Perturbational torsion about a finite extension of an elastic dielectric

Within the framework of finite electroelasticity theory of elastic dielectrics, a law is obtained which relates the longitudinal force necessary to produce a large simple extension of a hollow cylinder with the torsional modulus for a small torsion superimposed on that extension. The law depends on the strain-energy function which is not the case for a merely elastic material. In fact, the law is independent of the strain-energy function if and only if the electrical field is totally absent. In this case, the results reestablish those obtained by Green and Shield for an elastic hollow cylinder with no electric properties.

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Zusammenfassung Im Rahmen der endlichen elektro-elastischen Theorie der elastischen dielektrischen Materialien haben wir ein Gesetz, das die longitudinale Kraft, welche für eine große einfache Streckung eines hohlen Zylinders notwendig ist, mit dem Drehmodul für eine kleine Drehung, die der Streckung überlagert ist, in Beziehung bringt. Dieses Gesetz hängt jedoch — im Gegensatz zur endlichen Elastizitäts-Theorie — von der Verzerrungs-Energie Funktion ab. In der Tat, es ist unabhängig davon genau dann, wenn kein elektrisches Feld anwesend ist. In diesem Fall ergeben sich dieselben Resultate, welche Green und Shield für einen elastischen hohlen Zylinder ohne elektrische Eigenschaften erhalten haben.

     

The finite deformation of a pressurized circular tube for a class of compressible materials

This investigation is concerned with the problem of a hollow circular cylinder subjected to uniform internal and external pressure within the equilibrium theory of finite elasticity. The tube is composed of homogeneous, isotropic,compressible materials of special type, namely harmonic materials. Explicit closed-form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed as a function of the undeformed coordinates, is shown to be essentially independent of material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case, there is a critical value of the applied pressure at which the maximum hoop stress in the tube, occurring at the inner surface, becomes unbounded. Results appropriate for thin shells are also obtained. For the case of external pressure only, a critical value of the applied pressure exists for which closing of the cavity is predicted. For nearly solid cylinders, or equivalently, for a cavity in an unbounded medium, explicit results are provided for the corresponding stress concentration factor.Furthermore the stability of the preceding equilibrium states is investigated by employing a standard linearized perturbation scheme. It is found that an internally pressurized tube is always stable in the range of interest whereas an externally pressurized tube buckles prior to cavity closure.

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Zusammenfassung Die Untersuchung befaßt sich im Rahmen der Gleichgewichtstheorie fmiter Elastizität mit dem Problem eines Hohlzylinders, der einem gleichförmigen inneren und äußeren Druck ausgesetzt ist. Das Rohr besteht aus homogenen, isotropen,kompressiblen Materialien eines speziellen Typs, nämlich harmonischen Materialien. Explizite Lösungen in geschlossener Form werden für Verformung und Spannungsfelder erhalten. Es wird gezeigt, daß die Verteilung der wahren Spannung in Abhängigkeit von den unverformten Koordinaten im wesentlichen unabhängig von den Materialeigenschaften ist. Die beiden Fälle, nur innerer Druck und nur äußerer Druck, werden detailliert behandelt. Im ersten Fall gibt es einen kritischen Druck, bei dem die maximale Tangentialspannung im Rohr, und zwar an der inneren Oberfläche, über alle Grenzen wächst. Es werden außerdem Ergebnisse erhalten, die für dünne Schalen zutreffen. Im Fall eines ausschließlich äußeren Drucks existiert ein kritischer Druckwert, bei dem ein Schließen des Hohlraums vorhergesagt wird. Für annähernd massive Zylinder, oder damit gleichbedeutend, für einen Hohlraum in einem unendlichen Medium, werden explizite Ergebnisse für den entsprechenden Spannungskonzentrationsfaktor vorgelegt.Darüberhinaus wird die Stabilität vorausgehender Gleichgewichtszustände durch Anwendung eines üblichen linearisierten Störungsschemas untersucht. Es wird gefunden, daß das Rohr unter Innendruck im interessierenden Bereich immer stabil ist, während ein Rohr unter Außendruck einknickt, bevor sich der Hohlraum schließt.

