Strain Rate Tensor Research Articles

Numerical analysis of stress-state-dependent damage and failure behavior of ductile steel based on biaxial experiments

The paper deals with a numerical model to investigate the influence of stress state on damage and failure in the ductile steel X5CrNi18-10. The numerical analysis is based on an anisotropic continuum damage model taking into account yield and damage criteria as well as evolution equations for plastic and damage strain rate tensors. Results of numerical simulations of biaxial experiments with the X0- and the H-specimen presented. In the experiments, formation of strain fields are monitored by digital image correlation which can be compared with numerically predicted ones to validate the numerical model. Based on the numerical analysis the strain and stress quantities in selected parts of the specimens are predicted. Analysis of damage strain variables enables prediction of fracture lines observed in the tests. Stress measures are used to explain different stress-state-dependent damage and failure mechanisms on the micro-level visualized on fracture surfaces by scanning electron microscopy.

Vortex stretching and enstrophy production in high Reynolds number turbulence

An essential ingredient of turbulent flows is the vortex stretching mechanism, which emanates from the non-linear interaction of vorticity and strain-rate tensor and leads to formation of extreme events. We analyze the statistical correlations between vorticity and strain rate by using a massive database generated from very well resolved direct numerical simulations of forced isotropic turbulence in periodic domains. The grid resolution is up to $12288^3$, and the Taylor-scale Reynolds number is in the range $140-1300$. In order to understand the formation and structure of extreme vorticity fluctuations, we obtain statistics conditioned on enstrophy (vorticity-squared). The magnitude of strain, as well as its eigenvalues, is approximately constant when conditioned on weak enstrophy; whereas they grow approximately as power laws for strong enstrophy, which become steeper with increasing $R_\lambda$. We find that the well-known preferential alignment between vorticity and the intermediate eigenvector of strain tensor is even stronger for large enstrophy, whereas vorticity shows a tendency to be weakly orthogonal to the most extensive eigenvector (for large enstrophy). Yet the dominant contribution to the production of large enstrophy events arises from the most extensive eigendirection, the more so as $R_\lambda$ increases. Nevertheless, the stretching in intense vorticity regions is significantly depleted, consistent with the kinematic properties of weakly-curved tubes in which they are organized. Further analysis reveals that intense enstrophy is primarily depleted via viscous diffusion, though viscous dissipation is also significant. Implications for modeling are nominally addressed as appropriate.

Strain rate field on the Nigeria lithosphere derived from GNSS velocity

The Global Navigation Satellite System (GNSS) over the years has become an essential instrument in modelling geodynamic processes, because of its high spatio-temporal characteristics. Accurate velocity modelling of GNSS stations and strain rate analysis can provide an accurate mechanism for studying the deformation of the earth crust. Therefore, this study presents the strain rate tensor of the Nigeria crust from GNSS network, but first, the GAMIT/GLOBK was used to process and obtain accurate velocity solution of the Nigerian Permanent GNSS Network (NigNET). The surface deformation of Nigeria deduced from strain analysis was computed using splines in tension. The results revealed that the highest extensional and compressional strain rates of up to 2.67 to 2.89μstrain/year were observed around parts of the north-west (Jigawa, Kaduna and Kano) and north central region (parts Abuja and Jos) and some parts of the north-east (parts of Bauchi and Gombe), while the lowest minimal or low extensional and compressional strain rates of 0.09 to 0.534μstrain/year were observed around some parts of the north central (parts of Kwara and Niger) and parts of north-west (parts of Kebbi, Zamfara and Sokoto). The highest maximum shear strain rate of about 0.4μstrain/year which has its peak around some parts of Kaduna, Abuja, Plateau and Bauchi was observed. The highest areal dilatation rate observed in Nigeria is about 1.02μstrain/year. The study serves as a benchmark for further studies.

Frame-invariant formulation of novel generalized Newtonian fluid constitutive equation for polymer melts

In this work, the frame-invariant formulation of a new generalized Newtonian fluid (GNF) constitutive equation is proposed. Viscosity is considered as a specific function of the second and third invariant of the strain rate tensor and of the second invariant of the objective velocity gradient. The GNF model was successfully tested using experimental data taken from the open literature for different high and low density polyethylene melts with varying amounts of long-chain branches utilizing steady-state shear, uniaxial, and planar extensional viscosities plotted as a function of the strain rate.

