The Remarkable Nature of Cylindrically Anisotropic Elastic Materials Exemplified by an Anti-Plane Deformation

A material is cylindrically anisotropic when its elastic moduli referred to a cylindrical coordinate system are constants. Examples of cylindrically anisotropic materials are tree trunks, carbon fibers [1], certain steel bars, and manufactured composites [2]. Lekhnitskii [3] was the first one to observe that the stress at the axis of a circular rod of cylindrically monoclinic material can be infinite when the rod is subject to a uniform radial pressure (see also [4]). Ting [5] has shown that the stress at the axis of the circular rod can also be infinite under a torsion or a uniform extension. In this paper we first modify the Lekhnitskii formalism for a cylindrical coordinate system. We then consider a wedge of cylindrically monoclinic elastic material under anti-plane deformations. The stress singularity at the wedge apex depends on one material parameter γ. For a given wedge angle α, one can choose a γ so that the stress at the wedge apex is infinite. The wedge angle 2α can be any angle. It need not be larger than π, as is the case when the material is homogeneously isotropic or anisotropic. In the special case of a crack (2α=2π) there can be more than one stress singularity, some of them are stronger than the square root singularity. On the other hand, if γ < there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. The classical paradox of Levy [6] and Carothers [7] for an isotropic elastic wedge also appears for a cylindrically anisotropic elastic wedge. There can be more than one critical wedge angle and, again, the critical wedge angle can be any angle. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Pressurized Curvilinearly Orthotropic Circular Disk, Cylinder and Sphere Made of Radially Nonuniform Material

A unified analysis is presented for the elastic response of a pressurized cylindrically anisotropic hollow disk under assumed conditions of plane stress, or a hollow cylinder under plane strain conditions, and a spherically anisotropic hollow sphere, made of material which is nonuniform in the radial direction according to the power law relationship. The solution for a cylinder under generalized plane strain is also presented. Two parameters play a prominent role in the analysis: the material nonuniformity parameter m, and the parameter ?? which accounts for the combined effects of material anisotropy, represented by the specified parameters (??, ??, ??), and material nonuniformity, represented by the parameter m. The radial and circumferential stresses are the linear combinations of two power functions of the radial coordinate, whose exponents (n ₁ and n ₂) depend on the parameters m and ??. New light is added to the stress amplification and shielding under combined effects of curvilinear anisotropy and radial nonuniformity. Different loading combinations are considered, including the equal pressure at both boundaries, and the uniform pressure at the inner or the outer boundary. While the stress state for the equal pressure loading is uniform in the case of isotropic uniform material (m=0, ??=1), and for one particular radially nonuniform and anisotropic material, it is strongly nonuniform for a general anisotropic or nonuniform material. If the aspect ratio of the inner and outer radii decreases (small hole in a large disk/cylinder or sphere), the magnitude of the circumferential stress at the inner radius increases for n ₁>0 (stress amplification), and decreases for n ₁<0 (stress shielding). Both can be achieved by various combinations of the material parameters m, ??, ??, and ??. While the stress amplification in the case of a pressurized external boundary occurs readily, it occurs only exceptionally in the case of a pressurized internal boundary. The effects of material parameters on the displacement response are also analyzed. The approximate character of the plane stress solution of a pressurized thin disk is discussed and the results are compared with those obtained by numerical solution of the exact three-dimensional disk model.     

On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity. If they are equal at the center no stress singularity develops but if they are not equal then stress always develops a logarithmic singularity. In both cases, the energy density and strains are everywhere finite. As a related problem, we consider annular inclusions for which the eigenstrains vanish in a core around the center. We show that even for an anisotropic distribution of eigenstrains, the stress inside the core is always hydrostatic. We show how these general results are connected to recent claims on similar problems in the limit of small eigenstrains.     

Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid

Complex potentials are derived to describe the anti-plane singular shear stress fields around a kinked crack, the main portion of which is embedded along the interface of two dissimilar anisotropic elastic media. This is accomplished by formulating the problem as singular integral equations with generalized Cauchy kernels. The shear stress singularity at the kink differs from the familiar inverse square root of the local distance; it is found to influence the magnitude of the Mode III crack tip stress intensity factor, K₃. Numerical results of K₃ are obtained and displayed in graphical forms for different degree of material anisotropy and crack dimensions.     

Note on the Radial Deformation and Stresses in Anisotropic Nonhomogeneous Elastic Media

Abstract

In this paper we study the elasticity problem of a cylindrically anisotropic, elastic medium bounded by two axisymmetric cylindrical surfaces subjected to normal piessures (plane strain). The material of the structure is orthotropic with cylindrical anisotropy and, in addition, is continuously inhomogeneous with mechanical properties varying along the radius. General solutions in terms of Whittaker functions are presented. The results obtained by St. Venant for a homogeneous cylindrically anisotropic medium can be deduced from the general solutions. The problem of a solid cylinder of the same medium under the external pressure is also solved as a particular case of the above problem. Problems of the type covered in this paper are encountered in nuclear reactor design.     

Interaction of an anti-plane singularity with interfacial anti-cracks in cylindrically anisotropic composites

Anti-plane problem for a singularity interacting with interfacial anti-cracks (rigid lines) under uniform shear stress at infinity in cylindrically anisotropic composites is investigated by utilizing a complex potential technique in this paper. After obtaining the general solution for this problem, the closed solution for the interface containing one anti-crack is presented analytically. In addition, the complex potentials for a screw dislocation dipole inside matrix are obtained by the superimposing method. Expressions of stress singularities around the anti-crack tips, image forces and torques acting on the dislocation or the center of dipole are given explicitly. The results indicate that the anisotropy properties of materials may weaken the stress singularity near the anti-crack tip for the singularity being a concentrated force but enhance the one for the singularity being a screw dislocation and change the equilibrium position of screw dislocation. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some new or previously known results.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lamé problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.     

