Analysis of the wave solution of the elastokinetic equations of a Cosserat continuum for the case of bulk plane waves

A study is made of waves in a Cosserat continuum, whose strain state is characterized by independent displacement and rotation vectors. The propagation of longitudinal and transverse bulk waves is considered. Wave solutions are sought in the form of wave trains specified by a Fourier spectrum of arbitrary shape. It is shown that if the solution is sought in the form of three components of the displacement vector and three components of the rotation vector which depend on time and the longitudinal coordinate, the initial system is split into two systems, one of which describes longitudinal waves, and the other transverse waves. For waves of both types, dispersion relations and analytical solutions in displacement are obtained. The dispersion characteristics of the solutions obtained differ from the dispersion characteristics of the corresponding classical elastic solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 196–203, March–April, 2008.     

环形薄板二维驻波的研究

在理论上和实验上对环形薄板二维驻波波节图形(克拉尼图形) 进行了研究. 通过在极坐标下对垂直板面方向小振动方程进行分离变量, 求解出环形薄板小振动方程在外边界悬空时分别在两种内边界条件, 即内边界悬空和内边界简支下的解析解的简正模式, 并计算了在第一种边条件下几种共振模式的径向波速近似值, 以及两种边条件下的圆形驻波波节线的半径和薄板的弹性模量. 发现通过调节环形薄板上点振动源的频率, 可精确控制薄板上出现的克拉尼图形. 实验上观察到了仅有圆形波节线, 仅有辐射状波节线, 以及两种波节线同时存在3 种简正模式的情形, 且波节线的数量可严格控制. 理论结果跟实验符合得很好.      

模态分析与动态子结构方法新进展

综述在模态分析与动态子结构方法研究的一些最新进展.首先回顾经典的位移展开定理和模态叠加原理.为了加速经典方法的收敛速度、提高计算效率, 进一步介绍两个新的结构位移展开定理(采用固定界面模态的位移展开新定理, 给出采用低阶固定界面模态的高精度位移展开式;采用混合模态的位移展开新定理, 给出采用低阶混合模态表示的高精度位移展开式)和相应的动力学新解法.相应上述3个位移展开定理, 介绍采用解析推导的方法构造出3类动态精确子结构方法, 各种子结构模态综合法实质上都是它们的某种近似与变化形式, 从而形成系统的动态子结构分析技术.上述介绍的模态分析与动态子结构方法新进展与经典模态分析技术一起形成结构动力学分析技术的系统理论.      

薄膜涂层材料中的广义瑞利波

基于弹性动力学的线性理论，建立了涂层材料中广义瑞利波传播的理论分析模型，并 且由波动方程和边界条件推导了波的频散方程．分析了慢层和快层对相速度频散的影响，给 出了不同层厚-波长比和不同涂层-基体密度比情况下广义瑞利波相速度的理论解．算例分 析分别比较了慢层和快层结构中波的相速度、群速度，以及随深度衰减的位移与应力振 幅．另外，相速度曲线和位移振幅曲线与文献中给出的结果吻合，验证了理论模型和分析过 程的正确性．     

Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics

Anti-plane and plane-strain, time-harmonic, small-amplitude vibrations of an elastic layer on an elastic half space are considered, superimposed upon a state of finite, uniform stress and strain. A (compressible) elastic material is considered, orthotropic with orthotropy axes aligned parallel and orthogonal both to the layer and the prestress principal directions. A non-uniform mass density is assumed in the layer. A formal long-wave asymptotic solution is derived under the assumptions of high contrast between the stiffnesses of the layer and the half space and between certain prestress components and the current elastic shear modulus.It is shown that (i) the layer asymptotically behaves as a beam subject to transversal and axial vibrations; (ii) the response of the half space can be found in a closed-form, under the assumption of plane wave motion (which becomes consistent when the density of the layer is uniform), otherwise it is represented by a hypersingular integral equation; (iii) if the nonlocality introduced by the hypersingular integral equation is restricted to an influence area of finite extent, the integral can be analytically approximated, so that a Winkler-type spring model representing the half space is rigorously derived. For uniform density of the layer, the constants defining the spring model are given as functions of the prestress and anisotropy parameters of the half space; and, finally, (iv) the asymptotic solution provides new analytical expressions for incremental displacement of the layer, which, compared to the exact numerical solution (also included), are shown to perform quite well, even for values of parameters much beyond the limits imposed by the asymptotic analysis.The asymptotic analysis allows us to explore, for the first time, dynamic properties of a periodic layer bonded to an elastic half space and subject to a uniform prestress state. We find that the system exhibits band gaps (ranges of forbidden frequencies) and that the prestress can be used as a parameter tuning the filtering properties of the structure, an effect which may have important consequences in the design of resonant devices.     

