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.

     

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.

     

On the stability of a compressible Rivlin's cube made of transversely isotropic material

^** Email: kostas.soldatos{at}nottingham.ac.uk This paper considers a unit cube made of a compressible, transverselyisotropic elastic material, with the direction of transverseisotropy being aligned normal to one pair of the cube's faces,and investigates the stability of a dilatation equilibrium stateof that cube, with respect to superposed pure homogeneous deformationswith principal directions parallel to the cube edges. This dilatation,intermediate equilibrium state (state I) of the cube is assumedattainable in two different ways. Accordingly, in what is termedas the ‘principal stability problem’ under investigation,state I is considered to be that of uniform dilatation, whichis attained upon loading normally and uniformly the oppositefaces of the unit cube with certain pairs of equal and oppositelydirected forces having appropriately selected magnitudes. Inwhat is termed as the ‘modified stability problem’under investigation, the same compressible, transversely isotropicunit cube is loaded uniformly by three identical pairs of equaland oppositely directed forces acting normally to its faces,and, hence, it attains in state I the shape of a certain rectangularparallelepiped. The necessary and sufficient conditions forstability of state I of the cube deformation are obtained inthe form of three inequalities, which are found to hold regardlessof whether the intermediate equilibrium state I is that of uniformdilatation or that of the aforementioned rectangular parallelepiped(non-uniform dilatation). These, however, lead to quite differentspecific results and conclusions when applied in connectionwith, first, the principal and, then, the modified stabilityproblem of a unit cube made of a particular type of a transverselyisotropic extension of the Blatz–Ko (isotropic) material.     

Theory of the elastic stability of compressible and incompressible composite media

The present article considers general problems of the theory of the elastic stability of composite media in the presence of finite and small precritical deformations with an arbitrary elastic-potential form. Our investigation was conducted for homogeneous and piecewise-homogeneous anisotropic media. Numerical results were obtained for laminar media. We have elucidated the case in which there is internal loss of stability (in the material structure) for compressible materials with small deformations (plane problem) and for incompressible materials with highly elastic deformations (plane and three-dimensional problems) for the Treloar and Mooney potentials.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 267–275, March–April, 1972.     

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.     

Hold effect in finite torsion of a compressible elastic tube

We consider the application of complex variable method to elastic problem and investigate the nonlinear effect of finite torsion of a compressible elastic composite layer. We obtain that as a result of finite deformation approach, a tube subjected to torsion decreases in radius giving rise to a “hold effect”.     

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.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Asymptotic stability for one-dimensional motion of non-Newtonian compressible fluids

Rarefaction wave solutions for a one-dimensional model system associated with nomNewtonian compressible fluid are investigated in terms of asymptotic stability. The rarefaction wave solution is proved to be asymptotically stable, provided the initial disturbance is suitably small. The proof is given by the elemental L2 energy method.     

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

We study the possibility for an isotropic elastic body to support forms of instability induced by shear stress states which are reminiscent of the planar Couette and the Taylor–Couette patterns observed in the flow of viscous fluids. Here, we investigate the emergence of bifurcating periodic deformations for an infinitely long compressible elastic block confined between and attached to parallel plates which are subject to a relative shear displacement. We specialize our analysis by considering a generalized form of the Blatz–Ko strain energy function and show through numerical representative examples that planar Couette modes are always preferred with respect to the twisting Taylor–Couette modes. Finally, we introduce a suitably restricted form of the strong ellipticity condition for the incremental elasticity tensor and discuss its significance in this bifurcation problem.     

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).     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.