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This paper contains introductory notes to the special issue of the Journal of Engineering Mathematics about the life and scientific work of L. N. G. Filon (1875–1937). The objective of this paper is to provide a complete description of the influence of Filon's two fundamental papers [1, 2] cited in the classical texbooks on the theory of elasticity by Love and Timoshenko and to document related contributions in various branches of engineering mathematics and mechanics conducted over the past twenty years. The biharmonic equation, which is a central aspect of Filon's works, has provided engineers and scientists with a wealth of avenues for the investigation of a variety of problems in applied mechanics and engineering mathematics dealing with the theory of bending of plates, two-dimensional and axisymmetric problems of stress analysis in solids and two-dimensional problems of slow viscous flow. Through the celebration of Filon's classical works we are also given an opportunity for examining the role of the biharmonic equation in the formulation and solution of problems in mechanics and applied mathematics.

We study how the encasement of a growing elastic bulk within a possibly differently growing elastic coat may induce mechanical instabilities in the equilibrium shape of the combined body. The inhomogeneities induced in an incompressible bulk during growth are also discussed. These effects are illustrated through a simple example in which a growing elastic cylinder may undergo a shape transition towards a bent configuration., 17 pages, 3 figures

We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.

The steady mixed problem of the motion of a transversely isotropic elastic circular cylinder, compressed by a finite elastic shell, is solved by the method of piecewise-homogeneous solutions [1]. One of the relations of generalized orthogonality obtained for homogeneous solutions is used. Two special cases are considered: (1) a semi-infinite shell is placed on a movable cylinder with a specified negative allowance the edge of the shell is stress-free, and there is no preloading, and (2) a concentrated encircling load acts on the shell. The solution of the problem of a semi-infinite shell and the system of piecewise-homogeneous solutions are constructed in quadratures by the Wiener-Hopf method. (A similar problem was investigated in [2] in a static formulation. Steady mixed contact problems were investigated previously in [3–10]).

The issue of stress singularity in an elastic cylinder of cylindrically anisotropic materials is examined in the context of generalized plane strain and generalized torsion. With a viewpoint that the singularity may be attributed to a conflicting definition of anisotropy at r=0, we study the problem through a compound cylinder in which the outer cylinder is cylindrically anisotropic and the core is transversely isotropic. By letting the radius of the core go to zero, the cylinder becomes one with the central axis showing no conflict in the radial and tangential directions. Closed-form solutions are derived for the cylinder under pressure, extension, torsion, rotation and a uniform temperature change. It is found that the stress is bounded everywhere, and singularity does not occur if the anisotropy at r=0 is defined appropriately.

A new two-dimensional theory for the analysis of deep, doubly curved, multilayered shells is proposed. The theory is based on a kinematical approach in which the continuity conditions for displacements and shear stresses at layer interfaces and on the bounding surfaces of the shell are exactly satisfied, while, at the same time, refinements of the shear and membrane terms are taken into account, by means of trigonometric functions. The accuracy of the model is assessed in statics and dynamics through investigation of significant problems, for which an exact three-dimensional elasticity solution is known. In statics, the cylindrical bending of a cylindrical panel, of thickness h , and inner radius R , is examined. The results obtained are in good agreement with the analytical solution of Ren [Comput. Sci Technol. 29 (1987) 169–187], and prove that no shear-correction factors are requested. In dynamics, wave propagation in a three-layered elastic cylinder is explored. The accuracy of the theory is assessed by comparison with the exact three-dimensional elasticity solution of Armenakas [J. Acoust. Soc. Am. 49 (5, Part 2) (1971) 1511–1520]. The model is then applied to the exploration of the dispersive behavior of a viscoelastic cylinder.

The problem investigated herein concerns that of determining the stress-intensity factor (SIF) for a rigid circular disc-inclusion embedded into an infinitely long circular cylinder. The disc is given a constant translation along the direction perpendicular to its plane. An approximate closed-form expression for the SIF is deduced here using the method of plane sections.

