Finite elastic deformations of transversely isotropic circular cylindrical tubes

We consider the finite radially symmetric deformation of a circular cylindrical tube of a homogeneous transversely isotropic elastic material subject to axial stretch, radial deformation and torsion, supported by axial load, internal pressure and end moment. Two different directions of transverse isotropy are considered: the radial direction and an arbitrary direction in planes normal locally to the radial direction, the only directions for which the considered deformation is admissible in general. In the absence of body forces, formulas are obtained for the internal pressure, and the resultant axial load and torsional moment on the ends of the tube in respect of a general strain-energy function. For a specific material model of transversely isotropic elasticity, and material and geometrical parameters, numerical results are used to illustrate the dependence of the pressure, (reduced) axial load and moment on the radial stretch and a measure of the torsional deformation for a fixed value of the axial stretch.     

Finite deformations of fibre-reinforced vascular prosthesis

We present an mechanical study of a vascular prosthesis subjected to finite radial expansion, torsion and circular shearing. The material strain-energy function is expressed in the form of the fibre-reinforced neo-Hookean. The governing equations are solved analytically and the results present effects of the combined deformation on the stress distributions.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Mechanical performance and negative pressure instability for venous walls

Mechanical properties, such as the deformation and stress distributions for venous walls under the combined load of transmural pressure and axial stretch, are examined within the framework of nonlinear elasticity with one kind of hyper-elastic strain energy functions. The negative pressure instability problem of the venous wall is explained through energy comparison. First, the deformation equation of the venous wall under the combined loads is obtained with a thin-walled circular cylindrical tube. The deformation curves and the stress distributions for the venous wall are given under the normal transmural pressure, and the regulations are discussed. Then, the deformation curves of the venous wall under the negative transmural pressure or the internal pressure less than the external pressure are given. Finally, the negative pressure instability problem is discussed through energy comparison.     

Extension,torsion and expansion of an incompressible,hemitropic Cosserat circular cylinder

Constitutive equations for the stress and couple stres on an incompressible, hemitropic, constrained Cosserat material are derived, and the theory is applied to study the problem of finite extension, torsion and expansion of a circular cylinder. As in the theory of isotropic simple elastic materials, it is found that the deformation is controllable by application of only a normal force and a tosional moment at the cylinder ends. It is shown that in general the well known universal relation between the torsional stiffness and the axial force for incompressible, isotropic simple materials in the limit as the twist goes to zero does not exist for incompressible, hemitropic Cosserat materials. However, for a special and unusual class of hemitropic materials, the same universal formula is found to hold for a certain reduced torsional stiffness. The main problem is solved completely for incompressible, hemitropic, linearly elastic, Cosserat materials; and certain additional special features of the Kelvin-Poynting type, which here appear to the first order in the amount of twist of the cylinder, are derived and discussed in relation to experimentally observed composite material behavior.     

On the problem of axisymmetrically loaded shells of revolution with small elastic strains and arbitrarily large axial deflections

For the problem of axisymmetrically loaded shells of revolution with small elastic strains and arbitrarily large axial deflections, this paper suggests a group of state variable: radial displacement u, axial displacement w, angular, deflection of tangent in the meridian X, radial stress resultant H and meridional bending moment M_s, and derives a System of First-order Nonlinear Differential Equations under global coordinate system with these variables. The Principle of Minimum Potential Energy for the problem is obtained by means of weighted residual method, and its Generalized Variational Principle by means of identified Lagrange multiplier method.This paper also presents a Method of Variable-characteristic Nondimensionization with a scale of load parameter, which may efficientlky raise the probability of success for nonlinearity calculation. The obtained Nondimensional System of Differential Equations and Nondimensional Principle of Minimum Potential Energy could be taken as the theoretical basis for the numerical computation of axisymmetrical shells with arbitrarily large deflections.     

核电690合金传热管微动疲劳试验及损伤机理分析

核电蒸汽发生器传热管在微幅磨损与交变载荷的作用下形成微动疲劳,导致其表面裂纹萌生和扩展乃至破裂,从而影响反应堆的安全. 为研究径向载荷以及轴向交变应力对690合金管微动疲劳寿命的影响规律,开展690合金管管材的微动疲劳试验,获得690合金管管材的微动疲劳寿命曲线,并与相关研究数据进行对比分析,以便探讨材料在微动疲劳下的寿命模型. 对不同载荷下的690合金管试样的磨痕表面进行三维形貌和扫描电镜观测,分析磨损表面的损伤机理;对不同载荷下的690合金管试样断口的宏观与微观形貌进行表征,分析裂纹萌生、起裂过程及其失效机理. 结果表明690合金管与403不锈钢(SS)抗振条间的磨损机理为剥层及磨粒磨损;690合金管在径向载荷作用下于微动磨损处产生裂纹源,裂纹在轴向交变应力的作用下不断向内部扩展,最终导致断裂;其断裂形式为解理疲劳断裂.      

