Dual Integral Equations Research Articles

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Hot indentation of a FGM-coated thermoelastic half-space by a conical punch: Approximated analytical solution of the contact problem

Hot indentation of a thermoelastic coated half-space by a rigid punch is considered. The coating is assumed to be functionally-graded with all group of properties arbitrary varying with depth. The deformation of the half-space caused by the simultaneous mechanical and thermal loading is considered. The punch is subjected to a normal centrally applied force and preliminary heated to a uniformly distributed temperature. The contact problem is reduced to the solution of a system of dual integral equations over two unknown functions: Hankel transform of normal stresses and heat flux in the contact area. Heat flux is obtained using the bilateral asymptotic method. This method was sufficiently generalized to take into account additional function describing the displacements of the surface caused by thermal heating and obtain contact stresses. The solution is obtained in an analytical form which clearly demonstrates the dependence of the contact characteristics from the initial physical parameters of the problem. Numerical analysis is provided to illustrate the effect of the presence of the coating and its thickness on the results. Differences between the homogeneous and functionally-graded coatings is illustrated.

Griffith crack analysis in nonlocal magneto-elastic strip using Daubechies wavelets

The main focus of this work is to employ the Daubechies wavelet approximation to analyze the Griffith crack in nonlocal magneto-elastic horizontally shear (SH) wave propagation. It is assumed that the crack is located in a homogeneous isotropic nonlocal magneto-elastic strip of finite thickness and infinite length. The governing mixed boundary value problem is converted to a dual integral equation by employing Fourier transformation. This dual integral equation is further converted to a Fredholmn integral equation of the second kind by means of Abel transformation. A wavelet-based approximation scheme is employed to solve the integral equation as it is not solvable analytically. Griffith crack analysis using Daubechies wavelets has not been studied previously in the literature. The stress intensity factor (SIF) at the crack tip is determined numerically after solving the integral equation. In the end, the effect of the magnetic field, strip breath, incident angle of the shear wave and the nonlocal parameter on stress intensity factor (SIF) at the crack tip is explained through various graphical representations.

Axisymmetric frictionless contact between an elastic layer thickness and a circular base under a rigid punch

The study presented in this work deals with analytical methods for an axisymmetric problem of an elastic layer partially reposing on a rigid circular base, and is indented along the upper surface with a rigid punch. The contact between the medium and the base is smooth. This boundary value problem is transformed into a system of dual integral equations. In contrast to the classical approach consisting in resolving the corresponding Fredholm equation of the second kind, the latter equations are obtained from an infinite algebraic system of simultaneous equations, where the particular case [Formula: see text] is verified. The results of this system are also obtained numerically. The normal displacement, normal stress, and the stress singularity factor are given analytically and shown graphically with discussion. By comparison with those predicted by the finite element method, the accuracy of the numerical method is approved.

Simulation of a pre-deformed plate compression by two indenters of complex shape

Within the framework of linearized formulation of the elasticity theory problems, the stress-strain state of a pre-deformed plate, which is modeled by a pre-stressed layer, is analyzed in the case of its smooth contact interaction with a two rigid axisymmetric indenters. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

Фрикційний розігрів системи штамп-пружна півплощина при ковзанні вздовж твірної

Friction heating of system punch-elastic half plane when sliding along creative line is considered. Model of so-called “third body”, i.e., thin near-surface and intermediate layers, the physical and mechanical properties of which differ from those of the interacting bodies, and by the microgeometry of their surfaces in the contact zone, used for mathematical description of contact. The method of determination of thermal contact conductance in mathematical modelling of contact interaction with considering friction and hear generation by “third body” is presented. Using of modified conditions of heat contact in mathematical model of contact thermoelasticity, taking into account of friction and heat generation is proposed. The solution of the problem of thermoelasticity for a half-plane is obtained by means of the Fourier integral transformation. Heat conductivity problem for the punch is solved by method of straight lines. The system obtained of dual integral equations is reduced to the system of linear algebraic equations by means of points collocation method. Formulas for thermal fields, heat fluxes and contact stresses are proposed. In order to obtain the unknown contact area, the iterative scheme based on a control of a sign of normal stresses in the immediate contact interaction zones is used. Method of moving line of separation of boundary conditions is proposed.

