Thin Inclusions Research Articles

Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method

A model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal is discussed. The atomic arrangements of the matrix and the inclusion are both periodic along the $$x_{1}$$ -direction and quasiperiodic along the $$x_{2}$$ -direction in the $$ox_{1}x_{2}$$ -coordinate system. Using the hypersingular integral equation method, the inclusion problem is reduced to solving a set of hypersingular integral equations. Based on the exact analytical solution of the singular phonon and phason stresses near the inclusion front, a numerical method of the hypersingular integral equation is proposed using the finite-part integral method. Finally, the numerical solutions for the phonon and phason stress intensity factors of some examples are given.

Метод вирізування у задачі поздовжнього зсуву анізотропного півпростору з тріщиною

The previously developed direct cutting-out method in application to isotropic materials, in particular to bodies with thin inhomogeneities in the form of cracks and thin deformable inclusions is extended to the case of taking into account the possible anisotropy of the material. The basis of the method is to modulate the original problem of determining the stress state of a limited body with thin inclusions by means of a technically simpler to solve problem of elastic equilibrium of an infinite space with a slightly increased number of thin inhomogeneities, which in turn form the boundaries of the investigated body. By loaded cracks we model the boundary conditions of the first kind, and by absolutely rigid inclusions embedded into a matrix with a certain tension – the boundary conditions of the second kind. Using the method of the jump functions and the interaction conditions of a matrix with inclusion, the problem is reduced to a system of singular integral equations, the solution of which is carried out using the method of collocations. Approbation of the developed approach is carried out on the problem of elastic equilibrium of anisotropic (orthotropic in direction of shear) half-space with a symmetrically loaded very flexible inclusion (a crack) at jammed half-space boundary. The influence of inhomogeneity orientation and the half-space material on the generalized stress intensity factors were studied.

Optimal Control of the Location of a Thin Rigid Inclusion in the Equilibrium Problem of an Inhomogeneous Two-Dimensional Body with a Crack

Under study is some two-dimensional model describing equilibriumof a composite solid with a thin rigid inclusion and a crack. A boundary condition of Signorini’s type is prescribed on the crack curve. For a family of corresponding variational problems, the dependence is analyzed of their solutions on the parameter characterizing the location of the rigid inclusion. The existence of solution of the optimal control problem is proved. For this problem, the quality functional is defined with the help of an arbitrary continuous functional on the solution space, while the location of the inclusion is chosen as the control parameter.

Strength of Elastoplastic Plates Containing Square Holes (Inclusions) and Cuts (Thin Inclusions) Under Uniaxial Tension

We study the influence of stress concentrators (square hole, cut, or perfectly rigid inclusion) in a plate on its strength under uniaxial tension with the help of the numerical solution of boundary-value problems of the theory of small elastoplastic deformations for a linearly hardened material with regard for its unloading.

О напряжённо-деформационном состоянии упругой бесконечной пластины с трещиной, расширяющейся посредством вдавливания в неё гладкого тонкого включения

О напряжённо-деформационном состоянии упругой бесконечной пластины с трещиной, расширяющейся посредством вдавливания в неё гладкого тонкого включения

Mathematical model of SH-wave propagation in composites with distributed thin piezoelectric inclusions

The method to study the effective (averaged) parameters of SH-waves in composite media with stochastically distributed thin-walled non-contrast piezoelectric inclusions is proposed. The mathematical model of the wave phenomenon is based on the Foldy’s dispersion relation and solution for the scattering problem by the local obstacle. The influence of electromechanical properties of the composite on effective phase velocities and damping coefficients of longitudinal waves is analyzed. Cite as: R. V. Rabosh, Yu. I. Maksymiv, V. V. Porokhovs'kyi, V. O. Mishchenko, Ya. I. Kunets, “Mathematical model of SH-wave propagation in composites with distributed thin piezoelectric inclusions,” Prykl. Probl. Mekh. Mat. , Issue 16, 107–111 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.107-111

Optimal control of the location of a thin rigid inclusion in the equilibrium problem of an inhomogeneous two-dimensional body with a crack

Optimal control of the location of a thin rigid inclusion in the equilibrium problem of an inhomogeneous two-dimensional body with a crack

Conditions for the Jump of Stresses and Displacements on Thin Anisotropic Interlayers and Inclusions in Continuous Media

For problems of thin anisotropic interlayers, overlays, and inclusions in continuous three-dimensional media, we establish conditions that enable us to represent the action of the indicated heterogeneities by the presence of a surface with certain properties. In particular, in passing through this surface, the components of the vectors of displacements and stresses have discontinuities. In complicated problems of anisotropic heterogeneities in the medium, this enables us to decrease the dimensionality of defects in homogeneous structures, namely, the three-dimensional defects are reduced to two-dimensional defects and two-dimensional defects are reduced to one-dimensional. As a result, the problems are mathematically simplified, which makes it possible to obtain their efficient solutions.