Application of the method of current functions in stationary plastic deformation processes

A method of determining the components of the strain rate tensor based on the method of current functions is developed. It is assumed that, with an axisymmetric plastic deformation of the metal in a channel with curvilinear borders, the kinematics of the process are similar to a flat flow. Using the incompressibility equation and the differential equation of the current lines, it is established that the current function maintains a constant value along the current lines. Two infinitely close current lines are considered in the flow plane to explain the kinematic content of the current function. The expression for flow through the finite transverse-sized tube was obtained. In the absence of radial velocity components at the boundaries, we obtain the restrictions imposed on derivatives of the current functions at these boundaries. The method of calculation of kinematic deformation characteristics for established axisymmetric processes will allow to simplify the mathematical processing of the obtained results. Real processes of metal processing are always accompanied by uneven plastic molding, which affects the power and kinematic parameters, the quality of finished products. The main technological factors affecting the appearance of the inhomogeneity and the nature of its distribution are the form factor (geometric parameters of the original workpiece) and the coefficient of contact friction. The study of technological parameters affecting the distribution of non-uniformity of plastic molding will allow to create practical recommendations for reducing the influence of the above mentioned parameters to ensure more uniform plastic deformation. Due to elastic-plastic deformation and local heating, which occur during surface-plastic deformation processing, the stress-strain and physical state of the surface layer is formed. This changes the microgeometry of the surface, the physical and mechanical properties of the surface layer of the workpiece. The surface treated by methods of surface-plastic deformation, has high hardness, residual compression stresses in the surface layer, smoothed out the micro-roughness. Accordingly, increases wear resistance, fatigue strength, resistance to corrosion and more.

Modeling hydrate-bearing sediment with a mixed smoothed particle hydrodynamics

Marine gas hydrate is an important energy source while its extraction may induce environmental problems such as subsea landslide, which is usually challengeable for numerical simulation due to the marine environment with high pressure and the existence of gas hydrate. Smoothed particle hydrodynamics (SPH) is a Lagrangian particle method which is attractive in modeling problems with large deformations and fluid–solid interactions (FSI) for continuum and granular materials. However, the conventional SPH suffers from numerical instability when modeling the soil mechanics with the high confining stress. A mixed SPH is developed in this study to simulate hydrate-bearing sediment with the characteristics of high confining stress. In the mixed SPH model, the conventional SPH is used to discretize the momentum equations, and the kernel gradient correction (KGC) SPH is used to discretize strain and spin rate tensors. In order to consider the effect of hydrate saturation on the strength of soil-hydrate material, an existing linear model is applied into the mixed SPH to define the mechanic parameters of soil-hydrate material. The mixed SPH method is validated by several examples and the obtained numerical results are in close agreement with experimental observations. Finally, the mixed SPH method is used to model landslide of hydrate bearing sediments. The results show that the saturation of hydrate has important impact on the stability of hydrate bearing sediments.

Modal analysis of the behavior of inertial particles in turbulence subjected to Stokes drag

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence is known to be minimal at small and large Stokes number and substantial at $\rm St = \mathcal O(1)$. This non-monotonic trend, which has been shown computationally and experimentally, is yet to be explained analytically. In this study, we obtain an analytical expression for the Lyapunov exponents that quantitatively predicts this trend. The sum of the exponents, which is the normalized rate of change of the signed-volume of a small cloud of particles, is correctly predicted to be negative and positive at small and large Stokes numbers, respectively, asymptoting to $\tau Q$ as $\tau \to 0$ and $\tau^{-1/2} |Q|^{1/4}$ as $\tau \to \infty$, where $\tau$ is the particle relaxation time and $Q(\tau)$ is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory. Additionally, the trajectory crossing is predicted only in hyperbolic flows where $Q<0$ for sufficiently inertial particles with a $\tau$ that scales with $|Q|^{-1/2}$. Following the onset of crossovers, a transition from clustering to dispersion is predicted correctly. We show these behaviors are not unique to three-dimensional isotropic turbulence and can be reproduced closely by a one-dimensional mono-harmonic flow, which appears as a fundamental canonical problem in the study of particle clustering. Analysis of this one-dimensional canonical flow shows that the rate of clustering, quantified as the product of the Lyapunov exponent and particle relaxation time, is bounded by $-1/2$, behaving with extreme nonlinearity in the hyperbolic flows and always remaining positive in the elliptic flows. These findings, which are stemmed from our analysis, are corroborated by the direct numerical simulations.

