On the Modeling of Extension-Torsion Experimental Data for Transversely Isotropic Biological Soft Tissues

For the problem of torsion superimposed on extension of incompressible nonlinearly elastic transversely isotropic circular cylinders, a simple asymptotic analysis is carried out on using a small parameter that reflects the moderate twisting of slender cylinders, which corresponds to a typical testing regime for biological soft tissue. The analysis is carried out for a subclass of strain-energy densities that reflect transversely isotropic material response. On using a four-parameter polynomial expression for the strain-energy density in terms of certain classical invariants, this analysis is shown to be in excellent agreement with experimental data obtained by other authors for rabbit papillary muscles. An explicit condition on the strain-energy density is obtained that determines whether the stretched cylinder tends to elongate or shorten on twisting. For the special case of pure torsion where no extension is allowed, this condition determines whether the classical or reverse Poynting effect occurs. For the rabbit papillary muscles, the theoretical results predict and the experimental results confirm that a reverse Poynting-type effect occurs where the stretched rabbit muscle tends to shorten on twisting.     

On Extension and Torsion of Strain-Stiffening Rubber-Like Elastic Circular Cylinders

This paper is concerned with investigation of the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic nonlinearly elastic materials. Our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two particular phenomenological constitutive models for such materials that reflect limiting chain extensibility at the molecular level. The axial stretch γ and twist that can be sustained in cylinders composed of such materials are shown to be constrained in a coupled fashion. It is shown that, in the absence of an additional axial force, a transition value γ=γ _t of the axial stretch exists such that for γ<γ _t , the stretched cylinder tends to elongate on twisting whereas for γ>γ _t , the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. We also obtain results for materials modeled by the well-known exponential strain-energy widely used in biomechanics applications. This model reflects a strain-stiffening that is less abrupt than that for the limiting chain extensibility models. Surprisingly, it turns out that the results in this case are somewhat more complicated. For a fixed stiffening parameter, provided that the stretch is sufficiently small, the stretched bar always tends to elongate on twisting in the absence of an additional axial force. However, for sufficiently large stretch, the cylinder tends to shorten on undergoing sufficiently small twist but then tends to elongate on further twisting. These results are of interest in view of the widespread use of exponential models in the context of the mechanics of soft biological tissues. The special case of pure torsion is also briefly considered. In this case, the resultant axial force required to maintain pure torsion is compressive for all the models discussed here. In the absence of such a force, the bar would elongate on twisting reflecting the celebrated Poynting effect.      

Mullins Effect for a Cylinder Subjected to Combined Extension and Torsion

We study the Mullins effect for a circular cylinder of incompressible, isotropic material under loading cycles of combined extension and torsion. The analysis is based on the constitutive model recently proposed in De Tommasi et al. (J. Rheol. 50: 495–512, 2006). This model assumes that the mechanical response at each material point results as a homogenized effect of a mixture of different materials with variable activation and breaking thresholds. We show the feasibility of this approach to treat complex, inhomogeneous deformations. In particular, we obtain for the generic loading path the analytical expressions of the stress field, of the axial force, and of the twisting moment. The proposed model exhibits the Mullins stress softening effect in the case of simple extension, simple torsion, and combined extension and torsion. We analyze in detail the path dependent behavior and the preconditioning effects.      

Influence of thermally induced chemorheological changes on the torsion of elastomeric circular cylinders

When an elastomeric material is deformed and subjected to temperatures above some characteristic value T _cr (near 100^∘C for natural rubber), its macromolecular structure undergoes time and temperature-dependent chemical changes. The process continues until the temperature decreases below T _cr. Compared to the virgin material, the new material system has modified properties (reduced stiffness) and permanent set on removal of the applied load.A new constitutive theory is used to study the influence of the changes of macromolecular structure on the torsion of an initially homogenous elastomeric cylinder. The cylinder is held at its initial length and given a fixed twist while at a temperature below T _cr. The twist is then held fixed and the temperature of the outer radial surface is increased above T _cr for a period of time and then returned to its original value. Assuming radial heat conduction, each material element undergoes a different chemical change. After enough time has elapsed such that the temperature field is again uniform and at its initial value, the cylinder properties are now inhomogeneous. Expressions for the time variation of the twisting moment and axial force are determined, and related to assumptions about material properties. Assuming the elastomeric networks to act as Mooney-Rivlin materials, expressions are developed for the permanent twist on release of torque, residual stress, and the new torsional stiffness in terms of the kinetics of the chemical changes.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders

In the context of the theory of non-linear elasticity for rubber-like materials, the problem of finite extension and torsion of a circular bar or tube has been widely investigated. More recently, this problem has attracted considerable attention in studies on the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in non-linear elasticity was concerned specifically with the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic non-linearly elastic materials that undergo severe strain-stiffening in the stress–stretch response. For two specific material models that reflect limiting chain extensibility at the molecular level, it was shown that, in the absence of an additional axial force, a transition value γ=γ_t of the axial stretch exists such that for γ<γ_t, the stretched cylinder tends to elongate on twisting whereas for γ>γ_t, the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models for rubber such as the Mooney–Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. Here we investigate similar issues for fiber-reinforced transversely isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect limited fiber extensibility and in the biomechanics context model the stretch induced strain-stiffening of collagen fibers on loading. They have been shown to model the mechanical behavior of fiber-reinforced rubber and many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The results obtained here have important implications for extension–torsion tests for fiber-reinforced materials, for example in the development of accurate extension–torsion test protocols for determination of material properties of soft tissues.     

Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid

One of the most widely used constitutive models for compressible isotropic nonlinearly elastic solids is the generalized Blatz-Ko material for foam-rubber and its various specializations. For this model, a unified derivation of necessary and sufficient conditions for ellipticity of the governing three-dimensional displacement equations of equilibrium is provided. When the parameterf occurring in the generalized Blatz-Ko model is in the range 0f<1, it is shown that ellipticity is always lost at sufficiently large stretches, while forf=1, the equilibrium equations are globally elliptic. The implications of these results for a variety of physical problems are discussed.     

The torsion of a composite,nonlinear-elastic cylinder with an inclusion having initial large strains

This article considers a static problem of torsion of a cylinder composed of incompressible, nonlinear-elastic materials at large deformations. The cylinder contains a central, round, cylindrical inclusion that was initially twisted and stretched (or compressed) along the axis and fastened to a strainless, external, hollow cylinder. The problem statement and solution are based on the theory of superimposed large strains. An accurate analytical solution of this problem based on the universal solution for the incompressible material is obtained for arbitrary nonlinear-elastic isotropic incompressible materials. The detailed investigation of the obtained solution is performed for the case in which the cylinders are composed of Mooney-type materials. The Poynting effect is considered, and it is revealed that composite cylinder torsion can involve both its stretching along the axis and compression in this direction without axial force, depending on the initial deformation.     

Optimal Design of Thin-Walled Cylinders of Variable Wall-Thickness Subject to Flexural and Torsional Stiffness Constraints

ABSTRACT

This paper considers the problem of determining center-line shape and wall-thickness distribution of a thin-walled cylinder of given center-line length that uses the minimum possible amount of material to achieve prescribed minimum stiffnesses in torsion and bending in a given plane. Necessary optimality conditions are derived and the solution is found partly in closed form and partly by numerical methods. Optimal solutions are presented for various stiffness ratios and compared with other admissible designs.     

Stress Singularity in an Elastic Cylinder of Cylindrically Anisotropic Materials

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

Drag Reduction of a Circular Cylinder Using an Upstream Rod

Experimental studies on the drag reduction of the circular cylinder were conducted by pressure measurement at a Reynolds number of 82 000 (based on the cylinder diameter). A rod was placed upstream of and parallel to the cylinder to control the flow around the cylinder. The upstream rod can reduce the resultant force of the cylinder at various spacing between the rod and the cylinder for α < 5^∘(α defined as the staggered angle of the rod and the cylinder). For α > 10^∘, the resultant force coefficient has a large value, so the upstream rod cannot reduce the force on the cylinder any more. For α = 0^∘ and d/D = 0.5 (where d and D are the diameter of the rod and the cylinder, respectively), the maximum drag of the cylinder reduces to 2.34% that of the single cylinder. The mechanism of the drag reduction of the cylinder with an upstream rod in tandem was presented by estimating the local contributions to the drag reduction of the pressure variation. In the staggered arrangement, the flow structures have five flow patterns (they are the cavity mode, the wake splitting mode, the wake merge mode, the weak boundary layer interaction mode and the negligible interaction mode) according to the pressure distribution and the hydrogen bubble flow visualization. The half plane upwind of the cylinder can be divided to four regions, from which one can easily estimates the force acting on the circular cylinder with an upstream rod in staggered arrangement.     

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium

This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value p _max. At a value of pressure p _e (<p _max), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than p _e. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.This work was supported by the U.S. Army Research Office under Grant DAAG29-83-K-0145 (R.A. & C.O.H.) and by the U.S. National Science Foundation under Grant MEA 78-26071 (C.O.H.).     

FInite torsion of slightly compressible rubberlike circular cylinders∗

A regular perturbation procedure and the Rayleigh-Ritz method are used to study the finite torsion of cylinders made of slightly compressible neo-Hookean materials. Two cases are considered: 1. the length of the cylinder is not permitted to change during torsion and 2. the net axial force on the cylinder vanishes. The perturbation procedure fails for certain constitutive relations whereas, in principle, the Rayleigh-Ritz method has general applicability. When it works, the success of the perturbation procedure depends on prior knowledge of the problem for an incompressible material (the zeroth order nonlinear problem). The solution of problem 2. is considerably more complicated than the solution of problem 1. since the complete approximation of order n for problem 2. requires extensive work on the approximation of order (n + 1).     

Decay rates in a bimaterial circular cylinder

Decay rates in a bimaterial circular cylinder under axisymmetric torsion loading are considered via an eigen-expansion near the end of the cylinder. The decay rates depend on the shear modulus ratio of the materials and the radius ratio of inner and outer cylinders. Following the derivation of the traditional Saint-Venant end effect of an isotropic bimaterial cylinder, cases of anisotropic material (transversely isotropic material) and non-traditional Saint-Venant end effect (displacement prescribed on the side surface) are considered. This study sheds some light on the decay studies for other geometric configurations and the deformation modes of composite structures.     

