Search

Based on the theory of nonlinear piezoelectricity, an approximate solution for electrically forced nonlinear thickness-shear vibrations of piezoelectric plates is introduced. The model considers an infinite piezoelectric plate subjected to an alternative voltage, incorporating 3rd-order and 4th-order elastic constants and viscous damping under finite deformation. The system's differential equations for steady-state forced vibrations are derived from the nonlinear pure thickness-shear vibration problem, and transformed so that the new electrical boundary conditions are free. After the application of Galerkin's method, the transformed differential equations lead to a cubic equation, capturing the nonlinear current-frequency curves of AT-cut quartz plates without the need of the quality factor Q, due to the introduced viscous damping. When the damping effect is ignored, the frequency response curves of the AT-cut quartz plate under a specific voltage are confirmed with the existing literature. The study emphasizes the presence of the critical voltage and the importance of the fourth-order elastic constant (c 6666) in the stability, revealing lower c 6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability. • A cubic equation predicts the nonlinear behavior of piezoelectric resonators. • The decrease of c 6666 improves the stability of piezoelectric resonators. • The amplitude-frequency curve has a hysteresis region when voltage is that high. • An efficient algorithm for the critical voltage is suggested. [ABSTRACT FROM AUTHOR]

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective. [ABSTRACT FROM AUTHOR]

• Novel boundary-based ML method for 2D solid mechanics. • Requires only boundary discretization, enabling faster learning. • Replaces unstable differential operators by stable integral operators. • Higher precision and efficiency than traditional ML methods. In this paper, we make the first attempt to adopt the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. The proposed BINNs combine the artificial neural networks with the exact boundary integral equations (BIEs) to effectively solve the boundary value problems based on the corresponding partial differential equations (PDEs). The BIEs are utilized to localize all the unknown physical quantities on the boundary, which are approximated by using artificial neural networks and resolved via a training process. In contrast to many traditional neural network methods based on a domain discretization, the present BINNs offer several distinct advantages. Firstly, by embedding the analytical BIEs into the learning procedure, the present BINNs only need to discretize the boundary of the problem domain, which reduces the number of the unknowns and can lead to a faster and more stable learning process. Secondly, the differential operators in the original PDEs are substituted by integral operators, which can effectively eliminate the need for additional differentiations of the neural networks (high-order derivatives of the neural networks may lead to instabilities in the learning process). Thirdly, the loss function of the present BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighting functions, which are commonly used in most traditional methods to balance the gradients among the different objective functions. Extensive numerical experiments show that the present BINNs are much easier to train and can usually provide more accurate solutions as compared to many traditional neural network methods. [Display omitted] [ABSTRACT FROM AUTHOR]

Abstract Unified and explicit expressions of static three-dimensional generally anisotropic Green's functions and their first derivatives are presented. The application of residue calculus to the line integral form of the Green's functions and their derivatives yields the conventional explicit expressions. The numerical evaluations of these expressions become unstable in the nearly degenerate case. In contrast to the conventional explicit expressions, the novel unified explicit expressions have no factors like (p i − p j) in any denominator. The novel expressions remain valid for all cases. Their numerical evaluations, therefore, possess very high accuracy and efficiency. Following our previous works in generally anisotropic elasticity, the unified explicit expressions of the Green's functions and their derivatives for anisotropic piezoelectric materials are presented for the first time. The accuracy and stability of novel unified and explicit expressions are verified numerically. Furthermore, the newly derived explicit expressions are used to investigate the influences of the material anisotropy on the displacements and electrical potential in a piezoelectric solid, when a point force or an electrical charge is imposed. The numerical results suggest that the field quantities can be tuned by the proper choose of the material anisotropy or the orientation of the principal material axes. [ABSTRACT FROM AUTHOR]

