Viscoelastic Relaxation Research Articles

Pressure and frequency dependence of elastic moduli of fluid‐saturated dual‐porosity rocks

AbstractThe elastic moduli of subsurface rocks saturated with geofluid often depend on the wave frequency and confining pressure due to the wave‐induced fluid flow and their significant intrinsic compressibility. Therefore, the knowledge of the pressure‐dependent dispersion of elastic moduli is usually used in broad practical scenarios such as geofluid discrimination and in situ abnormal pressure detection. We propose a simple dual‐porosity model to describe the pressure and frequency dependence of elastic moduli of fluid‐saturated rocks. First, we follow the idea of the Shapiro dual‐porosity model to yield more accurate formulas for pressure‐dependent stiff porosity and crack porosity. Then the new formulas for stiff and crack porosities are utilized to express the bulk and shear moduli of dry rocks as a function of pressure. Further, the bulk and shear moduli of a modified frame (i.e. a rock skeleton with the dry stiff pores and fluid‐saturated cracks) are formulated with the pressure‐dependent elastic moduli of dry rock. In order to consider the wave energy attenuation induced by fluid in cracks, the frequency‐independent fluid modulus imbedded in the formulas for bulk and shear moduli of a modified frame is replaced with a frequency‐dependent one incorporating the effect of viscoelastic relaxation. The effective bulk and shear moduli of entire fluid‐saturated body can be computed by inserting the pressure‐dependent stiff porosity and elastic moduli of a modified frame into the Gassmann equation. Moreover, appropriate simplification is performed on the central equations in the case of low porosity and weak deformation of stiff pores to yield an approximate model. Modelling results show that the proposed model can reasonably account for the nonlinear variation of bulk and shear moduli with the elevating effective pressure and cautiously predict the wave dispersion and attenuation. We validate our model by comparing the predicted elastic moduli with the corresponding results given by Shapiro model–based equations and with the laboratory measurements of five different fluid‐saturated rock samples.

Triaxial Deformation of the Goldwyer Gas Shale at In Situ Stress Conditions—Part II: Viscoelastic Creep/Relaxation and Frictional Failure

To understand the geomechanical implications of long-term creep (time-dependent deformation) response of gas shale, short-duration creep was recorded from laboratory triaxial tests on ten Goldwyer gas shale samples in the onshore Canning Basin at in situ stress conditions under constant differential axial stress. A simple power-law function captures primary creep behaviour involving elastic compliance constant B and time-dependent factor n. Experimental creep data revealed larger axial creep strain in clay and organic-rich rocks, than those dominated by carbonates. Anisotropic nature of creep was observed depending upon the direction of constant axial stress application (perpendicular or parallel to the bedding plane). Upon the application of linear viscoelastic theory on laboratory creep fitting coefficients, differential horizontal stress accumulation over a geological time scale was estimated from the viscoelastic stress relaxation concept. Further, this model was used to derive lithology-dependent least principal stress (Shmin) magnitude at depth for two vertical wells intersecting the Goldwyer gas shale formations. This newly proposed Shmin model was found to have a profound influence on designing hydraulic fracture simulation. Further, pore size distribution and specific surface area value SN2 were derived from low-pressure gas adsorption experiments. These physical properties along with weak mineral components were linked with creep constitutive parameters to understand the physical mechanisms of creep. A strong correlation was noted between SN2 and creep parameters n and B. Finally, an attempt was made to investigate how gas shale composition and failure frictional properties can influence shear fracturing.

Unveiling the Rheological Control of Magmatic Systems on Volcano Deformation: The Interplay of Poroviscoelastic Magma‐Mush and Thermo‐Viscoelastic Crust

