The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

Exact and Asymptotic Solutions for the Field of a Point Source in a Transversely Isotropic Medium

Exact and asymptotic solutions are obtained for the acoustic field generated by an isotropic pulsed point source in an infinite transversely isotropic elastic medium. The exact solution for the displacement field is obtained in the form of a double integral over the horizontal slowness and the frequency by using the method of integral transforms. The calculation of the integral over the horizontal slowness by the method of stationary phase reduces the exact solution to an asymptotic solution that is convenient for numerical calculations. Formulas are given for calculating the spreading factors and the wave fronts of quasi-longitudinal qP-waves and quasi-transverse qSV-waves. With the formulas obtained, the displacement field of a point source is investigated for a particular transversely isotropic medium.     

Functional integral formulation of classical wave equations

It is shown in this paper that classical wave equations admit path integral formulations. For this, the evolution of the system is first set-up in terms of a fundamental solution or propagator. We choose this last name because it suggests a connection with functional integrals, which are exploited in this work. A functional integral in terms of non-singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations. One of the advantages of such formulation is that it provides an adequate framework for mesh-free numerical methods. This is demonstrated through a computational implementation that combines a simple second-degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations. Numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique.     

A fresh view of cosmological models describing very early universe: General solution of the dynamical equations

The dynamics of any spherical cosmology with a scalar field (‘scalaron’) coupling to gravity is described by the nonlinear second-order differential equations for two metric functions and the scalaron depending on the ‘time’ parameter. The equations depend on the scalaron potential and on arbitrary gauge function that describes time parameterizations. This dynamical system can be integrated for flat, isotropic models with very special potentials. But, somewhat unexpectedly, replacing the independent variable t by one of the metric functions allows us to completely integrate the general spherical theory in any gauge and with arbitrary potentials. In this approach, inflationary solutions can be easily identified, explicitly derived, and compared to the standard approximate expressions. This approach is also applicable to intrinsically anisotropic models with a massive vector field (‘vecton’) as well as to some non-inflationary models.     

Elastic wave field computation in multilayered nonplanar solid structures: a mesh-free semianalytical approach

Multilayered solid structures made of isotropic, transversely isotropic, or general anisotropic materials are frequently used in aerospace, mechanical, and civil structures. Ultrasonic fields developed in such structures by finite size transducers simulating actual experiments in laboratories or in the field have not been rigorously studied. Several attempts to compute the ultrasonic field inside solid media have been made based on approximate paraxial methods like the classical ray tracing and multi-Gaussian beam models. These approximate methods have several limitations. A new semianalytical method is adopted in this article to model elastic wave field in multilayered solid structures with planar or nonplanar interfaces generated by finite size transducers. A general formulation good for both isotropic and anisotropic solids is presented in this article. A variety of conditions have been incorporated in the formulation including irregularities at the interfaces. The method presented here requires frequency domain displacement and stress Green's functions. Due to the presence of different materials in the problem geometry various elastodynamic Green's functions for different materials are used in the formulation. Expressions of displacement and stress Green's functions for isotropic and anisotropic solids as well as for the fluid media are presented. Computed results are verified by checking the stress and displacement continuity conditions across the interface of two different solids of a bimetal plate and investigating if the results for a corrugated plate with very small corrugation match with the flat plate results.     

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

In this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.     

A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

On the flexural vibration of cylinders under axial loads: Numerical and experimental study

The flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper. Natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the Ritz method with terms of power series in the coordinate directions as approximating functions. The effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory, therefore, geometrically nonlinear strain–displacement relations with second-order terms are considered here. The natural frequencies of free–free, clamped–clamped, and sliding–sliding cylinders subjected to axial loads are calculated using the proposed three-dimensional Ritz approach and are compared with those obtained with the finite element method and the Bernoulli–Euler theory. Different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results. An approach based on the Ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression. The results show that for low compressions the cylinder behaves similarly to a sliding–sliding cylinder, whereas for high compressions the cylinder vibrates as a clamped–clamped one.     

