Homotopy Research Articles

High-Precision Vertical Deformation of the Chinese Mainland Constrained by Leveling and GNSS data

Summary A high-precision and high-resolution vertical velocity for the Chinese mainland is obtained by integrating precise leveling and GNSS data, using a Helmert joint adjustment method. The results show that the surface vertical rates range between -3.0 and 3.9 mm/yr with continuous deformation in most areas, except the obvious subsidence at the rates of -15.0 to -94.2 mm/yr induced by groundwater exploitation in the North China Plain. Particularly, the central and southern Tibet, Tien Shan, Alashan, Ordos, eastern Cathaysia, and Northeast China uplift at the rates of 0.5 – 3.9 mm/yr; the southeastern Tibetan Plateau, Sichuan basin, and Yangtze block are dominated by surface subsidence at the rates of -3.3 to -0.5 mm/yr. Furthermore, the vertical rates vary little between the eastern and western regions of the Chinese mainland despite their pronounced differences in horizontal deformations. The effects of gravity isostasy and non-tectonic factors, including the environmental mass loads, Glacier Isostatic Adjustment (GIA), poroelastic expansion/compression, and mining operations have partially contributed to the vertical deformation of the Chinese mainland. Overall, this velocity reflects the complicated deformation features induced by the multiple geodynamic processes of the Chinese mainland. These geodynamic processes include isostasy, orogenic processes, and geothermal anomalies associated with slab subduction/plate collision.

On multi-stage deformation and gradual energy absorption of 3D printed multi-cell tubes with varying cross-section

Thin-walled structures are widely used in passive safety systems due to their excellent lightweight and controllable progressive deformation performance. While designing to improve the energy absorption performance of thin-walled structures, it is also crucial to consider reducing the damage received by protected objects during accidental collisions. In this work, stepwise and continuous graded multi-cell tubes with varying cross-sections were proposed based on the fused deposition modeling technique to achieve a gradual energy absorption process. Compression responses and energy absorption characteristics of the graded multi-cell tubes were investigated by quasi-static compression tests. Experimental results showed that the stepwise graded multi-cell tubes produced obvious multi-stage deformation and energy absorption during axial compression. The specific energy absorption presented an upward trend with the increased number of stepwise segments. For the same ranges of cross-section variations, continuous graded multi-cell tubes exhibited more plastic hinges and better energy absorption performance due to the absence of discontinuous interfaces. In addition, continuous graded multi-cell tubes showed continuous folding deformation with the gradually decreasing folding wavelength. Thus, it exhibited a steadily increasing trend in crushing force and realized the gradual energy absorption mode.

Semi-analytical framework for modelling of surface heating and its impact on nonlinear stability of nanotube reinforced doubly curved porous fiber composite panels

In elevated temperatures, the stiffness of the panel deteriorates and can lead to loss of stability of structures. Hence, the stability analysis of thin-walled structures is a critical investigation in the thermal environment. Finding such criticality, the present semi-analytical study investigated the nonlinear stability behaviour of porous doubly curved thin-walled panels subject to surface heating like dome heating and localised heating. To improve the stiffness of the panel, the matrix is reinforced with carbon nanotubes (CNTs). Improper mixing of such nanoparticles can lead to agglomeration or bundling effect, which may reduce the structure's stiffness. Still, its investigation in surface heating can be an essential aspect that is found to be untouched by researchers for porous shell panels. The effective material properties of the three-phase composite panel are determined using the Eshelby-Mori-Tanaka (E-M-T) approach and the Chamis homogenisation technique. Using the variational principle, governing equations are derived and further simplified to non-linear algebraic equations using the Galerkin technique. Gibson and Ashby foam model is used to model cell structure. It is observed that a complete agglomerated state results in higher post-buckling strength, a continuous deformation path due to biaxial compression and curved geometry, and porosity of cellular structure distribution and cell walls are the important parameters to study the stability of panel in a thermal environment are the major conclusions drawn from the current study. Current investigation can be an important contribution towards modelling aircraft structure in supersonic airflow and furnace walls of thermal power plants subjected to localised heating.

