Displacement Potential Function Research Articles

Thermal-mechanical-electrical coupling isogeometric analysis for piezoelectric material structures

To solve the multi-physical coupling characteristics of piezoelectric structures, the thermal-mechanical-electrical coupling isogeometric analysis method (TMEC-IGA) is proposed to explore the structures of piezoelectric materials. Based on the basic equations and boundary conditions of piezoelectric materials, NURBS basis function is introduced to construct displacement and electrical potential functions, and the control equation of TMEC-IGA is derived. Numerical examples show that TMEC-IGA is effective in solving the thermal-mechanical-electro coupling problem of piezoelectric structures, and has broad application prospects in the design, development and performance analysis of piezoelectric devices.

Analytical technique for hygrothermo-electroelastic field in piezoelectric bodies with D ∞ symmetry

A general solution technique for hygrothermo-electroelastic problems in bodies with D ∞ symmetry is developed. The displacement and electric field are expressed in terms of their respective potential functions, and the hygrothermo-electroelastic field quantities are expressed in terms of elastic and piezoelastic potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two hygrothermoelastic displacement potential functions. As an application of this technique, a theoretical analysis of an infinite cylinder subjected to distributions of moisture content, temperature, and stress was performed, and the significance of the hygrothermo-electroelastic analyses was demonstrated.

Dispersive behavior of SH waves in a smart composite structure of viscoelastic media

This study focuses on the dispersive behavior of SH waves in a piezoelectric viscoelastic (PV) layer sandwiched between a magnetoelastic fiber-reinforced viscoelastic (MFRV) layer and a heterogeneous viscoelastic half-space. The linear variation of the downward-pointing space variable introduces inhomogeneity in the viscoelastic half-space. With the help of the variable separation technique, the governing equations of the present model are solved to obtain solutions for the mechanical displacement and electric potential function. A complex frequency equation for the SH waves has been derived by applying appropriate boundary conditions. The results are confirmed by comparison to the classical case of Love wave, and special cases of the problem are also discussed. In the fundamental analytical study, two cases of the PV layer are discussed, and the effects of physical parameters on the phase and damped velocity of SH waves are investigated through numerical calculations and presented graphically. A comparative study has also been carried out to analyze the effects of the viscoelastic parameters, reinforcement parameters, thickness ratio parameter, magnetoelastic coupling parameter, inhomogeneity parameter, and the angle at which the wave intersects the magnetic field on the phase and damped velocity by taking electrically open and short conditions.

Instantaneous thermal fracture behaviors of a bimaterial with a penny-shaped interface crack via generalized fractional heat transfer

In the high-temperature environment, a larger thermal expansion mismatch of a bimaterial leads to pronounced thermal stresses and interface crack growth. In this paper, a generalized fractional heat conduction model is used to determine the instantaneous thermal fracture behaviors of a bimaterial interface crack. An axisymmetric thermoelastic problem is solved with the aid of Goodier's thermoelastic displacement potential and Love's potential functions. A mixed initial-boundary value problem is then converted to a singular integral equation of second kind by using the Hankel and Laplace integral transforms. Numerical results of the intensity factors of heat flux and thermal stresses are evaluated using Stehfest's Laplace inversion scheme. The influence of fractional kernel functions with power and exponential law are analyzed, respectively, for aluminum-steel and steel-epoxy, respectively. The thermal stress intensity factors exhibit more wave-like behaviors based on Riemann-Liouville and Atangana-Baleanu models which is different from Caputo-Fabrizio and the complete diffusion model.

A Closed-Form Solution to the Mechanism of Interface Crack Formation with One Contact Area in Decagonal Quasicrystal Bi-Materials

Cracks and crack-like defects in engineering structures have greatly reduced the structural strength. An interface crack with one contact area in a combined tension–shear field of decagonal quasicrystal bi-material is investigated. Based on the deformation compatibility equation and displacement potential function, the complex representation of stress and displacement is given. Using the mixed boundary conditions, the closed-form expressions for the stresses and the displacement jumps in the phonon field and phason field on the material interface are obtained. The results show that the stress intensity factor at the crack tip is zero for the phason field. The variation in the stress intensity factor and the length of the contact zone in the phonon field is given, and the result is consistent with the properties of the crystal. The design of safe engineering structures and the formulation of reasonable quality acceptance standards may benefit from the theoretical research carried out here.

