Maxwell Time Research Articles

Viscosity, relaxation and elastic properties of photo-thermo-refractive glass

Abstract Viscometry, ultrasonic echography and mechanical spectroscopy were applied to explore the thermo-mechanical properties of a photo-thermo-refractive (PTR) glass. Newtonian shear viscosity and ultrasonic velocity were used to determine the Maxwell time of structural relaxation. Two fast relaxation modes are found in PTR glass below glass transition temperature, which are attributed to alkali and mixed cation mobility. Respective activation energies, Eγ = 72 ± 4 kJ mol−1 and Eβ = 117 ± 4 kJ mol−1, are considerably lower than that of viscous flow at the glass transition Eα ≈ 465 kJ mol−1 as quantified from mechanical spectroscopy. Decoupling ratios of Eγ/Eα = 0.155 and Eβ/Eα = 0.255 are similar to those of sodium trisilicate glass but at considerably longer timescales. These observations imply that β- and γ-relaxations are delayed as local structural arrangements are more complex for the multi-component PTR glass.

Displacement, strain and stress fields due to shear and tensile dislocations in a viscoelastic half-space

Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.

Large time step numerical modelling of the flow of Maxwell materials

SUMMARY Maxwell viscoelastic materials are commonly simulated numerically in order to model the stresses and deformations associated with large-scale earth processes, such as mantle convection or crustal deformation. Both implicit and explicit time-marching methods require that the time steps used be small compared with the Maxwell relaxation time if accurate solutions are to be obtained. For crustal tectonic modelling, where Maxwell times in a ductile lower crust may be of order of a decade or less, the large number of time steps required to model processes lasting many millions of years imposes a huge computational burden. This burden is avoidable. In this paper I show that, with the appropriate formulation of the problem, time steps may be taken which are much larger than the Maxwell time without loss of accuracy, as long as they are not large compared with the times over which strain rates vary significantly (‘tectonic’ timescales) in the model. The method relies on explicit analytic integration of the Maxwell constitutive relation for the stress over time intervals, which may be longer than the relaxation time as long as they are short compared with the timescale over which crustal stresses and geometries change. The validity of the formulation is also demonstrated numerically by comparison with the analytic solutions for three simple plane-strain models: extension of a uniform block, shear of a composite layer and development of a Rayleigh‐Taylor instability.

Then and now: James Clerk Maxwell and colour

A survey of the researches of James Clerk Maxwell into the production and perception of colour is presented, and links to a selection of present day applications of colour science are indicated. Although false colour photography was not known in Maxwell's time, aspects of his work, suitably interpreted, have a bearing on the wide applicability of the technique. An outline is given of one application of false colour to visualising and analysing data on fluid motion.

Sublinear dispersive conductivity in polymethyl methacrylate at temperatures above the glass transition

Dynamic electrical analysis shows that conductivity conditions the electrical properties of a polymer at high temperatures and low frequencies. In this paper we show the possibilities of the electric modulus formalism to study the properties of carrier transport and space charge relaxation processes in polymethyl methacrylate. Asymmetric Argand's plots are observed in the temperature range between 150 and 210 °C. This asymmetry is related to power-law dependencies in the real part of the conductivity, of the form ω n with n<1, as a result of correlated ion hopping. The complex part of the electric modulus exhibits a peak in the low frequency range that can be associated with these conductive processes. In the case of the relaxation time related to conductive processes, it has been observed that Maxwell time is higher than the characteristic time associated with the crossover frequency which determines the transition from dc to ac regime. This fact is explained by the presence of deep traps. Finally, a maximum in the value of n is observed between 180 and 190 °C which may be related to coupling between charge transport and chain segment motions.

Transfer matrix and full Maxwell time domain analysis of nonlinear gratings

In this paper we show that the transfer matrix method and the finite difference time domain technique applied to the full time-dependent Maxwell equations (in their one-dimensional form) permit to improve the description of nonlinear distributed feedback structures such as Bragg gratings. In fact, no simplifying hypothesis are introduced in the electromagnetic problem by these two numerical approaches. The two techniques validate qualitatively the results based on the standard coupled-mode theory for weakly nonlinear and shallow gratings, while significant quantitative deviations can appear in other cases.