AbstractUnderstanding the mechanical evolution of magmatic systems requires careful assessment of their rheological characteristics, particularly in light of the growing evidence that magma reservoirs are dominated by magma‐mush surrounded by thermally‐altered host‐rock. To address this complexity, we develop models for volcano deformation based on a poroviscoelastic source within a thermo‐viscoelastic host. We use Finite Element modeling to investigate the rheological and mechanical response to melt injection in a Maxwell poroviscoelastic reservoir hosted in a temperature‐dependent standard linear solid viscoelastic (“thermo‐viscoelastic”) crust. Our models consider the competing roles of poroelastic diffusion, and viscoelastic creep and relaxation. All cause time‐dependent post‐injection deformation. Post‐injection deformation of a poroviscoelastic reservoir in an elastic crust is dominated by poroelastic diffusion, a consequence of the short relaxation timescale of a hot and relatively low viscosity mush. For cooler, more viscous mush, the magnitude of viscoelastic deformation increases, supplementing the deformation caused by post‐injection poroelastic diffusion. Thermo‐viscoelasticity of the crust amplifies the poroelastic deformation response of the magma‐mush, leading to increased time‐dependent deformation both during and after melt injection. The rate of post‐injection surface deformation decreases at a rate proportional to the reservoir temperature. Crucially, our model sensitivity analysis demonstrates that the wall rock thermo‐viscoelastic response contributes more to surface deformation than the viscous effect of the magma‐mush. For this reason, neglecting the viscoelastic properties of the host rock and the poroelastic properties of the reservoir in interpretations of surface deformation data could produce errors in inferred processes (e.g., injection duration) and subsurface characteristics (e.g., reservoir compressibility, shape and depth).

The Investigation of Viscoelastic Relaxation Following the 2018 Mw 6.2 Situbondo, Indonesia, Earthquake

The Investigation of Viscoelastic Relaxation Following the 2018 Mw 6.2 Situbondo, Indonesia, Earthquake

A non-Fourier magneto-thermo-viscoelastic problem of a two-dimensional plate

Viscoelastic materials is a kind of damping materials which has been wildly applied in vibration control of ocean and civil engineering, and viscoelastic materials mostly work in a complex thermoelectric multi-field coupled environment. In this work, a two-dimensional viscoelastic plate is studied with stain relaxation time and viscoelastic relaxation time. The two-dimensional viscoelastic plate is placed in a magnetic field, and the upper surface is subjected to a thermal shock. The problem is investigated by a modified L-S thermo-viscoelastic model with stain, viscoelastic relaxation time, and magnetic field. Normal mode analysis is used to achieve exact analytical expressions of temperature distribution, displacement component and thermal stresses. Numerical computations are conducted for copper-like material and dimensionless results are illustrated graphically. Based on the results, the influence of magnetic field on the distributions of the nondimensional temperature, displacement and stresses are discussed.

PEA Signal Simulation Using Multiple Viscoelastic Relaxation Model for Determination of Acoustic Attenuation and Velocity Dispersion

Acoustic attenuation in polymer insulators prevents the accurate determination of space charge profiles measured by pulsed electroacoustic (PEA) measurements. A PEA signal simulation is developed on the basis of a multiple viscoelastic relaxation model, which can determine acoustic attenuation and sound velocity dispersion in polymer insulators. To implement the multiple relaxation model, we use an analogy between the viscoelastic and dielectric relaxations. The Kelvin–Voigt model is used as a single relaxation model, and the Cole–Cole model is used as a multiple relaxation model. The simulation based on the Cole–Cole model as a multiple relaxation model can accurately reproduce measured PEA signals. The distortion of a PEA measured space charge profile due to acoustic attenuation and velocity dispersion is also discussed. The charge profile calculated using the multiple relaxation model is in good agreement with the peak value, width, and tail shape obtained in experiments.

A Modular Platform for Cytocompatible Hydrogels with Tailored Mechanical Properties Based on Monolithic Matrices and Particulate Building Blocks.

We establish a versatile hydrogel platform based on modular building blocks that allows the design of hydrogels with tailored physical architecture and mechanical properties. We demonstrate its versatility by assembling (i) a fully monolithic gelatin methacryloyl (Gel-MA) hydrogel, (ii) a hybrid hydrogel composed of 1:1 Gel-MA and gelatin nanoparticles, and (iii) a fully particulate hydrogel based on methacryloyl-modified gelatin nanoparticles. The hydrogels were formulated to exhibit the same solid content and comparable storage modulus but different stiffness and viscoelastic stress relaxation. The incorporation of particles resulted in softer hydrogels with enhanced stress relaxation. Murine osteoblastic cells cultured in two-dimensional (2D) on hydrogels showed proliferation and metabolic activity comparable to established collagen hydrogels. Furthermore, the osteoblastic cells showed a trend of increased cell numbers, cell expansion, and more defined protrusions on stiffer hydrogels. Hence, modular assembly allows the design of hydrogels with tailored mechanical properties and the potential to alter cell behavior.

