Components Of Curvature Tensors Research Articles

Curvature analysis of concircular trajectories in doubly warped product manifolds

<p>The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.</p>

An accurate size-dependent sinusoidal shear deformable framework for GNP-reinforced cylindrical panels: Applications to dynamic stability analysis

In the current work, an accurate modified couple stress-based cylindrical micro-panel model is developed in the framework of sinusoidal shear deformation theory. The classical strain tensors and curvature tensor components in a modified couple stress-based model are given with consideration of the geometric curve of cylindrical panels. In conjunction with a general Lagrange procedure, the proposed model is implemented to derive a system of Mathieu-Hill equations of governing dynamic stability behaviors of the cylindrical micro-panels. The Navier-type solution and Bolotin's method are applied to predict the principle unstable regions of the micro-panels with simply-supported boundary condition. As an application of the proposed model, the size-dependent stability of a functionally graded graphene nanoplatelets reinforced composite (FG-GNPRC) cylindrical micro-panel under axial oscillation compression is investigated for the first time. It is assumed that the weight fraction of graphene nanoplatelets (GNPs) is graded in the panel thickness. The effective Young's modulus of FG-GNPRC is determined based on a modified Halpin–Tsai model, while the mixture rule used to evaluate the effective Poisson's ratio and the mass density. The parameter studies are performed to examine the effect of dimensionless length parameter, GPL distribution patterns, GPL size and shape, length-to-span ratio and radius-to-span ratio on the size-dependent dynamic stabilities of FG-GNPRC cylindrical micro-panels.

A unified modified couple stress model for size-dependent free vibrations of FG cylindrical microshells based on high-order shear deformation theory

The aim of present work is to propose a unified modified couple stress-based shear deformable shell model that has capability of analyzing size-dependent free vibrations of functionally graded (FG) cylindrical microshells. In the current model, the classical strain tensors and curvature tensor components are given within the framework of a unified high-order shear deformation theory. The trapezoidal shape factor is taken into account for achieving reasonable stress resultants over thickness and makes the proposed model more accurate for thick shell structures. Various classical boundary conditions of the cylindrical microshells are considered using the admissible functions of the displacements containing a beam modal function. The governing equations associated with free vibrations of microshells are derived using a standard Lagrange method. A few comparisons are conducted and verify that the results from the current model coincide favorably to the ones from three-dimensional elasticity solutions. The comparisons also prove that the proposed model has a sufficient capability of dealing with free vibration problems of the thick functionally graded cylindrical microshells. To the end, the size-dependent free vibrations of an FG cylindrical microshell are investigated. As new results, natural frequencies of the FG microshells with various boundary conditions are tabulated, and the effects of some important parameters on vibration characteristics of FG microshells are discussed.

Mathematical Treatise to Model Dihedral Energy in the Multiscale Modeling of Two-Dimensional Nanomaterials

An approximate mathematical treatise is proposed to improve the accuracy of multiscale models for nonlinear mechanics of two-dimensional (2D) nanomaterials by taking into account the contribution of dihedral energy term in the nonlinear constitutive model for the generalized deformation (three nonzero components of each strain and curvature tensors) of the corresponding continuum. Twelve dihedral angles per unit cell of graphene sheet are expressed as functions of strain and curvature tensor components. The proposed model is employed to study the bending modulus of graphene sheets under finite curvature. The atomic interactions are modeled using first- and second-generation reactive empirical bond order (REBO) potentials with the modifications in the former to include dihedral energy term for accurate prediction of bending stiffness coefficients. The constitutive law is obtained by coupling the atomistic and continuum deformations through Cauchy–Born rule. The present model will facilitate the investigations on the nonlinear mechanics of graphene sheets and carbon nanotubes (CNTs) with greater accuracy as compared to those reported in the literature without considering dihedral energy term in multiscale modeling.

Laue micro-diffraction and crystal plasticity finite element simulations to reveal a vein structure in fatigued Cu

The formation of a vein during cyclic shearing of a single copper crystal oriented for single slip can be followed in transmission Laue diffraction by analyzing the spatially resolved lattice rotation evolution. Because Laue transmission integrates the signal over the thickness of the sample, the structure of the vein in the beam direction is a priori believed to be inaccessible. Here we show that the vein geometry in the beam direction can be retrieved by comparing lattice curvature tensor components from crystal plasticity finite element simulations with those experimentally derived. Virtual sectional analysis facilitates the interpretation of the measured lattice curvatures of quasi-2D dislocation structures, allowing identifying a vein morphology that is slightly vertically and horizontally inclined in the through thickness direction.

SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE

In this paper, we discuss gravitational collapse of spherically symmetric spacetimes. We derive a general formalism by taking two arbitrary spherically symmetric spacetimes with g00 = 1. Israel's junction conditions are used to develop this formalism. The formulas for extrinsic curvature tensor are obtained. The general form of the surface energy–momentum tensor depending on extrinsic curvature tensor components is derived. This leads us to the surface energy density and the tangential pressure. The formalism is applied to two known spherically symmetric spacetimes. The results obtained show the regions for the collapse and expansion of the shell.

Tube dislocations in gravity

We consider static massive thin cylindrical shells (tubes) as the sources in Einstein’s equations. They correspond to δ- and δ′-function-type energy-momentum tensors. The corresponding metric components are found explicitly. They are not continuous functions, in general, and lead to ambiguous curvature tensor components. Nevertheless all ambiguous terms in Einstein’s equations safely cancel. The interplay between elasticity theory, geometric theory of defects, and general relativity is analyzed. The elasticity theory provides a simple picture for defect creation and a new look on general relativity.

Effective gravitational equations induced by quantum effects of a scalar field in the Bianchi type IX spaces

The effective gravitational equations induced by quantum effects of a scalar field in the Bianchi type IX space-time are derived in the present work. The curvature tensor components and their combinations are calculated in the Bianchi type IX spaces. To estimate a one-loop effective action, the asymptotic Schwinger-DeWitt perturbative formula with consideration of terms up to the second order is used. Quantum corrections involve the fourth-order derivative of the metric; as a consequence, the background gravitational action also includes terms up to the fourth-order derivatives of the metric.

Incompatible deformations – plastic intermediate configuration

AbstractThere are many approaches in continuum mechanics involving incompatible deformations, one well‐known example being the polar decomposition of the deformation gradient tensor. Such deformations have in the past been regarded rather in the context of local configurations. In the present paper, we discuss incompatible deformations among others, also from the point of view of material lines. Related covariant derivatives and spatial differential operators are introduced and investigated. In particular, it is shown that components of curvature tensors, present in many gradient constitutive theories, may be expressed in terms of the difference between different connections for the same manifold. These issues are illustrated with reference to multiplicative decompositions of deformation gradient tensors into elastic and plastic parts. Two classes of model materials are addressed, non‐polar and micropolar ones.

Sufficient conditions for a curvature tensor to be Riemannian and for determining its metric

A simple necessary and sufficient condition that a curvature tensor be Riemannian (i.e., that its symmetric connection be metric) is found, in very general conditions, for a four-dimensional space-time manifold. This enables existing uniqueness results for the metric tensor and an associated algebraic procedure for obtaining the metric tensor components from the curvature tensor components (when the curvature tensor is known to be a Riemannian tensor), to be extended to the situation where it is not known a priori whether the curvature tensor has come from a Lorentz metric.

On the determination of the energy-momentum tensor and Weyl tensor structure from the curvature in space-time

Consider the curvature tensor components Rbcda on a space-time M calculated from given space-time metric components gab. The number of alternative space-time metrics for M which yield the same curvature components is known to be heavily restricted, the exact ambiguity in the metric being dependent on the form of the curvature components. In this paper it is shown that, in spite of these ambiguities, the algebraic structure of the energy-momentum tensor and the Weyl tensor on M are essentially determined.

On tangent bundles with Sasakian metrics of Finslerian and Riemannian manifolds

On the basis of the so-called phase completion the notion of vertical, horizontal and complete objects is defined in the tangent bundles over Finslerian and Riemannian manifold. Such a tangent bundle is made into a manifold of almost Kaehlerian structure by endowing it with Sasakian metric. The components of curvature tensors with respect to the adapted frame are presented. This having been done it is shown possible to study the differential geometry of Finslerian spaces by dealing with that of their own tangent bundles.

A short proof of Berger’s curvature tensor estimates

A simple proof of Berger’s estimate of the curvature tensor components in terms of the pinching is given and the result is extended to nonorthonormal arguments of the curvature tensor.