Seasonal Seismicity of Northern California Before the Great 1906 Earthquake

During the late nineteenth century, coastal parts of northern California experienced seismicity which was more intense than at any other time in its historical record and was concentrated in Spring. The seasonal character of this anomalous seismicity was statistically significant from the 1880s, if not earlier, until the time of the great 1906 earthquake. It can be explained as a result of seasonal stress transients caused by the large seasonal sediment loads which accumulated in coastal bays that overlie active strike-slip faults, as a result of hydraulic mining of gold in inland areas. Such localised loading will increase the absolute vertical stress and reduce the absolute horizontal stress, thus facilitating shear failure by unclamping vertical fault planes. However, rather than simply influencing the timing of earthquakes which would otherwise have occurred anyway sometime later, this process appears to have affected the character of the region’s seismicity, causing a sequence of magnitude 5–6 earthquakes which propagated outward from these coastal depocentres. These results have three important implications. First, the proposed physical mechanism is only feasible provided the deepest continental crust in this region has a viscosity no greater than ~1017 Pa s, giving it a Maxwell time of the order of weeks. These observations of seasonal seismicity thus suggest that such low viscosities are reasonable, a conclusion which has been difficult to establish using other evidence. They indicate that future calculations of the stress field in this region, for instance for investigating the role of earthquake triggering, should incorporate this response by lower-crustal flow, not just the immediate elastic effect. Second, the conclusion that seasonal stress transients caused by seasonal changes in surface processes can affect seismicity, has important implications for hazard assessment. It raises the possibility that surface processes may influence the seasonal timing and character of seismicity in other regions also. Finally, it indicates that the decades of intense seismicity, widely interpreted as precursory activity to the 1906 earthquake, were atypical of the longer-term record. Analogous sequences of moderate-sized events cannot necessarily be expected to provide advance warning of future large earthquakes in this region.

Viscoelastic simulation of earthquake cycle using a simple spring‐Dashpot‐mass system with a friction law

We examine the viscoelastic effect on earthquake cycle using a simple spring‐dashpot‐mass system with a laboratory‐derived friction law. The temporal change of the frictional stress is large at the early interseismic stage and gradually decreases, in contrast to the linear change in an elastic medium. This temporal change depends on the Maxwell time. The smaller relaxation time leads to the larger change of the stress accumulation rate. Such a rate change in stress accumulation may affect the seismicity at subduction zones. Moreover, we find that the smaller relaxation time produces the smaller period of earthquake cycle and that, if the system relaxation time becomes smaller than the period of earthquake cycle, the bifurcation phenomenon, where the period splits into two, occurs. It is also found that the smaller relaxation time causes the earlier occurrence of the precursory stress decrease preceding the rapid stress drop.

High‐order, leapfrog methodology for the temporally dependent Maxwell's equations

A high‐order, leapfrog integrator is presented and analyzed for Maxwell's time domain equations. The integrator is shown to be quite robust in terms of its efficiency and memory demands. Moreover, the integrator retains the original simplicity of the traditional second‐order, leapfrog integrator. Rigorous Fourier analyses are presented to quantify the dispersion, dissipation, and stability properties of the integrator. To limit the scope of the presentation, the 2×2, 2×4, 4×2, and 4×4 schemes are examined and compared. Here the first digit denotes temporal accuracy; the second digit denotes spatial accuracy. Numerical demonstrations, using the three‐dimensional rectangular waveguide as the object under test, are provided to further substantiate the theoretical analysis.

Lithospheric Loading by the Northern Polar Cap on Mars

New topography data for the northern polar region on Mars, returned by the the Mars Orbiter Laser Altimeter (MOLA) during the aerobraking hiatus and science phasing orbits, allow characterization of the topography of the present northern polar cap and its environs. Models for loading of an elastic shell by an axisymmetric load approximating the present polar cap geometry indicate that the maximum deflection of the subice basement is in the range 1200 to 400 m for an elastic lithosphere of thickness 40 to 200 km overlain by a cap of pure H 2O ice. Corresponding model cap volumes increase from 1.5 to 1.8 × 10 6 km 3, as elastic lithosphere thickness decreases from 200 to 40 km. The presence of sediments in the polar cap increases the depth to basement and resulting cap volume for a given value of elastic lithosphere thickness. One-dimensional heat flow calculations indicate that the temperature at the base of the cap may approach the melting point of cap material if the lithosphere underlying the cap is thin. The basal temperature is 170 K for a 200-km-thick lithosphere overlain by pure ice but is as great as 234 K for a 40-km-thick lithosphere overlain by a cap with a high sediment/ice ratio. Constraints on elastic lithosphere thickness are weak, but geologic mapping and MOLA data suggest that a flexurally derived circumpolar depression filled with sediments is consistent with elastic lithosphere thickness values in the range 60–120 km. Gravity and topography over the whole cap are poorly correlated, possibly due to viscous relaxation of long-wavelength topography, but gravity and topography over the western portion of the main cap are consistent with an elastic lithosphere thickness of 120 km, for a crustal thickness of 50 km. Both MOLA data and geological information suggest a formerly larger northern polar cap. The relationship of time scales for changes in the polar cap volume and extent to time scales for viscous relaxation of topography has important implications for investigations of even present polar cap topography. Viscoelastic calculations show that the Maxwell time for an Earth-like mantle viscosity for Mars (10 21 Pa-s) is 10 5 yr. The Maxwell time scales directly with the martian mantle viscosity so that values as high as 10 7 yr are possible. Time scales for changes in polar cap volume are poorly constrained, but major changes in cap volume over periods of 10 6–10 8 yr are consistent with current understanding of polar cap processes.