Early postseismic deformation of the 2018 Lombok, Indonesia, earthquake sequence constrained by GPS data

Global Positioning System (GPS) stations in Lombok, Indonesia showed postseismic displacement following the 2018 Lombok earthquake sequence which occurred on 28 July 2018 (Mw 6.4), 5 August 2018 (Mw 6.9), 9 August 2018 (Mw 5.9), and the 19 August 2018 (Mw 6.3 and Mw 6.9). This research has investigated the contribution of afterslip and viscoelastic relaxation during the one-month period after the mainshock on 19 August 2018. This study utilizes continuous GPS data available from 20 August 2018–19 September 2018. The analysis reveals that the afterslip was distributed at the shallower portion of the fault, complementary to the deeper coseismic slip of the mainshock. Viscoelastic relaxation, involving a gravitational Maxwell viscoelastic response with viscosity of 9.0 × 1017 Pa s, was estimated. This research also found that the depth of elastic upper crust in this region is 50 km. Observations made during the one-month period after the mainshock revealed that afterslip was responsible for most of the ongoing postseismic deformation. Although the viscoelastic relaxation contributed only insignificantly during the one-month period, this research shows that viscoelastic relaxation is expected to increase with time.

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor.

The viscoelastic relaxation spectrum is vital for constitutive models and for insight into the mechanical properties of materials, since, from the relaxation spectrum, other material functions used to describe rheological properties can be uniquely determined. The spectrum is not directly accessible via measurement and must be recovered from relaxation stress or oscillatory shear data. This paper deals with the problem of the recovery of the relaxation time spectrum of linear viscoelastic material from discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation test. A two-level identification scheme is proposed. In the lower level, the regularized least-square identification combined with generalized cross-validation is used to find the optimal model with an arbitrary time-scale factor. Next, in the upper level, the optimal time-scale factor is determined to provide the best fit of the relaxation modulus to experiment data. The relaxation time spectrum is approximated by a finite series of power-exponential basis functions. The related model of the relaxation modulus is proved to be given by compact analytical formulas as the products of power of time and the modified Bessel functions of the second kind. The proposed approach merges the technique of an expansion of a function into a series of independent basis functions with the least-squares regularized identification and the optimal choice of the time-scale factor. Optimality conditions, approximation error, convergence, noise robustness and model smoothness are studied analytically. Applicability ranges are numerically examined. These studies have proved that using a developed model and algorithm, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and noise robust; small model errors are obtained for the optimal time-scale factors. The complete scheme of the hierarchical computations is outlined, which can be easily implemented in available computing environments.

Epoxy resins containing epoxy-modified polyrotaxanes

The toughening of epoxy resins is important for their applications in coatings, adhesives, and composite materials. In our previous study, we introduced polyrotaxane (PR), which comprises poly-ε-caprolactone (PCL)-grafted α-cyclodextrins (α-CDs) and a polyethylene glycol (PEG) axial chain, into epoxy resins as an additive and observed that the presence of PR causes an additional viscoelastic relaxation and enhances the mechanical toughness. In this study, we modified the side chain ends of PR with epoxy groups (PR-E) to enable incorporation into crosslinked epoxy networks. The chemical bonding of PR-E to the epoxy resins suppressed the local dynamics of the PCL graft chains and enhanced the mechanical properties.

Function Model Based on Nonlinear Transient Rheology of Rocks: An Analysis of Decadal GNSS Time Series After the 2011 Tohoku‐oki Earthquake

AbstractThe postseismic Global Navigation Satellite System (GNSS) time series after the 2011 Tohoku‐oki earthquake is often empirically explained by one or more logarithmic and exponential decay functions. However, a function model with these decay functions struggles to illuminate various deformation mechanisms. Here, we propose a new function model incorporating laboratory‐derived constitutive laws (power‐law flow and velocity‐strengthening friction) utilized in mechanical postseismic models of the last decade. We demonstrated that our function model accurately predicts the decade‐long time series of inland GNSS stations, including coastal areas where significant postseismic uplift continues now. Moreover, it decomposes them into displacements due to viscoelastic relaxation and afterslip, similar to results produced by previous stress‐dependent postseismic models of the 2011 Tohoku‐oki earthquake. This physics‐based function model is an effective tool to ease the process of forecasting long‐term GNSS time series by dividing them into two dominant mechanisms at the plate interface and the surrounding viscoelastic mantle.

