Maxwell Viscoelastic Model Research Articles

A magnetic field- and frequency-dependent dynamic shear modulus model for isotropic silicone rubber-based magnetorheological elastomers

Both experimental and modeling studies on the dynamic shear storage and loss moduli of isotropic magnetorheological elastomers (MREs) were carried out in this work. Isotropic MREs were prepared based on silicone rubber filled with different contents of micro-sized carbonyl iron particles (CIPs). The magnetization characteristics of the prepared MREs and their dynamic shear moduli at a linear viscoelastic (LVE) strain and under different magnetic flux densities were measured. Based on the magnetization model of MREs, the magnetic dipole theory, and the quantitative analysis on the frequency-dependence of magneto-induced dynamic shear moduli, a new magneto-induced shear modulus model that was dependent on the CIP content and frequency was proposed. The model parameters can be simply determined. Combining with the generalized Maxwell viscoelastic model, that was used to describe the zero-field shear moduli of MREs, a new macroscopic magneto-viscoelastic constitutive model was developed to describe the magnetic field- and frequency-dependent shear modulus of the isotropic MREs with different CIP contents. The developed magneto-viscoelastic constitutive model was verified by comparing the model calculations with experimental data. The results showed that in the LVE strain state, the proposed magneto-viscoelastic model could accurately predict the dynamic shear storage and loss moduli of the isotropic MREs with different CIP contents in broad ranges of frequency and magnetic flux density using the unified model parameters.

Thermoviscoelastic fracture analysis of a cracked orthotropic fiber reinforced composite strip by the dual-phase-lag theory

Fiber-reinforced polymer composites have gained tremendous popularities in various engineering applications due to their superior mechanical performances. This work is devoted to investigating the transient thermomechanical responses of an orthotropic fiber-reinforced strip containing a crack aligned along the fiber direction. The non-Fourier, dual-phase-lag theory is employed to account for the finite time required to complete the process of transporting heat and building local thermal equilibrium in composites, while the Generalized Maxwell viscoelastic model is utilized to characterize the constitutive law of the polymeric matrix and the fibers are treated to be elastic. By using Laplace transform and Fourier transform, the mixed, boundary-value problems are finally reduced to the Cauchy-type singular integral equations, which are solved by the Lobatto-Chebyshev technique numerically. The temperatures and thermal stress intensity factors in the time domain are evaluated through the numerical inversion of Laplace transform. Numerical results are displayed graphically based on graphite fiber reinforced epoxy composite, and detailed comparisons are made to investigate the effects of thermal lags, viscoelastic properties, and fiber volume fractions on the thermoviscoelastic responses.

Simulation of rotary friction welding using a viscoelastic Maxwell model

A 3-D finite element model of rotary friction welding was developed using a viscoelastic Maxwell model. The thermal softening of the material was accounted for in the viscous part of the material model as well as in an additional friction model. The thermal and mechanical material data were adapted from existing data of tempering steel AISI 4140. The final model was used to predict rotary friction welding of shafts using three different parameter settings. The simulation results showed that the melt temperature was never exceeded at the faying surfaces, which is seen as a main characteristic of friction welding. Subsequent validation also demonstrated that the shortening of the specimen as well as the resulting flash geometry could be predicted successfully.

The influence of elastic thickness non-uniformity on viscoelastic crustal response to magma emplacement: application to the Kutcharo caldera, eastern Hokkaido, Japan

SUMMARYAn elastic layer plays an important role in deformation of the crust. At active volcanoes, its thickness would be effectively thinned by a higher geothermal gradient, particularly in a region beneath which magmatic activity is relatively high. This study examines the influence of elastic thickness non-uniformity on viscoelastic crustal deformation by magma emplacement. A 3-D linear Maxwell viscoelastic model is employed, in which an elastic layer underlain by a viscoelastic layer with a spatially uniform viscosity is thinned to be hi in the volcano centre, compared with hi + Δh in the peripheral regions, and a sill-like magma emplacement occurs in the upper layer beneath the centre. It is found that the post-emplacement viscoelastic subsidence is diminished or enhanced by the elastic thickness non-uniformity, depending on whether or not the horizontal width of the magma emplacement (ωs) is greater than the horizontal width (ωe) over which the elastic layer is thinner. The available signature of the non-uniformity is explored by comparison with a model that has a spatially uniform elastic thickness (UET) of hi. If an apparent viscosity (ηa) of the UET model is adjusted so that the difference in post-emplacement subsidence is minimized at the deformation centre, the non-uniformity appears in the overall deformation field as a displacement anomaly over the perimeter of the sill in which viscoelastic subsidence is greater for the non-uniform model. The anomaly is, however, by no more than the magnitude of ∼15 per cent of the maximal syn-emplacement uplift, though ηa is necessarily modified to be ∼0.2–10 times the non-uniform model viscosity (ηc). If ωe is larger than a few times ωs, a weak signature is no longer expected in the deformation field, and ηa is not significantly deviated from ηc. Since the signature appears so faintly in a displacement field, the InSAR data in the Kutcharo caldera for a period from 1993 August 13 to 1998 June 9 do not allow us to capture the non-uniformity. However, it can be concluded that if ωe beneath the caldera is comparable with or greater than the topographic caldera diameter (ωc) as implied by the spatial variation of the geothermal gradient, the non-uniformity has no significant influence. Otherwise, if ωe < ωc, the non-uniformity influences the estimation of the crustal viscosity, but does not affect the overall deformation field. The elastic thickness non-uniformity can be theoretically captured in the deformation field, but in practice, its influence, particularly on estimating crustal viscosity, cannot be properly inferred without other geophysical data such as the geothermal gradient in and around the caldera.