Compressive behavior and deformation mechanisms of gradient microstructures in a dissimilar welded joint

In this study TC4/TC17 titanium alloy dissimilar joints were obtained via vacuum electron beam welding. The compressive behavior of base metals and joint at different strain levels was studied, along with microstructural and fracture surface observations. The true stress–strain curves of base metals and joint at different strains demonstrated smooth and continuous plastic deformation characteristics. The compressive yield strength of the joint (1065 MPa) was basically equivalent to that of TC17 alloy (1196 MPa), which was significantly higher than that of TC4 alloy (920 MPa) for the failed samples. However, the compressive fracture strain of the joint was lower than that of base metals. Comparison of the microstructures of base metals and joint deformed at different strains, it revealed that the equiaxed α phase in TC4 alloy mainly underwent the deformation, while the TC17 alloy mainly relied on the rotation of lamellar grains to resist the deformation. In the dissimilar joint with a large microstructural gradient, the compressive deformation of several zones, including the weld zone, far heat-affected zone and the base metal, was also related to the rotation of finer grains.

A geometric Elliott invariant and noncommutative rigidity of mapping tori

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C⁎-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C⁎-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

$L^{2}$-cohomology and quasi-isometries on the ends of unbounded geometry

The paper explores the minimal and maximal L^{2} -cohomology of oriented Riemannian manifolds, focusing on both the reduced and the unreduced versions. The main result is the proof of the invariance of the L^{2} -cohomology groups under uniform homotopy equivalences that are quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. Moreover, a map f between two Riemannian manifolds (X,g) and (Y,h) is quasi-isometric on the unbounded ends if X = M \cup E_{X} , where M is the interior of a manifold of bounded geometry with boundary, E_{X} is an open subset of X and the restriction of f to E_{X} is a quasi-isometry. Finally, some consequences are shown: the main ones are the definition of a mapping cone for L^{2} -cohomology and the invariance of the L^{2} -signature.

Constraints on the 2013 Saravan intraslab earthquake using permanent GNSS, InSAR and seismic data

SUMMARY On 16 April 2013, an Mw = 7.7 earthquake struck the border of Iran and Pakistan in the central part of the Makran subduction zone with a reported depth of 80 km by USGS. This rare event in this poorly instrumented region helps to shed light on the kinematics of the subducting slab. We investigate source parameters of the Saravan intraslab normal earthquake using RADARSAT-2 SAR images in three ascending tracks, nine permanent GNSS sites and teleseismic data. The maximum coseismic displacement occurred at the SRVN GNSS station with 54.1 mm southeast horizontal and 42.7 mm upward vertical displacements. The coseismic ascending InSAR displacement maps illustrate a continuous and smooth NE-trending elliptical shape deformation pattern with a maximum of ∼29 cm of displacement away from the satellite. We use 25 broad-band teleseismic P-waveforms to estimate the focal mechanism of the main shock. A joint uniform inversion of InSAR, GNSS and teleseismic data reveals a NW-dipping SW-striking fault and a primarily normal-faulting earthquake with a minor right-lateral strike-slip component. The static slip distribution of the InSAR coseismic maps localizes variable slip at depths between 50 and 81 km with a maximum amplitude >3 m at 60–75.5 km depth, rupturing the oceanic crust of the subducted slab. The kinematic slip distribution exhibits a well-constrained slip pattern with a nucleation depth of 65 km. The source time function indicates that the earthquake reaches its maximum moment tensor release at ∼8 and ∼16 s. The NE-trend of the Saravan earthquake slip pattern, the orientation of the volcanic arc, and the distribution of the intraslab intermediate-depth normal earthquakes provide new insights into slab geometry in the central Makran subduction zone. We suggest that the slab bending at the hinge of subducting Arabian Plate is oblique along a NE–SW direction parallel to the volcanic arc rather than the shoreline or deformation front, and it is likely to be the reason for an oblique volcanic arc in the Makran subduction zone. These new constraints on the Makran slab geometry will help further studies in establishing realistic coupling maps for seismic hazard assessment.

