Circular Inclusion Research Articles

Identifying inclusions in a non-uniform thermally conductive plate under external flows and internal heat sources using topological optimization

We propose applying topological optimization methods based on measuring temperature and heat fluxes to estimate the thermal conductivity of inhomogeneous thermally conductive plates and determine the shape and location of foreign inclusions. Examples of plates subjected to external heat fluxes and heat sources are considered. The solutions are obtained with the help of the finite element method combined with the method of moving asymptotes. The results show that identification accuracy depends on the defined boundary conditions, the source intensity value, heat fluxes, temperature distribution and the shape of the inclusions. For the problem of 18 circular inclusions under different heat fluxes, identification accuracy increases with increasing source intensity values.

Shear Wave Speed estimator using Continuous Wavelet Transform for Crawling Wave Sonoelastography.

Crawling Wave Sonoelastography (CWS) is an elastography ultrasound-based imaging approach that provides tissue stiffness information through the calculation of Shear Wave Speed (SWS). Many SWS estimators have been developed; however, they report important limitations such as the presence of artifacts, border effects or high computational cost. In addition, these techniques require a moving interference pattern which could be challenging for in vivo applications. In this study, a new estimator based on the Continuous Wavelet Transform (CWT) is proposed. This allows the generation of a SWS image for every sonoelasticity video frame. Testing was made with data acquired from experiments conducted on a gelatin phantom with a circular inclusion. It was excited with two vibration sources placed at both sides with frequencies ranging from 200 Hz to 360 Hz in steps of 20 Hz. Results show small variation of the SWS image across time. Additionally, images were compared with the Phase Derivative method (PD) and the Regularized Wavelength Average Velocity Estimator (R-WAVE). Similar SWS values were obtained for the three estimators within a certain region of interest in the inclusion (At 360 Hz, CWT: 5.01±0.2m/s, PD: 5.11±0.28m/s, R-WAVE: 4.51±0.62m/s) and in the background (At 360 Hz, CWT: 3.67±0.15m/s, PD: 3.69±0.23m/s, R-WAVE: 3.58±0.24m/s). CWT also presented the lowest coefficient of variation and the highest contrast-to-noise ratio for most frequencies, which allows better discrimination between regions.Clinical relevance-This study presents a new Shear Wave Speed estimator for Crawling Wave Sonoelastography, which can be useful to characterize soft tissue and detect lesions.

Representative volume elements for matrix-inclusion composites - a computational study on the effects of an improper treatment of particles intersecting the boundary and the benefits of periodizing the ensemble

We investigate volume-element sampling strategies for the stochastic homogenization of particle-reinforced composites and show, via computational experiments, that an improper treatment of particles intersecting the boundary of the computational cell may affect the accuracy of the computed effective properties. Motivated by recent results on a superior convergence rate of the systematic error for periodized ensembles compared to taking snapshots of ensembles, we conduct computational experiments for microstructures with circular, spherical and cylindrical inclusions and monitor the systematic errors in the effective thermal conductivity for snapshots of ensembles compared to working with microstructures sampled from periodized ensembles.We observe that the standard deviation of the apparent properties computed on microstructures sampled from the periodized ensembles decays at the scaling expected from the central limit theorem. In contrast, the standard deviation for the snapshot ensembles shows an inferior decay rate at high filler content. The latter effect is caused by additional long-range correlations that necessarily appear in particle-reinforced composites at high, industrially relevant, volume fractions. Periodized ensembles, however, appear to be less affected by these correlations.Our findings provide guidelines for working with digital volume images of material microstructures and the design of representative volume elements for computational homogenization.

Circular inclusion problem of two‐dimensional decagonal quasicrystals with interfacial rigid lines under concentrated force

AbstractThe plane problem of two‐dimensional decagonal quasicrystals with a rigid circular arc inclusion was investigated under infinite tension and concentrated force. Based on complex representations of stresses and displacements of two‐dimensional decagonal quasicrystals, the above problem is transformed into Riemann boundary problem by using the analytic continuation principle of complex functions. The general solutions of two‐dimensional decagonal quasicrystals under the action of plane concentrated force and infinite uniform tension are derived. The closed solutions of complex potential functions in several typical cases are obtained, and the formula of singular stress field at the tip of rigid line inclusions is given. The results show that the stress field at the tip of circular arc rigid line inclusions has singularity of oscillation under plane load. Numerical examples are given to analyze the effects of inclusion radius, different inclusions, the coupling coefficient and phason field parameter on stress singularity coefficients.

