Maxwell Viscoelastic Model Research Articles

Modified and generalized single-element Maxwell viscoelastic model.

In this Letter, the single-element Maxwell model is generalized with respect to the wave vector and extended with a correction function that measures the reduced viscous response. This model has only two free parameters and avoids the attenuation-frequency locking present in the original model. Through molecular simulations it is shown that the model satisfactory predicts the transverse dynamics of the binary Lennard-Jones system at different temperatures, as well as water and toluene at ambient conditions. The correction function shows that the viscous response is significantly reduced compared to the predictions of the original Maxwell model and that there exists a characteristic length scale of minimum dissipation.

Surface loading on a self-gravitating, linear viscoelastic Earth: moving beyond Maxwell

SUMMARY Constitutive laws are a necessary ingredient in calculations of glacial isostatic adjustment (GIA) or other surface loading problems (e.g. loading by ocean tides). An idealized constitutive law governed by the Maxwell viscoelastic model is widely used but increasing attention is being directed towards more intricate constitutive laws that, in particular, include transient rheology. In this context, transient rheology collectively refers to dissipative mechanisms activated in addition to creep modelled by the Maxwell viscoelastic model. Consideration of such viscoelastic models in GIA is in its infancy and to encourage their wider use, we present constitutive laws for several experimentally derived transient rheologies and outline a flexible method in which to incorporate them into geophysical problems, such as the viscoelastic deformation of the Earth induced by surface loading. To further motivate this need, we demonstrate, via the Love number collocation method, how predictions of crustal displacement depart significantly between Earth models that adopt only Maxwell viscoelasticity and those with transient rheology. Throughout this paper, we highlight the differences in terminology and emphases between the rock mechanics, seismology and GIA communities, which have perhaps contributed towards the relative scarcity in integrating this broader and more realistic class of constitutive laws within GIA. We focus on transient rheology since the associated deformation has been demonstrated to operate on timescales that range from hours to decades. With ice mass loss enhanced at similar timescales as a consequence of anthropogenically caused climate change, the ability to model GIA with more accurate constitutive laws is an important tool to investigate such problems.

Dynamic models for impact-initiated stress waves through snow columns

Abstract The objective of this research is to model snow's response to dynamic, impact loading. Two constitutive relationships are considered: elastic and Maxwell-viscoelastic. These material models are applied to laboratory experiments consisting of 1000 individual impacts across 22 snow column configurations. The columns are 60 cm tall with a 30 cm by 30 cm cross-section. The snow ranges in density from 135 to 428 kg m−3 and is loaded with both short-duration (~1 ms) and long-duration (~10 ms) impacts. The Maxwell-viscoelastic model more accurately describes snow's response because it contains a mechanism for energy dissipation, which the elastic model does not. Furthermore, the ascertained model parameters show a clear dependence on impact duration; shorter duration impacts resulted in higher wave speeds and greater damping coefficients. The stress wave's magnitude is amplified when it hits a stiffer material because of the positive interference between incident and reflected waves. This phenomenon is observed in the laboratory and modeled with the governing equations.

Timescales of glacial isostatic adjustment in Greenland: is transient rheology required?

SUMMARY The possibility of a transient rheological response to ice age loading, first discussed in the literature of the 1980s, has received renewed attention. Transient behaviour across centennial to millennial timescales has been invoked to reconcile apparently contradictory inferences of steady-state (Maxwell) viscosity based on two distinct data sets from Greenland: Holocene sea-level curves and Global Navigation Satellite System (GNSS) derived modern crustal uplift data. To revisit this issue, we first compute depth-dependent Fréchet kernels using 1-D Maxwell viscoelastic Earth models and demonstrate that the mantle resolving power of the two Greenland data sets is highly distinct, reflecting the differing spatial scale of the associated surface loading: the sea-level records are sensitive to viscosity structure across the entire upper mantle while uplift rates associated with post-1000 CE fluctuations of the Greenland Ice Sheet have a dominant sensitivity to shallow asthenosphere viscosity. Guided by these results, we present forward models which demonstrate that a moderate low viscosity zone beneath the lithosphere in Maxwell Earth models provides a simple route to simultaneously reconciling both data sets by significantly increasing predictions of present-day uplift rates in Greenland whilst having negligible impact on predictions of Holocene relative sea-level curves from the region. Our analysis does not rule out the possibility of transient deformation, but it suggests that it is not required to simultaneously explain these two data sets. A definitive demonstration of transient behaviour requires that one account for the resolving power of the data sets in modelling the glacial isostatic adjustment process.

Elastic/viscoelastic polymer bilayers: a model-based approach to stretch-responsive constructs.