The Gauss–Bonnet theorem

The Gauss–Bonnet theorem is a crowning result of surface theory that gives a fundamental connection between geometry and topology. Roughly speaking, geometry refers to the “local” properties—lengths, angles, curvature— of some fixed object, while topology seeks to identify the “global” properties that are unchanged by a continuous deformation, such as stretching or twisting. The theorem formalizes an intuitive idea: continuous changes to curvature on one region of a surface will be balanced out elsewhere, so the total curvature of the surface stays the same. Explicitly, the Gauss–Bonnet theorem says that a surface’s total curvature, defined using its local Gaussian curvature, is directly proportional to the number of holes in the surface, which comes from an invariant quantity called its Euler characteristic. The Euler characteristic is a way of classifying which surfaces can be continuously deformed into one another; as an informal example, the classic joke that “a topologist is a person who cannot tell the difference between a coffee mug and a doughnut” comes from the fact that the objects each have one hole. Even though a coffee mug and a doughnut have visibly different geometric shapes, according to the Gauss–Bonnet theorem, both objects will have the same total curvature. The proof itself is delightfully systematic: we first find the total curvature of a curve on a plane, extend that result to curves on three-dimensional surfaces, extend that result to “polygons” on surfaces, and finally the entire surface. In Section 2, we prove Hopf’s Umlaufsatz for the total curvature of a simple closed curve in R2. Sections 3, 4, and 5 introduce concepts from differential geometry to define Gaussian curvature. In Section 6, we prove the local Gauss–Bonnet theorem for the total curvature of a surface polygon. At last, in Section 7, we prove the global Gauss–Bonnet theorem for compact surfaces by covering the surface with polygons and applying the local Gauss–Bonnet theorem to each one. Our discussion focuses on exposition, and references will be given in place of tedious computations when reasonable. This paper assumes a somewhat rigorous understanding of multivariable calculus and linear algebra, as well as some elementary group theory.

On the path of spin-[formula omitted] Berry phase

This paper investigates the relation between the Berry phase of a spinor accumulated along a path corresponding to an adiabatically varying magnetic field and the conformal deformation along the path in the parameter space. The first part demonstrates that the accumulated phase can be controlled by continuous deformation characterized by a single conformal parameter. The source of the corresponding field strength responsible for the deformed phase path is shown to be a conformally deformed monopole located at the origin in the parameter space with the fundamental strength g=±1/2. In the second part, we model a rotating magnetic field coupled to a charged spinor with a time-varying amplitude subjected to the conformal deformation, realizing a single-variable deformation of the evolution process of the phase path.

Continuous Dynamic Recrystallization and Deformation Behavior of an AA1050 Aluminum Alloy during High-Temperature Compression

Continuous dynamic recrystallization (CDRX) forms a new recrystallized microstructure through the progressive increase in low-angle boundary misorientations (LAGBs) during the hot forming of metallic materials with high stacking fault energy (SFE), such as aluminum alloys. The present work investigates the effect of deformation parameters on the evolution of the dynamic recrystallization microstructures of an AA1050 aluminum alloy during compression at elevated temperatures. The alloy microstructure is investigated at deformation temperatures and strain rates in the range of 300 °C to 500 °C and 0.001 to 0.8 s−1. A well-defined substructure and subsequent DRX grains provide indication that recrystallization can proceed with continued strain under high-temperature compression. At a strain rate of 0.1 s−1, the DRX fraction is observed to be 0.25 at a temperature of 300 °C. This fraction increases to 0.32 as the temperature rises to 400 °C. The recrystallization mechanism is identified by analyzing the flow stress, the evolution of the subgrain misorientation angle, and the distribution of recrystallized grains. The observations of discontinuous dynamic recrystallization (DDRX) and CDRX under various deformation parameters are discussed. Moreover, the main substructure evolution laws observed from the high-temperature compression of an AA1050 Al alloy are summarized.

