Real Fundamental Solutions Research Articles

A penny-shaped crack in a piezoelectric layer sandwiched between two elastic layers with the electrical saturation and mechanical yielding zones

This paper deals with an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers. It is assumed that the crack faces are electrically impermeable, and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement. Three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. For such an axisymmetric penny-shaped crack problem, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by employing Hankel transform technique and the Copson method. The problem is solved in the real domain by constructing real fundamental solutions and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization. When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity, the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature. The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is discussed in details. The results of paper are believed to have some guiding references for the design of the piezoelectric sandwich plate.

Multilayered piezomagnetic/piezoelectric composites with periodic interfacial cracks subject to in-plane loading

A two-dimensional fracture problem of periodically distributed interfacial cracks in multilayered piezomagnetic/piezoelectric composites is studied under in-plane magnetic or electric loading. The magnetic permittivity of the piezoelectric material and the dielectric constant of the piezomagnetic material are considered. A system of singular integral equations of the second kind with a Cauchy kernel is obtained by means of Fourier transform and further solved by using Jacobi polynomials. The problem is solved in the real domain by constructing real fundamental solutions. The primary interfacial fracture mechanic parameters, such as the stress intensity factors (SIFs), the electric displacement intensity factors (EDIFs), the magnetic induction intensity factors (MIIFs) and the energy release rates (ERRs) are then obtained. It is found that a magnetic or electric loading normal to the crack surfaces can lead to a mixture of mode I and mode II type stress singularities at the crack tips. Numerical results show that increasing the thickness of the active layer will favor the crack initiation. Inversely, increasing the thickness of the passive layer will retard the crack initiation. Furthermore, the results indicate that the crack initiation can be inhibited by adjusting the direction of the applied magnetic or electric loading.

Fracture problem of a nonhomogeneous high temperature superconductor slab based on real fundamental solutions

To analyze the fracture problem of the nonhomogeneous high temperature superconductor (HTS) slab under electromagnetic force, we derive the real fundamental solutions based on eigenvalue and eigenvector analyses. The superconductor E-J constitutive law is characterized by the Bean model where the critical current density is independent of the flux density. Fracture analysis is performed by the methods of singular integral equations which are solved numerically by Lobatto–Chybeshev collocation method. Numerical results of the stress intensity factor (SIF) are obtained. Moreover, the crack opening displacement (COD) can be obtained by numerical integration dislocation density functions. The effects of the thickness ratio, HTS material nonhomogeneous parameters, applied magnetic field and critical current density on SIF and COD are discussed. The present work could theoretically provide quantitative predictions of the fracture mechanism of the nonhomogeneous HTS.

Fracture analysis of a transversely isotropic high temperature superconductor strip based on real fundamental solutions

Real fundamental solution for fracture problem of transversely isotropic high temperature superconductor (HTS) strip is obtained. The superconductor E–J constitutive law is characterized by the Bean model where the critical current density is independent of the flux density. Fracture analysis is performed by the methods of singular integral equations which are solved numerically by Gauss–Lobatto–Chybeshev (GSL) collocation method. To guarantee a satisfactory accuracy, the convergence behavior of the kernel function is investigated. Numerical results of fracture parameters are obtained and the effects of the geometric characteristics, applied magnetic field and critical current density on the stress intensity factors (SIF) are discussed.

Closed-form solutions for the contact problem of anisotropic materials indented by two collinear punches

The present paper performs an exact analysis of the contact problem of anisotropic materials indented by two collinear punches. By considering the eigenvalue properties, the real fundamental solutions are given for the roots of the complex conjugate case and the real number case. A singular integral equation reduced from the stated problem is solved exactly for the case of two flat punches and two semi-cylindrical punches. Numerical results are presented to reveal how the interaction of the two punches and the elastic coefficient ratio affect the contact behaviors under the two collinear punches. When two individual rigid punches mingle as one rigid punch, classical results for one rigid punch are obtained, which validates the present derivation.

