Circular Inclusion Research Articles

On Galvanomagnetic Properties of the 2D Rayleigh Model

periodic arrangement of circular inclusions) with the metal–insulator phase transition are considered. The expression obtained for the corresponding effective conductivity tensor σ^e is valid in the entire subthreshold critical region (up to the contacts of circles) in the vicinity of the transition point. The behavior of components of tensor σ^e in weak and strong magnetic fields is determined.

Bending of a plate with circular rigid inclusion and system of cracks on the assumption of strip crack face contact

The article presents a solution to the bending problem for an infinite isotropic plate with a circular rigid inclusion and arbitrarily oriented straight through cracks. It is assumed that under the action of an external load at infinity, the faces of all cracks are in smooth contact along a region of constant width near the upper surface of the plate. The solution is obtained using methods of the theory of functions of complex variable and complex potentials and is reduced to a system of singular integral equations, which is numerically solved by the mechanical quadrature method. On the basis of numerical analysis, the graphical dependences of the contact force, forces and moments intensity factors are constructed at various parameters of the problem.

An Eshelby inclusion in a nonlinearly coupled thermoelectric bi‐material and tri‐material

AbstractWe first derive an analytic solution to Eshelby's problem concerning a circular inclusion within one of two perfectly bonded nonlinearly coupled thermoelectric half‐planes. The inclusion undergoes a prescribed uniform electric current‐free thermoelectric potential gradient and a prescribed uniform energy flux‐free temperature gradient. Closed‐form expressions for the six analytic functions characterizing the thermoelectric fields of electric current density and energy flux in all three phases of the composite are obtained primarily with the aid of analytic continuation. A general method is then proposed for the analytic solution of Eshelby's problem of an inclusion of arbitrary shape in a nonlinearly coupled thermoelectric bi‐material. Finally, we extend our ideas to the case of a tri‐material by developing an analytic solution to the thermoelectric problem associated with a circular Eshelby inclusion in a nonlinearly coupled thermoelectric tri‐material composed of two semi‐infinite thermoelectric media bonded together through an intermediate thermoelectric layer of finite thickness.

DeepBND: A machine learning approach to enhance multiscale solid mechanics

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude. A few test cases accounting for random circular and elliptical inclusions are reported aiming at proving the potentials of the DeepBND method.

Correction to: Spectrum of the Neumann–Poincaré Operator and Optimal Estimates for Transmission Problems in the Presence of Two Circular Inclusions

Abstract

Pixel-based boundary element method for computing effective thermal conductivity of heterogeneous materials

In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.

A novel quantitative explanation for the fracture mechanism of sandstone containing a circular inclusion

AbstractIn this study, uniaxial compressive tests combined with the digital image correlation (DIC) method are used to investigate the mechanical properties, fracture process, and failure pattern of sandstone containing a circular inclusion. A DIC‐based methodology is proposed to quantitatively explain the crack initiation and propagation mechanism. The identification criterion of crack type is presented based on the nature of cracks and the development trends of relative displacement. Three crack types, including tensile cracks, tensile–shear cracks, and shear cracks, with different cracking mechanisms are identified. The inhibitory effect of the inclusions on crack initiation and propagation is revealed. The inclusion–matrix debonding mechanism is further investigated, and it is found that the inclusion interface consists of tensile concentration zones on the left/right sides and compressive concentration zones on the top/bottom sides. The tensile concentration zone is the main reason for debonding of the inclusions from the sandstone matrix.

Geometric multipole expansion and its application to semi-neutral inclusions of general shape

We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.

Many-body interactions between curvature-inducing membrane inclusions with arbitrary cross-sections.

By means of a multipolar expansion, we study analytically and numerically the interaction, in tensionless membranes, between multiple identical curvature-inducing membrane inclusions having arbitrary cross sections but uniform small detachment angles. In particular, for N circular inclusions forming regular polygons, we obtain analytical expressions for the total asymptotic interaction, up to N = 6, and we numerically compute the different multi-body contributions at arbitrary separations. We find that the latter are comparable to the sum of the two-body contributions. For N = 5 inclusions, the analytical asymptotic interaction scales as the inverse sixth power of the nearest neighbors distance d, weaker than the d-4 power for N ≠ 5. The analytical interactions are always repulsive and in good agreement with the numerical results. In the case of noncircular cross sections, we consider the case of two identical inclusions having a given number of equally shaped lobes. Depending on the number of lobes and their amplitude, we find that the interaction is asymptotically either repulsive as d-4 or attractive as d-2, and always repulsive at short distances. We also characterize how the interaction depends on the inclusion rotation angles in the membrane plane.

Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio.