Reconciling rapid strain accumulation with deep seismogenic fault planes in the Ventura Basin, California

Global Positioning System measurements across the east central Ventura basin, Transverse Ranges, southern California, before the nearby 1994 Northridge earthquake show high strain rates. Interpreting this rapid strain accumulation using the usual model of deep slip on a dislocation in a uniform elastic half‐space requires slip to extend to within 5 km of the surface. Such shallow slip is difficult to reconcile with the substantial coseismic displacement at depths from 7 to 19 km during the Northridge earthquake. Here we model the displacement and velocity fields throughout the earthquake cycle using a two‐dimensional finite element model with a viscoelastic rheology. Displacements are driven by far‐field and basal velocity boundary conditions and by imposed periodic earthquakes on the thrust faults bounding the basin. The thrust faults rupture through an elastic upper crust to a depth of 15 km. After a transient stage, during which stresses and strains build up to quasi‐equilibrium values, the behavior of the model becomes periodic. The sum of the coseismic displacement divided by the repeat interval, plus the average interseismic velocity, is equal to the geologic velocity. The temporal variation in surface velocity depends mainly on the Elsasser relaxation time (proportional to the product of the Maxwell time of the lower crust and the ratio of the thicknesses of the entire crust and viscoelastic lower crust). We are able to match the observed high strain rate only if we include the observed variations in elastic modulus associated with the deep basin sediments. The model reconciles geologic, geodetic, and seismological observations of deformation. There are trade‐offs among the far‐field convergence rate, the Elsasser time, the earthquake repeat time, and the time into the earthquake cycle. Acceptable convergence rates range from 8 mm/yr, for a relaxation time of the lower crust of 300 years, to 12 mm/yr, for a 30‐year relaxation time.

Simulation of postseismic deformations caused by the 1896 Riku‐u Earthquake, northeast Japan: Re‐evaluation of the viscosity in the upper mantle

We examine the postseismic deformations which are induced by an inland earthquake at a subduction zone, and investigate especially the effect of the viscoelastic heterogeneity through Finite Element Method (FEM). We estimate the viscosity of the upper mantle beneath the northeast Japan with realistic structures based on the leveling data which includes the postseismic deformations due to the 1896 Riku‐u earthquake. For this purpose, we construct two 3‐D FEM models with different viscoelastic structures, namely Layered Model and Plate Model. The results show that spatial patterns of the postseismic deformations differ considerably, depending on the two viscoelastic structures. Postseismic deformations are strongly controlled by the effective thickness of the viscoelastic medium above descending plate. At the surface, if the structure is complicated, the relaxation time is outwardly longer and varies significantly with position. Considering the plate structure, the profile of surface deformation and decaying subsidence rate constrain the Maxwell time of the upper mantle to be 5 years, which is shorter than the value of 10 years estimated by Thatcher et al. [1980] assuming the layered structures.

Nonlinear response of photorefractive lithium tantalate and niobate at acoustic frequencies

An amplification of the intensity of pump oscillations is observed experimentally at frequencies from 100 Hz to 1 kHz during photoinduced light scattering and holographic-type parametric scattering in photorefractive lithium tantalate and niobate. Possible ways are analyzed for explaining the existence of a photorefractive response in these crystals over times of 10−2–10−3 s, which are five orders of magnitude shorter than the Maxwell time.

