Lekhnitskii Formalism Research Articles

Closed-form methodology for the structural analysis of stiffened composite plates with cutouts and non-uniform lay-up

Building upon the methodology developed in prior studies, which utilized the Lekhnitskii formalism to address composite plates featuring cutouts, varying thicknesses, and different stacking sequences, the present work introduces the incorporation of stiffeners. The structural configuration comprises plates and stringers, with each stringer positioned between two plate regions and subjected to both stretching and bending loads. It is important to note that no coupling between the bending and stretching responses is accounted for; therefore, lay-ups must be symmetric, and stringers must be embedded and bisymmetric. The stringers are modeled using the Euler–Bernoulli formulation with the free torsion hypothesis, while the plates are modeled using the Kirchhoff–Love formulation. Several benchmark problems are analyzed, and the results are compared with those obtained using finite element analysis (utilizing Abaqus software), demonstrating a satisfactory agreement while also showcasing competitive computational efficiency. Thus, the present methodology provides the industry with a novel tool that enables efficient parametric analysis and facilitates the most promising configurations during the initial phases of the design to be selected by engineers.

Micromechanical approach to determine the effects of surface and interfacial roughness in materials and structure under cosinusoidal normal pressure

Contact mechanics is a topic that performs the investigation of the deformation of solids that touch each other at one or more points. A principal distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces and stresses acting tangentially between the surfaces. This study focuses mainly on the normal stresses that are caused by applied forces. As a case study, the present work aims at investigating the bi-dimensional contact mechanics of wavy cosinusoidal anisotropic finite planes. To achieve this objective, results on the displacement and stress component are first calculated with the help of the Lekhnitskii formalism. Then, with the application of normal pressure at plane surface and by applying boundary conditions at depth h of solid we obtain solution for the contact pressure in closed form. In case of infinite anisotropic plane where the depth h tends to infinite, by using results obtained with finite h we derive the analytical solution for vertical displacement at the surface. As an illustration, behaviour of a monoclinic material under consinusoidal pressure is analyzed

Multiscale Failure Analysis of Cylindrical Composite Pressure Vessel: A Parametric Study

Abstract This paper deals with a multiscale approach to model thick-walled laminate cylinder with internal pressure. Micromechanics defines material homogenization considering two steps: determination of equivalent properties of each lamina from matrix and fiber properties according to the Mori-Tanaka model for elastic properties and to the Bridging model for strengths; and determination of anisotropic homogeneous properties of the laminate built with a set of laminae using asymptotic homogenization. On the other hand, macromechanics determines stress and failure analysis. Lekhnitskii formalism is used to obtain the elastic solution of the stress and strain distributions and failure is analyzed employing the Tsai-Wu criterion. Three different pressure vessel configurations are analyzed according to end conditions: restrained-ends, open-ends and closed-ends. Angle-ply laminates made of carbon fibers and epoxy matrix are considered to evaluate the influence of lay-up angle, fiber volume fraction, wall thickness and end-conditions. The optimum angles as well as the maximum internal pressure are obtained and a parametric analysis is presented. The main results indicate that the optimum angle is almost constant for restrained and closed-ends. On the other hand, for open-end, angle varies in a significant way. Besides, results show that the increase the fiber volume fraction is more effective to increase vessel strength than the increase of the number of layers.

Closed-form approximations to borehole stresses for weak transverse isotropic elastic media

We have derived new closed-form approximations to borehole stresses for weak tilted-transverse-isotropic media with boreholes oriented in the plane of isotropy but not necessarily along a principal stress direction. By introducing Green's three anisotropic parameters, ω1,ω2 and ω3, into Lekhnitskii's formalism for anisotropic elastic media to calculate stresses around boreholes subjected to internal pressure and in situ stress, we first have recasted the general borehole stresses into more compact expressions that are useful to highlight explicitly the impact of the elastic anisotropy. We have shown the equivalence between Lekhnitskii's and Green's equations for the in-plane components and extended Green's expressions to the anti-plane problem. By linearizing the expressions for weak anisotropy degree, i.e. ∣ωi∣≪1, we have shown that the borehole stress expressions are the sum of the isotropic “Kirsch” solution and additional terms that depend on the anisotropic parameters ω1+ω2 and ω3. The additional terms have azimuthal/radial functional dependence of higher orders (cos4θ,sin4θ) and (1/r2,1/r4,1/r6) for in-plane components, and, (cos3θ,sin3θ) and (1/r2,1/r4) for anti-plane components. Our verification examples show that the new borehole stress expressions provide good approximations not only for weak anisotropy but also moderate and strong anisotropy in certain cases.

遠方一様荷重下でのだ円孔を有する圧電材料および異方性材料に対する類似的解析

This paper presents general solutions for piezoelectric materials and anisotropic materials with elliptical cavity subjected to uniform loads at infinity. General solutions are provided by using complex analytical functions based on Lekhnitskii formalism. In the analysis, similarities of fundamental equation between piezoelectric materials and anisotropic materials are shown. By defining new similarity-based complex functions, expressions of mechanical and electrical quantities are derived by analogous form. And distributions of stress, electric displacement, strain and electric field around elliptical cavity are shown by contour plot. Furthermore, it is shown that theory of anisotropic material is applicable to piezoelectric material by adequately replacing material constants.