Stress state of two collinear stamps over the surface of orthotropic materials

An exact contact analysis of two collinear stamps over the surface of orthotropic materials is performed. Eigenvalue analyses related to the governing equations are conducted, and real fundamental solutions are obtained for each eigenvalue distribution. Using Fourier sine or cosine transforms, a singular integral equation of Cauchy type is obtained. In particular, the exact solution of the reduced singular integral equation is obtained in the case of two collinear flat or cylindrical stamps. Explicit expressions for various stresses are obtained in terms of elementary functions. Numerical results are presented, and interesting observations are found. For two collinear flat stamps, detailed calculations are provided to show influences of distance between two flat stamps on the contact stress and in-plane stress. For a cylindrical stamp, influences of distance between two cylindrical stamps, mechanical loading P, and the cylindrical stamp radius R on the surface normal stress and surface in-plane stress are detailed.

Multi-field Response of Anisotropic Magneto-Electro-Elastic Materials at a Rigid Conducting Cylinder

Contact problem of anisotropic magneto-electro-elastic materials indented by a perfectly conducting cylindrical punch is investigated based on a complete coupling theory. The 12 cases of the distinctive eigenvalue distribution of the related governing equations are detailed. For 3 available eigenvalue distribution cases involving semi-infinite anisotropic magneto-electro-elastic materials, real fundamental solutions are provided. A system of singular integral equations is obtained and solved exactly. The explicit expressions for the coupled magneto-electro-elastic fields in the half-plane are presented in the form of elementary functions. Figures are plotted to show the effects of various parameters, such as the volume fraction of the piezoelectric phase, on the contact behaviors. In-depth analyses are given to explain how the various parameters cause the contact properties to change and develop. Connections between the present study and practical application are presented. This article may greatly benefit the experimental and numerical test involving magneto-electro-elastic materials.

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.

Contact problem for magneto-electro-elastic half-plane materials indented by a moving punch. Part I: Closed-form solutions

A theoretical model is developed for the exact contact analysis of magneto-electro-elastic half-plane materials indented by a moving rigid punch in this paper, which is Part I of a series of papers. A numerical analysis based on this theoretical model will be presented in Part II. The Galilean transformation and the Fourier sine and cosine transforms are applied to make the transient problem tractable. Detailed analyses of the eigenvalue distributions of the double-biquadrate order characteristic equation related to the magneto-electro-elastic governing equations are performed. Real fundamental solutions are derived for each eigenvalue distribution. The punch may have a flat or cylindrical profile and may be electrically and magnetically conducting, electrically conducting and magnetically insulating, electrically insulating and magnetically conducting, or electrically and magnetically insulating. For each type of punch, the singular integral equations are derived with the surface contact stress, surface electric charge, and/or surface magnetic induction inside the contact region as the unknown functions. Exact solutions to the system of integral equations are obtained. In particular, closed-form expressions for the stresses, electric displacements, and magnetic inductions in terms of fundamental functions are derived, which provide a scientific basis for the interpretation of the contact behaviors of multiferroic materials as will be shown in Part II of this series of papers.

Exact Two-Dimensional Contact Analysis of Piezomagnetic Materials Indented by a Rigid Sliding Punch

The aim of the present paper is to investigate the two-dimensional moving contact behavior of piezomagnetic materials under the action of a sliding rigid punch. Introduction of the Galilean transformation makes the constitutive equations containing the inertial terms tractable. Eigenvalues analyses of the piezomagnetic governing equations are detailed, which are more complex than those of the commercially available piezoelectric materials. Four eigenvalue distribution cases occur in the practical computation. For each case, real fundamental solutions are derived. The original mixed boundary value problem with either a flat or a cylindrical punch foundation is reduced to a singular integral equation. Exact solution to the singular integral equation is obtained. Especially, explicit form of the stresses and magnetic inductions are given. Figures are plotted both to show the correctness of the derivation of the exact solution and to reveal the effects of various parameters on the stress and magnetic induction.

