Displacement Potential Function Research Articles

Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

• A set of analytical Green's functions for a cross anisotropic inhomogeneous tri-material solid is given. • The response of a graded solid sandwiched between two homogenous transversely isotropic layers is studied numerically. • The analysis of an finite bi-material layer has been given in terms of Green's functions and analyzed numerically. • The results may be used as benchmarks for numerical studies or developing numerical methods like boundary element methods . This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.

An analytical Solution for Bending Problem of micro Plate using New Displacement Potential Function

In this paper to include small scale effect, the augmented Love displacement potential functions (DPF) are developed for isotropic micro or Nano scales medium based on couple stress theory. By substituting the new DPF in equilibrium equations, governing equations are simplified to two linear partial differential equations of sixth and fourth order. Then the governing differential equations are solved for simply supported rectangular plate using the separation of variable method with satisfying exact boundary conditions without any simplification assumptions. Displacements, bending and torsional moments of rectangular plate are obtained for different length scale parameters, aspect and Poisson’s ratios. The obtained results are compared with other studies which show excellent agreement between them.

Dispersion Waves in a Submerged Thermo-Piezoelectric Membrane

Abstract Wave propagation in a thermo piezoelectric membrane immersed in an infinite fluid medium is discussed using three-dimensional linear theory of elasticity and thermos piezoelectricity. Three displacement potential functions are introduced to uncouple the equations of motion, heat and electric conduction equations. The frequency equations are obtained for longitudinal and flexural modes at the solid fluid interfacial boundary conditions. The numerical results are analyzed for PZT-4 material and the computed stress, strain, electric displacement and temperature distribution are presented in the form of dispersion curves and its characteristics are studied.

An exact three dimensional solution for bending of thick rectangular FGM plate

This study uses the displacement potential function, DPF, to present an exact solution for bending of functionally graded, FGM, thick plates. The plates are simply-supported rectangular, isotropic, and nonhomogeneous, with exponential variation of Young's modulus along their thickness Using the DPF, the governing equations are simplified to second and fourth order partial differential equations solved by using the separation of variables method and applying exact boundary conditions. The solution is applicable to any plate with no restriction on its thickness as well as to all FGM plates. The results are compared with other existing analytical and numerical results for thin, moderately thick, and thick plate. Comparisons show excellent agreement making the method applicable and reliable for various kinds of material properties, thickness ratios and loading conditions without any simplifying assumptions such as for strain or stress distribution in plate thickness. Various inhomogeneity, thicknesses, and aspect ratios are also investigated with respect to plate responses with results showing that increase in inhomogeneity, and decrease of aspect ratio and FGM plate thickness result in decrease in the dimensionless transverse displacement of the plate. Moreover, the effects of increasing inhomogeneity on transverse displacement are more significant for thicker plates and higher aspect ratios.

Analysis of thermal effect around an underground storage cavern with a combined three-dimensional indirect boundary element method

This paper combines a displacement discontinuity method (DDM) for elasticity and an indirect boundary element method (IBEM) for heat conduction to investigate three dimensional thermal displacements and stresses, such as those around an underground storage cavern. The thermal displacements and stresses due to temperature change are expressed in terms of a thermal displacement potential function. For a constant point heat source, we present a derivation for solution of the thermal displacement potential function and the solution is the same as found in literature. A hybrid scheme with semi-analytical integration and a direct numerical integration are used to integrate the thermal displacement potential function due to a constant heat source on a triangular region. The hybrid scheme is used to overcome a difficulty arising with the semi-analytical integration for points whose projection on the integration plane is close to the vertices of the triangle. The combined IBEM with the hybrid integration scheme is first verified with the analytical solution for an infinite elastic body with a spherical cavity. The results agree well, although differences are observed in the early stages of simulation. Then the method is used to simulate the deformation around a liquefied natural gas underground storage cavern.

