Mixed Boundary Value Problem Research Articles

On the indentation of a poroelastic halfspace

This paper re-examines the problem of the indentation of a halfspace region with Biot poroelastic properties by considering the displacement and pore fluid pressure boundary conditions that are consistent for adhesive contact at an impermeable interface. The mixed initial boundary value problems associated with the adhesive-impermeable indentation are reduced to a set of coupled Fredholm integral equations of the second-kind in the Laplace transform domain. These equations are solved to establish the time-dependant indentation response of the indenter. The results are supplemented by sets of bounds obtained by maintaining a zero radial displacement condition within the contact zone and either zero pore fluid pressure boundary conditions or impermeable conditions on the entire surface of the halfspace or impermeable conditions within the contact region but free-draining exterior to the indenter. The accuracy of the numerical schemes used in the solution of these poroelastic contact problems is also discussed.

Mixed- and crack-type dynamical problems of electro-magneto-elasticity theory

Abstract We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide.

A Study of Deformations in a Thermoelastic Dipolar Body with Voids

In this paper, we consider the mixed initial boundary value problem in the context of a thermoelastic porous body having a dipolar structure. We intend to analyze the rate of decay of solutions to this problem to ensure that in a finite time, they become null. In our main result, we find that the combined contribution of the dipolar constitution of the body together with voids dissipation and thermal behavior cannot cause vanishing of the deformations in a finite time.

Biot number effect and non-Fourier effect on temperature field and stress intensity factor of a cracked strip under thermal shock loading

In this paper, we considered a thermal shock problem of a strip with an inner crack parallel to the heated or cooled surface. The effect of the surface heat transfer properties and the non-Fourier effect, which are modified by effective Biot’s number and the non-Fourier factor, were considered to investigate the transient temperature field and the dynamic stress intensity factors (SIFs). Fourier transform and Laplace transform were used to reduce this mixed boundary value problem to singular integral equations (SIEs). The Gauss-Chebyshev numerical integration method was applied to the SIEs and linear equations are obtained. Finally, numerical inverse Laplace transform was used to obtain the temperature field and SIFs. The numerical results showed that Biot’s number not only affects the uniformity of the temperature field and the magnitude of the SIFs strongly, but also makes contributions to the non-Fourier effect. This finding can help better understanding the fracture behaviors of advanced materials and making a progress to determine the applicable conditions of the non-Fourier effect.

Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems**Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant agreement No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Natural Science Foundation of Guangxi under Grant No. 2018JJA110006, and

This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge’s fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p -Laplace operator and an implicit obstacle.

Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory

Summary The exact solutions of a class of hypersingular integral equations with kernels $\left( {s-x} \right)^{-2}$, $\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\left( {\sinh \frac{s-x}{2}} \right)^{-2},\cos \frac{s-x}{2}\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\cosh \frac{s-x}{2}\left( {\sinh \frac{s-x}{2}} \right)^{-2}$ are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.

Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.

ON INCOMPLETE FACTORIZATION IMPLICIT TECHNIQUE FOR 2D ELLIPTIC FD EQUATIONS

A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

Nonlinear contact force law for spherical indentation of FGM coated elastic substrate: An extension of Hertz's solution

This paper develops a nonlinear contact force law for the indentation of an elastic substrate with thin functionally graded material (FGM) coating by a rigid smooth sphere. The nonlinear contact force law is explicitly expressed in form of extended Hertz's solution with a correction factor, which is related to the coating thickness, radius of the indenter, contact interference and the modulus ratio. The explicit expression of the correction factor for arbitrary coating modulus gradation can be determined statistically with extensive numerical results. For computational efficiency and accuracy, an analytical contact model based on multilayered half space is developed and the associated mixed boundary value problem is converted to a Fredholm integral equation of the second kind. This model discretizes the thin FGM coating into n dissimilar and fully bonded sub-layers. Each sub-layer is a homogeneous elastic seam of finite thickness and constant elastic modulus and Poisson's ratio. Variation of the coating elastic properties along the thickness direction is accurately approximated by the multilayered system. The non-linear contact force laws in closed-form are specifically given for both linear and exponential modulus gradations. The load-displacement relations predicted by these force laws are shown to be in exact agreement with the numerical results from the Fredholm integral equation. Additionally, this nonlinear contact force law for FGM coating is further incorporated into the Greenwood and Williamson model, which provides a feasible and effective way for modeling the contact of FGM coated rough surfaces.

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. Boundary-domain integral equations are formulated in terms of parametrix-based potential type integral operators defined on the boundary and the domain. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

Floating Structures in Shallow Water: Local Well-posedness in the Axisymmetric Case

The floating structure problem describes the interaction between surface water waves and a floating body, generally a boat or a wave energy converter. As recently shown by Lannes, the equations for the fluid motion can be reduced to a set of two evolution equations on the surface elevation and the horizontal discharge. The presence of the object is accounted for by a constraint on the discharge under the object; the pressure exerted by the fluid on this object is then the Lagrange multiplier associated with this constraint. Our goal in this paper is to prove the well-posedness of this fluid-structure interaction problem in the shallow water approximation under the assumption that the flow is axisymmetric without swirl. We write the fluid equations as a quasilinear hyperbolic mixed initial boundary value problem and the solid equation as a second order ODE coupled to the fluid equations. Finally we prove the local in time well-posedness for this coupled problem, provided some compatibility conditions on the initial data are satisfied.