Elasticity imaging is a technique that discovers the spatial distribution of mechanical properties of tissue using deformation and force measurements under various loading conditions. Given the complexity of this discovery, most existing methods approximate only one material parameter while assuming homogeneous distributions for the others. We employ physics-informed neural networks (PINN) in linear elasticity problems to discover the space-dependent distribution of both elastic modulus (E) and Poisson's ratio (ν) simultaneously, using strain data, normal stress boundary conditions, and the governing physics. We validated our model on three examples. First, we experimentally loaded hydrogel samples with embedded stiff inclusions, representing tumorous tissue, and compared the approximations against ground truth determined through tensile tests. Next, using data from finite element simulation of a rectangular domain containing a stiff circular inclusion, the PINN model accurately localized the inclusion and estimated both E and ν. We observed that in a heterogeneous domain, assuming a homogeneous ν distribution increases estimation error for stiffness as well as the area of the stiff inclusion, which could have clinical importance when determining size and stiffness of tumorous tissue. Finally, our model accurately captured spatial distribution of mechanical properties and the tissue interfaces on data from another computational model, simulating uniaxial loading of a rectangular hydrogel sample containing a human brain slice with distinct gray matter and white matter regions and complex geometrical features. This elasticity imaging implementation has the potential to be used in clinical imaging scenarios to reliably discover the spatial distribution of mechanical parameters and identify material interfaces such as tumors. STATEMENT OF SIGNIFICANCE: Our work is the first implementation of physics-informed neural networks to reconstruct both material parameters - Young's modulus and Poisson's ratio - and stress distributions for isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using experimental measurements and synthetic data generated using finite element modeling. Our method can be implemented in clinical elasticity imaging scenarios to improve diagnosis of tumors and for mechanical characterization of biomaterials and biological tissues in a minimally invasive manner.

Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions

Numerical treatment of the problem of two-dimensional viscous fluid flow in and around circular porous inclusions is considered. The mathematical model is described by Navier–Stokes equation in the free flow domain Ωf and nonlinear convective Darcy–Brinkman–Forchheimer equations in porous subdomains Ωp. It is well-known that numerical solutions of the problems in such heterogeneous domains require a very fine computational mesh that resolve inclusions on the grid level. The size alteration of the relevant system requires model reduction techniques. Here, we present a multiscale model reduction technique based on the Generalized Multiscale Finite Element Method (GMsFEM). We discuss construction of the multiscale basis functions for the velocity fields based on the solution of the local problems with and without oversampling strategy. Three test cases are considered for a given choice of the three key model parameters, namely, the Reynolds number (Re), the Forchheimer coefficient (C) and the Darcy number (Da). For the test runs, the Reynolds number values are taken to be Re=1,10,100 while the Forchheimer coefficient and Darcy number are chosen as C=1,10 and Da=10−5,10−4,10−3, respectively. We numerically study the convergence of the method as we increase the number of multiscale basis functions in each domain, and observe good performance of the multiscale method.

Combined diffused material interface and hybrid phase-field model for brittle fracture in heterogeneous composites

In this article, we propose a novel approach for modeling brittle fracture in heterogeneous composites using a combined diffused material interface method and a hybrid phase-field model (PFM). We have used the diffused material interface method to derive analytical expressions for the regularized material properties across the material interfaces that are treated as an input to the PFM for the simulation of brittle fractures in composites. The use of diffused material interface method regularizes the material properties in such a way that one can avoid the discontinuity in stress across the material interface and thus bypass the requirement of explicit tracking of material interfaces for finite element implementation of the hybrid PFM. We have implemented the proposed combined model using a recently developed open-source finite element toolbox, Gridap in Julia. We have considered tests on a single circular fiber inclusion under tension, a bi-material rectangular plate under tension, single square inclusion under tensile loading, a notched composite beam with a central notch under symmetric three-point bending, and a rectangular plate with multiple fibers under tensile loading to show how the proposed combined model can be used to simulate brittle fracture in composites.

Stress concentration around a circular rigid inclusion in incompressible nonlinearly elastic transversely isotropic and orthotropic thin sheets under uniaxial extension

In this paper, we study the stress concentration around a circular rigid inclusion in an incompressible anisotropic homogeneous nonlinearly elastic thin sheet subjected to in-plane finite deformation uniaxial extension. Two classes of material symmetry are considered: transverse isotropy and orthotropy. For the case of transverse isotropy, a three-parameter constitutive relation that under infinitesimal deformations reduces appropriately to the classical incompressible linearized elastic transversely isotropic constitutive relation is studied. Similarly, a six-parameter constitutive relation is considered for the case of orthotropic material symmetry. These constitutive relations may be suitable to describe the response of transversely isotropic and orthotropic bodies subjected to moderate deformations.

Machine learning-enabled development of high performance gradient-index phononic crystals for energy focusing and harvesting

Gradient-index (GRIN) phononic crystals (PnCs) offer an excellent platform for various applications, including energy harvesting via wave focusing. Despite its versatile wave manipulation capability, the conventional design of GRIN PnCs has thus far been limited to relatively simple shapes, such as circular holes or inclusions. In this study, we propose a GRIN PnC comprising of unconventional unit cell designs derived from machine learning-based optimization for maximizing elastic wave focusing and harvesting. A deep neural network (NN) is trained to learn the complicated relationship between the hole shape and intensity at the focal point. By leveraging the fast inference of the trained NN, the genetic optimization approach derives new hole shapes with improved focusing performance, and the NN is updated by augmenting the new dataset to enhance the prediction accuracy over a gradually extended range of performance via active learning. The optimized GRIN PnC design exhibits 3.06 times higher wave energy intensity compared to the conventional GRIN PnC with circular holes. The performance of the best GRIN PnC within the allowable range of our machining tools was validated against experimental measurements, which shows 1.35 and 2.35 times higher focused wave energy intensity and energy harvesting output, respectively.