Fur die elastische Scheibe werden die vollen Integralgleichungen des Kontaktproblems aufgestellt. Mit Hilfe der Methode des Riemannschen Randwertproblems erhalt man die exakte Losung fur den Fall mit vollkommenem Haften und fur die Einteilung der Beruhrungsflache in ein Haftgebiet und ein Gleitgebiet. In Anlehnung an die Frommsche Theorie zur Abhangigkeit der Losung vom Belastungsverlauf werden einige wichtige Anwendungen fur gebremste Scheiben behandelt. The exact integral equations of the contact problem for the elastic disk are represented. The exact solutions in case of rigid adhesion and for the division of the contact region for one adhesion zone and one sliding zone are obtained by the method of the Riemannian boundary value problem. A number of important applications for the deterred disk are treated, based on Fromm's theory about the dependence of the solution on the load history.

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors \(\hat b,b\) separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force \(F\left( {on \hat b} \right)\) is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation \(\hat b\) does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction.

Resume Nous etudions la convergence de la solution d'un probleme d'elasticite lineaire anisotrope heterogene pose dans un cylindre dont le diametre tend vers zero. Nous montrons en particulier que uɛ − (u + ɛv + ɛ2w) converge fortement vers zero (dans un sens que nous precisons), ou (u, v, w) est la solution unique d'un systeme elliptique d'equations aux derivees partielles.

The eversion problem of incompressible elastic cylinders is studied in the context of Finite Elasticity theory with non-smooth internal strain, LAZOPOULOS and OGDEN [1]. Considering Mooney-Rivlin like materials with radially symmetric internal strain, non-classical eversion solutions of incompressible cylinders are obtained exhibiting the coexistence of phases phenomenon, ERICKSEN [2].

Charts were prepared for estimating local maxima or minima of dimensionless, principal stresses of axial, radial, and tangential orientations formed in an elastic, cylindrical food during a drying or hydrating process. For this, analytical solutions derived previously were used. These charts showed that tangential stresses were the most critical for stress cracking during drying processes and radial stresses for hydrating processes. Sample applications of the charts were presented for adjusting process conditions to prevent stress cracking.

In a semi‐infinite cylinder composed of anisotropic linearised elastic material, loaded on the base and clamped along the lateral surface, it is known that the solution as measured, for example, by the strain‐energy flux through a plane cross‐ection decays longitudinally at most exponentially with respect to the axial distance from the base. There is, however, also a transverse radial decay of the solution, again measured for example by the strain‐energy, occurring from the region close to the cylinder's axis to the region near the lateral surface, where the energy vanishes.

A finite element model is presented which takes careful account of the three components of the cold, flat rolling process: the work roll, the rolled strip and their interface. The work roll is treated as an elastic cylinder. The elastic and plastic regions of the rolled strip are considered. The events at their interface are modelled by a coefficient of friction which depends on the relative velocity between the contacting bodies. The model calculates the roll separating force, the roll torque and the forward slip. These calculations compare well to experimental data. Following the comparisons, the model is used to examine the distributions of the stresses, strains and rates of strain in the work roll and in the rolled strip.

The present paper concerns the investigation of the stress, temperature and magnetic fields in an isotropic elastic cylinder of infinite length placed in a primary magnetic field when the curved surface of the cylinder is subjected to periodic loading. The generalised theory of thermo-elasticity proposed by Lord and Shulman [ J. Mech. Phys. Solids , 1967, 15 , 2991 is applied to account for finite velocity of heat propagation. Numerical values of the stress and temperature fields at various points of the cylindrical medium and numerical values of the perturbed magnetic field in the surrounding free space are computed for an appropriate material and the results are displayed graphically to illustrate the problem.

This paper concerns the steady state roll contact between elastic cylinder and rigid plane with modified Coulomb's law of friction. A three-dimensional contact algorithm with the help of the finite element method is used for this problem. The stresses on the contact surface are displayed graphically. From the results we know that the friction will influence the distribution of pressure on the contact area and the contact region may be divided into multiphase stick-slip zones.

Elastohydrodynamic lubrication analysis for an elastomeric layer bonded to an elastic substrate and indented by a rotating elastic cylinder is introduced. The surface elastic deformations are computed from full elasticity analysis of layered elastic half-space. The dry-contact problem was solved first, and the iterative Newton-Raphson method was used to solve the elastohydrodynamic lubrication problem. The lubricant rheological equations were considered in the present analysis. The significant effects of the elastomeric layer thickness on the pressure profiles and film shapes at different operating conditions are presented and discussed.