The scalar differential equation in the thickness eigendisplacement for the doubly-rotated quartz plates is applied to analyze thickness vibrations of an unelectroded circular plate with free edges at the neighborhood of the pure thickness vibration mode. The scalar differential equation is transformed into an elliptical coordinate system. With the boundary conditions of free edges, the frequencies and the modes are solved in terms of the Mathieu function and the Modified Mathieu function. The results of frequencies of the fundamental harmonic and its third overtone of an AT-cut quartz circular plate by the present approach agree well with the existing theoretical results and the experiment results. The frequencies and modes of an SC-cut quartz circular plate are investigated by the present approach. The frequencies are close to each other when the order of the harmonics is the same. A rotation angle of the symmetric axes of the vibration modes are observed that is dependent on the anisotropic material constants and the order of the harmonics. This approach has potential applications in the design of the doubly-rotated quartz circular resonators. [ABSTRACT FROM AUTHOR]

It is well known that the Green’s function for anisotropic magnetoelectroelastic materials can be written as a line integral by solving the governing equations and thereafter explicit expressions can be obtained by applying the Cauchy residue calculus to the line integral. Alternatively, the Green’s function can be evaluated explicitly by constructing the integral in terms of the solutions of a standard eigen-system known as the Stroh eigen-relation. In this paper, explicit expressions of the Green’s function for the magnetoelectroelastic materials are successfully constructed by the solutions of the Stroh eigen-relation associated with the oblique plane perpendicular to the position vector x . With the distinctness assumption on the eigenvalues, the first and second derivatives of the Green’s function are expressed as linear combinations of the eigenvectors, where the corresponding coefficients are determined explicitly. As a special case, the numerical results of these explicit expressions are validated by analytical results for the transversely isotropic magnetoelectroelastic materials. Furthermore, the present explicit expressions of the Green’s function and its derivatives are applied to calculate the generalized displacement and stress fields due to a buried quantum dot in an infinite magnetoelectroelastic solid. [ABSTRACT FROM AUTHOR]

Three advanced methods are presented for constructing and evaluating Green's function and its derivatives for three-dimensional anisotropic elastic solids. The first one is obtained from a line integral followed by Cauchy's residue theorem, and expressed in terms of Stroh's eigenvalues. The second one is obtained by using the explicit solutions of Stroh's eigenvector matrix available in two-dimensional anisotropic elasticity. Whereas the last one is derived from the solution obtained from 2D Radon transform, and expressed in terms of the fundamental elasticity matrix in vertical planes, which makes the derivatives simpler than those obtained from the line integral operated on an oblique plane. The novelty of the first two methods lies in the construction and computation of the first and second derivatives of the Green's function, though the Green's function itself in the first two methods is rather known. In contrast, the third method is quite new for both the Green's function itself and its first and second derivatives. The advantages and disadvantages of these three solution methods are investigated and discussed through their derivations and numerical implementations for isotropic, transversely isotropic, and fully anisotropic elastic solids. Good agreement between the results calculated by these three methods is shown in all numerical examples. Among these three methods, the last one is proved to be the most efficient one. [ABSTRACT FROM AUTHOR]

The Rayleigh–Ritz method is widely employed to analyze free vibrations of elastic solids and structures. For a simple formulation and efficient evaluation of vibrations of elastic ellipsoids, which also include spheres, the Cartesian coordinate system is utilized. It is hoped that the formulation with lengthy expressions can be solved with fewer terms of displacement functions in Chebyshev polynomials for simple evaluations of stiffness and mass matrices of the elastic ellipsoids with the procedure. The vibrations of elastic ellipsoids are calculated with geometric parameters for the validation of the procedure and formulation with known results and the analysis from this study. [ABSTRACT FROM AUTHOR]

Due to the inherent viscosity of polymer, piezoelectric response in the thickness direction (d33) of cellular ferroelectret films usually depends on the time of measurement. In this letter, the micromechanical theory of viscoelastic composite was extended to predict the time dependence of the overall piezoelectric d33 coefficient of voided charged polymer foam. Experiments were carried out to find the time spectra of piezoelectric d33 coefficient of voided charged polypropylene film. Theoretical simulation agrees well with experiment data. [ABSTRACT FROM AUTHOR]

