Closed form solutions for small deformations superimposed upon the simultaneous inflation and extension of a cylindrical tube

For small deformations of isotropic incompressible hyperelastic materials which are superimposed upon the simultaneous inflation and extension of a cylindrical tube, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the load-deflection relation for small radial deformations of pre-compressed long bonded cylindrical rubber bush mountings. They are also used to formulate the n=1 buckling criterion for long cylindrical tubes which are subjected to uniform external pressure.

---

Zusammenfassung Für kleine Deformationen, die der gleichzeitigen Weitung und Streckung eines zylindrischen Rahmens aus isotropischem, kompressiblem und hyperelastischem Material überlagert sind, werden Lösungen in geschlossener Form abgeleitet, ohne einschränkende Bedingungen für die Deformationsenergiefunktion. Mit Hilfe dieser Lösungen wird die Last-Deflektions-Beziehung für kleine radiale Deformationen vorgespannter zylindrischer Gummibuchsen abgeleitet. Weiterhin werden die Lösungen benutzt, um die Knickbedingung (für n=1) langer, zylindrischer Rohre zu formulieren, die gleichförmigem äusserem Druck ausgesetzt sind.

     

Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure

For isotropic incompressible hyperelastic materials, the problem of determining the critical external pressure at which a long thick-walled circular cylindrical shell will buckle involves solving a fourth-order system of highly non-homogeneous, ordinary differential equations. Closed-form solutions of this system are derived here for plane-strain conditions and for the particular case of the Varga material. These solutions are used to derive the buckling criterion and numerical values are obtained for the resulting critical pressures. They are found to be in good agreement with existing theoretical and experimental results for the neo-Hookean material.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

A family of solutions describing plane strain cylindrical inflation in finite compressible elasticity

Within the context of finite, compressible, isotropic elasticity, a family of solutions describing plane strain cylindrical inflation of cylindrical shells is obtained for a class of materials that includes both the harmonic and Varga materials. Additionally it is shown that the class of materials chsen is the largest class of materials for which the family of solutions is possible.     

Stability analysis of radial inflation of incompressible composite rubber tubes

The inflation mechanism is examined for a composite cylindrical tube composed of two incompressible rubber materials, and the inner surface of the tube is subjected to a suddenly applied radial pressure. The mathematical model of the problem is formulated, and the corresponding governing equation is reduced to a second-order ordinary differential equation by means of the incompressible condition of the material, the boundary conditions, and the continuity conditions of the radial displacement and the radial stress of the cylindrical tube. Moreover, the first integral of the equation is obtained. The qualitative analyses of static inflation and dynamic inflation of the tube are presented. Particularly, the effects of material parameters, structure parameters, and the radial pressure on radial inflation and nonlinearly periodic oscillation of the tube are discussed by combining numerical examples.     

Finite Deformations and Motions of Radially Inextensible Hollow Spheres

Radial inflation–compaction and radial oscillation solutions are presented for hollow spheres of isotropic elastic material that are radially inextensible. The solutions for radial inflation–compaction and radial oscillation are obtained also for everted radially inextensible hollow spheres of isotropic elastic material. The static and dynamic results for everted and uneverted radially inextensible hollow spheres are then compared. Harmonic and compressible Varga materials are used to demonstrate the solutions.      

Asymptotic Axially Symmetric Deformations for Perfectly Elastic Neo-Hookean and Mooney Materials

For axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal series solutions are determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar coordinate for the material coordinates. Remarkably, for both the neo-Hookean and Mooney materials, the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From the incompressibility condition and the equilibrium equations, the six unknown deformation functions, appearing in the first three terms can be reduced to five formal integrations involving in total seven arbitrary constants A, B, C, D, E, H and k ², and a further five integration constants, making a total of 12 integration constants for the deformation field. The solutions obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which has bonded metal end-plates. The solution so determined is approximate in two senses; namely as an approximate solution of the governing equations and for which the stress free and displacement boundary conditions are satisfied in an average manner only. The resulting load-deflection relation is shown graphically. The solution so determined, although approximate, attempts to solve a problem not previously tackled in the literature.      

