Thin Inhomogeneities Research Articles

A crack in a confocal elliptical inhomogeneity embedded in an infinite matrix subjected to uniform heat flux

We derive a series-form analytical solution to the thermoelastic problem of an insulated and traction-free crack in a confocal elliptical isotropic elastic inhomogeneity embedded in an infinite isotropic elastic matrix subjected to uniform remote heat flux. Using complex variable techniques such as conformal mapping, analytic continuation and Laurent series expansions, the original thermoelastic problem is reduced to an infinite system of linear algebraic equations, the solution of which will yield the mode I and mode II stress intensity factors at the crack tip. An exact closed-form solution is derived when the inhomogeneity and the matrix have identical shear moduli. Detailed numerical results are presented to demonstrate the series-form and closed-form solutions with an emphasis on the particular case of a vanishingly thin inhomogeneity.

Thermomagnetoelectroelasticity of finite bimaterial bodies in the presence of a bonding interlayer of high thermal conductivity and internal thin inhomogeneities

Thermomagnetoelectroelasticity of finite bimaterial bodies in the presence of a bonding interlayer of high thermal conductivity and internal thin inhomogeneities

Method of Direct Cutting-Out in Modeling Orthotropic Solids with Thin Elastic Inclusions Under Longitudinal Shear

By the method of direct cutting-out, the problems of longitudinal shear of an orthotropic half space, a layer, and a wedge with thin elastic orthotropic inclusions are reduced to the basic problem of interaction of thin inhomogeneities in the orthotropic space. We establish the conditions of interaction of loaded elastic anisotropic inclusions with the matrix of the body. We study the influence of elastic moduli both of the inclusion and of the body, as well as of the geometric parameters of the problems on the generalized stress intensity factors. The level lines of stresses are plotted in the vicinity of the inclusion.

Deformation and Strength Parameters of a Composite Structure with a Thin Multilayer Ribbon-like Inclusion.

Within the framework of the concept of deformable solid mechanics, an analytical-numerical method to the problem of determining the mechanical fields in the composite structures with interphase ribbon-like deformable multilayered inhomogeneities under combined force and dislocation loading has been proposed. Based on the general relations of linear elasticity theory, a mathematical model of thin multilayered inclusion of finite width is constructed. The possibility of nonperfect contact along a part of the interface between the inclusion and the matrix, and between the layers of inclusion where surface energy or sliding with dry friction occurs, is envisaged. Based on the application of the theory of functions of a complex variable and the jump function method, the stress-strain field in the vicinity of the inclusion during its interaction with the concentrated forces and screw dislocations was calculated. The values of generalized stress intensity factors for the asymptotics of stress-strain fields in the vicinity of the ends of thin inhomogeneities are calculated, using which the stress concentration and local strength of the structure can be calculated. Several effects have been identified which can be used in designing the structure of layers and operation modes of such composites. The proposed method has shown its effectiveness for solving a whole class of problems of deformation and fracture of bodies with thin deformable inclusions of finite length and can be used for mathematical modeling of the mechanical effects of thin FGM heterogeneities in composites.

The topological ligament in shape optimization: a connection with thin tubular inhomogeneities

In this article, we propose a formal method for evaluating the asymptotic behavior of a shape functional when a thin tubular is added between two distant regions of the boundary of a domain. In the contexts of the conductivity equation and the linear elasticity system, we relate this issue to a perhaps more classical problem of thin tubular inhomogeneities: we analyze the solutions to versions of the physical partial differential equations which are posed inside a fixed background medium, and whose material coefficients are altered inside a tube with vanishing thickness. Our main contribution from the theoretical point of view is to propose a heuristic energy argument to calculate the limiting behavior of these solutions with a minimum amount of effort. We retrieve known formulas when they are available, and we manage to treat situations which are, to the best of our knowledge, not reported in the literature (including the setting of the 3d linear elasticity system). From the numerical point of view, we propose three different applications of the formal ligament approach derived from these expansions. At first, it is an original way to account for variations of a domain, and it thereby provides a new type of sensitivity for a shape functional, to be used concurrently with more classical shape and topological derivatives in optimal design frameworks. Besides, it suggests new, interesting algorithms for the design of the scaffold structure sustaining a shape during its fabrication by a 3d printing technique, and for the design of truss-like structures. Several numerical examples are presented in two and three space dimensions to appraise the efficiency of these methods.

