Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables： time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.     

Bi-Lipschitz homogeneous curves in are quasicircles

We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.

     

Finite element modeling of micropolar-based phononic crystals

The performance of a Cosserat/micropolar solid as a numerical vehicle to represent dispersive media is explored. The study is conducted using the finite element method with emphasis on Hermiticity, positive definiteness, principle of virtual work and Bloch–Floquet boundary conditions. The periodic boundary conditions are given for both translational and rotational degrees of freedom and for the associated force- and couple-traction vectors. Results in terms of band structures for different material cells and mechanical parameters are provided.     

Homogeneous continua in 2-manifolds

In 1960 R.H. Bing [2] proved that every homogeneous plane continuum that contains an arc is a simple closed curve. At that time Bing [2, p. 228] asked if every 1-dimensional homogeneous continuum that contains an arc and lies on a 2-manifold is a simple closed curve. We prove that no 2-manifold contains uncountably many disjoint triods. We use this theorem and decomposition theorems of F.B. Jones [10] and H.C. Wiser [19] to answer Bing's question in the affirmative. We also prove that every homogeneous indecomposable continuum in a 2-manifold can be embedded in the plane. It follows from this result and another theorem of Wiser [20] that every homogeneous continuum that is properly contained in an orientable 2-manifold is planar.     

A new class of flexure-free torsional vibrations of annular rods

It is shown that the system of partial differential equations governing small amplitude vibrations of an elastic ring has solutions describing motions in which the axial curve C of the ring remains rigid, but executes a rocking motion while the cross-sections undergo torsional rotations about C that vary periodically both in time and in distance along C. This type of flexure-free torsional vibration can occur both in rings that are stress-free in a circular equilibrium configuration and in rings formed by bringing together and sealing, with or without the addition of twist, the ends of rods that are stress-free when straight.

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Sommario Si mostra che le equazioni che governano le oscillazioni di piccola ampiezza di travi elastiche anulari hanno soluzioni che corrispondono a moti nei quali l'asse C della trave è sottoposto a un'oscillazione rigida e le sezioni trasversali subiscono una rotazione torsionale che varia periodicamente nel tempo e con l'ascissa lungo C. Questo tipo di vibrazione torsionale, non accompagnata da flessione dell'asse, si presenta sia quando la configurazione anulare è priva di sforzi sia quando tale configurazione può pensarsi ottenuta inflettendo una trave che in uno stato privo di sforzi è ad asse rettilineo, imprimendo ad essa una eventuale deformazione torsionale, e collegando fra loro le sue estremità.

     

How to lasso a plane gravitational wave

Beginning with the stress-energy tensor of an elastic string this paper derives a relativistic string and its form in a parallel transported Fermi frame including its reduction to a Cosserat string in the Newtonian limit. In a Fermi frame gravitational curvature is seen to induce three dominant relative acceleration terms dependent on: position, velocity and position, strain and position, respectively. An example of a string arranged in an axially flowing ring (a lasso) is shown to have a set of natural frequencies that can be parametrically excited by a monochromatic plane gravitational wave. The lasso also exhibits, in common with spinning particles, oscillations about geodesic motion in proportion to spin magnitude and wave amplitude when the spin axis lies in the gravitational wave front.     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

The Space of subcontinua of a 2-dimensional continuum is infinite dimensional

Let be a metric continuum and let denote the space of subcontinua of with the Hausdorff metric. We settle a longstanding problem showing that if then . The special structure and properties of hereditarily indecomposable continua are applied in the proof.

     

The equations of continuum mechanics and the laws of thermodynamics

With appropriate constitutive assumptions on the stress tensor, the heat flux vector, and the frictional heating associated to a process, we derive for a fluid media the existence of internal energy and entropy as well as the classical energy balance equation and the Clausius-Duhem inequality.

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Sommario Con riferimento ad un continuo fluido, si dimostra-sotto appropriate ipotesi costitutive sul tensore delle tensioni, il vettore del flusso termico e il riscaldamento per altrito associato a un processo-l'esistenza della energia interna e dell'entropia di un continuo fluido. Si derivano inoltre la classica equazione di bilancio dell'energia e la diseguaglianza di Clausius-Duhem.

     

A Semi-Analytical Semi-Experimental Solution for Flow Simulations in COSSERAT Continuum Mechanical Approach to Pulsatile Blood Flow in a Tapered Femoral Artery of a Dog

Flow simulations (investigation of velocity and microrotation fields) were carried out by solving the mass, linear momentum, and angular momentum equations in Cosserat continuum mechanics with a semi-analytical semi-experimental method; for unsteady, pulsatile, laminar, and locally fully developed blood flow and validation, using experimental pressure distribution in a mildly tapered femoral artery of a living dog. Finally, we present a time-dependent profile and an approximated Gaussian equation for k _v (a material quantity that shows influence of microrotation field on the stress tensor) in this article.