Explicit expressions of S(v), H(v) and L(v) for anisotropic elastic materials

The three matrices L(v), S(v) and H(v), appearing frequently in the investigations of the two-dimensional steady state motions of elastic solids, are expressed explicitly in terms of the elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with a plane of symmetry at x₃ = 0, x₁ = 0, and x₂ = 0 are all deduced. Results for orthotropic materials appearing in the literature may be recovered from the present explicit expressions.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(), H(), L() appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x ₃=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product –AB ^-1, whose real and imaginary parts are SL ^-1 and L ^-1, respectively. The tensors S, H, L are then determined from SL ^-1 and L ^-1. For S(), H(), L() we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x ₃=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.     

Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u ₃ and the inplane displacements u ₁, u ₂ are uncoupled but the antiplane stresses σ₃₁, σ₃₂ and the inplane stresses σ₁₁, σ₁₂, σ₂₂ remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x ₃ = 0, we show that the unnormalized Stroh eigenvectors a _k for k = 1, 2, 3 are all real. The matrix A =[a ₁, a ₂, a ₃] is a unit matrix when the material has a symmetry plane at x ₂ = 0. Thus any one of the u ₁, u ₂, u ₃ can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u ₁, u ₂ or u ₃. One can also have a two-displacement field polarized on a plane other than the (x ₁, x ₂)-plane. The material that uncouples u ₁, u ₂, u ₃ is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the generalized Barnett–Lothe tensors for monoclinic piezoelectric materials

The three generalized Barnett–Lothe tensors L, S and H, appearing frequently in the investigations of the two-dimensional deformations of anisotropic piezoelectric materials, may be expressed in terms of the material constants. In this paper, the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x₃ = 0 are constructed based on the extended Stroh formalism. Then the three generalized Barnett–Lothe tensors are calculated from these eigenvectors and are expressed explicitly in terms of the elastic stiffness instead of the reduced elastic compliance. The special case of transversely isotropic piezoelectric materials is also presented.     

Supersonic responses induced by point load moving steadily on an anisotropic half-plane

Supersonic responses of an anisotropic half-plane solid induced by a point load moving steadily on the half-plane boundary are investigated. Analytic expressions for the responses of the displacements and stresses for field points either inside or on the surface of the half-plane solid are given for general anisotropic materials. For the special cases of monoclinic materials with symmetry plane at $x_3=0$ and orthotropic materials, the supersonic as well as subsonic responses of the displacements and stresses are further expressed explicitly in terms of elastic stiffnesses. Responses for the case of isotropic materials known in the literature are recoverable from present results.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

Electrostatic stress analysis of an anisotropic piezoelectric half-plane under surface electromechanical loading

The generalized stress components on an anisotropic piezoelectric half-plane boundary under surface electromechanical loading are investigated. It is found that the behaviors of generalized stress components are related to matrices Γ and Ω, which have the same form as those for the purely elastostatic problem. Matrices Γ and Ω contain all the electro-mechanical coupling phenomena of the generalized stress components. All elements of matrices Γ and Ω are expressed explicitly in terms of generalized elastic stiffness for monoclinic piezoelectric materials with the plane of symmetry at x₃ = 0 and for transversely isotropic piezoelectric materials in which the coupled effects between the mechanical (electrical) deformations induced by electrical (mechanical) loadings are studied analytically. A numerical example of the electro-mechanical coupling behavior for PZT-4 is also given.     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

The Remarkable Nature of Radially Symmetric Deformation of Spherically Uniform Linear Anisotropic Elastic Solids

