General Anisotropic Materials Research Articles

Methods for calculating stress intensity factors in anisotropic materials: Part I— z = 0 is a symmetric plane

The problem of a crack in general anisotropic material under LEFM conditions is presented. In Part I, three methods are presented for calculating stress intensity factors for various anisotropic materials in which z = 0 is a plane of symmetry. All of the methods employ the displacement field obtained by means of the finite element method. The first one is known as displacement extrapolation and requires the values of the crack face displacements. The other two are conservative integrals based upon the J-integral. One employs symmetric and asymmetric fields to separate the mode I and II stress intensity factors. The second is the M-integral which also allows for calculation of K I and K II separately. All of these methods were originally presented for isotopic materials. Displacement extrapolation and the M-integral are extended for orthotropic and monoclinic materials, whereas the J I- and J II-integrals are only extended for orthotropic material in which the crack and material directions coincide. Results are obtained by these methods for several problems appearing in the literature. Good to excellent agreement is found in comparison to published values. New results are obtained for several problems. In Part II, the M-integral is extended for more general anisotropies. In these cases, three-dimensional problems must be solved, requiring a three-dimensional M-integral.

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 Poisson’s ratios for all (n,m) pairs are called completely auxetic while those with positive Poisson’s 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.

A brief note on elastic T-stress for centred crack in anisotropic plate

The stress intensity factors (SIFs) and the T-stress for a planar crack with anisotropic materials are evaluated by the fractal finite element method (FFEM). The FFEM combines an exterior finite element model and a localized inner model near the crack tip. The mesh geometry of the latter is self-similar in radial layers around the tip. A higher order displacement series derived from Laurent series and Goursat functions is used to condense the large numbers of nodal displacements at the inner model near the crack tip into a small set of unknown coefficients. In this study, the variations of the SIFs and the T-stress with material properties and orientations of a crack are presented. The separation of the analytical displacement series into four fundamental cases has shown to be necessary in order to cover all the material variations and the orientations of a crack in the plate with general rectilinear anisotropic materials.

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.

Mechanics of Composite Structures

5R4. Mechanics of Composite Structures. - LP Kollar and GS Springer. Cambridge UP, Cambridge, UK. 2003. 480 pp. ISBN 0-521-80165-6. $95.00.Reviewed by M-A Erki (Dept of Civil Eng, Royal Military Col of Canada, PO Box 17000 Station Forces, Kingston ON, K7K 7B4, Canada).This comprehensive book presents traditional topics of structural mechanics that target the topics of interest for designers of aerospace, land, and marine structures, namely plates, beams, and shells, including where appropriate the analyses for their buckling, vibrations, deformations, and displacements. While the topics are familiar to designers using isotropic materials, in this book they are presented for anisotropic composite materials of all types, which are long-fiber composites, short-fiber composites, particulate composites, unidirectional lamina, biaxial weave woven fabric, and triaxial weave woven fabric. Chapter 1 illustrates these materials. It describes the structure of laminate composites and how the individual plies contribute to the laminate behavior. The chapter also defines the various levels of analysis for a structure made of a laminated composite, from the micro (analysis of the matrix and fiber) to macro (analysis of the composite), ending with structural analysis of components, herein plates, beams, and shells. The foundation for all subsequent derivations and analyses in the book are given in Chapter 2. It presents the generalized 3D relationships between displacements and strains and between strains and stresses, via equilibrium equations. The stress-strain relationship is developed first for general anisotropic material, followed by the simplifications that can be made for monoclinic material, orthotropic material, transversely isotropic material, and isotropic material. The differences in analytical approaches for plane strain and plane stress analyses are carefully explained using the specific examples for all these materials for the conditions of free end and built-in ends. The effect of temperature and moisture content on strains and stresses is discussed for both the plane strain and plane stress conditions.A section on boundary conditions and how these are applied to obtain solutions to the equilibrium, stress-strain, and strain displacement equations is given. Continuity conditions between laminated layers that are assumed perfectly bonded are described. Stress and strain transformations using direction cosines and transformation of the stiffness and compliance matrices are discussed for the plane-strain and plane-stress conditions. Finally the strain energy equations are derived, and the Ritz Method is summarized as a solution method for these. The chapter ends with a comprehensive summary of the chapter contents and a numerical example. Chapter 3 discusses the analysis of laminated composites and the contribution of the behavior of individual plies to the overall behavior of the laminate. Stiffness matrices for thin laminates, an example of symmetrical laminate, balanced laminate and orthotropic laminate, isotropic laminate, and quasi-isotropic laminate are illustrated with numerical examples. The next five chapters present the analyses for the basic structural members composed of composite materials, namely thin plates, sandwich plates, solid and thin-walled beams without and with shear deformations, and shells, all with numerical examples as appropriate. Unique considerations for the finite element analysis of composite materials and composite material structures are described in Chapter 9. Chapter 10 discusses failure criteria for the basic types of composite materials, and Chapter 11 summarizes composite material micromechanics. Three valuable appendices give supplementary material of value to the designer. They are the cross-sectional properties of thin-walled composite beams; the buckling loads and natural frequencies of orthotropic beams with shear deformations; and typical composite material properties. Owing to the clarity and breadth of its content, Mechanics of Composite Structures makes a valuable contribution to the field of analysis and design using composite materials. It is well illustrated throughout. Of note, the examples given throughout are truly excellent, because they are both relevant to the development of the concepts presented and practical for designers. The authors have made a special effort to make the book self-contained so that it can be equally suitable as a text for self-study, senior undergraduate, or for a graduate course.

