Strength Envelope Research Articles

Case study: 3D mobilized strength of compacted fill

Water infiltration can cause softening of compacted structural fill and a reduction of the shear strength from the peak compacted strength to the fully softened strength (FSS) with an accompanying reduction in drained factor of safety (FoS). This study presents two-dimensional (2D) and 3D stability analyses of a compacted fill slope failure that occurred 6 years after construction due to water leaking from a connection between the main and lateral water pipes in the water supply system. The compacted fill material primarily consists of high plasticity fine-grained soil. The 3D FoS at the end of construction is 2.44 using the peak compacted strength envelope. However, the 3D FoS is close to unity (1.0) when the FSS is assigned to the compacted fill material with the appropriate piezometric surface, which means the 2 H:1 V compacted fill slope softened to the FSS within 6 years. This is an interesting FSS case because the failure surface is 4 m deep and semi-circular, which differs from infiltration cases that exhibit a shallower and more planar surface.

Continuum-based heterogenous models for capturing brittle damage and failure of hard rocks under loading and unloading conditions

Grain-scale heterogeneities play a crucial role in the failure process of hard brittle rocks. Previous numerical investigations have often overlooked certain sources of heterogeneity and the inherent variability in laboratory test data during model calibration. This study aims to assess the impact of grain-scale heterogeneities on macro-property variability, internal stress variations, and unloading-induced brittle damage using continuum-based heterogeneous models. For this purpose, ten realizations of homogeneous (consisting of one mineral type) and heterogeneous (consisting of four mineral types) Voronoi Tessellated Models (VTMs) were generated. These models were calibrated to match the mean strength of undamaged Lac du Bonnet (LdB) granite. The heterogeneous VTM demonstrated a stronger agreement with LdB granite in terms of the shape of the strength envelope and peak strength variability compared to the homogeneous VTM. Analysis of internal elastic stresses revealed higher induced tensile stresses within the heterogeneous VTM than the homogeneous VTM during compressive loading. Furthermore, it was found that, in contrast to the homogeneous VTM, unloading-induced brittle damage in the heterogeneous VTM led to a reduction in its peak strength and Young’s modulus, aligning with laboratory test results. Therefore, the heterogeneous VTM is deemed a more representative model for hard rocks than the homogeneous VTM.

Inelastic analysis of octagonal concrete‐filled steel tubular short columns under eccentric loading

AbstractOctagonal concrete‐filled steel tubular (OCFST) columns combine the benefits of circular and square concrete‐filled steel tubular (CFST) columns so that they not only possess higher strength and ductility but also provide the ease of connection to composite beams. However, research studies have been very limited on the performance analysis of OCFST short beam‐columns subjected to eccentric loading. In this study, a fiber‐based numerical model is developed for the performance simulation of high‐strength OCFST short beam‐columns under eccentric loading. The simulation model takes into account material nonlinearities and concrete confinement induced by the octagonal steel tube. Computational methods are given that predict the axial load–moment interaction curves and moment–curvature responses of OCFST beam‐columns. The developed fiber model is verified against available test data with good accuracy. The influences of important parameters on the responses of high‐strength OCFST short beam‐columns are studied by means of utilizing the computational model. It is found that the behavior of OCFST beam‐columns is significantly influenced by the diameter‐to‐thickness ratio of the cross‐section, concrete strength, steel yield stress, and axial load ratio. Interaction equations are proposed for expressing the axial load–moment strength envelopes of the cross‐sections of OCFST beam‐columns and validated against numerical results.

Anisotropy in the Liquefaction Resistance of Fibre Reinforced Sand.

