Exact dynamic solutions to piezoelectric smart beams including peel stresses Part II: Numerical results,comparison and approximate solution

This work presents numerical results for the exact dynamic solution of piezoelectric (PZT) smart beams including peel stresses, which was developed in Part I. Numerical results are presented in details for frequency spectra, natural frequencies, normal mode shapes, harmonic responses of the shear and peel stresses, and sensing electric charges for a cantilever beam with a bonded PZT patch to the clamped end. The exact dynamic solution can provide useful data for benchmarking other methods. The numerical results of the present model including peel stresses (PSM) are also compared with those obtained using the shear lag beam model and the shear lag rod model. On the basis of the equivalent forces derived in the static analysis, simple approximate dynamic solutions are obtained and compared with the exact solutions, and then the application and limitation of the simple approximate solutions are investigated. By comparing numerical results predicted by the present PSM model with the shear lag models and the approximate solutions based on the static equivalent forces, effects of the dynamic shear and peel stresses on natural frequencies and dynamic responses of the smart structures are examined.     

Exact dynamic solutions to piezoelectric smart beams including peel stresses I: Theory and application

This work presents exact dynamic solutions to piezoelectric (PZT) smart beams including peel stresses. The governing equations of partial differential forms are firstly derived for a PZT smart beam made of the identical adherends, and then general solutions of the governing equations are studied. The analytical solutions are applied to a cantilever beam with a partially bonded PZT patch to the fixed end. For the given boundary conditions, exact solutions of the steady state motions are obtained. Based on the exact solutions, frequency spectra, natural frequencies, normal mode shapes, harmonic responses of the shear and peel stresses are discussed for the PZT actuator. The details of the numerical results and sensing electric charges will be presented in Part II of this work. The exact dynamic solutions can be directly applied to a PZT bimorph bender. To compare with the classic shear lag model whose numerical demonstrations will be given in Part II, the related equations are also derived for the shear lag rod model and shear lag beam model.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅱ)——ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

IntroductionThis paper is a continuation of Ref.[1],in which a series of orthotropic piezoelectricplane problems was solved and the corresponding exact solutions were obtained with the trial-and-error method,on the basis of the general solution expressed …     

A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials. II. Numerical results and discussion

Based on the investigation performed in Part I of this series, numerical results for the interaction between a semi-infinite interface crack and multiple subinterface matrix microcracks in three kinds of material combinations are given in Part II. The major interaction behaviors are discussed in detail. Special attention is focused on the influences of the different material combinations, the T-stress, the orientation angles, and the location angles of the microcracks on the local stress intensity factor at the interface crack tip. In addition, the variable tendencies of the interaction effect induced from change of the distance between the interface crack tip and the centers of the microcracks are studied. It is concluded that the different material combinations introduced in this paper have little influence on the variable tendencies of the effect, but have significant influence on the effect in magnitude. Detailed comparisons of the results with those in a homogeneous orthotropic material show that the dissimilar materials shift the maximum amplification angle, the maximum shielding angle, the neutral shielding angle, and the neutral T-stress angle, respectively.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

Exact Relations for Effective Tensors of Polycrystals. II. Applications to Elasticity and Piezoelectricity

An important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination. This requirement is so restrictive that it is possible (if not always feasible) to find all such relations explicitly. In order to do this one needs to combine the results developed in Part I of this paper and the representation theory of the rotation groups SO(2) and SO(3). More precisely, one needs to know all rotationally invariant subspaces of the space of material moduli. This paper presents an algorithm for finding all such subspaces. We illustrate the workings of the algorithm on the examples of 3‐dimensional elasticity, where we get all the exact relations, and the examples of 2‐dimensional and 3‐dimensional piezoelectricity, where we get some (possibly all) of them. (Accepted September 24, 1997)     

Elastic behaviour of shrink-fitted forming die assemblies. Part II: Numerical application and results

Summary A study is presented on the elastic behaviour of a set of assemblies consisting of a cylinder prestressed by a shrink-fitted ring of different length as is customary in extrusion toolsets. An accurate analysis is carried out of the individual and interactive effects pertaining both to the bulk support provided by the relatively unstressed parts of the die, and the external compressive stresses induced by radial support of the shrinking ring.Stresses and displacements in the two cylindrical components are evaluated by a numerical application of the three-dimensional elastic theory and the point-matching method.Again a discussion is presented of the limitations involved by the numerical approach.

