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Quartz crystals are widely used in sensors, auctors, filters, and resonators due to their excellent piezoelectric properties and operational stability. As electronic devices continue to miniaturize, understanding the nonlinearity in quartz crystal structures becomes increasingly important. This study aims to support the design of high-sensitivity piezoelectric sensors by analyzing the thickness-mode nonlinear vibration of randomly cut quartz crystals, incorporating the effects of initial stress. Specifically, a theoretical framework is developed to determine the nonlinear vibration frequencies of the fast and slow thickness-shear modes, as well as the thickness-stretch mode in a randomly cut quartz crystal. The study explores the dependence of vibration frequencies on nonlinear vibration amplitude and initial stress. To ensure that frequency variations are attributed to initial stress, the nonlinear vibration amplitude is maintained at a reasonable value. Furthermore, the effects of cut orientation on frequency variation under a given initial stress are examined to identify the optimal cut for frequency sensitivity. Our results demonstrate that quartz crystals exhibit high sensitivity to initial stress, with the fundamental vibration mode showing the largest frequency shift despite having the lowest frequency. This mode proves particularly suitable for sensor applications. The study identifies the cut orientation with the optimal frequency sensitivity and provides insights that could guide the design of piezoelectric sensors and expand their application.

This work aims to assess the response of viscoelastic Kelvin–Voigt microscale beams under initial stress. The microbeam is photostimulated by the light emitted by an intense picosecond pulsed laser. The photothermal elasticity model with dual-phase lags, the plasma wave equation and Euler–Bernoulli beam theory are utilized to construct the system equations governing the thermoelastic vibrations of microbeams. Using the Laplace transform technique, the problem is solved analytically and expressions are provided for the distributions of photothermal fields. Taking aluminum as a numerical example, the effect of the pulsed laser duration coefficient, viscoelasticity constants and initial stress on photothermal vibrations has been studied. In addition, a comparison has been made between different models of photo-thermoelasticity to validate the results of the current model. Photo-microdynamic systems might be monolithically integrated on aluminum microbeams using microsurface processing technology as a result of this research.

This research aims to study the different phenomena of generalized Rayleigh waves (RWs) in composite structures with piezoelectric material and corrugated surfaces. A thin piezoelectric layer that is precisely attached to an elastic substrate with initial stresses makes up the structure under consideration. By resolving the related electromechanical field equations, expressions for the displacement potential function are produced for both the elastic substrate and the piezoelectric layer. The frequency equations of the studied wave have been discovered as determinants for electrically open and short conditions. There have been both numerical illustrations and graphical demonstrations. On the stage speed of the combined RW, the impact of the grooved border, initial pressure, piezoelectric constant, dielectric constant, and thickness of the piezoelectric layer have been studied. Additionally, the analytical solution to the issue is compared to and shown to be in good agreement with the stiffness matrix approach result. This study offers a theoretical foundation for creating and developing piezoelectric composite-based devices. It has been found that the generalized RW velocity declines with the increase of the values of $\widetilde{e_{15}}$ and ${\epsilon }_{11}$ and an inverse relationship has been found with the increase of the value of $H$. In addition, it has also been noticed that the velocity profile of the generalized RW is greatly influenced by the size of the corrugation (measured at the upper layer).

The paper discusses the problem of a two-dimensional static deformation as the result of normal line-load acting inside an irregular initially stressed isotropic half-space. The eigenvalue approach method has been used. The irregularity is expressed by a rectangle shape. Further, the results for the displacements and stresses have been derived in the closed form. The effect of initial stress and irregularity are shown graphically. It was found that the initial stresses as well as irregularity have a notable effect on this deformation.

The initial stress is induced during film formation and is partially counterbalanced through curvature changes. Therefore, it is commonly evaluated by the measured residual stress. Initial stress may directly affect the film formation rather than residual stress in progressively deposited films. In the present work, we introduced a multiple layer model for progressively deposited films to obtain a quantitative solution for estimating the initial stress. The results showed that residual stress in the last layer is equal to the initial stress when layer number approaches infinity. In particular, the initial stress, σ_i in deposited film could be determined using the equation, σ_i = σ_St/β, in which σ_St is the averaged residual stress in films calculated by Stoney formula and β is the correction factor. The value of β varied between 0 and 1, depending on relative modulus and relative thickness of film and substrate. Finally, using element birth and death technique, a finite element model was presented to verify the analytical multiple layer model. Good agreement was obtained between the analytical and FE results.

