Becker and Lomnitz rheological models: a comparison

The viscoelastic material functions for the Becker and the Lomnitz rheological models, sometimes employed to describe the transient flow of rocks, are studied and compared. Their creep functions, which are known in a closed form, share a similar time dependence and asymptotic behavior. This is also found for the relaxation functions, obtained by solving numerically a Volterra equation of the second kind. We show that the two rheologies constitute a clear example of broadly similar creep and relaxation patterns associated with neatly distinct retardation spectra, for which analytical expressions are available.Comment: 7 pages, 4 figure

A generalization of the Becker model in linear viscoelasticity: Creep, relaxation and internal friction

We present a new rheological model depending on a real parameter $\nu \in [0,1]$ that reduces to the Maxwell body for $\nu=0$ and to the Becker body for $\nu=1$. The corresponding creep law is expressed in an integral form in which the exponential function of the Becker model is replaced and generalized by a Mittag-Leffler function of order $\nu$. Then, the corresponding non-dimensional creep function and its rate are studied as functions of time for different values of $\nu$ in order to visualize the transition from the classical Maxwell body to the Becker body. Based on the hereditary theory of linear viscoelasticity, we also approximate the relaxation function by solving numerically a Volterra integral equation of the second kind. In turn, the relaxation function is shown versus time for different values of $\nu$ to visualize again the transition from the classical Maxwell body to the Becker body. Furthermore, we provide a full characterization of the new model by computing, in addition to the creep and relaxation functions, the so-called specific dissipation $Q^{-1}$ as a function of frequency, which is of particularly relevance for geophysical applicationsComment: 18 pages, 8 figures. arXiv admin note: text overlap with arXiv:1701.0306

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $\nu \in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $\nu=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.Comment: 15 pages, 2 figures, to appear in Chaos, Solitons and Fractals (2017

Generalized Maxwell Love numbers

By elementary methods, I study the Love numbers of a homogeneous, incompressible, self–gravitating sphere characterized by a generalized Maxwell rheology, whose mechanical analogue is represented by a finite or infinite system of classical Maxwell elements disposed in parallel. Analytical, previously unknown forms of the complex shear modulus for the generalized Maxwell body are found by algebraic manipulation, and studied in the particular case of systems of springs and dashpots whose strength follows a power–law distribution. We show that the sphere is asymptotically stable for any choice of the mechanical parameters that define the generalized Maxwell body and analytical forms of the Love numbers are always available for generalized bodies composed by less than five classical Maxwell bodies. For the homogeneous sphere, “real” Laplace inversion methods based on the Post–Widder formula can be applied without performing a numerical discretization of the n–th derivative, which can be computed in a “closed–form” with the aid of the Fa`a di Bruno formula

Modeling sea level changes and geodetic variations by glacial isostasy: the improved SELEN code

We describe the basic features of SELEN, an open source Fortran 90 program for the numerical solution of the so-called "Sea Level Equation" for a spherical, layered, non-rotating Earth with Maxwell viscoelastic rheology. The Sea Level Equation was introduced in the 70s to model the sea level variations in response to the melting of late-Pleistocene ice-sheets, but it can be also employed for predictions of geodetic quantities such as vertical and horizontal surface displacements and gravity variations on a global and a regional scale. SELEN (acronym of SEa Level EquatioN solver) is particularly oriented to scientists at their first approach to the glacial isostatic adjustment problem and, according to our experience, it can be successfully used in teaching. The current release (2.9) considerably improves the previous versions of the code in terms of computational efficiency, portability and versatility. In this paper we describe the essentials of the theory behind the Sea Level Equation, the purposes of SELEN and its implementation, and we provide practical guidelines for the use of the program. Various examples showing how SELEN can be configured to solve geodynamical problems involving past and present sea level changes and current geodetic variations are also presented and discussed