     

Controllable states in compressible elastic dielectrics

Zusammenfassung Ein kontrollierter Zustand ist definiert als einer, in welchem eine endliche statische Verformung und entweder das elektrische Feld oder das dielektrische Verschiebungsfeld anfänglich vorgeschrieben sind. Es kann gezeigt werden, dass der resultierende Zustand ohne Volumkräfte oder Ladungen in jedem homogenen, isotropen und zusammendrückbaren Dielektrikum hervorgerufen werden kann.Es wird gezeigt, dass die einzig möglichen kontrollierten Zustände aus reinen homogenen Verschiebungen bestehen, kombiniert mit einheitlichen Feldern.     

Some stability conditions for a compressible elastic material

Two sets of restrictions on the strain-energy function for a compressible isotropic elastic material are obtained which are necessary conditions for stability of the material. These arise from the following considerations. (i) A rectangular block is subjected to a finite pure homogeneous deformation, and an infinitesimal pure homogeneous deformation with arbitrary principal directions is superposed. The dimensions in two of these principal directions are held constant. Then the incremental modulus associated with the third principal direction must be positive for stability to obtain. (ii) In the initial pure homogeneous deformation one pair of faces of the block is force-free. The superposed inifinitesimal pure homogeneous deformation has one of its principal directions normal to these faces, which remain force-free, and the principal extension ratio corresponding to another is unity. The incremental modulus corresponding to the third principal direction must be positive to obtain stability.

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Zusammenfassung Für ein kompressibles isotropes elastisches Material werden zwei Sätze von Einschränkungen der Funktion der Verformungsenergie als notwendige Bedingungen für die Materialstabilität hergeleitet. Diese entstehen aus den folgenden Überlegungen: (i) Ein rechteckiger Block wird einer endlichen homogenen Verformung, und einer darauf überlagerten infinitesimalen homogenen Verformung mit beliebigen Hauptachsen ausgesetzt. Die Längen in zwei dieser Hauptachsen-Richtungen werden festgehalten. Dann muß der inkrementale Modul in der dritten Hauptrichtung positiv sein, damit die Stabilität gewährleistet ist. (ii) Bei der endlichen homogenen Verformung sind zwei parallele Begrenzungsflächen des Blockes kräftefrei. Die eine Hauptrichtung der überlagerten infinitesimalen homogenen Verformung sei senkrecht zu diesen Flächen, die kräftefrei bleiben, und die Hauptstreckung in einer der anderen Hauptrichtungen sei gleich 1. Der inkrementale Modul in der dritten Hauptrichtung muß zur Gewährleistung der Stabilität positiv sein.

     

Resonances of an elastic plate in a compressible confined fluid

We present a theoretical study of the resonances of a fluid–structureproblem, an elastic plate placed in a duct in the presence ofa compressible fluid. The case of a rigid plate has been largelystudied. Acoustic resonances are then associated to resonantmodes trapped by the plate. Due to the elasticity of the plate,we need to solve a quadratic eigenvalue problem in which theresonance frequencies k solve the equations (k) = k², where are the eigenvalues of a self-adjoint operator of the formA + kB. First, we show how to study the eigenvalues locatedbelow the essential spectrum by using the min–max principle.Then, we study the fixed-point equations. We establish sufficientconditions on the characteristics of the plate and of the fluidto ensure the existence of resonances. Such conditions are validatednumerically.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

The incremental shear modulus in a compressible isotropic elastic material

A compressible isotropic elastic material is subjected to a finite pure homogeneous deformation. An infinitesimal simple shear is superposed on the deformation. Necessary and sufficient conditions are obtained for the shear modulus to be positive for all choices of the direction and plane of shear.

