Half-space Problem Research Articles

Thermoelastic fracture of two-dimensional hexagonal quasicrystal media weakened by a penny-shaped crack subjected to uniformly antisymmetric heat fluxes

This study focuses on the thermoelastic fracture behavior of quasicrystals, a class of solids that exhibit unique properties between traditional crystals and amorphous materials. The complex atomic structure of quasicrystals poses challenges in understanding their fracture mechanisms under thermoelastic loading conditions. Central to our investigation is an analytical examination of a penny-shaped crack in an infinite three-dimensional body composed of two-dimensional hexagonal quasicrystals, where the upper and lower crack surfaces are applied to a pair of uniformly antisymmetric heat fluxes. This crack problem requires simultaneous consideration of thermal-phason-phonon multiple field coupling. According to the symmetry of field variables with respect to the crack plane, the thermal-phason-phonon coupled crack problem is transformed into a mixed boundary value problem in the upper half space. The extended displacement discontinuities, encompassing both phonon and phason displacement discontinuities, as well as the temperature discontinuity, are chosen as the basic unknown variables to construct the boundary integral-differential equations governing the mixed boundary value problem. Based on these boundary governing equations and Fabrikant's potential theory method, the problem with the crack surface subjected to uniform antisymmetric heat fluxes is solved. The solutions of thermal-phason-phonon fields on the crack plane, and in the full space are given in closed-form. Numerical results are employed to validate the obtained analytical solutions and visually illustrate the spatial distribution of thermal-phonon-phason coupling fields in the vicinity of the crack. The study provides fundamental insights into the behavior of cracks in quasicrystals under thermal loading, with potential implications for the design of new materials and structures.

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

Analysis of reflection of wave propagation in magneto-thermoelastic nonlocal micropolar orthotropic medium at impedance boundary

PurposeThis study aims to examine the plane wave reflection problem in micropolar orthotropic magneto-thermoelastic half space, considering the influence of impedance as a boundary in a nonlocal elasticity.Design/methodology/approachThis study presents the novel formulation of governing partial differential equations for micropolar orthotropic medium with impact of nonlocal thermo-elasticity under magnetic field.FindingsThis study provides the numerical results validation for a particular numerical data and expression for the amplitude ratios of reflected waves and identifies the existence of four different waves, namely, quasi longitudinal displacement qCLD-wave, quasi thermal wave qCT-wave, quasi transverse displacement qCTD-wave and quasi-transverse micro-rotational qCTM-wave. The study derives the velocity equation giving the speed and phase velocity of these waves. The study also shows that the small-scale size effect gives significant impact on phase velocity.Research limitations/implicationsThe graphical analysis examines the variation of speeds and coefficients of attenuation of these waves due to frequency, magnetic field and nonlocal parameters. Also, significant conclusions on the variation of reflection coefficient against nonlocal parameter, frequency, impedance parameter and angle of incidence are provided graphically.Practical implicationsThe creation of more effective micropolar orthotropic anisotropic materials which are very useful in the daily life and their applications in earth science are greatly impacted by the findings of this study.Originality/valueThe authors of the submitted document initiated and produced it collectively, with equal contributions from all members.

Rayleigh waves in viscoelastic piezoelectric half-space with cladding structures: An analytic Legendre-Laguerre polynomial method

Piezoelectric materials are widely used in surface acoustic wave devices. Many piezoelectric materials themselves have viscoelastic properties, and their surface wave characteristics, especially the attenuation characteristics still needs to be explored. This article proposes a Legendre-Laguerre orthogonal polynomial method to solve the Rayleigh wave problems in a viscoelastic piezoelectric half space with a covering layer. The proposed method compensates for the inherent shortcomings of traditional Laguerre polynomials in solving layered half spaces: the normal stress and electric displacement are discontinuous. The correctness of the method was verified through literature comparison and finite element simulation. At the same time, by utilizing the orthogonality of the Legendre-Laguerre polynomial, the integral analytical formula encountered in the solution process is derived, which improves the computational efficiency by more than ten times. Through the analysis and discussion of the dispersion and attenuation curves, it is found that the piezoelectric effect can suppress the attenuation of Rayleigh waves; the piezoelectric and viscous properties of the covering layer mainly affect attenuation at high frequencies, while those of the half space layer mainly affect attenuation at low frequencies of high-order modes.

Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Estimating potential functions with applications to elliptic problems in half space

We first derive sharp estimates on some potential functions on the half space 0 \\} $ ]]> R + n := { x = ( x 1 , … , x n ) ∈ R n ; x n > 0 } , n ≥ 3 . Then, these estimates are applied to demonstrate the existence of a positive solution with an explicit asymptotic behavior for a semi-linear elliptic problem in the half space. Our approach is founded on Karamata's theory for regularly varying functions, complemented by the use of monotonicity methods.

A stable and efficient infinite meshfree approach for solving half-space heat conduction problems

A stable and efficient infinite meshfree approach for solving half-space heat conduction problems

Effect of non-local thermoelasticity on diffusion under hydrostatic pressure

AbstractIn this paper, thermoelastic diffusion theory is demonstrated on a two-dimensional half-space problem. The surface is subjected to thermal shock and hydrostatic loads. The chemical potential must be a known function of time. The solution is obtained by the direct method using the Laplace technique and the Fourier exponential transform. Local and non-local thermoelasticity are discussed, and the influence of non-local parameters on the general theory of thermoelastic diffusion is investigated. Chemical potential, concentration, temperature, stress and displacement distributions are calculated numerically and displayed graphically.

