under a shear force. The results of the numerical model are compared with experimental results which show that the source term cannot be temperature independent. For a more accurate description of the real behavior of the material the source term is then identified from experimental results via an inverse method. The numerical results agree well with the experimental ones.

Generalized viscoelastic models: their fractional equations with solutions

Recently fractional calculus (FC) has encountered much success in the description of complex dynamics. In particular FC has proved to be a valuable tool to handle viscoelastic aspects. In this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models, especially the Maxwell, the Kelvin-Voigt, the Zener and the Poynting-Thomson model. To this end we introduce a fractional element, in addition to the standard purely elastic and purely viscous elements. As we proceed to show, many of the fractional differential equations which we obtain by this construction method admit closed form, analytical solutions in terms of Fox H-functions of the Minag-Leffler type.

Local piezoelectric polarization of human cortical bone as a function of stress frequency

Using a suitable model for the orientation of the collagen fibrils in the osteon system, the piezoelectric constants of the mineralized fibril, including the groundsubstance in dried bone can be calculated from approximately local charge measurements. These frequency-dependent constants are substantially greater than those determined by integrating the charge of areas greater that the osteon system. The measured frequency response of the piezoelectric effect can be described by a Poynting-Thomson model. So it seems very likely that the dielectric displacement is proportional to the strain, independent of deformation frequency.