The purpose of the paper is to discuss the influence of the cross effects arising from the coupling of the fields of temperature, mass diffusion and that of strain in an elastic cylinder. By application of the Laplace and Fourier integral transforms we have solved the initial-boundary value problem for an infinite cylinder. The influence of Dufour's and of Soret's cross effects, dependent on time at particular points of the cylinder, have been discussed and presented on diagrams.

In this work, the finite extension and torsion of a swollen elastic cylinder bonded to an inner rigid cylindrical core are investigated in the context of the theory of interacting continua. The boundary value problem is solved numerically in order to determine the variation of the stretch-ratios, the stresses and also the volume fraction of the solid. The results of these investigations not only demonstrate significant gradients in the stretch ratios and severe stress concentrations but also stress reversals under certain conditions. These stress reversals which occur in the neighborhood of the interface of the inner and outer cylinders can have a significant impact on damage studies pertaining to fiber/matrix debonding in fiber-reinforced composites, for example.

By using the finite element method, the propagation of Lamb waves in sandwich phononic crystal plates is investigated. This structure is composed of a two-dimensional phononic crystal plate coated with two homogeneous substrates on both surfaces. The phononic crystal plate is constituted by a square array of elastic cylinder embedded in a solid matrix. It is shown that some complete band gaps are opened by changing the thickness of the phononic crystal plate, and that the width of the first band gap becomes larger as the thickness increased. The influence of the thickness and material properties of the substrates on the width of the complete band gap is discussed as well. It is noted that the thickness of either phononic crystal plate or substrates plays a key role in tuning the band gaps of the sandwich phononic crystal plates.

With reference to the end-on penetration of a cylinder, an attempt is made to deal with the water entry problem as a whole and present a picture of the various processes involved and their interrelationships. The penetration process is considered from the moment of initial contact of the cylinder with the surface of the fluid to transition to the quasisteady regime.

It is shown that the coupled singular integral equations with trigonometric kernels appearing in the problem of adhesive contact between an elastic circular cylinder and two identical rigid compressive rollers may be reduced to a problem of Muskhelishvili type and may be explicitly solved. The solution is applied to the cases when the rollers are either flat or circular, and the results are compared with those found by Hill and Tordesillas [2].

The problem considered is that of the torsion of a finite elastic cylinder which is embedded in an elastic medium of different shear modulus. By the use of integral transforms and the theory of dual integral equations, the problem is reduced to that of solving a Fredholm integral equation of the second kind. Numerical results have been displayed graphically.

This paper presents two elastic systems which protect the pipe by reducing the celerity. We propose two options for protection: the first solution named “barrel jacket”, where the wall pipe consists of an elastic inner layer soldered in the steel pipe (rubber) which has no role in resistance and the second solution named “elastic cylinder”, where in the ordinary pipe, is disposed longitudinally a cilinder with small section, fabricated from an elastic material (rubber). The common idea is that the overpressure strain the elastic element and so the flow area increase while the celerity decrease. We give the formula for celerity in the both cases and two graphical representation which showes the dependence of the celerity on the given parameters.

A Fourier transform solution for a small source on or in the vicinity of an elastic cylinder is developed. The case of acoustic radiation from the source and cylinder and, also, the direct mechanical coupling to an elastic shell are treated in a consistent manner that allows coupling through the transducer. Numerical results are given for a sample case of a thin-walled elastic tube and these results are compared with the limiting cases of a rigid and soft cylinder.

We study asymptotic expansions for the displacement field of a long elastic cylinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.

In this paper, we investigate the problem of radial diffusion of fluids through a transversely isotropic hollow non-linearly elastic cylinder. The transversely isotropic cylinder is both sheared and stretched. We study in detail how shearing and stretching the cylinder affects the diffusion process. The influence of the anisotropy of the solid on diffusion is also determined. A comparison is made with previous work (cf. Gandhi, Rajagopal and Wineman [11]) on the radial diffusion of fluids through isotropic non-linearly elastic solids.