Slow Transients and Metastability in Wormlike Micelle Rheology

The steady-state nonlinear rheology of wormlike micellar systems is thought to be subject to shear banding (the underlying shear stress vs. strain rate curve σ(γ) is nonmonotonic). Shear banding may result in a plateau (σ(γ) = σp) in the measured flow curve (at controlled mean strain rate γ). We present new rheological data for aqueous CPyCl/NaSal (100 mM /60 mM). Steady-state flow curves published previously for this system (Rehage H. and Hoffmann H., Mol. Phys. 74 (1991) 933) have since been interpreted as shear-banded flow with top-jumping, in which the steady-state shear rate γ1 in the low shear band is the largest possible (γ 1 = γ 1 max , σ p = σ max ). That would rule out the existence of a metastable branch with a stress larger than σp. We show that such a branch does, however, exist (for temperatures in the range 20 - 25 °C). Similar results are found for a 100 mM/75 mM system. The time scale for relaxation of a metastable state onto true steady state flow, τ ss , is far longer than the Maxwell time of the fluid; this is consistent with shear banding. We observe τ ss ∼ (γ - γ c ) -P in the metastable regime (γ > γ 1 ), with p an exponent that depends on composition and temperature. The critical shear rate γ c is in some cases less than γ 1 so that no actual divergence of τ ss occurs. In at least one case, though, there is evidence for a physical divergence (γ c > γ 1 ) accompanied by a small window of shear rates, γ 1 < γ < γ c , for which τ ss is effectively infinite. In some respects the observed behaviour resembles that reported previously (Berret J.-F., Roux D.C. and Porte G., J. Phys. II Prance 4 (1994) 1261) for equimolar CPyCl/NaSal in 0.5 M NaCl. Those results were interpreted in terms of nucleation and growth of a shear-induced nematic phase. However the same explanation is unlikely for the low weight fractions (Φ ≤ 5%) used in our study.

Some analytical solutions for the viscoelastic gravitational relaxation of a two-layer non-self-gravitating incompressible spherical earth

SUMMARY In the study of the postglacial isostatic readjustment process, deformation of the Earth is usually expressed as the superposition of normal modes. In order further to understand the properties of these normal modes in a stratified earth, analytical solutions for incompressible, non-self-gravitating, two-layer spherical Maxwell earth models are derived and presented. Since the main effect of non-self-gravitation is to increase the influence of gravity for small harmonics of deformation (n) and the number of eigenmodes is not affected, these analytical solutions are used to establish theoretically the number of eigenmodes for a particular earth model. It is found that a jump in density will introduce a buoyancy mode, whereas a discontinuity in the value of Maxwell time will introduce two viscoelastic (or ‘transition‘) modes simultaneously. However, for an elastic lithosphere, only one extra mode is introduced. These analytical solutions also demonstrate that unstable eigenmodes exist whenever there is a density inversion. However, complex eigenmodes (which correspond to damped oscillations) are not found. Negative excitation strengths are also shown to be possible for the horizontal displacements—their magnitudes are generally small. A notable exception is when an elastic lithosphere is present, in which case the excitation strength for the dominant mode can be negative. Finally, a very simple method has been introduced to remove the problem of dense singularities during the numerical search for the eigenspectrum when the earth model is composed of a large number of uniform layers.

Cyclic stressing and seismicity at strongly coupled subduction zones

We use the finite element method to analyze stress variations in and near a strongly coupled subduction zone during an earthquake cycle. Deformation is assumed to be uniform along strike (plane strain on a cross section normal to the trench axis), and periodic earthquake slip is imposed consistent with the long‐term rate of plate convergence and degree of coupling. Simulations of stress and displacement rate fields represent periodic fluctuations in time superimposed on an average field. The oceanic plate, descending slab, and continental lithosphere are assumed here to respond elastically to these fluctuations, and the remaining mantle under and between plates is assumed to respond as Maxwell viscoelastic. In the first part of the analysis we find that computed stress fluctuations in space and time are generally consistent with observed earthquake mechanism variations with time since a great thrust event. In particular, trench‐normal extensional earthquakes tend to occur early in the earthquake cycle toward the outer rise but occur more abundantly late in the cycle in the subducting slab downdip of the main thrust zone. Compressional earthquakes, when they occur at all, have the opposite pattern. Our results suggest also that the actual timing of extensional outer rise events is controlled by the rheology of the shallow aseismic portion of the thrust interface. The second part of the analysis shows the effects of mantle relaxation on the rate of ground surface deformation during the earthquake cycle. Models without relaxation predict a strong overall compressional strain rate in the continental plate above the main thrust zone, with the strain rate constant between mainshocks. However with significant relaxation present, a localized region of unusually low compressional, or even slightly extensional, strain rate develops along the surface of the continental plate above and somewhat inland from the downdip edge of the locked main thrust zone. The low strain rate starts in the middle or late part of the cycle, depending on position. This result suggests that the negligible or small contraction measured on the Shumagin Islands, Alaska, during 1980 to 1991, may not invalidate an interpretation of that region as being a moderately coupled subduction zone. In contrast, mantle relaxation causes only modest temporal nonuniformity of uplift rates in the overriding plate and of extensional stress rates in the subducting plate, even when the Maxwell time is an order of magnitude less than the recurrence interval.