In this paper, on the basis of gradient elasticity theory with one gradient parameter, wave propagation in rectangular nanoplates is studied. In the governing equation, the influences of initial stresses and elastic foundation are also considered. An analytical approach is used to solve the governing equation. The effects of different parameters such as gradient parameter on the circular and cut-off frequencies are presented. One can see that the initial stress and gradient parameter play an important role in investigating the wave propagation in nanoplates.

The vibration responses of a pre-stressed rotating cylindrical shell under stationary constant point load are analyzed by numerical method. The cylindrical shell is modeled by a nine-node super-parametric finite element which includes the effects of transverse shear deformation, axial deformation and rotary inertia. The geometric nonlinearity due to large deformation is also considered in the model. The numerical integer method is applied to attain the vibration responses. The effects of initial axial stress and rotating speed on the vibration of cylindrical shell are investigated in detail.

This paper is concerned with the vibration problem of initially stressed micro/nano-beams. The vibration problem is formulated on the basis of Eringen's nonlocal elasticity theory and the Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of initial stress, transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. These equations are solved analytically for the vibration frequencies of micro/nano-beams with different initial stress values and boundary conditions. The effect of the initial stress on the fundamental frequency and vibration mode shape of the beam is investigated. The solutions obtained provide a better representation of the vibration behavior of initially stressed micro/nano-beams which are stubby and short, since the effects of small scale, transverse shear deformation and rotary inertia are significant and cannot be neglected.

This paper deals with the nonlinear dynamic response of elastically supported stiffened plates with initial stresses under impact loads. A stiffened plate is assumed to be composed of a plate with some stiffeners, which are treated separately. The plate is modeled by the thin plate theory, whereas the stiffeners are considered as geometrically nonlinear Euler–Bernoulli beams. First, the equations of both the kinetic energies and strain energies of the plate and stiffeners are established. Then, the dynamic equilibrium equations for the stiffened plate are derived as the Lagrange’s equation of the functional. A parametric analysis is performed to evaluate how initial stresses, initial geometric imperfections, elastic supports, impact loads and configuration of stiffeners affect the time-history responses of the stiffened plates. Some useful nonlinear dynamic properties are obtained, which serve as references for engineering design and application.

The objective of this paper is to study the effect of loosely bonded interface on torsional surface wave propagation in a fiber reinforced composite medium constrained between dry sandy layer and an anisotropic gravitating poroelastic substrate. All the media are assumed to be under initial stress. The dispersion relation on this proposed multilayer ground structure has been derived in closed form under certain boundary conditions, which contain Whittaker function and its derivative, which is further expanded asymptotically, retaining up to only the linear terms. The numerical solution for the limiting case of torsional surface waves is also discussed. As a special case of the problem, when the entire medium is isotropic and one of the upper layer vanishes and removing the initial stress and gravity, the dispersion relation obtained is in agreement with the classical Love type wave equation. The influence of various technical constants, such as sandy parameter, reinforcement parameter, porosity parameter, Biot’s gravity parameter, loosely bonded parameters, initial stress of both the layers and half spaces on the phase velocity of torsional surface wave has been pointed out by means of graphs.

This paper aimed at studying the effect of initial stress and impedance boundary surface on the elastic waves propagation in transversely isotropic materials. The governing equations of transversely isotropic materials with initial stress are treated in the $xy$-plane to obtain the phase velocities of two waves, quasi-longitudinal ($qP$) and quasi-transverse ($qS$) waves. The normal mode operation is employed to obtain the expression of the displacement components and stress components. The problem of reflection of incident waves at the impedance boundary of transversely isotropic material is also investigated in the presence of initial stress to obtain the amplitude ratios of reflected waves. The variations of displacement and force stress are illustrated with respect to the axis of propagation by employing amplitude ratio expression. The effect of initial stress and impedance parameters on the expression of displacement and force stress as well as amplitude ratios of transversely isotropic materials are illustrated graphically for a particular model.

