Viscoelastic Relaxation Research Articles

GNSS-Based Modeling and Study of Postseismic Crustal Movement of the May 24, 2013, MW 8.3 Sea of Okhotsk Deep-Focus Earthquake

The mechanisms of preparation and occurrence of the strongest deep-focus earthquakes with MW≥8, as well as their surface manifestations, remain insufficiently studied because of the lack of the relevant data. There are but three seismic events of this kind which have so far been instrumentally recorded. This paper describes the identification and analysis of the changes in the characteristics of modern crustal movement of the 2013, MW 8.3 Sea of Okhotsk deep-focus earthquake based on the data from long-term continuous geodetic-class GNSS stations in the Sea of Okhotsk region on the Kamchatka Peninsula, the Sakhalin Island, and the coast of the Sea of Okhotsk and the Sea of Japan. There has been found temporal stability of variations in the average annual geodetic site velocities. The coordinates of GNSS-stations do not show non-linear changes typical of strong shallow earthquakes in the initial post-seismic period. The Maxwell rheology for modeling of viscoelastic relaxation of the asthenosphere/upper mantle as a result of seismic impact allows for a first approximation to qualitatively and quantitatively reproduce the displacement patterns of GNSS-sites of the Kamchatka Peninsula observed in the initial postseismic period (2–3 years after the mainshock). After that, the model estimates of postseismic movements of the peninsula become systematically lower than the observed. The values calculated for the OKHT station motion on the western coast of the Sea of Okhotsk are in good agreement with those recorded for postseismic displacements over the entire measurement interval. The observed directions of the Sakhalin Island postseismic movements systematically deviate to the northeast from the model directions and are oriented almost orthogonally to the Kuril-Kamchatka Trench. Besides the viscoelastic relaxation process, another possible reason for this issue could be an enhanced viscous friction in the bottom of the subducting Pacific plate, leading to the intense deformation of the Sakhalin Island and the western coast of Kamchatka.

Deep versus shallow emplacement of sills and dykes: new insight from thermo-visco-elastic modelling

Rapid emplacement of a mafic dyke or sill at mid-crustal depth heats and possibly melts the felsic wall rock followed by solidification. Associated volume changes generate stresses, possibly enforcing brittle failure and melt migration. We model the evolution of melting, solidification, temperature, and stress including visco-elastic relaxation in 1D - dykes or -sills using realistic rock rheologies of the Weschnitz pluton (Odenwald). For deep emplacement (Case 1, 15.3 km) extensive contact melting of the wall rock occurs, for shallow emplacement (Case 2, 10 km) it is negligible. The stresses are zero at high melt fractions, but increase during solidification and cooling: The intrusion orthogonal stress is always zero. The intrusion parallel stress σ∥ within the intrusion is tensile (O(200 MPa)). It relaxes on a time scale between a few years (Case 1) and 0.6 m.y. (Case 2). Within the wall rock σ∥ is compressive during heating, but becomes tensile under solidification and cooling. Wall rock stresses relax on a time scale of months to 100 years. A Deborah number is defined based on viscous to thermal relaxation allowing generalization of our results. Adding lithostatic stresses, the total stresses of Case 1 remain below the brittle strength, while for Case 2 they may exceed it. Adding the lithostatic pressure to the melt pressure, the effective stresses exceed the brittle strength and intrusion orthogonal tensile fractures are predicted. Combined with the pressure gradient within the over-pressurized felsic melts generated in the wall rock, this explains the migration of felsic contact melt into shrinkage cracks of the mafic sill in the Weschnitz pluton.

A Numerical Model to Predict the Relaxation Phenomena in Thermoset Polymers and Their Effects on Residual Stress during Curing, Part II: Numerical Evaluation of Residual Stress.

This article proposes a numerical routine to predict the residual stresses developing in an epoxy component during its curing. The scaling of viscoelastic properties with the temperature and the degree of conversion is modeled, adopting a mathematical formulation that considers the concurrent effects of curing and structural relaxation on the epoxy's viscoelastic relaxation time. The procedure comprises two moduli: at first, the thermal-kinetical problem is solved using the thermal module of Ansys and a homemade routine written in APDL, then the results in terms of temperature and the degree of conversion profiles are used to evaluate the viscoelastic functions, and the structural problem is solved in the mechanical module of Ansys, allowing the residual stresses calculation. The results show that the residual stresses mainly arise during cooling and scale with the logarithm of the Biot number.

