Thin Spherical Shell Research Articles

Analysis of Fracturing Above Block Caving Back: A Spherical Shell Theory Approach and BEM Numerical Simulation

ABSTRACTExperiments and field monitoring have revealed that in block caving, fractures over the cave crown tend to form in narrow curved bands that are parallel or subparallel to the cave back surface. These fractures delineate curved shells of orebody between the bands and the cave back. The effectiveness of block caving hinges on the subsequent fracturing and fragmentation of these orebody shells. This study adopts a dual approach, combining thin spherical shell theory and full 3D numerical simulations along with principles of linear elastic fracture mechanics, to investigate the fracturing behaviour of these shells. Analytical analysis indicates that under axisymmetric loading, latitudinal tensile fractures predominantly initiate across the most part of the shell, occurring on both the upper and lower surfaces, except at a localised area. Additionally, longitudinal tensile fractures may initiate at the central area of the upper surface, while shear fractures tend to occur around the edge of the shell. Consequently, the shells become susceptible to fracturing, leading to the collapse or cave‐in of the orebody. Numerical simulations agree with these findings, illustrating that fracturing points within the shell region are longitudinally dispersed throughout the entire shell. Most of these fracturing points satisfy the criteria for tensile fracturing, particularly within the middle portion of the shell, aligning with the analytical results. Furthermore, simulations considering nonaxisymmetric loading patterns demonstrate that regions surrounding the caving cavity, aligned with the minimum principal in situ stress, exhibit heightened susceptibility to fracture initiation. This insight holds potential significance for optimising the design of the caving process.

Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

SN 2021adxl: A luminous nearby interacting supernova in an extremely low-metallicity environment

SN 2021adxl is a slowly evolving, luminous, Type IIn supernova with asymmetric emission line profiles, similar to the well-studied SN 2010jl. We present extensive optical, near-ultraviolet, and near-infrared photometry and spectroscopy covering ∼1.5 years post discovery. SN 2021adxl occurred in an unusual environment, atop a vigorously star-forming region that is offset from its host galaxy core. The appearance of Lyα and O II, as well as the compact core, would classify the host of SN 2021adxl as a “Blueberry” galaxy, analogous to higher redshift, low-metallicity, star-forming dwarf “Green Pea” galaxies. Using several abundance indicators, we find a metallicity of the explosion environment of only ∼0.1 Z⊙, the lowest reported metallicity for a Type IIn SN environment. SN 2021adxl reaches a peak magnitude of Mr ≈ −20.2 mag and since discovery, SN 2021adxl has faded by only ∼4 magnitudes in the r band with a cumulative radiated energy of ∼1.5 × 1050 erg over 18 months. SN 2021adxl shows strong signs of interaction with a complex circumstellar medium, seen by the detection of X-rays, revealed by the detection of coronal emission lines, and through multi-component hydrogen and helium profiles. In order to further understand this interaction, we model the Hα profile using a Monte Carlo electron scattering code. The blueshifted high-velocity component is consistent with emission from a radially thin spherical shell resulting in the broad emission components due to electron scattering. Using the velocity evolution of this emitting shell, we find that the SN ejecta collide with circumstellar material of at least ∼5 M⊙ assuming a steady-state mass-loss rate of ∼4 − 6 × 10−3 M⊙ yr−1 for the first ∼200 days of evolution. SN 2021adxl was last observed to be slowly declining at ∼0.01 mag d−1, and if this trend continues, SN 2021adxl will remain observable after its current solar conjunction. Continuing the observations of SN 2021adxl may reveal signatures of dust formation or an infrared excess, similar to that seen for SN 2010jl.

Investigation on dynamic response of thin spherical shells impacted by flat-nose projectile based on a novel damage model

Thin-shell structures are widely used in various engineering applications. It is essential to investigate the impact resistance of thin-shell structures, to provide theoretical support for engineering applications. Numerous impact tests have been conducted on thin spherical shells using ballistic guns. The effects of the impact velocity and shell thickness on the deformation and fracture of thin spherical shells are summarized. Moreover, a novel damage model based on statistic damage mechanics is proposed to better predict dynamic responses of thin shells impacted by projectiles. Considering that fracture surfaces are formed by void evolution and are affected by the stress states, the damage level is defined as the ratio of the statistical cross-sectional area of the voids to the cross-sectional area of the representative elements. Utilizing statistical methods, the incorporation of continuous void nucleation, ellipsoidal void growth, and the acceleration of dynamic void evolution are introduced into the novel damage model. Subsequently, numerical investigations of the dynamic response of spherical shells under impact are conducted based on the proposed damage model. The numerical results are consistent with the experimental results in terms of the depression deformations and strain signals. The effects of shell thickness and double-layer structures on the dynamic response of spherical shells are investigated via numerical simulations considering the novel damage model in detail. The results demonstrate that the proposed model can accurately predict the dynamic response of spherical shells impacted by flat-nose projectiles, thus serving as a valuable reference for engineering design.