Frictional Properties and Rheological Structure at the Ecuadorian Subduction Zone Revealed by the Postseismic Deformation Due To the 2016 Mw 7.8 Pedernales (Ecuador) Earthquake

AbstractPostseismic deformation following subduction earthquakes includes the combined effects of afterslip surrounding the coseismic rupture areas and viscoelastic relaxation in the asthenosphere and provides unique and valuable information for understanding the rheological structure. Because the two postseismic mechanisms are usually spatiotemporally intertwined, we developed an integrated model combining their contributions, based on 5 years of observations following the 2016 Pedernales (Ecuador) earthquake. The results show that the early, near‐field postseismic deformation is dominated by afterslip, both updip and downdip of the coseismic rupture, and requires heterogeneous interface frictional properties. Viscoelastic relaxation contributes more to far‐field displacements at later time periods. The best‐fit integrated model favors a 45‐km thick lithosphere overlying a Burgers body viscoelastic asthenosphere with a Maxwell viscosity of 3 × 1019 Pa s (0.9–5 × 1019 Pa s at 95% confidence), assuming the Kelvin viscosity equal to 10% of that value. In addition to the postseismic afterslip, the coastal displacement of sites north and south of the rupture clearly require extra slip in the plate motion direction due to slow slip events that may be triggered by the coseismic stress changes (CSC) but are not purely driven by the CSC. Spatially variable afterslip following the Pedernales event, combined with the SSEs during the interseismic period, demonstrate that spatial frictional variability persists throughout the whole earthquake cycle. The interaction of adjacent fault patches with heterogeneous properties may contribute to the clustered large earthquakes in this area.

Multiscale characterization of left ventricle active behavior in the mouse.

The myocardium possesses an intricately designed microarchitecture to produce an optimal cardiac contraction. The contractile behavior of the heart is generated at the sarcomere level and travels across several length scales to manifest as the systolic function at the organ level. While passive myocardial behavior has been studied extensively, the translation of active tension produced at the fiber level to the organ-level function is not well understood. Alterations in cardiac systolic function are often key sequelae in structural heart diseases, such as myocardial infarction and systolic heart failure; thus, characterization of the contractile behavior of the heart across multiple length scales is essential to improve our understanding of mechanisms collectively leading to depressed systolic function. In this study, we present a methodology to characterize the active behavior of left ventricle free wall (LVFW) myocardial tissues in mice. Combined with active tests in papillary muscle fibers and conventional in vivo contractility measurement at the organ level in an animal-specific manner, we establish a multiscale active characterization of the heart from fiber to organ. In addition, we quantified myocardial architecture from histology to shed light on the directionality of the contractility at the tissue level. The LVFW tissue activation-relaxation behavior under isometric conditions was qualitatively similar to that of the papillary muscle fiber bundle. However, the maximum stress developed in the LVFW tissue was an order of magnitude lower than that developed by a fiber bundle, and the time taken for active forces to plateau was 2-3 orders of magnitude longer. Although the LVFW tissue exhibited a slightly stiffer passive response in the circumferential direction, the tissues produced significantly larger active stresses in the longitudinal direction during active testing. Also, contrary to passive viscoelastic stress relaxation, active stresses relaxed faster in the direction with larger peak stresses. The multiscale experimental pipeline presented in this work is expected to provide crucial insight into the contractile adaptation mechanisms of the heart with impaired systolic function. STATEMENT OF SIGNIFICANCE: Heart failure cause significant alterations to the contractile-relaxation behavior of the yocardium. Multiscale characterization of the contractile behavior of the myocardium is essential to advance our understanding of how contractility translates from fiber to organ and to identify the multiscale mechanisms leading to impaired cardiac function. While passive myocardial behavior has been studied extensively, the investigation of tissue-level contractile behavior remains critically scarce in the literature. To the best of our knowledge, our study here is the first to investigate the contractile behavior of the left ventricle at multiple length scales in small animals. Our results indicate that the active myocardial wall is a function of transmural depth and relaxes faster in the direction with larger peak stresses.