Experimental detection and shape reconstruction of inclusions in elastic bodies via a monotonicity method

We deal with the detection and shape reconstruction of inclusions in elastic bodies based on a monotonicity method and aim to reconstruct them with experimental measurements. Thus, we base our studies on the rigorously proven theory of the linearized monotonicity tests for noisy measurement data, where the so-called Neumann-to-Dirichlet operator, its Fréchet derivative and the corresponding monotonicity properties play an essential role. Further on, we give an insight into the lab experiment itself. More specifically, we take a look at Makrolon plates with one or two circular aluminium inclusions. Concerning the realization of the measurements, we have to deal with missing data which cannot be measured due to the set-up of the experiment. Hence, we take a look at a modified spline interpolation in order to determine this data. In doing so, we state the required steps for the implementation of the linearized monotonicity tests. Finally, we present our reconstructions based on experimental data and compare them with the simulations obtained from artificial data, where we want to highlight that all inclusions can be detected from the noisy experimental data, thus, we obtain accurate results. This paper combines the rigorously proven theory of the monotonicity methods developed for linear elasticity with the explicit application of the methods, i.e. the implementation and simulation of the reconstruction of inclusions in elastic bodies for both artificial and experimental data.

Block structured adaptive mesh refinement and strong form elasticity approach to phase field fracture with applications to delamination, crack branching and crack deflection

Fracture is a ubiquitous phenomenon in most composite engineering structures, and is often the responsible mechanism for catastrophic failure. Over the past several decades, many approaches have emerged to model and predict crack failure. The phase field method for fracture uses a surrogate damage field to model crack propagation, eliminating the arduous need for explicit crack meshing. In this work a novel numerical framework is proposed for implementing hybrid phase field fracture in heterogeneous materials. The proposed method is based on the “reflux-free” method for solving, in strong form, the equations of linear elasticity on a block-structured adaptive mesh refinement (BSAMR) mesh. The use of BSAMR enables highly efficient and scalable regridding, facilitates the use of temporal subcycling for explicit time integration, and allows for ultra-high refinement at crack boundaries with minimal computational cost. The method is applied to a variety of simple heterogeneous structures: laminates, wavy interfaces, and circular inclusions. In each case a non-dimensionalized parameter study is performed to identify regions of behavior, varying both the geometry of the problem and the relative fracture energy release rate. In the laminate and wavy interface cases, regions of delamination and fracture correspond to simple analytical predictions. For the circular inclusions, the modulus ratio of the inclusion is varied as well as the delamination energy release rate and the problem geometry. In this case, a wide variety of behaviors was observed, including deflection, splitting, delamination, and pure fracture.

Method of image charges for describing deformation of bounded two-dimensional solids with circular inclusions.

We present a method for predicting the linear response deformation of finite and semi-infinite 2D solid structures with circular holes and inclusions by employing the analogies with image charges and induction in electrostatics. Charges in electrostatics induce image charges near conductive boundaries and an external electric field induces polarization (dipoles, quadrupoles, and other multipoles) of conductive and dielectric objects. Similarly, charges in elasticity induce image charges near boundaries and external stress induces polarization (quadrupoles and other multipoles) inside holes and inclusions. Stresses generated by these induced elastic multipoles as well as stresses generated by their images near boundaries then lead to interactions between holes and inclusions and with their images, which induce additional polarization and thus additional deformation of holes and inclusions. We present a method that expands induced polarization in a series of elastic multipoles, which systematically takes into account the interactions of inclusions and holes with the external field, between them, and with their images. The results of our method for linear deformation of circular holes and inclusions near straight and curved boundaries show good agreement with both linear finite element simulations and experiments.

Elastic multipole method for describing deformation of infinite two-dimensional solids with circular inclusions.