Mixed boundary value problems of the system for steady flow of heat-conducting incompressible viscous fluids with dissipative heating

Mixed boundary value problems of the system for steady flow of heat-conducting incompressible viscous fluids with dissipative heating

Trapping Elastic Waves by a Semi-Infinite Cylinder with Partly Fixed Surface

We consider the three-dimensional mixed boundary value problem in elasticity about time harmonic oscillations of a semi-infinite anisotropic cylinder. We show that for certain position and shape of the clamping zone of the surface the elastic wave is trapped; i.e., the problem admits a nontrivial solution with exponential decay at infinity or, conversely, the absence of the trapped wave is guaranteed on all frequencies. We state some open questions that concern similar spectral problems.

Решение обратной краевой задачи аэрогидродинамики решеток без дополнительных условий

Решение обратной краевой задачи аэрогидродинамики решеток без дополнительных условий

On a mathematical model of dynamics of the elastic wedge-shaped medium with radiating defect

In the paper the mixed boundary value problem of antiplane vibrations is considered in the elastic wedge-shaped medium containing the radiating defect J2. Radiating generators are assumed to be located on defect boundaries and on the interval J1 of the wedge free boundary as well. The problem of reconstructing the wave field in the whole wedgeshaped region with its boundary is stated. A number of problems of analyzing acoustic emission signals by radiating defect are reduced to the problem considered in connections with using non-destructive testing elements of the technological equipment under exploitation. The problem in question is reduced to studying the solvability problems of the equivalent boundary integral equation system both for stress saltus on the defect J2and contact stresses on the interval J1 of the upper plane of the wedge.

Асимптотическое решение одной смешанной краевой задачи анизотропной двухслойной цилиндрической оболочки

Асимптотическое решение одной смешанной краевой задачи анизотропной двухслойной цилиндрической оболочки

Holomorphic approximation and mixed boundary value problems for $\overline \partial $

Holomorphic approximation and mixed boundary value problems for $\overline \partial $

Semi-analytical solution to the problem of frequency dependent anisotropy of porous media with an aligned set of slit cracks

Fractures control fluid flow in geological formations because their presence can greatly affect effective permeability. Existence of cracks in a porous rock can induce seismic dispersion attenuation and frequency-dependent anisotropy, due to wave-induced fluid flow (WIFF) between the fractures and pores. Thus, dispersion and attenuation can be valuable seismic signatures for fracture characterization. In previous studies these effects were modeled by considering cracks as low aspect-ratio inclusions embedded in a porous background medium modeled by Biot's poroelasticity equations. However, previous works are limited to normal wave incidence. In this research, we study the P-wave dispersion and attenuation for oblique incidence in a saturated porous medium with aligned slit (two-dimensional) fractures, through solving a mixed boundary value problem for Biot's dynamic poroelasticity equations. The semi-analytical solution of this problem gives dispersion and attenuation as functions of the incidence angle. The strongest dispersion and attenuation occur when the P-wave propagates along the fracture normal and decrease with increasing incident angle (as measured from the fracture normal). For the frequency-dependent anisotropy at low frequencies, the sample behaves as nearly elliptical, and at high frequencies the medium exhibits a strong anellipticity. The attenuation anisotropy parameters reach their maximum when the fluid diffusion length is comparable to the crack radius. Comparison of the heuristic solution, which is s based on an interpolation between low- and high frequencies using a relaxation function, to the semi-analytical solution of the rigorous equations shows excellent agreement, demonstrating the validity of the heuristic approach.

An Axisymmetric Contact Problem of a Thermoelastic Layer on a Rigid Circular Base

We study the thermoelastic deformation of an elastic layer. The upper surface of the medium is subjected to a uniform thermal field along a circular area while the layer is resting on a rigid smooth circular base. The doubly mixed boundary value problem is reduced to a pair of systems of dual integral equations. The both system of the heat conduction and the mechanical problems are calculated by solving a dual integral equation systems which are reduced to an infinite algebraic one using a Gegenbauer’s formulas. The stresses and displacements are then obtained as Bessel function series. To get the unknown coefficients, the infinite systems are solved by the truncation method. A closed form solution is given for the displacements, stresses and the stress singularity factors. The effects of the radius of the punch with the rigid base and the layer thickness on the stress field are discussed. A numerical application is also considered with some concluding results.

Effective properties of a porous medium with aligned cracks containing compressible fluid

SUMMARYUnderstanding the wave propagation in fluid-saturated cracked rocks is important for detecting and characterizing cracked reservoirs and fault zones with applications in geomechanics, hydrogeology, exploration geophysics and reservoir engineering. In sedimentary rocks, microscopic-scale pores are usually filled with fluid. One logical means of modelling the essential features of such rocks is to use poroelasticity theory. But previous models of wave propagation in cracked porous medium are either restricted to low frequencies at which effects of the elastic scattering (scattering into fast-P and S waves via mode conversion at the crack faces) are negligible or to the case that the crack-filling fluid is assumed to be incompressible. To overcome these restrictions, we consider the effects of crack fluid compressibility by extending spring condition into poroelasticity and derive exact solutions of the scattering problem of an incident P wave by a circular crack containing compressible fluid in a porous medium. Based on the solutions, we develop two different effective medium models to estimate frequency-dependent effective velocity and attenuation in a fluid-saturated porous rock with a set of aligned cracks. The mixed-boundary value problem reveals that both the wave-induced fluid flow (WIFF) and elastic wave scattering can cause important velocity dispersion and attenuation. The diffusion-type WIFF dominates the velocity change and attenuation for the low frequency range, while the elastic scattering dominates them for the relatively higher frequency range. The dependences of the P-wave velocity on the crack fluid compressibility are different at different frequencies. For the WIFF-dominated frequency range and Rayleigh-scattering frequency range, the P-wave velocity decreases with the crack fluid compressibility. In contrast, for the Mie scattering frequency range, the opposite occurs (the P-wave velocity increases with the crack fluid compressibility).