Antman and Negron-Marrero [1] have shown the remarkable nature of a sphere of nonlinear elastic material subjected to a uniform pressure at the surface of the sphere. When the applied pressure exceeds a critical value the stress at the center r=0 of the sphere is infinite. Instead of nonlinear elastic material, we consider in this paper a spherically uniform linear anisotropic elastic material. It means that the stress-strain law referred to a spherical coordinate system is the same for any material point. We show that the same remarkable nature appears here. What distinguishes the present case from that considered in [1] is that the existence of the infinite stress at r=0 is independent of the magnitude of the applied traction σ0 at the surface of the sphere. It depends only on one nondimensional material parameter κ. For a certain range of κ a cavitation (if σ0>0) or a blackhole (if σ0<0) occurs at the center of the sphere. What is more remarkable is that, even though the deformation is radially symmetric, the material at any point need not be transversely isotropic with the radial direction being the axis of symmetry as assumed in [1]. We show that the material can be triclinic, i.e., it need not possess a plane of material symmetry. Triclinic materials that have as few as two independent elastic constants are presented. Also presented are conditions for the materials that are capable of a radially symmetric deformation to possess one or more symmetry planes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

We investigate the equations of anisotropic incompressible viscous fluids in , rotating around an inhomogeneous vector B(t, x ₁, x ₂). We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniform local existence result with respect to the Rossby number in the same functional spaces under the additional assumption that B = B(t, x ₁) or B = B(t, x ₂). We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.     

On Saint-Venant's principle in linear elastodynamics

We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x ₃ >ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x ₃ , an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaKazaaiacaGGOaGccaaIXaGaaeiiaiabgkHiTiaabccadaWcaaqa% amXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hha4naaBaaaleaacaqGZa% aabeaaaOqaaiaabogacaqG0baaaKazaakacaGGPaaaaa!4BB0!\[(1{\text{ }} - {\text{ }}\frac{{x_{\text{3}} }}{{{\text{ct}}}})\]. Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragmén-Lindelöf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end.     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Rectilinear dislocation in an anisotropic plate

A solution has been found to the problem of calculating the stress and displacement fields caused by a rectilinear dislocation in an anisotropic elastic plate. Special cases of anisotropy have been found with solutions represented by elementary functions.Certain problems in describing crystal plastic deformation phenomena make it vital to know the fields of the elastic stresses and displacements caused by an individual dislocation in a bounded crystal. It is interesting to study the effect of crystal boundaries on these fields with a simple model which approximates fairly closely to experimental conditions.The model selected is shown in Fig, 1. A dislocation with a Burgers vector (b₁, b₂, b₃) is situated in an infinite elastic anisotropic plate of thickness 2h. The dislocation line is parallel to the plate boundaries. The following restriction is introduced in relation to the plate's elastic properties: the medium has a plane of elastic symmetry perpendicular to the dislocation line. The selection of the coordinate system and position of the dislocation are shown in Fig. 1. The requirement is to find the stresses and displacements at an arbitrary point in the plate.One limited special form of this problem has been solved by Kroupa [1]. The limitations which he introduced are as follows: the medium is isotropic, the dislocation is at the precise center of the band and the Burgers vector has only one component b₂ differing from zero (the same coordinates were chosen in [1] as in Fig. 1).Thus Kroupa's results can be obtained from the results of the present work as a special case. Other special cases arising from this problem are those concerning the elastic stress and displacement fields caused by a dislocation in anisotropic semi-bounded [2] and bounded [3] media.It is immediately apparent that the problem is a plane one, in the sense that the fields to be found do not depend on coordinate z. Since the medium has a plane of elastic symmetry perpendicular to the dislocation line, it is clear from [4] that the system of stresses and strains in such a medium can be divided into two independent subsystems. The first of these is plane deformation with stress components _xx, _yy and _xy differing from zero and displacement vector components u_x and u_y, the second is antiplane deformation with stress components _xz and _yz differing from zero and the displacement vector component u_z.In the case under examination, the plane deformation is caused by the Burgers vector edge components b_x and b_y and the antiplane deformation by the screw component b_z. The solution is therefore divided into two stages, corresponding to edge and screw dislocations.In conclusion, I wish to thank A. M. Kosevich for his valuable advice and L. A. Pastur for his constant vigilance and assistance in the work.     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

