SECOND ORDER EFFECTS IN AN ELASTIC HALF－SPACE ACTED UPON BY A NON－UNIFORM NORMAL LOAD

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The general solution of second order effects in elastic half-space acted upon by a non-uniform shear load

I.IntroductionInfiniteelasticitytheorythemathematicalequationsgoverningthedeformationofanisotropiccompressibleelasticmaterialarehighlynonlinear.Asaresult,theexactsolutionsoftheboundaryvalueproblemshavebeenpossibleonlyinsomerestrictedcasesand,oftenapproxim…     

The illustration calculations of second order effects in elastic half-space acted upon by a non-uniform shear load

I.IllustrationsThegeneralsolutionofsecondordereffectsinelastichalfspaceacteduponbyanon-uniformshearloadpresentedbypaperIl]areapplicableforallvalueof6>-l,butareverycomplicated.Asani1lustrationofthemethodwegivebelowthelinearandsecondordersolutionsfor8=l/2.T…     

Second order elastic effects in a homogeneous isotropic compressible infinite medium with an elliptic hole and acted upon by a tensile force at infinity

Analytic expressions for the second order elasticity case of the problem of an isotropic, infinite medium of compressible material having an elliptic hole and acted upon by a tensile force at infinity are obtained. These results are improvements on the classical elasticity theory in which the elastic constants do not play any role for determination of the stresses. The expressions for the second order effects in the cases of a circular hole and a slit along the major axis have been obtained as particular cases.     

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.     

On a 3D moving load problem for an elastic half space

The paper deals with 3D dynamic response of an elastic half-space loaded by a point force moving at a constant speed along a straight line on the surface. The problem is formulated within the framework of the asymptotic hyperbolic–elliptic model developed earlier by two of the authors. The validity of the model is restricted to the range of speeds close to the Rayleigh wave speed. Steady-state near-field solutions are derived in terms of elementary functions. Transient analysis of surface motion illustrates peculiarities of the resonance associated with the Rayleigh wave.     

Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load

Complex potentials in common form for compressible and incompressible elastic bodies are used to formulate and solve the problem of stationary motion of a prestressed two-layer elastic half-space under a moving surface load. The results presented are similar to those obtained earlier using the Fourier transform __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 3–15, May 2008.     

The influence of the third order elastic constants on the dynamical interface stress field in a half-space covered with a pre-stretched layer

The influence of the third order elastic constants on the dynamical (time-harmonic) axisymmetric interface stress field in the system which comprises the half-space and the pre-stretched covering layer is investigated within the framework of the piecewise homogeneous body model by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies. The elasticity relations for the layer and half-space materials are given through the Murnaghan potential. It is assumed that the force acting on the free face plane of the covering layer is a time-harmonic point-located normal force. The influences due to both of the quantities of the pre-stretching of the layer and the third order elastic constants of the layer material on the interface normal stress are analysed. The numerical results are presented for concrete selected materials such as steel, aluminium and acrylic plastic.     

Plane waves in a fractional order micropolar magneto-thermoelastic half-space

This paper deals with the problem of magneto-thermoelastic interactions in an unbounded, perfectly conducting half-space whose surface suffers a time harmonic thermal source in the context of micropolar generalized thermoelasticity with fractional heat transfer allowing the second sound effects. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The Laplace–Fourier double transform technique has been used to solve the resulting non-dimensional coupled field equations. Expressions for displacements, stresses and temperature in the physical domain are obtained using a numerical inversion technique. The effects of fractional parameter, magnetic field and micropolarity on the physical fields are noticed and depicted graphically. For a particular model, these fields are found to be significantly affected by the above mentioned parameters. Some particular cases of interest have been deduced from the present problem. Numerical results predict finite speed of propagation for thermoelastic waves.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

GENERAL SECOND ORDER FLUID FLOW IN A PIPE

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非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Incremental elastic motions superimposed on a finite deformation in the presence of an electromagnetic field

The equations describing the interaction of an electromagnetic sensitive elastic solid with electric and magnetic fields under finite deformations are summarized, both for time-independent deformations and, in the non-relativistic approximation, time-dependent motions. The equations are given in both Eulerian and Lagrangian form, and the latter are then used to derive the equations governing incremental motions and electromagnetic fields superimposed on a configuration with a known static finite deformation and time-independent electromagnetic field. As a first application the equations are specialized to the quasimagnetostatic approximation and in this context the general equations governing time-harmonic plane-wave disturbances of an initial static configuration are derived. For a prototype model of an incompressible isotropic magnetoelastic solid a specific formula for the acoustic shear wave speed is obtained, which allows results for different relative orientations of the underlying magnetic field and the direction of wave propagation to be compared. The general equations are then used to examine two-dimensional motions, and further expressions for the wave speed are obtained for a general incompressible isotropic magnetoelastic solid.     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

Steady-state vibrations of an elastic beam on a visco-elastic layer under moving load

Summary  The steady-state response of an elastic beam on a visco-elastic layer to a uniformly moving constant load is investigated. As a method of investigation the concept of “equivalent stiffness” of the layer is used. According to this concept, the layer is replaced by a 1D continuous foundation with a complex stiffness, which depends on the frequency and the wave number of the bending waves in the beam. This stiffness is analyzed as a function of the phase velocity of the waves. It is shown that the real part of the stiffness decreases severely as the phase velocity tends to a critical value, a value determined by the lowest dispersion branch of the layer. As the phase velocity exceeds the critical value, the imaginary part of the equivalent stiffness grows substantially. The dispersion relation for bending waves in the beam is studied to analyze the effect of the layer depth on the critical (resonance) velocity of the load. It is shown that the critical velocity is in the order of the Rayleigh wave velocity. The smaller the layer depth, the higher the critical velocity. The effect of viscosity in the layer on the resonance vibrations is studied. It is shown that the deeper the layer, the smaller this effect. Received 22 March 1999; accepted 26 July 1999     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Deformation produced by a simple tensile load in an isotropic elastic body

It is shown that a simple tensile load produces a simple extension provided the empirical inequalities (Truesdell and Noll [1], eqn. 51.27) hold.