Calculation of low-frequency sound fields in irregular waveguides with strong backscattering

An approach is developed for calculating the sound fields in a non-stratified sea medium with irregularities that are not weak. The method of cross sections for horizontal parts of acoustic modes is used to obtain first-order causal equations that are equivalent to the boundary-value problem. A matrix equation describing the backscattered field of modes is analyzed, and the conditions that determine the weakness of the irregularities of the medium and the validity of the known approximate methods of sound field calculations are considered. The approximation of unidirectional propagation is represented in the form of quadratures. The example of a 2D shallow-water waveguide with a strongly irregular profile of a perfectly rigid bottom is considered to illustrate the advantages of the proposed approach in comparison with the approximate methods for specific low frequencies. The qualitative and quantitative differences that arise because of taking into account the backscattering between the curves of propagation losses corresponding to the exact solution and the conventional approximate methods are discussed.     

Forced responses of solid axially polarized piezoelectric ceramic finite cylinders with internal losses

A method is presented to determine the forced responses of piezoelectric cylinders using weighted sums of only certain exact solutions to the equations of motion and the Gauss electrostatic conditions. One infinite set of solutions is chosen such that each field variable is expressed in terms of Bessel functions that form a complete set in the radial direction. Another infinite set of solutions is chosen such that each field variable is expressed in terms of trigonometric functions that form a complete set in the axial direction. Another solution is used to account for the electric field that can exist even when there is no vibration. The weights are determined by using the orthogonal properties of the functions and are used to satisfy specified, arbitrary, axisymmetric boundary conditions on all the surfaces. Special cases including simultaneous mechanical and electrical excitation of cylinders are presented. All numerical results are in excellent agreement with those obtained using the finite element software ATILA. For example, the five lowest frequencies at which the conductance and susceptance of a stress-free cylinder, of length 10 mm and radius 5 mm, reach a local maximum or minimum differ by less than 0.01% from those computed using ATILA.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

Bianchi Type-V Cosmological Model with Two Interacting Scalar Fields

The open anisotropic cosmological model of the early Universe is considered. Two interacting scalar fields with special form of potential energy are a source of matter fields. Analytic solutions for inflationary and scalaron stages are found. The exact solutions to the corresponding field equations are obtained in quadrature form. The cosmological parameters have been discussed in detail and it is also shown that the solutions tend asymptotically to isotropic Friedmann-Robertson-Walker cosmological model.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Inhomogeneous mixmaster universes: Some exact solutions

Algorithms for generating new exact solutions of the Einstein-Klein-Gordon field equations, which describe inhomogeneous universes with S³ topology of spatial sections, are developed. The known exact vacuum and stiff-fluid solutions with S³ topology are used as an input. The methods developed are further applied to derive inhomogenous generalizations of Bianchi type IX solutions and inhomogeneous S³ Gowdy models with gravitational and scalar waves. It is shown that the new solutions, which are generalizations of the Bianchi type IX models, permit identification of the scalar field with the velocity potential of the stiff irrotational fluid. The latter result is further used to study the growth rate of density perturbations of the isotropic and anisotropic Bianchi type IX universes in a fully nonlinear relativistic regime. The role of anisotropy of the rate of growth of density perturbations is studied in detail.     

Equations of equilibrium for a strongly heterogeneous shallow shell

A linear set of equations is proposed for a strongly thickness-heterogeneous (in particular, multilayer) shallow shell. The model unifies the equations of the Mushtary?Donnell?Vlasov technical-theory and the Timoshenko?Reissner equations, which take into account transverse shear. The thickness-heterogeneous shell is replaced with an equivalent homogeneous transversally isotropic shell, the elasticity modula of which are chosen just as the previously determined elasticity modula for heterogeneous plates. In the test example for a multilayered cylindrical shell, the approximate solution according to the proposed model is compared with the exact solution of the three-dimensional problem. The model gives good results in accuracy for a reasonably wide level of inhomogeneity.     

Ideal fluid with short-range scalar interactions in the field of a plane gravitational wave

An exact solution of the self-consistent equations of relativistic hydrodynamics and the scalar field equation is obtained. The solution describes motion of a fluid with short-range scalar interactions in the field of a plane gravitational wave.