Improving the differentiability of generalized fourier series solutions of boundary value problems of mechanics by using boundary functions

We prove a theorem on conditions for the differentiation of generalized Fourier series. We show that Fourier series solutions of boundary value problems can in general be differentiated term by term only once. To improve the differentiability properties of such series, we suggest to use pth-order boundary functions. We suggest an algorithm for constructing boundary functions for classical domains. This approach is illustrated by a new solution, with improved differentiability properties, of the problem on the torsion of an elastic rod of rectangular cross-section.     

Problem of the impact interaction of an elastic rod with a Uflyand-Mindlin plate

Conclusions A problem on the impact of an elastic rod of circular cross-section onto an Uflyand-Mindlin isotropic elastic plate is considered. The radial method, which allows construction of an approximate solution behind fronts of discontinuity surfaces up to the boundary of the contact region by means of segments of power series with variable coefficients, is used as a method of solution. It is, however, established that segments of radial series are not uniformly suitable throughout the region of existence of wave motion. A method of multiple scales is used for their regularization. The method, based on asymtotics of the radial series, permits construction of a uniformly suitable expansion containing a smaller number of terms than segments of radial series, but equal to them in accuracy.Voronezh Structural Engineering Institute, Russia. Translated from Prikladnaya Mekhanika, Vol. 29, No. 2, pp. 39–46, February, 1993.     

On the solution of boundary value problems of mechanics for the Poisson equation and related equations

To a boundary value problem, we assign an auxiliary problem of determining the spectrum of eigenfunctions and eigenvalues (EFV). After the main problem has been reduced to a form with homogeneous boundary conditions, it becomes possible to prove theorems about the formulas for the solution of the boundary value problem with linear equations of elliptic type for multidimensional multiply connected domains by using the spectral expansion in the Fourier series. We find conditions under which the action of second-order differential operators on the obtained solutions in the Fourier series can be computed not only in the interior of the domain but also on its boundary. But if these conditions are not satisfied, then the series for second-order differential operators do not converge on the boundary. The proposed method for the expansion in the EFV can be used not only in plane but also in spatial problems if the domain of complicated shape can be represented as a combination of bounded domains with known EFV spectra. As one of the examples, we consider the problem of torsion of an elastic rod whose cross-section consists of a rectangle and a half-disk.     

Generalized thermoelastic functionally graded solid with a periodically varying heat source

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity without energy dissipation (TEWOED). The governing equations of generalized thermoelasticity without energy dissipation (GN model type II) for a functionally graded materials (FGM) (i.e. material with spatially varying material properties)are established. The governing equations are expressed in Laplace–Fourier double transform domain and solved in that domain. Now, the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand is obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking nonhomogeneity parameter suitably. Finally the results obtained are presented graphically to show the effect of nonhomogeneity on displacement, temperature, stress and strain.     

Stress concentration due to an array or hemispherical cavities at the surface of an elastic half-space

The paper investigates the perturbation in an otherwise uniform stress field in an elastic half-space due to a doubly-periodic array of small hemispherical holes at the free surface. The solution is obtained using three potential functions of double Fourier series form in Galerkin's strain potential solution, the coefficients of which are determined using the collocation method. The unperturbed field is taken to be one of uniform plane stress parallel to the free surface. Two special cases are studied—uniform tension and uniform shear stress. Numerical results for these cases can be generalized by superposition to give solutions for a general state of biaxial plane stress. It is found that, for both tension and shear, the maximum stress concentration occurs at the bottom of the holes. The stress concentration factor increases with the ratio of hole spacing to radius, approaching the known solution for a single hemispherical hole at large ratios.     

Scattering of plane SH waves by an isosceles trapezoidal hill

This paper presents a theoretical approach to study the surface motion of an isosceles trapezoidal hill impacted by incident SH waves. A rigorous solution has been derived by applying an accurate region-matching technique. The solution region is divided into three parts by an appropriate partitioning method. Based on complex function method and multipolar coordinates, a fractional factor is introduced to construct suitable wave functions which satisfy the governing equation and zero-stress condition on the free surface in each sub-region. According to the continuity condition at the auxiliary boundary, surface displacements are expressed in series of infinite algebraic equations, and the unknown coefficients of the series can be determined by Fourier series expansion technique in complex domain. Numerical results demonstrate the analytical results depend on the following parameters: The slope, the height and the width of the trapezoidal hill, the frequency content of the excitation and the incidence angle.     