The use of polymers in the fabrication of bilayers for stimuli-responsive systems is well-known, yet viscoelasticity and viscoelastic models representing bilayer behavior have received surprisingly little attention. Of particular recent interest to us are simple polymeric bilayers in which one material, such as styrene-ethylene-propylene-styrene (SEPS) or styrene-isobutylene-styrene (SIBS), shows typical rubbery elastic response upon extension and retraction, and the other, an unvulcanized, low-Tg polymer such as butyl rubber (butyl), exhibits a viscoelastic response. When such a bilayer strip is extended to a fixed strain and held for several seconds followed by sudden release of this strain, rapid curling is observed, achieving a maximum curvature within 1 second, with a gradual uncurling, typically taking 300-600 seconds to eventually return to a flat strip. Attention has been directed to modeling the observed bilayer behavior. We compare predicted curvature and relaxation time constants from finite element analysis (FEA) simulations using Maxwell, Zener, Generalized Maxwell, and Parallel Rheological Framework (PRF) viscoelastic models to the experimentally measured values. We find that the Generalized Maxwell model predicts curvature over time with the lowest overall mean absolute scaled error (MASE) of 0.519, corresponding to a 4.9% difference from the second lowest error model and a 76.8% difference from the highest error model. Building upon an understanding of the material mechanics in simple bilayer strips, more complex bilayer systems can be designed. Samples of cross and weave geometries were fabricated from bilayer films and initial testing demonstrates how these materials can be used in potential applications.

Understanding stress relaxation behavior of amorphous polystyrene based on microstructural heterogeneity

The relationship between stress relaxation behavior and inherent microstructural heterogeneity in amorphous polystyrene materials is studied in this work. Starting from the basic Maxwell viscoelastic model and extending to the three-parameter stretched exponential equation, the nature of the distribution of characteristic timescales and the segmental effects during polymer stress relaxation are discussed. The results indicate that the stress relaxation behavior of amorphous polymers exhibits non-exponential characteristics. Neither a single characteristic time with exponential decay nor a finite spectrum method with finite characteristic time can adequately describe the stress relaxation behavior of polystyrene due to the continuous distribution of characteristic timescales resulting from microstructural heterogeneity in amorphous polymers. In addition, the changes in stress relaxation behavior caused by physical aging are explored. Aging leads to a transition of the system towards a more stable energy state, making it difficult to activate the relaxation of the individual units, thus slowing down the stress relaxation process and increasing the characteristic time.

Unraveling the rheology of inverse vulcanized polymers

Multiple relaxation times are used to capture the numerous stress relaxation modes found in bulk polymer melts. Herein, inverse vulcanization is used to synthesize high sulfur content (≥50 wt%) polymers that only need a single relaxation time to describe their stress relaxation. The S-S bonds in these organopolysulfides undergo dissociative bond exchange when exposed to elevated temperatures, making the bond exchange dominate the stress relaxation. Through the introduction of a dimeric norbornadiene crosslinker that improves thermomechanical properties, we show that it is possible for the Maxwell model of viscoelasticity to describe both dissociative covalent adaptable networks and living polymers, which is one of the few experimental realizations of a Maxwellian material. Rheological master curves utilizing time-temperature superposition were constructed using relaxation times as nonarbitrary horizontal shift factors. Despite advances in inverse vulcanization, this is the first complete characterization of the rheological properties of this class of unique polymeric material.

A general approach to the mechanical analysis of continuous local inhomogeneity using continuum mechanics theory and a new general energy-based-model

While usually we don't know complete geometrical and physical information on the occurred inhomogeneity, but in order to study the related phenomena, continuum mechanics theory as well as exiting energy-based models need direct information from the desired system in order to A General Approach to the Mechanical Analysis of Continuous Local Inhomogeneity Using Continuum Mechanics Theory and A New General Energy-Based- Modelapply the field equations. Of course, theories and ideas based on the unification of mechanics and thermodynamics can offer other solutions. This paper establishes a general energy based model to study effects of local inhomogeneity on the mechanical behavior of materials using thermodynamic laws with a new deviation as well as new approach to the total energy. In fact, the main goal is that the established model can be used with a wide range of applications and appropriate accuracy, both theoretically and experimentally, while not getting involved with excessive computational complexity. It can be noted that the extracted formulations develop possibility to study inhomogeneity effects on the mechanical behavior without any more limiting conditions. In addition, due to that study of the stored and dissipated energy, usually, has the main role in the investigating of the inhomogeneity effects on the mechanical behavior, also the point of view in the presented model is in complete agreement to provide the conditions for this study directly. Therefore, the prediction possibility of the inhomogeneity effects on the mechanical behavior will be provided with the smallest volume of needed calculations. Also, due to that the structure and properties of the inhomogeneity are unknown usually, the formulations aren't dependent to these knowledge directly, and can be analyzed theoretically and experimentally to study the homogeneity part, as a known material. In the following, feasible processes are studied using extracted formulations. To validate the equations, a rectangular material shape with local inhomogeneity is considered, and extracted equations are developed for that. To consider our mean on the stored and dissipated energies, it is assumed that the homogeneity part of material has viscoelastic behavior, and the equations are developed to Maxwell and Kelvin viscoelastic models in different homogeneity parts of the body. In the following, classical continuum mechanic theory due to elasticity theory is used, and the field equations are developed to the considered body. Finally, results are discussed, compared as well as their differences, and corresponding capabilities for functional completion are discussed. Finally, the fundamental equivalence of the results is studied, and matching of the results between continuum mechanics theory and extracted formulations is shown.