In-vivo blood pressure sensing with bi-filler nanocomposite

Conductive elastomers present desirable qualities for sensing pressure in-vivo, such as high piezoresistance in tiny volumes, conformability and, biocompatibility. Many electrically conductive nanocomposites however, are susceptible to electrical drift following repeated stress cycles and chemical aging. Here we propose an innovative approach to stabilize nanocomposite percolation network against incomplete recovery to improve reproducibility and facilitate sensor calibration. We decouple the tunnelling-percolation network of highly-oriented pyrolytic graphite (HOPG) nanoparticles from the incomplete viscoelastic recovery of the polydimethylsiloxane (PDMS) matrix by inserting minute amounts of insulating SiO2 nanospheres. SiO2 nanospheres effectively reduce the number of nearest neighbours at each percolation node switching off the parallel electrical pathways that might become activated under incomplete viscoelastic relaxation. We varied the size of SiO2 nanospheres and their filling fraction to demonstrate nearly complete piezoresistance recovery when SiO2 and HOPG nanoparticles have equal diameters (≈400 nm) and SiO2 and HOPG volume fractions are 1 % and 29.5 % respectively. We demonstrate an in-vivo blood pressure sensor based on this bi-filler composite.

An interpenetrating-network theory of the cytoskeletal networks in living cells

Cells undergo dramatic deformations during many physiological and pathological processes such as division and migration, and their mechanical integrity is supported by cytoskeletal networks (i.e. intermediate filaments, F-actin, and microtubules). Recent observations of cytoplasmic microstructure indicate interpenetration among different cytoskeletal networks, and micromechanical experiments have shown evidence of complex characteristics in the mechanical response of the interpenetrating cytoskeletal networks of living cells including viscoelastic, nonlinear stiffening, microdamage, and healing characteristics (Hu et al., 2019). However, a theoretical framework describing such a response is missing, and thus it is not clear how different cytoskeletal components with distinct mechanical properties come together to build the overall complex mechanical features of the cytoskeletal networks. In this work, we address this gap by developing a finite-deformation continuum-mechanical theory with a multi-branch visco-hyperelastic constitutive relation coupled with phase-field damage and healing. The proposed interpenetrating-network model elucidates the coupling among interpenetrating cytoskeletal components, and the roles of finite elasticity, viscoelastic relaxation, damage, and healing in the experimentally-observed mechanical response of interpenetrating cytoskeletal networks in living eukaryotic cells.

Direct integration of measured viscoelastic relaxation data in time-domain finite element simulations

The current approach to modeling viscoelastic materials in most commercial finite element packages is based on the General Maxwell Model, which views these materials as combinations of spring and dashpot elements. However, the data can be incorporated more directly into a transient finite element study by direct interpolation of the relaxation function. This work explores a linear interpolation scheme to the inclusion of viscoelastic relaxation functions on an example problem. The results show several benefits over the General Maxwell Model for transient studies. Included in the analysis are displacement solutions utilizing both approaches, relaxation function error calculations for both approaches, and parametric runtime studies comparing speed of calculation. The variation in computational flop counts is considered and an argument is made for the preference of the proposed approach.

Investigations of cardiac fibrosis rheology by in vitro cardiac tissue modeling with 3D cellular spheroids

Cardiac fibrosis refers to the abnormal accumulation of extracellular matrix within the cardiac muscle, leading to increased stiffness and impaired heart function. From a rheological standpoint, knowledge about myocardial behavior is still lacking, partially due to a lack of appropriate techniques to investigate the rheology of in vitro cardiac tissue models. 3D multicellular cardiac spheroids are powerful and versatile platforms for modeling healthy and fibrotic cardiac tissue in vitro and studying how their mechanical properties are modulated. In this study, cardiac spheroids were created by co-culturing neonatal rat ventricular cardiomyocytes and fibroblasts in definite ratios using the hanging-drop method. The rheological characterization of such models was performed by Atomic Force Microscopy-based stress-relaxation measurements on the whole spheroid. After strain application, a viscoelastic bi-exponential relaxation was observed, characterized by a fast relaxation time (τ1) followed by a slower one (τ2). In particular, spheroids with higher fibroblasts density showed reduction for both relaxation times comparing to control, with a more pronounced decrement of τ1 with respect to τ2. Such response was found compatible with the increased production of extracellular matrix within these spheroids, which recapitulates the main feature of the fibrosis pathophysiology. These results demonstrate how the rheological characteristics of cardiac tissue vary as a function of cellular composition and extracellular matrix, confirming the suitability of such system as an in vitro preclinical model of cardiac fibrosis.

