Modeling the Viscoelastic Behavior of Quartz and Clay Minerals in Shale by Nanoindentation Creep Tests

The viscoelastic behavior of minerals in shales is important in predicting the macroscale creep behavior of heterogeneous bulk shale. In this study, in situ indentation measurements of two major constitutive minerals (i.e., quartz and clay) in Longmaxi Formation shale from the Sichuan Basin, South China, were conducted using a nanoindentation technique and high-resolution optical microscope. Firstly, quartz and clay minerals were identified under an optical microscope based on their morphology, surface features, reflection characteristics, particle shapes, and indentation responses. Three viscoelastic models (i.e., three-element Voigt, Burger’s, and two-dashpot Kelvin models) were then used to fit the creep data for both minerals. Finally, the effects of peak load on the viscoelastic behavior of quartz and clay minerals were investigated. Our results show that the sizes of the residual imprints on clay minerals were larger than that of quartz for a specific peak load. Moreover, the initial creep rates and depths in clay minerals were higher than those in quartz. However, the creep rates of quartz and clay minerals displayed similar trends, which were independent of peak load. In addition, all three viscoelastic models produced good fits to the experimental data. However, due to the poor fit in the initial holding stage of the three-element Voigt model and instability of the two-dashpot Kelvin model, Burger’s model is best in obtaining the regression parameters. The regression results indicate that the viscoelastic parameters obtained by these models are associated with peak load, and that a relatively small peak load is more reliable for the determination of viscoelastic parameters. Furthermore, the regression values for the viscoelastic parameters of clay minerals were lower than those of quartz and the bulk shale, suggesting the former facilitates the viscoelastic deformation of shale. Our study provides a better understanding of the nanoscale viscoelastic properties of shale, which can be used to predict the time-dependent deformation of shale.

RESEARCH AND MATHEMATICAL MODELING OF FRACTIONAL-DIFFERENTIAL RHEOLOGICAL MODELS

Deformation processes in media with fractal structure have been studied. At present, research on the construction of mathematical methods and models of interconnected deformation-relaxation and heat-mass transfer processes in environments with a fractal structure is at an early stage. There are a number of unsolved problems, in particular, the problem of correct and physically meaningful setting of initial and boundary conditions for nonlocal mathematical models of nonequilibrium processes in environments with fractal structure remains unsolved. To develop adequate mathematical models of heat and mass transfer and viscoelastic deformation in environments with fractal structure, which are characterized by the effects of memory, self-organization and spatial nonlocality, deterministic chaos and variability of rheological properties of the material, it is necessary to use non-traditional approaches. -differential operators. The presence of a fractional derivative in differential equations over time characterizes the effects of memory (eridity) or non-marking of modeling processes. The implementation of mathematical models can be carried out by both analytical and numerical methods. In particular, in this paper the integral form of fractional-differential rheological models is obtained on the basis of using the properties of the non-integer integral-differentiation operator and the Laplace transform method. The obtained analytical solutions of mathematical models of deformation in viscoelastic fractal media made it possible to obtain thermodynamic functions, creep nuclei and fractal-type relaxation. Developed software to study the effect of fractional differentiation parameters on the rheological properties of viscoelastic media.

Evaluation of Linear Deformation and Unloading Stiffness Characteristics of Asphalt Mixtures Incorporating Various Aggregate Gradations