变截面梁动力响应的频域传递矩阵计算方法

为提高变截面梁振动分析的计算效率,提出了基于频域传递矩阵法的动力计算算法．首先,选择线速度、角速度、弯矩和剪力作为求解变量,通过Laplace变换将变截面梁的动力响应时域偏微分方程转换为频域常微分方程;然后,通过求解频域方程并结合协调和边界条件建立变截面梁的频域传递矩阵;通过构造傅里叶级数展开形式的时域响应函数,对变截面梁传递矩阵方法求解的频响函数进行Laplace逆变换,建立了变截面梁的固有特性计算和时域瞬态响应计算方法,最后,借助数值仿真软件,开发了变截面梁动力响应分析的计算程序．完成对算例的仿真计算和分析,并与有限元计算结果进行对比,数值结果验证了该方法的正确性和有效性．     

An exact and asymptotic analysis of a diffraction problem

The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being ‘integer +1/2’, is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.     

Impact of an elastic rod on a deformable barrier: analytical and numerical investigations on models of a valve and a rod-shaped stamping tool

The longitudinal motion of an elastic rod is studied for the case that the rod is suddenly elastically fixed at one end and is hit by a mass at its other end. This configuration represents real settings e.g. as a valve impacts an elastic valve-seat or as a stamping device used in forging is hit by a large mass. The solution of the problem is formulated in the Laplace transformation space. The inverse transformation into the time domain is performed by engaging the so-called Laguerre polynomial technique. This method allows to calculate exact solutions for finite times from a finite number of series elements. Rigorous mathematical proofs not established up to now are given with respect to the convergence of the series encountered and the validity of exchanging the order of inversion of the Laplace transformation and summation of the established series. For comparison also a numerical solution of the problem is presented. An analysis of the energy transfer between rod, impacting mass and elastic barrier elucidates the marked influence of the deformability of the elastic barrier on the stress state in the rod.     

Extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry

An extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading. The plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials. Both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior. In this extended meshfree method, a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it. A particular solution in the form of an analytic generalized Navier solution (a compound double Fourier series) is used to capture the response due to the loading within the rectangular domain. Then, a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate. These augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate’s contour.Concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized Navier solution representation. When a meshfree portion is added to account for the boundary conditions, such solutions constitute the Green’s functions for the plate. The viability of these double Fourier series representations is shown by the convergence rates for the kinematic and force/moment fields. An additional example of a two layer ±30° angleply circular plate is given to illustrate the capability of this extended meshfree method.     

An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials

This paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber-reinforced composites and perforated materials. The material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions. The rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated Fourier series. The analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method. The boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation. All of the integrals arising in the method can be evaluated analytically. As a result, the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions. Several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials.     

SH波绕界面孔的散射

用波函数展开方法研究了SH波绕界面孔的散射问题。由入射、反射和透射波组成的自由波场与孔的散射场叠加成总波场。按照一定方式将两个半平面散射波场延拓于全平面，通过Hankel-Fourier展开方法求得了任意形状孔散射场的级数解。以椭圆形孔为例计算了孔边缘的动应力集中系数。     

Stresses in a transversely isotropic,short hollow cylinder subjected to an outer band load

Summary An axi-symmetric stress analysis of a transversely isotropic, short hollow cylinder subjected to an outer band load is presented in a series form. The generalized Elliott's solution is used for the analysis. The solution consists of five independent potential functions which yield two kinds of elasticity solution. The boundary conditions for the shearing stress on the four surfaces are exactly satisfied. Other boundary conditions are numerically satisfied with the aid of a Fourier series expansion or a Fourier-Bessel series expansion. Numerical results for stresses in magnesium and cadmium crystals, as examples of transversely isotropic materials, and in an isotropic material are illustrated. The effect of anisotropy on the stresses is investigated by comparison with the stresses in the isotropic material.

