In the present paper, the problem of a functionally graded piezoelectric cantilever beam subjected to different loadings is studied. The piezoelectric beam is characterized by continuously graded properties for one elastic parameter and the material density. A pair of stress and induction functions in the form of polynomials is proposed and determined. Based on these functions, a set of analytical solutions for the beam subjected to different loadings is obtained. As particular cases, series of solutions for some canonical problems can be directly obtained from the solutions of the present paper, such as for the problems of a piezoelectric cantilever beam with constant body force or without body forces, etc.
The paper presents the results of the research on modal parameter estimation based alone on output measurements. A system is excited randomly and random decrement functions are used to separate the random responses from the determination free vibrations. It is shown that on estimation of the natural frequencies and damping ratios of the system is possible using the wavelet transform of the system’s free response. A particular form of the son wavelet function improves the results compared to those obtained with the Morlet wavelet function. An optimal son wavelet function is obtained by minimisation of the wavelet transform entropy. The accuracy of this new technique is confirmed in a numerical example and by applying it to ambient vibration measurements of a bridge excited by traffic.
Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.
The problem of non-scalability of structures under impact loads caused by strain-rate effects is solved in this article by properly correcting the impact velocity. The technique relies on the use of an alternative dimensionless basis, together with a mathematical model which allows the calculation of a correction factor for the impact velocity. This new velocity, when applied to the model, makes it to assure the satisfaction of the scaling laws. The indirect similitude method detailed here is applied to two strain-rate sensitive structures, a double plate under in-plane impact and a beam subjected to a blast load. The results show a very good agreement so that the model and a prototype made from strain rate sensitive materials behave the same.
The paper presents the results of experiments on the technical fabric "Panama" carried out with the purpose of identification of inelastic properties of the warp and weft as well as the identification techniques based on the least-squares method. The material parameters are calculated on the basis of the uniaxial tension test in the warp and weft directions. The dense net type of a finite element is proposed to express the behaviour of the technical fabric in the FEM analysis. The Bodner-Partom and Chaboche viscoplastic models are applied to the description of the warp and weft properties. The results are verified by numerical simulation of the laboratory tests.
The objective of the present work is to reexamine the dynamic phenomenon of sprag-slip instability. To that purpose, a model consisting of an elastic beam sliding over a rigid belt at constant speed is set up and investigated. It turns out that there are parameter combinations for which the system does not posess a static solution corresponding to a steady sliding state, neither stable nor unstable, a phenomenon first discovered by Painlevé considering sliding rigid bodies, sometimes referred to as the Painlevé paradox. The nonexistence of a steady sliding state is a sufficient criterion for the appearance of sprag-slip oscillations, since in the corresponding configurations the fundamental behavior of the system is intrinsically dynamic in nature.
The interface crack problem for a piezoelectric bimaterial based on permeable conditions is studied numerically. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical and electrical loads, the complex stress intensity factor for an interface crack is obtained. The influence of the applied loads on the electromechanical fields near the crack tip is also studied. For a particular case of a short crack with respect to the bimaterial size, the numerical results are compared with the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.
Diffraction of water waves by a small cylindrical elevation of the bottom of a laterally unbounded ocean covered by an ice sheet is investigated by the perturbation analysis. The ice sheet is modelled as a thin elastic plate. The reflection and transmission coefficients are evaluated up to the first order in terms of integrals involving the shape function representing the bottom elevation. Three particular forms of the shape function are considered for which explicit expressions for these coefficients are obtained. For the particular case of a patch of sinusoidal undulations at the bottom, the reflection coefficient up to first order is found to be an oscillatory function of the ratio of the wavelength of the bottom undulations and that of the incident wave train. When this ratio approaches 0.5, the reflection coefficient becomes a multiple of the number of undulations and high reflection of the incident wave energy occurs if this number is large. Reflection coefficient is depicted graphically to visualize the effect of the presence of ice-cover and the number of undulations.
The paper presents a thermodynamically consistent constitutive model for elasto-plastic analysis of orthotropic materials at large strain. The elastic and plastic anisotropies are assumed to be persistent in the material but the anisotropy axes can undergo a rigid rotation due to large plastic deformations. The orthotropic yield function is formulated in terms of the generally nonsymmetric Mandel stress tensor such that its skew-symmetric part is additionally taken into account. Special attention is focused on the convexity of the yield surface resulting in the nine-dimensional stress space. Of particular interest are new convexity conditions which do not appear in the classical theory of anisotropic plasticity. They impose additional constraints on the material constants governing the plastic spin. The role of the plastic spin is further studied in simple shear accompanied by large elastic and large plastic deformations. If the plastic spin is neglected, the shear stress response is characterized by oscillations with an amplitude strictly dependent on the degree of the plastic anisotropy.
A novel technique for the determination of the pose and the twist of rigid bodies using point-acceleration data is proposed. These data are collected from an accelerometer array, which is a kinematically redundant set of triaxial accelerometers. Because orientational error in the installation of the accelerometers can be fatal to the accuracy of the results, a calibration procedure based on the consistency of the point accelerations is outlined. The formulation developed is then utilized in the simulation analysis of two sample motions. The relations required to estimate the pose and the twist are derived in a body-fixed frame. The body angular acceleration and angular velocity, in this order, are determined directly from the acceleration data; the body attitude is then computed through integration.