Optimum stiffness and linear deformation in the unloading phase are fundamental properties of asphalt mixtures required for the durability of flexible pavements. In this research, blends of six different aggregate gradations were used for two base course (BC) and four wearing course (WC) asphalt mixtures. Stability and indirect tensile strength of resulting asphalt mixtures were evaluated to relate to viscoelastic unloading deformation and resilient moduli (instantaneous (MRI) and total (MRT)) at 25 °C using a 40/50 binder for 0.1 and 0.3 s load durations. Results indicated that an increase in coarse aggregate proportion from 48 to 70% for BC has shown a 12% and 14% increase in MRT for 0.1 and 0.3 s load durations, respectively, and an increase in coarse aggregate proportion from 41 to 57.5% for WC has caused a 26% and 20% increase in MRI for 0.1 and 0.3 s load durations, respectively. The same coarse aggregate proportions showed an increase in linear viscoelastic deformation at 0.1 s load duration from 54.6 to 68.2% for WC and from 53.0 to 62.7% for BC, whereas for 0.3 s load duration linear viscoelastic deformation increased from 58.1 to 69.1% for WC and 64.3 to 69.2% for BC. The findings of this study will assist in the selection of aggregate gradations to be used in wearing and base course asphalt mixtures for pavement design, construction and maintenance.

Application of Fractional Calculus to Modeling the Non-Linear Behaviors of Ferroelectric Polymer Composites: Viscoelasticity and Dielectricity

Ferroelectric polymer composites normally show non-linear mechanical and electrical behaviors due to the viscoelastic and dielectric relaxation of polymer matrixes. In this paper, a fractional calculus approach is used to describe the non-linear behavior of ferroelectric polymer composites from both viscoelastic and dielectric perspectives. The fractional elements for viscoelasticity and dielectricity are “spring-pot” and “cap-resistor”, which can capture the intermediate properties between spring and dashpot or capacitor and resistor, respectively. For modeling the viscoelastic deformation, the “spring-pot” equation is directly used as the fractional mechanical model. By contrast, for the dielectricity of ferroelectric polymer composites, which is usually characterized by dielectric constants and dielectric losses, the “cap-resistor” equation is further formulated into the frequency domain by Fourier transform to obtain the fractional order dielectric model. The comparisons with experimental results suggest that the proposed models can well describe the viscoelastic deformation as well as the frequency dependence of the dielectric constant and dielectric loss of ferroelectric polymer composites. It is noted that the fractional order dielectric model needs to be separated into two regions at low and high frequencies due to the polarization effect. Additionally, when the dipole relaxations occur at higher frequencies, the proposed model cannot describe the rise of the dielectric loss curve.

Spatio-temporal evolution of the Greenland ice sheet and associated deformation of the Earth: a multi-technic geodetic approach

<p>The evolution of the Greenland Ice Sheet (GIS) is an important indicator of climate change and driver of sea level rise. However, providing accurate GIS ice mass balance remains a challenge today. Here, we propose to combine a unique set of geodetic measurements to improve our knowledge of the GIS spatial and temporal evolution. We attempt at reconciling satellite observations of ice volume with regional GNSS velocities estimates and time variable space gravity measurements over the 2003-2009 and 2011-2015 periods. The GIS mass variations are inferred from satellite altimetry for large ice sheets (IceSat and CryoSat-2; Sorensen et al.,2018, Simonsen et al.,2017) and digital elevation models (DEMs) generated from multiple satellite archives for peripheral glaciers (Hugonnet et al.,2020), associated with IMAU-FDM firn model (Ligtenberg et al., 2011). The spatial and temporal variations of the gravity field are given by the GRACE mission for which we use a solution where smaller wavelength signals are preserved (Prevost et al., 2019).</p><p>To resolve short wavelengths load variations affecting the displacement of nearby GNSS stations, we use Green’s functions for vertical crustal displacements assuming purely elastic Earth properties (Martens et al., 2019). We first assume that the deformation is entirely due to recent ice melting and show that vertical elastic displacements predicted by our refined ice loading model, while in good agreement with observations in some regions, cannot explain observations overall. In particular, observations and model disagree in the Southeastern and the Northern parts of Greenland.</p><p>We then explore potential viscoelastic deformation associated with short-term rheology of the asthenosphere induced by recent ice melting that could explain the observed GNSS displacements. We define a history of ice loading from 1900 to 2009 using both in situ and satellite altimetric measurements, compute today’s associated viscoelastic deformation for various mantle rheologies and discuss the potential contribution of ice melting since the little ice age to current observations. Remaining differences between observations and viscoelastic models may reflect a viscoelastic deformation induced by glacial isostatic adjustment. We discuss implications in terms of regional rheological constraints, and impact on estimates of present-day GIS ice mass budget.</p><p>Hugonnet, R. (2020). A globally complete, spatially, and temporally resolved estimate of glacier mass change: 2000 to 2019. In EGU 2020. </p><p>Ligtenberg, S. R. M., et al (2011). An improved semi-empirical model for the densification of Antarctic firn. The Cryosphere, 5, 809-819.</p><p>Martens, H. R.,et al (2019). LoadDef: A Python‐based toolkit to model elastic deformation caused by surface mass loading on spherically symmetric bodies. Earth and Space Science, 6(2), 311-323.</p><p>Prevost, P., et al (2019). Data-adaptive spatio-temporal filtering of GRACE data. Geophysical Journal International, 219(3), 2034-2055.</p><p>Simonsen, S. B., & Sørensen, L. S. (2017). Implications of changing scattering properties on Greenland ice sheet volume change from Cryosat-2 altimetry. Remote Sensing of Environment, 190, 207-216.</p><p>Sørensen, L. S., et al (2018). 25 years of elevation changes of the Greenland Ice Sheet from ERS, Envisat, and CryoSat-2 radar altimetry. Earth and Planetary Science Letters, 495, 234-241.</p>