Analysis of Love-type wave in a piezoelectric layer bonded between fiber-reinforced viscoelastic and dual porous media

Love-type wave propagation in a piezoelectric layer bonded by a layer of fiber-reinforced viscoelastic and a heterogeneous fluid-saturated cracked dual porous half-space is investigated in this paper. The heterogeneity of fracture double porous medium is assumed to be exponential distribution functions. With the help of the variable separation technique, the governing equations of each layer and half-space are solved to get the solutions of mechanical displacement and electric potential function. The results are confirmed by comparison with the classical case of Love wave, and special cases of the problem are also discussed. The phase velocity is calculated numerically and plotted graphically for various material characteristics of the layers and half-space. The analysis reveals that the phase velocity rises with an increase in viscoelastic parameters and the dielectric constant, as well as the reinforcement parameters, while it decreases with an increase in the inhomogeneity parameter, piezoelectric constant, and thickness ratio parameter. A comparative study has also been conducted to analyze the effect of these parameters on the phase velocity using two piezoelectric materials, PZT-4 and PZT-7. The importance of parameter effects increases for Love-type waves with shorter wavelengths and small guiding layer thicknesses.

On generalized Rayleigh waves in a pre-stressed piezoelectric medium

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 [Formula: see text] and [Formula: see text] and an inverse relationship has been found with the increase of the value of [Formula: see text]. 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).

Electromechanical coupling enriched finite element method for dynamic characteristic of piezoelectric materials structures

To improve the application depth and breadth of intelligent unit devices made from piezoelectric structures, their kinetic properties shall be urgently studied. Herein, based on the basic equations and boundary conditions of piezoelectric materials, an interpolation coverage function of node displacement was introduced into the displacement shape function and potential shape function of coupling electromechanical finite element method (FEM), forming an electromechanical coupling enriched FEM generalized shape function. Together with the variation principle, an electromechanical coupling enriched FEM was put forward, and its motion equations were deduced. Next, the free vibration and transient response of the piezoelectric structure were analyzed, and the accuracy and validity of the new method were validated with numerical cases. Results show this method has high application prospects for analyzing the dynamic properties of piezoelectric material structures.

Saint-venant Torsion of Functional Graded Orthotropic Piezoelectric Hollow Circular Cylinder

This paper gives an analytical solution to the Saint-Venant torsion of the hollow and solid circular cylinder made of orthotropic functionally graded piezoelectric material. The elastic flexibility coefficients and piezoelectric constants and permittivities have only radial dependence. The considered material non-homogeneity is described by power function. The solution of the Saint-Venant torsion problem is presented for Prandtl’s stress function, shearing stresses, electric displacement potential function, torsion function and electric potential function. An example illustrates the application of the formulated analytical method.

Scattering of plane SH waves by a circular tunnel in nonlocal fractional-order viscoelastic half-space

This paper proposed a complex function solution to the scattering of plane SH waves by circular tunnels in nonlocal fractional-order viscoelastic half-space. The nonlocal theory and the fractional order viscoelastic model are used to describe the viscoelastic properties and nonlocal effects of the soil, respectively. The basic equations are expressed in the complex domain based on the theory of complex functions, and then the scattering displacement potential function of plane SH waves is obtained using the wave function expansion method. The correctness of the presented method is verified by degrading it to the classical local elastic solution. Finally, the effects of non-local parameters, soil viscosity and fractional order on the distribution of dynamic stress concentration factors inside the tunnel were analyzed, and it was found that they had significant effects on the dynamic stress concentration factors under high-frequency wave incidence.

Acoustic Radiation From Stiffened Double Concentric Large Cylindrical Shells: Part I Circumferential Harmonic Waves

Abstract Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.