Assessment of the wall-adapting local eddy-viscosity model in transitional boundary layer

The main idea of this manuscript is to assess the wall-adapting local eddy-viscosity (WALE) model, which is designed for large-eddy simulation (LES) of turbulent boundary layer, in transitional flow. In contrast to many other sub-grid-scale (SGS) models, the WALE model demonstrates the asymptotic decay of the eddy viscosity in the vicinity of a solid wall in turbulent boundary layer without relying on a dynamic procedure on the SGS model coefficient. Furthermore, the WALE model yields zero eddy viscosity in pure shear flow. Such features are attractive for LES of laminar-to-turbulent transition, yet the WALE model has not been thoroughly investigated in transitional boundary layer flow. Well-resolved LES is conducted for canonical boundary layer transition triggered by sub-harmonic resonance. The model formulation is thoroughly analyzed in the transition process. The asymptotic behavior of the two major tensors, i.e., the strain rate tensor and the traceless tensor of the velocity gradient squared, are confirmed in the current well-resolved LES. The cubic decay of the eddy viscosity in the wall distance is confirmed from the asymptotic analysis on the transitional flow. The presence of the strain rate in the model formulation generates practically zero eddy viscosity in the pre-transition region, allowing interactions of small-amplitude instabilities. The traceless tensor of the velocity gradient squared escalates if small-scale eddies appear, leading to sizable eddy viscosity particularly in the log-law layer in turbulent flow. The response of the WALE model to grid resolution is also discussed. The performance of the WALE model is compared to two other SGS models, Smagorinsky and Vreman models with a constant coefficient, in the current transitional wall-bounded flow.

Applying Conservation of Energy to Estimate Earthquake Frequencies from Strain Rates and Stresses

AbstractEstimating earthquake occurrence rates from the accumulation rate of seismic moment is an established tool of seismic hazard analysis. We propose an alternative, fault‐agnostic approach based on the conservation of energy: the Energy‐Conserving Seismicity Framework (ENCOS). Working in energy space has the advantage that the radiated energy is a better predictor of the damage potential of earthquake waves than the seismic moment release. In a region, ENCOS balances the stationary power available to cause earthquakes with the long‐term seismic energy release represented by the energy‐frequency distribution's first moment. Accumulation and release are connected through the average seismic efficiency, by which we mean the fraction of released energy that is converted into seismic waves. Besides measuring earthquakes in energy, ENCOS differs from moment balance essentially in that the energy accumulation rate depends on the total stress in addition to the strain rate tensor. To validate ENCOS, we exemplarily model the energy‐frequency distribution around Southern California. We estimate the energy accumulation rate due to tectonic loading assuming poroelasticity and hydrostasis. Using data from the World Stress Map and assuming the frictional limit to estimate the stress tensor, we obtain a power of 0.8 GW. The uncertainty range, 0.3–2.0 GW, originates mainly from the thickness of the seismogenic crust, the friction coefficient on preexisting faults, and models of Global Positioning System (GPS) derived strain rates. Based on a Gutenberg‐Richter magnitude‐frequency distribution, this power can be distributed over a range of energies consistent with historical earthquake rates and reasonable bounds on the seismic efficiency.