An infinite system of one-dimensional differential equations is derived from the two-dimensional theory of elasticity by expanding the displacement field in a series of trigonometrical functions together with a linear term. Since the trigonometrical functions are pure thickness-vibration modes of infinite plates or beams with the top and bottom surfaces being free, the differential equations and the corresponding boundary conditions serve as the basis of an accurate beam theory for vibration analysis, named Lee's beam theory (LBT). Naturally, a high-order set of the infinite system should be quite useful in the analysis of beams at high frequencies. With the objective of vibration analysis of beams, this paper focuses on the first-order approximation, which leads to a first-order shear deformation beam theory for flexural vibrations (LBT1st). The differential equations in LBT1st are equivalent to those in Timoshenko's beam theory (TBT). The most important difference between LBT1st and TBT is the different field displacements. For the assessment of the accuracy of LBT1st, the numerical results of frequencies of free vibrations, frequency spectra and mode shapes of beams with classical boundary conditions are obtained and compared with those by TBT and plane stress problem of elasticity. Considering the plane stress problem as a reference, LBT1st is slightly more accurate in describing the field shapes of beams than TBT. Therefore, LBT1st, as well as LBT, is an addition to the existing beam theories with improved accuracy for the vibration analysis of beams and their combinations. [ABSTRACT FROM AUTHOR]

• A new explicit expression of Green's function of MEE material is present. • The expression is derived from non-degenerate cases but valid in degenerate cases. • No additional numerical treatment is required except in obtaining Stroh's eigenvalues. The paper presents a novel explicit expression of three-dimensional Green's function of magnetoelectroelastic solids with general anisotropy. The novel explicit expression of Green's function is an alternative of the simple explicit expression in the non-degenerate case. The most impressive property of the novel explicit expression is its applicability in degenerate cases. The explicit expression is assumed to be unified since it is valid in both degenerate and non-degenerate cases. The numerical results of Green's function of a transversely isotropic magentoelectroelastic material by using the unified explicit expression show the high accuracy of the numerical evaluation of unified explicit expression in degenerate, nearly-degenerate and non-degenerate cases. [ABSTRACT FROM AUTHOR]

With large accelerations, the mechanical vibrations of device structures can be driven to nonlinear states yielding strong variations of vibration characteristics such as frequency change and modes shape distortions. The exact equation of motion with an external acceleration of the thickness-shear vibrations of an infinite isotropic plate is approximated through the coupling in frequencies and solved with harmonics matching, and the approximate solutions of frequency and displacements due to the acceleration are obtained. The analytical solution with good accuracy is obtained and the frequency variation is evaluated with the simple relationship between deformation, acceleration, and frequency. [ABSTRACT FROM AUTHOR]

With the advances of material processing technology and miniaturization of mechanical devices and components, it is clear now that curvilinear coordinate systems in dealing with special configurations of elastic solids including cylinders are also naturally needed in the analytical process. By adopting the cylindrical coordinates, it is found that the Love wave in semi-infinite solids possess the same velocity as in the Cartesian coordinates, but the displacement is dependent on radius near the origin and decaying slowly with the radius by exhibiting a strong contrast of the uniform displacement in the Cartesian formulation. Numerical examples show that the asymptotic approximation is accurate in one wavelength away from the origin, implying that solutions will be different only in the vicinity of the point of excitation. [ABSTRACT FROM AUTHOR]

Circular and annular elastic plates have wide applications as essential elements in various engineering structures and products demanding accurate analysis of their vibrations. At higher frequencies, the analysis of vibrations needs appropriate equations, as shown by the Mindlin plate equations for rectangular plates with tailored applications for the analysis of quartz crystal resonators. Naturally, there are equally strong demands for the equations and applications in circular and annular plates with the consideration of higher-order vibration modes. By following the procedure established by Mindlin based on displacement expansion in the thickness coordinate, a set of higher-order equations of vibrations of circular and annular plates are derived and truncated for comparisons with classical and first-order plate equations of circular plates. By utilizing these equations, coupled thickness-shear and flexural vibrations of circular and annular plates are analyzed for vibration frequencies and mode shapes. [ABSTRACT FROM AUTHOR]

