Curvature Function Research Articles

Semi-Analytical Solution for Elastoplastic Deflection of Non-Prismatic Cantilever Beams with Circular Cross-Section

A solution for the elastoplastic deflection of cantilever beams with linearly variable circular cross-section loaded by shear force at the free end, which is suitable for practical use, has not yet been developed. A semi-analytical solution for such a problem is proposed in this paper. The solution involves beams made of homogenous and isotropic materials with bilinear elastoplastic strain hardening behavior. The Bernoulli–Euler formula is used for determining the elastic deflection. However, for the plastic domain of material behavior, the differential equation of beam bending does not have a solution in closed form. Therefore, an incremental procedure for determining the curvature of the plastified region of the beam is suggested. Deflection of the cantilever beam is calculated via integration of the approximated function of the beam curvature. The proposed semi-analytical solution is validated using experimental results of the seismic energy dissipation device components which have been selected as a sample of a real engineering system. Also, validation is done via finite element analysis of six different cantilever beam models with varying geometric and material characteristics. A satisfying agreement between the proposed semi-analytical results and the subsequent experimental and numerical results is herein achieved, confirming its reliability.

Catastrophe optics theory unveils the localised wave aberration features that generate ghost images

Monocular polyplopia (ghost or multiple images) is a serious visual impediment for some people who report seeing two (diplopia), three (triplopia) or even more images. Polyplopia is expected to appear if the point spread function (PSF) has multiple intensity cores (a dense concentration of a large portion of the radiant flux contained in the PSF) relatively separated from each other, each of which contributes to a distinct image. We present a theory that assigns these multiple PSF cores to specific features of aberrated wavefronts, thereby accounting optically for the perceptual phenomenon of monocular polyplopia. The theory provides two major conclusions. First, the most likely event giving rise to multiple PSF cores is the presence of hyperbolic, or less probably elliptical, umbilic caustics (using the terminology of catastrophe optics). Second, those umbilic caustics formed on the retinal surface are associated with certain points of the wave aberration function, called cusps of Gauss, where the gradient of a curvature function vanishes. However, not all cusps of Gauss generate those umbilic caustics. We also provide necessary conditions for those cusps of Gauss to be fertile. To show the potential of this theoretical framework for understanding the nature and origin of polyplopia, we provide specific examples of ocular wave aberration functions that induce diplopia and triplopia. The polyplopia effects in these examples are illustrated by depicting the multi‐core PSFs and the convolved retinal images for clinical letter charts, both through computer simulations and through experimental recording using an adaptive optics set‐up. The number and location of cores in the PSF is thus a potentially useful metric for the existence and severity of polyplopia in spatial vision. These examples also help explain why physiological pupil constriction might reduce the incidence of ghosting and multiple images of daily objects that affect vision with dilated pupils. This mechanistic explanation suggests a possible role for optical phase‐masking as a clinical treatment for polyplopia and ghosting.

Uniqueness of convex ancient solutions to hypersurface flows

Abstract We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

Curvature homogeneous critical metrics in dimension three

We study curvature homogeneous three-manifolds modeled on a symmetric space which are critical for some quadratic curvature functional. If the Ricci operator is diagonalizable, critical metrics are 1-curvature homogeneous Brinkmann waves and are critical for one specific functional. Otherwise, critical metrics are modeled on Cahen-Wallach symmetric spaces and they are Kundt spacetimes which are critical for all quadratic curvature functionals.

Highly sensitive bending sensor based on Mach-Zehnder interferometer built in Sagnac loop with specified offset

A high sensitivity bending sensor based on Mach-Zehnder interferometer (MZI) built in Sagnac loop is proposed and experimentally demonstrated. One stress zone in Panda type polarization maintaining fiber (PMF) was intentionally collimated to the core of single mode fiber (SMF) and then the discharge was performed by fusion splicer. A commercial fiber Bragg grating (FBG) acting as phase shift monitor was fusion spliced to the PMF. The prepared fiber optical configuration was connected to a 3 dB coupler to construct a Sagnac loop. The spectral characteristics of the proposed bending sensor as a function of curvature were investigated experimentally. The bending sensitivity of 12.5 nm/m−1 was obtained experimentally. Furthermore, the π phase shift was detected when the curvature was 0.587 m−1. Such bending sensor with simple fabrication and cost-efficient could be a good candidate in the field of curvature measurements.

