Arbitrary Inclusion Shapes Research Articles

Role of interface tension in the thermoelastic analysis of inclusions: Unified formulation and closed-form results

We study the thermoelastic problem of an inclusion of general shape surrounded by a foreign elastic matrix under plane deformation and the influence of an in-plane far-field heat flux. The small-scale interface effects, related to not only interface conductivity, interface stretching rigidity and interface thermal expansion but also (residual) interface tension, are incorporated in the corresponding analysis. We establish general boundary value formulations in both heat conduction and thermoelasticity for an arbitrary inclusion shape, allowing for analytic or at least semi-analytic treatment of the problem. Specially, we derive closed-form solutions for the case of circular inclusion and obtain concise expressions for describing the remote heat flux-induced thermal stress near the interface (at both the inclusion side and matrix side). Surprisingly, it is shown that for the case of circular inclusion, interface tension plays the same role as interface stretching rigidity in determining the full thermal stress field, or in other words, the full thermal stress field depends on only the sum of interface tension and interface stretching rigidity but not on the specific value of either of them. Numerical examples are also given to illustrate the influence of interface tension on the von Mises stress distribution in the vicinity of the interface.

Automated computation of topological derivatives with application to nonlinear elasticity and reaction–diffusion problems

While topological derivatives have proven useful in applications of topology optimization and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the numerical computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction–diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behavior also for other, previously unknown problems, indicating the correctness of the procedure.

A computationally efficient approach for generating RVEs of various inclusion/fibre shapes

A computationally efficient method for generating virtual periodic representative volume element (RVE), capable of handling arbitrary inclusion shapes, is developed. A universal collision/overlap detection and repair method is proposed, where each inclusion shape is represented as a union of n-Spheres (UnS). A constrained optimization problem is formulated and solved to remove inclusion overlaps; a closed-form solution is derived for calculating the degree of inclusions overlap and its gradient vector with respect to inclusion position. RVE generation is illustrated with circular, spherical, four non-circular and four non-spherical inclusion shapes. Computational efficiency is demonstrated using an elaborate RVE generation time study. The generated RVEs are evaluated using various statistical metrics; results confirm the random distribution of inclusions. Effective properties of RVEs, representing unidirectional composites, are determined using homogenization with various fibre cross-section shapes; obtained mechanical properties have shown transverse isotropy.

Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host

Abstract. Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and temperature (P–T) conditions of mineral inclusion entrapment. Existing modelling methods in petrology either adopt an assumption of a spherical, isotropic inclusion embedded in an isotropic, infinite host or use numerical techniques such as the finite-element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusions. Here, we use the Eshelby solution to develop an analytical framework for calculating the residual stress and strain state of an elastically anisotropic, ellipsoidal inclusion in an infinite, isotropic host. The analytical solution is applicable to any class of inclusion symmetry and an arbitrary inclusion aspect ratio. Explicit expressions are derived for some symmetry classes, including tetragonal, hexagonal, and trigonal. The effect of changing the aspect ratio on residual stress is investigated, including quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Quartz is demonstrated to be the least affected, while rutile is the most affected. For prolate quartz inclusion (c axis longer than a axis), the effect of varying the aspect ratio on Raman shift is demonstrated to be insignificant. When c/a=5, only ca. 0.3 cm−1 wavenumber variation is induced as compared to the spherical inclusion shape. For oblate quartz inclusions, the effect is more significant, when c/a=0.5, ca. 0.8 cm−1 wavenumber variation for the 464 cm−1 band is induced compared to the reference spherical inclusion case. We also show that it is possible to fit an effective ellipsoid to obtain a proxy for the averaged residual stress or strain within a faceted inclusion. The difference between the volumetrically averaged stress of a faceted inclusion and the analytically calculated stress from the best-fitted effective ellipsoid is calculated to obtain the root-mean-square deviation (RMSD) for quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Based on the results of 500 randomly generated (a wide range of aspect ratio and random crystallographic orientation) faceted inclusions, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond, which has a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.

A three-dimensional model for rain-wind induced vibration of stay cables in cable-stayed bridges

The most rigorous effective medium approximations for elastic moduli are elaborated for matrix composites made from an isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres. In this paper, we focus specially on the effective elastic moduli of the heterogeneous composites with arbitrary inclusion shapes. The main idea of this paper is to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified elastic moduli. Available simple approximations for the equivalent circular (spherical) inclusion media then can be used to estimate the effective properties of the original medium. The data driven technique is employed to estimate the properties of equivalent inclusions and the Extended Finite Element Method is introduced to modeling complex inclusion shapes. Robustness of the proposed approach is demonstrated through numerical examples with arbitrary inclusion shapes.
 Keywords:
 data driven approach; equivalent inclusion, effective elastic moduli; heterogeneous media; artificial neural network.

Consideration of arbitrary inclusion shapes in the framework of isotropic continuum micromechanics: The replacement Eshelby tensor approach

The effective behavior of matrix-inclusion materials is governed by the properties and volume fractions of involved material phases as well as by the morphology of the inclusion domain. Eshelby’s solutions on the ellipsoidal inclusion problem (Eshelby, 1957) paved the way to the development of a vast number of continuum micromechanics based homogenization schemes for evaluation of effective properties. This paper proposes an approach for utilizing these homogenization schemes also for non-ellipsoidal inclusions with particular focus on macroscopic isotropy. The key concept is to replace the originally-used Eshelby tensor by an analogous representation capturing the effect of non-spherical inclusions on strain fluctuation within the material domain. The so-called replacement Eshelby tensor is obtained by numerical evaluation of two adjustment factors being functions of the inclusion morphology and the stiffness contrast between the matrix and inclusion phase. These factors are evaluated and discussed for selected inclusion morphologies. Finally, the replacement Eshelby tensor approach is implemented in a homogenization procedure for effective elastic properties (cascade continuum micromechanics model (Timothy and Meschke, 2016)) as well as in a scheme for determining effective yield surfaces in case of plastic material behavior (Traxl and Lackner, 2015).