Potential Flow Past a Slightly Deformed Porous Circular Cylinder

A uniform potential flow past a porous circular cylinder with a core of different permeability is discussed. The porous circular cylinder is slightly deformed whose radius is r=r₁(1+ecosm q){r=r_1(1+\epsilon \cos m \theta)} , where | e | << 1{\mid\epsilon\mid\ll 1} and m is a positive integer. Here r, θ are the polar coordinates and r ₁ is the characteristic radius of the cylinder. The drag force exerted by the exterior flow on the surface of the cylinder is calculated and it depends on the thickness of the porous material and on the permeabilities of the two porous regions. As special cases, porous cylinder with hollow core, rigid core, and deformed cylinder is discussed.     

Stress singularities at crack corners

In this paper the stress and displacement fields near an embedded crack corner in a linear elastic medium are analytically computed. The conical-spherical coordinate system is introduced to solve this problem. It is observed that the strength of the stress singularity depends on the angle of the crack corner. The singularity becomes weaker, varying from r ^-1 to r ⁰, as the angle of the crack corner varies from 360° to 0°. Both symmetric and skew-symmetric loadings give the same variation of the behavior of the stress singularity. It is also found that the order of the singularity is independent of the Poisson's ratio, unlike the corner cracks at a free surface where Poisson's ratio affects the results.     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

On singularities associated with the curvilinear anisotropic elastic symmetries

The objective of this paper is to describe a different approach to modeling the material symmetry associated with singularities that can occur in curvilinear anisotropic elastic symmetries. In this analysis, the intrinsic non-linearity of a cylindrically anisotropic problem is demonstrated. We prove that a simple homogenization process applied to a representative volume element containing the cylindrical anisotropic singularity removes the singularity. This geometric and interpretive approach is an aid to better modeling of material symmetry associated with these singularities.     

Bending and flexure of cylindrically monoclinic elastic cylinders

We consider a circular cylinder of linearly elastic material with cylindrically monoclinic material symmetry. This represents a model for a helically wound composite cable or wire rope. The elastic moduli are allowed to be arbitrary functions of the radius r. The cylinder undergoes deformation in which the axis of the cylinder is bent into a plane quartic curve. For the resulting stress field, we obtain exact integrals of the equilibrium equations, and derive simplified expressions for the shear stress resultants and bending moments.     

A moving dislocation in a magneto–electro-elastic solid

This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, E_rand H_r, of the electric and magnetic fields as well as the hoop components, D_θ and B_θ, of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r⁻¹ near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail.     

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.     

On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder

Here the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. At time t=0 the fluid is at rest and the motion is produced by the rotation of the cylinder around its axis. The solutions that have been obtained are presented under series form in terms of the generalized G-functions. The similar solutions for ordinary second grade and Newtonian fluids are obtained from general solution for β→1, respectively, β→1 and α ₁→0. Finally, the influences of the pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.     

On the r-value of textured sheet metals

The r-value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f(σ, w), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.     

The Saint-Venant torsion of a circular bar consisting of a composite cylinder assemblage with cylindrically orthotropic constituents

We consider the Saint-Venant torsion of a cylindrical rod of a circular cross section which is filled up by an assemblage of composite circular cylinders. The constituent cylinders consist of a core and a coating both of which are cylindrically orthotropic with the volume fraction of the core being the same in every composite cylinder. The described microstructure is the composite cylinder assemblage of Hashin and Rosen [J. Appl. Mech. 29 (1964) 143] which is now subjected to torsion. The main results are (a) the warping function on the lateral surface of the host rod is zero, (b) an exact expression for the torsional rigidity of the host rod is derived which depends on the size distribution of the composite cylinders but not on their position and (c) there are two circumstances in which the torsional rigidity becomes size distribution independent: The first one is that in which the sizes of the composite cylinders are much smaller than the size of the host rod; the second one is that in which a certain specific relation holds between the properties of the composite cylinder and the volume fraction of the core. If the coating disappears and the core is cylindrically orthotropic, we get the configuration of a polycrystalline rod. Simple bounds for the torsional rigidity of the constructed composite rod are obtained.     

Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties

In this paper, we present the homogenization of an anisotropic hollow layered tube with discontinuous elastic coefficients. We focus on some aspects of technological importance, such as the effective coefficients of anisotropic materials, the behavior of the homogenized displacements and stresses, the discontinuities of in-plane shear, hoop and longitudinal stresses, the homogenization-induced anisotropy in the isotropic case. We conclude that the problem of cylindrically anisotropic tubes under extension, torsion, shearing and pressuring is stable by homogenization and we define the effective tensor of the material elastic coefficients. Some numerical examples confirm the theoretical results.     

Effect of orthotropy on crack propagation

The tendency of moving cracks to spread along the preferred directions of material anisotropy is treated. Depending on the velocity of crack propagation, the change of material properties in orthogonal planes is shown to affect the bifurcation characteristics. The problem is reduced to a system of dual integral equations that can be solved in a standard fashion. Of particular interest is the dynamic stress field near the tip of a moving crack in an orthotropic material. Although the 1√r stress singularity is preserved with r being the radial distance measured from the crack tip, the angular variations of the stresses are dependent on crack speed and material anisotropy. The possibility of crack bifurcation is examined by application of the strain energy density criterion for several composite systems. Crack branching is found to be enhanced by material anisotropy, a phenomenon that is not uncommon in composite materials.