T.C.T. Ting

Explicit solution and invariance of the singularities at an interface crack in anisotropic composites

Assuming that stress distribution near the tip of an interface crack in an anisotropic composite is proportional to rδ, application of the interface and boundary conditions yields ||K(δ)|| = 0, where K is a 12 × 12 complex matrix. The surfaces of the crack can be free-free, fixed-fixed or free-fixed. For the cases of free-free and fixed-fixed cracks, explicit solutions for all δs are obtained. For the case of free-fixed crack, the determinant of K is reduced to a 3 × 3 determinant which yields a sextic equation. Explicit solutions are obtained only for isotropic composites. The special cases of a homogeneous anisotropic material with a semi-infinite crack and the half-plane problems are also considered. Explicit solutions for δs are obtained for all three boundary conditions. Finally, it is shown that δ is invariant with respect to the orientation of the plane boundary (in the case of half-plane problems), the semi-infinite crack (in the case of a crack in a homogeneous material) and the crack and interface (in the case of a composite with an interface crack) relative to the materials. This is a somewhat surprising result not expected of anisotropic materials.

Piezoelectric solid with an elliptic inclusion or hole

The two-dimensional problem of an elliptic hole in a solid of anisotropic piezoelectric material is studied. The Stroh formalism is adopted here. Real form solutions are obtained along the hole boundary in the case of an arbitrarily prescribed vector field on the hole surface. For an elliptic rigid inclusion of electric conductor subjected to a line force, a torque, and a line charge, a real form solution at the interface is obtained. Finally, general solutions for an elliptic piezoelectric inclusion with uniform loading at infinity are investigated.

Edge singularities in anisotropic composites

Abstract The stress singularity at the vertex of an anistropic wedge has the form r − ϵ F ( r , θ ) as r → 0 where 0 F is a real function of the polar coordinates ( r , θ). In many cases, F is independent of r . The explicit form of F ( r , θ ) depends on the eigenvalues of the elasticity constants, called p here and on the order of singularity k . When k is real, ξ = k If k is complex, ξ is the real part of k . The p s are all complex and consist of 3 pairs of complex conjugates which reduce to ± i when the material is isotropic. The function F depends not only on p and k , it also depends on whether p and k are distinct roots of the corresponding determinants. In this paper we present the function F ( r , θ ) in terms of p and k for the cases when p and k are single roots as well as when they are multiple roots. The relationship between the complex variable Z introduced in the analysis and the polar coordinates ( r , θ ) is interpreted geometrically. After presenting the form of F for individual cases, a general form of F is given in eqn (74). We also show that the stress singularity at the crack tip of general anisotropic materials has the order of singularity ξ=1/2 which is at least a multiple root of order 3. The implication of this on the form F ( r , θ ) is discussed.

Effects of change of reference coordinates on the stress analyses of anisotropic elastic materials

Abstract When the displacements are independent of the x 3 -coordinate, the eigenvalues and eigenvectors of the anisotropic elasticity constants depend on the orientation of the ( x 1 , x 2 ) axes. It is shown that the components of the eigenvectors (which are complex values) transform according to the law of transformation for tensors of order one. The transformation of the eigenvalues is more complicated. The effects of change of the reference coordinates on the form of general solution are discussed. Also discussed is the form of general solution when the eigenvalue p is a multiple root. Finally, we show that as the angle of rotation φ of the coordinate axes varies from 0 to 2π, each p traverses a circle in the complex plane which is orthogonal to the unit circle with center at the origin. A graphical solution of the eigenvalue p for a given φ is presented. Some functions of p which are invariant to the rotation of the coordinate axes are obtained.