Investigation of bituminized waste products swelling behavior due to water uptake under free leaching conditions: Experiments and modeling

AbstractBituminized waste products (BWPs) were produced by conditioning in bitumen the co‐precipitation sludge resulting from industrial reprocessing of spent nuclear fuel. Underground geological disposal is a solution for the long‐term disposal of some intermediate level long‐lived (ILW‐LL) categorized BWPs in France. After one or several hundred thousand years, the water from the host rock will fully saturate the disposal cells. By an osmotic phenomenon enabled by the semi‐permeable capacity of the bituminous matrix, water in direct contact with BWPs could cause them to swell, which would lead to pressure on the host rock and may damage the disposal facility. Therefore, the BWPs’ swelling behavior has to be taken into account in safety studies for disposal facilities. This paper presents an experimental and numerical investigation of BWPs’ swelling behavior due to water uptake. Experimentally, water uptake, swelling and released salts were monitored for about 4 years during free leaching tests of simplified BWPs. The numerical model presented is extended from an existing one that incorporates transport mechanisms (diffusion, permeation, osmosis) with mechanics via Maxwell's viscoelastic model. An original coupled homogenization of transport terms is proposed to better capture the role of the semi‐permeable membrane played by the bitumen and the porosity dependency of coupled transport coefficients during water uptake. The presented numerical model is able to reproduce the experimental results of free leaching tests and shows its validity for the entire leaching process.

Fractional Maxwell viscoelastic model to explain dynamic magneto-viscoelastic properties of an isotropic magnetorheological elastomer containing flake-shaped magnetic particles

ABSTRACT A modified fractional Maxwell viscoelastic model was developed for the dynamic magneto-viscoelastic behavior of an isotropic magnetorheological elastomer (MRE) having flake-shaped iron particles. This model can capture the low-frequency and high magnetic field static friction contribution with frequency, both in storage and loss modulus. The magnetic dipole-dipole interaction model is used to find the influence of the external magnetic field on viscoelastic behavior. The variation of field-induced modulus with the magnetic field is well described using the Langevin function. The predicted and experimental measured storage and loss modulus agree with the present model with an accuracy of ≥99%. The present model is compared with the results of spherical particle-based MRE.

Stabilized mixed finite element method for a quasistatic Maxwell viscoelastic model

This paper considers a stabilized mixed finite element method (MFE) for a quasistatic Maxwell viscoelastic model based on the L2(Ω)×H1(Ω) variational framework. The spatial discretization uses arbitrary degree piecewise polynomials Pl−Pk(l≥0,k≥1) to approximate stress and velocity. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We give a unified analysis to show the stability of the semi-discrete and fully discrete solutions and derive the corresponding priori error estimates. Numerical examples are provided to support the theoretical analysis.

A new mathematical model of compressive stress-strain behaviour of low viscosity and high viscosity bone cement with different strain rates

A new mathematical model of compressive stress-strain behaviour of low viscosity (LV) and high viscosity (HV) bone cement has been proposed to capture large uniaxial deformation under constant applied strain rate by incorporating three-term power law. The modeling capacity of the proposed model has been validated using uniaxial compressive test under eight different low strain rates ranging from 1.39 × 10-4 s-1 to 3.53 × 10-2 s-1 for low viscosity and high viscosity bone cement. The well agreement between the model and experimental response suggests that the proposed model can successfully predict rate dependent deformation behavior for Poly(methyl methacrylate) (PMMA) bone cement. Additionally, the proposed model was compared with the generalized Maxwell viscoelastic model and found to be in good agreement. The comparison of compressive responses over low strain rates for LV and HV bone cement reveals their rate-dependent compressive yield stress behaviour along with a higher value of compressive yield stress of LV bone cement compared to HV bone cement. For example, at the strain rate of 1.39 × 10-4 s-1 the mean value of compressive yield stress of LV bone cement was found to be 64.46 MPa, whereas for HV bone cement it was 54.00 MPa. Moreover, the modeling of experimental compressive yield stress with the Ree-Eyring molecular theory suggests that the variation of yield stress of PMMA bone cement can be predicted using two processes Ree-Eyring theory. The proposed constitutive model might be useful to characterize large deformation behaviour with high accuracy for PMMA bone cement. Finally, both variants of PMMA bone cement also exhibit ductile-like compressive behaviour below the strain rate of 2.1 × 10-2 s-1, whereas above this threshold strain rate, brittle-like compressive failure behavior is observed.