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Zusammenfassung Ein kompressibles isotropes elastisches Material wird einer endlichen homogenen Verformung ausgesetzt. Ein einfacher, infinitesimaler Schub wird dieser Verformung überlagert. Für alle Schubrichtungen und -ebenen werden notwendige und hinreichende Bedingungen für einen positiven Schubmodul hergeleitet.

     

Plane isentropic large displacement simple waves in a compressible elastic solid

Zusammenfassung Es gibt drei einfache Wellen, die sich in gegebener Richtung in gegebenem Sinn in einem kompressiblen elastischen Körper fortpflanzen können. In einem isotropen Körper ist eine dieser Wellen zirkular, die andern eben polarisiert. Bei kleinen Amplituden kann eine von den eben polarisierten Wellen rein dilatatorisch sein, in der anderen hat die longitudinale Komponente des Verschiebungsgradienten die Grössenordnung des Quadrats der transversalen Komponente. Es wird ein Beispiel einer solchen quasi-transversalen einfachen Welle angeführt. Die Theorie der einfachen Welle wird kurz in einem Anhang behandelt.     

Centralizers in residually finite torsion groups

Let be a residually finite torsion group. We show that, if has a finite 2-subgroup whose centralizer is finite, then is locally finite. We also show that, if has no -torsion, and is a finite 2-group acting on in such a way that the centralizer is soluble, or of finite exponent, then is locally finite.

     

The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((₀, ₀), 0, P₀) in the solution-parameter space of ((₀, ₀),0, P₀) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((₀, ₀), 0, P₀) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P₀ and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((₀, ₀), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((₀, ₀),0).     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

Dead loading of a unit cube of compressible isotropic elastic material

A unit cube of compressible isotropic elastic material undergoes a homogeneous dilatation by dead loading forces applied to its faces. Conditions are obtained for stability of the resulting equilibrium state. The physical nature of these conditions is described and the results are illustrated for a compressible Blatz-Ko foamed rubber material.     

On the speed of propagation of waves in a deformed compressible elastic material

A sinusoidal wave of infinitesimal amplitude is propagated in an isotropic compressible elastic material subjected to a static pure homogeneous deformation. Necessary and sufficient conditions are obtained for the speed of propagation to be real in the case when the direction of propagation is parallel to a principal plane of the static deformation.

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Zusammenfassung Eine Sinuswelle mit infinitesimaler Amplitude pflanzt sich in einem isotropen kompressiblen elastischen Material fort, das einer rein homogenen statischen Deformation unterworfen ist. Es werden notwendige und hinreichende Bedingungen dafür erhalten, dass die Fortpflanzungsgeschwindigkeit bei Fortpflanzung parallel zu einer Hauptebene der statischen Deformation reell ist.

     

Modelling torsion in an elastic cable in space

When an imperfectly flexible, elastic cable is suspended in a vertical plane under the sole influence of gravity and boundary conditions involving axial twist are then applied to the endpoints of the cable, its shape extends from the original vertical plane to a three-dimensional configuration in space. The aim of this paper is to develop a mathematical model (consisting of differential equations) for the configuration of and tension in such a twisted cable. The model is solved numerically for different boundary conditions. Although the final model delivers satisfactory results for small amounts of twist (inside the elastic deformation domain of the cable), realistic bounds for the amount of twist that may be applied (before plastic deformation of the cable causes model inaccuracies) are yet unknown. However, the effect of an increase in the amount of torsion applied at the endpoints of the cable is investigated numerically. The model seems capable of capturing first points of twisting bifurcation, when so much twist is applied at the endpoints that the cable jumps from un unstable equilibrium configuration to a more stable one.     

On a finite element method for unsteady compressible viscous flows

In this article we present an analysis of a finite element method for solving two‐dimensional unsteady compressible Navier‐Stokes equations. Under the time‐stepping size restriction Δt ≤ Ch, we prove the existence and uniqueness of the numerical solution and obtain an a prior error estimate uniform in time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 152–166, 2003