Thermoelastic fields for a heat exchanger of arbitrary shape in a bi-material infinite plane

When a bi-material with two jointed dissimilar half-planes containing an arbitrarily shaped polygonal inclusion is subjected to heat flow, the thermoelastic fields, including temperature and displacement, can be derived by the Green’s function technique with the integral of the source over the inclusion. Using Hadamard’s regularization, the two-dimensional (2D) thermal, elastic, and thermoelastic Green’s functions of two-jointed dissimilar half-planes are firstly derived from the 3D Green’s functions as the corresponding fundamental solutions. The fundamental solutions for semi-infinite and infinite domains can be recovered by adjusting the material constants. Eshelby’s tensors are derived in terms of the biharmonic, harmonic, and two Boussinesq’s displacement potential functions. When a heat exchanger of arbitrary shape is embedded in a matrix with different thermal and mechanical properties, combining a continuously distributed eigen-temperature gradient and eigenstrain field, the dual equivalent inclusion method (DEIM) is applied to handle the material mismatch of thermal conductivity, stiffness, and thermal expansion coefficient, respectively. Therefore, the full thermoelastic fields can be obtained by the integral over the heat exchanger only. The eigen-fields are expanded in the Taylor series referred to the center of the particle, which exhibits tailorable accuracy with uniform, linear or quadratic terms in comparison with the analytical solution for a circular inhomogeneity in the infinite domain. An exact thermoelastic solution of a circular inhomogeneity embedded within the infinite domain is present. The case study of an electric heat cable in the concrete block demonstrates the capability and exactness of the model. The method can be used for a thin film containing a heat exchanger of arbitrary shape as either a heat sink or source.

An analytical method for free vibration analysis of multi-layered transversely isotropic cylindrical shells

This work presents an effective analytical method based on displacement potential functions (DPF) for solving 3-D free vibration problem of thick and multi-layered transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. These equations are analytically solved using the separation of variables method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. One of the advantages of the present work is the analysis of thick cylindrical shells using an exact method without any approximations and simplifying assumptions in the distribution of stress or strain along the thickness of the shells. The analysis is extended to multi-layered cylindrical shells by applying appropriate boundary conditions at the interfacial surfaces of the layers. Numerical comparisons to other analytical works shows the accuracy of the present work and also due to the lack of study for multi-layered transversely isotropic cylindrical shells, 3-D finite element analysis has been performed for comparison. In addition, various more involved problems are studied and solved analytically for single and multi-layered shells of transversely isotropic material and the effect of various parameters such as shell dimensions, mechanical properties, number of layers and other parameters on shell behavior are evaluated. The results of the present study can be used as benchmark solutions for other studies.

Validating bond-based peridynamic model using displacement potential approach

Although peridynamics is widely used to investigate mechanical responses in materials, the ability of peridynamics to capture the main features of realistic stress states remains unknown. Here, we present a procedure that combines analytic investigation and numerical simulation to capture the elastic field in the mixed boundary condition. By using the displacement potential function, the mixed boundary condition elasticity problem is reduced to a single partial differential equation which can be analytically solved through Fourier analysis. To validate the peridynamic model, we conduct a numerical uniaxial tensile test using peridynamics, which is further compared with the analytic solution through a convergence study. We find that, when the parameters are carefully calibrated, the numerical predicted stress distribution agrees very well with the one obtained from the theoretical calculation.

Hydrodynamic pressures analysis of submerged floating tunnel under coaction of seismic SV wave and ocean wave

Submerged floating tunnel (SFT) is an innovative traffic structure for crossing long waterway or deep straits. To investigate the hydrodynamic pressure acting on the SFT under coaction of seismic SV wave and ocean wave, a theoretical model is developed for considering the effect of marine sediment. According to the displacement potential functions and linear wave theory, the reflection and refraction coefficients of wave action in different media are derived. Numerical examples are illustrated to analyze the hydrodynamic pressure distribution. The results show that the hydrodynamic pressure has a spatial difference along the SFT longitudinal direction when the structure operating in the water subjects different incident phase ocean wave. Wave incidence frequency not only changes the loading action mode but also affects the amplitude of the hydrodynamic pressure. Besides, due to the ocean wave loading combined, the effect of saturation on the hydrodynamic pressure load on the structure is not obvious. And the dense arrangement of tethers is recommended to the structure construction.