Thermoelastic diffusion in a half space due to fractional order theory

Based on the theory of fractional thermoelastic diffusion, a two dimensional problem for thermoelastic half space is considered. The upper surface of the half space is taken to be traction free and subjected to ramp type heating. Integral transform technique and a direct approach method are used to obtain the solution in the transformed domain. Numerical inversion techniques are used to obtain the inverse double transforms. The effect of Tempered-Caputo fractional derivative is studied. Numerical results are discussed and represented graphically.

Calculation of Sommerfeld Integrals in Dipole Radiation Problems

This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

<abstract><p>For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.</p></abstract>

A time-domain soil structure interaction analysis for wave scattering in 3D layered half-space

A direct time-domain Soil Structure Interaction (SSI) model is proposed to analyze the scattering problems in 3D horizontal layered half-space. This model uses the scaling surface based Scaled Boundary Finite Element Method (SBFEM) to simulate the infinite domain. Additionally, it employs a combined approach, merging the Domain Reduction Method (DRM) with ray tracking algorithm to calculate the effective nodal forces of seismic input. Compared to the original SBFEM, the scaling surface based SBFEM overcomes the limitation of scaling requirements, and the acceleration unit-impulse response function computed by this method satisfies the radiation damping of 3D horizontal layered half-space. For the seismic input, seismic waves are introduced to the system through effective nodal forces, which are applied exclusively to a single layer of elements adjacent to the truncated boundary in the near field. Importantly, these forces only depend on the free field motion. The accuracy and applicability of the proposed SSI analysis model in solving scattering problems in 3D horizontal layered half-space are demonstrated through various examples.

Analysis of three-dimensional cracks horizontally placed in transversely isotropic FGMs under interior compressive stresses

This paper develops the dual boundary element method (DBEM) using a new fundamental singular solution. This new solution is the elastic fields of transversely isotropic multilayered materials of infinite extent under point concentrated loads. The discretization technique is utilized to approximate arbitrary variations of functionally graded materials (FGMs) with transverse isotropy in a graded direction. The proposed DBEM is used to analyze the three-dimensional crack problems in FGM halfspaces. The penny-shaped and elliptical cracks are horizontally placed on the interface between the FGM layer and homogeneous halfspace and are subjected to uniform compressive stresses on crack surfaces. The stress intensity factors are provided to understand the fracture behaviors of transversely isotropic FGMs. An influential factor is defined to describe the influence of the anisotropy of the FGMs on stress intensity factors. Results show that the anisotropy and heterogeneity of materials exert an obvious influence on the fracture properties of crack problems.

Linear half-space problems in kinetic theory: Abstract formulation and regime transitions

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.

Surface motion of P/SV wave scattering by a semicircular canyon in saturated half-space by complex function and conformal mapping

Based on the complex function theory of elastic dynamics, a complex function solution of P/SV-wave scattering by a semicircular canyon in an elastic porous saturated half-space is obtained. Firstly, through Biot theory and Helmholtz decomposition, the wave potential functions with unknown coefficients in the model are obtained based on the wave function expansion method. Secondly, the complex function expressions of stress and displacement in the model are obtained by using the complex function method and conformal mapping. According to the boundary conditions of the horizontal ground surface and semicircular canyon surface, this boundary value problem can be converted into a series of algebraic equations and solved numerically by truncation. Then, the correctness of the complex function solution in this paper is verified by comparing it with the published results through degenerating the scattering problem in the saturated porous elastic half-space in this paper into that in the elastic half-space. Finally, the influence of the boundary drainage conditions, the medium porosity, and the incident angle and frequency of the incident wave on the ground surface displacement is investigated. Although the P/SV-wave scattering problem is a traditional problem, there is currently no accepted analytical solution that strictly satisfies the zero-stress condition on the horizontal ground surface. This paper is an attempt at this problem, which may provide insight into the search for the strictly analytical solution for P/SV-wave scattering.

Discrete Quantum Kinetic Equation

A semi-classical approach to the study of the evolution of bosonic or fermionic excitations is through the Nordheim—Boltzmann- or, Uehling—Uhlenbeck—equation, also known as the quantum Boltzmann equation. In some low ranges of temperatures—e.g., in the presence of a Bose condensate—also other types of collision operators may render in essential contributions. Therefore, extended— or, even other—collision operators are to be considered as well. This work concerns a discretized version—a system of partial differential equations—of such a quantum equation with an extended collision operator. Trend to equilibrium is studied for a planar stationary system, as well as the spatially homogeneous system. Some essential properties of the linearized operator are proven, implying that results for general half-space problems for the discrete Boltzmann equation can be applied. A more general collision operator is also introduced, and similar results are obtained also for this general equation.

Existence of a Solution to Lamb’s Initial-Boundary Value Problem with a Limiting Value of Poisson’s Ratio

The paper considers a Lamb’s initial-boundary value problem for an elastic half-space in the case when Poisson’s ratio takes the limiting value of 1/2. The existence of a classical solution in the form of an iterated improper integral in the case of axial symmetry is proved.

Persistence of regularity of the solution to a hyperbolic boundary value problem in domain with corner

This article is about the question of the persistence of regularity for the solution to hyperbolic boundary value problem in the quarter-space. More precisely we will both consider the pure boundary value problem and the initial boundary value problem and we propose a functional space, based upon the high order Sobolev space in which a control of the data of the problem leads to a control of the solution (in the same space). The space proposed here contains the tangential Sobolev space. The analysis borrows some ideas of the study of characteristic boundary value problems in the half-space for which the good derivative to consider is known to be the tangential derivatives x1∂1 instead of the normal derivative ∂1. For quarter-space problems the good quantity to consider will be the radial derivative x1∂1+x2∂2 and then we recover the control of tangential derivatives x1∂1 and x2∂2 using explicit formulas in polar coordinates. The regularity of the solution is then established intrinsically by adapting the method introduced by the author to deal with half-space problems without using regularization methods.