Learning the constitutive relation of polymeric flows with memory

We develop a learning strategy to infer the constitutive relation for the stress of polymeric flows with memory. We make no assumptions regarding the functional form of the constitutive relations, except that they should be expressible in differential form as a function of the local stress- and strain-rate tensors. In particular, we use a Gaussian Process regression to infer the constitutive relations from stress trajectories generated from small-scale (fixed strain-rate) microscopic polymer simulations. For simplicity, a Hookean dumbbell representation is used as a microscopic model, but the method itself can be generalized to incorporate more realistic descriptions. The learned constitutive relation is then used to perform macroscopic flow simulations, allowing us to update the stress distribution in the fluid in a manner that accounts for the microscopic polymer dynamics. The results using the learned constitutive relation are in excellent agreement with full Multi-Scale Simulations, which directly couple micro/macro degrees of freedom, as well as the exact analytical solution given by the Maxwell constitutive relation. We are able to fully capture the history dependence of the flow, as well as the elastic effects in the fluid. We expect the proposed learning/simulation approach to be used not only to study the dynamics of entangled polymer flows, but also for the complex dynamics of other Soft Matter systems, which possess a similar hierarchy of length- and time-scales.

Spatial distribution of earthquake potential along the Himalayan arc

To determine the spatial distribution of earthquake potential along the active Himalayan arc, we utilize GPS measurements and earthquake data. We derive horizontal velocity field and 2-D strain rates from a new set of 41 regional GPS stations along with 446 published velocities. We convert these strain rate tensors to geodetic moment rate build-up within 24 contiguous segments and compare to the seismic moment rate release derived from a reassessed earthquake catalog of 900 years. The geodetic to seismic moment rate ratio, an indicator of stored strain energy, varies from below unity to more than 50 in different segments. The estimated geodetic moment rate ranges from 1.7 × 1018 Nm/yr to 10.2 × 1018 Nm/yr, whereas the seismic moment rate ranges from 3.7 × 1016 Nm/yr to 5.1 × 1019 Nm/yr. This variation between the geodetic and seismic moment rate corresponds to a moment deficit rate of ~1.15×1017 Nm/yr to 7.97 × 1018 Nm/yr along various segments of the study region. The above moment deficit rate provides an equivalent earthquake potential of magnitude 5.7 − 8.2 in different segments. Specifically, the higher earthquake potential (Mw≥8.0) corresponds to the segments in the central seismic gap and the northeast part of Himalaya, whereas the lower earthquake potential (Mw<7.0) corresponds to the segments encompassing the rupture areas of recent large events. The present findings not only provide input constraints on the contemporary crustal deformation but also contributes to the time-dependent seismic hazard analysis along the Himalaya.

Seismogenic potential of the High Durance Fault constrained by 20 yr of GNSS measurements in the Western European Alps

SUMMARYDue to the steady moderate seismicity observed along the Briançon seismic arc, in the south-western French Alps, three temporary GNSS (Global Navigation Satellite System) surveys took place in 1996, 2006 and 2011, across a ∼50 × 60 km² wide area, to investigate the surface deformation field. The horizontal velocity field computed from these three surveys showed an east–west extension in the network. A fourth campaign was led in 2016, creating a 20 yr observation span, resulting in measurements which reach a sufficient accuracy to assess whether extension found within the Briançon network is localized onto any particular tectonic feature. Several faults in this area are known to be active normal faults. Assessing the localization of the deformation may lead to a better understanding of the active tectonics of the Alpine belt. To address this issue, a robust velocity field was computed from the combination of the different campaign and permanent GNSS data. Strain rate tensors were derived for the first time in this area on a 0.1 × 0.1 deg grid to assess the distribution of the deformation. The regional deformation appears localized in the Briançon area and reaches up to 20 ± 5 nanostrain yr−1 in the centre of the network. The observed velocities were projected on a profile across the network and compared with modelled interseismic deformation to characterize the behaviour of the major active faults known in the study zone. While a two-fault model provides the best fit to the data, a single fault model has only marginally higher residuals, with parameters which are more consistent with the seismotectonics of the region. The localization of the single modelled fault is consistent with the location of the High Durance Fault (HDF). Therefore, we used the known geological location of this structure as a priori information in a block model to compute a fault slip rate at the interface between the two blocks. The velocities on the interface indicate 0.4–0.5 mm yr−1 of extension, and therefore strain accumulates along the HDF throughout the seismic cycle. The geodetically derived fault slip rate is converted into an equivalent seismic moment release rate, which is consistent within its uncertainty bounds with the known historical and instrumental seismicity of the Briançon area.