Equilibrium of an Elastic Body with Closely Spaced Thin Inclusions

Problems with unknown boundaries describing an equilibrium of two-dimensional elastic bodies with two thin closely spaced inclusions are considered. The inclusions are in contact with each other, which means that there is a crack between them. On the crack faces, nonlinear boundary conditions of the inequality type that prevent the interpenetration of the faces are set. The unique solvability of the problems is proved. The passages to the limit as the stiffness parameter of thin inclusions tends to infinity are studied, and limiting models are analyzed.

Antiplane Deformation of a Bimaterial with Thin Interfacial Nonlinear Elastic Inclusion

Abstract The problem of longitudinal shear of bimaterial with thin nonlinear elastic inclusion at the interface of matrix materials is considered. Solution of the problem is constructed using the boundary value problem of combining analytical functions and jump functions method. The model of the thin inclusion with nonlinear resilient parameters is built. Solution of the problem is reduced to a system of singular integral equations with variable coefficients. The convergent iterative method for solving such a system is offered for various nonlinear strain models, including Ramberg-Osgood law. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analysed for the tensely-deformed matrix under loading a uniformly distributed shear stresses and for a balanced system of the concentrated forces.

Plastic Exfoliation of a Thin Stiff Inclusion Parallel to the Boundary of Half Space in the Case of its Unilateral Contact with the Medium

We obtain a numerical-analytic solution of the antiplane problem of stress-strain state of the elastoplastic half space with a thin rigid tunnel inclusion parallel to the boundary of the half space. It is assumed that, prior to loading, the inclusion is in unilateral mechanical contact with the medium. The specific features of plastic exfoliation of the inclusion are analyzed. Some partial cases are investigated.

On modeling thin inclusions in elastic bodies with a damage parameter

In this paper, we analyze equilibrium problems for 2D elastic bodies with two thin inclusions in the presence of damage. A delamination of the inclusions from the matrix is assumed, thus forming a crack between the elastic body and the inclusions. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. Weak and strong formulations of the problems are discussed. The paper provides an asymptotic analysis with respect to the damage parameter. An optimal control problem is analyzed with a cost functional to be equal to the derivative of the energy functional with respect to the crack length, and the damage parameter being a control function.

Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.

Mathematical modeling, analysis and numerical approximation of second-order elliptic problems with inclusions

Many biological and geological systems can be modeled as porous media with small inclusions. Vascularized tissue, roots embedded in soil or fractured rocks are examples of such systems. In these applications, tissue, soil or rocks are considered to be porous media, while blood vessels, roots or fractures form small inclusions. To model flow processes in thin inclusions, one-dimensional (1D) models of Darcy- or Poiseuille type have been used, whereas Darcy-equations of higher dimension have been considered for the flow processes within the porous matrix. A coupling between flow in the porous matrix and the inclusions can be achieved by setting suitable source terms for the corresponding models, where the source term of the higher-dimensional model is concentrated on the center lines of the inclusions. In this paper, we investigate an alternative coupling scheme. Here, the source term lives on the boundary of the inclusions. By doing so, we lift the dimension by one and thus increase the regularity of the solution. We show that this model can be derived from a full-dimensional model and the occurring modeling errors are estimated. Furthermore, we prove the well-posedness of the variational formulation and discuss the convergence behavior of standard finite element methods with respect to this model. Our theoretical results are confirmed by numerical tests. Finally, we demonstrate how the new coupling concept can be used to simulate stationary flow through a capillary network embedded in a biological tissue.

Volume integral equations of the scattering problem of poroelasticity and their properties

Scattering of monochromatic waves on an isolated inhomogeneity (inclusion) in an infinite poroelastic medium is considered. Wave propagation in the medium and the inclusion are described by Biot's equations of poroelasticity. The problem is reduced to 3D‐integro‐differential equations for displacement and pressure fields in the region occupied by the inclusion. Properties of the integral operators in these equations are studied. Discontinuities of the fields on the inclusion boundary are indicated. The case of a thin inclusion with low permeability is considered. The corresponding scattering problem is reduced to a 2D integral equation on the middle surface of the inclusion. The unknown function in this equation is the pressure jump in the transverse direction to the inclusion middle surface. An inclusion with a thin layer of low permeability on its interface is considered. The appropriate boundary conditions on the inclusion interface are pointed out. Methods of numerical solution of the volume integral equations of the scattering problems of poroelasticity are discussed.

