Non-smooth solutions in the azimuthal shear of an anisotropic nonlinearly elastic material

In the context of nonlinear anisotropic elasticity the model plane-strain problem of azimuthal shear of a circular cylindrical tube is considered. In particular, the effect of anisotropy associated with preferred material directions on the emergence and disappearance of non-uniqueness of solution is examined. The non-smooth character of the global energy-minimizing solution of the boundary-value problem in which the inner boundary is fixed and the outer boundary is subject to a prescribed shear traction is highlighted and illustrated graphically.     

Nonlinear magnetoelastic deformations of elastomers

Summary Magneto-sensitive (MS) elastomers are smart materials whose mechanical properties may be changed rapidly by the application of a magnetic field. Such materials typically consist of micron-sized ferrous particles dispersed within an elastomeric matrix. The equations governing deformations of these materials were discussed in a recent paper by the present authors and applied in a particular specialization of the constitutive model to the problem of axial shear of a circular cylindrical tube subject to a radial magnetic field. In the present paper we develop the governing equations for a more general form of constitutive model and provide alternative forms of the equations, including a Lagrangian formulation. To illustrate the theory the problem of azimuthal shear of a circular cylindrical tube is formulated and then solved for a specific constitutive law with a magnetic field that is initially radial. The results, which show the stiffening of the azimuthal shear stress/strain response with increasing magnetic field strength, are illustrated graphically.     

Finite axisymmetric deformations of elastic tubes: An approximate method

Finite axisymmetric deformation of a hollow circular cylinder with a finite length, composed of a neo-Hookean material, is studied. The inner surface of the tube is subjected to both normal and tangential tractions, while the outer surface is free of tractions. The cylinder will undergo both radial and axial deformations. An asymptotic-expansion method is used to determine the stress and shape of the deformed tube. The deformed radial and axial coordinates, the stress tensor and the surface tractions are expanded into a power series of an appropriate thickness parameter. A hierarchy of equilibrium equations, boundary conditions and constitutive equation are derived following the usual procedure. The theories corresponding to the lowest two order members in this hierarchy are studied in detail. It is shown that the zeroth-order theory corresponds to the membrane theory. The shape of the deformed tube, up to the second-order in the thickness parameter, is determined in terms of the zeroth-order radial and axial deformations. The zeroth-order radial and axial deformations are governed by a coupled pair of nonlinear ordinary differential equations, both of which are of second order. For illustrative purposes the present approach is then applied to a simple representative problem: simultaneous extension and inflation of a cylindrical elastic tube. Finally, the solutions corresponding to the zeroth and first-order approximations of the present theory and the exact solutions obtained from finite elasticity theory are compared for the above-mentioned problem.     

Analysis of shear bands in a dynamically loaded viscoplastic cylinder containing two rigid inclusions

Summary We study plane strain thermomechanical deformations of a hollow circular cylinder containing two rigid non-heat-conducting ellipsoidal inclusions placed on a radial line symmetrically with respect to the center. These inclusions can be viewed as precipitates or second phase particles in an alloy. The material of the cylinder is presumed to exhibit thermal softening, but strain and strain-rate hardening. The impact load applied on the inner surface of the cylinder is modeled by prescribing a radial velocity and zero tangential tractions at material particles situated on the inner surface. Rigid body motion of the inclusion is considered and no slip condition between the inclusion and the cylinder material is imposed.It is found that shear bands initiate from points adjacent to inclusion tips near the inner surface of the cylinder and propagate toward this surface. At inclusion tips near the outer surface of the cylinder, the maximum principal logarithmic strain and the temperature are high and the effective stress is low, but severe deformations there do not propagate outward.     

Second-gradient plane deformations of ideal fibre-reinforced materials: implications of hyper-elasticity theory

The influence that some new, second-gradient, effects introduced in a recent publication (Spencer and Soldatos, Int J Non-linear Mech 42:355-368, 2007) have on finite plane deformations of ideal fibre-reinforced hyper-elastic solids is investigated. The second-gradient effects are due to the ability of the fibres to resist bending but, in the present case, the constraints of material incompressibility and fibre inextensibility associated with this ideal class of materials offer considerable theoretical simplification. In agreement with its conventional counterpart, where inextensible fibres are perfectly flexible, the present new theoretical development is still associated with kinematics and reaction stresses that are largely independent of the specific type of material behaviour considered. Static equilibrium considerations reveal therefore a manner in which relevant, non-symmetric stress distributions can be determined by solving two simultaneous, first-order linear differential equations. However, the principal interest of this investigation remains within the class of hyper-elastic materials for which two sets of relatively simple constitutive equations are obtained. At this stage of early theoretical development, immediate interest is directed towards the simplest of those sets, namely the set associated with problems where only gradients relevant to the change of the deformed fibre direction are of principal importance. These developments are applied (i) to the classical problem of plane-strain bending of a rectangular block reinforced by a family of straight fibres running parallel to one of its sides; and (ii) to the problem of “area-preserving” azimuthal shear strain of a circular cylindrical tube having its cross-section reinforced by a family of strong fibres. In the particular case in which the fibres are initially straight and aligned with the radii of the tube cross-section, the solution of the latter problem, which is new in the literature, reveals that fibres resist local bending completely. Instead, they remain straight during deformation and force the tube cross-section to undergo area-preserving azimuthal shear by changing their direction.     