Pulsatory patterns in active viscoelastic fluids with distinct relaxation time scales

Developing tissues need to pattern themselves in space and time. A prevalent mechanism to achieve this are pulsatile active stresses generated by the actin cytoskeleton. Active gel theory is a powerful tool to model the dynamics of cytoskeletal pattern formation. In theoretical models, the influence of the viscoelastic nature of the actin cytoskeleton has so far only been investigated by the incorporation of one viscoelastic relaxation time scale. Here, using a minimal model of active gel theory, we show that distinct shear and areal relaxation times are sufficient to drive pulsatile dynamics in active surfaces with only a single molecular regulator.

Spatiotemporal Distribution of Afterslip following the 2014 Yutian Mw 6.9 Earthquake Using COSMO-SkyMed and Sentinel-1 InSAR Data

Spatiotemporal distribution of early afterslip is essential for seismic hazard evaluation and determination of fault friction properties. In this study, we used early post-seismic COSMO-SkyMed (19 February 2014–08 April 2014) and long-term Sentinel-1 (16 October 2014–17 June 2020) observations from multiple platforms over different periods to create a rate decay model driven by post-seismic afterslip. The combined observations provide full coverage of the post-seismic deformation following the 2014 Yutian Mw 6.9 earthquake that occurred at the southwestern end of the Altyn Tagh Fault. The observation and modeling results showed that post-seismic deformation was characterized by left-lateral strike-slip movement with minor normal slip, which was consistent with that of co-seismic rupture. The maximum early afterslip (7–55 days) was as large as approximately 0.09 m with a depth of 7 km in the west of co-seismic rupture, and the maximum long-term afterslip was about 0.24 m. The simulated post-seismic deformation caused by poroelastic rebound and viscoelastic relaxation suggests that the afterslip mechanism controls the post-seismic deformation. The coupling pattern of the aftershock and afterslip indicates that the aftershock was mainly caused by the afterslip. The post-seismic spatiotemporal features of the 2014 Yutian earthquake have significant implications for analyzing seismic hazards at the southwestern end of the Altyn Tagh Fault.

Poro-viscoelastic material parameter identification of brain tissue-mimicking hydrogels.

Understanding and characterizing the mechanical and structural properties of brain tissue is essential for developing and calibrating reliable material models. Based on the Theory of Porous Media, a novel nonlinear poro-viscoelastic computational model was recently proposed to describe the mechanical response of the tissue under different loading conditions. The model contains parameters related to the time-dependent behavior arising from both the viscoelastic relaxation of the solid matrix and its interaction with the fluid phase. This study focuses on the characterization of these parameters through indentation experiments on a tailor-made polyvinyl alcohol-based hydrogel mimicking brain tissue. The material behavior is adjusted to ex vivo porcine brain tissue. An inverse parameter identification scheme using a trust region reflective algorithm is introduced and applied to match experimental data obtained from the indentation with the proposed computational model. By minimizing the error between experimental values and finite element simulation results, the optimal constitutive model parameters of the brain tissue-mimicking hydrogel are extracted. Finally, the model is validated using the derived material parameters in a finite element simulation.

Wood as a hydrothermally stimulated shape-memory material: mechanisms of shape-memory effect and molecular assembly structure networks

Abstract This study aimed to evaluate the shape-memory effect (SME) of wood (Populus x beijingensis W. Y. Hsu) and identify the net-points and switches in its molecular and morphological structures. During several cycles of deformation and subsequent recovery, a high shape recovery rate and ratio were maintained. The transverse compression tests of wet and dry wood reveal that the hydrothermal coupling stimulation can considerably reduce the strength of wood. The X-ray diffraction characterization of wood under hydrothermal stimulation shows that the role of network nodes in the SME of wood is influenced by temperature. The wavenumber shifting and changes in the intensity ratio of the characteristic Fourier transform infrared peaks showed that hydrogen bonds acted as switches for the water-stimulated shape-memory behavior. By taking into account viscoelastic relaxation, a kinetic model derived from nonequilibrium thermodynamic fluctuation theory was used to describe the shape recovery process. The effects of hydration on recovery kinetics, activation, and dynamic mechanical behaviors were also studied. To explain the shape-memory mechanism of wood under hydrothermal stimulation, a hybrid-structure network model based on a single three-dimensional switch network was proposed in this study.