In this paper, the effect of initial stress on the propagation of Love waves in a layered structure with a thin piezoelectric film bonded perfectly to an elastic substrate has been investigated. General dispersion equations, describing the properties of Love waves in both cases, electrically open case and electrically shorted case of the piezoelectric layer, have been obtained. The effects of inhomogeneity parameters in the substrate and the initial stress in both, the layer and the substrate on the phase velocity of Love waves, are analyzed and presented graphically. The analytical method and obtained results may find applications for designing the resonators and sensors.

In this paper, we study Scholte’s waves propagating along the interface of micropolar solid and liquid media, considering the effect of initial stress and impedance boundary surface. We obtained the phase velocities of basic elastic waves in both media and used them to derive the secular equation of Scholte’s wave. We found that the initial stress in the solid material does not affect the speed of basic longitudinal waves. After solving the secular equation, we obtained the speed of Scholte’s wave and the corresponding attenuation. We observed that Scholte’s surface wave obtained at the micropolar solid–liquid interface is inhomogeneous and dispersive. We investigated the effects of initial stress, Tiersten’s impedance boundary condition, and micropolar solid parameters on Scholte’s wave and basic shear waves. Numerically, these effects are computed for a particular model using the appropriate value of the parameters, and the results are illustrated graphically. This paper also contributes to not only understanding the behavior of Scholte’s waves in micropolar solid–liquid interfaces but also the factors that affect the propagation.

The basic of the initial stress in an elastic continuum is revisited, then this investigation studies micropolar elastic solid subjected to initial stress using the classical method. The strain energy density function was developed to derive the constitutive relations and field equations when the influence of initial stress is taken into account. The case of linear-homogeneous isotropic and anisotropic condition of field and constitutive relations of the solid is obtained.

The objective of this study is to develop a theory to study the propagation of Rayleigh-type waves in an inhomogeneous layer having yielding surface. A detailed study of a Rayleigh-type wave propagating in an exponentially graded incompressible layer resting on yielding surface is considered. The frequency equation being a function of phase velocity, wave number and heterogeneity parameter associated with the yielding parameter and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by the above-stated parameters. In particular cases, the dispersion relation has been discussed for stress-free foundation by taking yielding parameter tending to zero. In numerical and graphical computation, the significant effects of distortional velocity have been carried out. Moreover, the obtained dispersion relation is found in well agreement to the classical case in homogeneous isotropic layer resting on a yielding foundation.

This chapter has two objectives: (1) to present the micro-mechanics model of Piezoelectric Fiber-Reinforced Composites (PFRCs) and illustrate some of its advantages and (2) to analytically study the impacts of normal/shear initial stresses and rotation on energies carried by different reflected/transmitted waves at the interface of two dissimilar PFRCs. Numerical studies are performed on PFRCs comprised of PZT-5A-epoxy combination and CdSe-epoxy combination, which are modeled employing Strength of Materials (SM) technique with Rule of Mixtures (RM). Some electro-mechanical advantages of PFRC over monolithic piezoelectric materials are demonstrated. Due to incidence of a quasi-longitudinal (qP) wave, three reflected/transmitted waves, viz. quasi-longitudinal (qP), quasi-transverse (qSV), and electro-acoustic (EA) waves are generated in the PFRCs. The propagation directions of all reflected/transmitted waves are graphically demonstrated. The closed-form expressions of amplitude ratios of all reflected/transmitted waves are derived utilizing appropriate electro-mechanical boundary conditions. As the amplitude ratios cannot be used exclusively to validate the numerical results, the expressions of energy ratios of all reflected/transmitted waves and interaction energy are derived, which exhibit the influence of existing parameters, and the law of conservation of energy is established. This work presents a novel effort to develop a connection between deriving the PFRC’s micro-mechanical model and analyzing the wave reflection/transmission phenomenon in it.

The propagation behaviors of surface acoustic waves in an isotropic semi-infinite solid and infinite plate with initial stress field are investigated, and we obtain the phase velocity equations of Rayleigh waves in a semi-infinite solid and infinite plate. By comparing surface acoustic waves velocity in plate under initial stress with that without initial stress, it is clear that velocity of surface acoustic waves are evidently affected by initial stress. When the stress is in the same direction of the propagation of the surface acoustic waves, the change of the wave velocity of the surface acoustic waves has certain relation to the stress within small numerical value of stress.