Mode I fracture analysis of a piezoelectric strip based on real fundamental solutions

In existing papers, mode I crack problems of piezoelectric ceramics are generally solved in complex domain because of the complex fundamental solutions of in-plane piezoelectric governing equations. In fact, these problems can alternatively be analyzed in real number field by recasting the solutions in real form instead. The main purpose of the present work is to develop such real fundamental solutions by detailed eigenvalue and eigenvector analyses. As an example of application, the widely studied fracture problem of a piezoelectric strip with a center-situated crack under mode I loading condition is then revisited based on the real fundamental solutions. Mixed boundary value conditions of the crack are transformed into Cauchy singular integral equations, which are then solved numerically to get fracture parameters including the energy release rate and intensity factors. Convergence behaviors of the kernel functions are surveyed. Theoretical derivation and computation are validated by the exact solution in a special case. The effect of a combined geometrical parameter on the crack is also investigated.     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

Frictional moving contact problem between a conducting rigid cylindrical punch and a functionally graded piezoelectric layered half plane

Meccanica - In this study, a frictional moving contact problem between an electrically conducting rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to a...     

Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.     

The transient piezothermoelastic problem of a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply

This paper is concerned with the theoretical treatment of the transient piezothermoelastic problem involving a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply in the width direction. The thermal, thermoelastic and piezoelectric constants of the strip are assumed to vary exponentially in the thickness direction. The transient two-dimensional temperature is analyzed by the methods of Laplace and finite sine transformations. We obtain the exact solution for a simply supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement, the stress and electric potential distributions are presented in figures and table. Furthermore, the influence of the nonhomogeneity of the material and that of the electric boundary conditions are investigated.     

Thermo-electro-mechanical contact behavior of a finite piezoelectric layer under a sliding punch with frictional heat generation

The thermal contact problem of a piezoelectric strip with heat supply generated by the frictional tangential traction under the action of a rigid sliding punch is investigated. The inertial effects are considered. It is convenient to introduce the Galilean transform. Whole cases of the root distribution of the corresponding characteristic equation are detailed. Appropriate fundamental solutions that can lead to real solutions of the thermo-electro-mechanical quantities are derived for the piezoelectric governing equation. The stated problem is reduced to Cauchy singular integral equation of the second kind finally. Numerical results are also presented. The solutions have a reduced dependence on the material properties. The singular behaviors at the edges of the punch are revealed. The stress distribution and temperature distribution above the punch with the variations of the relative sliding speed, the frictional coefficient and the thickness are plotted. The effects of the material constants on the stress distribution and temperature distribution above the punch are presented.     

Effects of the moving speed of a rigid stamp on contact behaviors of anisotropic materials based on real fundamental solutions

The paper studies contact problem of a rigid stamp moving at a constant speed over the surface of anisotropic materials. The solution method is based on Galilean transformation, Fourier transform and singular integral equation. The stated mixed boundary value problem is reduced to a Cauchy type singular integral equation based on real fundamental solutions, which is solved exactly in the case of a rigid flat or cylindrical stamp. Explicit expressions for various stresses are obtained in terms of elementary functions. In particular, explicit formula is derived to determine the unknown contact region for the cylindrical stamp. For a flat stamp, detailed calculations are provided to show the influences of dimensionless moving speed on the normal and in-plane stress. For a cylindrical stamp, the effects of dimensionless moving speed, the mechanical loading and the radius on the contact region, the normal and in-plane stress are analyzed in detail.     

The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

Critical velocity in a contact elasticity problem for the case of transonic punch velocity

The paper deals with a dynamic contact problem in the presence of friction forces in the transonic range of punch velocities, where the punch velocity exceeds the transverse wave velocity but is still less than the longitudinal wave velocity. It is shown that there exists a critical velocity at which the solution structure and the character of its behavior on the boundary of the contact region change. This velocity is $\sqrt 2 $ times the transverse wave velocity. The existence of this velocity is possibly related to the surface wave velocity under restricted deformation conditions.     

Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem

Uniqueness and continuous dependence on the initial temperature are established for the solution of a multidimensional, quasistatic thermoelastic contact problem. The proof of this result does not depend on the ability to decouple the system of governing equations as required in the technique used by Shi and Shillor [European J. Appl. Math., 1990, 371–387] in the one dimensional analogue of this problem. Some extensions to other contact problems are suggested.     

New analytic solutions to 2D transient heat conduction problems with/without heat sources in the symplectic space

The two-dimensional (2D) transient heat conduction problems with/without heat sources in a rectangular domain under different combinations of temperature and heat flux boundary conditions are studied by a novel symplectic superposition method (SSM). The solution process is within the Hamiltonian system framework such that the mathematical procedures in the symplectic space can be implemented, which provides an exceptional direct rigorous derivation without any assumptions or predetermination of the solution forms compared with the conventional inverse/semi-inverse methods. The distinctive advantage of the SSM offers an access to new analytic heat conduction solutions. The results obtained by the SSM agree well with those obtained from the finite element method (FEM), which confirms the accuracy of the SSM.     

One approach to solving the problem of the contact of a cylindrical shell with a rigid body

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Sverdlovsk. Translated from Prikladnaya Mekhanika, Vol. 26, No. 5, pp. 36–42, May, 1990.     

Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate

In this study, the frictional contact problem for a layer bonded to a homogeneous substrate is considered according to the theory of elasticity. The layer is indented by a rigid cylindrical stamp which is subjected to concentrated normal and tangential forces. The friction between the layer and the stamp is taken into account. The problem is reduced to a singular integral equation of the second kind in which the contact pressure function and the contact area are the unknown by using integral transform technique and the boundary conditions of the problem. The singular integral equation is solved numerically using both the Jacobi polynomials and the Gauss?CJacobi integration formula, considering equilibrium and consistency conditions. Numerical results for the contact pressures, the contact areas, the normal stresses, and the shear stresses are given, for both the frictional and the frictionless contacts.     

Numerical solution to a nonlinear,one-dimensional problem of thermoelasticity with volume force and heat supply in a half-space

A numerical solution is presented for a nonlinear, one-dimensional boundary-value problem of thermoelasticity with variable volume force and heat supply in a half-space. The surface of the body is subjected to a given periodic displacement. The volume force and bulk heating simulate the effect of a beam of particles infiltrating the medium. No phase transition is considered and the domain of the solution excludes any shock wave formation. The basic equations are formulated in material coordinates, making them adequate for dealing with moving boundaries. The used numerical scheme reproduces correctly the process of coupled thermomechanical wave propagation. The presented figures display the process of propagation of the coupled nonlinear thermoelastic waves. They also show the effects of volume force and heat supply on the distributions of the mechanical displacements and temperature inside the medium. Moreover, the interplay between these two factors and the applied boundary disturbance is outlined. The presented solutions, however, is not meant to capture the expected process of shock formation at the breaking distance.     

Elastic solutions of the end problem for a nonhomogeneous semi-infinite strip in anti-plane shear

The end problem referring to anti-plane shear deformation of a nonhomogeneous semi-infinite strip is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in plane problem. Two types of nonhomogeneity are considered: (i) the shear modulus varies with the thickness coordinate x exponentially; (ii) it varies with the length coordinate y exponentially. The closed elastic solutions in trigonometric series form are derived by the eigenfunctions expansion, and the completeness of the solutions is also proved. Therefore, the elastic field caused by a self-equilibrating traction on the end could be solved in an arbitrary accuracy by taking a necessary number of terms in the series, approximatively, which is usually neglected by invoking Saint-Venant principle. By considering the biggest negative eigenvalue, the Saint-Venant Decay rates of the problem is also estimated in the last section.