Elastic materials with holes and inclusions are important in a large variety of contexts ranging from construction material to biological membranes. More recently, they have also been exploited in mechanical metamaterials, where the geometry of highly deformable structures is responsible for their unusual properties, such as negative Poisson's ratio, mechanical cloaking, and tunable phononic band gaps. Understanding how such structures deform in response to applied external loads is thus crucial for designing novel mechanical metamaterials. Here we present a method for predicting the linear response of infinite 2D solid structures with circular holes and inclusions by employing analogies with electrostatics. Just like an external electric field induces polarization (dipoles, quadrupoles, and other multipoles) of conductive and dielectric objects, external stress induces elastic multipoles inside holes and inclusions. Stresses generated by these induced elastic multipoles then lead to interactions between holes and inclusions, which induce additional polarization and thus additional deformation of holes and inclusions. We present a method that expands the induced polarization in a series of elastic multipoles, which systematically takes into account the interactions of inclusions and holes with the external stress field and also between them. The results of our method show good agreement with both linear finite element simulations and experiments.

Residual stress as a fracture toughening mechanism: A Phase-Field study on a brittle material

Recent engineering design practice for materials and structures relies more and more on damage-tolerant criteria. Such a design approach is attained mainly by employing materials showing a certain level of fracture toughness.This work aims to explore a way to generate fracture toughness in materials that intrinsically shows no toughness at all, i.e. brittle materials. The key idea lies in the introduction of inelastically deformed sub-regions (e.g. circular inclusions) in the base material, which inevitably generate a residual stress field.To accomplish this purpose, the advanced Phase-Field method coupled with the eigenstrain theory is employed, respectively to simulate the crack propagation behavior and to introduce a residual stress field in a pre-notched sample. Information about crack propagation and displacement externally imposed is used to obtain the resistance curve (R-curve) for several configurations.One of the main findings of this research regards the possibility of originating fracture toughness in intrinsically brittle materials upon appropriate positioning of one inclusion - containing a certain amount of inelastic deformation – with respect to a notch tip. This result demonstrates that accurate design of residual stress is crucial to attaining unprecedented material or structure performance, and the method shown here represents a valid tool to exploit this advanced design capability.

Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusion

We establish the uniformity of stresses inside both a non-parabolic open inhomogeneity and a non-elliptical closed inhomogeneity interacting with a nearby circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Our procedure involves the introduction of a conformal mapping function for the doubly connected domain occupied by the matrix and the circular Eshelby inclusion. Two conditions are established in order to achieve the uniformity property inside each of the two inhomogeneities. Our results indicate that: (a) the internal uniform stresses are independent of the specific shapes of the two inhomogeneities and the existence of the nearby circular Eshelby inclusion; (b) the open and closed shapes of the respective inhomogeneities are significantly affected by the presence of the circular Eshelby inclusion. We also consider the two more complex cases involving: (a) an arbitrary number of circular Eshelby inclusions undergoing uniform eigenstrains; (b) a circular Eshelby inclusion undergoing linear eigenstrains. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed theory.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

The Emergence of Sequential Buckling in Reconfigurable Hexagonal Networks Embedded into Soft Matrix

The extreme and unconventional properties of mechanical metamaterials originate in their sophisticated internal architectures. Traditionally, the architecture of mechanical metamaterials is decided on in the design stage and cannot be altered after fabrication. However, the phenomenon of elastic instability, usually accompanied by a reconfiguration in periodic lattices, can be harnessed to alter their mechanical properties. Here, we study the behavior of mechanical metamaterials consisting of hexagonal networks embedded into a soft matrix. Using finite element analysis, we reveal that under specific conditions, such metamaterials can undergo sequential buckling at two different strain levels. While the first reconfiguration keeps the periodicity of the metamaterial intact, the secondary buckling is accompanied by the change in the global periodicity and formation of a new periodic unit cell. We reveal that the critical strains for the first and the second buckling depend on the metamaterial geometry and the ratio between elastic moduli. Moreover, we demonstrate that the buckling behavior can be further controlled by the placement of the rigid circular inclusions in the rotation centers of order 6. The observed sequential buckling in bulk metamaterials can provide additional routes to program their mechanical behavior and control the propagation of elastic waves.

Electro-elastic field of a piezoelectric quasicrystal medium containing two cylindrical inclusions

Considering the piezoelectric effect, the electro-elastic field of an infinite one-dimensional quasicrystal medium with two circular cylindrical inclusions is derived under antiplane shear and inplane electric loading. The boundary value problem of the composite material with circular cylindrical inclusions is analytically solved by the use of the conformal mapping technique and analytical continuation theory. The stresses in the phonon and phason fields and the electric displacements are obtained explicitly in the form of a power series both for the matrix and the inclusions. Some typical examples are analyzed to show the effect of the geometric parameters, material properties, and electro-mechanical loading on the electro-elastic fields in the matrix, inclusions, and interfaces. The limiting cases of circular cavities and rigid circular inclusions have also been investigated and discussed.