Transitional and turbulent flows in rectangular ducts: budgets and projection in principal mean strain axes

We carry out Direct Numerical Simulation (DNS) of flows in closed rectangular ducts with several aspect ratios. The Navier–Stokes equations are discretised through a second-order finite difference scheme, with non-uniform grids in two directions. The duct cross-sectional area is maintained constant as well as the flow rate, which allows to investigate which is the appropriate length scale in the Reynolds number for a good scaling in the laminar and in the fully turbulent regimes. We find that the Reynolds number based on the half length of the short side leads to a critical Reynolds number which is independent on the aspect ratio (), for ducts with . The mean and rms wall-normal velocity profiles are found to scale with the local value of the friction velocity. At high friction Reynolds numbers, the Reynolds number dependence is similar to that in turbulent plane channels, hence flows in rectangular ducts allow to investigate the Reynolds number dependency through a reduced number of simulations. At low Re, the profiles of the statistics differ from those in the two-dimensional channel due to the interaction of flow structures of different size. The projection of the velocity vector and of the Reynolds stress tensor along the eigenvectors of the strain-rate tensor yields reduced Reynolds stress anisotropy and simple turbulence kinetic energy budgets. We further show that the isotropic rate of dissipation is more difficult to model than the full dissipation rate, whose distribution does not largely differ from that of turbulence kinetic energy production. We expect that this information may be exploited for the development of advanced RANS models for complex flows.

Revisiting turbulence small-scale behavior using velocity gradient triple decomposition

Turbulence small-scale behavior has been commonly investigated in literature by decomposing the velocity-gradient tensor (Aij) into the symmetric strain-rate (Sij) and anti-symmetric rotation-rate (Wij) tensors. To develop further insight, we revisit some of the key studies using a triple decomposition of the velocity-gradient tensor. The additive triple decomposition formally segregates the contributions of normal-strain-rate (Nij), pure-shear (Hij) and rigid-body-rotation-rate (Rij). The decomposition not only highlights the key role of shear, but it also provides a more accurate account of the influence of normal-strain and pure rotation on important small-scale features. First, the local streamline topology and geometry are described in terms of the three constituent tensors in velocity-gradient invariants’ space. Using direct numerical simulation (DNS) data sets of forced isotropic turbulence, the velocity-gradient and pressure field fluctuations are examined at different Reynolds numbers. At all Reynolds numbers, shear contributes the most and rigid-body-rotation the least toward the velocity-gradient magnitude (A2 ≡ AijAij). Especially, shear contribution is dominant in regions of high values of A2 (intermittency). It is shown that the high-degree of enstrophy intermittency reported in literature is due to the shear contribution toward vorticity rather than that of rigid-body-rotation. The study also provides an explanation for the non-intermittent nature of pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. The study further investigates the alignment of the rotation axis with normal strain-rate and pressure Hessian eigenvectors. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence.

Subgrid-scale model based on the vorticity gradient tensor for rotating turbulent flows

A new subgrid-scale (SGS) stress model is proposed for rotating turbulent flows, and the new model is based on the traceless symmetric part of the square of the velocity gradient tensor and the symmetric part of the vorticity gradient tensor (or the so-called vorticity strain rate tensor). The new subgrid-scale stress model is taken into account the effect of the vortex motions in turbulence, which is reflected on the anti-symmetric part of the velocity gradient tensor. In addition, the eddy viscosity of the new model reproduces the proper scaling as O(y3) near the wall. Then, the new SGS model is applied in large-eddy simulation of the spanwise rotating turbulent channel flow. Different simulating cases are selected to test the new model. The results demonstrate that the present model can well predict the mean velocity profiles, the turbulence intensities, and the rotating turbulence structures. In addition, it needs no a second filter, and is convenient to be used in the engineering rotational flows.