Revisit Birkhoff’s Theorem: The Post-Newtonian Metric of a Self-Gravitating and Collapsing Thin Spherical Shell

We calculate the metric of a self-gravitating and collapsing infinitely thin spherical shell in the weak-field and slow-motion limits, and we demonstrate that Birkhoff’s theorem is not consistent with the theory of the post-Newtonian approximation. More importantly, it is illustrated that performing a coordinate transformation in solving Einstein field equations may change the matter energy-momentum tensor, making the resultant solution not correspond to the original problem.

A reliability-based design against post-buckling load drop in spherical shell cap with stochastic imperfections

Elastic buckling is a critical design consideration for deep spherical shell caps. Significant reduction is observed in the experimental critical load from the linearized eigen-buckling analysis, for reasons extensively discussed in the past and relooked recently through the energy barrier concepts. The Knock Down Factors (KDFs) were proposed to account for such a reduction. The highly conservative KDFs constitute the lower bound of a large experimental dataset; that can be reconciled by nonlinear stability analysis of the imperfect shell. This study models the KDFs as Uncertain-but-Bounded (UBB) variables to present a Reliability-Based Design (RBD) for spherical shell caps having random geometric imperfections in radius (R) and thickness (t). The RBD furnishes the pertinent design variables ((R/t) ratio and the dome height-to-base radius (H/r0) ratio) for a target reliability index (β) and the respective KDFs. A metamodelling approach is employed to reduce the computational burden of the proposed RBD for a thin spherical shell cap having zero-mean, stationary, homogeneous, gaussian stochastic geometric imperfections. The Riks algorithm is employed in the commercial code ABAQUS for the finite element-based nonlinear stability analysis of shells. The KDFs and the reliability bounds are presented for varying degrees of imperfections. Remarkably, the margin of conservatism in the KDFs is shown to be narrowed down by incorporating the dependency on (H/r0).

CON-quest

Context. Nuclear regions of ultraluminous and luminous infrared galaxies (U/LIRGs) are powered by starbursts and/or active galactic nuclei (AGNs). These regions are often obscured by extremely high columns of gas and dust. Molecular lines in the submillimeter windows have the potential to determine the physical conditions of these compact obscured nuclei (CONs). Aims. We aim to reveal the distributions of HCN and HCO+ emission in local U/LIRGs and investigate whether and how they are related to galaxy properties. Methods. Using the Atacama Large Millimeter/submillimeter Array (ALMA), we have conducted sensitive observations of the HCN J = 3−2 and HCO+J = 3−2 lines toward 23 U/LIRGs in the local Universe (z < 0.07) with a spatial resolution of ∼0.3″ (∼50−400 pc). Results. We detected both HCN and HCO+ in 21 galaxies, only HCN in one galaxy, and neither in one galaxy. The global HCN/HCO+ line ratios, averaged over scales of ∼0.5−4 kpc, range from 0.4 to 2.3, with an unweighted mean of 1.1. These line ratios appear to have no systematic trend with bolometric AGN luminosity or star formation rate. The line ratio varies with position and velocity within each galaxy, with an average interquartile range of 0.38 on a spaxel-by-spaxel basis. In eight out of ten galaxies known to have outflows and/or inflows, we found spatially and kinematically symmetric structures of high line ratios. These structures appear as a collimated bicone in two galaxies and as a thin spherical shell in six galaxies. Conclusions. Non-LTE analysis suggests that the high HCN/HCO+ line ratio in outflows is predominantly influenced by the abundance ratio. Chemical model calculations indicate that the enhancement of HCN abundance in outflows is likely due to high-temperature chemistry triggered by shock heating. These results imply that the HCN/HCO+ line ratio can aid in identifying the outflow geometry when the shock velocity of the outflows is sufficiently high to heat the gas.