Stress change in southwest Japan due to the 1944–1946 Nankai megathrust rupture sequence based on a 3-D heterogeneous rheological model

The Nankai trough has repeatedly experienced large megathrust earthquakes at intervals of 100–200 years. The inland region of southwest (SW) Japan has a seismically active period from 50 years before to 10 years after megathrust earthquakes. To assess the activities of inland earthquakes after megathrust earthquakes, we need to quantitatively evaluate the postseismic stress accumulation on the inland source faults considering plausible viscoelastic relaxation. Recent studies have shown the importance of low-viscosity layers along the lithosphere-asthenosphere boundary (LAB layer) in postseismic deformation. In the present study, we focus on the most recent ruptures, the 1944 M7.9 Tonankai and the 1946 M8.0 Nankai earthquakes, estimating the 4-year stress change on the source faults in SW Japan in a forward modeling approach. The 1944–1946 megathrust rupture sequence was followed by severe ~ M7 inland earthquakes, such as the 1945 M6.8 Mikawa and 1948 M7.1 Fukui earthquakes. For stress calculation, we used a highly detailed finite element model incorporating the actual topography and the plausible viscoelastic underground structure from past studies. The calculated inland stress field shows the dominance of the coseismic change during the 1944 and 1946 earthquakes and little contribution from viscoelastic relaxation. In contrast, viscoelastic relaxation has a significant effect on stress in the slab, indicating the importance of quantifying the viscosity of the LAB layer. Based on the calculated stress, we evaluated the change in the Coulomb failure stress (ΔCFS) on each source fault. The ΔCFS is generally positive on the strike-slip faults east of 135°E due to the 1944 rupture. In contrast, the ΔCFS on the faults west of 135°E, including the Median Tectonic Line segments, became positive due to the 1946 rupture. The occurrence of the damaging earthquakes in 1945 and 1948 can be explained by the calculated ΔCFS. The ΔCFS on the recent earthquake faults of the recent damaging earthquakes such as the 2016 M7.2 Kumamoto earthquake is generally negative, suggesting the delay in stress accumulation. The ΔCFS on the source faults of the intra-slab earthquakes differ significantly as large as tens of kilopascals depending on the viscosity of the LAB layer.Graphical

The Time-Temperature Superposition of Polymeric Rubber Gels Treated by Means of the Mode-Coupling Theory.

Viscoelastic relaxation measurements on styrene-butadiene rubbers (SBRs) doped with carbon nanotube (CNT) at different concentrations around the sol-gel transition show the time-temperature superposition (TTS). This process is described in terms of the mode coupling theory (MCT) approach to viscoelasticity by considering the frequency behavior of the loss modulus E″(ω) and showing that the corresponding TTS is linked to ω1/2 decay. From the analysis of the obtained data, we observe that the interaction between SBRs and CNT determines different levels of decay according to their concentration. Systems with the lowest CNT concentration are only characterized in the studied T-range by their fragile glass-forming behavior. However, at a specific temperature TL, those with the highest CNT concentration show a crossover towards pure Arrhenius that, according to the MCT, indicates the presence of kinetic glass transition (KGT), where system response functions are characterized by scaling behaviors.

Rheological behavior of brain tissue: Experiments vs theory and forensic applications

Experimental data describing the uniaxial compression and relaxation of brain tissue are compared to the predictions from a rheological model developed by Yarin and Kosmerl [“Rheology of brain tissue and hydrogels: A novel hyperelastic and viscoelastic model for forensic applications,” Phys. Fluids 35, 101910 (2023)]. A qualitative agreement between the model and experiments with swine brain tissue is confirmed, and the uniformly valid values (i.e., valid in all rheometric experiments without any change) of the rheological parameters are established. These are the values of the following four parameters: G (the shear modulus), κ (the bulk modulus), α (the dimensionless degree of hyperelasticity), and θ (the viscoelastic relaxation time). In addition, the present rheological model with the established rheological parameters is incorporated into a dynamic model of bullet penetration into brain tissue after a short-range shooting, when muzzle gases and/or air fill the bullet channel leading to its widening, wave propagation, fragmentation, and backspatter of brain tissue. This problem is of significant interest in forensic science because there is an urgent need to provide physics-informed models to reconstruct and analyze crime scenes.