Oblique fault movement during the 2016 Mw 5.9 Zaduo earthquake: insights into regional tectonics of the Qiangtang block, Tibetan Plateau

The present east–west crustal extension of the Tibetan Plateau has been demonstrated through field investigations, satellite imagery, geodetic deformation, and earthquake focal mechanisms. Normal faulting earthquakes in the interior Tibetan Plateau are almost entirely confined to regions at elevations over 4000 m. However, our knowledge of the eastward extent of normal faulting in the plateau is still uncertain due to the limited occurrence of well-documented earthquakes. Based on a retrospective analysis of the 2016 Mw 5.9 Zaduo earthquake in the Tibetan Plateau, we consider the NE trending Zaduo-Shanglaxiu fault as the most likely rupture fault through a comprehensive analysis of relocated aftershock sequences, mapped active faults, and newly acquired strain rate tensor. We further determine seismogenic fault geometry using a Bayesian approach and sample with a Markov Chain Monte Carlo method. We interpret the Zaduo earthquake to reflect the release of slowly accumulated elastic strain accumulated mainly by gravitational forces rather than a delay triggering event from the 2010 Yushu earthquake. The viscoelastic calculations to estimate Coulomb stress changes over time indicate that long-term viscous flow in a weak mid-crust can load adjacent faults far more than static stress changes alone. Our results show that the Zaduo earthquake was a Mw 5.9 oblique normal faulting event that occurred in the easternmost part of the Tibetan Plateau, suggesting that the Qiangtang block at longitude ~ 95° E accommodates east–west extensional crustal deformation by small-scale oblique normal faults, which may act as the boundary of micro-blocks. This may also mean that the normal faulting in the Qiangtang block is expanding outwards, and a new rifting system may be formed, which requires more geological evidence.

Strain calculations of active tectonic blocks in northeastern Venezuela from GNSS analysis

We study translation, rotation, and strain of active tectonic blocks in Northeastern Venezuela from the Global Navigation Satellite System (GNSS) observations. Since the installation of the geodetic network in 2003, one of the goals was to place at least three observation sites at each tectonic block to study the deformation of each one. Based on this premise, we define at least seven blocks: Bergantín and Caripe blocks south of the El Pilar Fault (EPF), and Cariaco Gulf, Land bridge, Paria, North Peninsula, and Margarita Island blocks north of the EPF. Our preferred block modeling shows angular rotations from 0.02 to 0.29° Ma−1. It is known that the EPF concentrates the active deformation in this region of the Caribbean-South American plate boundary. However, the existent rotation could accommodate part of the motion. The strain rate tensors (SRT) indicate NW-SE compression and NE-SW extension for the western blocks. To the east, the 4-Land bridge block keeps the NW-SE compression but shows a decrease in the extensional component. The 5-Paria blocks show a complete inversion in the sense of semi-axis. Additionally, we evaluate the possibility of different motions in Margarita block calculating translational vector, rotational velocity, and strain from three GNSS sites at each side thereof. Our results show remarkable similarities for the Macanao Peninsula and Eastern Margarita Island, pointing to both belonging to a single block.

Parametric numerical study of passive scalar mixing in shock turbulence interaction

Turbulent mixing of passive scalars is studied in the canonical shock–turbulence interaction configuration via shock-capturing direct numerical simulations, varying the shock Mach number ( ), turbulence Mach number ( ), Taylor microscale Reynolds number ( ) and Schmidt number ( , 1, 2). The shock-normal evolution of scalar variance and dissipation transport equations, spectra and probability density functions (PDFs) are examined. Scalar dissipation, its production and destruction increase across the shock with higher , lower and lower . Mixing enhancement for different flow topologies across the shock is studied from changes in the PDFs of velocity gradient tensor invariants and conditional distributions of scalar dissipation. The proportion of the stable-focus-stretching flow topology is the highest among all the topologies in the flow both before and after the shock. Unstable-node/saddle/saddle topology is the most dissipative throughout the flow domain, despite variations across the shock. Preshock and postshock distributions of the alignment between the strain-rate tensor eigenvectors and the scalar gradient, vorticity and the mean streamwise vector conditioned on flow topology are studied. A novel barycentric map representation is introduced for a more direct visualization of the alignments and conditioned scalar dissipation distributions. Interaction with the shock increases alignment of the scalar gradient with the most extensive eigenvector, decreasing it with the most compressive, which is still dominant. The barycentric map of the passive scalar gradient also reveals that, across the shock, the most probable alignment between scalar gradient and strain eigendirections converges towards the alignment that provides the most dissipation. This also leads to an enhancement of scalar dissipation immediately downstream of the shock.