Asymptotic profiles of mean zonal flows generated by thermal convection of Boussinesq fluid in a rapidly rotating thin spherical shell

The asymptotic behavior of the profiles of mean zonal flows generated by thermal convection of a Boussinesq fluid in a rapidly rotating thin spherical shell was studied by extending the integration time of the numerical experiments of a previous study under similar conditions. Calculations were performed for the whole globe, as well as for a one-eighth sector region, with various values of the hyper-diffusion parameters. For sufficiently weak hyper-diffusion, a prograde zonal jet and alternating narrow zonal jets appeared at the equator and in the middle and high latitudes, respectively, as reported in the previous study. However, further integrations showed that the middle and high latitude flows are asymptotically accelerated eastward, jet peaks merge, and the banded zonal flow structure eventually disappears. Thus, the rotating thin spherical shell convection model of a Boussinesq fluid used in the previous studies seems inadequate to explain the surface banded structures of Jovian planetary atmospheres without modification of the standard setups. To maintain the alternating jet structure in the middle and high latitudes over the long-term, some sort of scale-independent dissipation process should be specified that prevents the evolution of zonal flows toward the long-term asymptotic state.

The binding of cosmological structures by massless topological defects

ABSTRACT Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is mitigated, at least in part.

On the Generalized Lemaitre Tolman Bondi Metric: Classical Sensitivities and Quantum Einstein‐Vaz Shells

AbstractIn this paper, in the classical framework, the lower bounds for the sensitivities of the generalized Lemaitre Tolman Bondi metric are evaluated. The calculated lower bounds via the linear dynamical systems , , and are , and respectively. The sensitivities and the lower sensitivities via are zero are also shown. In the quantum framework, the properties of the Einstein‐Vaz shells which are the final result of the quantum gravitational collapse arising from the Lemaitre Tolman Bondi discussed by Vaz in 2014 are analyzed. In fact, Vaz showed that continued collapse to a singularity can only be obtained if one combines two independent and entire solutions of the Wheeler‐DeWitt equation. Forbidding such a combination leads naturally to matter condensing on the Schwarzschild surface during quantum collapse. In that way, an entirely new framework for black holes (BHs) has emerged. The approach of Vaz was also consistent with Einstein's idea in 1939 of the localization of the collapsing particles within a thin spherical shell. Here, following an approach of oned of us (CC), we derive the BH mass and energy spectra via a Schrodinger‐like approach, by further supporting Vaz's conclusions that instead of a spacetime singularity covered by an event horizon, the final result of the gravitational collapse is an essentially quantum object, an extremely compact “dark star”. This “gravitational atom” is held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. Finally, the time evolution of the Einstein‐Vaz shells is discussed.

Spontaneous crumpling of active spherical shells.

The existence of a crumpled phase for self-avoiding elastic surfaces was postulated more than three decades ago using simple Flory-like scaling arguments. Despite much effort, its stability in a microscopic environment has been the subject of much debate. In this paper we show how a crumpled phase develops reliably and consistently upon subjecting a thin spherical shell to active fluctuations. We find a master curve describing how the relative volume of a shell changes with the strength of the active forces, that applies for every shell independent of size and elastic constants. Furthermore, we extract a general expression for the onset active force beyond which a shell begins to crumple. Finally, we calculate how the size exponent varies along the crumpling curve.

Experimental and finite element analysis of the buckling response of thin spherical shells under hyperbaric pressure

Experimental and finite element analysis of the buckling response of thin spherical shells under hyperbaric pressure

Acoustic scattering from spherical shells and its active control

The scattering of sound by a thin spherical shell is considered using an analytic formulation involving spherical harmonics. The integral of the far field scattered intensity, termed the scattered power, can then be expressed in a simple form. At low frequencies the scattered power can be minimised by an appropriate choice of material properties and shell thickness, which is illustrated for both a steel shell and for one in which the mass and stiffness of the shell are equal to those of the displaced fluid. Simulations of feedforward active control are then used to investigate the best possible performance in attenuating the scattered power, although this approach requires knowledge of the incident and scattered sound fields. Feedback control of the shell vibration using structural actuators and sensors is a more practical control strategy since it does not involve the need to separate the incident and scattered contributions. Direct velocity feedback control is considered using collocated and distributed actuators and sensors that spread the applied force and sensed velocity over spherical caps on the surface of the shell. This approach shows effective suppression of the structural shell modes that give rise to significant scattering at their resonant frequencies.