Land surface response to groundwater drawdown and recovery in Taiyuan city, Northern China, analyzed with a long-term elevation change measurements from leveling and multi-sensor InSAR

In recent years, a comprehensive program of groundwater management was adopted in Taiyuan city, Northern China, with the aim to prevent the lowering of groundwater level and manage land subsidence due to excessive groundwater pumping in the past decades. Groundwater recovery has been observed in this city following the termination of groundwater pumping. While the link between land subsidence and groundwater pumping is well documented, how the land surface responds to the recovery of groundwater has not been studied in detail. In this study, we analyzed the leveling data from 1956 to 2000 to investigate changes in ground elevation in response to groundwater drawdown prior to the termination of groundwater pumping and combined them with InSAR (Interferometric Synthetic Aperture Radar) displacement data during 2003–2020 to evaluate the effect of groundwater recharge after the termination of groundwater pumping. A method was proposed to merge InSAR time series of displacement derived from multiple satellites (ENVISAT, COSMO-SkyMed, TerraSAR-X, and Sentinel-1) to achieve a consistent, long-term continuous deformation. We found that previous groundwater depression cones in Taiyuan (Xizhang, Wanbailin, Wujiabao, and Xiayuan) have turned into groundwater recovery and the corresponding subsidence centers have switched to land uplift. The subsidence center of Xizhang, north of Taiyuan city, experienced a total subsidence of 749 mm by Aug 2003; it then turned into uplift with an amount of rebound of 9.2 % of previous subsidence by the end of 2020. In central area of Taiyuan, the cumulated subsidence is 1127 mm in Wanbailin, 1817 mm in Xiayuan, and 3343 mm in Wujiabao. Beginning in Aug 2010, they all turned into land uplift, with a rebound of 7.9 %, 4.8 % and 3.6 % of the previous subsidence in each center, respectively. An analysis of the correlation between groundwater level and ground displacement suggests that the ground uplift is related to the poro-elastic rebound mechanism in the porous aquifer system, which is caused by the rise in pore pressure with groundwater recovery, and the amount of subsidence/uplift is controlled by the stratigraphic properties. The adverse impacts on the built environment due to the rising groundwater level were discussed. The findings improve our understanding of anthropogenic ground deformation in complex urban aquifers influenced by groundwater pumping and recharge and provides essential information that can contribute to future groundwater management and groundwater-related hazard mitigation over the Taiyuan city.

Experimental investigation on the anti-detonation performance of composite structure containing foam geopolymer backfill material

The compression and energy absorption properties of foam geopolymers increase stress wave attenuation under explosion impacts, reducing the vibration effect on the structure. Explosion tests were conducted using several composite structure models, including a concrete lining structure (CLS) without foam geopolymer and six foam geopolymer composite structures (FGCS) with different backfill parameters, to study the dynamic response and wave dissipation mechanisms of FGCS under explosive loading. Pressure, strain, and vibration responses at different locations were synchronously tested. The damage modes and dynamic responses of different models were compared, and how wave elimination and energy absorption efficiencies were affected by foam geopolymer backfill parameters was analyzed. The results showed that the foam geopolymer absorbed and dissipated the impact energy through continuous compressive deformation under high strain rates and dynamic loading, reducing the strain in the liner structure by 52% and increasing the pressure attenuation rate by 28%. Additionally, the foam geopolymer backfill reduced structural vibration and liner deformation, with the FGCS structure showing 35% less displacement and 70% less acceleration compared to the CLS. The FGCS model with thicker, less dense foam geopolymer backfill, having more pores and higher porosity, demonstrated better compression and energy absorption under dynamic impact, increasing stress wave attenuation efficiency. By analyzing the stress wave propagation and the compression characteristics of the porous medium, it was concluded that the stress transfer ratio of FGCS-ρ-579 was 77% lower than that of CLS, and the transmitted wave energy was 90% lower. The results of this study provide a scientific basis for optimizing underground composite structure interlayer parameters.