Adding discrete fibres to sand has been seen as a feasible technique to improve sand's strength as well as liquefaction resistance. Considering the anisotropic distribution of fibre orientations, the anisotropy in the liquefaction resistance of the reinforced sand is also introduced using fibres. Here, the triaxial compression and extension test results of unreinforced and fibre-reinforced sand in different density states are provided, from which the anisotropy in the liquefaction resistance of fibre-reinforced sand is demonstrated. Fibre reinforcement improves the liquefaction resistance of sand by introducing both the densifying effect and the confining effect. The inclusion of fibres increases both the slope and the intercept of the strength envelope in comparison with the unreinforced sand under triaxial compression, while the strength envelope is not affected by fibres under triaxial extension. Stress contribution of fibres makes the ESP of the composite under undrained loading reverse its direction to develop even though the phase transformation is absent. The stress ratio initiating the ESP reversal is irrespective of the fibre content but dependent on the density state under triaxial compression. Under triaxial extension, the stress ratio initiating the ESP reversal remains the same in the samples with varied density states and fibre contents. The mechanism correlating to the strength envelope and ESP reversal of the fibre-reinforced sand was demonstrated following a rule of mixture based constitutive modelling framework. By introducing an alternatively defined pore pressure ratio that incorporates the stress contribution of fibres, the liquefaction state of the fibre reinforced sand is reasonably assessed. Liquefaction remains absent in the sand once the fibres are mixed. The anisotropy in the liquefaction resistance of fibre-reinforced sand arises, as the predominant role played by the fibres to suppress the liquefaction is different when varied loading paths are involved, which is sourced from the anisotropic distribution of fibre orientations.

Lithospheric contraction concentric to Tharsis: 3D structural modeling of large thrust faults between Thaumasia highlands and Aonia Terra, Mars

Large thrust faults on Mars are caused by lithospheric planetary contraction. The geometry of these faults is linked with the mechanical behavior of the lithosphere. Tharsis, the largest volcano-tectonic province on Mars, controls the global tectonic pattern of the planet. Here, we present a study of five large thrust faults concentric to Tharsis, located between the Thaumasia Highlands and the Argyre impact basin. We applied a 3D structural modeling, using a combination of fault-parallel flow and trishear algorithms to estimate the geometry and kinematics of the faults at depth. The modeled faults show an upper planar part dipping 33° to 40°, rooting with a listric geometry into horizontal levels at 13–27 km depth, with fault slips of 801–3366 m. The general out-of-Tharsis vergence, the listric fault geometries and the deepening of the depth of faulting toward Thaumasia outline an incipient thrust wedge architecture. Assuming that the largest faults rooted at the Brittle-Ductile Transition, we calculate a heat flow at the time of faulting of 24–54 mW m−2. The obtained strength envelopes for dry and wet conditions show that all the strength of the lithosphere was located in the upper half of the crust.

Revisiting the face stability of circular tunnels driven in strength nonlinearity soils

The face stability of shield-driven tunnels in soil materials with distinct strength nonlinearity is re-evaluated, based on the kinematic limit analysis theorem. The nonlinear Power-Law failure criterion is piecewise approximated by a multi-tangent technique to improve the conventional single straight-line substitution. A new multi-cone rotational mechanism formed by n curvilinear cones is proposed to accommodate the nonlinearity of the strength envelope considering the associative flow rule. The analytical solution of the critical support pressure on tunnel faces is derived from the energy equilibrium equation. Calculation results show that the multi-tangent method can improve the calculated values by up to 20% compared to the generalized tangent method. According to the back-calculated stress vectors, it is interestingly found that the stress level is not the critical factor influencing the tunnel face stability, while the shear strength parameters are the direct factor. Although the upper-bound analysis is a kinematic method without involving stress analysis, the parametric form of the strength criterion gives a chance to infer the possible stress distribution on the rupture surface and explain the variational trend of the result based on the distribution of back-calculated stress vectors.

Research on meso-equivalence simulation method of freeze–thaw concrete under multiaxial loading

In view of the influence of freeze–thaw cycles (FTCs) on the spatial strength of principal stress was not considered, a meso-equivalence simulation method was developed, considering the heterogeneous characteristics of FT concrete, to accurately depict frost damage behavior, which can be used to simulate the spatial strength distribution on principal stress of concrete with FTCs. Additionally, a meso-calculation method was employed to verify the loading boundary conditions and material constitutive models. Subsequently, the effectiveness of the meso-equivalence simulation method was established by its application to the mechanical analysis of FT concrete under multiaxial loads, alongside the formulation of meso-equivalence calculation models for FT-reinforced concrete (RC) beams. The results underscore the efficacy of the meso-calculation and meso-equivalence models in capturing damage mode within concrete, subjected to FTCs under multiaxial loads. Importantly, the strength envelope derived from the calculation model and the corresponding strength criterion for the multiaxial loading limit state of FTC-affected concrete aligns consistently with test findings. Moreover, the validation of the meso-equivalence simulation method is established by comparative analysis of the mechanical behavior of RC beams under FTCs, compared with test results, which not only enhances understanding of FT response but also contributes to the advancement of durability assessment.