The phenomenon of cavitation in an elastic cylinder subjected to a combined axial stretch λz and radial traction P is examined. One possible response of the cylinder to this loading, for all λz; and P, is a pure homogeneous deformation. However, for some materials, the homogeneous deformation becomes unstable at certain critical values of (λz, P), and a second deformation bifurcates from the homogeneous one; this latter deformation involves a cylindrical cavity co-axial with the elastic cylinder. We determine the values of (λz, P) at which this happens. The results are displayed by constructing a curve in the (λz, P)-plane which divides this plane into regions where the homogeneous deformation is stable and unstable.

The purpose of this study is to compare the mathematical formalism obtained by using linear elasticity with the formalism obtained by using large elastic deformation in the calculation of the stress in the myocardium. This study is motivated by previous results based on the theory of linear elasticity that have shown consistency in the calculation of the stress expressed as a sum of an active and a passive component. The myocardium is represented as a thick-walled elastic cylinder contracting symmetrically in which the active force is expressed as a force/unit volume of the myocardium.

We investigate the influence of viscous damping on the shape and stability of travelling waves induced by Coulomb friction between a rotating rigid shaft and an annular elastic cylinder. As expected, the damping causes the travelling wave solutions to become smoother and more stable. It also decreases the amplitude and range of separation solutions and may destroy the travelling waves by grazing bifurcations at large values of the damping parameters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Recent experiment shows that an elastic strain gradient field can be utilized to transport spherical particles on a stretchable substrate by rolling, inspired by which a generalized plane-strain Johnson-Kendall-Roberts model is developed in this paper in order to verify possible rolling of an elastic cylinder adhering on an elastic substrate subject to a strain gradient. With the help of contact mechanics, closed form solutions of interface tractions, stress intensity factors, and corresponding energy release rates in the plane-strain contact model are obtained, based on which a possible rolling motion of an elastic cylinder induced by strain gradients is found and the criterion for the initiation of rolling is established. The theoretical prediction is consistent well with the existing experimental observation. The result should be helpful for understanding biological transport mechanisms through muscle contractions and the design of transport systems with strain gradient.

In the present work we consider the inverse problem of identification the spherical cavity of small relative size in the isotropic elastic cylinder. Based on the methods of perturbation theory, analytical formulas for determining the parameters of the cavity have been obtained using information concerning corrections to the natural frequencies. A series of numerical experiments has been carried out.

In this work, we construct a new model of a fluid inside an elastic cylinder with two chambers. Mathematical modeling and numerical simulations of flows driven by pumping are presented. A periodic compression of the asymmetric part of the elastic tube generates a unidirectional flow. Both size and the direction of the generated net flow highly depend on the pumping frequency.

The acoustic scattering from elastic objects features in many tasks stretching from the use of sonar to search for underwater objects such as submarines and mines to the inspection of shipping containers and cargo screening. The properties of canonical objects, such as cylindrical and spherical shells, which are completely filled with fluid, or immersed in fluid, have been widely studied, but typically stop short of studying shells containing different levels of filling. Not knowing the extent of the effect of filling on the scattering from different objects can limit how well one can discriminate between similar objects or inspect the interior of objects and, for example, find contraband or other suspicious materials. This paper looks at the two‐dimensional response of a partially filled elastic cylinder. A computational method to model non‐uniform domains is introduced and the use of perfectly matched layers is discussed. Results are shown for combinations of cylinders and different levels of filling. In particular, we show how the response of the cylinder depends on the level of the filling. The results are used to comment on the usefulness, or otherwise, of the response to estimate the nature and level of filling based on short‐ or long‐range observations.

In this paper, a hollow elastic cylinder with an inner–surface circular crack under an impacting tensile load has been analysed by experimental method, the varying curve of dynamic SIF with time has been put forward. This results is helpful in theoretical research of the near–tip field of symmetric three–dimensional problem.