The influence of withdrawal-recharging pattern on the deformation characteristics of sand in confined aquifer

Land subsidence caused by excessive groundwater withdrawal has become a global hazard, which demands further researches and the potential measures to control. Using the FlowTrac Ⅱ consolidation test system, six compression tests were designed to investigate the stress state and stress paths of sand within confined aquifers under conditions of withdrawal and recharging. The deformation characteristics of aquifer sand were studied under different withdrawal-recharging patterns. During pumping and recharge processes, sand deformation responses were observed to lag behind changes in applied stress. The characteristics of this hysteresis effect on deformation were summarized. The alternating phenomenon of rebound and compression of sand deformation under the recharging process is analyzed. When the recharging effect was relatively small than withdrawing effect under the stable withdrawal-recharging pattern, the compression deformation was observed in the recharging process. The research results provide a rational explanation for the continuous compression deformation of the aquifer during groundwater level recovery and offer experimental evidence for the rational design of artificial groundwater recharge in engineering construction.

Homotopy Classification of Loops of Clifford Unitaries

Homotopy Classification of Loops of Clifford Unitaries

Mechanical characterization of spectrin at the molecular level

Spectrin, a large cytoskeletal protein, consists of a heterodimeric structure comprising α and β subunits. Here, we have studied the mechanics of spectrin filament as a major constituent of dendrites and dendritic spines. Given the intricate biological details and compact biological construction of spectrin, we've developed a constitutive model of spectrin that describes its continuous deformation over three distinct stages and it’s progressive failure mechanisms. Our model closely predicts both the force at which uncoiling begins and the ultimate force at which spectrin fails, measuring approximately 93 ~ 100 pN. Remarkably, our predicted failure force closely matches the findings from AFM experiments focused on the uncoiling of spectrin repeats, which reported a force of 90 pN. Our theoretical model proposes a plausible pathway for the potential failure of dendrites and the intricate connection between strain and strain rate. These findings deepen our understanding of how spectrin can contribute to traumatic brain injury risk analysis.

Projective modules and the homotopy classification of (G,n)–complexes

Projective modules and the homotopy classification of (G,n)–complexes

Microstructure evolution and mechanical properties of Cu–9Ni–6Sn alloy during continuous cold deformation process

A Cu–9Ni–6Sn alloy bar with pronounced axial columnar grains and a diameter of 10.5 mm was fabricated by using directional solidification technology, which was directly cold-drawn into a superfine alloy wire with a diameter of 40 μm without intermediate annealing. The strength and microstructure evolutions of the Cu–9Ni–6Sn alloy during the cold drawing were studied. The results indicate that twinning is more likely to occur in fiber structures ranging from 60 to 120 nm. A dual-fiber structure consisting of <111> and <100> orientations formed after cold-drawing. Unexpectedly, a large number of deformation twins were discovered within the <111> fibrous texture and later transformed into detwinning structures, disrupting the axial fibrous texture. The generation of deformation twinning primarily results from Ni and Sn elements disrupting crystal symmetry and reducing stacking fault energy of the alloy and the formation of <111> texture. The combined effects of detwinning structures and dislocation wall induced work-softening of the alloy, enabling high-strain processing and ultimately yielding nanocrystalline grains, i.e., the Cu–9Ni–6Sn alloy underwent cold drawing with a high train of 11.23 and obtained an average grain size of ∼38 nm, which exhibited a tensile strength of 1453 MPa.