The 8th Victor de Mello lecture: role played by viscosity on the undrained behaviour of normally consolidated clays

Phenomena that do not obey Terzaghi’s principle of effective stress (PES) are related to strain rate and time effects. This issue led the author to refer to early articles in soil mechanics, which used to consider the shear resistance of clays as a combination of two components: a frictional and a viscous one. In these articles the viscous component was assigned to the distortion of highly viscous adsorbed water layer in the contact points between grains along the plane where shearing takes place. Assuming the shear resistance of plastic soils comprises frictional and viscous resistance components, a shear stress equation can be added to the PES. It is shown that Mohr’s circle of effective stress is the sum of two ellipses: the viscosity and the friction ellipses. The ordinates of the viscosity and the friction ellipses represent the viscous and the frictional components of shear resistance in different planes, respectively. This approach leads to a failure criterion considering strain rate, according to which failure takes place whenever the friction ellipse touches the strength envelope, which is the 'e φ sloped straight line passing through the origin, 'e φ being the Hvorslev’s true angle of friction. By adding such shear stress equation to the PES, a model that explains strain rate and time effects is developed. Predictions of the proposed model are compared to results from tests carried out on San Francisco Bay Mud specimens.

Seismic design and experimental testing of precast tolerance concrete shear wall subject to cyclic load

A precast tolerance concrete shear wall (PTCW) characterized by proper-length steel connectors is proposed to adjust the tolerance and improve the joint failure model of the precast wall. The design method for this wall is also presented in this study and verified by test results. Four full-scale walls, including two cast-in-situ walls (RCWs) and two precast walls (PCWs) with axial compression ratios of 0.2 and 0.3, respectively, were tested under lateral loading. The seismic behavior of the panels, including the failure modes, hysteretic behavior, strength envelope curves, stiffness degradation, energy dissipation, and steel strains, was analyzed and discussed herein. The test results showed that all four walls were in a flexural-shear failure mode, and the PCWs showed better performance in terms of hysteretic behavior, strength envelope, stiffness, and energy dissipation, particularly under an axial compression ratio of 0.3. Furthermore, the peak load of PCW1 (axial compression ratio is 0.2) and PCW2 (axial compression ratio is 0.3) increased by 5.93% and 15.51%, and the yield load increased by 12.1% and 11.39% compared with those of RCW1 (axial compression ratio of 0.2) and RCW2 (axial compression ratio of 0.3), respectively. The equivalent viscous damping coefficients of all four panels were higher than 0.05. The lateral bearing capacity of the walls increased with an increase in the axial compression ratio. Moreover, no visible damage or slippage was observed when the PCWs were demolished after testing. The main nonlinear strain was concentrated on the vertical bar and not on the connectors, indicating that the plastic hinge position was transferred from the joint interface to the wall panel. The results indicated that the use of steel connectors to strengthen the joints can achieve the seismic design concept of strong joints and weak members.

Shear behavior and off-fault damage of saw-cut smooth and tension-induced rough joints in granite