Problems of Thin Inclusions in a Two-Dimensional Viscoelastic Body

Under study are the equilibrium problems for a two-dimensional viscoelastic body with delaminated thin inclusions in the cases of elastic and rigid inclusions. Both variational and differential formulations of the problems with nonlinear boundary conditions are presented; their unique solvability is substantiated. For the case of a thin elastic inclusion modelled as a Bernoulli–Euler beam, we consider the passage to the limit as the rigidity parameter of the inclusion tends to infinity. In the limit it is the problem about a thin rigid inclusion. Relationship is established between the problems about thin rigid inclusions and the previously considered problems about volume rigid inclusions. The corresponding passage to the limit is justified in the case of inclusions without delamination.

On a class of problems on interaction of stress concentrators of different types with an elastic semi-infinite plate

A class of mixed boundary-value problems of mathematical theory of elasticity dealing with interaction between stress concentrators of different types (such as cracks, absolutely rigid thin inclusions, punches, and stringers) and an elastic semi-infinite plate is considered. The method of Mellin integral transformation is used to reduce solving these problems to solving singular integral equations (SIE). After the governing SIE are solved, the following characteristics of the problem are determined: tangential contact stresses under stringers, dislocation density on the crack edges, breaking stresses outside the cracks on their line of location, the stress intensity factor (SIF), crack openings, jumps of contact stresses on the edges of inclusions.

Impact of sequence stratigraphy, depositional facies, diagenesis and CO2 charge on reservoir quality of the lower cretaceous Quantou Formation, Southern Songliao Basin, China

Abstract The fourth member of the Lower Cretaceous Quantou Formation in Southern Fuxin Uplift zone of the Southern Songliao Basin is typical tight oil reservoir which has recently been explored and produced. For effective exploration and development, the evaluation and control factors of the reservoir quality of the K1q4 sandstones should be thoroughly studied. The study focused on the control of the combination of sequence stratigraphy, depositional facies and diagenesis on the reservoir quality, and identified the impact of CO2 charging, based on the core and thin section observation, XRD, SEM, fluid inclusion and isotope testing and analysis. The fourth member of the Lower Cretaceous Quantou Formation consists of lowstand system tracts (LST), transgressive system tracts (TST) and highstand system tracts (HST). The K1q4 sandstones were mostly deposited in distributary channels and crevasse splays of a delta plain within the LST and TST, and underwater distributary channels and sheet sands of a delta front within the HST. The K1q4 sandstones, mostly silt to medium grained, medium to well sorted, and are mostly lithic arkoses and feldspathic litharenites, have undergone significantly diagenetic alterations, such as compaction, quartz cementation, unstable grains (feldspar and volcanic fragments) dissolution, carbonate cementation and clay minerals cementation. The reservoir quality of the K1q4 sandstones are poor with low porosity and permeability, showing variations among the different system tracts and depositional facies, and are influenced by the charging of CO2 and controlled by the combination of sequence stratigraphy, depositional facies and diagenesis. The sandstones with high reservoir quality are mostly deposited in high energy environments, including the (underwater) distributary channel sandstones, especially the LST distributary channel sandstones with the best reservoir quality among the LST, TST and HST sandstones, which can be ascribed to larger grain size and better sorting resulting in larger intergranular pore space after compaction, and less content of carbonate cements, illite and mixed-layer I/S, and more pore-lining chlorite and more effectively dissolution. The poor reservoir quality of sandstones from crevasse splay and sheet sand is due to the small grain size and bad sorting which resulted in extensive compaction and the resulting weak dissolution. The charging of CO2 has dual impact on the reservoir quality of the K1q4 sandstones. The CO2 charging can induce the dissolution of unstable detrital grains and the precipitation of dawsonite. The former create new secondary pore space, while the latter occupy the intergranular pore space. The CO2 charging is harmful to the reservoir quality when the dawsonite content exceeds a threshold of 5%.

Evaluation of the Stress Concentration Near Thin Elastic and Viscoelastic Inclusions

We present a survey of the efficient approximate methods for the evaluation of the stress concentration near notches, slits, and cavities filled with elastic or viscoelastic materials. Our main attention is given to a method based on the use of solutions of the corresponding problems for bodies with mathematical cuts along the median surfaces of the concentrators. The idea of the method is to replace an inclusion with a certain volume by a surface with unknown discontinuities of stresses and/or displacements. We describe a procedure of construction of the integral equations of the corresponding boundary-value problems used to determine the indicated discontinuities. By an example of the inhomogeneities bounded by canonical surfaces, we illustrate the efficiency of construction of approximate solutions of these problems.

Nonlinear Deformation of a Thin Interface Inclusion

We develop a model of thin inclusion with nonlinear anisotropic mechanical properties of the general form. By using this model and the methods of the problem of conjugation of the limit values of analytic and jump functions, we construct a system of singular integral equations with variable coefficients (functions). The solution of the system enables us to describe any changes (monotonic or nonmonotonic) in the quasistatic load and its influence on the stress-strain state of the body with inhomogeneity on the basis of the incremental approach. For the numerical solution of the system, we propose an iterative numerical-analytic method.