Effects of tissue degradation by collagenase and elastase on the biaxial mechanics of porcine airways

BackgroundCommon respiratory illnesses, such as emphysema and chronic obstructive pulmonary disease, are characterized by connective tissue damage and remodeling. Two major fibers govern the mechanics of airway tissue: elastin enables stretch and permits airway recoil, while collagen prevents overextension with stiffer properties. Collagenase and elastase degradation treatments are common avenues for contrasting the role of collagen and elastin in healthy and diseased states; while previous lung studies of collagen and elastin have analyzed parenchymal strips in animal and human specimens, none have focused on the airways to date.MethodsSpecimens were extracted from the proximal and distal airways, namely the trachea, large bronchi, and small bronchi to facilitate evaluations of material heterogeneity, and subjected to biaxial planar loading in the circumferential and axial directions to assess airway anisotropy. Next, samples were subjected to collagenase and elastase enzymatic treatment and tensile tests were repeated. Airway tissue mechanical properties pre- and post-treatment were comprehensively characterized via measures of initial and ultimate moduli, strain transitions, maximum stress, hysteresis, energy loss, and viscoelasticity to gain insights regarding the specialized role of individual connective tissue fibers and network interactions.ResultsEnzymatic treatment demonstrated an increase in airway tissue compliance throughout loading and resulted in at least a 50% decrease in maximum stress overall. Strain transition values led to significant anisotropic manifestation post-treatment, where circumferential tissues transitioned at higher strains compared to axial counterparts. Hysteresis values and energy loss decreased after enzymatic treatment, where hysteresis reduced by almost half of the untreated value. Anisotropic ratios exhibited axially led stiffness at low strains which transitioned to circumferentially led stiffness when subjected to higher strains. Viscoelastic stress relaxation was found to be greater in the circumferential direction for bronchial airway regions compared to axial counterparts.ConclusionTargeted fiber treatment resulted in mechanical alterations across the loading range and interactions between elastin and collagen connective tissue networks was observed. Providing novel mechanical characterization of elastase and collagenase treated airways aids our understanding of individual and interconnected fiber roles, ultimately helping to establish a foundation for constructing constitutive models to represent various states and progressions of pulmonary disease.

Modelling crack propagation during relaxation of viscoelastic material

A viscoelastic material which is stretched and is then held at constant elongation, normally results in decreasing stresses till the equilibrium has been reached. With the decreasing stresses a crack propagation is not expected as the energy of the system is decreasing. However, an initial damage could lead to an increase in the mechanical load on the undamaged chains during relaxation, leading to material degradation and crack propagation. While experimental investigations have been presented in the literature, modelling such an effect has not been thoroughly investigated. In this work, an initial framework for modelling the damage evolution during relaxation is presented. A mechanical model is coupled with a phase field to model the crack propagation. For simplicity, a linear viscoelastic model is implemented for the mechanical part. A mobility constant is employed to model the evolution of the phase field with the changing mechanical energy during relaxation. The evolution of phase field can be interpreted as the evolution with which the polymer chains get damaged. Different load conditions and geometries are simulated, which shows that the proposed framework is able to model the damage evolution during viscoelastic relaxation. Thus, with the help of the numerical model a physical explanation for the failure during relaxation is presented.Graphical abstract

The formation and evolution of Pluto's Sputnik basin prior to nitrogen ice fill

We simulate Sputnik Planitia (SP), Pluto's largest impact basin, from formation through cooling and relaxation prior to loading of the basin with N2 ice. To assess potential conditions for Pluto's interior that permit the formation of an SP-like basin in terms of depth and diameter, we consider impacts into targets that possess a subsurface ocean with variable ice shell thicknesses and thermal structures. We use a shock physics code to model excavation and transient crater collapse. Then, using the final temperature and density structure from the shock physics code as initial conditions, we simulate the subsequent cooling and viscoelastic relaxation of basin topography using a finite element model (FEM). We show that a thin ice shell (on the order of 100 km) overlying a thick ocean (on the order of 228 km) produces an SP-like basin in terms of diameter and pre-N2 depth. A basin formed in a much thicker ice shell (200 km) overlying a thin ocean (128 km) can reproduce SP if the ice shell is relatively warm and possesses a strong rheology. Our results suggest that SP was not a mascon basin prior to subsequent loading by N2 ice, but rather close to isostatically compensated. We find that SP could have developed into a mascon basin following N2 loading if the ice shell is thin, conductive and possesses a lithosphere capable of supporting the N2 load.