Enhancing the mechanical properties and thermal conductivity of Ti/AZ91 composites through integrated extrusion and rolling continuous deformation processing

With the increasing demand for highly integrated circuits to optimize energy efficiency, reduce weight, and improve mechanical properties, finding a Magnesium (Mg) matrix material with excellent mechanical strength and high thermal conductivity poses a significant challenge. In this work, titanium (Ti) particles were introduced into AZ91 (Mg-9Al-1Zn) alloy to fabricate Ti/AZ91 composites, employing a combination of stir casting, extrusion and rolling processes. This strategy simultaneously enhances the mechanical properties and thermal conductivity of the Ti/AZ91 composites compared with AZ91 alloy. The incorporation of Ti particles serves to refine grains, expediting the formation of Mg17Al12 phase within the Mg matrix. Optimal comprehensive mechanical properties are attained in 10 wt% Ti/AZ91 composite, exhibiting an ultimate tensile strength of 365 MPa, an elongation of 11.9 %, and a significantly low wear rate of 4.548×10−3 mm3/m. Notably, the 10 wt% Ti/AZ91 composite shows a 41.1 % reduction in wear rate relative to the AZ91 alloy, attributed to the incorporation of hard Ti particles. Furthermore, the thermal conductivity of the 2.5 wt% Ti/AZ91 composite exhibits a significant enhancement over that of the AZ91 alloy. Detailed discussions are provided on the fundamental mechanisms responsible for strengthening, wear resistance, and thermally conductive in Ti/AZ91 composites. These results highlight the substantial potential of Ti/AZ91 composites for lightweight structural applications, such as highly integrated circuits requiring excellent mechanical properties and high thermal conductivity.

Objects at Work: Cybersecurity Certificates Making Topological Expertise

AbstractObjects do work creating expertise. This article parses how. It works with, revisits, and develops the notion of “boundary-work” located at the core of science and technology studies by Thomas Gieryn, Susan Leigh Star, and others and adds to it the notion of “orienting work” borrowing from the work of materialist feminists such as Sarah Ahmed and Maria do Mar Pereira. Working with these concepts, it shows that cybersecurity certificates make expertise topological. They demarcate, connect, create affective attachments to, and prefigure shapeshifting boundaries of cybersecurity expertise. They orient toward a maze of routes to cybersecurity expertise, a multiplicity of shifting locations in it, and toward fences that are stretched and straddled. Departing from conventional assumptions about the role of certificates in the formation of expertise, I show that rather than close off and stabilize expertise, the work of cybersecurity certificates opens and unsettles it. The cybersecurity expertise they make is preserved in continuous deformation. I also show that rather than fragment and fracture expertise, the work of certificates stretches, bends, and twists an expertise that remains connected. The work makes expertise of a distinctly topological quality. Engaging this moving, shapeshifting terrain politically is both exceptionally challenging and fundamentally important. The shifting expert terrain is generating a steadily expanding range of cybersecurity concerns that are inscribed ever more firmly in the socio-technical infrastructures of cybersecurity with ramifications for everyday life.

Construction of meta-dynamic recrystallization dynamic model of 316L stainless steel based on grain orientation spread during continuous variable rate thermal deformation

Because the deformation parameters (temperature, strain rate) in the actual manufacturing process are non-constant. We investigate meta-dynamic recrystallization behavior under the transient mutation state. First, based on a single-pass thermal deformation experiment and electron backscatter diffraction (EBSD) technique, this study establishes a quantitative calculation method for the volume fraction of meta-dynamic recrystallization under different deformation conditions, and find the GOS threshold basis for judging the occurrence of meta-dynamic recrystallization. Comparing the experimental results with the numerical simulation results, it is found that the MDRX behavior of the material could be accurately described. The meta-dynamic recrystallization evolution law of the material in the thermal deformation insulation stage under the variable strain rate state is investigated by continuous variable rate thermal deformation experiment and numerical simulation. It is observed that an increase of the average strain rate in the continuous variable strain rate state would promote the meta-dynamic recrystallization softening behavior of the material in comparison with the steady state deformation stage, and vice versa.