The Effect of Adhesion on Indentation Behavior of Various Smart Materials

The nanoindentation technique plays a significant role in characterizing the mechanical properties of materials at nanoscale, where the adhesion effect becomes very prominent due to the high surface-to-volume ratio. For this paper, the classical adhesion theories were generalized to study the contact behaviors of various piezoelectric materials indented by conical punches with different electric properties. With the use of the Hankel integral transform, dual integral equations, and superposing principle, the closed-form solutions of the physical fields for the Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models were obtained, respectively. The contribution of the electrical energy to the energy release rate under the conducting punch was taken into consideration. The relationships between the contact radius, the indentation load, and the indentation depth were set up using the total energy method for the JKR model and the Griffith energy balance for the M-D model, respectively. Numerical results indicate that increasing the half cone angle of the conical punch enhances the adhesion effect, which can significantly affect the accuracy of the results of characterization in nanoindentation tests. It was found that the effect of electric potential on adhesion behaviors is sensitive to different material properties, which are not revealed in the existing studies of axisymmetric adhesive contact of piezoelectric materials and multiferroic composite materials. The load-displacement curves under conical punches with different half cone angles have very different slopes. These results indicate that the half cone angle has a prominent effect on the characterization of mechanical properties of piezoelectric solids in nanoindentation tests.

Axisymmetric torsion problem by a rigid disc of an elastic half-space weakened by an annular crack

This investigation is devoted to an axisymmetric torsion of a circular disc subjected to a half-space, containing an annular crack. The stress–strain state and the stress intensity factors pertaining to the inner and outer crack edges are investigated. The proposed problem is solved by applying the Hankel integral transformation method. The mixed boundary-value problem is reduced to a system of coupled dual and triple integral equations. We developed a new method to solve this system. By using the Gegenbauer formulas, we obtain a system of infinite algebraic equations for getting the unknown functions. The solution is analysed and discussed. Numerical computations are carried out and the results are presented in the form of tables and graphs. Some special cases are considered.

Scattering of SH wave by a crack in a functionally graded one dimensional hexagonal piezoelectric quasicrystals

AbstractIn this paper, the dynamic interaction between SH wave and a crack in functionally graded one‐dimensional hexagonal piezoelectric quasicrystals (PQCs) is studied by using integral transform technique. To make the analysis executable, the material properties are assumed to vary partially in the direction perpendicular to the crack face following an exponential function. The problem was transformed through Fourier transform into three pairs of dual integral equations, which are solved numerically by the Copson method. The explicit expressions of phonon field, phason field and electric field on crack surface are determined. The dynamic field intensity factors of crack tip are given and some special cases are discussed. Numerical examples are then conducted to reveal the influences of amplitude, incident angle and wave number of SH wave and crack length, gradient parameter on the normalized dynamic stress intensity factors (DSIFs). The results of this paper lay a theoretical foundation for the design and nondestructive testing of QCs structures.

On the adhesive nanocontact of a graded coating

This paper analyzes the adhesive nanocontact properties between a rigid cylinder and an exponentially graded coating-substrate structure. While adhesion between the indenter and the graded coating is accommodated by Johnson–Kendall–Roberts adhesive model, surface effects are addressed by the full version of Steigmann–Ogden surface mechanical theory. In the presence of surface effects alone, the fundamental Flamant solution of the graded coating-substrate structure is determined first, with the aid of Fourier integral transforms. The displacement boundary condition under the rigid cylinder and the static equilibrium equation lead to dual Fredholm integral equations of the first kind. The adhesion is subsequently introduced in terms of the principle of minimum potential energy, by which both the adhesive nanocontact length and pressure are uniquely determined. In order to clarify the effects of adhesion and surface elasticity, parametric studies are carried out with respect to adhesive energy density, surface material constants, shear modulus gradation of the coating and indenter size. Surface effects tend to reduce contact length, maximum contact pressure and subsidence displacement. On the other hand, adhesion increases these three properties. Nonetheless, both effects result in higher pull-off forces. In general, the relation between adhesion and surface elasticity can be qualified as competitive. The current study systematically reveals the feature of adhesive nanocontact of a graded coating-substrate system in the presence of both adhesion and surface effects. It helps the community to extend adhesive nanocontact mechanics from homogeneous to graded materials and structures.