Difference between the longitudinal Frenkel-Biot waves in water-and gas-saturated porous media

The problem of the propagation of longitudinal Biot waves in a porous medium saturated with a weakly compressible liquid (water) or a gas is considered theoretically. The frequency dependence of the phase velocities and damping coefficients is investigated numerically. It is shown that for a certain relationship between the parameters of the porous medium and the saturating fluid there is a “critical” frequency at which the properties of longitudinal waves of both kinds are identical. An analytical expression for this “critical” frequency is obtained. It is shown that for a gas-saturated porous medium, at a certain frequency, in both longitudinal waves the relative gas-matrix motion changes type. Assuming that the saturating-gas behavior corresponds to an adiabatic equation of state, an estimate is obtained for the threshold pore pressure necessary for the restructuring of the relative motion. The wave associated with matrix deformation is shown to have a high damping coefficient in a porous medium saturated with a weakly compressible liquid (water in the case considered) but to be only weakly damped in a gas-saturated porous medium.     

u-w方程的有限差分法分析饱和土的两个压缩波

给出了饱和土u-w方程的差分格式和稳定性条件。用此方法计算了两个不同的一维初边值问题：土柱受突加水压作用和土柱受突加位移作用，并用求导的方法解决了位移边界条件的输入。与理论解比较，证明了此方法结果在变化趋势和关键数值上的正确性。并讨论了渗透系数以及流体与土骨架压缩模量比对两类压缩波的影响。     

Solitary waves in initially stressed thin elastic tubes

In the present work, we study the propagation of non-linear waves in an initially stressed thin elastic tube filled with an inviscid fluid. Considering the physiological conditions of the arteries, in the analysis, the tube is assumed to be subjected to a uniform inner pressure P₀ and an axial stretch ratio λ_z. It is assumed that due to blood flow, a finite dynamical displacement field is superimposed on this static field and, then, the non-linear governing equations of the elastic tube are obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the longwave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. It is observed that the present model equations give two solitary wave solutions. The results are also discussed for some elastic materials existing in the literature.     

Minimum energy consumption of soil working cyclic processes

Analysis of some experimental studies concerning the processes of soil penetration and cutting indicates that the loading stages of both these processes as well as their unloading stages are respectively analogous. This means that the relationships between the frontal resistance forces of soil and tool displacement for penetration and cutting can be characterized by an identical mathematical model of soil.An analytical investigation of energy consumption for a generalized cyclic soil working process is carried out. This investigation shows that the minimum energy consumption can be reached if, on the loading stage, the tool displacement is equal to a so-called optimum displacement per cycle, which can be calculated according to the formulas derived in this analysis.The necessary and sufficient conditions of existence of optimum displacements per cycle are determined. These conditions are expressed by a definite correlation between some mechanical characteristics of soil.It is shown that the energy consumption of a corresponding quasistatic soil working process considerably exceeds the minimum energy consumption of a cyclic process.     

Wave motion in a thick cylindrical rod undergoing longitudinal impact

The paper presents a new formulation and the comprehensive analytical solution to longitudinal impact of thick elastic rods. In contrast to previously published works, the solution is derived based on the exact three-dimensional theory without using the classic Skalak’s decomposition. This new direct approach makes the analytical solution more transparent and much easier to obtain. The resulting formulas for basic mechanical quantities are derived using the residue theorem and their evaluation is made in such a way that the accuracy of presented results is significantly higher than those previously published. Based on these results, the transient wave phenomena occurring in the rods are discussed in detail. Additionally, the solution in time domain is obtained also by semi-analytical approach making use of numerical inverse Laplace transform. It is shown that the selected FFT based algorithm is accurate and robust enough, such that the analysis of wave motion in spatial and time domain can be done effectively preserving the results precision. Presented solution can be used as a benchmark for verification of numerical and experimental methods applied to elastodynamics problems.     

Wave reflection by a submerged cycloidal breakwater in presence of a beach with different depth profiles

In this work, formulas for the reflection and transmission coefficients of one-dimensional linear water waves propagating over a submerged structure with a cycloidal cross section in presence of a sloping beach are determined. In the specialized literature, the previous coefficients are obtained mainly for the limit of linear water waves, considering that the water depth upstream and downstream of the structure is flat. For the analysis, we have obtained an approximate analytical solution to the dimensionless Modified Mild-Slope Equation, which models the interactions of a wide range of water waves, from short waves to long waves. The results shown that the presence of small breakwaters not always generate increments on the reflection coefficients, but on the contrary case they contribute to the reflection of the waves decreasing, which is due to the interference of energy that exists between the inclined beach and the structure. To validate the approximate analytical solution, we present a comparison against analytical solutions reported in the specialized literature, obtained with the aid of linear long wave theory, and a numerical solution, all the solutions adjust properly. Results of this study are expected to be used by coastal engineers for preliminary feasibility and desk design of submerged cycloidal breakwaters.     