This investigation is concerned with the dynamic displacements of a beam on a poroelastic half space under a periodic oscillating load of constant velocity. The governing equations for the proposed analysis are solved using Fourier transform. The expression for the vertical displacement is obtained according to the contact condition between a beam and a half space. The effects of the moving velocity and vibration frequency of the load on the dynamic displacement are considered in the numerical examples. The results show that the load velocity has significant influence on dynamic displacement. It is also noted that large differences exist between the dynamic responses for a beam on a poroelastic half space and on an elastic half space when the load velocity is larger than the shear wave speed of the medium.
The subject of analysis is the bending of elastic plates exhibiting a nonhomogeneous periodic structure and/or a periodically variable thickness in a certain direction parallel to the plate’s midplane. The fundamental modelling problem is how to obtain an effective 2D-model of a plate under consideration, i.e., a 2D-model represented by PDEs with constant coefficients. This problem for periodic plates has been solved independently in  and , using asymptotic homogenization. However, homogenization neglects dynamic phenomena related to the plate’s rotational inertia and cannot be applied to the analysis of higher-order vibration frequences. The main aim of this contribution is to formulate a new non-asymptotic effective 2D-model of a periodic plate which is free from the mentioned drawbacks and describes the dynamic behaviour of plates having the thickness of the order of the period length. The proposed model is applied to the analysis of some vibration problems.
In this paper, the reflection and refraction of a plane wave at an interface between two half-spaces composed of triclinic crystalline material is considered. It is shown that due to incidence of plane wave three types of waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH), will be generated governed by the propagation condition involving the acoustic tensor. A simple procedure has been presented for the calculation of all the three phase velocities of the quasi waves. It has been established that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. Relations are established between directions of motion and propagation, respectively. The expressions for reflection and refraction coefficients of qP, qSV and qSH waves are obtained. Numerical results of reflection and refraction coefficients are presented for different types of anisotropic media and for different types of incident waves. Graphical representations have been made for incident qP waves, and for incident qSV and qSH waves numerical data are presented in tables.
A heuristic constitutive equation and an FE formulation for rubber under small oscillatory loads superimposed on large static deformations are presented. The viscoelastic constitutive model, , is implemented in an FE code to analyze the dynamic characteristics of rubber elements under general loading conditions. Dynamic tests in which the rubber specimens endure steady-state harmonic motion superimposed on large static deformation are performed in order to verify the proposed model. Compression and complex-stress tests are included in the tests to check the proposed model under multi-axial stress states. The FEA results are compared with the experimental results. The proposed model successfully predicts dynamic stiffness peak in the complex-stress test, which cannot be explained by conventional models. The model shows a better performance than existing models in predicting the behavior of rubber specimens that are subject to complex pre-strain.
Under creeping flow conditions, we consider the steady Couette flow of a newtonian fluid between two plates, one of them a plane, the other one with a sinusoidal profile. The streamlines of the flow are obtained from an exact analytical treatment of the Stokes equations by making use of complex function theory. This nonconventional approach, which has been recently developed for film flows, [16, 17], allows for a reduction of the problem to the solving of ordinary differential and integral equations for functions of one variable. Based on our calculations, we visualize the formation and evolution of vortices in the flow. The influence of the plate undulations and vortex structure on the drag force and flow rate is shown by parameter studies. We also discuss the classical lubrication approximation and its limitations. Taking the drag flow as a simple model for lubricant friction, we also discuss resulting friction coefficients and material transport of the lubricant.
Explicit forms of the first-order approximate boundary conditions are derived for a 2D problem of SH waves scattering by a thin, curvilinear, elastic, rigidly supported inclusion in a uniform background. The effects of varying elastic modulus and geometrical forms of the inclusion on the stress and strain states of the body near and far from the ends of the inhomogeneity are examined. The method of investigation is based on the matching of asymptotic expansions with the thickness-to-length ratio as the perturbation parameter.
In the present paper, we consider the behavior of nonlinear piezoelectric materials by generalization for this case of the Hashin-Shtrikman variational principles. The new general formulation used here differs from others, because, it gives the possibility to evaluate the upper and lower Hashin-Shtrikman bounds for specific physical nonlinearities of piezoelectric materials. Geometrical nonlinearities are not considered.
Monitoring by vibration measurement and analysis is largely used in the industry for detection of defects in revolving parts of machines. The determination of good sensor positions is one of the main research goals in the field of predictive maintenance. This paper proposes a numerical methodology based on the FEM and spectral analysis in order to find the optimum sensor positions. Bearings are key components in the vibration propagation from moving parts to immobile ones. Two existing nonlinear models of bearings are recalled and implemented in a FE code. The obtained tangent stiffness matrices of bearings are then put in the global system to study the dynamic behaviour. The dynamic response of the whole system under defect excitations is used to determine the optimum sensor placements for the defect detection in the predictive maintenance.
This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r- and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in [24, 25] are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.
The elastodynamic solution of a multilayered hollow sphere for spherically symmetric problems is obtained by decomposition into two parts, one being the quasi-static and the other the dynamic solution. The quasi-static solution is firstly derived by means of the state-space method, and the dynamic solution is obtained by utilizing the separation of variables method and the orthotropic expansion technique. The solutions for displacement and stresses are obtained in result. The present method is suitable for a multilayered spherically isotropic hollow sphere, with arbitrary thickness of each of the layers and arbitrary initial conditions, subjected to arbitrary form of a spherically symmetric dynamic load at the internal and external surfaces. Numerical results are finally presented.