Octet Formalism Research Articles

A three‐phase laminated anisotropic elliptical inhomogeneity with internal uniform stress resultants

AbstractWe use the Stroh octet formalism for the coupled stretching and bending deformations of Kirchhoff laminated anisotropic thin plates together with the complex variable formulations of the decoupled stretching and bending deformations of Kirchhoff homogeneous isotropic thin plates to study the internal stress resultants inside a laminated anisotropic elliptical inhomogeneity bonded to a surrounding infinite laminated anisotropic matrix through a confocal interphase layer composed of a homogeneous isotropic plate when the matrix is subjected to uniform remote membrane stress resultants and bending moments. We prove that for given geometric and material parameters of the composite, the internal stress resultants inside the elliptical inhomogeneity are uniform when four specific conditions on the remote loading are satisfied. Furthermore, the internal uniform elastic field is determined in real‐form in terms of the two 8 × 8 fundamental elastic plate matrices for both the inhomogeneity and the matrix and the three 4 × 4 real matrices S, H, and L for the matrix.

The electroelastic fields inside and outside a piezoelectric parabolic inclusion with uniform eigenstrains and eigenelectric fields

AbstractUsing the Stroh octet formalism, we obtain the electroelastic fields in an infinite homogeneous piezoelectric medium containing a parabolic Eshelby inclusion undergoing uniform eigenstrains and eigenelectric fields. We prove that the internal electroelastic field defining stresses, total strains, electric displacements, total electric fields and rigid‐body rotation inside the parabolic inclusion is uniform. The internal uniform electroelastic field is expressed explicitly in terms of the reduced generalized compliances and the imposed eigenstrains and eigenelectric fields. The decaying non‐uniform electroelastic field in the exterior of the parabolic inclusion is also obtained.

Deformations near an elliptical hole with surface effect in a laminated anisotropic thin plate

This work is concerned with the coupled stretching and bending deformation around an elliptical hole with surface energy in a laminated and inhomogeneous anisotropic elastic thin plate within the context of the Kirchhoff theory. A closed‐form full‐field solution is derived by using the octet formalism recently developed by Cheng and Reddy (2002, 2003, 2004, 2005) and by incorporating a simplified version of the surface elasticity model. In particular, explicit real‐form expressions of the hoop membrane stress resultant, hoop bending moment, in‐plane displacements and slopes on the mid‐plane along the edge of the elliptical hole are obtained.

Eshelby’s problem for infinite, semi-infinite and two bonded semi-infinite laminated anisotropic thin plates

We consider an Eshelby’s inclusion of arbitrary shape with prescribed uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. The inclusion has the same extensional, coupling and bending stiffnesses as the surrounding material. The boundary of the semi-infinite plate can be described by free, rigidly clamped and simply supported edges. We derive solutions of simple form by using the new Stroh octet formalism for the coupled stretching and bending deformations of anisotropic thin plates and the method of analytic continuation. In particular, real solutions of the far-field elastic fields induced by an inclusion of arbitrary shape are obtained. Specific examples of an elliptical inclusion in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite anisotropic plates are presented to demonstrate the obtained general solutions.

Interface Cracks in Kirchhoff Anisotropic Thin Plates of Dissimilar Materials

This paper investigates the problem of stretching and bending deformations of a Kirchhoff anisotropic thin plate composed of two dissimilar materials bonded along a straight interface containing a crack. Our analysis makes use of the Stroh octet formalism developed recently by Cheng and Reddy (Cheng and Reddy, 2002, “Octet Formalism for Kirchhoff Anisotropic Plates,” Proc. R. Soc. Lond., A458, pp. 1499–1517; Cheng and Reddy, 2003, “Green’s Functions for Infinite and Semi-Infinite Anisotropic Thin Plates,” ASME J. Appl. Mech., 70, pp. 260–267; Cheng and Reddy, 2004, “Laminated Anisotropic Thin Plate With an Elliptic Inhomogeneity,” Mech. Mater., 36, pp. 647–657; Cheng and Reddy, 2005, “Structure and Properties of the Fundamental Elastic Plate Matrix,” J. Appl. Math. Mech., 85, pp. 721–739) for Kirchhoff anisotropic plates. It is found that the interfacial crack-tip field consists of a pair of two-dimensional oscillatory singularities, which are explicitly determined. Two complex intensity factors are proposed to evaluate the two oscillatory singularities.