Three-dimensional rolling/sliding contact on a viscoelastic layered half-space

This paper proposes an efficient contact model for a viscoelastic layered half-space where coating and substrate have different creep functions (i.e. different viscoelastic behaviours). The problem is formulated in three dimensions for an imposed pressure and shear field. The influence coefficients are calculated with the Papkovich-Neuber potentials and computation performed using Discrete Convolution and Fast Fourier Transform (DC-FFT) algorithms. From there, an elastic/viscoelastic correspondence is applied to move the elastic solution to a viscoelastic one. The Generalized Maxwell viscoelastic model is used and presented in the form of Prony series. A validation is performed using not only results within the literature but also by comparison with a finite element analysis. Next, a parametric study is done to analyse the influence of different parameters such as the elastic modulus ratio, the relaxation time ratio, the coating thickness or the rolling velocity. The model allows the analysis of the transient regime for which most of the known models struggle. It allows also to calculate 3D stresses and so to extract the stresses at the interface between the coating and the substrate so one can estimate the risk of failure such as delamination.

Research on Anisotropic Viscoelastic Constitutive Model of Compression Molding for CFRP

The carbon-fiber-reinforced polymer (CFRP) is a mainstream material for lightweight products from the end of the 20th century to the present day. Its compression molding process has obvious advantages in mass production. This paper attempts to establish the constitutive models of compression molding of the CFRP materials and study their mechanism. Based on anisotropic linear elastic mechanics, viscoelastic mechanics, and thermodynamics, as well as the Maxwell viscoelastic constitutive model, we first establish the constitutive model of thermorheologically simple CFRP materials (TSMs). Then, considering the influence of temperature on the initial stiffness and equilibrium stiffness, the concept of temperature stiffness coefficient is introduced, and the Cartier coordinate system is converted into a cylindrical coordinate system, thereby establishing the constitutive model of thermorheologically complex materials (TCMs) using the tensor form. Finally, by comparing to the structure of the Zocher model, the two constitutive models established in this study are verified. The research findings have important theoretical research significance for studying the compression molding mechanism of carbon fiber and further improving the quality of product molding.

Measurement of junctional tension in epithelial cells at the onset of primitive streak formation in the chick embryo via non-destructive optical manipulation.

ABSTRACTDirectional cell intercalations of epithelial cells during gastrulation has, in several organisms, been shown to be associated with a planar cell polarity in the organisation of the actin-myosin cytoskeleton and is postulated to reflect directional tension that drives oriented cell intercalations. We have characterised and applied a recently introduced non-destructive optical manipulation technique to measure the tension in individual epithelial cell junctions of cells in various locations and orientations in the epiblast of chick embryos in the early stages of primitive streak formation. Junctional tension of mesendoderm precursors in the epiblast is higher in junctions oriented in the direction of intercalation than in junctions oriented perpendicular to the direction of intercalation and higher than in junctions of other cells in the epiblast. The kinetic data fit best with a simple viscoelastic Maxwell model, and we find that junctional tension, and to a lesser extent viscoelastic relaxation time, are dependent on myosin activity.

Study on Coulomb Stress Evolution Along Altyn Fault

Taking the altyn fault zone as the study area, the maxwell layered viscoelastic medium model and the PSGRN / PSCMP program were used to study the evolution of the cumulative coulomb stress over the altyn fault from 1900 to 2020, analyze the future seismic hazards of faults in the altyn fault zone. The results show that: the minfeng earthquake and changma earthquake were mainly caused by the long-term tectonic loading of the altyn fault. The wuzunxiaoer S8 section and the shulehe 4 S18 south section were affected by the combined effects of the changma earthquake and long-term earthquake loading. In particular, the maximum cumulative coulomb stress of shulehe 4 S18 south section is 2.58Mpa,which is a great danger to strong earthquakes.