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.     

Torsional Swelling of a Hyperelastic Tube with Helically Wound Reinforcement

This paper examines the combination of radial deformation with torsion for a circular cylindrical tube composed of a transversely isotropic hyperelastic material subject to finite deformation swelling. The stored energy function involves separate matrix and fiber contributions such that the fiber contribution is minimized when the fiber direction is at a natural length. This natural length is not affected by the swelling. Hence swelling preferentially expands directions that are orthogonal to the fiber. The swelling itself is described via a swelling field that prescribes the local free volume at each location in the body. Such a treatment is a relatively simple generalization of the conventional incompressible theory. The direction of transverse isotropy associated with the fiber reinforcement is described by a helical winding about the tube axis. The swelling induced preferential expansion orthogonal to this direction induces the torsional aspect of the deformation. For a specific class of strain energy functions we find that the twist increases with swelling and approaches a limiting asymptotic value as the swelling becomes large. The fibers reorient such that fibers at the inner portion of the tube assume a more circumferential orientation whereas, at least for small and moderate swelling, the fibers in the outer portion of the tube assume a more axial orientation. For large swelling the fibers in the outer portion of the tube reorient beyond the axial orientation, and so are described by helices with orientation in the opposite sense to that in the reference configuration.      

Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model. This revised version was published online in August 2006 with corrections to the Cover Date.     

Strain gradient elasticity and stress fibers

Since stress fibers have micro-size dimensions, their biomechanical behavior should demand mechanical models conforming with gradient strain deformation theories. In particular, the torsion and the stretching of stress fibers are discussed into the context of strain gradient elasticity theory and their size effects. It is proven for the torsion problem that the torsion moment varies with the axial length of the bar for constant twist angle, whereas for the simple tension problem, the strain is non-uniform along the stress fiber. The proposed theory is supported by experimental evidence.     

Buckling of embedded multi-walled carbon nanotubes under combined torsion and axial loading

This paper describes an investigation into elastic buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading, which takes account of the radial constraint from the surrounding elastic medium and van der Waals force between two adjacent tube walls. Depending on the ratio of radius to thickness, the multi-walled carbon nanotubes discussed here are classified as thin, thick, and nearly solid. Critical buckling load with the corresponding mode is obtained for multi-walled carbon nanotubes under combined torsion and axial loading, with various values of the radius to thickness ratio and surrounded with different elastic media. The study indicates that the buckling mode (m, n) of an embedded multi-walled carbon nanotube under combined torsion and axial loading is unique and it is different from that with axial compression only. New features for the buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading and the meaningful numerical results are useful in the design of nanodrive device, nanotorsional oscillator and rotational actuators, where multi-walled carbon nanotubes act as basic elements.     

Lüders bands propagation of 1045 steel under multiaxial stress state

Inhomogeneous plastic deformation of 1045 steel under monotonic loading was experimentally studied. Thin-walled tubular specimens were used in the experiments and custom-made small strain gages were bonded on the specimen surface to characterize the local deformation. Experiments were conducted under tension, torsion, and combined tension–torsion. During the propagation of Lüders bands, the local deformation experienced two-stage deformation: an abrupt plastic deformation stage followed by a slower deformation process. In some area of the gage section of the specimen, a small amount of initial plastic deformation occurred before the Lüders front reached. During the propagation of Lüders bands, multiple Lüders fronts can be formed. Under tension, torsion, and combined tension–torsion with a constant axial load, the Lüders front was approximately parallel to the material plane of maximum shear stress. When the combined axial-torsion followed a proportional fashion, the stress–extensometer strain responses were dependent on the axial/torsional loading ratio, and the Lüders fronts were oriented differently and propagated along the specimen axis at a different velocity. The local strain was inhomogeneous even at the work-hardening stage. The relationships between the equivalent stress and the equivalent plastic strain were found to be practically identical for all the loading cases studied.     

Analytical model for a rigid, linear-hardening plastic tube subjected to impulsive torsional moment

An analytical model of a rigid, linear-hardening plastic tube subjected to impulsive torsion at its free end is suggested. It is shown that an impulsive torsional moment at the free end of the tube gives rise to a torsional disturbance of strong discontinuity that propagate at a constant speed and the plastic deformation is concentrated at the front of the disturbances. Two examples are given to illustrate the characterization of the structure under the intense dynamic pure torsional loading condition.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.