3-D analytical solution of non-homogeneous transversely isotropic thick closed cylindrical shells

This paper presents an effective analytical method based on displacement potential functions (DPF) for solving 3D static problem of thick and multilayer transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. The governing equations are solved by the separation of variable method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. The analysis covers a straightforward solution process for handling problems on multilayered cylindrical shells of transversely isotropic material, adopting all boundary and continuity conditions. Extensive sets of general radial loads located on the inner and outer faces of the cylindrical shell may be stated and examined with in a systematic manner. Comparisons are performed to other existing analytical results for one and multilayered cylindrical shells, and show excellent agreement for different materials, thicknesses and aspect ratios of the shell. In addition, various more involved problems are studied and solved analytically for single and three-layered shells of transversely isotropic material with different sets of radial loading functions at the outer and inner shell surfaces. The results of the present study can be used as benchmark solutions for other studies.

Dynamic analysis of piezoelectric materials with an elliptical hole under the action of shear horizontal waves

The dynamic mechanical properties of piezoelectric materials are of interest because of their brittle nature. Previous studies have mainly focused on cylindrical cavities or cracks. However, these are specific forms of elliptical cylindrical cavities. This study aims to develop a semi-analytical approach for analyzing the anti-plane dynamics of piezoelectric materials containing elliptical cavities by exploiting the geometrical generality of ellipses. The Helmholtz and Laplace equations were identified as governing equations by decoupling the displacement and electric potential functions. The wave function expansion method was employed to express the displacement and electric potential functions in the form of a Mathieu function series containing unknown coefficients in the elliptical coordinate system. The unknown coefficients were determined using the traction-free boundary and electric displacement continuous condition. Although the method of intercepting a finite-term series was used, the solutions are reliable because the wave function exhibits convergence. In addition, a parametric study was conducted. The results demonstrated the importance of using large axial ratios, small wave numbers, and large piezoelectric characteristic parameters in the dynamic analysis of piezoelectric materials with elliptical defects.

Remarks on the scattering phenomena of love-type wave propagation in a layered porous piezoelectric structure containing surface irregularity

The present analysis adduces the scattering of Love-type wave in a composite structure comprised of an irregular porous piezoelectric layer nested on a porous piezoelectric substrate. The dispersion equation for the scattering of Love-type waves in the aforesaid composite structure have been derived. The impacts of the parameters associated with the piezoelectricity of the upper layer on the phase and group velocities are also observed. The effect of vertical irregularity parameter, piezoelectricity of upper layer, and porosity of the upper layer on the reflected mechanical displacement and electrical potential functions (for both solid and fluid phases) have also been examined.

Propagation of leaky Rayleigh waves along a curved fluid-solid interface.

A characteristic equation is derived for a leaky Rayleigh wave (LRW), propagating on a curved fluid-solid interface. The equations of motion for the curved solid and fluid are formulated using the constitutive equations of a homogenous isotropic curved solid and an inviscid fluid, respectively. The displacement potential functions are used to simplify the derivation. The interface conditions are used to ensure continuities of the mass, momentum, and energy across the interface. Then, with the consideration of the interface radius of the curvature, the characteristic equation for the LRW is established and solved numerically by Muller's method. One important outcome is that there is weaker directional dependence for the velocity of the LRWs on the radius of curvature in comparison with the Rayleigh waves at an air-solid interface. However, the numerical results show a strong directional dependence for the attenuation due to the LRW leakage on the complex curvatures. Moreover, a quantitative relation between the curvature and attenuation caused by the leakage for different materials is shown. The results are significant especially with respect to relevant future applications of ultrasonic testing.

EFFECT OF SEDIMENT MODEL ON DYNAMIC PRESSURES OF SUBMERGED FLOATING TUNNEL DUE TO SV WAVE INCIDENCE

To study the dynamic pressure acting on the submerged floating tunnel (SFT) under the seismic SV wave in complex seawater environment, a theoretical analysis model is proposed considering the marine sediment effect. Based on the wave equations and displacement potential functions of different media materials, the theoretical equations of wave transmission and reflection coefficients at different interfaces are deduced and solved on MATLAB through the seismic SV wave propagation boundary. Numerical examples are employed to illustrate the effects of the saturation of the sediment, the incident SV-wave angle and the layout of tether on the dynamic pressure on the SFT. The results show that the unsaturated sediment has greater effect on the dynamic pressure on the SFT than that of the saturated sediment. Thicker sediment is more benefit to the SFT under the action of seismic SV wave. And increasing the tether stiffness or decreasing the tether spacing can reduce the dynamic pressure on the SFT.