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Sommario L'articolo riferisce sul comportamento elastico di accoppiamenti realizzati per forzamento di un anello di grosso spessore su un cilindro cavo di lunghezza finita e diversa da quello dell'elemento forzante, del tipo di quelli impiegati nelle attrezzature di estrusione.Si analizzano individualmente e nella loro interazione le influenze che su tale comportamento hanno l'irrigidimento fornito da quelle porzioni dell'anello interno che sono relativamente scariche di tensioni e la pressurizzazione esterna indotta dall'elemento forzante.Il calcolo delle tensioni e degli spostamenti nei due componenti è operato mediante un'applicazione numerica della teoria dell'elasticità tridimensionale e del metodo delle collocazioni.Si discutono inoltre le limitazioni connaturate con l'approccio di tipo numerico.

     

Existence and comparison results of solutions for nonlinear random integral and differential equations relative to weak topology

In this paper, we prove several existence theorems of random solutions to nonlinear random Volterra integral equations under the weak topology of Banach spaces. Then, as applications, we obtain the existence theorems of weak random solutions to random differential equations. Existence of extremal random solutions and a random comparison theorem for these random equations are also obtained. Our theorems improve and extend the corresponding results in [4,5,10,11,12]. Projects supported by the Science Fund of the Chinese Academy of Sciences.     

The dynamics of coupled planar rigid bodies. II. Bifurcations,periodic solutions,and chaos

We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.     

A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part II,numerical solution and applications

This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electric and mechanical behaviors. In Part I, we have obtained a system of governing integro-differential equations for the structure via a variational principle. This part presents a numerical solution algorithm of the integro-differential equations and the numerical results of some applications. A numerical algorithm for solving the system of four integro-differential equations with strongly singular kernels is developed. The convergence of the numerical algorithm is discussed. The numerical results suggest that the fully coupled electromechanical analysis is helpful for a better understanding of the performance of the piezoelectric sensor and actuator. The interfacial normal stress is much higher than the interfacial shear stress, suggesting that the interfacial normal stress causes a delamination initiation.     

Forces, moment and wave pattern for surface combatant in regular head wavesPart II. Measurement results and discussions

Unsteady resistance, heave force, pitch moment and free-surface elevations for DTMB 5512 at steady forward speed and in regular head waves are investigated with different measurement systems for a fairly wide range of test conditions in the IIHR towing tank. Test data are procured for validation of Reynolds-averaged Navier–Stokes computational fluid dynamics codes and for understanding the physics of unsteady ship hydrodynamics. The second part of the paper presents the measurement results and discussions. Received: 4 October 2000 / Accepted: 23 April 2001 Published online: 29 November 2001     

An investigation of the heat transfer and static pressure on the over-tip casing wall of an axial turbine operating at engine representative flow conditions. (II). Time-resolved results

This article reports the measurements of time-resolved heat transfer rate and time-resolved static pressure that have been made on the over-tip casing of a transonic axial-flow turbine operating at flow conditions that are representative of those found in modern gas turbine engines. This data is discussed and analysed in the context of explaining the physical mechanisms that influence the casing heat flux. The physical size of the measurement domain was one nozzle guide vane-pitch and from −20% to +80% rotor axial chord. Additionally, measurements of the time-resolved adiabatic wall temperature are presented. The time-mean data from the same set of experiments is presented and discussed in Part I of this article. The nozzle guide vane exit flow conditions in these experiments were a Mach number of 0.93 and a Reynolds number of 2.7 × 10⁶ based on nozzle guide vane mid-height axial chord. The data reveal large temporal variations in heat transfer characteristics to the casing wall that are associated with blade-tip passing events and in particular the blade over-tip leakage flow. The highest instantaneous heat flux to the casing wall occurs within the blade-tip gap, and this has been found to be caused by a combination of increasing flow temperature and heat transfer coefficient. The time-resolved static pressure measurements have enabled a detailed understanding of the tip-leakage aerodynamics to be established, and the physical mechanisms influencing the casing heat load have been determined. In particular, this has focused on the role of the unsteady blade lift distribution that is produced by upstream vane effects. This has been seen to modulate the tip-leakage flow and cause subsequent variations in casing heat flux. The novel experimental techniques employed in these experiments have allowed the measurement of the time-resolved adiabatic wall temperature on the casing wall. These data clearly show the falling flow temperatures as work is extracted from the gas by the turbine. Additionally, these temperature measurements have revealed that the absolute stagnation temperature within the tip-gap flow can be above the turbine inlet total temperature, and indicates the presence of a work process that leads to high adiabatic wall temperatures as a blade-tip passes a point on the casing wall. It is shown that this phenomena can be explained by consideration of the flow vectors within the tip-gap, and that these in turn are related to the local blade loading distribution. The paper also assesses the relative importance of different time-varying phenomena to the casing heat load distribution. This analysis has indicated that up to half of the casing heat load is associated with the over-tip leakage flow. Finally, the implications of the experimental findings are discussed in relation to future shroudless turbine design, and in particular the importance of accounting for the high heat fluxes found within the tip-gap.     

