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.     

On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.     

Stress singularities of anisotropic plates with angular cuts

Translated from Prikladnaya Mekhanika, Vol. 32, No. 1, pp. 48–52, January, 1996.     

On the simple tension of a nonhomogeneous anisotropic elastic plate with cracks

Summary A two-dimensional solution expressed in finite terms is given to the problem of the extension of two anisotropic semi-infinite plates, which have different elastic properties and are bonded to each other along a finite number of straight-line segments on their boundaries. The method of solution is based on the reduction to a type of dual nonhomogeneousHilbert problems for two functions.To explain the effect of anisotropy on the stresses near the common edge, numerical computations are carried out for the case where two semi-infinite plates bonded along a single segment are subjected to some special loading conditions. Stress distributions on this common edge and isochromatic lines in its neighborhood are shown in several figures.

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Übersicht Es wird eine zweidimensionale geschlossene Lösung für die Zugbeanspruchung zweier anisotroper Halbscheiben gegeben, die längs eines Stücks ihrer geraden Berandung miteinander verbunden sind. Die Lösung wird auf ein duales inhomogenesHilbertsches Problem für zwei unbekannte Funktionen zurückgeführt. Um den Einfluß der Anisotropie auf den Verlauf der Spannungen in der Nähe der miteinander verbundenen Randteile zu erkennen, werden numerische Berechnungen für verschiedene Belastungsfälle durchgeführt. Der Verlauf dieser Spannungen ist in einigen Diagrammen dargestellt.

     

On the influence of cohesive stress-separation laws on elastic stress singularities

The nature of the stress field occurring at the vertex of an angular elastic plate under in-plane loading is reconsidered. An additional boundary condition is introduced. This boundary condition reflects the action of cohesive stress-separation laws. Companion asymptotic analysis proceeds routinely on employing coupled eigenfunction expansions. Results show that a number of configurations that had previously contained stress singularities become singularity free.     

The degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half-plane

The degeneration of image singularities from an anisotropic material to an isotropic material for a half-plane is discussed in this study. The Green’s functions for anisotropic and isotropic half-planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors. It was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions. However, we believe that this possibility exists as we will demonstrate in this paper. Anisotropic materials include only image singularities of order O(1/r) (i.e., forces and dislocations) existing on image points. There are many image points for anisotropic materials and the locations of these image points depend on the material constants. However, isotropic materials have only one image point with higher order image singularities (O(1/r²), O(1/r³)). From the analysis provided in this study, it is found that the higher order image singularities for an isotropic half-plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material. The solutions of higher order image singularities for isotropic material are dependent. Therefore, these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material.     

On the theory of plane inhomogeneous waves in anisotropic elastic media

In the theory of plane inhomogeneous elastic waves, the complex wave vector constituted by two real vectors in a given plane may be described with the aid of two complex scalar parameters. Either of those parameters may be taken as a free one in the characteristic condition assigned to the wave equation. This alternative underlies the two fundamental approaches in the theory, namely, one associated with the Stroh eigenvalue problem and the other with the generalized Christoffel eigenvalue problem. The two approaches are identical insofar as a partial nondegenerate wave solution (partial mode) is concerned, but they differ in the fundamental solution (wave packet) assembling, and their dissimilarity is also revealed in the presence of degeneracies, which may involve either of the two governing parameters or both of them. Therefore, use of both approaches is essential for studying the degeneracy phenomenon in the theory of inhomogeneous waves. The criteria for different types of degeneracy, related to a double eigenvalue of the Stroh matrix or the Christoffel matrix and at the same time to a repeated root of the characteristic condition, are formulated by appeal to the matrix algebra and to the theory of polynomial equations. On this basis, dimensions of the manifolds, associated with degeneracy of different types in the space of variables, are established for elastic media of unrestricted anisotropy. The relation to the boundary-value problems is discussed.     

On the representation of anisotropic elastic materials by symmetric irreducible tensors

In this paper we will derive a general framework for nonlinear anisotropic elastic materials. The method developed is based on concepts from the theory of symmetric irreducible tensors and group representation theory. Different expansions of constitutive functions with respect to specific basis functions will be introduced. It will be shown that these basis functions satisfy certain orthogonality relations which allow to establish an effective procedure for parameter identification. Finally the treatment of various symmetry groups will be discussed. Received July 12, 1999     

Strong flexure of an elastic curvilinear rod

St. Petersburg Military School of Structural Engineering. Translated from Prikladnaya Mekhanika, Vol. 28, No. 4, pp. 48–56, April, 1992.     

On minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

Piezoelectric study on singularities interacting with interfaces in an anisotropic media

In this work, the singularity problem of a three-phase anisotropic piezoelectric media is studied using the extended Stroh formalism. Based on the method of analytical continuation in conjunction with alternating technique, the general expressions for the complex potentials are derived in each medium of a three-phase anisotropic piezoelectric media. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution for the corresponding homogeneous problem. The presented series solutions have rapid convergence which is guaranteed numerically. Stress and electric fields which are dependent on the mismatch in the material constants, the location of singularities and the magnitude of electromechanical loadings are studied in detail. Numerical results demonstrate that the continuity conditions at the interfaces are indeed satisfied and show the effects of material mismatch on the stress and electric displacement fields. The image forces exerted on a dislocation due to the interfaces are also calculated by means of the generalized Peach–Koehler formula.     

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.