Anisotropic elastic materials capable of a three-dimensional deformation (static or dynamic) with only one displacement component and uncoupling of all three displacement components

It is shown that there are anisotropic elastic materials that are capable of a non-uniform three-dimensional deformation with only one displacement component. For wave propagation, the equation of motion can be cast in the form of the differential equation for acoustic waves. For elastostatics, the equation of equilibrium reduces to Laplace’s equation. The material can be monoclinic, orthotropic, tetragonal, hexagonal or cubic. There are also anisotropic elastic materials that uncouple all three displacement components. The governing equation for each of the uncoupled displacement can be cast in the form of the differential equation for acoustic waves in the case of dynamic or Laplace’s equation in the case of static. The material can be orthotropic, tetragonal, hexagonal or cubic.     

Numerical simulation of internal gravity waves using a lattice gas model

Internal waves are modelled in two different circumstances: in a continuously stratified fluid and at the interface between two immiscible fluids. This is done using the lattice gas approach. The standard single phase model and an immiscible two-phase model are both modified to incorporate gravitational interactions. Standing internal waves are set up in both models and are seen to oscillate under the action of the gravitational interaction. The results obtained suggest that the lattice gas approach can be a useful tool in the modelling of such phenomena. © 1998 John Wiley & Sons, Ltd.     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Absolute versus relative formulations of the moving oscillator problem

In the general framework of the bridge-vehicle dynamic interaction, the so-called “moving oscillator” problem is revisited in order to provide a deeper insight into some analytical and physical aspects not specifically analyzed in previous investigations. Without lack of generality, the case of a stream of moving oscillators crossing a simply supported beam with arbitrary time law is considered. The formulations in terms of both absolute and relative displacements of the moving oscillators are critically reviewed and compared, and alternative sets of differential equations with time-dependent coefficients are derived. The study enlightens, both theoretically and numerically, that impulsive contributions to the dynamic response appear in the relative displacement formulation at the time instants in which each vehicle enters or exits the bridge. It is demonstrated that such contributions, somehow “hidden” in the absolute displacement formulation, may have a significant influence on the vibration of the moving oscillators, and thus cannot be a priori neglected in the analysis. It is also shown that the analytical and computational difficulties associated with these additional impulses make preferable the use of the absolute displacement formulation. Far from being restricted to the case of simply supported beams, these findings are valid for any type of bridge structure which induces a discontinuity in the slope of the road profile experienced by the vehicles.     

On the Measurement and Prediction of the Out-of-Plane Displacement Surrounding Cold-Expanded Holes

Experimental measurements of the out-of-plane displacement surrounding cold-expanded holes in a 6082-T6 aluminum alloy were made with a 3D optical scanner using the technique of the encoded light-pattern projection in white light. The measured surface profiles have shown the thickness discontinuity along the hole edge due to the effect of the split in the sleeve. An analytical–numerical solution of the out-of-plane displacement is presented based on existing analytical models. Thus, the results given by the analytical model were then compared with the experimental data and with a finite element (FE) model that simulates the cold-expansion process. The location of the elastic–plastic boundary was estimated as the point at which no change in thickness was observed; a good agreement was found in the comparison of measured, FE and analytical results. The measured surface profiles agreed with those predicted by the FE model and analytical solution. The proposed experimental approach can be used together with FE analysis for predicting the radial and circumferential residual stresses in cold-expanded hole. It is quite versatile and can also be used as quality-control technique in the manufacturing processes of cold-expanded holes.     

Complementary mild-slope equations in a two-layer fluid

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry.     

An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.     

Effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces

The effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces are studied in this paper. First, the secular equations in the traverse isotropic piezoelectric half space are derived from the general dynamic equation with initial stress taken into consideration. Then, the interface conditions that displacement, stress, electric potential, and electric displacement are continuous across interface are required to be satisfied by three sets of coupled waves, namely, quasi-longitudinal wave, quasi-transverse wave and the electric–acoustic wave. The algebraic equations resulting from the interface conditions are solved to obtain the amplitude ratio of various waves and furthermore the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and the effects of initial stress are discussed.