The damage of rock joints or fractures upon shear includes the surface damage occurring at the contact asperities and the damage beneath the shear surface within the host rock. The latter is commonly known as off-fault damage and has been much less investigated than the surface damage. The main contribution of this study is to compare the results of direct shear tests conducted on saw-cut planar joints and tension-induced rough granite joints under normal stresses ranging from 1 MPa to 50 MPa. The shear-induced off-fault damages are quantified and compared with the optical microscope observation. Our results clearly show that the planar joints slip stably under all the normal stresses except under 50 MPa, where some local fractures and regular stick-slip occur towards the end of the test. Both post-peak stress drop and stick-slip occur for all the rough joints. The residual shear strength envelopes for the rough joints and the peak shear strength envelope for the planar joints almost overlap. The root mean square (RMS) of asperity height for the rough joints decreases while it increases for the planar joint after shear, and a larger normal stress usually leads to a more significant decrease or increase in RMS. Besides, the extent of off-fault damage (or damage zone) increases with normal stress for both planar and rough joints, and it is restricted to a very thin layer with limited micro-cracks beneath the planar joint surface. In comparison, the thickness of the damage zone for the rough joints is about an order of magnitude larger than that of the planar joints, and the coalesced micro-cracks are generally inclined to the shear direction with acute angles. The findings obtained in this study contribute to a better understanding on the frictional behavior and damage characteristics of rock joints or fractures with different roughness.

Single-flux-quantum-based qubit control with tunable driving strength

Single-flux-quantum (SFQ) circuits have great potential in building cryogenic quantum-classical interfaces for scaling up superconducting quantum processors. SFQ-based quantum gates have been designed and realized. However, current control schemes are difficult to tune the driving strength to qubits, which restricts the gate length and usually induces leakage to unwanted levels. In this study, we design the scheme and corresponding pulse generator circuit to continuously adjust the driving strength by coupling SFQ pulses with variable intervals. This scheme not only provides a way to adjust the SFQ-based gate length, but also proposes the possibility to tune the driving strength envelope. Simulations show that our scheme can suppress leakage to unwanted levels and reduce the error of SFQ-based Clifford gates by more than an order of magnitude.

A phase‐field model for hydraulic fracture nucleation and propagation in porous media

AbstractMany geo‐engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase‐field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase‐field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy‐based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase‐field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, for example, rocks. Built on the variational formulation of previous phase‐field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two‐dimensional test cases with existing analytical solutions demonstrates that the proposed phase‐field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three‐dimensional domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.

Effects of Particle Shape and Packing Density on the Mechanical Performance of Recycled Aggregates for Construction Purposes

This paper employs the discrete element method (DEM) to study the mechanical properties of artificial crushed stone. Different grain shapes and gradations are considered, and three types of 3D artificial stone models are generated based on the statistical conclusions in the relevant literature and the observed data. Concurrently, the 3D models of the artificial stones are divided into three groups by their shape parameters (elongation index and flatness index). Furthermore, three types of gradation with different Cu (coefficient of uniformity) and Cc (coefficient of curvature) are also considered. Then, several 3D triaxial compression tests are conducted with the numerical methods to determine the relationship between the grain shapes and their mechanical characteristics. The test results showed that there was a positive correlation between a particles’ angularities and the maximum deviatoric stress in the triaxial compression tests when there were obvious distinctions between the particles. In addition, gradations had a conspicuous impact on the stiffness of the sample. The stress–strain curve possessed a larger slope when the coefficient of curvature was bigger. In terms of shear strength, the results in this paper align well with the traditional shear strength envelope which are convincing for the dependability of the methods used in this paper. The radial deformation capacity and volume strain of the specimen during the triaxial compression tests are also examined. It is believed that there were great differences in deformability between different samples. At the mesoscopic level, the change in coordination number is identified as the fundamental reason for the change in volume strain trend.

A low-dimensional energy model to distinguish strength and stability in a jammed column

A low-dimensional “phase space” energy model is presented that outlines contributions to relative strength and stability of two jammed granular+fluid columns: one of small, fine, nearly monodisperse grains and one of larger, coarse, more polydisperse grains. Both granular (dry) structure and pore-fluid (wet) structure contributions to the strength and stability are determined by experimental parameters obtained for bulk measurements from previously published studies. Physical quantities that are controlled by granular-dominant parameters or controlled by fluid-dominant parameters can be fit to sigmoid functions that produce a phase space envelope of strength and stability for a particular column. The novel model presented here replicates this phase space envelope by defining a single granular temperature based on the sigmoid fits. The model demonstrates the overall role that granular temperature plays in both the relative strength and stability of the system in a manner that allows the two characteristics to be distinguished.