New,real fundamental solutions to the transient thermal contact problem in a piezoelectric strip under the coupling actions of a rigid punch and a convective heat supply

A thermo-electro-mechanical contact analysis has been performed for a finite piezoelectric strip, which is subjected to the joint actions of a rigid, flat punch and a transient convective heat supply. The Laplace transform and Fourier sine and cosine transforms were applied in solving the governing equations. A detailed analysis of the characteristic roots of the corresponding characteristic equation was made. Real fundamental solutions were derived, which can readily lead to real solutions to the thermo-electro-mechanical quantities. A Cauchy-type singular integral equation was obtained for the stated problem and then solved numerically. Closed form solutions of a special case were obtained. To obtain the accurate solution in the time domain, an effective numerical inversion algorithm of the Laplace transform was applied. Detailed analyses were performed to reveal the variation law of temperature, contact stress beneath the punch, stress intensity factor at the punch edge and strain with time. Parametric studies were performed to discover the effects of the layer thickness on the distribution of temperature, contact stress beneath the punch and stress intensity factor at the punch edge.     

Material degradation due to moisture and temperature. Part 1: mathematical model,analysis, and analytical solutions

The mechanical response, serviceability, and load-bearing capacity of materials and structural components can be adversely affected due to external stimuli, which include exposure to a corrosive chemical species, high temperatures, temperature fluctuations (i.e., freezing–thawing), cyclic mechanical loading, just to name a few. It is, therefore, of paramount importance in several branches of engineering—ranging from aerospace engineering, civil engineering to biomedical engineering—to have a fundamental understanding of degradation of materials, as the materials in these applications are often subjected to adverse environments. As a result of recent advancements in material science, new materials such as fiber-reinforced polymers and multi-functional materials that exhibit high ductility have been developed and widely used, for example, as infrastructural materials or in medical devices (e.g., stents). The traditional small-strain approaches of modeling these materials will not be adequate. In this paper, we study degradation of materials due to an exposure to chemical species and temperature under large strain and large deformations. In the first part of our research work, we present a consistent mathematical model with firm thermodynamic underpinning. We then obtain semi-analytical solutions of several canonical problems to illustrate the nature of the quasi-static and unsteady behaviors of degrading hyperelastic solids.     

Planar laser-induced fluorescence (PLIF) measurements of liquid film thickness in annular flow. Part II: Analysis and comparison to models

Film thickness distributions in upward vertical air–water annular flow have been determined using planar laser-induced fluorescence (PLIF). Film thickness data are frequently used to estimate interfacial shear and pressure loss. This film roughness concept has been used in a number of models for annular flow of varying complexity. The PLIF data are presently applied to the single-zone interfacial shear correlation of Wallis; the more detailed model of Owen and Hewitt; and the two-zone (base film and waves) model of Hurlburt, Fore, and Bauer. For the present data, these models all under-predict the importance of increasing liquid flow on pressure loss and interfacial shear. Since high liquid flow rates in annular flow induce disturbance wave and entrainment activity, further modeling in these areas is advised.     

Comments on “Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material” by Yi-Shyong Ing, Mau-Jung Wang [Int. J. Solids Struct. 41 (2004) 3849–3864]

In this comment it is pointed out that the analysis of the dynamic stress intensity factor, dynamic electric displacement intensity factor and dynamic energy release rate conducted by Ing and Wang [Ing, Y.S., Wang, M.J., 2004. Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material. International Journal of Solids and Structures 41, 3849–3864] is incorrect. The correct analysis and corresponding correct plots are presented.     

Modeling of cementitious materials exposed to isothermal calcium leaching,considering process kinetics and advective water flow. Part 2: Numerical solution

The second part of the paper presents numerical solutions of the mathematical model of hydro-chemo-mechanical behavior of cementitious materials exposed to contact with deionized water of part 1. The model defines kinetics of the calcium leaching process instead of a direct application of a curve describing equilibrium between solid calcium in the material skeleton and the calcium dissolved in the pore solution. It further takes into account the advective flux of calcium ions. Both aspects are new as compared to previous models. The weak form of the governing equations of the model is derived first using the Galerkin method. Then, the equations are discretized in space with finite elements and in time domain with finite differences, and finally the procedures used for numerical solution of their discretized form are presented. Three numerical examples are solved to test the numerical solution procedure proposed and demonstrate its robustness for solution of 1D and 2D problems concerning fast and slow leaching of cement-based materials. The effect of various factors on the results concerning chemical degradation of structures made of cementitious materials is analyzed as well.