Singularities at the tip of a crack normal to the interface of an anisotropic layered composite

Abstract The order of stress singularities at the tip of a crack which is normal to and ends at an interface between two anisotropic elastic layers in a composite is studied. Assuming that the stress singularities have the form r − κ , equations are derived for determining the order of singularities κ . If the materials on both sides of the interface are identical, κ = ½ is a root of multiplicity three each of which can be identified with the singularity due to, respectively, a symmetric tensile stress applied at infinity, an antisymmetric plane shear stress and an antiplane shear stress applied at infinity. When the materials on both sides of the interface are not the same, there are in general three distinct roots for k . Numerical examples for a typical high modulus graphite/epoxy and for a special T300/5208 graphite/epoxy show that k has three positive roots all of which are close to ½ for most combinations of ply-angles in the two materials.

Recent developments in anisotropic elasticity

Anisotropic elasticity has been an active research subject for the last thirty years due to its applications to composite materials. There are essentially two formalisms for two-dimensional deformations of a general anisotropic elastic material. The Lekhnitskii formalism [Lekhnitskii, S.G., 1950. Theory of Elasticity of an Anisotropic Elastic Body. Gostekhizdat, Moscow (in Russian)] has been the favorite among the engineering community, while the newer Stroh formalism is well-known in the material sciences, applied mathematics and physics community. The Stroh formalism (Stroh, A.N., 1958. Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3, 625–646.) is mathematically elegant and technically powerful. It began to be noticed by the engineering community in recent years, specially among the younger researchers. A comprehensive treatment of both formalisms and applications of the theory have been presented in a book by Ting. Since the appearance of the book in 1996, there have been several new developments in the theory and applications of anisotropic elasticity. We present here new results that have appeared since 1996. Only linear anisotropic elasticity is considered here; for nonlinear elasticity, the reader is referred to the book by Antman (Antman, S.S., 1995. Nonlinear Problems in Elasticity. Springer–Verlag, New York).

Interface cracks in anisotropic bimaterials

Abstract It is known that the displacement at the interface crack surface in a bimaterial under a two-dimensional deformation may be oscillatory. The existence of oscillation depends on the material properties and the orientations of the two materials on both sides of the crack-interface. We show in this paper that the existence of oscillation depends on the material properties only and is otherwise independent of the individual orientation of the two materials. This means that. if the crack surface displacement is oscillatory for one choice of orientations of the two materials, no other orientations obtained by rotating about the x3-axis can alleviate the oscillation. Conversely, if the crack surface displacement is not oscillatory for one choice of orientations of the two materials, no other orientations can generate oscillation. An exception is the Type B bimaterials defined in the paper for which the crack surface displacement is always oscillatory except for a particular Type B bimaterial at a particular choice of relative orientation. For bimaterials which consist of two different orthotropic materials, it is shown that whether the crack surface displacement is oscillatory or not depends on whether C 11 C 22 +C 12 of the two materials are different or not.

Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N

Abstract The sextic formalism of Stroh for anisotropic elasticity leads to the eigen-relation N ξ = pξ in which N is a 6 × 6 real matrix. The orthogonality and closure relations as well as many other relations involving the eigenvalues p and the eigenvectors ξ x are based on the assumption that N is simple or semisimple so that the six eigenvectors ξ α span a six-dimensional space. Problems arise when N is non-semisimple. In fact there are problems even when N is almost non-semisimple. We present a modified formalism which is valid regardless of whether N is simple, almost non-semisimple or non-semisimple. The modified formalism does not apply when N is semisimple.

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2α satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for α not equal but very close to 235-3, the classical solution can still be very large as α approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as α approaches 235-5 and reproduces Dempseys solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempseys solutions when 2α=π and 2π. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r-λ and ln r are replaced by R-gl and ln R where R=r/r0is the dimensionless radial distance and r0 is an arbitrary constant having the dimension of length.

Negative Poisson's Ratios in Anisotropic Linear Elastic Media

Poisson s ratio for an anisotropic linear elastic material depends on two orthogonal directions n and m. Materials with negative Poissons ratios for all (n,m) pairs are called completely auxetic while those with positive Poissons ratios for all (n,m) pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some (n,m) pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.