Propagation characteristics of vibration waves induced in surrounding rock by tunneling blasting

The effect of blasting vibration waves on surrounding rock and supporting structures is an important field in underground engineering. In this paper, the separation variable method is used to solve the displacement potential function for the propagation of the blasting vibration waves. In the axis coordinate system, the particle motion and stress change with axial distance, radial distance and time is obtained in surrounding rock. The peak particle velocity law in surrounding rock under different blast loads and surrounding rock parameters is discussed. In addition, the particle vibration characteristics in the surrounding rock are studied using numerical simulations method. The results shows that the peak particle velocity in surrounding rock appears negative exponent attenuation with the increase of axial distance, but it appears positive and negative fluctuations in radial direction. This phenomenon is a new discovery and it has been rarely investigated before. Moreover, the peak particle velocity attenuates more quickly and intensely in the near blasting field, which means that the supporting structure in a shorter distance away from the heading face is vulnerable to the impact of blasting vibration. The attenuation of blasting vibration velocity is closely related to charge length, blasting load amplitude, attenuation index and rock elastic modulus. The numerical simulation accomplishes the same results and then demonstrates the validity of theoretical results.

Analytical technique for thermoelectroelastic field in piezoelectric bodies with D∞ symmetry in cylindrical coordinates

ABSTRACTA general solution technique for non-axisymmetric thermoelectroelastic problems in cylindrical domains with D∞ symmetry is constructed. The displacement and electric field are expressed in terms of the respective potential functions, and the thermoelectroelastic field quantities are expressed in terms of the elastic and piezoelastic potential functions, each of which essentially satisfies a Laplace equation with respect to the spatial coordinates, combined with the two thermoelastic displacement potential functions. As an application of the technique, the theoretical analysis of a solid cylinder subjected to combined mechanical and thermal loading is performed, and the necessity of the thermoelectroelastic analyses is demonstrated.

Displacement potentials for functionally graded piezoelectric solids

• Solutions of Functionally Graded Transversely Isotropic Piezoelectric Media studied. • Displacement potential functions for the media are introduced. • The system of governing differential equations has been reduced and decoupled. • Limiting cases of the problem are discussed in detail. Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.

Benchmark solution for free vibration analysis of transversely isotropic thick rectangular plates

In this research, by using displacement potential functions, the exact solution is presented for free vibration analysis of simply supported rectangular transversely isotropic plates with constant thickness. The displacement components of the plate are written in terms of displacement potential functions, and the governing differential equations are derived by the substitution of the displacement components in Navier’s equations. The two governing partial differential equations of fourth and second order are solved using the separation of variables method and satisfying the exact boundary conditions. Having no simplifying assumptions for the strain or stress distribution in the plate thickness, the obtained results in this paper are applicable to any arbitrary plate thickness with no limitation on its thickness ratio. Due to absence of available research on thick transversely isotropic plates, the obtained results are compared with other analytical works for thin and moderately thick plates and the finite element method for thick plates, showing remarkable agreement. Accurate natural frequencies are presented for different ranges of thickness ratio and aspect ratio and different materials. It is observed that with increasing thickness the ratio and aspect ratio of the plate, the non-dimensional natural frequencies decrease and increase, respectively. In addition, comparative results of isotropic material and four different transversely isotropic materials also illustrate that the shear modulus in transverse direction has important influence on the vibrational behavior of plates.

Benchmark solution for buckling of thick rectangular transversely isotropic plates under biaxial load

This paper presents an analytical solution for elastic buckling problems of thick rectangular transversely isotropic simply supported plates subjected to uniaxial or biaxial uniformly distributed in-plane load, one of which could also be tensile. Governing equations are simplified to two partial differential equations using displacement potential functions (DPF) which are solved using the separation of variables method with exact satisfaction of boundary conditions. The presented solution is applicable to any plate with no restriction on its thickness ratio and also to any composite plates which have the same in-plane mechanical properties in all directions of the plate. Obtained results for critical buckling loads are compared with other analytical results for thin and moderately thick, and with numerical results for thick plates which show excellent agreement between them. Then, the effects of loading conditions aspect and thickness ratios for four transversely isotropic materials and also the engineering constants on the buckling of plates are discovered. The results show that the presented method is applicable and reliable for various kinds of material properties, thickness ratios, and loading conditions. Results also show that in comparison with a uniaxially loaded plate, adding a compressive load in second direction decreases the critical buckling load and the plate buckles in lower modes, while for adding a tensile load it is vice versa. Among materials engineering constants, in-plane modulus of elasticity and transverse shear module have the most effects on the critical buckling load of plates.

