Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.     

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L ¹ distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the “hydrodynamical” equations for this kinetic system. This remains true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globally in time. (Accepted September 3, 1996)     

Electrorheologial properties of hematite/silicone oil suspensions under DC fields

Electrorheological (ER) fluids composed of α-Fe₂O₃ (hematite) particles suspended in silicone oil are studied in this work. The rheological response has been characterized as a function of field strength, shear rate and volume fraction. Rheological tests under DC electric fields elucidated the influence of the electric field strength, E, and volume fraction, ϕ, on the field-dependent yield stress, τ_y. It was found that this quantity scales with E and ϕ with a linear and parabolic dependence, respectively. The viscosities of electrified suspensions were found to increase by several orders of magnitude as compared to the unelectrified suspension at low shear rates, although at high-shear rates hydrodynamic effects become dominant and no effects of the electric field on the viscosity are observed. The work is completed with the analysis of microscopic observations of the structure acquired by the ER fluid upon application of a constant electric field. Electrohydrodynamic convection is found to be the origin of the ER response rather than the commonly admitted particle fibrillation. This fact can provide an explanation to the relationship between yield stress and electric field strength as well as the pattern of periodic structures observed in the measurement geometries.     

Crack problems of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in piezoelectrics or on the interfaces of piezoelectric bimaterials. A class of boundary problems involving many cracks is also solved. For homogeneous materials it is found that the normal electric displacementD ₂ induced by the crack is constant along the crack faces which depends only on the applied remote stress field. Within the crack slit, the electric fields induced by the crack are also constant and not affected by the applied electric field. For the bimaterials with realH, the normal electric displacementD ₂ is constant along the crack faces and electric fieldE ₂ has the singularity ahead of the crack tip and a jump across the interface. The project is supported by the National Natural Science Foundation of China(No. 19704100) and the Natural Science Foundation of Chinese Academy of Sciences(No. KJ951-1-201).     

Nonlinear Longitudinal Vibrations of Transversally Polarized Piezoceramics: Experiments and Modeling

Nonlinear behavior of piezoceramics at strong electric fields is a well-known phenomenon and is described by various hysteresis curves. On the other hand, nonlinear vibration behavior of piezoceramics at weak electric fields has recently been attracting considerable attention. Ultrasonic motors (USM) utilize the piezoceramics at relatively weak electric fields near the resonance. The consistent efforts to improve the performance of these motors has led to a detailed investigation of their nonlinear behavior. Typical nonlinear dynamic effects can be observed, even if only the stator is experimentally investigated. At weak electric fields, the vibration behavior of piezoceramics is usually described by constitutive relations linearized around an operating point. However, in experiments at weak electric fields with longitudinal vibrations of piezoceramic rods, a typical nonlinear vibration behavior similar to that of the USM-stator is observed at near-resonance frequency excitations. The observed behavior is that of a softening Duffing-oscillator, including jump phenomena and multiple stable amplitude responses at the same excitation frequency and voltage. Other observed phenomena are the decrease of normalized amplitude responses with increasing excitation voltage and the presence of superharmonics in spectra. In this paper, we have attempted to model the nonlinear behavior using higher order (quadratic and cubic) conservative and dissipative terms in the constitutive equations. Hamilton's principle and the Ritz method is used to obtain the equation of motion that is solved using perturbation techniques. Using this solution, nonlinear parameters can be fitted from the experimental data. As an alternative approach, the partial differential equation is directly solved using perturbation techniques. The results of these two different approaches are compared.     

Two general integrals of singular crack tip deformation fields

The Eshelby tensor E has vanishing divergence in a homogeneous elastic material, whereas the invariance of the crack tip J integral suggests, in accord with known solutions, that the product rE will have a finite limit at the tip. Here r is distance from the tip. These considerations are shown to lead to two general integrals of the equations governing singular crack tip deformation fields. Some of their consequences are discussed for analysis of crack tip fields in linear and nonlinear materials.     

Numerical evaluation of the equivalent properties of Macro Fiber Composite (MFC) transducers using periodic homogenization

This paper focuses on the evaluation of the homogeneous properties of the active layer in Macro Fiber Composite (MFC) transducers using finite element periodic homogenization. The proposed method is applied to both d₃₁ and d₃₃-MFCs and the results are compared to previously published analytical mixing rules, showing a good agreement. The main advantages of the finite element homogenization is the possibility to take into account local details in the representative volume element such as complicated electrode patterns or local variations of the poling direction due to curved electric field lines. Although these influences have been found to be rather small in the present study, the method presented is useful for a better understanding of the behavior of piezocomposite transducers.     