Stroh-like formalism for Kirchhoff anisotropic thermoelastic plates

A Stroh-like formalism is developed for the heat conduction and the coupled stretching and bending deformations of a laminated anisotropic thermoelastic thin plate based on Kirchhoff theory. For the heat conduction problem, a Stroh-like quartic formalism is developed. Twodimensional generalized temperature and heat flux function vectors are introduced. The structure of the introduced 4x4 fundamental plate matrix for heat conduction is the same as that of the 8x8 fundamental elasticity matrix in the Stroh sextic formalism for generalized plane strain elasticity. Consequently, the orthogonality and closure relations for heat conduction in thin plates is established. For the thermoelastic problem, an inhomogeneous particular solution is derived rigorously. We obtain an octet formalism in which the general solution is composed of the well-known homogeneous solution developed by Cheng and Reddy (isothermal case) and the inhomogeneous particular solution arising from the thermal effect.

Lekhnitskii’s formalism of one-dimensional quasicrystals and its application

By generalizing the complex potential approach developed by Lekhnitskii, plane problems of one-dimensional quasicrystals are solved first by using an octet formalism for which there are four pairs of complex roots. The approach uses a representation of stresses and proceeds by integration of the expressions for deformations and application of the anisotropic constitutive law and the compatibility of displacements. To illustrate its utility, the generalized Lekhnitskii’s formalism is used to analyse the coupled phonon and phason fields in an infinite quasicrystal medium containing an elliptic rigid inclusion.

Acoustic wave transmission through piezoelectric structured materials

This paper deals with the transmission of acoustic waves through multilayered piezoelectric materials. It is modeled in an octet formalism via the hybrid matrix of the structure. The theoretical evolution with the angle and frequency of the transmission coefficients of ultrasonic plane waves propagating through a partially depoled PZT plate is compared to finite element calculations showing that both methods are in very good agreement. The model is then used to study a periodic stack of 0.65PMN-0.35PT/0.90PMN-0.10PT layers. The transmission spectra are interpreted in terms of a dispersive behavior of the critical angles of longitudinal and transverse waves, and band gap structures are analysed. Transmission measurements confirm the theoretical calculations and deliver an experimental validation of the model.

Acoustic waves in the vicinity of the normal to the surface of piezoelectric crystals

The acoustic wave propagation in the vicinity of the normal to the plane surface confining a piezoelectric crystal of arbitrary symmetry is theoretically studied. An octet formalism and a perturbation theory have been put forward to describe the wave fields in the region of concern. The developed mathematical approach has been applied to several problems. Specifically, the derivation of the transfer matrix for the normal direction to the surface has been discussed. Furthermore, we have discussed how to estimate the electric potential induced outside the piezoelectric material by a normally incident wave. In addition, an analytical expression has been derived for the numerical factor in the function describing the asymptotic behavior of quasi-electrostatic Green's function for half-infinite piezoelectric substrates at small values of the wave vector.

Structure and properties of the fundamental elastic plate matrix

AbstractThis work presents further development of the octet formalism established by the authors for the classical Kirchhoff anisotropic plate theory. The structure of the fundamental elastic plate matrix is fully explored and the explicit expression is provided. The matrices N2 and N3 are proved to be positive semi‐definite. Thus, H and L are positive definite. Further studies are concerned with a rotated coordinate system. The transform relation between the eigenvectors in the original and the rotated coordinate system is given. The fundamental elastic plate matrix associated with the eigenrelation referring to the dual coordinate systems, NΘ, is studied. The major properties that hold in the Stroh sextic formalism for generalized plane strain problems are also valid in the octet formalism for thin plate bending problems. In particular, we generalize a property in Stroh's formalism for any non‐semisimple matrix NΘ. We show a new property in the octet formalism. The non‐semisimple cases of NΘ are discussed. Finally, we make it transparent that the mixed/hybrid formalism of others is precisely one of sixteen permuted forms of the octet formalism.