The frequency-temperature behavior has long been analyzed to ensure the smoothness of functioning frequency of a quartz crystal resonator during the temperature variation and maintain its operations in a stable state. It has been proven that vibrations of quartz crystal plates in the thickness-shear mode can be well described by the theory of incremental thermal field in conjunction with the Mindlin plate equations, demonstrating the possibility that vibration frequencies and mode shapes can also be well predicted with the temperature as a targeted variable. By studying the vibrations with temperature changes in a given range, the frequencies and vibration modes can be calculated as a function of temperature, then they can be used in the calculations of both strain and kinetic energies for vibration mode identification or characterization based on energy proportion and changes. The trends of changes of vibration frequency, amplitudes, and energy with temperature variation make it possible for characterization the variations of vibrations and particularly the weakening or enhancement of the dominance of specific vibration modes. With such relations between vibration properties and temperature known, possible analytical tools can be developed to aid the optimal design by avoiding significant mode changes or occurrences of mode conversion. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear thickness-shear vibration model for one-dimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity. The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle. The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode. Only the shear strain gradient through the thickness is considered in the present model. With geometric nonlinearity, the governing equations are converted into differential equations as the function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation. Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent, and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates. The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly. The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices. [ABSTRACT FROM AUTHOR]

Quantifying the temporal and spatial patterns of impervious surfaces (IS) is important for assessing the environmental and ecological impacts of urbanization. In order to better extract IS, and to explore the divergence in urbanization in different regions, research on the regional differentiation features and regional change difference features of IS are required. To extract China's 2013 urban impervious area, we used the 2013 night light (NTL) data and the Moderate Resolution Imaging Spectroradiometer Normalized Difference Vegetation Index and enhanced vegetation index (EVI) temporal series data, and used three urban impervious surface extraction indexes—Human Settlements Index, Vegetation-Adjusted NTL Urban Index, and the EVI-adjusted NTL index (EANTLI)—which are recognized as the best and most widely used indexes for extracting urban impervious areas. We used the classification results of the Landsat-8 images as the benchmark data to visually compare and verify the results of the urban impervious area extracted by the three indexes. We determined that the EANTLI index better reflects the distribution of the impervious area. Therefore, we used the EANTLI index to extract the urban impervious area from 2003 to 2013 in the study area, and researched the spatial and temporal differentiation in urban IS. The results showed that China's urban IS area was 70,179.06 km2, accounting for 0.73% of the country's land area in 2013, compared with 20,565.24 km2 in 2003, which accounted for 0.21% of the land area, representing an increase of 0.52%. On a spatial scale, like economic development, the distribution of urban impervious surfaces was different in different regions. The overall performance of the urban IS percentage was characterized by a decreasing trend from Northwest China, Southwest China, the Middle Reaches of the Yellow River, Northeast China, the Middle Reaches of the Yangtze River, Southern Coastal China, and Northern Coastal China to Eastern Coastal China. On the provincial scale, the urban IS expansion showed considerable differences in different regions. The overall performance of the Urban IS Expansion index showed that the eastern coastal areas had higher values than the western inland areas. The cities or provinces of Beijing, Tianjin, Jiangsu, and Shanghai had the largest growth in impervious areas. Spatially and temporally quantifying the change in urban impervious areas can help to better understand the intensity of urbanization in a region. Therefore, quantifying the change in urban impervious area has an important role in the study of regional environmental and economic development, policy formulation, and the rational use of resources in both time and space. [ABSTRACT FROM AUTHOR]

In this paper, we present a machine learning (ML) method to search for geometric patterns of magnetoelectric multi-phase composites with optimal magnetoelectric coupling properties. The 2D finite element method is used to calculate the coupling coefficients and build the database for the training and testing of ML algorithms. Both the convolution neural network (CNN) and artificial neural network (ANN) algorithms are used as ML algorithms in this work. By investigating the effects of network parameters, such as training data density, iteration number and batch size, we construct the networks with proper parameters and good prediction accuracy. We present the predicted geometric patterns by two methods, CNN and ANN, and compare them with the FEM patterns. Two types of magnetoelectric composites, two -volume fractions of magnetostrictive phase and two system sizes are considered to investigate and improve the prediction efficiency. The presented results show that by using the ML methods, it can well predict the coupling effect and rank optimal patterns. The results also prove the feasibility that by using the ML method, we can accurately predict the coupling performance with very limited data. We aim to predict the optimal patterns instead of the best pattern. Therefore, this work demonstrates the feasibility and offers a new perspective way for the design and optimization of magnetoelectric multi-phase composites by using the ML algorithm. [ABSTRACT FROM AUTHOR]