Three dimensional solution of thick rectangular simply supported plates under a moving load

Dynamic behavior of continuous systems such as beams and plates, under a moving load is an important engineering subject. In this paper, 3D elasticity equations are solved by use of the displacement potential functions and the exact solution of a simply supported thick rectangular plate under moving load is presented. For this purpose, the governing equations in terms of displacements, Navier’s equations, are converted to two linear partial differential equations of forth and second order using displacement potential functions. Then the governing equations in terms of the potential functions are solved using the separation of variables and Laplace integral transform, satisfying exact initial and boundary conditions. In order to validate the present approach, the obtained results of this study are compared with the results of the classical theory of plates for thin and existing solutions for moderately thick plates. Also, it is observed that the speed of a moving load has an important effect on the dynamic response of plate.

An efficient multiscale computational formulation for geometric nonlinear analysis of heterogeneous piezoelectric composite

In this contribution, an efficient multiscale computational formulation is developed for the geometric nonlinear analysis of heterogeneous piezoelectric materials. In this formulation, the relations between the microscopic heterogeneous properties and macroscopic behaviors of the piezoelectric composite are established by the numerically constructed displacement and electric potential base functions. The heterogeneous and complex microstructure can be equivalent to a simple macroscopic piezoelectric coarse element through those constructed multiscale base functions. Then the equivalent tangent stiffness matrix and internal force vector of the macroscopic coarse element are deduced based on the co-rotational approach, which can describe the motion of macroscopic coarse element clearly and efficiently. Thus the original electro-mechanical coupling geometric nonlinear problem could be solved iteratively on the macroscopic scale, which will save a tremendous amount of computing time and cost. After all the macroscopic calculations, the microscopic mechanical and electrical responses could be retrieved and calculated from the macroscopic solutions by using the above-mentioned multiscale shape functions. To verify the validation and high-efficiency of the proposed multiscale computation formulation, several typical numerical examples are carried out. All the computation results indicate that the developed multiscale formulation not only could provide high precision solutions but also has high efficiency.

The Rotation Effect on Shear Horizontal Surface Wave in a Piezoelectric Semi-Space Covered by a Semiconductor Film

In this study, a mathematical model of the rotation effect on Shear Horizontally wave (SH-wave) propagation in a piezoelectric semi-space covered by a semiconductor film is investigated. The semiconducting layer is rotating with a constant angular velocity and the interface between the piezoelectric substrate and the semiconductor layer is imperfectly bonded. Furthermore, the surfaces of the bilayer system assumed free of traction and electrically shorted or open. The governing equations of the dynamical displacement and electrical potential function under the effect of rotation are driven by solving the coupled electromechanical field equations of the piezoelectric half-space and the semiconductor film. In addition, the exact frequency equations of SH waves are derived. Next, the numerical examples are considered to clarify the effects of rotation and electromagnetic boundary conditions for the different values of the film thickness and wave number on the dispersion behaviors. Finally, the effect of rotation on the frequency equation is investigated in detail for piezoelectric material PZT-5H and the semiconductor silicon. The obtained results provide a predictable and theoretical basis for applications of piezoelectric and semiconductor structures to surface acoustic wave equipments.

Fractional Heat Conduction in a Thin Circular Plate With Constant Temperature Distribution and Associated Thermal Stresses

In this work, a fractional-order theory of thermoelasticity by quasi-static approach is applied to the two-dimensional problem of a thin circular plate whose lower surface is maintained at zero temperature, whereas the upper surface is insulated and subjected to a constant temperature distribution. Integral transform technique is used to derive the solution in the physical domain. The corresponding thermal stresses are found using the displacement potential function.

Hydrodynamic Pressure on Submerged Floating Tunnel under the P-Wave

Based on the displacement potential functions, considering submarine rock and soil as elastic half-space, seawater as ideal fluid, anchors as springs, the formulae for determining the hydrodynamic pressure on submerged floating tunnel were deduced while the P-wave was incident. Subsequently, effects of different submarine rock and soil parameters (such as shear modulus, Poisson ratio) and different spring constants and spaces of the anchors on the hydrodynamic pressures were discussed. It could be concluded that for normal incidence, the shear modulus and the Poisson ratio of submarine rock and soil have no influence on the amplitude of the hydrodynamic pressure on SFT for. The peak value of the amplitude of the hydrodynamic pressure on SFT increases as the Poisson ratio of submarine rock and soil and the spring constant of anchor increases, decreases as the spacing of the anchor increases.