A spectral target signature for thin surfaces with higher order jump conditions

<p style='text-indent:20px;'>In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb R}^m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ m = 2, 3 $\end{document}</tex-math></inline-formula> from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open <inline-formula><tex-math id="M3">\begin{document}$ m-1 $\end{document}</tex-math></inline-formula> dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [<xref ref-type="bibr" rid="b20">20</xref>]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach</p>

Mechanical deformation of atomically thin layers during stamp transfer

The way transition metal dichalcogenide (TMD) strains during its transfer from one substrate to another is very interesting and holds a special place in the creation of heterostructures. In our work we observe the spectrum of photoluminescence in TMD during the transfer. For this we use a specially designed transfer system with inverted geometry. During transfer we observe a modification of exciton photoluminescence linewidth and resonance shift in atomically thin layers of TMD. We believe that our results lay grounds for the future work on the assessment of the atomically thin layer inhomogeneity introduced by the typical mechanical transfer.

Role of nanoscale surface defects on Sn adsorption and diffusion behavior on oxidized Nb(100)

Nanoscale structural defects such as grain boundaries, atomic dislocations, and surface roughness inhibit the stoichiometrically homogeneous growth of Nb3Sn on Nb. This is a critical technological bottleneck for the implementation of next-generation Nb3Sn superconducting radio frequency cavities, as thin film inhomogeneities are known to degrade superconducting properties that are essential for reaching optimal cavity performance. To determine the influence of structural defects on Nb3Sn film growth, low and moderate surface defect densities were intentionally induced onto a (3 × 1)-O Nb(100) substrate, which serves as a model system to study atomic-scale Sn adsorption and diffusion. Scanning tunneling microscopy shows that, while initial Sn adsorption behavior at room temperature differs between the low and moderate defect density Nb(100) surfaces, the overall diffusion pathways at elevated temperatures are guided by the underlying oxide structure with variations resulting from increased nanoscale surface defects. The (3 × 1)-O Nb(100) surface with a moderate defect density also demonstrates enhanced Sn thermal stability, with the Sn desorption threshold occurring between 850 and 900 °C, approximately 50 °C higher than desorption from both the low defect density and pristine thin oxide surfaces. This suggests that structural surface defects may stabilize adsorbed Sn species on oxidized Nb at the elevated temperatures utilized in Nb3Sn alloy growth procedures. Auger electron spectroscopy shows no significant difference in surface composition following Sn deposition at varying coverages on the pristine and defect-induced (3 × 1)-O Nb(100) surfaces. This indicates that the surface and near-surface composition are not influenced by the presence of nanoscale surface defects despite slight attenuations in Sn diffusion pathways on defected substrates. These results provide the first in situ visualization of Sn adsorption and diffusion behavior on oxidized Nb at the nanoscale, revealing the significance of the underlying Nb oxide surface structure and defect density on Nb3Sn film growth and, ultimately, cavity performance.

Method of Direct Cutting-Out in the Problems of Elastic Equilibrium of Anisotropic Bodies with Cracks Under Longitudinal Shear

In the earlier developed method of direct cutting-out, we take into account the anisotropy of material. This method is based on the procedure of modeling of finite or bounded bodies with thin structural defects of any type and boundary conditions on its contour by an infinite space with the same inhomogeneities as in the original problem and additional thin inhomogeneities (cracks or absolutely rigid inclusions), which form the boundary of the investigated body. Thus, loaded cracks are used to model boundary conditions of the first kind, whereas absolutely rigid inclusions embedded in the matrix with certain tension model boundary conditions of the second kind. The developed approach is verified for several problems of longitudinal shear of an anisotropic half space, a layer, and a wedge in the presence of an internal crack under given boundary conditions of the first kind.