On the viscoelastic characterization of the Jeffreys-Lomnitz law of creep

In 1958 Jeffreys proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys-Lomnitz law of creep by allowing its power law exponent $\alpha$, usually limited to the range [0,1] to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotonic derivative, with a related spectrum of retardation times. The complete range $\alpha \le 1$ yields a continuous transition from a Hooke elastic solid with no creep ($\alpha \to -\infty$) to a Maxwell fluid with linear creep ($\alpha=1$) passing through the Lomnitz viscoelastic body with logarithmic creep ($\alpha=0$), which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys-Lomnitz creep law extended to all $\alpha \le 1$.Comment: 23 pages, 3 figures (5 files .ps

SELEN 4 (SELEN version 4.0): a Fortran program for solving the gravitationally and topographically self-consistent sea-level equation in glacial isostatic adjustment modeling

Abstract. We present SELEN4 (SealEveL EquatioN solver), an open-source program written in Fortran 90 that simulates the glacial isostatic adjustment (GIA) process in response to the melting of the Late Pleistocene ice sheets. Using a pseudo-spectral approach complemented by a spatial discretization on an icosahedron-based spherical geodesic grid, SELEN4 solves a generalized sea-level equation (SLE) for a spherically symmetric Earth with linear viscoelastic rheology, taking the migration of the shorelines and the rotational feedback on sea level into account. The approach is gravitationally and topographically self-consistent, since it considers the gravitational interactions between the solid Earth, the cryosphere, and the oceans, and it accounts for the evolution of the Earth's topography in response to changes in sea level. The SELEN4 program can be employed to study a broad range of geophysical effects of GIA, including past relative sea-level variations induced by the melting of the Late Pleistocene ice sheets, the time evolution of paleogeography and of the ocean function since the Last Glacial Maximum, the history of the Earth's rotational variations, present-day geodetic signals observed by Global Navigation Satellite Systems, and gravity field variations detected by satellite gravity missions like GRACE (the Gravity Recovery and Climate Experiment). The "GIA fingerprints" constitute a standard output of SELEN4. Along with the source code, we provide a supplementary document with a full account of the theory, some numerical results obtained from a standard run, and a user guide. Originally, the SELEN program was conceived by Giorgio Spada (GS) in 2005 as a tool for students eager to learn about GIA, and it has been the first SLE solver made available to the community

Ground motion and stress accumulation driven by density anomalies in a viscoelastic lithosphere. Some results for the Apennines

SUMMARY We provide the analytical formulation for calculating the displacement and stress field forced by internal sources in a stratified, self-gravitating, viscoelastic earth. This model is specialized to study the rate of vertical motion and shear stress accumulation produced by lithospheric density anomalies. These sources are allowed to vary in the lateral direction. We show that sphericity plays a crucial role for elongated lithospheric anomalies while self-gravitation produces minor deviations from a gravitating Earth. When the model is applied to the Apennines we get, for lithospheric viscosity ranging between 10" and loz3 Pas, the subsidence of the plate underlying the active front of the overthrusting load to be around 0.5-1.0 mm yr-'. This is consistent with the amount of sedimentation in the Adriatic foredeep. The deformation pattern is very peculiar, with the largest subsidence localized beneath the active front of the topography. Our model enlightens the impact of discontinuities of tectonic phases on vertical motions in collision zones. If lithospheric viscosity is around 10'' Pa s, vertical motions decay drastically on time scales of 105yr if lateral migration of density anomalies comes to an end. For higher viscosities, deformation rates are maintained longer. This correlation between horizontal and vertical motions suggests that altimetric geodetic surveying along levelling lines of a few hundred kilometers can be an important tool to constrain the tectonics of the studied region. Results are also shown for vertical motions along a transect perpendicular to the Apennines, when the crustal structure inverted from Bouguer gravity data is considered. Analysis of the stress field induced by an overthrusting load shows that principal stress differences of a few bar (or a few tenths of MPa) can be accumulated on time scales of 102-103yr. These low values agree with the average stress drop of earthquakes in the Appalachians and northern Apennines where our modelling can be applied. We find that lateral density variations certainly contribute to intraplate stresses, but they are less efficient in triggering earthquakes than other mechanisms, such as transcurrent motions along active plate margins. Seismicity induced by lateral variations of crustal and lithospheric density must be moderate, characterized by long return times. These results are in agreement with the recorded seismicity in the northern Apennines where the largest earthquakes have return times of lo2 yr. If shear stress is forced by an overthrusting load, we find that the largest rate of stress accumulation in the crust is concentrated beneath the active front, close to the boundary with the ductile lithosphere. Discontinuities of tectonic phases play an important role in controlling the amount of shear stress due to density anomalies