Effect of blockage on critical parameters for flow past a circular cylinder

The effect of location of the lateral boundaries, of the computational domain, on the critical parameters for the instability of the flow past a circular cylinder is investigated. Linear stability analysis of the governing equations for incompressible flows is carried out via a stabilized finite element method to predict the primary instability of the wake. The generalized eigenvalue problem resulting from the finite element discretization of the equations is solved using a subspace iteration method to get the most unstable eigenmode. Computations are carried out for a large range of blockage, 0.005?D/H ?0.125, where D is the diameter of the cylinder and H is the lateral width of the domain. A non‐monotonic variation of the critical Re with the blockage is observed. It is found that as the blockage increases, the critical Re for the onset of the instability first decreases and then increases. However, a monotonic increase in the non‐dimensional shedding frequency at the onset of instability, with increase in blockage, is observed. The increased blockage damps out the low‐frequency modes giving way to higher frequency modes. The blockage is found to play an important role in the scatter in the data for the non‐dimensional vortex shedding frequency at the onset of the instability, from various researchers in the past. Copyright © 2005 John Wiley & Sons, Ltd.     

Second-order strain accumulation in workhardening media

Summary Second-order or cross effects are the result of quadratic tensor terms in the constitutive equations of isotropic elastic, viscous and visco-elastic media, which are required by the condition of tensor invariance of those relations. These effects are most pronounced when they are clearly separable from the first-order deformation, as in the case of second-order elongation and volume change of an elastic cylinder subject to a twisting moment (Poynting effect, dilatancy) or of second-order normal stress in the case of shear flow of polymeric liquids (Weissenberg effect).An accumulating second-order effect (Ronay effect) has been discovered in experiments on strain-hardening metal specimens in reversed torsion. While thePoynting effect vanishes at zero strain in elastic solids and theWeissenberg effect at zero velocity in polymeric fluids, the second-order strain increments accumulate in strainhardening media with the number of repeated torsion cycles. Hence their observation is simple and does not require the elaborate procedures necessary for the observation od second-order effects in elastic solids, viscous fluids and visco-elastic substances.It can be shown that the observed second-order strain accumulation (Ronay effect) is implied by thePrager-Hill stress strain-increment relation for strain-hardening media, combined with theKadshevich-Novozhilov formulation of kinematic hardening, provided that the arbitrary condition that strain-increment and stress change sign simultaneously is not imposed.Paper read at the Annual Meeting of the German Rheologists, Berlin-Dahlem June 7–10, 1966.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

The shape of the strongest column

The problem of determining that shape of column which has the largest critical buckling load is solved, assuming that the length and volume are given and that each cross section is convex. The strongest column has an equilateral triangle as cross section, and it is tapered along its length, being thickest in the middle and thinnest at its ends. Its buckling load is 61.2% larger than that of a circular cylinder. For columns all of whose cross sections are similar and of prescribed shape-not necessarily convex—the best tapering is found to increase the buckling load by one third over that of a uniform column. This result, which was independently obtained by H. F. Weinberger, is originally due to Clausen (1851). For a uniform column, triangularizing is shown to increase the buckling load by 20.9% over that of a circular cylinder. The results lead to isoperimetric inequalities for the buckling loads of arbitrary columns. The research reported in this paper has been sponsored by the Office of Naval Research under Contract No. (285) 46.     

Stresses distributions in a rotating functionally graded piezoelectric hollow cylinder

An analytic solution to the axisymmetric problem of a long, radially polarized, hollow cylinder composed of functionally graded piezoelectric material (FGPM) rotating about its axis at a constant angular velocity is presented. For the case that electric, thermal and mechanical properties of the material obey different power laws in the thickness direction, distributions for radial displacement, stresses and electric potential in the FGPM hollow cylinder are determined by using the theory of electrothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material nonhomogeneity, and adopting suitable graded indexes and applying suitable geometric size and rotating velocity ω may optimize the rotating FGPM hollow cylindrical structures. This will be of particular importance in modern engineering application.     

Limit-analysis of a circular cylinder obeying the Green plasticity criterion and loaded in combined tension and torsion

This short paper presents the limit-analysis of a cylinder with circular basis, made of an ideal-plastic material obeying Green’s yield criterion and subjected to combined tension and torsion. The exact solution of the problem is provided in the form of a statically and plastically admissible stress field and a kinematically admissible velocity field, associated via the normality rule. The overall yield locus, that is the set of pairs [tension force, torsion torque] for which unrestrained plastic flow occurs, is expressed first in parametric form, then explicitly upon elimination of the parameter involved. The explicit expression of this yield locus also entails that of the overall flow rule via the overall normality property. The impact of these results is two-fold. First, they provide a fresh example of a solution to a limit-analysis problem exceptionally combining three generally mutually exclusive features: be non-trivial, exact and explicit. Second, they provide a way of using simple experiments of combined tension and torsion of cylinders to determine the parameter characterizing the influence of the mean stress in Green’s criterion.