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.      

On Non-Symmetric Deformations of an Incompressible Nonlinearly Elastic Isotropic Sphere

A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearly elastic sphere are investigated. It is found via the semi-inverse method that, to satisfy the governing three-dimensional equations of equilibrium and the incompressibility constraint, only three special cases of the class of deformation fields are possible. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation describing inflation and stretching. The implications of these results for cavitation phenomena are also discussed. In the course of this work, we also present explicitly the spherical polar coordinate form of the equilibrium equations for the nominal stress tensor for a general hyperelastic solid. These are more complicated than their counterparts for Cauchy stresses due to the mixed bases (both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, but more useful in considering general deformation fields. This revised version was published online in August 2006 with corrections to the Cover Date.     

Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

The classical problem of simple shear in nonlinear elasticity has played an important role as a basic pilot problem involving a homogeneous deformation that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects. Here our focus is on certain ambiguities in the formulation of simple shear arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses for the case of an incompressible isotropic hyperelastic material. A new formulation in terms of the principal stretches is given. An alternative approach to the determination of the hydrostatic pressure is proposed here: it will be required that the stress distribution for a perfectly incompressible material be the same as that for a slightly compressible counterpart. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. For the particular case of a neo-Hookean material, the different stress distributions are compared and contrasted.     

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.     

Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells II. Incompressible Shells

This paper treats the eversion of axisymmetric, strictly convex, incompressible nonlinearly elastic shells within a general geometrically exact theory in which the shell can suffer flexure, shear, and both midsurface and transverse extension. The governing equations differ considerably from those for compressible shells. We first formulate the governing equations carefully, showing how to handle the 3-dimensional notion of incompressibility, and paying special attention to the constitutive equations. We prove that when a thickness parameter is sufficiently small, there is an everted state, having a lip near the edge, that can be approximated effectively by an asymptotic series whose error we estimate.     

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.     

Loss of Ellipticity in Plane Deformation of a Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solid

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving 2 deformation parameters and the strength of reinforcing parameter. Orientation normals for the associated surfaces of weak discontinuity at incipient breakdown do not in general conform to the reinforcing direction. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.      

Paradoxical Bending Behavior of Shearable Nonlinearly Elastic Rods

In nonlinear elasticity the exact geometry of deformation is combined with general constitutive relations. This allows a very sophisticated interaction of deformations in different material directions. Based on the Cosserat theory for planar deformations of nonlinearly elastic rods we demonstrate some paradoxical bending effects caused by a nontrivial interaction of extension, flexure, and shear. The analytical results are illustrated by numerical examples.     

Polyconvexity and Existence Theorem for Nonlinearly Elastic Shells

We present an existence theorem for a large class of nonlinearly elastic shells with low regularity in the framework of a two-dimensional theory involving the mean and Gaussian curvatures. We restrict our discussion to hyperelastic materials, that is to elastic materials possessing a stored energy function. Under some specific conditions of polyconvexity, coerciveness and growth of the stored energy function, we prove the existence of global minimizers. In addition, we define a general class of polyconvex stored energy functions which satisfies a coerciveness inequality.     

Contact with a Corner for Nonlinearly Elastic Rods

In elastic contact problems it is usually required that the contact force has to be directed normally to the contact surface in the absence of friction. For an obstacle with nonsmooth surface this gives infinitely many normal directions at an edge or at a corner. For the case where a nonlinearly elastic rod under terminal loads is hanging over a needle, it is shown that the balance equations supplemented with such a normality condition have a continuum of solutions. Moreover, an additional contact condition is derived from a corresponding variational problem by means of special inner variations that preserve the shape of the rod. This way one is finally lead to a unique solution at least locally.     

Euler-Lagrange Equations for Nonlinearly Elastic Rods with Self-Contact

 We derive the Euler-Lagrange equations for nonlinearly elastic rods with self-contact. The excluded-volume constraint is formulated in terms of an upper bound on the global curvature of the centre line. This condition is shown to guarantee the global injectivity of the deformation of the elastic rod. Topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed, e.g., in DNA-molecules, are included by using the notion of isotopy and Gaussian linking number. The bound on the global curvature as a nonsmooth side condition requires the use of Clarke's generalized gradients to obtain the explicit structure of the contact forces, which appear naturally as Lagrange multipliers in the Euler-Lagrange equations. Transversality conditions are discussed and higher regularity for the strains, moments, the centre line and the directors is shown. (Accepted December 20, 2002) Published online April 8, 2003 Communicated by S. S. Antman     

Propagation of the Energy of Nonlinearly Elastic Plane Waves

The propagation of the energy of nonlinearly elastic plane waves in a Murnaghan material is simulated on a computer. The velocity of energy propagation is found in an explicit form. A procedure of determining the critical values of the time and space coordinates for the given material is described. The resultant plots are discussed and analyzed     

Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells

In this paper we study the eversion of axisymmetric, strictly convex, nonlinearly elastic shells within a general, geometrically exact theory in which the shell can suffer flexure, extension, and shear. Each such theory is endowed with a thickness parameter ε². For such shells with free or with fixed hinged edges, we give conditions on ε and the data ensuring that there is an everted state under zero applied load, we show how to approximate it effectively with an asymptotic series in ε whose error we can estimate, we determine the qualitative properties of the everted state, paying particular attention to the formation of a lip near the edge, and we give specific formulas for the shape, the strains, and the stress resultants. This revised version was published online in August 2006 with corrections to the Cover Date.     

Chirality in the Torsion of Cylinders with Trigonal Symmetry

Solutions are obtained for a class of torsion problems for cylinders of a material with trigonal material symmetry. In particular, the solutions for elliptical, circular and equilateral triangular cross-sections are presented. These solutions show that the stress distributions are non-chiral and the same as they would be if the material were isotropic; however the in-plane displacements are chiral and different from the isotropic case. The results show that there will be transverse, in-plane stress interactions between the layers of a composite cylinder composed of concentric cylinders of different trigonal materials in torsional loading. Such composite cylinders are structural designs used by nature and by man. This revised version was published online in July 2006 with corrections to the Cover Date.     

等边布置三圆柱绕流的数值分析

用有限元方法对于等边三角形布置的三个相同直径的二维圆柱绕流问题进行了数值模拟，分别求得了在不同间距比下的流场分布和各圆柱的升、阻力系数以及斯托哈罗数。计算结果表明，小间距比情形下三圆柱之间的干扰是严重的，流动并不对称于中心轴线，而是偏向下游的某个圆柱。数值计算结果与有关文献和实验进行了对比，结果令人满意。     

粘性不可压缩流体定常层流管流及柱体的扭转

本文用加权残数法求得了任意等腰三角形及矩形截面的柱体的扭转问题以及和柱体形状相同的等截面长管的不可压缩粘性流体的定常层流问题的近似解.近似解有较高的精确度.另外,文中还提出了变率配域法的概念.     

Regularity for Shearable Nonlinearly Elastic Rods in Obstacle Problems

Based on the Cosserat theory describing planar deformations of shearable nonlinearly elastic rods we study the regularity of equilibrium states for problems where the deformations are restricted by rigid obstacles. We start with the discussion of general conditions modeling frictionless contact. In particular we motivate a contact condition that, roughly speaking, requires the contact forces to be directed normally, in a generalized sense, both to the obstacle and to the deformed shape of the rod. We show that there is a jump in the strains in the case of a concentrated contact force, i.e., the deformed shape of the rod has a corner. Then we assume some smoothness for the boundary of the obstacle and derive corresponding regularity for the contact forces. Finally we compare the results with the case of unshearable rods and obtain interesting qualitative differences. (Accepted January 21, 1998)