Versatile polydopamine-mediated growth of N and C co-doped hollow porous TiO2 spheres with oxygen vacancies for visible-light-driven oxidation of acetaldehyde

Developing visible-light-responsive TiO2-based photocatalysts provides a promising avenue for addressing VOCs pollution. Herein, polydopamine (PDA) template pyrolysis waste was in-situ up-cycled for simultaneous growth of N and C co-doped hollow porous TiO2 microreactors (H-NT-0.2) in air. The resulting H-NT-0.2 with a highly intact hollow structure, thin spherical shell (∼ 13 nm) and high porosity possesses the highest N and C co-doping ratio and largest oxygen vacancies (OVs) amount, leading to the highest concentration of reactive oxygen species (ROS) among all H-NT-x. A crucial role of OVs in enhancing H2O adsorption for the generation of hydroxyl radicals (⋅OH) has been revealed by DFT calculations. As a result, H-NT-0.2 shows the highest photocatalytic oxidation (PCO) efficiency of flowing acetaldehyde gas (90%) under visible light irradiation, which exceeds 3.5 times of commercial TiO2 (P25)), and outperforms most of the reported photocatalysts. The growth mechanism, as well as the photocatalytic reaction pathways, were analyzed based on TG-MS and in-situ DRIFTS results, respectively. This work provides an economical, green and feasible strategy to develop advanced visible-light-responsive photocatalysts for aldehydes VOCs abatement in practical applications.

Mechanical fatigue testing in silico: Dynamic evolution of material properties of nanoscale biological particles

Biological particles have evolved to possess mechanical characteristics necessary to carry out their functions. We developed a computational approach to "fatigue testing in silico", in which constant-amplitude cyclic loading is applied to a particle to explore its mechanobiology. We used this approach to describe dynamic evolution of nanomaterial properties and low-cycle fatigue in the thin spherical encapsulin shell, thick spherical Cowpea Chlorotic Mottle Virus (CCMV) capsid, and thick cylindrical microtubule (MT) fragment over 20 cycles of deformation. Changing structures and force-deformation curves enabled us to describe their damage-dependent biomechanics (strength, deformability, stiffness), thermodynamics (released and dissipated energies, enthalpy, and entropy) and material properties (toughness). Thick CCMV and MT particles experience material fatigue due to slow recovery and damage accumulation over 3-5 loading cycles; thin encapsulin shells show little fatigue due to rapid remodeling and limited damage. The results obtained challenge the existing paradigm: damage in biological particles is partially reversible owing to particle's partial recovery; fatigue crack may or may not grow with each loading cycle and may heal; and particles adapt to deformation amplitude and frequency to minimize the energy dissipated. Using crack size to quantitate damage is problematic as several cracks might form simultaneously in a particle. Dynamic evolution of strength, deformability, and stiffness, can be predicted by analyzing the cycle number (N) dependent damage, [Formula: see text] , where α is a power law and Nf is fatigue life. Fatigue testing in silico can now be used to explore damage-induced changes in the material properties of other biological particles. STATEMENT OF SIGNIFICANCE: Biological particles possess mechanical characteristics necessary to perform their functions. We developed "fatigue testing in silico" approach, which employes Langevin Dynamics simulations of constant-amplitude cyclic loading of nanoscale biological particles, to explore dynamic evolution of the mechanical, energetic, and material properties of the thin and thick spherical particles of encapsulin and Cowpea Chlorotic Mottle Virus, and the microtubule filament fragment. Our study of damage growth and fatigue development challenge the existing paradigm. Damage in biological particles is partially reversible as fatigue crack might heal with each loading cycle. Particles adapt to deformation amplitude and frequency to minimize energy dissipation. The evolution of strength, deformability, and stiffness, can be accurately predicted by analyzing the damage growth in particle structure.

Gravastar-like black hole solutions in q-theory

We present a stationary spherically symmetric solution of the Einstein equations, with a source generated by a scalar field of q-theory. In this theory Riemannian gravity, as described by the Einstein–Hilbert action, is coupled to a three—form field that describes the dynamical vacuum. Formally it behaves like a matter field with its own stress—energy tensor, equivalent to a scalar field minimally coupled to gravity. The asymptotically flat solutions obtained to the field equations represent black holes (BHs). For a sufficiently large horizon radius the energy density is localized within a thin spherical shell situated just outside of the horizon, analogous to a gravastar. The resulting solutions to the field equations, which admit this class of configurations, satisfy existence conditions that stem from the BH no—hair theorem, thanks to the presence of a region in space in which the energy density is negative.