Athermal pulsational mode dynamics with negative ions

The stability dynamics of the pulsational mode of gravitational collapse (PMGC), excitable in athermal (nonthermal) spherical complex dust molecular clouds (DMCs) in the presence of active role of negative ions, is semi-analytically investigated. The adopted partially ionized DMC model consists of five constitutive plasma species, well-coupled via the gravito-electrostatic Poisson formalism, depicting the corresponding potential-density correlations. The nonthermal uncorrelated lighter (inertialess) species are electrons, positive ions, and negative ions. The thermal correlated heavier (inertial) species are neutral and charged dust grains. The application of a standard spherical wave analysis results in a generalized linear cubic dispersion equation involving a new set of multi-factorial dispersion coefficients dependent on diverse equilibrium plasma parameters. A numerical illustrative platform is broadly created to analyze the DMC stabilization and destabilization factors relative to the non-local inward Newtonian self-gravity. It is interestingly seen that the DMC stabilizing agents are the radial cloud size, dust charge, neutral dust viscoelastic relaxation time, Tribeche thermostatistical parameters, etc. In contrast, the dust mass and equilibrium neutral dust density show destabilizing effects, and so forth. The results systematically explored here are fairly consistent and reliably correlated with the previous results on structure formation mechanism in the DMCs as heretofore reported extensively in the literature.

Measuring the viscoelastic relaxation function of cells with a time-dependent interpretation of the Hertz-Sneddon indentation model

The Hertz-Sneddon elastic indentation model is widely adopted in the biomechanical investigation of living cells and other soft materials using atomic force microscopy despite the explicit viscoelastic nature of these materials. In this work, we demonstrate that an exact analytical viscoelastic force model for power-law materials, can be interpreted as a time-dependent Hertz-Sneddon-like model. Characterizing fibroblasts (L929) and osteoblasts (OFCOLII) demonstrates the model's accuracy. Our results show that the difference between Young's modulus EY obtained by fitting force curves with the Hertz-Sneddon model and the effective Young's modulus derived from the viscoelastic force model is less than 3%, even when cells are probed at large forces where nonlinear deformation effects become significant. We also propose a measurement protocol that involves probing samples at different indentation speeds and forces, enabling the construction of the average viscoelastic relaxation function of samples by conveniently fitting the force curves with the Hertz-Sneddon model.

Polytropic spherical astrocosmic fluid stability in the EiBI gravity with strong collective correlative effects

We herein present a dynamical analysis of the Jeans (gravitational) instability in a viscoelastic polytropic astrocloud within the Eddington-inspired Born-Infeld (EiBI) gravity framework in spherical geometry. The zeroth-order material density curvature corrections to the effective self-gravitational potential are properly incorporated. The applied spherical normal mode analysis yields a unique form of monic cubic dispersion relation with multiparametric coefficients dependent on the diverse equilibrium astrofluidic conditions without invoking any kind of traditional quasi-classic approximation. It is seen that the positive EiBI gravity parameter (χ>0) acts as a stabilizing agent and a negative EiBI gravity parameter (χ<0) acts as a destabilizing agent against the inward non-local self-gravity action. Additionally, we find that larger astroclouds are more prone to gravitational collapse, irrespective of the χ-polarity, and vice-versa. It is further investigated that the cloud temperature plays a stabilizing role at larger wavenumbers and destabilizing at smaller wavenumbers for χ>0. However, for χ<0, the fluid temperature consistently exhibits a destabilizing effect. The viscoelastic relaxation time stabilizes the astrofluid against the self-gravity irrespective of the χ-polarity. The fluid density acts as a stabilizer for χ>0, destabilizer for χ<0, and so forth. This study could be potentially applicable in the EiBI-modified gravitational theory in exploring gravity-driven large-scale astrocosmic phenomena towards structure formation mechanism via the canonical Jeans condensation in diverse astrocosmic conditions.