Torsional wave propagation on a penny-shaped crack in an orthotopic layer sandwiched between two rigid discs bonded by an orthotropic elastic half-space

The mechanical stability of the interface of two materials determines the stress behavior of the interface. So, observation of failure analysis in orthotropic composites with penny-shaped crack and circular disc in orthotropic materials due to the presence of torsional waves play a major role in structural design. The present article concerns the study of the torsional wave propagation of a penny-shaped crack in an orthotropic layer and two circular discs bonded between the layer and half-spaces. A general solution for the system is presented as a set of dual integral equations using the Hankel transform technique. Using Abel's transform method, the equations have been transformed into Fredholm integral equations of the second kind, which have been solved numerically to compute the stress intensity factors (SIFs) near the rims of crack and discs. Numerical results are obtained using material constants of two orthotropic mediums to demonstrate the impact of material non-homogeneity, normalized disc radius, and layer depth on SIFs and portrayed by virtue of graphs. The analysis of the physical quantity SIF in the present model leads to speculation about the stability of composites against the propagation of cracks in layered engineering solids by surveilling geometric parameters of orthotropic materials and layer depth.

Formula omitted]-fractional Askey–Wilson integrals and related semigroups of operators

We introduce three one parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey–Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey–Wilson polynomials, and solving certain dual integral equations.

Antiplane anisotropic elastodynamics: Babinet’s principle

Babinet’s principle is established for certain problems in antiplane anisotropic elasticity. For example, the scattered field behind a finite straight crack is exactly cancelled by the total field behind an aperture in an infinite flat rigid screen, where the crack and the aperture have the same length and the incident wave is the same for both problems. The method used involves Fourier integrals and dual integral equations.

Axisymmetric slow motion of a non-deformable spherical droplet or slip particle toward an orifice in a plane wall

This paper reports the axisymmetric motion of a viscous droplet or solid spherical particle with a slip-flow surface that moves perpendicular toward an orifice in a plane wall. The motion is studied in the quasi-steady limit under a low Reynolds number. To maintain the spherical shape of the droplet, we assumed that the interfacial tension is very large. The radius of the droplet/particle may be either smaller or larger than the radius of the orifice. A general solution is established from fundamental solutions in both spherical and cylindrical coordinate systems. A semi-analytical approach based on dual integral equations and a collocation scheme is used. Numerical results show that the normalized drag coefficient acting on the droplet/particle is obtained with good convergence for different values of slip parameter, viscosity ratio, and spacing parameters. The findings demonstrate that the collocation results of the drag coefficient are consistent with the limiting cases available in the literature.

Effects of a tilted flat-ended punch on the receding contact between a graded and a homogeneous layer

This paper considers the receding contact problem between an exponentially graded layer and a homogeneous layer under the indentation of a tilted rigid flat-ended punch. In the presence of an offset load deviated from the symmetry axis of the rigid punch, the contact pressures both under the rigid punch and along the receding contact interface between the layers lose symmetry. Depending on the magnitude of eccentricity, either complete or incomplete contact may take place under the rigid flat-ended punch, whereas a receding type of contact always occurs along the interface separating the graded and the homogeneous layers. Fourier integral transforms help to convert the governing equations and mixed boundary conditions of the double-contact problem into dual Fredholm integral equations of the second kind with Cauchy-type singular kernels. Gauss–Chebyshev and Gauss–Jacobi numerical quadratures are subsequently employed to discretize and collocate the dual singular integral equations, together with the four force and moment equilibrium equations, for the complete and incomplete contact, as well as the transitional status corresponding to the critical load eccentricity. Extensive parametric studies indicate that the critical load eccentricity depends on the property gradation of the graded layer and the layer-thickness ratio, but not on the magnitude of the indentation load. The extent of contact both under the rigid flat-ended punch and along the receding contact interface is also independent of the magnitude of the indentation load. The results suggest reasonable departures from the contact properties of a homogeneous half-plane under a tilted rigid flat-ended punch, indicating the valuable information on optimizing the double-contact properties in terms of the property gradation and layer-thickness ratio under a given eccentricity.