Interaction of Physicomechanical Fields in Bodies with Thin Structural Inhomogeneities: A Survey

We present a survey of investigations aimed at the analysis of thermally, magnetically, and electrically stressed states of the bodies containing thin inclusions and systems of inclusions, as well as at the analysis of their stress-strain states.

A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities I. The case of an open mid-curve

Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the

A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result

A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result

Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

A connection between topological ligaments in shape optimization and thin tubular inhomogeneities

In this note, we propose a formal framework accounting for the sensitivity of a function of the domain with respect to the addition of a thin ligament. To set ideas, we consider the model setting of elastic structures, and we approximate this question by a thin tubular inhomogeneity problem: we look for the sensitivity of the solution to a partial differential equation posed inside a background medium, and that of a related quantity of interest, with respect to the inclusion of a thin tube filled with a different material. A practical formula for this sensitivity is derived, which lends itself to numerical implementation. Two applications of this idea in structural optimization are presented.

Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

General solution for inhomogeneous line inclusion with non-uniform eigenstrain

The inhomogeneous line inclusion problem has various backgrounds in practical application such as graphene sheet-reinforced composites, and hydrogen embrittlement, grain boundary segregation in metallic materials. Due to the long-standing mathematical difficulty, there is no explicit analytical solution obtained except for the thin ellipsoidal inhomogeneity and rigid line inhomogeneity. In this paper, to find the deformation state due to the presence of such kind of elastic inhomogeneities, the inhomogeneous line inclusion problem is tackled in the framework of plane deformation. Firstly, the fundamental solution for a point-wise residual strain is presented and its deformation strain field is derived. By using Green’s function method, the homogeneous line inclusion problem with non-uniform eigenstrain is formulated and an Eshelby tensor-like line inclusion tensor is derived. From the line inclusion concept, the classical edge dislocation is revisited. Also, by virtue of this model, some elementary line homogenous inclusion problems are explored. Secondly, based on the homogeneous line inclusion solution, the inhomogeneous line inclusion problem is formulated using the equivalent eigenstrain principle, and its general solution is derived. Then, an inhomogeneous edge dislocation model is proposed and its analytical solution is presented. Furthermore, to demonstrate the application of the proposed inhomogeneous line inclusion model, a typical thin inclusion under remote load is studied. This study provides a general solution for inhomogeneous thin inclusion problems. The models and their solutions introduced here will also find application in the mechanics of composites analysis, heterogeneous material modeling, etc.

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.

Investigation of reflectometry for in situ process monitoring and characterization of co-evaporated and stacked Cu-Zn-Sn-S based thin films

During preparation of compound semiconductor thin films undesired secondary phases and inhomogeneities may occur. Therefore, process control and monitoring are important aspects towards thin film optimization. In this study, white light reflectometry (WLR) was investigated as an in situ non-destructive optical technique to monitor thin films during thermal vacuum evaporation (PVD) of kesterite-type Cu-Zn-Sn-S thin films.The impact of composition on optical properties was studied ex situ by Raman spectroscopy and reflectometry to identify possibilities for in situ detection of secondary phases. A WLR setup was designed and integrated with the PVD system. Four Cu-Zn-Sn-S films were prepared by evaporation and direct reflection was monitored in situ. Reflection spectra were analyzed to identify the influence of process parameters and phases such as CuS and ZnS. Transfer matrix simulations were performed to further explain experimental observations.It is shown that the occurrence of different phases, growth rate/thickness are well related to reflection spectra. Time-dependent reflection spectra reveal specific properties that could serve as characteristic fingerprints for process control and monitoring. Therefore, this study extends the possibilities for reflectometry as a straightforward, low cost method for in situ process control of Cu-Zn-Sn-S based films.

Reconstruction of thin electromagnetic inhomogeneity without diagonal elements of a multi-static response matrix

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.