Material tailoring and moduli homogenization for finite twisting deformations of functionally graded Mooney-Rivlin hollow cylinders

We analytically analyze finite plane strain twisting deformations of a hollow cylinder made of an isotropic and inhomogeneous Mooney-Rivlin material with material moduli varying in the radial direction. The cylinder is deformed by applying either tangential tractions on the inner surface and tangential displacements on the outer surface or vice versa. The radial variation of the moduli is found that will minimize the tangential displacement of the bounding surface where tangential traction is specified. Furthermore, the modulus of a homogeneous neo-Hookean cylinder is found that is energetically equivalent to the inhomogeneous cylinder.     

Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a pre-assigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r². The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r². For a hollow cylinder fixed on the inner surface and subjected to tangential tractions on the outer surface (or vice versa) the through-the-thickness in-plane shear stress distribution is also universal and is determined by surface tractions and the outer radius of the cylinder; it is independent of the spatial variation of the shear modulus.     

Effect of omitting terms involving tube radii difference in shell models on buckling solutions of DWNTs

Continuum cylindrical shell models have been widely applied in the buckling analysis of carbon nanotubes. An explicit expression for the critical buckling strain of double-walled carbon nanotubes (DWNTs) may be obtained based on cylindrical shell models. The expression is usually simplified by neglecting the terms involving outer and inner tube radii difference. In this brief note, we present the critical buckling strains of DWNTs with the inclusion of these terms and investigate the quantitative effect of neglecting these terms on the critical strain. It was found that the omission of the terms related to outer and inner tube radii difference leads to an overprediction of the critical buckling strain as well as a possible change in the buckling mode shape. It is also observed that the effect of the terms is especially significant for DWNTs with small inner radius but is negligible when the inner radius is relatively large.     

离散多层圆筒在热冲击载荷下的弹性动力响应

离散多层圆筒由薄内筒和倾角错绕的钢带层组成,具有制造简便、成本低等优点。预测筒体在热冲击载荷下的热应力对强度设计和安全操作具有重要的应用价值。该文首次研究了离散多层圆筒在热冲击载荷作用下的热弹性动态响应。将内筒和钢带层的径向位移分别分解为满足给定应力边界条件的准静态解和满足初始条件的动态解,准静态解通过齐次线性方法确定,热弹性动态解通过有限Hankel积分变换和Laplace变换确定。根据内外层界面处位移连续条件,得到层间压力关于时间的第二类Volterra积分方程,利用Hermit二次三项式插值方法可求得该层间应力。最后将离散多层圆筒的热弹性动力响应与单层厚壁圆筒的响应进行了比较,并分析了钢带缠绕倾角和材料参数对热弹性动力响应的影响。     

The eigenvalue problem for hollow circular cylinders

In three-dimensional elasticity the solution of the biharmonic equation for a hollow circular cylinder can be presented in terms of Bessel functions. If there are no surface tractions on either of the radial faces and no thermal effects, an eigenvalue problem arises. A method of establishing these eigenvalues and tables of them for various types of hollow cylinders are presented. Two special cases are investigated, namely, as the ratio of radii tends to unity, that is, a ‘thin shell’, and as the ratio tends to infinity, which can either be regarded as the inner radius tending to zero for a fixed outer radius or as the outer radius tending to infinity for a fixed inner.The eigenvalues are subsequently used for the calculation of the effect of end loading on a semi-infinite length cylinder. From this a quantitative comparison can be made with thin shell theory in the transition region between thin and thick shells of this type.     

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction. The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements, and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum. Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.     

Shear induced loss of saturation in a fluid infused swollen hyperelastic cylinder

We discuss a continuum model for the absorption and redistribution of fluid in swollen elastomers due to mechanical loading; and in particular distinguish between systems that are saturated with liquid and those that are not. To this end we consider a boundary value problem of radial displacement combined with azimuthal shear for an annular cylinder consisting of a fluid infused hyperelastic media. In the absence of load the elastomer deforms by free swelling, giving a homogeneous expansion in which the imbibed fluid is uniformly distributed. This free swelling is described by the theory of Flory and Rehner. We then consider the effect of various boundary displacements and tractions so as to study how this alters the uniform fluid distribution. This problem was previously considered by Wineman and Rajagopal for the case in which the lateral surfaces maintained the radius associated with free swelling. Here we consider certain generalizations in which the lateral surfaces may undergo not only a prescribed relative twist, but also radial displacement. This permits fluid to either enter or exit the cylinder. A numerical method is invoked to solve these boundary value problems using representative material parameters. Certain boundary tractions generate an overall volume increase after the free swelling. If the amount of available fluid is limited, this gives rise to the possibility of complete fluid absorption, whereupon the system is no longer saturated. It is found that the overall mechanical response after loss of saturation is stiffer than it would be if the system had remained saturated.     