Existence of supersonic zones in three-dimensional separation flows

Up till now the region of three-dimensional separation flows which occur with supersonic flow past obstacles has received insufficient study. Supersonic flow with a Mach number of 2.5 past a cylinder mounted on a plate was studied in [1]. A local zone with supersonic velocities was found in the reverse subsonic flow region ahead of the cylinder. Its presence is explained by the three-dimensional nature of the flow. Similar supersonic zones are not observed in the case of supersonic flow over plane and axisymmetric steps.The present paper presents the results of experimental studies whose objective was refinement of the flow pattern ahead of a cylinder on a plate and the study of the local supersonic zones.The experiments were performed in a supersonic wind tunnel with a freestream Mach number M₁=3.11. The 24-mm-diameter cylinder with pressure taps along the generating line was mounted perpendicular to the surface of a sharpened plate. The distance from the plate leading edge to the cylinder axis wasl ₀=140 mm. The plate was pressure tapped along the flow symmetry axis. The Reynolds number was Rl ₀=u₀ l ₀/v ₁, Rl ₀=1.87.10⁷, where u₁ andv ₁ are the freestream velocity and the kinematic viscosity, respectively. The pressures were measured using a Pilot probe with internal and external diameters of 0.15 and 0.9 mm, respectively.The probe was displaced in the flow symmetry plane at a distance of 1.6 mm from the plate surface and at a distance of 1.1 mm along the leading generator of the cylinder. The flow on the surface of the plate and cylinder was studied with the aid of a visualization composition and the flow past the model was photographed with a schlieren instrument. Typical patterns of the visualization composition distribution and the pressure distribution curves over the plate surface, and also photographs of the flow past the model, are shown in [1].     

Wind-driven nonlinear oscillations of cables

The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity.Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.Notation A=diag(a _i ,i=1, 2, 3) amplitude matrix (diagonal) - A _n,A undeformed area, deformed area - B span of hanging cables - D sag for static conditions - E Young's modulus - vector of external force - diagonal matrix - symmetric coefficient matrix - H ^* =HR I unit matrix - diagonal matrix - L original length of cables before hanging - M the symmetric stiffness matrix - N integer - P damping constant matrix (diagonal) - R linear mode shape matrix (diagonal) - S sway of hanging cables - T tension of cables - T _o tension of cables for static conditions - T _o(0) tension of the lowest point for static conditions - V eigenfunction matrix - b=y ^T R coefficient vector - b - c,c ₁,c ₂,c ₃ vector, and the components in thex ₁,x ₂,x ₃ directions respectively, in terms of cosine functions. - e, e _o strain, and static strain of elongation - e ₁ time-dependent perturbation ine - f wind force in the sway direction - f, f ₀,f ₁ vector of external force - g gravity constant - h time-dependent amplitude vector - m mass density per unit length of the undeformed cable - r=(R ₁,R ₂,R ₃) ^T vector of modal shapes - s undeformed arc length - t time - u ₁ linear scalar in z - u ₂ quadratic scalar in z - v ₁,v ₂,v ₃ eigenfunctions inx ₁,x ₂, andx ₃ directions, respectively - x=(x ₁,x ₂,x ₃) ^T Cartesian position vector and components - y=(y ₁,y ₂,y ₃) ^T static position vector and components - error vector - matrix operator - =diag[₁, ₂, ₃] internal frequency matrix and components - excitation frequency - global matrix of coordinate functions - T _o(0)/mgL - mgL/EA _o - yy ^T - s/L - = diag[₁, ₂, ₃] phase angle matrix and components of characteristic modes - phase angle of excitation force - ₁, ₂ time-dependent amplitude vectors in timet _o and timet ₁ - _ij,i=1, 2...N,j=1, 2, 3 theith coordinate function of thejth component - _i = diag[_i1, _i2, _i3] theith matrix of coordinate functions - global vector of modal amplitudes - ₁ external detuning parameter - _i,i=2, 3 internal detuning parameter - _i,i=1, 2, 3 phase angles