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Radial deformation of a plane sheet containing a circular hole or inclusion

Seth's generalized measure of strain is used in the determination of the deformation field produced by radial forces acting on a thin plane sheet of incompressible isotropic elastic material, containing a circular hole or circular rigid inclusion, at its outer edge. Results calculated are comparable with the experimental results obtained by Rivlin and Thomas, employing a vulcanized natural rubber.     

Critical stretch for formation of a cylindrical void in a compressible hyperelastic material

Formation of a cylindrical void in an infinitely long compressible hyperelastic cylinder under axial and radial stretches is examined. The cavitation phenomenon is viewed here as a bifurcation of a solution with a cavity from the homogeneously deformed configuration, taking place when the applied radial stretch reaches a certain critical value. This amounts to modeling the underlying phenomenon as a kind of elastic instability. A formulation of shooting-method type is presented to derive an equation which gives the critical radial stretch for a prescribed axial stretch. For a special class of hyperelastic solids, called modified Blatz-Ko material, the obtained equation leads to an explicit expression for the critical stretches and stresses. Some analytical as well as numerical calculations are carried out to explicitly obtain the critical values for cavitation. The results are summarized in the form of cavitation curves in the two-dimensional space of axial and radial stretches or stresses. Influence of the axial stretch on the critical radial stretch is discussed. Throughout the paper, the corresponding results for a spherically symmetric void formation are referred to when appropriate and compared with the cylindrical case of the present interest. It is then indicated that in the state of equitriaxial stretch, cavitation into a cylindrical shape is likely to occur at lower stretch than into a spherical one.     

Existence and stability of bushes of vibrational modes for octahedral mechanical systems with Lennard-Jones potential

The special non-linear dynamical regimes, “bushes of normal modes”, can exist in the N-particle Hamiltonian systems with discrete symmetry (Physica D 117 (1998) 43). The dimension of the bush can be essentially less than that of the whole mechanical system. One-dimensional bushes represent the similar non-linear normal modes introduced by Rosenberg. A given bush can be excited by imposing the appropriate initial conditions, and the energy of the initial excitation turns out to be trapped in this bush.In the present paper, we consider all possible vibrational bushes in the simple octahedral mechanical system and discuss their stability under assumption that the interactions between particles are described by the Lennard-Jones potential.     

Dynamic crack-tip fields in rate-sensitive solids

The asymptotic stress and strain fields near the tip of a crack which propagates dynamically in a rate-sensitive solid are obtained under anti-plane shear and plane strain conditions. The problem is formulated within the context of a small-strain theory for a solid whose mechanical behavior under high strain rates is described by an elastic-viscoplastic constitutive relation. It is shown that, if the stresses are singular at the crack-tip, the viscoplastic relation is equivalent asymptotically to an elastic-non-linear viscous relation. Furthermore, for a certain range of the material parameter which characterizes the rate-sensitivity of the material, the elastic strain-rates near the propagating crack tip are shown to have the same asymptotic radial dependence near the propagating crack-tip as the inelastic strain-rates. This determines the order of the stress singularity uniquely. The governing equations for anti-plane shear and plane strain are then derived. The numerical results for the stress and strain fields are presented for anti-plane shear and plane strain. For the present model, the results suggest that under small-scale yielding conditions, there exists a minimum velocity for stable steady crack propagation. The implication that a terminal velocity for a running crack may exist is also discussed.     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

On Computational Issues in Large Deformation Analysis of Rubber Bushings*

ABSTRACT

Accurate bushing analysis requires a locking free finite element formulation, an appropriate selection of the strain energy density function, and an adequate use of bulk modulus to assure numerical stability and accuracy. In this paper, the pressure projection finite element method is employed. The method projects displacement-calculated pressure onto a lower order pressure field, based on the Babuska-Brezzi condition, to avoid volumetric locking and pressure oscillation. Mooney-Rivlin and Cubic strain energy density functions are used to study the material effect on the predicted rubber behavior in tension-compression and shear deformation modes, and the need to use a higher order strain energy density function for bushing analysis is identified. The effect of bulk modulus on bonded rubber behavior in bushings with respect to bushing shape factor is studied, and the minimum allowable bulk modulus to impose incompressibility in bushing analysis is characterized. The load-deflection response of annular bushings subjected to axial, torsional, and radial deformations are analyzed and results are compared to linear approximations. An effort is made to demonstrate how a Mooney-Rivlin model cannot capture load-displacement nonlinearities in bushing axial and torsional deformations. Two- and three-dimensional results are compared and the applicability of two-dimensional analysis is discussed.     