Steady and dynamic shear properties of non-aqueous drag-reducing polymer solutions

The steady and dynamic shear properties of two non-aqueous drag-reducers (a medium molecular weight polyisobutylene and a commercial organic drag-reducer) in kerosene solutions over a wide range of temperature and concentration were presented. The intrinsic and zero-shear viscosity results were used to identify the concentrate regimes of these solutions. A characteristic time constant λ0, which was based on the spring-bead model for dilute solutions, was employed as the scaling parameter for both steady-shear and dynamic data over a wide range of concentration and temperature. The inadequacy of the Graessley reduced-variable method in the dilute region was illustrated. The shear-thinning behaviour of these polymer solutions could be described by the Carreau model. The dynamic data followed the Zimm and Rouse-like behaviour in the low and high frequency limits. The Cox-Merz rule was obeyed in the low shear rate and frequency regions. The Carreau and the zero-frequency Maxwell time constants appeared to be related to λ0 by a constant factor over a wide range of polymer concentrations. The finding provides a method for extrapolating viscoelastic information into the drag reduction regime, and could be useful for interpretation of drag reduction results.

FD-TD analysis of Vivaldi flared horn antennas and arrays

This paper summarizes a detailed computational study of Vivaldi flared horn antenna designs including single element, double element crossed-pair subarray elements, and linear arrays using the finite-difference time-domain (FD-TD) method. FD-TD, which numerically solves Maxwell's time dependent curl equations, accounts for the complex geometrical and electrical characteristics associated with this antenna design and array implementation. Validations have been made relative to a moment method (MM) model of an electrically large linearly tapered slot antenna. Also, good correlation is shown to exist in the primary features of the antenna patterns between computed and measured data for all cases. This work has achieved a number firsts for the FD-TD methodology. It represents the first exact computational model of a single quad Vivaldi slot antenna; the first exact computational model of a phased array of Vivaldi quad elements; and the first FD-TD model to demonstrate grating lobes for a phased array antenna of any sort. Lastly, this research represents an extensive study of the largest grid-based antenna models conducted to date. >

Nonequilibrium transport and slow relaxation in hopping conductivity.

We present experimental results that demonstrate the nonergodic nature of charge transport in the insulating regime of indium oxide samples. These results include an anomalous field effect described in detail by Ben-Chorin et al. and persistent photoconductivity created by exposure to light. The similarity of the temporal dependence of the conductance after excitation due to a burst of light and that due to charging the sample by a nearby gate suggests that in both processes the electronic system is excited and the time it takes the system to reach thermal equilibrium is much longer than the Maxwell time. We offer a simple theoretical model that ascribes all of these effects to nonequilibrium transport phenomena peculiar to the hopping regime. It is argued that exciting a hopping system out of thermal equilibrium leads to a conductivity that is higher than in equilibrium. The excited state is long lived and similar in nature to that observed in the phenomenon of persistent photoconductivity of various semiconductors measured at low temperatures. The sluggish equilibration process of the electronic system is ascribed to the inhomogeneous nature of charge transport and to the slow energy relaxation which are inherent features of disordered solids.

Orientational ordering as a possible mechanism for viscosity-enhancement of supercooled liquids

A supercooled liquid is viewed to have regions of local orientational order which can be pictured in terms of cages that restrict single-particle diffusion. The mismatch in the orientation of two locally ordered neighbouring regions causes an internal stress which is added to the stress that appears in the Maxwell model of viscoelasticity. This leads to a renormalized Maxwell time which is related to the relaxation time and the susceptibility associated with the orientational order. Hence, when either the relaxation time or the susceptibility becomes very large, one obtains a large enhancement of the viscosity.