Effect of couple stresses on the rheology and dynamics of linear Maxwell viscoelastic fluids

The constitutive analog of the linear Maxwell viscoelastic fluid model with the explicit addition of couple stresses is derived from thermodynamic first principles. Furthermore, the concepts of the recently proposed “Consistent Couple Stress theory” are considered to fully determine the rheological description of the fluid model. Finally, the resulting linear momentum equations are used to investigate the couple stress modulated flow dynamics of a few simple flow configurations. The plots of the variations in the elastic behavior of the fluid due to the presence of couple stresses show that the couple stress acts in addition to the conventional force-stress to increase the diffusion of momentum and thereby enhance the dampening of the elastic deformations in the equivalent spring-dashpot interpretation of the linear Maxwell viscoelastic fluid.

Perturbation Analysis of a Multiple Layer Guided Love Wave Sensor in a Viscoelastic Environment.

Surface acoustic wave sensors have the advantage of fast response, low-cost, and wireless interfacing capability and they have been used in the medical analysis, material characterization, and other application fields that immerse the device under a liquid environment. The theoretical analysis of the single guided layer shear horizontal acoustic wave based on the perturbation theory has seen developments that span the past 20 years. However, multiple guided layer systems under a liquid environment have not been thoroughly analyzed by existing theoretical models. A dispersion equation previously derived from a system of three rigidly coupled elastic mass layers is extended and developed in this study with multiple guided layers to analyze how the liquid layer’s properties affect the device’s sensitivity. The combination of the multiple layers to optimize the sensitivity of an acoustic wave sensor is investigated in this study. The Maxwell model of viscoelasticity is applied to represent the liquid layer. A thorough analysis of the complex velocity due to the variations of the liquid layer’s properties and thickness is derived and discussed to optimize multilayer Surface acoustic wave (SAW) sensor design. Numerical simulation of the sensitivity with a liquid layer on top of two guided layers is investigated in this study as well. The parametric investigation was conducted by varying the thicknesses for the liquid layer and the guided layers. The effect of the liquid layer viscosity on the sensitivity of the design is also presented in this study. The two guided layer device can achieve higher sensitivity than the single guided layer counterpart in a liquid environment by optimizing the second guided layer thickness. This perturbation analysis is valuable for Love wave sensor optimization to detect the liquid biological samples and analytes.

Dynamic characterization of elastomer buffer under impact loading by low-velocity drop test method

Abstract An approach is proposed to establish a discrete model to characterize the dynamic behavior of a thermoplastic polyurethane elastomer buffer under impact loading. To this end, a series of low velocity drop tests with initial impact velocities 1.77–4 m/s was carried out. The impact force versus time histories for the samples under impact loading were measured with varying initial impact velocities. To obtain acceleration, velocity and displacement of the projectile, Newton's second law of motion and the numerical integration method were applied. The stress-strain, coefficient of restitution, loss factor and dissipated energy of samples were determined. Following, the experimental data were used to establish a mathematical model for the elastomer buffer. To reach this aim, Maxwell and Kelvin-Voigt viscoelastic impact models and one-dimensional viscoelastic free decay oscillation model were employed. It was observed that Kelvin-Voigt and one-dimensional viscoelastic free decay oscillation models underestimated the buffer dynamic characteristics. An approach was proposed to calculate the calibrated nonlinear Maxwell viscoelastic impact model. The proposed model consists of a nonlinear dashpot and nonlinear spring. The calibrated nonlinear Maxwell model predicted the elastomer buffer impact behavior in terms of peak force, maximum deflection, dissipated energy and coefficient of restitution with high accuracy.

A Study on the Influences of Abrasive Media's Viscoelasticity on Entrance Effect in Abrasive Flow Machining

Abrasive flow machining (AFM) is a nontraditional surface finishing method that finishes complex surface by pushing the abrasive media flow through the workpiece surface. The entrance effect that the material removal increases at the entrance of changing the cross-sectional flow channel is a difficult problem for AFM. In this paper, the effects of media rheological properties on the entrance effect are discussed. To explore the effects of the media's viscoelasticity on the entrance effect, two sets of media with different viscoelasticity properties are adopted to study their rheological and machining performances in the designed flow channel with a contraction area. The rheological properties are tested by frequency sweep and characterized by the Maxwell viscoelastic model and the Carreau viscous model. In the experiment, the variation of the profile height (ΔH) and the variation ratio of the roughness (ΔRa) on the workpiece surface are measured. Moreover, numerical simulation results under different constitutive equations are compared with the experimental results. It shows that the numerical simulation results of a viscoelastic model have a better agreement with the experimental results than the viscous model, and the increase of the viscoelasticity makes the entrance effect be exacerbated, which can be predicted by the viscoelastic numerical simulation.