Surface tension-induced interfacial stresses around a nanoscale inclusion of arbitrary shape

Investigated in this paper is the surface tension-induced stress field around a nanoscale inclusion of arbitrary shape embedded in an infinite elastic plane, with an emphasis on the combined effects of surface tension and arbitrary inclusion shape. Muskhelishvili’s complex variable method is employed to formulate the basic equations that are solved with the aid of conformal mapping and series expansion methods. Accuracy of the present solution is verified by comparing its predictions with known results of an arbitrarily shaped hole. The surface tension-induced stress field is demonstrated for four shapes of inclusions (ellipse, approximately regular triangle, square and regular pentagon with round corners). The numerical results show that as the inclusion changes from a “soft” to a “hard” one (compared to the matrix), hoop and normal stresses along the inclusion–matrix interface on the inclusion side will increase, while those on the matrix side will decrease. However, shear stress along the inclusion–matrix interface does not considerably change as the inclusion changes from a “soft” to a “hard” one. It is also found that the maximum hoop stress on the matrix side of a “soft” inclusion and the maximum shear stress always occur nearby, but not exactly at the round corners where all other stresses for all four discussed inclusion shapes attain their maximums. Besides, for the four inclusion shapes discussed, shear stress along the inclusion–matrix interface vanishes at all corners as a consequence of the symmetry.

A statistical descriptor based volume-integral micromechanics model of heterogeneous material with arbitrary inclusion shape

A continuing challenge in computational materials design is developing a model to link the microstructure of a material to its material properties in both an accurate and computationally efficient manner. In this paper, such a model is developed which uses image-based data from characterization studies combined with a newly developed self-consistent volume-integral micromechanics model (SVIM) for linear elastic material. It is observed that SVIM is able to capture the effective stress/strain distribution inside the inclusion, as well as effects of volume fraction and nearest inclusion distance on the effective properties of heterogeneous material. More importantly, SVIM can be applied to inclusions with arbitrary shape through discretizing the inclusion domain. For both 2-dimensional and 3-dimensional problems with circular and spherical inhomogeneities, SVIM's capability of predicting effective elastic properties is validated against experiments and direct numerical simulations using the finite element method. Finally, the effect of inclusion shape is predicted by SVIM.

Dielectric responses of anisotropic graded granular composites having arbitrary inclusion shapes

The transformation field method (TFM) originated from Eshelby's transformation field theory is developed to estimate the effective permittivity of an anisotropic graded granular composite having inclusions of arbitrary shape and arbitrary anisotropic grading profile. The complicated boundary-value problem of the anisotropic graded composite is solved by introducing an appropriate transformation field within the whole composite region. As an example, the effective dielectric response for an anisotropic graded composite with inclusions having arbitrary geometrical shape and arbitrary grading profile is formulated. The validity of TFM is tested by comparing our results with the exact solution of an isotropic graded composite having inclusions with a power-law dielectric grading profile and good agreement is achieved in the dilute limit. Furthermore, it is found that the inclusion shape and the parameters of the grading profile can have profound effect on the effective permittivity at high concentrations of the inclusions. It is pointed out that TFM used in this paper can be further extended to investigate the effective elastic, thermal, and electroelastic properties of anisotropic graded granular composite materials.

Transformation field method for calculating the effective properties of isotropic graded composites having arbitrary shapes

A method of transformation field is developed to estimate the effective properties of graded composites whose inclusions have arbitrary shapes and gradient profiles by means of a periodic cell model. The boundary-value problem of graded composites having arbitrary inclusion shapes is solved by introducing the transformation field into the inclusion region. As an example, the effective dielectric response of isotropic graded composites having arbitrary shapes and gradient profiles is handled by the transformation field method (TFM). Moreover, TFM results are validated by the exact solutions of isotropic graded spherical inclusions having a power-law profile and good agreement is obtained in the dilute limit. Furthermore, it is found that the inclusion shapes and the parameters of the gradient profiles can have profound effect on the effective properties of composite systems at high concentration of inclusions.

Estimation of Effective Elastic Properties of Random Structure Composites for Arbitrary Inclusion Shape and Anisotropy of Components Using Finite Element Analysis

Estimation of Effective Elastic Properties of Random Structure Composites for Arbitrary Inclusion Shape and Anisotropy of Components Using Finite Element Analysis

Debonding of rigid curvilinear inclusions in longitudinal shear deformation

The problem of two symmetrically placed interface cracks at rigid curvilinear inclusions under longitudinal shear deformation is considered. A solution valid for arbitrary inclusion shapes is found. It depends on a parameter β describing the cracks. For β = e iα where α is an angle, the cracks lie in the interface. For β real and greater than unity, we have two radial cracks emanating from a curvilinear cavity. The solution for β = 1 corresponds to a completely debonded inclusion. Examples of elliptic, square with rounded corners, and rectangular inclusions are worked out in detail. It is shown that the crack tip stress intensity factor becomes infinite for interface cracks terminating at cusps and corners. This phenomenon is attributed to the change in the nature of the singularity as the crack tip approaches a cusp or corner. The singularity is three-quarter power at a cusp and two-thirds power at a corner of a rectangular inclusion. Finally, the application of the results to composite materials is indicated.