The influence of the first non-singular stress terms on crack initiation direction in an orthotropic bi-material plate

A bi-material plate composed of two orthotropic parts leads to free edge stress singularity. This is considered as a bi-material notch with ω1=ω2=90° under general loading. Besides the singular term the first non-singular term is also included in the stress distribution at the notch tip. The Stroh–Eshelby–Lekhnitskii formalism for plane elasticity is used to express stress and displacement fields and the strain energy density factor. The exponents of the singular and non-singular stress terms and corresponding eigenvectors are the solution of the eigenvalue problem resulting from the prescribed boundary and compatibility conditions at the notch tip. The potential direction of crack initiation is determined from the local minimum of the mean value of the strain energy density factor in both materials. The necessary knowledge of the generalized stress intensity factor and generalized T-stress is realized by means of the employment of the two state conservation integrals. Following the assumption of the same mechanism of rupture in the case of a crack and a notch, an expression for the critical values of the generalized stress intensity factor is obtained.

側面に任意荷重を受ける異方性だ円柱の解析

This paper presents general solutions for anisotropic cylinder using complex stress functions based on Lekhnitskii formalism. Up to now, an analytical solution of isotropic cylinder has been widely used whereas solutions of anisotropic elliptic cylinders subject to arbitrary surface loads have not been derived. The proposed complex stress functions, which are developed by Laurent series, enable to solve conditions under distributed loads like hydro pressures and concentrated loads. The function can also analyze a shear load parallel to longitudinal axis of a cylinder. Analyzed stress distributions are graphically represented.

Two‐dimensional analysis of a crack in quasicrystal materials

AbstractThe two‐dimensional problem of a crack in three‐dimensional quasicrystals subject to far field loadings is studied. The analysis is based on the generalized Lekhnitskii's formalism. The analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The fracture quantities of quasicrystals, i.e., field intensity factors, energy release rates and so on, is a prerequisite. Numerical results for a Griffith crack under phason loading Mode I and II conditions are poltted. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

G010015 側面に任意の表面荷重が作用する異方性リングの解析([G01001]計算力学部門一般セッション(1):固体力学の基礎的な数値解析)

Analytical solution for anisotropic hollow cylinder subjected any loadings at lateral surfaces has been not derived so far whereas solution in case of an isotropy was obtained. Even in the case of isotropy, antiplane loading is not yet treated. In this study, the general solution for anisotropic hollow cylinder is shown using a complex stress functions based on Lekhnitskii formalism. Stress functions are derived using a constraint release technique, which is a kind of the convergence numeration applied for the problems of solid cylinder and infinite medium with cavity. Any loadings, for example, distributed load such as hydro pressure and concentrated load are taken into account a stress function developed a Laurent series. Example calculation is shown in several graphical representations.

Extraordinary degeneracy and space of degeneracy in transversely isotropic elastic media

A large variety of degeneracy exist in both static and dynamic regimes in anisotropic elastic media. Extraordinary degeneracies occur mostly in the static regime and they are of great importance in understanding the existence of the so-called space of degeneracy in the dynamic regime. This paper deals with the interconnection between degeneracies in the static and dynamic regimes. Both the modified Lekhnitskii formalism and the Stroh formalism are employed in exploring the degeneracies in the transversely isotropic media. It is demonstrated that the space of degeneracy can be regarded as an extension of the static degeneracy, and the development of the static extraordinary degeneracy into the dynamic regime is via a number of spaces of degeneracy of various types. The static degeneracy can also span a space of degeneracy.

Trefftz solutions for piezoelectricity by Lekhnitskii's formalism and boundary‐collocation method

AbstractIn this paper, a solution procedure for plane piezoelectricity is developed by Trefftz boundary‐collocation method. Starting with the general plane piezoelectricity solution derived by Lekhnitskii's formalism, the basic sets of Trefftz functions which satisfy the homogeneous governing equations are derived. Moreover, special sets of Trefftz functions are derived for impermeable elliptical voids, impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction. The functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects. By adopting Trefftz functions as the trial functions, multi‐domain Trefftz boundary‐collocation method is formulated. Numerical examples are presented to illustrate the efficacy of the formulation. Copyright © 2005 John Wiley & Sons, Ltd.

Analysis for Three-Dimensional Anisotropic Elastic Medium with Multi-Layered Elliptic Ring Inclusions under In-Plane and Out-of-Plane Loadings

This paper presents numerical solutions for anisotropic elastic medium with multi-layered elliptic ring inclusions. An arbitrary direction of principal axes of elasticity is considered in the matrix. General solutions of this study are obtained by using complex stress functions. The new formulation to improve problems of Lekhniskii's Formalism is proposed for anisotropic medium. Using the formulation, it is possible to determine a solution without arbitrary exchange of anisotropic parameters μ<k (k=1, 2, 3). The formulation is compared with Lekhnitskii's Formalism as well as Stroh's Formalism. Several results of calculation are shown by numerical representations.

Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems

We will derive the fundamental generalized displacement solution, using the Radon transform, and present the direct formulation of the time-harmonic boundary element method (BEM) for the two-dimensional general piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM, which is formulated following the physical interpretation of Somigliana’s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the Stroh–Lekhnitskii (SL) formalism. The time-harmonic BEM is obtained by adding the boundary integrals for the dynamic regular part which, from the original double integral representation over the boundary element and the unit circle, are reduced to simple line integrals along the unit circle. The BEM will be applied to the determination of the eigen frequencies of piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. A comparative study will be made of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. It is found that the IRAM is faster and has more control over the solution procedure than the QZ algorithm. The use of the time-harmonic fundamental solution provides a clean boundary only formulation of the BEM and, when applied to the eigenvalue problems with IRAM, provides eigen frequencies accurate enough to be used for industrial applications. It supersedes the dual reciprocity BEM and challenges to replace the FEM designed for the eigenvalue problems for piezoelectricity.

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

Passing stiffness anisotropy in multilayers and its effects on nanoscale surface self-organization

A binary monolayer adsorbed on a solid surface can separate into distinct phases that further self-assemble into various two-dimensional patterns. The surface stresses in the two phases are different, causing an elastic field in the substrate. The self-organization minimizes the combined free energy of mixing, phase boundary, and elasticity. One can obtain diverse patterns by using substrates with various crystalline symmetries. Consider the pattern of a set of periodic stripes. The stripe orientation depends on the anisotropy in surface stress, substrate stiffness, and phase boundary energy. A more powerful and flexible way is to use a layered substrate. Surface properties designed for the applications of those patterns can be obtained by choosing appropriate materials and structures for the monolayer and the top layer of the substrate. The subsequent layers of the substrate provide the required stiffness anisotropy, the effect of which is passed to the monolayer patterns through the elastic field. We solve the elastic field in the anisotropic, heterogeneous, three-dimensional half-space by using the Eshelby–Stroh–Lekhnitskii formalism and the Fourier transformation. Depending on the thicknesses and the degrees of the stiffness anisotropy of the substrate layers, the lowest energy stripes can have tunable equilibrium size and orientation. We also discuss other possibilities of manipulating the phase patterns by engineering the elastic field.

Fracture Mechanics of Piezoelectric Materials

Fracture Mechanics of Piezoelectric Materials

Boundary element modeling of cracks in piezoelectric solids

This study focuses on the application of boundary element methods for linear fracture mechanics of two-dimensional piezoelectric solids. A complete set of piezoelectric Green's functions, based on the extended Lekhnitskii's formalism and distributed dislocation modeling, are presented. Special Green's functions are obtained for an infinite medium containing a conducting crack or an impermeable crack. The numerical solution of the boundary integral equation and the computation of fracture parameters are discussed. The concept of crack closure integral is utilized to calculate energy release rates. Accuracy of the boundary element solutions is confirmed by comparing with analytical solutions reported in the literature. The present scheme can be applied to study complex cracks such as branched cracks, forked cracks and microcrack clusters.

Applications of an explicit solution for the transversely isotropic circular disc compressed diametrically

Abstract This paper focuses on the Brazilian test configuration of anisotropic rocks. The proposed methodology follows that of Pinto (Proceedings of the Fourth ISRM Cong, Montreux, 1979); however, it is more accurate since we adopt Amadei's method of analysis (Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33 (1996) 293) that accounts for the influence of anisotropy on the stress distribution. In a first step, explicit expressions for stresses and strains for the anisotropic circular disc compressed diametrically are presented by employing Lekhnitskii's formalism. It is shown that the proposed analytical solution can be effectively used as a “back-analysis” tool for the characterization of rock elasticity and strength properties. This is illustrated both in the case of published experimental results on schist and gneiss rock types, and finally in the case of specially designed Brazilian tests on Dionysos marble.

A new modified Lekhnitskii formalism àla Stroh for steady-state waves in anisotropic elastic materials

For the analysis of a two-dimensional steady-state motion such as the surface wave in an anisotropic elastic half-space, the Stroh formalism has always been employed. The solutions are in terms of the elastic stiffnesses C αβ . The Lekhnitskii formalism for elastostatics that provides the solutions in terms of the reduced elastic compliances s αβ ′ is not applicable for two-dimensional steady-state motion. We present a new modified Lekhnitskii formalism in the style of Stroh that can be employed for analyzing two-dimensional steady-state motion. In contrast to the Stroh formalism for which one computes the eigenvector b in terms of the eigenvector a, the new modified Lekhnitskii formalism can compute the eigenvector b without computing the vector a. This feature is attractive in the study of surface waves because the vector b is related to the surface traction. The vanishing of the surface traction at the boundary of the half-space is the key in the surface wave theory. Application to one-component surface waves shows that the conditions for such waves are easily deduced. Motivated by the new modified Lekhnitskii formalism we show that an eigenrelation for the vector b can also be derived for the Stroh formalism.

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).