Coseismic and Early Postseismic Deformation Mechanism Following the 2021 Mw 7.4 Maduo Earthquake: Insights from Satellite Radar Interferometry and GPS

Exploring the deformation mechanism of the 2021 Mw 7.4 Maduo Earthquake is crucial for better understanding the seismic hazard of the faults with low strain rates inside the Bayan Har block. This study leverages deformation information derived from Sentient-1 A/B images and GPS data to investigate in detail the co- and postseismic deformation mechanisms using multiple methods. The main results are as follows. First, the postseismic InSAR time series robustly identified the reactivation of the Changmahe fault, indicating the impact of the Maduo event on surrounding active faults. Second, the joint inversion of Interferometric Synthetic Aperture Radar and GPS revealed that (1) there was a complementary and partially overlapping relationship between the coseismic slip and postseismic afterslip of the main rupture; and (2) the Changmahe fault exhibited thrust compression dislocation in the early stage and experienced a sustained compressive effect from afterslip in the one year after the mainshock. Third, modeling the processes of viscoelastic relaxation and poroelastic rebound revealed that the postseismic deformation was probably caused by a combination of afterslip (near-field) and viscoelastic relaxation (near and far field). Fourth, the stress changes driven by the Maduo event revealed that the seismic gaps inside the Maqin-Maqu segment and the Kunlun Pass-Jiangcuo fault will be potential seismic risks in the future.

Coupling between viscoelasticity and soft elasticity in main-chain nematic Liquid Crystal Elastomers

Liquid crystal elastomers (LCEs) are a class of smart elastomers exhibiting unusual mechanical behavior, including large energy dissipation and soft elasticity under uniaxial tensile loading. LCEs are composed of liquid crystal molecules, called mesogens, linked by a network of polymer chains. During deformation, the mesogens orient in the direction of the loading, leading to soft elasticity, which is an increase in strain at constant stress. The combination of mesogen rotation and intrinsic polymer viscoelasticity leads to a nonlinear viscoelastic soft elastic behavior. The aim of this paper is to investigate the coupling between the viscoelastic mechanisms and soft elasticity in main chain LCEs. We propose a rheological model in which the mesogen rotation during deformation is represented by a reversible slider while viscoelastic relaxation mechanisms are modeled as series of Maxwell elements coupled or decoupled with mesogen rotation. Fitting this model to experimental data demonstrate that the coupling between polymer chain viscoelasticity and mesogen rotation is partial, i.e. the long-time relaxation mechanisms are coupled and the short-time relaxation mechanisms are decoupled from mesogen rotation. Furthermore, we show that the viscosity of mesogen rotation is not necessary to properly predict the elastic modulus during the soft elasticity but it is needed to properly predict the initiation of the phenomenon.

Viscoelastic relaxation and topological fluctuations in glass-forming liquids.

A method for characterizing the topological fluctuations in liquids is proposed. This approach exploits the concept of the weighted gyration tensor of a collection of particles and permits the definition of a local configurational unit (LCU). The first principal axis of the gyration tensor serves as the director of the LCU, which can be tracked and analyzed by molecular dynamics simulations. Analysis of moderately supercooled Kob-Andersen mixtures suggests that orientational relaxation of the LCU closely follows viscoelastic relaxation and exhibits a two-stage behavior. The slow relaxing component of the LCU corresponds to the structural, Maxwellian mechanical relaxation. Additionally, it is found that the mean curvature of the LCUs is approximately zero at the Maxwell relaxation time with the Gaussian curvature being negative. This observation implies that structural relaxation occurs when the configurationally stable and destabilized regions interpenetrate each other in a bicontinuous manner. Finally, the mean and Gaussian curvatures of the LCUs can serve as reduced variables for the shear stress correlation, providing a compelling proof of the close connection between viscoelastic relaxation and topological fluctuations in glass-forming liquids.

Unsteady wetting of soft solids

HypothesisSpreading of liquids on soft solids often occurs intermittently, i.e., the liquid's wetting front switches between sticking and slipping. Studies of this so-called stick-slip wetting on soft solids mostly are confined within quasi-static or forced spreading conditions. In these situations, because the sticking duration is set much larger than the viscoelastic relaxation time of the solid, a ridge is persistently and fully developed at the wetting front as the soft solid yields to the liquid's surface tension. The sticking duration and spreading velocity, therefore, were shown to have little impact to the contact angle change required for stick-to-slip transitions. For unsteady wetting of soft solids, a commonly encountered but largely unexplored situation, we hypothesize that the stick-to-slip transition is controlled not only by a combination of sticking duration and the spreading velocity, but also by an increasing depinning threshold caused by the growing ridge at the wetting front. ExperimentWe performed unsteady wetting experiment on soft solids by letting water droplets spread freely on soft solid surfaces of various stiffness. We capture both the stick-slip spreading behavior and growing wetting ridges using synchronous high-speed imaging and high-speed interferometry. Recorded data of liquid spreading and solid deforming at the wetting front were analyzed to shed light on the relation between stick-slip characteristics and the growing wetting ridge. FindingsWe find that intermittent wetting on a soft solid surface results from a competition between three key factors: liquid inertia, capillary force change during sticking, and growing pinning force caused by the solid's viscoelastic response. We theoretically formulate their quantitative contributions to predict how stick-to-slip transitions occur, i.e., how the contact angle change and sticking duration depend on the liquid's spreading velocity and the solid's viscoelastic characteristics. This provides a mechanistic understanding and methods to control unsteady wetting phenomena in diverse applications, from tissue engineering and fabrication of flexible electronics to biomedicine.