The Dde l'Op´ erateur de Retournement Temporel method applies to scattering analysis with arrays of transducers. It comprises the study of the time-reversal invariants which correspond to the eigenvectors of the time- reversal operator or to the singular vectors of the array response matrix K.I n this paper, the decomposition of the scattered pressure into normal modes of vibrations is used to determine the theoretical time-reversal invariants for a large elastic object such as a cylinder or sphere. For an N transducers one- dimensional array, the time-reversal operator is dimension N. It is shown that the dimension is reduced to 2k0a for a cylinder or a sphere, where a is the scatterer radius and k0 is the wave number in the surrounding fluid. Furthermore, this approach provides analytical expressions of symmetric and anti-symmetric singular values and vectors in the sub-resolution limit, i.e. when the scatterer diameter is smaller than the array resolution cell. These results are verified experimentally and in good agreement with the original point of view: for a small scatterer, one singular value dominates and the associated singular vector focuses isotropically on the scatterer. (Some figures in this article are in colour only in the electronic version)

In this study, a new analytical expression of the radiation force function—which is the radiation force per unit energy density and unit cross sectional surface—for an elastic cylinder immersed in water and placed in a plane progressive acoustic wave is presented. The calculation is based on the far-field derivation approach. Furthermore, two solutions for the radiation force function are compared. The first is based on the far-field derivation and the second on the near-field. It is shown that for an ideal fluid both results are commensurate with the same result. Moreover, an experiment is conducted to verify the theoretical predictions for an elastic cylinder as well. The experimental results confirm the theory for elastic cylinders but show a quite different deviation from the rigid solution. These results are the first to confirm the theory for the radiation force on an elastic cylinder placed in a plane progressive wave-field.

A T‐matrix formalism developed for plane‐wave scattering from a fluid‐loaded elastic spherical shell near a rough fluid elastic interface [J. Acoust. Soc. Am. 101, 767–788 (1997)] is generalized to include scattering from targets for which a T‐matrix is known and constructed on a spherical wave basis, and from the interface of penetrable sediments for which a T‐matrix is known and constructed on a Weyl plane‐wave basis. A variety of numerical results are obtained for plane‐wave scattering from elastic spheres and elastic spherical shells, as well as a finite elastic cylinder with hemispherical end caps near rigid, soft, fluid, elastic, poroelastic, and layered fluid sediments with planar boundaries that demonstrate some of the effects’ target‐boundary scattering on resonant and nonresonant scattering.

Submarine hulls are often modeled as stiffened elastic plates at frequencies above the plate coincidence frequency. At lower frequencies, hull curvature and stiffener characteristics become more important. At these frequencies, cylindrical shell and circular beam models may be more appropriate in predicting the acoustic scattered field from stiffened hulls. A theoretical approach is presented describing the bistatic acoustic field scattered from a stiffened elastic cylinder. A finite number of stiffeners are treated as nonuniform translational and rotational modal constraints upon the elastic motion of the fluid‐loaded cylindrical shell. The stiffeners are modeled as curved beams that constrain the shell's three degrees‐of‐freedom at variable locations. This formulation allows the lighter hull rib stiffeners and more massive bulkheads to be considered as an elastic system. For M variably spaced stiffeners, the modal solution depends upon the calculation of seven sets of M(M − 1)/2 cylindrical, fluid‐loaded, cross‐admittance functions and the inversion of a 4M by 4M matrix.

The authors achieve a parametrical study of the acoustical resonances frequencies of an elastic cylinder versus the elastic parameters of the scatterer (longitudinal sound velocity cl, shear sound velocity ct, density ρ). They use numerical computations because of the complexity of the equations obtained in the resonance scattering theory, even for this simple geometry. The result obtained is quite surprising. Most of the resonance of a given mode n belong to specific planes in the 3‐D space (x1, cl, ct), where x1 is the dimensionless frequency (x1 = k1a). These planes obey very simple equations and belong to two families. The implications of these results are discussed.

Frictionless static contact between two layered elastic cylinders is examined. Local curvature of the the contacting surfaces is taken into account. Green’s function is derived and utilized to reduce the contact problem to Fredholm integral equation of the first kind with unknown contact width, unknown contact pressure distribution and unknown normal approach. Solution of this integral equation is obtained in the form of an infinite system of linear algebraic equations with the aid of Fourier transformation. The resulting equations are solved to give the pressure distribution and the normal approach for assumed contact width. Numerical results of two examples are presented: one in which an elastic layer bonded to a rigid cylinder is indented by another rigid cylinder, and, a second in which a layered elastic cylinder is indented by another layered elastic cylinder. The results obtained in the first example are compared with analytical solutions and with experimental data known from the literature.