Mechanical effects of circular liquid inclusions inside soft matrix: role of internal pressure change and surface tension

The mechanical effects of dilute liquid inclusions on the solid-liquid composite are explored, based on an analytical circular inclusion model incorporating the internal pressure change of the liquid and the surface tension of the interface. Several simple explicit dependences of the stress field and effective stiffness on the bulk modulus and the size of the liquid, the surface tension, and Poisson’s ratio of the matrix are derived. The results show that the stresses in the matrix are reduced, and the stiffness of the solid-liquid composite is enhanced with the consideration of either the surface tension or the internal pressure change. Particularly, the effective Young’s modulus predicted by the present model for either soft or stiff matrices agrees well with the known experimental data. In addition, according to the theoretical results, it is possible to stiffen a soft solid by pressured gas with the presence of the surface tension of the solid-gas interface.

Limit transitions in plane homogenization problems for two-phase dielectric composites with extreme material properties of one phase

The plane problem of calculating the effective dielectric constant of two-phase composite consisting of main material with one circular inclusion is considered. To solve the homogenization problem, the method of effective moduli with the support of the energy balance between the composite and the homogeneous comparison medium was used. In the solution obtained, the passage to the limit was made for the case of an inclusion with zero dielectric constant and for the case of a conductive inclusion. The limiting solutions are compared with the solutions of homogenization problems for a medium with a hollow and for a medium with a conducting inclusion boundary.

Conductivity in the 2D Rayleigh Model for the Critical Concentration (Percolation Threshold)

The conductivity in the 2D Rayleigh model (isotropic matrix with a periodic distribution of circular inclusions) is considered for the critical concentration (percolation threshold). The effective conductivity of the model with the metal–ideal conductor phase transition is calculated using the binary approximation. The corresponding effective conductivity for an alternative model with a metal–insulator transition is determined from the Keller–Dykhne reciprocity relation.

Derivation of FFT numerical bounds of the effective properties of composites and polycristals

In this paper, we provide exact fast Fourier transform (FFT)-based numerical bounds for the elastic properties of composites having arbitrary microstructures. Two bounds, an upper and a lower, are derived by considering usual variational principles based on the strain and the stress potentials. The bounds are computed by solving the Lippmann-Schwinger equation together with the shape coefficients which allow an exact description of the microstructure of the composite. These coefficients are the exact Fourier transform of the characteristic functions of the phases. In this study, the geometry of the microstructure is approximated by polygonals (two-dimensional, 2D objects) and by polyhedrons (three-dimensional, 3D objects) for which exact expressions of the shape coefficients are available. Various applications are presented in the paper showing the relevance of the approach. In the first benchmark example, we consider the case of a composite with fibers. The effective elastic coefficients ares derived and compared, considering the exact shape coefficient of the circular inclusion and its approximation with a polygonal. Next, the homogenized elastic coefficients are derived for a composite reinforced by 2D flower-shaped inclusions and with 3D toroidal-shaped inclusions. Finally, the method is applied to polycristals considering Voronoi tessellations for which the description with polygonals and polyhedrons becomes exact. The comparison with the original FFT method of Moulinec and Suquet is provided in order to show the relevance of these numerical bounds.

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

A mixed boundary value problem of a cracked elastic medium under torsion

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method

Low- and mid-frequency range vibrations usually present issues in their insulation. The development of periodic metamaterials (artificially engineered structures) has been thoroughly studied since their periodicity and geometric properties may yield good attenuation properties in such frequency ranges. Several works investigated periodic structures presenting binary or ternary material configurations using the Kirchhoff plate theory or binary material configurations using the Mindlin plate theory. In this paper, a ternary periodic metamaterials made of concentric circular inclusions of a rigid material coated with a soft material embedded in a matrix made of a low-cost recyclable polymer is investigated using the Mindlin plate theory combined with the plane wave expansion method. Computed band diagrams are compared to those obtained using the Kirchhoff plate theory and three-dimensional models. Additionally, the effect of different boundary conditions is investigated on finite structures using the finite element method. Our results assess the validity of the Mindlin plate model in such case and indicate its use in practical applications.