DEVELOPMENT OF THE METAL RHEOLOGY MODEL OF HIGH-TEMPERATURE DEFORMATION FOR MODELING BY FINITE ELEMENT METHOD

It is shown that when modeling the processes of forging and stamping, it is necessary to take into account not only the hardening of the material, but also softening, which occurs during hot processing. Otherwise, the power parameters of the deformation processes are precisely determined, which leads to the choice of more powerful equipment. Softening accounting (processes of stress relaxation) will allow to accurately determine the stress and strain state (SSS) of the workpiece, as well as the power parameters of the processes of deformation. This will expand the technological capabilities of these processes. Existing commercial software systems for modeling hot plastic deformations based on the finite element method (FEM) do not allow this. This is due to the absence in these software products of the communication model of the component deformation rates and stresses, which would take into account stress relaxation. As a result, on the basis of the Maxwell visco-elastic model, a relationship is established between deformation rates and stresses. The developed model allows to take into account the metal softening during a pause after hot deformation. The resulting mathematical model is tested by experiment on different steels at different temperatures of deformation. The process of steels softening is determined using plastometers. It is established experimentally that the model developed by 89 ... 93 % describes the rheology of the metal during hot deformation. The relationship between the components of the deformation rates and stresses is established, which allows to obtain a direct numerical solution of plastic deformation problems without FED iterative procedures, taking into account the real properties of the metal during deformation. As a result, the number of iterations and calculations has significantly decreased.

On nonlinear strain theory for a viscoelastic material model and its implications for calving of ice shelves

ABSTRACTIn the current ice-sheet models calving of ice shelves is based on phenomenological approaches. To obtain physics-based calving criteria, a viscoelastic Maxwell model is required accounting for short-term elastic and long-term viscous deformation. On timescales of months to years between calving events, as well as on long timescales with several subsequent iceberg break-offs, deformations are no longer small and linearized strain measures cannot be used. We present a finite deformation framework of viscoelasticity and extend this model by a nonlinear Glen-type viscosity. A finite element implementation is used to compute stress and strain states in the vicinity of the ice-shelf calving front. Stress and strain maxima of small (linearized strain measure) and finite strain formulations differ by ~ 5% after 1 and by ~ 30% after 10 years, respectively. A finite deformation formulation reaches a critical stress or strain faster, thus calving rates will be higher, despite the fact that the exact critical values are not known. Nonlinear viscosity of Glen-type leads to higher stress values. The Maxwell material model formulation for finite deformations presented here can also be applied to other glaciological problems, for example, tidal forcing at grounding lines or closure of englacial and subglacial melt channels.

Simulation of wave propagation in linear thermoelastic media

We have developed a numerical algorithm for simulation of wave propagation in linear thermoelastic media, based on a generalized Fourier law of heat transport in analogy with a Maxwell model of viscoelasticity. The wavefield is computed by using a grid method based on the Fourier differential operator and two time-integration algorithms to cross-check solutions. Because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods, first, a second-order time-splitting algorithm solves the unstable part analytically and a Runge-Kutta method the regular equations. Alternatively, a first-order explicit Crank-Nicolson algorithm yields more stable solutions for low values of the thermal conductivity. These time-stepping methods are second- and first-order accurate, respectively. The Fourier differential provides spectral accuracy in the calculation of the spatial derivatives. The model predicts three propagation modes, namely, a fast compressional or (elastic) P-wave, a slow thermal P diffusion/wave (the T-wave), having similar characteristics to the fast and slow P-waves of poroelasticity, respectively, and an S-wave. The thermal mode is diffusive for low values of the thermal conductivity and wave-like for high values of this property. Three velocities define the wavefront of the fast P-wave, i.e., the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies.