The Frequency Equations for Shear Horizontal Waves in Semiconductor/Piezoelectric Structures Under the Influence of Initial Stress

In this paper, we investigated analytically the frequency equations for shear horizontal wave propagation in a piezoelectric half space covered by a semiconductor film with initial stress effect. The semiconducting layer is influenced by initial stress and the interface between the piezoelectric substrate and the semiconductor layer. The governing equations of the mechanical displacement and electrical potential function under the effect of initial stress are obtained by solving the coupled electromechanical field equations of the piezoelectric half-space and the semiconductor film. Next, the numerical examples are presented to illustrate the influence of initial stress and electromagnetic boundary conditions for the different values of the film thickness and wave number. Furthermore, we studied in more details the effect of initial stress on the frequency equation for piezoelectric Barium Titanate (BaTiO3) and semiconductor silicon. The obtained results provide a predictable and theoretical basis for applications of piezoelectric and semiconductor composites to acoustic wave devices.

Analytical technique for thermoelectroelastic field in piezoelectric bodies with D∞ symmetry

ABSTRACTA general solution technique for thermoelectroelastic problems in bodies with D∞ symmetry is constructed. The displacement and electric field are expressed in terms of the respective potential functions, and the thermoelectroelastic field quantities are expressed in terms of the elastic and piezoelastic potential functions, each of which satisfies the Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two thermoelastic displacement potential functions. As an application of the technique, the theoretical analysis of a semi-infinite body subjected to a temperature distribution is performed, and the significance of the thermoelectroelastic analyses is demonstrated.

Electroelastic response of a piezoelectric cylinder with D∞ symmetry under axisymmetric mechanical loading

To understand the electroelastic field in fibers made of biodegradable polymers such as poly-l-lactic acid, we theoretically analyze the axisymmetric electroelastic field in a solid and infinitely long cylinder with D∞ symmetry, when subjected to a combined pressure and torsional shear stress. By introducing two kinds of displacement potential functions and an electric potential function and applying a Fourier transform technique, we derive the formulae for the electroelastic field quantities inside the body. Moreover, by performing numerical calculations, we elucidate the field both qualitatively and quantitatively, and investigate the possibility of using fibers made of poly-l-lactic acid for medical engineering applications.

Axisymmetric time-harmonic response of a surface-stiffened transversely isotropic half-space

This study deals with the elastodynamic response of a surface-stiffened transversely isotropic half-space subjected to a buried time-harmonic normal load. The half-space is reinforced by a Kirchhoff thin plate on its surface. By virtue of a displacement potential function and appropriate time-harmonic Green’s functions of transversely isotropic half-spaces, a robust solution corresponding to two plate-medium bonding assumptions, namely (a) frictionless interface, and (b) perfectly bonded interface is obtained for the first time. All elastic fields of the problem are expressed explicitly in the form of semi-infinite line integrals. Results of some limiting cases including isotropic materials, static loading, and surface loading are recovered from the obtained solutions and subsequently have been verified with those available in the literature. Effects of anisotropy, depth of loading, bonding assumption, and frequency of excitation on the results are precisely discussed. Based on the proposed numerical scheme, some plots of practical importance are depicted.

A general solution technique for electroelastic fields in piezoelectric bodies with D ∞ symmetry in cylindrical coordinates

To elucidate the electroelastic field in bodies with D ∞ symmetry such as wooden materials, we constructed a general solution technique for electroelastic problems in such bodies in a cylindrical coordinate system. We introduced the displacement and electric potential functions to express the displacement and electric field in cylindrical coordinates; their governing equations were obtained using the fundamental equations for the electroelastic field. The electroelastic field quantities could be expressed in terms of two elastic displacement potential functions and two piezoelastic displacement potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed cylindrical coordinates. As an application of the technique, we analyzed the problem of an infinitely long cylinder subjected to a non-axisymmetrically distributed electric surface potential. Using numerical calculations, we elucidated the electroelastic field quantities within the cylinder and found unique electroelastic coupling behaviors, which clearly demonstrate the necessity for the analytical technique presented. Moreover, we confirmed the possibility of the nondestructive evaluation (NDE) techniques by use of the piezoelectric effects.