Bearing capacity of shallow foundations on Florida limestone: A study of single layer and rock-over-sand subsurface

This study investigated the bearing capacity of shallow foundations on Florida limestone using 3D finite element modeling (FEM) with a bilinear strength envelope based on rock formations and bulk dry unit weights. Most Florida limestones at shallow depths are highly porous and have a low density and strength, thereby resulting in a ductile stress–strain response and contractive volumetric behavior under service loads, both of which are not well characterized by conventional bearing capacity methodologies.In this study, a bilinear Mohr–Coulomb elastoplastic model with a non-associated flow rule (negative dilation angle) was integrated into a 3D FEM code from a series of laboratory triaxial strength test results. Numerical simulations were performed for different footing sizes, shapes, and embedment depths using homogeneous rock and rock-over-sand subsurface. Beginning with the numerical results from a strip footing on a homogeneous rock deposit, a bearing capacity equation was developed for bilinear strength conditions. Further, the width, shape, and embedment factors were developed, and a reduction factor was developed based on the rock layer thickness and moduli of rock and sand for the rock-over-sand condition often encountered in South Florida. The study concludes with a comparison between developed equations and conventional bearing capacity methodologies.

Design and Analysis of a New Multi-Part Composite Frangible Cover.

In this paper, a new multi-part composite frangible cover (MCFC) was designed and fabricated. The frangible cover, manufactured with a traditional manual lay-up method, is designed to conduct a simulated missile launch test using a specially developed test device. A weak zone structure of the composite multi-part frangible cover was designed, and the separation process of the cover was studied by numerical simulation. Based on the strength envelope of the weak zone and the equal-strength design principle, a design method for the weak zone structure of the composite multi-part frangible cover was proposed. A finite element model of the composite multi-part frangible cover was established, and the separation process was numerically simulated and analyzed. Afterward, the verification experiments were carried out. Close agreements between the numerical and experimental results are observed.

Mechanical Properties of Composite Silty Soil Modified with Cement and Zirconia-Based Nanopowder.

This study assessed the modification effects of zirconia-based nanopowder and cement contents and curing age on the mechanical properties of silty soil. The orthogonal test design was applied to derive the best combination of each influencing factor using the lateral unconfined compressive test. Two-dimensional particle flow code (PFC2D) distinct-element modeling software was also used to fit and analyze the test curves, as well as simulate the triaxial test with the derived parameters. The test results reveal the optimal combination of 20% cement, 2% zirconia-based nanopowder, and 28 d curing age. The extreme difference table was used to plot the orthogonal trend diagram, and cement content was found to be the most significant factor controlling the silty soil strength. The maximum peak stress was 2196.33 kPa under the optimum combination of factors, which could be obtained through the index estimation, and these results were experimentally verified. According to the predicted strength envelope, the cohesive force of nanopowder-cement-modified silty soil in the optimal proportion was 717.11 kPa, and the internal friction angle was 21.05°. Nano zirconium dioxide will accelerate the hydration reaction of cement, the flocculent structure produced by the hydration of cement and soil particles connected to each other, play the role of filling and anchoring, and thus increase the strength of the nano-zirconium dioxide, and the optimal dosage of nano-zirconium dioxide is 2%.

Elasto-plastic and post-yield weakening jointed rockmass response in a comparison of equivalent-continuum and explicit structural models

The strength and stiffness of a composite rockmass (intact material, defects, joints, bedding, etc.) are primary inputs for the engineering analysis of rock slopes and underground excavations. The Geological Strength Index (GSI) has served rock engineering for three decades as a rockmass assessment system based on the blockiness of the rockmass and discontinuity condition. GSI is used to factor the Hoek–Brown strength envelope for intact rock to represent jointed rockmasses in conventional equivalent-continuum numerical modelling. Modern numerical tools can represent networks of discrete (explicit) structure with assigned discontinuity properties. This provides an opportunity to compare explicit structural modelling with the conventional implicit equivalent-continuum approach. In this paper, explicit structural models are developed using elasto-plastic (constant-strength) constitutive models for intact rock and realistic parametric ranges for explicit structure properties. Explicit model results are compared to equivalent-continuum results, validating the classical implicit approach while also identifying key limitations. Explicit models with post-yield weakening of the intact and structural elements are then developed and compared to post-yield weakening implicit models that use post-yield dilation and empirical relationships between peak and residual GSI. The results provide guidance for practical modelling in strain-weakening rockmasses, including recommendations for post-yield dilation in the traditional implicit approach.