Evaluation of shock migration performance for a multi-stable mechanical metamaterial

Studies on multi-stable metamaterials mainly focus on the quasi-static performance and the viscoelastic properties of the substrate are ignored, which could cause great deviations in shock migration performance evaluation. In the study, a multi-stable mechanical metamaterial prototype with 2 × 4 cells are investigated and fabricated by thermoplastic polyurethanes (TPU). An analytical, experimental and numerical model was developed to evaluate the dynamic characteristics of the TPU substrate from 0.001 s−1 to 33 s−1, which illustrates significant higher stress levels at higher strain rates. A generalized Maxwell viscoelastic constitutive model was constructed for numerical analysis. Experiments involving quasi-static compression and drop-impact were done to evaluate the prototype's energy absorption and shock reduction capabilities. Quasi-static compression tests and simulations revealed that the peak force of the prototype is about 2.20kN, which presented great repeatability at different loading velocities. The prototype showed significant shock mitigation ability, which could reduce shock acceleration amplitude from 147.02 g to 22.54 g. But the peak reaction force obtained through the acceleration response curve was 3.21 kN, which was 50 % larger than those obtained by quasi-static experiments and simulations. Numerical simulation considering viscoelasticity of the substrate could accurately predict the response of this type of prototype with different shock amplitude, which demonstrates an effective method for the design of protective facilities with specified requirements.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

Study on Mechanical Properties and Penetration Stability of DNAN Based Explosives Under Additive Effects

A series of mechanical experiments and penetration tests, for investigating the effects of additives, on DNAN-based explosives were carried out. At various strain rates, the data result that the stress-strain curves range 10−5s−1-102s−1, were obtained in mechanical experiments by the application of quasi-static and dynamic apparatus. Utilizing the visco-elastic Maxwell model which is calibrated with the experiment data,the dynamic behavior of the explosives could be simulated in penetration process. The results show that the peak stress and displacement of the explosive with binder 1# is smaller than those of the explosive with plasticizer 2# at the same positions. It indicates that the explosive with binder 1# undertake a less severe stress condition than the other, and the explosive with binder 1# may perform more stable. The conclusion of penetration tests which show that both of the additives stabilize the explosive are more effective, coincides with the simulation results.

H∞ Optimization of a Novel Maxwell Dynamic Vibration Absorber with Lever, Inerter, and Grounded Stiffness

In this paper, we propose a novel Maxwell dynamic vibration absorber (DVA) with lever, inerter, and grounded stiffness. Firstly, the governing equation of the coupled system is established. The analytical formula of the amplitude amplification factor of the primary system and the natural frequencies of the coupled system are derived. There are three fixed points in the amplitude–frequency response curve of the primary system, which are independent of damping. Then, based on H∞ optimization criterion, two possible optimal parameter designs of the proposed model are obtained. Considering the practical engineering application and ensuring the stability of the system, the optimal grounded stiffness ratio is selected, and six working ranges of inerter–mass ratio are calculated. Furthermore, the performance of the vibration reduction is compared for six cases. It is found that when the values of the mass ratio, lever amplification ratio, and inerter–mass ratio change in different intervals, and the optimal grounded stiffness ratio has different cases of negative, zero, and positive results. Especially when the stiffness coefficient of the viscoelastic Maxwell model and another grounded stiffness are positive at the same time, the vibration absorption effect is better theoretically. Finally, comparing with the traditional DVAs, the performance of the novel DVA is better under harmonic excitation and random excitation. The results could provide theoretical guidance for the design of inerter-based Maxwell-type DVA with a lever component.