Propagation of SH-waves in inhomogeneous piezoelectric/piezomagnetic half-space with circular inclusion

Propagation of SH-waves in inhomogeneous piezoelectric/piezomagnetic half-space with circular inclusion

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

AbstractPaper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants.

Thermal conductivity in spherical and circular inclusion composites with highly- and lowly-conducting imperfect interfaces

The variational three-point correlation results are specified to the problem of conductivity of doubly-coated sphere (circle) assemblage. Limiting procedure is developed to deduce the respective explicit expressions, to calculate and analyse the macroscopic conductivity and the microscopic thermal fields in the assemblages with highly- and lowly-conducting imperfect interfaces under the imposed averaged temperature gradient or averaged flux fields, in comparisons with numerical results for some periodic and random micro-geometries of imperfectly-coated inclusions reported in the literature. Equivalent-inclusion approach based on comparisons of analytical dilute results is combined with available effective medium approximations to extend the results for other typical composites with additional three-point correlation information about their micro-geometry, and the complex mixtures. The theoretical models are also correlated with some available experimental data.

Interaction of a rigid line inclusion with various discontinuities using experimental and numerical techniques

Using boundary element method (BEM) and photoelasticity, the interaction effects of a rigid line inclusion with various discontinuities embedded in a matrix are investigated. The interaction effect of discontinuities on a rigid line inclusion has not received much attention when compared to the studies on the interaction effect of discontinuities on a crack. The following discontinuities are considered: rigid circular inclusion, rigid elliptical inclusion, circular hole, elliptical hole, and crack when a tensile load is applied along the inclusion length. A circular hole approaching normal to the line inclusion strengthens the strain fields at the rigid line inclusion tip, whereas a hole along the line inclusion alleviates the strain level. However, in the case of rigid circular inclusion, the opposite trends are observed. The influence of the aspect ratio of the elliptical hole on the elastic fields of a rigid line inclusion is also brought out. The influence of a crack near the vicinity of a rigid line inclusion is also studied. The interaction effects of a circular inclusion, a circular hole, and a crack on the rigid line inclusion are further experimentally analyzed for the first time using the photoelastic technique towards validating the boundary element model. The magnitude of singularity at the inclusion tip is quantified using the strain intensity factor. Also, the magnitude of the stress intensity factor (SIF) at the crack tip is reported. Both the strain intensity factor for rigid line inclusion and the SIF for the crack tip are estimated using the domain integral method.

Fracture mechanism and energy evolution of sandstone with a circular inclusion

Understanding the stress distribution, deformation and fracture behavior around a circular hole or inclusion is critical for the stability assessment of rock structures such as tunnels and backfilled openings. In this study, the analytical solutions of stress and displacement distribution around a circular inclusion of different elastic properties were first derived based on the theory of elasticity. Then prismatic sandstones with a circular hole in different filling modes were prepared and tested in uniaxial compression tests . Progressive deformation and fracturing behavior around holes were analyzed by digital image correlation technique . Moreover, the energy evolution characteristics of specimens were studied to reveal its deformation and failure mechanism. The results show that after filling, the radial and tangential stress and displacement of the inclusion increase, while the stress and displacement around the surrounding rock of the hole decreases, thereby improving the strength and deformation properties . In addition, the crack types and initiation positions around the unfilled and filled holes are quite different due to the stress redistribution. Based on the energy evolution characteristics and the major principal strain fields, the stress-strain behavior is divided into five stages. It is found that the inclusion not only has passive supporting and energy absorption effects on the surrounding rock, but also has a similar crack arrest effect on the initiation and propagation of new cracks. Finally, the effects of inclusion elastic properties and lateral stress coefficient on the stress concentration characteristics around the filled hole were discussed.

Advective Trapping in the Flow Through Composite Heterogeneous Porous Media

We study the mechanisms of advective trapping in composite porous media that consist of circular inclusions of distributed hydraulic conductivity embedded in a high conductivity matrix. Advective trapping occurs when solute enters low velocity regions in the media. Transport is analyzed in terms of breakthrough curves measured at the outlet of the system. The curve’s peak behavior depends on the volume fraction occupied by the inclusions, while the tail behavior depends on the distribution of hydraulic conductivity values. In order to quantify the observed behaviors, we derive two equivalent upscaled transport models. First, we derive a Lagrangian trapping model using the continuous-time random walk framework that is parameterized in terms of volume fraction and the distribution of conductivities in the inclusions. Second, we establish a non-local partial differential equation for the mobile solute concentration by volume averaging of the microscale transport equation. We show the equivalence between the two models as well as (first-order) multirate mass transfer models. The upscaled approach parameterized by medium and flow properties captures all features of the observed solute breakthrough curves and sheds new light on the modeling of advective trapping in heterogeneous media.