Optical properties of anisotropic materials: an experimental approach

In this contribution, we review the analysis of ellipsometric spectra of general anisotropic materials from an experimental point of view. The basic framework of data analysis is the numerical solution of the exact ellipsometric equations by using the 4×4 transfer matrix algorithm. The flexibility of our approach allows undertaking the study of the dielectric tensor in general systems. We illustrate the practical use of our method with different examples. In particular we discuss the study of systems of reduced symmetry where principal axes of the dielectric tensor are a priori unknown, as monoclinic α-PTCDA and anthracene.

A Rough Surface Contact Model for General Anisotropic Materials

A method for solving the two-dimensional (2-D) isothermal rough surface contact problem of general anisotropic materials with friction is presented. By using Stroh’s formalism, the surface displacements of an elastic half-space due to uniform distributions of traction over a strip are derived from the surface Green’s function. The surface displacement and subsurface stresses of the anisotropic half-space due to the distributed contact pressure may then be calculated by superposition. The real contact area and the contact pressure are determined via an iteration scheme using the conjugate gradient method.

The polarization vector and secular equation for surface waves in an anisotropic elastic half-space

The displacement at the free surface of an anisotropic elastic half-space x 2>0 generated by a surface wave propagating in the direction of the x 1-axis traces an elliptic path. It is represented by the polarization vector a R= e 1+ i e 2 , where e 1 , e 2 are the conjugate radii of the ellipse on the polarization plane. The displacement traces the ellipse in the direction from e 1 to e 2 . We present explicit expressions of e 1 , e 2 and the secular equation without computing the Stroh eigenvalues p and the associated eigenvectors a and b . After presenting the expressions for a general anisotropic elastic material, the special cases are studied separately. For monoclinic materials with the symmetry plane at x 3=0, the secular equation and the conjugate radii e 1 , e 2 are identical to that for orthotropic materials when s ′ 16=0 but s ′ 26 need not vanish. For monoclinic materials with the symmetry plane at x 1=0, e 1 is along the x 1-axis while e 2 is on the plane x 1=0. If the symmetry plane is at x 2=0, e 1 is on the plane x 2=0 while e 2 is along the negative x 2-axis. In both cases, e 1 , e 2 are the principal radii of the ellipse. We also present the derivative of a R with respect to x 2, the depth from the free surface, that provides information on (i) whether the conjugate radii of the ellipse increase as x 2 increases and (ii) whether the polarization plane rotates as x 2 increases. New secular equations are obtained for monoclinic materials with the symmetry plane at x 1=0 or x 2=0.

Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H , M , and N . With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P i ( i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.

Modeling of optical waveguide structures with general anisotropy in arbitrary orthogonal coordinate systems

An extended algorithm for the analysis of arbitrary waveguide structures with general anisotropic materials is proposed and substantiated. This algorithm is based on generalized transmission line (GTL) equations which are developed here for numerical algorithms. For the purpose of analysis, a general modal matrix for the discretized transverse electric and magnetic fields is calculated. Furthermore, impedance/admittance transformation formulas are developed with the help of the GTL equations for longitudinal sections. The proposed algorithm is verified by numerical results.