Gent models for the inflation of spherical balloons

We revisit an iconic deformation of non-linear elasticity: the inflation of a rubber spherical thin shell. We use the 3-parameter Mooney and Gent-Gent (GG) phenomenological models to explain the stretch–strain curve of a typical inflation, as these two models cover a wide spectrum of known models for rubber, including the Varga, Mooney–Rivlin, one-term Ogden, Gent-Thomas and Gent models. We find that the basic physics of inflation exclude the Varga, one-term Ogden and Gent-Thomas models. We find the link between the exact solution of non-linear elasticity and the membrane and Young–Laplace theories often used a priori in the literature. We compare the performance of both models on fitting the data for experiments on rubber balloons and animal bladder. We conclude that the GG model is the most accurate and versatile model on offer for the modelling of rubber balloon inflation.     

A closed-form solution procedure for circular cylindrical shell vibrations

A systematic procedure for obtaining the closed-form eigensolution for thin circular cylindrical shell vibrations is presented, which utilizes the computational power of existing commercial software packages. For cylindrical shells, the longitudinal, radial, and circumferential displacements are all coupled with each other due to Poissons ratio and the curvature of the shell. For beam and plate vibrations, the eigensolution can often be found without knowledge of absolute dimensions or material properties. For cylindrical shell vibrations, however, one must know the relative ratios between shell radius, length, and thickness, as well as Poissons ratio of the material. The mode shapes and natural frequencies can be determined analytically to within numerically determined coefficients for a wide variety of boundary conditions, including elastic and rigid ring stiffeners at the boundaries. Excellent agreement is obtained when the computed natural frequencies are compared with known experimental results.     

Vibration analysis of nonhomogeneous orthotropic cylindrical shells including combined effect of shear deformation and rotary inertia

This paper is the result of an investigation on the vibration of non-homogeneous orthotropic cylindrical shells, based on the shear deformation theory. Assume that the Young’s moduli, shear moduli and density of the orthotropic material are continuous functions of the coordinate in the thickness direction. The basic equations of non-homogeneous orthotropic cylindrical shells with the shear deformation and rotary inertia are derived in the framework of Donnell-type shell theory. The ends of a non-homogeneous orthotropic cylindrical shell are considered as simply supported. The basic equations are reduced to the sixth-order algebraic equation for the frequency using the Galerkin method. Solving this algebraic equation, the lowest values of non-dimensional frequency parameters for non-homogeneous orthotropic cylindrical shells with and without shear deformation and rotary inertia are obtained. Calculations, effects of shear stresses and rotary inertia, orthotropy, non-homogeneity and shell geometry parameters on the lowest values of non-dimensional frequency parameter are described. The results are verified by comparing the obtained values with those in the existing literature.     

理想正交各向异性弹塑性材料平面应变条件下静止裂纹尖端场

以Hill唯象理论为基础,建立正交各向异性弹塑性材料的本构关系,给出理想正交各向异性弹塑性材料在平面应变条件下混合型静止裂纹尖端的弹塑性场.与J.Pan的解不同,采用自相似假定,可以用解析方法求得不存在应力间断的应力场.对满塑性区条件和应变的奇异性加以讨论,这些为建立断裂准则提供了理论的依据.     

On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes

The finite amplitude, free radial oscillations of a thick-walled circular cylindrical tube are studied for an arbitrary incompressible, isotropic and homogeneous rubber-like material having limiting molecular chain extensibility. First, based on classical results for hyperelastic tubes, some results for thick-walled Mooney-Rivlin tubes are described graphically in the phase plane. Then the periodicity of the finite amplitude, free oscillations of a general limited elastic, thick-walled tube is studied; and some analytical results for the Gent model are illustrated in several numerical examples. Results for thick-walled Gent tubes are compared with those for corresponding Mooney-Rivlin tubes; and the motion of thin-walled Gent tubes is illustrated in the phase plane. Physical conclusions are presented. The period of small amplitude oscillations of an arbitrary elastic or limited elastic tube is derived from relations obtained by a linearization of a general class of equations of which the tube problem is a special case. Classical results of the linear theory are thereby recovered and compared with results for Mooney-Rivlin and Gent tubes.