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Spannungen in einem transversal-isotropen, kurzen Hohlzylinder unter einer äußeren Bandlast

Übersicht Vorgestellt wird eine axialsymmetrische Spannungsanalyse mittels Reihen für einen transversalisotropen, kurzen Hohlzylinder unter einer äußeren Bandlast. Für die Analyse wird die verallgemeinerte Elliottsche Lösung benutzt. Die Lösung besteht aus fünf unabhängigen Potentialen, welche zwei Arten der Elastizitätslösung abgeben. Die Randbedingungen der Schubspannung auf den vier Flächen werden genau erfüllt. Die anderen Randbedingungen werden durch eine Fouriersche Reihenentwicklung oder eine Fourier-Besselsche Reihenentwicklung numerisch erfüllt. Die numerischen Resultate der Spannungen in einem Magnesiumkristall und einem Kadmiumkristall als Beispiele transversal-isotroper Materialien und in einem isotropen Material werden illustriert. Der Einfluß der Anisotropie auf die Spannungen wird durch Vergleich mit den Spannungen im isotropen Material untersucht.

     

A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.     

Time-dependent motion of a two-dimensional floating elastic plate

Three methods are presented to determine the motion of a two-dimensional finite elastic plate floating on the water surface, which is released from rest and allowed to evolve freely. The first method is based on a generalized eigenfunction expansion and it is valid for all water depths. The second method is based on an integral equation derived from the Fourier transform, and it is valid for all water depths, although computations are made only for water of infinite depth. These two methods are both based on the frequency-domain solution—however no obvious connection exists between the two methods. The third method is valid only for shallow water, and it expresses the solution as the sum over decaying modes. We present a new derivation of the integral equation for a floating plate based on the Fourier transform of the equations of motion in the time domain. The solution obtained by each method is compared in the appropriate regime, and excellent agreement is found, thereby providing benchmark solutions. We also investigate the regime of validity of the infinite and shallow-depth solutions, and show that both give good results for a quite wide range of depths.     

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Interaction Between an Oscillating Sphere and a Thin Elastic Cylindrical Shell Filled with a Compressible Liquid. Internal Axisymmetric Problem

The problem on the interaction between a spherical body that oscillates in a prescribed manner and a thin elastic cylindrical shell filled with an ideal compressible liquid is formulated. It is assumed that the geometrical center of the sphere is located on the cylinder axis. The problem is solved based on the possibility of representing a partial solution of the Helmholtz equation written in cylindrical coordinates in terms of partial solutions in spherical coordinates, and vice versa. By satisfying the boundary conditions on the surfaces of the sphere and the shell, we obtain an infinite system of linear algebraic equations to determine the coefficients of expansion of the liquid-velocity potential into a Fourier series in terms of Legendre polynomials. The hydrodynamic characteristics of the liquid filling the cylindrical shell are determined and compared with the cases where a sphere oscillates in an infinite liquid and in a rigid cylindrical vessel     

扇形截面杆扭转的应力分析

柱体扭转的基本方程为非齐次偏微分方程,在极坐标系下,利用分离变量法及傅立叶级数展开法,求出了扭转应力函数,进一步即可计算出扇形截面杆在外力偶作用下,扭转角和横截面上剪应力的精确解答。这种方法为精确解法,在各种机械及其他工程设备中,对受扭转作用的扇形截面杆设计,有一定实用价值。     

Acceleration wave propagation in hyperelastic rods of variable cross-section

It is shown that when an acceleration wave propagates in hyperelastic rod with slowly varying cross-section, the transport equation for the wave intensity is a generalized Riccati equation. The three coefficients in the equation all depend on the material properties, but only the coefficient of the quadratic term is independent of the effect of area change. Three theorems are proved, based on the use of comparison equations, which establish that in general the acceleration wave intensity will become infinite (escape) after the wave has propagated only a finite distance along the rod. The existence of thresholds for the initial intensity are also established in certain cases, with their most notable property being that as the intial intensity decreases towards the threshold, so the distance the wave propagates to escape increases without bound.     

Wave scattering by a floating porous elastic plate of arbitrary shape: A semi-analytical study

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by a porous elastic plate with arbitrary shape floating in water of finite depth. The water domain is divided into the interior and exterior regions, corresponding to the domain beneath the plate and the rest extending towards infinite distance horizontally, respectively. The unknown coefficients in the potential expressions are determined by satisfying the continuity conditions for pressure and velocity at the interface of the two regions, together with the conditions for the motion/force at the edge of the plate, where the Fourier series expansion method is employed to deal with the terms associated with the radius function. A plate with three shapes – circular, cosine and elliptical – and three edge conditions are considered. We find that the largest deflection of the plate with a simply supported edge and a clamped edge is more sensitive to the change in porosity when the porosity is small. The power dissipated by an elliptical plate with its major axis perpendicular to the incident wave direction is the largest among the case studies for the majority of the porosity values tested.