On the Kramers-Kronig relations

We provide a new derivation of the Kramers-Kronig relations on the basis of the Sokhotski-Plemelj equation with detailed mathematical justifications. The relations hold for a causal function, whose Fourier transform is regular (holomorphic) and square-integrable. This implies analyticity in the lower complex plane and a Fourier transform that vanishes at the high-frequency limit. In viscoelasticity, we show that the complex and frequency-dependent modulus describing the stiffness does not satisfy the relation but the modulus minus its high-frequency value does it. This is due to the fact that despite its causality, the modulus is not square-integrable due to a non-null instantaneous response. The relations are obtained in addition for the wave velocity and attenuation factor. The Zener, Maxwell, and Kelvin-Voigt viscoelastic models illustrate these properties. We verify the Kramers-Kronig relations on experimental data of sound attenuation in seabottoms sediments.

Geodetically Inferred Locking State of the Cascadia Megathrust Based on a Viscoelastic Earth Model

AbstractIn a viscoelastic Earth, stresses slowly built up due to fault locking are relaxed concurrently during the entire interseismic period. This interseismic stress relaxation causes crustal deformation much farther away from the locked fault than can be explained using elastic models that neglect the relaxation. Here we develop a viscoelastic geodetic inversion model to address this problem at Cascadia. We invert ~500 horizontal velocity vectors based on continuous and campaign geodetic measurements over the past two decades. Ambiguities arising from long‐term rotation of upper‐plate crustal blocks are addressed by test‐correcting the geodetic velocities with two different block‐motion models. Fault back slip (i.e., slip deficit) Green's functions are derived using a Maxwell viscoelastic finite element model with realistic subduction zone structure and megathrust geometry. The preferred model features a narrow and shallow megathrust locked zone, consistent with earlier thermorheological reasoning. For an elastic model to fit the data to the same fidelity, megathrust locking has to extend to much greater depths. However, even with the viscoelastic model, the land‐based geodetic data still cannot resolve whether there is some creep (incomplete locking) in the shallowest part of the megathrust far offshore. Neither can the land data fully resolve along‐strike variations of the locking state. These ambiguities can be resolved only when adequate seafloor geodetic data are obtained.

The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.

Investigation on indentation rolling resistance of belt conveyor based on Hertz contact theory compared with one-dimensional Winkler foundation

Based on Hertz contact theory and one-dimensional Winkler foundation combination with viscoelastic theory, the author derived theoretical formulas of indentation rolling resistance, respectively. Using the laboratorial apparatus of indentation rolling resistance, the author mainly concentrates on the error analysis about two kinds of theoretical formula which bear on indentation rolling resistance compared with experimental result. The reason why author employs Hertz contact theory to discuss indentation rolling resistance is that indentation rolling resistance is a sort of contact resistance. As a result, Hertz contact theory is generally applicable to study it. On the other hand, because conveyor belt has viscoelastic property, it is appropriate to use viscoelastic theory by the aid of three-parameter Maxwell viscoelastic model combination with one-dimensional Winkler foundation. Ultimately, this article infers that theoretical formula based on the Hertz contact is brief and clear compared with one-dimensional Winkler foundation in principle. However, it is noticeable that when the belt is at high speed, the reliability of formula based on Hertz theory has decreased obviously. This conclusion can give a beneficial reference for the energy saving of belt conveyor.

Effect of pressure and temperature on viscosity of a borosilicate glass

AbstractDuring industrial glass production processes, the actual distribution of stress components in the glass during scribing remains, to date, poorly quantified, and thus continues to be challenging to model numerically. In this work, we experimentally quantified the effect of pressure and temperature on the viscosity of SCHOTT N‐BK7® glass, by performing in situ deformation experiments at temperatures between 550 and 595°C and confining pressures between 100 and 300 MPa. Experiments were performed at constant displacement rates to produce almost constant strain rates between 9.70 × 10−6 and 4.98 × 10−5 s−1. The resulting net axial stresses range from 81 to 802 MPa, and the finite strains range from 1.4% to 8.9%. The mechanical results show that the SCHOTT N‐BK7® glass is viscoelastic near the glass transition temperature at 300 MPa of confining pressure. To elucidate the data, we incorporated both 1‐element and 2‐element generalized Maxwell viscoelastic models in an inversion approach, for which we provide MATLAB scrips. Results show that the 2‐element Maxwell model fits the experimental data well. The stress decreases with increasing temperature at 300 MPa and the temperature dependence yields a similar activation energy (601 ± 10 kJ mol−1 or ∆H/R = 7.2 × 104 K) to a previously reported value at 1‐atm (615 kJ mol−1 or ∆H/R = 7.4 × 104 K). The SCHOTT N‐BK7® glass shows a limited linear increase in viscosity with increasing pressure of ~0.1 log10 (Pa·s)/100 MPa, which is in agreement with the most recent 2‐internal‐parameter relaxation model (based on experiments).