Relationship between the moment of inertia and the k2k_2 Love number of fluid extra-solar planets

Context: Tidal and rotational deformation of fluid giant extra-solar planets may impact their transit light curves, making the $k_2$ Love number observable in the upcoming years. Studying the sensitivity of $k_2$ to mass concentration at depth is thus expected to provide new constraints on the internal structure of gaseous extra-solar planets. Aims: We investigate the link between the mean polar moment of inertia $N$ of a fluid, stably layered extra-solar planet and its $k_2$ Love number, aiming at obtaining analytical relationships valid, at least, for some particular ranges of the model parameters. We also seek a general, approximate relationship useful to constrain $N$ once observations of $k_2$ will become available. Methods: For two-layer fluid extra-solar planets, we explore the relationship between $N$ and $k_2$ by analytical methods, for particular values of the model parameters. We also explore approximate relationships valid over all the possible range of two-layer models. More complex planetary structures are investigated by the semi-analytical propagator technique. Results: A unique relationship between $N$ and $k_2$ cannot be established. However, our numerical experiments show that a `rule of thumb' can be inferred, valid for complex, randomly layered stable planetary structures. The rule robustly defines the upper limit to the values of $N$ for a given $k_2$, and agrees with analytical results for a polytrope of index one and with a realistic non-rotating model of the tidal equilibrium of Jupiter.Comment: Accepted for publication on Astronomy & Astrophysic

Constraining the Internal Structures of Venus and Mars from the Gravity Response to Atmospheric Loading

The gravity fields of celestial bodies that possess an atmosphere are periodically perturbed by the redistribution of fluid mass associated with the atmospheric dynamics. A component of this perturbation is due to the gravitational response of the body to the deformation of its surface induced by the atmospheric pressure loading. The magnitude of this effect depends on the relation between the loading and the response in terms of geopotential variations measured by the load Love numbers. In this work, we simulate and analyze the gravity field generated by the atmospheres of Venus and Mars by accounting for different models of their internal structure. By precisely characterizing the phenomena that drive the mass transportation in the atmosphere through general circulation models, we determine the effect of the interior structure on the response to the atmospheric loading. An accurate estimation of the time-varying gravity field, which measures the atmospheric contribution, may provide significant constraints on the interior structure through the measurement of the load Love numbers. A combined determination of tidal and load Love numbers would enhance our knowledge of the interior of planetary bodies, providing further geophysical constraints in the inversion of internal structure models.The gravity fields of celestial bodies that possess an atmosphere are periodically perturbed by the redistribution of fluid mass associated with the atmospheric dynamics. A component of this perturbation is due to the gravitational response of the body to the deformation of its surface induced by the atmospheric pressure loading. The magnitude of this effect depends on the relation between the loading and the response in terms of geopotential variations measured by the load Love numbers. In this work, we simulate and analyze the gravity field generated by the atmospheres of Venus and Mars by accounting for different models of their internal structure. By precisely characterizing the phenomena that drive the mass transportation in the atmosphere through general circulation models, we determine the effect of the interior structure on the response to the atmospheric loading. An accurate estimation of the time-varying gravity field, which measures the atmospheric contribution, may provide significant constraints on the interior structure through the measurement of the load Love numbers. A combined determination of tidal and load Love numbers would enhance our knowledge of the interior of planetary bodies, providing further geophysical constraints in the inversion of internal structure models