A necessary condition for prescribing mean curvature equations in 𝔹ⁿ

Let ( B n , g 0 ) (\mathbb {B}^n,g_0) be the n n -dimensional Euclidean unit ball with n ⩾ 3 n\geqslant 3 . We obtain a necessary condition on the existence of conformal metric in B n \mathbb {B}^n with zero scalar curvature and prescribed mean curvature function on the boundary.

Polyharmonic hypersurfaces into space forms

In this paper we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic hypersurfaces), where r ≥ 3 is an integer, into a space form Nm+1 (c) of curvature c. For this class of hypersurfaces we shall prove that, if c ≤ 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is $${\mathbb{S}^{m + 1}}$$ , we shall obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper r-harmonic isoparametric hypersurfaces in $${\mathbb{S}^{m + 1}}$$ for suitable values of m and r. Finally, we shall show that all these r-harmonic hypersurfaces are also ES-r-harmonic, i.e., critical points of the Eells-Sampson r-energy functional.

Magnetic Curves in C-manifolds

In this paper, we study normal magnetic curves in $C$-manifolds. We prove that magnetic trajectories with respect to the contact magnetic fields areindeed $\theta _{\alpha }$-slant curves with certain curvature functions. Then, we give the parametrizations of normal magnetic curves in $\mathbb{R}^{2n+s}$ with its structures as a $C$-manifold.

Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$

In this present paper, rectifying curves are re-characterized in a shorter and simpler way using harmonic curvatures and some relations between rectifying curves and focal curves are found in terms of their harmonic curvature functions in $n-$dimensional Euclidean space. Then, a rectifying Salkowski curve, which is the focal curve of a given space curve is investigated. Finally, some figures related to the theory are given in the case $n=3$.

Curved Magnetism in CrI_{3}.

Curved magnets attract considerable interest for their unusually rich phase diagram, often encompassing exotic (e.g., topological or chiral) spin states. Micromagnetic simulations are playing a central role in the theoretical understanding of such phenomena; their predictive power, however, rests on the availability of reliable model parameters to describe a given material or nanostructure. Here we demonstrate how noncollinear-spin polarized density-functional theory can be used to determine the flexomagnetic coupling coefficients in real systems. By focusing on monolayer CrI_{3}, we find a crossover as a function of curvature between a magnetization normal to the surface to a cycloidal state, which we rationalize in terms of effective anisotropy and Dzyaloshinskii-Moriya contributions to the magnetic energy. Our results reveal an unexpectedly large impact of spin-orbit interactions on the curvature-induced anisotropy, which we discuss in the context of existing phenomenological models.

A pinching estimate for convex hypersurfaces evolving under a non-homogeneous variant of mean curvature flow

AbstractWe study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a non-homogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.

Modeling and Design of Periodic Polygonal Lattices Constructed from Microstructures with Varying Curvatures

Lattice structures can exhibit unusual mechanical properties by appropriate design and arrangement of the microstructures, and have already found applications in areas such as tissue engineering, stretchable electronics, and soft robotics. However, the designed microstructures generally follow simple geometric patterns with constant curvatures and theoretical models are developed for each geometry particularly. It lacks a universal method to model and design the periodic lattice constructed from microstructures with varying curvatures. In this paper, we develop a universal approach to design polygonal lattices with a wide range of microstructures with varying curvatures using instant curvature and parametric functions. A finite deformation model is proposed for the microstructures and the corresponding lattices, which considers the wavy or curled microstructure, hierarchical geometry, and large deformation. Experiments and finite-element simulations are conducted to validate the theoretical model. All of Young's modulus, stretchability, and Poisson's ratio can be inversely designed using the developed theoretical model. Results show that the stretchability of the highly stretchable lattice can be further increased by more than 4 times using densely curled microstructure. The signs of Poisson's ratio can also change with the axial strain by rationally designing the microstructures. By combining curve fitting and curvature and parametric functions, the mechanical behaviors of lattices constructed by bioinspired microstructures can also be predicted. The lattice structures are then demonstrated to reinforce a hydrogel with modulus increased by around 2 orders and act as a biomimetic soft-strain sensor. This work could aid the design of a hierarchical lattice and have potential applications in tissue engineering, stretchable electronics, and soft robotics.