Green’s functions for an anisotropic thin plate with a crack or an anticrack

Based on the octet formalism established by the authors, Green’s functions for an infinite anisotropic elastic thin plate with a Griffith crack or an anticrack (rigid line inclusion) are studied. The plate can be laminated or consist of a continuously inhomogeneous material in the thickness direction. Two systems of problems of the plate under non-self-equilibrated loads are solved. The first is associated with discontinuous in-plane displacements and slopes, in-plane concentrated forces and out-of-plane concentrated moments. The second is associated with a transverse concentrated force. The Green functions for an infinite plate with a crack or an anticrack and the surface Green functions for an infinite plate with a crack are obtained in an exact closed form. The surface Green functions and associated stress resultants at any point throughout the line where the crack is located are presented in a real form. The intensity factor of membrane stress resultants and bending moments are obtained in a real form for the Griffith crack subjected to concentrated loads.

Laminated anisotropic thin plate with an elliptic inhomogeneity

This work is concerned with a through-thickness elliptic elastic inhomogeneity in a laminated anisotropic elastic thin plate within the context of the Kirchhoff theory. By means of the octet formalism recently established by the authors, an exact closed-form solution is obtained, for the first time, for coupled stretching and bending deformations of the plate subjected to remote uniform membrane stress resultants and bending moments. The stress resultants inside the elastic elliptic inhomogeneity are uniform, which is consistent with the uniformity property of the Eshelby inclusion solution in three-dimensional elasticity. In two special limit cases where the elliptic inhomogeneity becomes an elliptic rigid inclusion or hole, the corresponding solutions are also obtained. A relation between the rotational moment and the rotation is given for the elliptic rigid inclusion problem. The concentration factors of hoop membrane stress resultants and hoop bending moments are given for the elliptic hole problem. The intensity factors of membrane stress resultants and moments are obtained for a Griffith crack. Displacements, slopes, membrane stress resultants and bending moments along the elliptic boundary for the elliptic inhomogeneity, rigid inclusion and hole problems are all presented in a real form.

Green’s Functions for Infinite and Semi-infinite Anisotropic Thin Plates

This paper presents fundamental solutions of an anisotropic elastic thin plate within the context of the Kirchhoff theory. The plate material is inhomogeneous in the thickness direction. Two systems of problems with non-self-equilibrated loads are solved. The first is concerned with in-plane concentrated forces and moments and in-plane discontinuous displacements and slopes, and the second with transverse concentrated forces. Exact closed-form Green’s functions for infinite and semi-infinite plates are obtained using the recently established octet formalism by the authors for coupled stretching and bending deformations of a plate. The Green functions for an infinite plate and the surface Green functions for a semi-infinite plate are presented in a real form. The hoop stress resultants are also presented in a real form for a semi-infinite plate.

Octet formalism for Kirchhoff anisotropic plates

An octet formalism is established for stretching and bending deformations of an anisotropic elastic plate based on the Kirchhoff theory. The plate is inhomogeneous in the thickness direction and thus includes the laminated plate as a special case. By defining a 4–eigenvector associated with two in–plane displacements and two rotational angles of a normal of the mid–plane, and a 4–eigenvector associated with four stress functions, the three equilibrium equations of the plate reduce to a standard eight–dimensional eigenrelation. The fundamental elastic plate matrix defined in this work is found to place the two eigenvectors in reverse order to form the left and right 8–eigenvectors of the matrix. The same capability has been observed for the fundamental elasticity matrix in Stroh's sextic formalism for generalized plane strain elasticity. Thus our octet formalism symbolically preserves extensive elegant properties and identities pertaining to the capability, e.g. the orthogonality and closure relations, which have been established in the Stroh sextic formalism. Having shown that the eigenvalues cannot be real, we provide the eigenvalue representations for special cases of material and geometry. The non–semisimple case including an isotropic material is briefly discussed.