Thermomechanical interactions due to mode-I crack under modified temperature-rate dependent two-temperature thermoelasticity theory

Temperature-rate dependent two-temperature (TRDTT) thermoelasticity is reformulated by Shivay and Mukhopadhyay [On the temperature-rate dependent two-temperature thermoelasticity theory. J Heat Transfer. 2020;142(2):022102.] and it is suggested that the effect of the temperature-rate term should not be neglected in the two-temperature relation. As a result, an improved two-temperature theory has been proposed that generalize the TRDTT theory suggested by Youssef [Theory of two-temperature-generalized thermoelasticity. IMA J Appl Math 2006;71(3):383–390.]. The present work aims to investigate thermo-mechanical interactions due to the presence of crack under this modified TRDTT theory. We consider a homogeneous and isotropic 2D infinite medium weakened by mode-I crack of finite length. The boundary of the crack is subjected to time-dependent thermal and stress distributions. Laplace and Fourier transformations are used to solve the present problem in the transformed domain. Two different sets of dual integral equations are derived, which are solved by reducing them to the Fredholm integral equation of the first kind. The regularization method, along with a numerical integration technique, is used to solve this integral equation. Present results are validated with the results derived under GL (temperature-rate dependent) theory. Some interesting points highlighting the effects of a crack in the present context are concluded.

Vertical vibration of a rigid strip footing on a half-space of unsaturated porous soil

This study presents an analytical solution to the compliance coefficient of rigid strip footings lying on an unsaturated porous soil half-space when subjected to vertical vibration. Using the Fourier transform techniques, the complex interaction between unsaturated soils and rigid strip foundations is transformed into a mixed boundary-value problem based on the developed framework of Biot’s model. The wavenumber domain auxiliary function and boundary conditions are used to convert the governing equations to a pair of dual integral equations. To yield the compliance function, the integral equations are transformed into a system of inhomogeneous linear equations by utilizing hypergeometric polynomials and Bessel functions. The proposed analytical solution is verified against existing solutions to interactions between foundations and saturated soil or single-phase soil by adjusting the parameters in the unsaturated soil model. Additionally, the impacts of soil skeleton saturation, intrinsic permeability, and Poisson’s ratio of unsaturated soil on the dynamic compliance coefficient of rigid strip footings are examined. It is shown that the Poisson’s ratio of the soil skeleton and the soil saturation at critical saturation have a significant effect on the dynamic compliance coefficient, but have a negligible effect on the intrinsic permeability.

Dynamic analysis of eccentrically loaded rigid strip foundation on layered transversely isotropic poroelastic media

This paper presents an analytical method for analyzing vertical and rocking motions of a rigid strip foundation on layered transversely isotropic poroelastic media. The extended precise integration method is employed to obtain the plain strain dynamic responses of layered poroelastic media due to the reaction force from the rigid strip. Then, the contact stresses for dynamic soil-structure interactions (SSI) are modelled by the Bessel series expansions, and a set of linear equations can be developed from the dual integral equations of mixed boundary value problems. The explicit expressions for the contact stresses are further obtained by solving these linear equations. Finally, the presented method is verified, and numerical examples are provided to demonstrate the effects of stratifications, transverse isotropy and permeability of media on the dynamic responses of eccentrically loaded rigid strip foundation.

Fourier Series And Fourier Integral Equations And Their Applications In Elasticity

The role of Fourier Series and Fourier Integral Equations in both theoretical and applied mathematics is extremely important. The aim of this paper is to introduce the reader to some of the most critical aspects of historical development of Fourier series and Fourier integrals theory and applications. The paper is based on the work of Sneddon, Srivastva et al. on solving Dual Fourier series, Triple Fourier series and Quadruple Fourier series Equations by using various methods. The different methods for solving Dual integral equations, Triple integral equations and Quadruple integral equations given by Parihar, Tranter et al. will also be mentioned. Finally, some recent applications of these equations to the mathematical theory of elasticity also stated.

Dynamic interaction between complex defect and crack in functionally graded magnetic-electro-elastic bi-materials

This paper aims to develop an effective theoretical method for analyzing the interaction of interfacial crack and complex defect in functionally graded magnetic-electro-elastic bi-materials with exponential variation under the action of anti-plane incident SH-wave. The boundary value problem of interest is solved by Green’s function method. The mechanical model of the cracks is constructed through interface conjunction and crack deviation techniques, thereby simplifying the crack problem to solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tips of the left crack are expressed. The validity of the present method is verified by comparing with a single crack model in previous work. Through the analyses of numerical cases, it can be concluded that the DSIF is related with 3 factors：the geometry of complex defect and crack, the characteristics of incident wave and the inhomogeneity of materials. Comparing with dual integral equations in dealing with asymmetric defects, the method in this paper is relatively more flexible and applicable, also it comes up with a new way to study fracture problems in functionally graded magnetic-electro-elastic materials with more complex defects.