DSMC approach to nonstationary Mach reflection of strong incoming shock waves using a smoothing technique

The direct simulation Monte Carlo (DSMC) method is applied to simulation of nonstationary Mach reflection of strong shock waves. Normally the DSMC method is very time consuming in solving unsteady flow field problems especially for high Mach numbers, because of the necessity of iterative calculations to eliminate the inherent statistical fluctuation caused by a finite sample size. A central weighted smoothing technique is introduced to process the DSMC results, so that the iteration time can be significantly reduced. In spite of some relaxations of the shock wave structure, the smoothing technique is verified to be useful to estima te the flow fields qualitatively and even quantitatively by using a relatively small sample size. The comparison between the present approach and the kineticmodel approach (Xu et al. 1991a, 1991b) on the application to unsteady rarefied flow fields was also carried out.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Nonlinear behaviour of piezoceramics under weak electric fields. Part-II: Numerical results and validation with experiments

When excited near resonance in the presence of weak electric fields, piezoceramic materials exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In an accompanying paper, a generalized nonlinear 3D finite element formulation has been developed incorporating quadratic and cubic terms in the electric enthalpy density function and the virtual work done by damping forces. In this paper, the formulation has been validated by conducting experiments on test pieces of various geometries and of three different materials (in all, four case studies). Both proportional damping and nonlinear damping formulations have been used to predict the frequency response of these systems. Newmark-β method has been used to obtain the dynamic response of the systems using FE analysis. It is demonstrated that the nonlinear finite element model is able to predict the responses of the various test cases studied and the results match very well with those of experimental observations.     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

T–S fuzzy control design for a class of nonlinear networked control systems

This paper is devoted to controlling a class of nonlinear systems over a communication network in the presence of packet transmission delays, packet losses, quantization errors, and sampling-related phenomena. Specifically, based on the Takagi–Sugeno (T–S) fuzzy approach, this paper presents a way of designing a quantized controller that allows all the states of nonlinear NCSs to converge exponentially to a bounded ellipsoid. In particular, this paper provides a method capable of exploiting fully the time-varying delay term d(t), induced by Jensen inequality, for nonlinear NCSs.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ?. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal ${\bar 4}$ crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d ₀,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d ₀. In the limit as d ₀→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal ${\bar 4}$ class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.     

On the interaction between two fixed spherical particles

The variation of the drag (C_D) and lift coefficients (C_L) of two fixed solid spherical particles placed at different positions relative each other is studied. Simulations are carried out for particle Reynolds numbers of 50, 100 and 200 and the particle position is defined by the angle between the line connecting the centers of the particles and the free-stream direction (α) and the separation distance (d₀) between the particles. The flow around the particles is simulated using two different methods; the Lattice Boltzmann Method (LBM), using two different computational codes, and a conventional finite difference approach, where the Volume of Solid Method (VOS) is used to represent the particles. Comparisons with available numerical and experimental data show that both methods can be used to accurately resolve the flow field around particles and calculate the forces the particles are subjected to. Independent of the Reynolds number, the largest change in drag, as compared to the single particle case, occurs for particles placed in tandem formation. Compared to a single particle, the drag reduction for the secondary particle in tandem arrangement is as high as 60%, 70% and 80% for Re = 50, 100 and 200, respectively. The development of the recirculation zone is found to have a significant influence on the drag force. Depending on the flow situation in-between the particles for various particle arrangements, attraction and repulsion forces are detected due to low and high pressure regions, respectively. The results show that the inter-particle forces are not negligible even under very dilute conditions.     

Numerical-experimental method for elastic parameters identification of a composite panel

A hybrid numerical-experimental approach to identify elastic modulus of a textile composite panel using vibration test data is proposed and investigated. Homogenization method is adopted to predict the initial values of elastic parameters of the composite, and parameter identification is transformed to an optimization problem in which the objective function is the minimization of the discrepancies between the experimental and numerical modal data. Case study is conducted employing a woven fabric reinforced composite panel. Three parameters (E₁₁, E₂₂, G₁₂) with higher sensitivities are selected to be identified. It is shown that the elastic parameters can be accurately identified from experimental modal data.     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.