Lekhnitskii Formalism Research Articles

Green's function for thermopiezoelectric plates with holes of various shapes

Thermoelectroelastic problems for holes of various shapes embedded in an infinite matrix are considered in this paper. Based on Lekhnitskii's formalism, the technique of conformal mapping and the exact electric boundary conditions on the hole boundary, the thermoelectroelastic Green's function has been obtained analytically in terms of a complex potential. As an application of the proposed function, the problem of an infinite plate containing a crack and a hole is analysed. A system of singular integral equations for the unknown temperature discontinuity and the discontinuity of elastic displacement and electric potential (EDEP) defined on crack faces is developed and solved numerically. Numerical results for stress and electric displacement (SED) intensity factors of the crack-hole system are presented to illustrate the application of the proposed formulation.

Analytical solution for an arbitrarily oriented void/crack and fracture of piezoceramics

The solutions for a piezoelectric plane with an arbitrarily oriented elliptical void and a crack are derived by the method of extended Lekhnitskii's formalism. Analytical solutions are obtained for all electroelastic field variables around the void and at the crack tip. The generalized solutions presented in this study can be reduced to the results reported in the literature for special cases of defect orientation, i.e. defects parallel or perpendicular to the poling direction. It is found that the effect of defect orientation on the electroelastic fields around a void and at the crack tip is rather subtle. Solutions for defects oriented parallel or perpendicular to polarization cannot be always considered as the critical case. A change of crack orientation may alter the trend of toughness variation, and initiate failure along different planes or cause crack closure depending on the direction of electrical field and ratio of electrical to mechanical loading. It is shown that a positive electric field may retard or enhance crack growth depending on the orientation of a crack with respect to the polarization direction, and a negative one tends to close a crack except for the crack parallel to the poling direction.

Green function and its application for a piezoelectric plate with various openings

For a two-dimensional piezoelectric plate subjected to mechanical and electric load, a Green function satisfying traction free and exact electric boundary conditions along a hole is developed using Lekhnitskii's formalism and the technique of conformal mapping. The critical points for the mapping function used is investigated numerically, and the study indicates that the transformation of a polygonal hole in a piezoelectric plate into a unit circle is nonsingle-valued. A simple approach is presented to treat such a situation. Based on the Green function developed in this paper, a system of singular integral equations for the unknown dislocation defined on crack faces is presented to study the interaction between cracks and holes. Numerical results are presented to elucidate the effects of crack orientation on stress and electric displacement (SED) intensity factors and to illustrate the application of the proposed formulation.

A modified Lekhnitskii formalismà laStroh for anisotropic elasticity and classifications of the 6 × 6 matrixN

The Stroh formalism for two-dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation. The vector b is then determined from a . Depending on the number of repeated eigenvalues p and the number of independent eigenvectors ( a , b ) the system has, it can be classified into six groups. The Lekhnitskii formalism has no eigenrelation to speak of. We present a modified Lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation. The vector a is then determined from b . Thus the modified Lekhnitskii formalism is a dual to the Stroh formalism. Not only does the modified formalism enable us to do the classifications, it is much simpler than using the Stroh formalism. The six groups are the SP, SS, D1, D2, ED and ES groups. The ES group does not exist for a real material. The SS group (that has p 1 = p 2 ≠ p 3) and the D2 group (that has p 1 = p 2 ≠ p 3) can be identified without computing the eigenvalues p and the eigenvectors ( a , b ). We show that the repeated eigenvalue p 1 = p 2 in the SS and D2 groups is simply a root of the quadratic equation l 2 = 0. We present an explicit expression of p 3 for the SS group. The ED group that has three identical p can also be identified without computing p and ( a , b ); however, we do present an explicit expression of p . We show that monoclinic materials with the symmetry plane at x 3 = 0 cannot belong to the ED group. The identification of the SP and D1 groups is the only one that requires computation of p but not ( a , b ). For special classes of materials, however, they can be identified without computing p . In all cases, the eigenvectors and the generalized eigenvectors are obtained explicitly.

Piezoelectric Analogy of Generalized Torsion in Anisotropic Elasticity

We consider a piezoelectric body bounded by a cylindrical surface in which all cross sections are of the same geometry. Suppose on the lateral surface the body is loaded in such a way that the stress and electric displacement do not vary along the axial direction. In addition the end of the cylinder is also subjected to forces reducing to bending moments, twisting moment, axial force and electric charge. We follow Lekhnitskii's formalism to characterize the deformation of the considered problem. It is found that, when the solid possesses a material symmetry plane normal to the axial direction, the simplified field equations together with suitably chosen boundary conditions are entirely analogous to those of a generalized torsion problem in anisotropic elasticity. In particular, we show that by setting a linkage between two sets of 21 material constants, any problem in one field can be resolved as another problem in the other area.

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 < \( - \frac{1}{2}\) 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.

Anisotropic Elasticity: Theory and Applications

1. Matrix Algebra 2. Linear Anisotropic Elastic Materials 3. Antiplane Deformations 4. The Lekhnitskii Formalism 5. The Stroh Formalism 6. The Structures and Identities of the Elasticity Matrices 7. Transformation of the Elasticity Matrices and Dual Coordinate Systems 8. Green's Functions for Infinite Space, Half-space, and Composite Space 9. Particular Solutions, Stress Singularities, and Stress Decay 10. Anisotropic Matrials with an Elliptic Boundary 11. Anisotropic Media with a Crack or a Rigid Line Inclusion 12. Steady State Motion and Surface Waves 13. Degenerate and Near Degenerate Materials 14. Generalization of the Stroh Formulism 15. Three-Dimensionsal Deformations