Elastic guided waves in a layered plate with rectangular cross section.

Guided waves in a layered elastic plate of rectangular cross section (finite width and thickness) has been studied in this paper. A semianalytical finite element method in which the deformation of the cross section is modeled by two-dimensional finite elements and analytical representation of propagating waves along the length of the plate has been used. The method is applicable to arbitrary number of layers and general anisotropic material properties of each layer, and is similar to the stiffness method used earlier to study guided waves in a laminated composite plate of infinite width. Numerical results showing the effect of varying the width of the plate on the dispersion of guided waves are presented and are compared with those for an infinite plate. In addition, effect of thin anisotropic coating or interface layers on the guided waves is investigated.

A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials

For a two-dimensional deformation of an anisotropic elastic material, the Lekhnitski formalism applies to elastostatics only. It does not apply to steady state motion such as surface waves or a moving line dislocation. The Stroh formalism applies to elastostatics as well as steady state motion. However, it does not apply to an incompressible material because some of the elastic stiffnesses become unbounded in the incompressible limit. Modifying a new formalism studied recently, a more explicit and remarkably simple expression of the matrices in the new formalism is given, and a unified formalism that is applicable to elastostatics as well as steady state motion of a compressible or incompressible anisotropic elastic material is obtained. As an application, the secular equation is presented for surface waves propagating in a monoclinic material with the symmetry plane at x 3=0 that has a more explicit expression than what is available in the literature. For a general anisotropic elastic material, explicit expressions are obtained for the sextic equation for the eigenvalues p and the eigenvectors b and a for any steady state wave speed. The explicit expressions recover results related to one-component surface waves. When the material is incompressible, it is shown that the eigenvalues p remain complex for elastostatics. We also show that, when the material is incompressible, a monoclinic material with the symmetry plane at x 3=0 behaves like that of an orthotropic material.

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.

Micromechanical modeling of smart compositestructures

Effective elastic, actuation, thermal expansion andhygroscopic expansion coefficients for periodic smart compositestructures are derived through the application of asymptotichomogenization models. The actuation coefficients characterizethe intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a controlledmanner. The pertinent mathematical framework is that ofasymptotic homogenization. Differential equations with rapidlyoscillating coefficients which govern the behavior of a generalanisotropic (composite) material with a regular array ofreinforcements and/or actuators are transformed into simplerones that are characterized by some effective coefficients; itis implicit, of course, that the physical problem based onthese effective coefficients should give predictions differingas little as possible from those of the original problem. Thegoverning equations pertaining to a generalized model of asmart structure with non-homogeneous boundary conditions arederived and are shown to differ from those of a correspondingproblem with homogeneous boundary conditions by what amounts toa boundary layer solution. The effective properties aredetermined by means of so-called `unit cell' problems andcalculated for the case of periodic laminates. The use of theseeffective coefficients is illustrated by means of two- and three-dimensional examples.

An Explicit Secular Equation for Surface Waves in an Elastic Material of General Anisotropy

An explicit secular equation for surface waves in an elastic half-space was obtained by Rayleigh in 1885 (Proc. London Math. Soc. 17) for isotropic materials and by Stoneley in 1963 (Geophys. J. Astron. Soc. 8) for orthotropic materials. Since then no explicit secular equation was found for a material more general than orthotropic materials. Recently, Destrade (J. Acoust. Soc. Amer. 109, 2001) employed the first integrals (originally proposed by Mozhaev) to obtain an explicit secular equation for monoclinic materials with the symmetry plane at x 3 = 0, which is the plane of motion. The first integral approach does not work for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. Ting (Proc. R. Soc. 32, 2002) proposed a new approach that not only recovers Destrade's secular equation easily, but also provides an explicit expression of the Stroh eigenvalues p and the eigenvectors a, b without solving the quartic equation for p. These eigenvalues and eigenvectors are needed in the surface wave solution. Employing this new approach, Ting also obtained explicit secular equations for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. This new approach is extended here to derive an explicit secular equation for surface waves in a general anisotropic elastic material. The secular equation for a general anisotropic material is then specialized to monoclinic materials with the symmetry plane at x 1 = 0, x 2 = 0 or x 3 = 0. It does recover the secular equations of Destrade, Stoneley and Rayleigh.