Solution Behavior in the Vicinity of Characteristic Envelopes for the Double Slip and Rotation Model

The boundary conditions significantly affect solution behavior near rough interfaces. This paper presents general asymptotic analysis of solutions for the rigid plastic double slip and rotation model in the vicinity of an envelope of characteristics under plane strain and axially symmetric conditions. This model is used in the mechanics of granular materials. The analysis has important implications for solving boundary value problems because the envelope of characteristics is a natural boundary of the analytic solution. Moreover, an envelope of characteristics often coincides with frictional interfaces. In this case, the regime of sticking is not possible independently of the friction law chosen. It is shown that the solution is singular in the vicinity of envelopes. In particular, the profile of the velocity component tangential to the envelope is described by the sum of the constant and square root functions of the normal distance to the envelope in its vicinity. As a result, some components of the strain rate tensor approach infinity. This finding might help to develop an efficient numerical method for solving boundary value problems and provide the basis for the interpretation of some experimental results.

Contemporary crustal deformation of Northeast Tibet from geodetic investigations and a comparison between the seismic and geodetic moment release rates

We processed the data recorded by 32 continuous GPS stations from the Crustal Movement Observation Network of China and combined them with published GPS observations in a common geodetic reference frame to better depict the ongoing crustal velocities and deformation field of Northeast Tibet. The derived velocity field indicates that the deformation of Northeast Tibet is characterized mainly by crustal shortening coupled with the eastward extrusion of crustal material and clockwise block rotation. We further used the GPS velocity field to evaluate the strain rate tensors by the method of Sandwell and Wessel (2016). The strain rate field shows an obvious transition of the contraction rate from NE-SW to NEE-SWW, and a decrease in the maximum shear strain rate from west to east was observed along the East Kunlun fault and Haiyuan fault, indicating that the block rotation and lateral extrusion therein may be partitioned by NE-SW-oriented compression to some extent. The inhomogeneity of the rotation strain rate implies that Northeast Tibet does not behave like a single rigid block but instead resembles a left-lateral shear zone, supporting the hypothesis that the northward transfer of motion along the East Kunlun fault may play a significant role in the deformation style of this area. The dilatation rate verified that most of Northeast Tibet experiences crustal thickening, which coincides with the average uplift rate of ~1 mm/y revealed by GPS velocities and channel steepness indices. We also estimated the geodetic moment rates and compared them with the seismic moment rates estimated using the truncated Gutenberg-Richter earthquake distribution. The ratio of the seismic strain rate to the geodetic strain rate in the regions near the Altyn Tagh fault, the Qaidam Basin adjoining the Qilian Mountains, the Haiyuan fault and the central part of the East Kunlun fault are near unity, indicating that the deformation in these areas is dominated by seismic release and may represent fully coupled seismogenic zones. The occurrence of large earthquakes in the past (estimated magnitudes M ≥ 6) coupled with a high geodetic deformation rate in these regions could indicate overdue M ≥ 6 earthquakes. However, other factors, including the occurrence of creep and the long recurrence time of Mw ≥ 8.0 earthquakes, should be taken into consideration along the Haiyuan fault. By contrast, in other regions, including the southern and eastern parts of the Haiyuan fault, along the Elashan fault, and the northern Qilian Mountains, the geodetic moment rate is higher than the seismic moment rate by a factor between 3 and 9. We propose that this phenomenon is related to the undersampling of long-term earthquake rates within the seismic catalogue, long-term aseismic crustal deformation, or a combination of these factors. Our study has implications for better comprehending the current seismotectonic pattern of Northeast Tibet.