Three dimensional axisymmetric problems in piezoelectric media: Revisited by a real fundamental solutions based new method

The purpose of the present work is to establish a set of real fundamental solutions for the differential governing equations of three dimensional axisymmetric problems in piezoelectric media. Firstly, conventional complex fundamental solutions are derived by analysis on the eigenvalue problem, and then, Euler’s formula is used to transform them into equivalent real fundamental solutions. As an example of application, the fracture problem of an axisymmetric penny-shaped crack in a piezoelectric layer is resolved by the real fundamental solutions based new method. Theoretical derivation and numerical computation are validated in the special case of a penny-shaped crack in an infinite piezoelectric body. Effects of geometrical parameters and electric-loading coefficient on energy release rates are surveyed and their agreement with the results of existing papers is also indicated. The advantage of such a real fundamental solutions based new method is that it can effectively help to avoid the difficult complex analysis in mixed boundary value problems.

Theory of moving contact of anisotropic piezoelectric materials via real fundamental solutions approach

Abstract A general theory for the moving contact behaviors of anisotropic piezoelectric materials under the action of a rigid flat or cylindrical punch is proposed. It is assumed that the punch is either a perfectly electric conductor or a perfectly electric insulator. The Galilean transformation, Fourier sine and cosine transforms are employed to solve the piezoelectric governing equations containing the inertial terms. The characteristic equation is a double-biquadrate equation. A detailed analysis is performed for the eigenvalue distribution and real fundamental solutions are derived for each eigenvalue distribution. The originally mixed boundary value problem is reduced to the Cauchy integral equations and then exact solutions to these integral equations are obtained for the conducting or insulated punch with the flat or cylindrical punch profile. Finally, closed-form expressions for the stresses and electric displacements are derived. The present analysis provides a scientific basis for the interpretation of contact behaviors of anisotropic piezoelectric materials.

Thermal response of a partially insulated interface crack in a graded coating-substrate structure under thermo-mechanical disturbance: Energy release and density

Consider the thermal fracture problem of a functionally graded coating-substrate structure of finite thickness with a partially insulated interface crack subjected to thermal-mechanical supply. A new model is proposed that the heat conduction through the crack region occurs and the temperature drop across the crack surfaces is the result of the thermal resistance. For the first time, real fundamental solutions are derived for the fracture analysis of functionally graded materials. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The asymptotic expressions with higher order terms for the singular integral kernels are considered to improve the accuracy and efficiency of the numerical integration. Explicit expressions of various failure modes including stress intensity factors, energy release rate and strain energy density, are provided. Numerical results are presented to illustrate the effects of non-homogeneity parameters and the dimensionless thermal resistance on the temperature distribution along the crack surfaces and extended crack line, the thermal stress intensity factors and minimum strain energy density.

New, real fundamental solutions to the transient thermal contact problem in a piezoelectric strip under the coupling actions of a rigid punch and a convective heat supply

A thermo-electro-mechanical contact analysis has been performed for a finite piezoelectric strip, which is subjected to the joint actions of a rigid, flat punch and a transient convective heat supply. The Laplace transform and Fourier sine and cosine transforms were applied in solving the governing equations. A detailed analysis of the characteristic roots of the corresponding characteristic equation was made. Real fundamental solutions were derived, which can readily lead to real solutions to the thermo-electro-mechanical quantities. A Cauchy-type singular integral equation was obtained for the stated problem and then solved numerically. Closed form solutions of a special case were obtained. To obtain the accurate solution in the time domain, an effective numerical inversion algorithm of the Laplace transform was applied. Detailed analyses were performed to reveal the variation law of temperature, contact stress beneath the punch, stress intensity factor at the punch edge and strain with time. Parametric studies were performed to discover the effects of the layer thickness on the distribution of temperature, contact stress beneath the punch and stress intensity factor at the punch edge.

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

Abstract 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.

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.

Collinear unequal crack series in magnetoelectroelastic materials: Mode I case solved via new real fundamental solutions

For the first time, real fundamental solutions are proposed for in-plane magnetoelectroelastic governing equations. Dislocation simulation method is used to derive singular integral equations for collinear unequal crack series under mode I loadings. Special cases of a single crack and two collinear equal cracks are used to validate the proposed solution. The effects of crack space, crack length and magnetic/electric loading are discussed. Real fundamental solutions have an advantage over complex fundamental solutions, because real analyses are more straightforward than complex analyses. Moreover, the model of collinear unequal cracks is more practical than that of collinear equal cracks.