The early postseismic phase of Tohoku-Oki earthquake (2011) from kinematics solutions: implication for subduction interface dynamics

<p>Earthquakes are usually followed by a postseismic phase where the stresses induced by the earthquakes are relaxed. It is a combination of different processes among which aseismic slip on the fault zone (called afterslip), viscoelastic deformation of the surrounding material, poroelastic relaxation and aftershocks. However, little work has been done at the transition from the co- to the postseismic phase, and the physical processes involved.</p><p>We study the 2011 Mw 9.0 Tohoku-Oki earthquake, one of the largest and most instrumented recent earthquake, using GEONET GPS data. We focus on the few minutes to the first month following the mainshock, a period dominated by afterslip.</p><p>Based on the method developed by Twardzik et al. (2019), we process 30-s kinematic position time series and we use it to characterize the fast displacements rates that typically occur during the early stages of the postseismic phase. We quantify precisely the co-seismic offset of the mainshock, without including early afterslip, and we also characterize the co-seismic offset of the Mw 7.9 Ibaraki-Oki aftershock, which occurred 30 minutes after the mainshock. We analyze the spatial distribution of the co-seismic offsets for both earthquakes. We also use signal induced by the postseismic phase over different time windows to investigate the spatio-temporal evolution of the postseismic slip. We determine the redistribution of stresses to estimate the regional influence of the mainshock and aftershock on postseismic slip.</p><p>From a detailed characterization of the first month of postseismic kinematic time series, we find that the best-fitting law is given by an Omori-like decay. The displacement rate is of the type v<sub>0</sub>/(t+c)<sup>p</sup> with spatial variation for the initial velocity v<sub>0</sub> and for the time constant c. We find a consistent estimate of the p-value close to 0.7 over most of the studied area, apart from a small region close to the aftershock location where higher p values (p~1) are observed. This p value of 0.7 shows that the evolution of the Tohoku-Oki early afterslip is not logarithmic. We discuss about the implications of these observations in terms of subduction interface dynamics and rheology. We also discuss about the different time-scales involved in the relaxation, and how this model, established for the early postseismic phase over one month, performs over longer time scales (by comparison with daily time series lasting several years).</p><p>Twardzik Cedric, Mathilde Vergnolle, Anthony Sladen and Antonio Avallone (2019), doi.org/10.1038/s41598-019-39038-z</p><p><strong>Keywords: </strong>Early Postseismic, Afterslip, GPS, Kinematic, Omori Law</p>