On the dynamical thermoelasticity problem for a hollow funtionally graded cylinder

A thermoelasticity problem of a pressurized infinite cylinder made of functionally graded material is solved analytically where material properties vary with radial position. Time dependent thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. For thermal boundary conditions, temperature is prescribed on both surfaces, whereas for mechanical boundary conditions, tractions are prescribed on the boundaries. Obtaining distribution of temperature throughout the cylinder, the dynamical structural problem is solved and closed form relations are extracted for distributions of stress components.     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

Sequentially linearizable initial‐boundary value problems for a neo‐hookean cylinder

Abstract

The sequential linearizabilities of some initial‐boundary value problems for the plane‐strain equations of motion of the neo‐Hookean solid are studied. The initial‐boundary value problems are formulated for a cylinder in plane‐strain condition and subjected to shear and radial deformations or tractions on its inner and outer surfaces. The results obtained extend the previous concept about the sequential linearizability of the governing equations for the neo‐Hookean solid to the sequential linearizability of initial‐boundary value problems for the governing equations.     

Investigation of the problem of a plane axisymmetric cylindrical tube under internal and external pressures

An analysis has been presented in linear elasticity for the investigation of the problem of a plane axisymmetric cylindrical tube under internal and external pressures. Stress concentration coefficients are introduced and the loci where the stress, strain and displacements fields become nul, are determined. Combinations of the lateral loads are studied and the limit values of the thickness of the tube are also examined. Applications are made for a half circle cross-tunnel and for the case of zero tensile stresses at the inner or the outer ring of a cylindrical tube.     

Dynamic elastic response of an infinite discrete multi-layered cylindrical shell subjected to uniformly distributed pressure pulse

A discrete multi-layered cylindrical shell (DMC) consists of a thin inner cylindrical shell and helically cross-winding flat steel ribbons. The dynamic elastic response of such cylindrical shell under uniformly distributed pressure pulse is studied in this paper. Under the axisymmetric plane strain assumption, the solution of the problem is divided into two parts: a quasi-static part satisfying inhomogeneous stress boundary conditions and a dynamic part complying with homogeneous stress boundary conditions. The quasi-static part is determined by homogeneous linearity method and boundary conditions, and the dynamic part was worked out by means of finite Hankel transform and Laplace transform. The dynamic response of a DMC is compared with the response of monobloc cylindrical shell. Parametric analyses with the consideration of major influential factors, such as thickness ratio of the inner shell to that of the complete shell, material parameters and winding-angle, are discussed.     

A transverse shear deformation theory for homogeneous monoclinic plates

Summary A general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any type of shear tractions at its lateral planes, is presented. Developed on the basis of Hamilton's principle, in conjunction with the method of Lagrange multipliers, this new theory accounts for an unlimited number of choices of through-thickness displacement distributions, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. For the particular case of a theory operating with five degrees of freedom, special attention is given to displacement expansions producing symmetric, through thicknes, distributions of transverse shear strain. For the cylindrical bending problem of a specially orthotropic plate, the governing equations of that five-degrees-of-freedom theory are solved and for three different choices of symmetric, through tickness, transverse shear deformation, numerical results are obtained and compared with corresponding results based on the exact three-dimensional solution existing in the literature. The comparisons made show clearly, that the multiple options offered by the new theory, by either suitably altering the displacement expansions or gradually increasing the degrees of freedom involved, will be found useful in future studies dealing with the static and/or dynamic analysis of homogeneous plates.     

Finite azimuthal shear motions of a transversely isotropic compressible elastic and prestressed tube

An exact solution for the case of tube undergoing a dynamic azimuthal shear deformation is given. The tube is prestressed and made of hyperelastic, anisotropic and compressible material. The strain energy density is expressed in a general form of Ogden’s model, decomposed in an isotropic mechanical response augmented with unidirectional reinforcing. The proposed solution is used to study the effects of the prestress on the stress distributions.     

On deforming a sector of a circular cylindrical tube into an intact tube: Existence, uniqueness, and stability

Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of estimating the radial and circumferential residual stress present in an intact tube, which is a problem of particular concern in dealing with the mechanical response of arteries. The initial sector is assumed to be unstressed and the stress distribution resulting from the closure of the sector is then calculated in the absence of loads on the cylindrical surfaces. Conditions on the form of the elastic strain-energy function required for existence and uniqueness of the deformed configuration are then examined. Finally, stability of the resulting finite deformation is analyzed using the theory of incremental deformations superimposed on the finite deformation, implemented in terms of the Stroh formulation. The main results are that convexity of the strain energy as a function of a certain deformation variable ensures existence and uniqueness of the residually-stressed intact tube, and that bifurcation can occur in the closing of thick, widely opened sectors, depending on the values of geometrical and physical parameters. The results are illustrated for particular choices of these parameters, based on data available in the biomechanics literature.