Predicting the effects of degradation on viscoelastic relaxation time using model transient networks

In this study, we utilized tetra-armed polyethylene glycol (Tetra-PEG) slimes, which are model transient networks with well-controlled structures, to predict the effects of long-term degradation on the viscoelastic properties of liquids. Viscoelastic liquids, such as sodium hyaluronate, are frequently used in biomedical applications within the human body. However, precisely controlling the viscoelastic properties of these liquids in the long-term is challenging, as the main chains of the liquids undergo stochastic degradation. To establish a predictable model for studying long-term degradation effects, we employed Tetra-PEG slime, and modifications were performed to introduce specific cleavage sites in areas with connections. The Tetra-PEG slimes were characterized by single relaxation modes, and these modes were independent from the degree of degradation, which was determined by hydrolyzing the cleavage sites. Overall, this work provides a universal design for viscoelastic liquids with precisely-controllable degradation.

Postseismic deformation mechanism of the 2021 Mw7.3 Maduo earthquake, northeastern Tibetan plateau, China, revealed by Sentinel-1 SAR images

The postseismic deformation mechanism is crucial for understanding the evolution of postseismic crustal stress, strain, and aftershocks. Here, we used ∼ 1 year Sentinel-1 data to obtain the time-dependent postseismic deformation of the 2021 Mw 7.3 Maduo earthquake and inverted afterslip distribution. We calculated the frictional parameter of the causative fault based on the time series afterslip. A wide range of viscoelastic relaxation models with varying lower crust viscosities were also tested to constrain the rheological structure. Our findings indicated that (1) Time-dependent postseismic deformation conforms to a logarithmic function. In the north of the fault, the attenuation time decreases first and then increases from east to west, while the south of the fault is the opposite. (2) The afterslip mainly occurred in the downdip direction of the coseismic slip, and there is an excellent complementary relationship between them in space. The partial shallow afterslip overlapped with the coseismic ruptures, mainly caused by the low frictional parameter (a-b = 10-3 ∼ 10-2). (3) Considering the postseismic viscoelastic relaxation, when the viscosity coefficient is small, the maximum deformation in 1 year after the earthquake is only about 6 mm, which only accounts for about 6 % of the InSAR observed deformation. It indicates that afterslip plays a significant role, and viscous relaxation plays a secondary role.

Resolving a ramp-flat structure from combined analysis of co- and post-seismic geodetic data: an example of the 2015 Pishan Mw 6.5 earthquake

SUMMARY Previous studies have shown that it is difficult to determine whether the 2015 Pishan earthquake occurred on a uniform fault or a ramp-flat fault with variable dip angles due to the similar goodness of data fit to coseismic and afterslip models on these two fault models. Here, we first present the InSAR deformation obtained from both ascending and descending orbits, covering the coseismic period and cumulative 5-yr period after the 2015 Pishan earthquake. We then determine the preferred fault geometry by the spatial distributions between the positive Coulomb failure stress change triggered by main shock and the afterslip. Based on the preferred fault model, we finally use a combined model to determine the contributions of elastic and viscoelastic deformation in the post-seismic deformation. We find that the Pishan earthquake prefers to occur on a ramp-flat fault, and the coseismic slip is mainly distributed at a depth of 9–13 km, with a maximum slip of about 1.3 m. The post-seismic deformation is primarily governed by afterslip, as the poroelastic rebound-induced deformation fails to account for the observed post-seismic deformation and the contributions from the viscoelastic relaxation mechanism can be considered negligible in the combined model. Moreover, the modelled stress-driven afterslip and observed kinematic afterslip have good consistency, and the difference between the root mean square error of the two afterslip models is only 4.3 mm. The results from the afterslip model indicate that both of the updip and downdip directions distribute the afterslip, and slip in the updip direction is greater than that of the downdip direction. Meanwhile, the maximum cumulative afterslip after 5 yr is approximately 0.26 m which is equivalent to a released seismic moment of a Mw 6.47.