Black Hole Spectra from Vaz's Quantum Gravitational Collapse

AbstractIn 2014, in a famous paper Hawking strongly criticized the firewall paradox by claiming that it violates the equivalence principle and breaks the CPT invariance of quantum gravity. He proposed that the final result of the gravitational collapse should not be an event horizon, but an apparent horizon instead. On the other hand, Hawking did not give a mechanism for how this could work. In the same year, Vaz endorsed Hawking's proposal in a quantum gravitational model of dust collapse by winning the Second Prize in the 2014 Gravity Research Foundation Essay Competition. He indeed showed that continued collapse to a singularity can only be obtained if one combines two independent and entire solutions of the Wheeler‐DeWitt equation. Vaz's interpretation of the paradox was in terms of simply forbidding such a combination. This leads naturally to matter condensing on the apparent horizon during quantum collapse. In that way, an entirely new framework for black holes (BHs) has emerged. The approach of Vaz was also consistent with Einstein's idea in 1939 of the localization of the collapsing particles within a thin spherical shell. In this work we derive the BH mass and energy spectra via a Schrodinger‐like approach, by further supporting Vaz's conclusions that instead of a spacetime singularity covered by an event horizon, the final result of the gravitational collapse is an essentially quantum object, an extremely compact “dark star”. This “gravitational atom” is held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. Finally, by evoking the generalized uncertainty principle, the maximum value of the density of Vaz's shell will be estimated.

AdS to dS phase transition mediated by thermalon in Einstein-Gauss-Bonnet gravity from Rényi statistics

In this work, we present the possible existence of a thermalon phase transition between anti-de Sitter (AdS) to de Sitter (dS) vacua in Einstein-Gauss-Bonnet gravity by considering the R\'{e}nyi statistics. A thermalon changes the asymptotic structure of spacetimes via the bubble nucleation of spherical thin-shells which host a black hole in the interior. All relevant thermodynamical quantities are computed in terms of the Renyi statistics in order to demonstrate the possible existence of the AdS to dS phase transition. In addition, we also comment on the behaviors of the phase transitions in the Renyi statistics.

A Non-Destructive Method for Predicting Critical Load, Critical Thickness and Service Life for Corroded Spherical Shells under Uniform External Pressure Based on NDT Data

A pressurized spherical shell that is continuously corroded will likely buckle and lose its stability. There are many analytical and numerical methods to study this problem (critical load, critical thickness, and service life), but the friendliness (operability) in engineering test applications is still not ideal. Therefore, in this paper, we propose a new non-destructive method by combining the Southwell non-destructive procedure with the stable analysis method of corroded spherical thin shells. When used carefully, it can estimate the critical load (critical thickness) and service life of these thin shells. Furthermore, its procedure proved to be more practical than existing methods; it can be easily mastered, applied, and generalized in most engineering tests. When used properly, its accuracy is acceptable in the field of engineering estimations. In the context of the high demand for non-destructive analysis in industry, it may be of sufficient potential value to be used as a reference for existing estimating methods based on NDT data.

Active Damping, Vibration Isolation, and Shape Control of Space Structures: A Tutorial

This tutorial reviews the author’s contributions to the active control of precision space structures over the past 35 years. It is based on the Santini lecture presented at the IAC-2022 Astronautical Congress in Paris in September 2022. The first part is devoted to the active damping of space trusses with an emphasis on robustness. Guaranteed stability is achieved by using decentralized collocated actuator–sensor pairs. The so-called integral force feedback (IFF) is simple, robust, and effective, and the performances can be predicted easily with simple formulae based on modal analyses. These predictions have been confirmed by numerous experiments. The damping strategy for trusses has been extended to cable structures, and also confirmed experimentally. The second part addresses the problem of vibration isolation: isolating a sensitive payload from the vibration induced by the spacecraft (i.e., the unbalanced mass of attitude control reaction wheels and gyros). A six-axis isolator based on a Gough–Stewart platform is discussed; once again, the approach emphasizes robustness. Two different solutions are presented: The first one (active isolation) uses a decentralized controller with collocated pairs of the actuator and force sensor, with IFF control. It is demonstrated that this special implementation of the skyhook, unlike the classical one, has guaranteed stability, even if the two substructures it connects are flexible (typical of large space structures). A second approach (passive) discusses an electromagnetic implementation of the relaxation isolator where the classical dash-pot of the linear damper is substituted by a Maxwell unit, leading to an asymptotic decay rate of −40 dB/decade, similar to the skyhook (although much simpler in terms of electronics). The third part of the lecture summarizes more recent work done on the control of flexible mirrors: (i) flat mirrors for adaptive optics (AO) controlled by an array of piezoelectric ceramic (PZT) actuators and (ii) spherical thin shell polymer reflectors controlled by an array of piezoelectric polymer actuators (PVDF-TrFE) aimed at being deployed in space.