Inference of the Timescale‐Dependent Apparent Viscosity Structure in the Upper Mantle Beneath Greenland

AbstractContemporary crustal uplift and relative sea level (RSL) change in Greenland is caused by the response of the solid Earth to ongoing and historical ice mass change. Glacial isostatic adjustment (GIA) models, which seek to match patterns of land surface displacement and RSL change, typically employ a linear Maxwell viscoelastic model for the Earth's mantle. In Greenland, however, upper mantle viscosities inferred from ice load changes and other geophysical phenomena occurring over a range of timescales vary by up to two orders of magnitude. Here, we use full‐spectrum rheological models to examine the influence of transient deformation within the Greenland upper mantle, which may account for these differing viscosity estimates. We use observations of shear wave velocity combined with constitutive rheological models to self‐consistently calculate mechanical properties including the apparent upper mantle viscosity and lithosphere thickness across a broad spectrum of frequencies. We find that the contribution of transient behavior is most significant over loading timescales of 102–103 years, which corresponds to the timeframe of ice mass loss over recent centuries. Predicted apparent lithosphere thicknesses are also in good agreement with inferences made across seismic, GIA, and flexural timescales. Our results indicate that full‐spectrum constitutive models that more fully capture broadband mantle relaxation provide a means of reconciling seemingly contradictory estimates of Greenland's upper mantle viscosity and lithosphere thickness made from observations spanning a range of timescales.

ON THE INFLUENCE OF VISCOSITY ON THE DEVELOPMENT OF INSTABILITY IN ANOMALOUSLY STRATIFIED GEOSYSTEMS

The paper investigates the question of the influence of viscosity on the rate of development of instability in heavy viscoelastic geosystems with an anomalous density distribution over depth. Geomaterials are considered as incompressible, and the Maxwell model of viscoelasticity (that makes it possible to adequately describe the dynamics of geosystems even in the case of huge viscosity of geomaterials) is adopted for them. Along with viscoelastic systems, the elastic comparison systems are considered, which are characterized by the same anomalous stratification and the same values of shear stiffnesses as the viscoelastic systems studied. Bilateral estimates for the increment of growth of the fastest mode of loss of stability (the “main increment of growth”) are obtained, which are valid for any viscosity. It has been found that the main growth increment always exceeds the main growth increment for the unstable elastic comparison system, which is determined by the inertial properties of the latter together with the degree of supercriticality of the parameters characterizing its unstable equilibrium state. As viscosity increases, the main growth increment for the viscoelastic system tends from above to the corresponding increment for the unstable elastic comparison system. In the case of stability of elastic comparison systems, the corresponding viscoelastic systems are unstable, but the rate of development of their instability is very low due to the great viscosity of geomaterials. Precisely and only in the case of stability of comparison systems, the viscosity slows down this rate, and for geosystems with extremely great viscosity, the manifestation of their instability during real observation times is very small. The analysis carried out confirms the fact that the study of stability and instability of elastic comparison systems can serve as a methodological basis for studying stability and instability of viscoelastic geosystems.

Non-Maxwellian viscoelastic stress relaxations in soft matter.

Viscoelastic stress relaxation is a basic characteristic of soft matter systems such as colloids, gels, and biological networks. Although the Maxwell model of linear viscoelasticity provides a classical description of stress relaxation, it is often not sufficient for capturing the complex relaxation dynamics of soft matter. In this Tutorial, we introduce and discuss the physics of non-Maxwellian linear stress relaxation as observed in soft materials, the ascribed origins of this effect in different systems, and appropriate models that can be used to capture this relaxation behavior. We provide a basic toolkit that can assist the understanding and modeling of the mechanical relaxation of soft materials for diverse applications.

Using asymptotic homogenization to determine effective thermo-viscoelastic properties of fibrous composites with interphase layer

In this paper, we apply the method of asymptotic homogenization in parametric space in order to determine thermo-viscoelastic properties of materials composed of cylindrical fibers surrounding by an intermediate layer separating them from viscoelastic matrix. We use the Maxwell model of viscoelasticity. The volumetric fraction and physical properties of the layer can vary. There is a possibility of a contrast between the properties of fibers, the layer, and the matrix. Our approach allows us to treat temperature as a parameter and study the response of mechanical characteristics on the change in temperature. We perform the Fourier transform of governing equations and then implement asymptotic homogenization of the resulting equations in parametric space. Microscopic boundary value problem is defined on a periodic cell consisting of a fiber surrounded by a layer and embedded in the matrix. Macroscopic equations are derived for the complex Young’s and shear moduli. Effective viscoelastic properties of the material are computed for different values of the frequency of harmonic oscillations and for various temperatures. This paper discusses how effective storage and loss Young’s and shear moduli depend on volumetric fraction and mechanical properties of the material of the layer. The obtained results can provide information on how concentration, structure and properties of inclusions can affect the effective storage and loss Young’s and shear moduli.