Two-dimensional face stability analysis in rock masses governed by the Hoek-Brown strength criterion with a new multi-horn mechanism

The face stability problem is a major concern for tunnels excavated in rock masses governed by the Hoek-Brown strength criterion. To provide an accurate prediction for the theoretical solution of the critical face pressure, this study adopts the piecewise linear method (PLM) to account for the nonlinearity of the strength envelope and proposes a new multi-horn rotational mechanism based on the Hoek-Brown strength criterion and the associative flow rule. The analytical solution of critical support pressure is derived from the energy-work balance equation in the framework of the plastic limit theorem; it is formulated as a multivariable nonlinear optimization problem relying on 2m dependent variables (m is the number of segments). Meanwhile, two classic linearized measures, the generalized tangential technique (GTT) and equivalent Mohr-Coulomb parameters method (EMM), are incorporated into the analysis for comparison. Surprisingly, the parametric study indicates a significant improvement in support pressure by up to 13% compared with the GTT, and as expected, the stability of the tunnel face is greatly influenced by the rock strength parameters. The stress distribution on the rupture surface is calculated to gain an intuitive understanding of the failure at the limit state. Although the limit analysis is incapable of calculating the true stress distribution in rock masses, a rough approximation of the stress vector on the rupture surface is permitted. In the end, sets of normalized face pressure are provided in the form of charts for a quick assessment of face stability in rock masses.

An intrusive cryomagmatic origin for northern radial labyrinth terrains on Titan and implications for the presence of crustal clathrates

Titan's labyrinth terrains are an organic-rich, topographically elevated, highly dissected and puzzling geomorphic unit. How these features came to be composed of organics and remain elevated may hold clues about Titan's complicated history, and in particular the dynamics and composition of Titan's crust. One subtype of labyrinth terrains, the radially networked labyrinth terrains, is found in Titan's mid-latitudes. They are dome-shaped with radial drainage patterns and appear to be a clustering of uplifted, organic-rich dissected plateaux. We use scaling relationships to determine whether they formed as elevated surfaces that were uplifted by solid-state diapirs or cryomagmatic laccoliths at depth. Based on the large, variable spacing between features, we find it unlikely that they formed via density-driven diapirism. Instead, their dimensions suggest that they are cryomagmatic intrusions that formed near the most prominent rheological contrast in the ice shell, the brittle–ductile transition (BDT). At that location, we argue that intrusions spread horizontally and inflate, forming large cryomagmatic laccoliths (upturned, large saucer-shaped sills). The intrusions would flex the overlying lithosphere and surficial sedimentary layers (likely undifferentiated plains), resulting in prominent domed features that are susceptible to erosion and incision by methane rain and wind. This process then leads to the highly dissected, dome-shaped labyrinth terrains. To determine whether the laccoliths formed near Titan's BDT, we calculate lithospheric strength envelopes for a pure water ice shell with and without an insulating methane clathrate crust using two conductive heat flows: 4 and 7 mW/m2. The plausible range for the BDT for an ice shell with a 1 km thick methane clathrate crust is 12–34 km for the 4 mW/m2 heat flow. This agrees well with the expected intrusion depths of the laccoliths (21–28 km) associated with a cluster of radial labyrinths in Titan's northern mid-latitudes, as derived from a scaling relationship relating intrusion depth to dome width. We find that methane clathrate thicknesses of 2 km or greater result in a BDT that is generally too shallow (3–24 km) to match our observations. If we consider the higher heat flow, these BDTs are even shallower. Although methane clathrate is known to be stronger than ice, its low thermal conductivity significantly raises the underlying ice shell's temperature, which results in a significantly thinner lithosphere that is not able to support these large plateaux. We conclude that in the location of this radial labyrinth terrain cluster there may be intrusive cryomagmatic activity approximately 21 km deep within the water ice shell, with a putative surface methane clathrate crust, if it exists atop a conductive ice shell, that is constrained to be <2 km thick.