Darboux Vector in Four-Dimensional Space-Time

As the space-time model of the theory of relativity, four-dimensional Minkowski space is the basis of the theoretical framework for the development of the theory of relativity. In this paper, we introduce Darboux vector fields in four-dimensional Minkowski space. Using these vector fields, we define some new planes and curves. We find that the new planes are the instantaneous rotation planes of rigid body moving in four-dimensional space-time. In addition, according to some characteristics of Darboux vectors in geometry, we define some new space curves in four-dimensional space-time and describe them with curvature functions. Finally, we give some examples.

Curvature–matter couplings in modified gravity: From linear models to conformally invariant theories

In this paper, we review modified theories of gravity with a curvature–matter coupling between an arbitrary function of the scalar curvature and the Lagrangian density of matter. This explicit nonminimal coupling induces a nonvanishing covariant derivative of the energy–momentum tensor, that implies nongeodesic motion and consequently leads to the appearance of an extra force. Here, we explore the physical and cosmological implications of the nonconservation of the energy–momentum tensor by using the formalism of irreversible thermodynamics of open systems in the presence of matter creation/annihilation. The particle creation rates, pressure, and the expression of the comoving entropy are obtained in a covariant formulation and discussed in detail. Applied together with the gravitational field equations, the thermodynamics of open systems lead to a generalization of the standard [Formula: see text]CDM cosmological paradigm, in which the particle creation rates and pressures are effectively considered as components of the cosmological fluid energy–momentum tensor. Furthermore, we also briefly present the coupling of curvature to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form [Formula: see text], where [Formula: see text] is the ordinary matter Lagrangian, and [Formula: see text] is the Weyl scalar. The coupling explicitly satisfies the requirement of the conformal invariance of the theory. Expressing [Formula: see text] with the use of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized in the Ricci scalar, leading in the Riemann space to a conformally invariant [Formula: see text]-type theory, with the matter Lagrangian nonminimally coupled to geometry.

Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

In this paper, we give a generalization of the osculating curves to the $n$-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the ( n − 3 ) (n−3) th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type ( n − 3 ) (n−3) ”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n En . In particular, we characterize the osculating curves in E n En in terms of their curvature functions. Finally, we show that the ratio of the ( n − 1 ) (n−1) th and ( n − 2 ) (n−2) th curvatures of the osculating curve is the solution of an ( n − 2 ) (n−2) th order linear nonhomogeneous differential equation.

Rotational Hypersurfaces in Euclidean 4-Space with Density

In this paper, the Euclidean 4-space with a positive density function e^(x^2+y^2+z^2+t^2 ) is studied. Firstly, the weighted mean and weighted Gaussian curvature functions of a rotational hypersurface in 4-dimensional Euclidean space with density are obtained. The rotational hypersurfaces are constructed by solving these obtained functions which are second-order non-linear ordinary differential equations. Besides, the examples of rotational hypersurfaces are given with the aid of the weighted Gaussian and weighted mean curvatures in E^4 with density.

Prescribed Webster Scalar Curvatures on Compact Pseudo-Hermitian Manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

Bakry-Émery curvature on graphs as an eigenvalue problem

In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

Gaussian curvature estimates for the convex level sets of minimal graphs revisited

The main aim of the present paper is to give a simplified and easy-to-follow proof of the convexity estimates of level sets of minimal graphs. This proof greatly simplifies the original one in Wang and Zhang (2012) [36]. The innovation lies in that we find new curvature functions which are exactly supersolutions of the minimal surface equations.

Quantifying the contributions of ENTH to membrane spontaneous curvature using coarse grained molecular dynamics

The ability of cells to produce and maintain membrane curvature is critical for cellular function. Experiments throughout history have revealed a diverse set of mechanisms for how proteins can bend membranes. Models have been developed to shed insight on the mechanics behind these mechanisms. As the quality and fidelity of experimental data improves, using these models we can ask how changes to cellular regulatory pathways influence membrane ultrastructure and membrane trafficking processes. For continuum modeling of the membrane shape, if we select the spontaneous curvature model, which represents the bending energy as a function of membrane curvature, the key parameters to define are the bending modulus and spontaneous curvature.