Modeling of smart composites on account of actuation, thermal conductivity and hygroscopic absorption

The asymptotic homogenization models for smart composite materials are derived and effective elastic, actuation, thermal expansion and hygroscopic expansion coefficients for smart structures are obtained. The actuation coefficients characterize the intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a coordinated fashion. Examples of such actuators employed with smart composite material systems are derived from piezoelectric, magnetostrictive, and some other materials. The pertinent mathematical framework is that of asymptotic homogenization. The objective is to transform a general anisotropic composite material with a regular array of reinforcements and/or actuators into a simpler one that is characterized by some effective coefficients; it is implicit, of course, that the physical problem based on these homogenized coefficients should give predictions differing as little as possible from those of the original problem. The effectiveness of the derived models is illustrated by means of two- and three-dimensional examples.

Three‐dimensional time‐harmonic Green's functions for a triclinic full‐space using a symbolic computation system

AbstractTime‐harmonic Green's functions for a triclinic anisotropic full‐space are evaluated through the use of a symbolic computation system.This procedure allows evaluation of the Green's functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions. Copyright © 2001 John Wiley & Sons, Ltd.

The scaled boundary finite-element method – a fundamental solution-less boundary-element method

In this boundary-element method based on finite elements only the boundary is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary and thus no singular integrals must be evaluated and general anisotropic material can be analysed. For an unbounded (semi-infinite or infinite) medium the radiation condition at infinity is satisfied exactly. No discretization of free and fixed boundaries and interfaces between different materials is required. The semi-analytical solution inside the domain leads to an efficient procedure to calculate the stress intensity factors accurately without any discretization in the vicinity of the crack tip. Body loads are included without discretization of the domain. Thus, the scaled boundary finite-element method not only combines the advantages of the finite-element and boundary-element methods but also presents appealing features of its own. After discretizing the boundary with finite elements the governing partial differential equations of linear elastodynamics are transformed to the scaled boundary finite-element equation in displacement, a system of linear second-order ordinary differential equations with the radial coordinate as independent variable, which can be solved analytically. Introducing the definition of the dynamic stiffness, a system of nonlinear first-order ordinary differential equations in dynamic stiffness with the frequency as independent variable is obtained. Besides the displacements in the interior the static-stiffness and mass matrices of a bounded medium and the dynamic-stiffness and unit-impulse response matrices of an unbounded medium are calculated.

Anisotropic Elastic Materials With a Parabolic or Hyperbolic Boundary: A Classical Problem Revisited

When an anisotropic elastic material is under a two-dimensional deformation that has a hole of given geometry Γ subjected to a prescribed boundary condition, the problem can be solved by mapping Γ to a circle of unit radius. It is important that (i) each point on Γ is mapped to the same point for the three Stroh eigenvalues p1,p2,p3 and (ii) the mapping is one-to-one for the region outside Γ. In an earlier paper it was shown that conditions (i) and (ii) are satisfied when Γ is an ellipse. The paper did not address to the case when Γ is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the mappings employed by Lekhnitskii for a parabola and hyperbola, and show that while the mapping for a parabola satisfies conditions (i) and (ii), the mapping for a hyperbola does not satisfy condition (i). Nevertheless, a valid solution can be obtained for the problem with a hyperbolic boundary, although the prescription of the boundary condition is restricted. We generalize Lekhnitskii’s solutions for general anisotropic elastic materials and for more general boundary conditions. Using known identities and new identities presented here, real form expressions are given for the displacement and hoop stress vector at the parabolic and hyperbolic boundary.

Kirchhoff evaluation of scattered elastic wavefields in anisotropic media

Three-dimensional elastic wave scattering is addressed for the case of general anisotropic materials. For traction-free scatterers the elastodynamic Kirchhoff approximation is applied, which is valid at high frequencies and near specular angles. Based on a mathematical formulation including Green’s triadic stress tensor function, asymptotic evaluation yields the displacement vector of the scattered wavefield in a form which is convenient for effective numerical computation. The (point source) far-field approximation applied to Green’s triadic function is valid in noncaustic regions.