Ellipsoidal Inclusions Research Articles

A Layered-Shell Model of Anisotropic Composites: Extension of the Milgrom and Shtrikman Model

In this work, the extension of the Milgrom and Shtrikman model to anisotropic composite materials containing n-layered hollow ellipsoidal inclusions, is presented. The effective properties of such materials are determined using the Green function techniques and interfacial operators. Here, the basic unit of the microstructure is a hollow system of contacting concentric ellipsoidal shells, each of which is made of one of the components. Space is packed with such units of different sizes, but the same proportions; the cavity within each such shell system is then packed with similar systems and this continues in an infinite nesting sequence. In the final configuration, the effective properties are inside and outside the basic unit of layered shells (n+1). For n=2 and in the case of isotropic material, it is shown that the effective compressibility covers all ranges of the Hashin-Strikman bounds.

Anisotropic induced polarization modeling with neural networks and effective medium theory

Interpreting three-dimensional induced polarization (IP) data requires rock physics models that reflect the omnipresent anisotropy of the Earth’s crust. The Generalized Effective Medium Theory of Induced Polarization (GEMTIP) can model the IP signatures of rocks with polarizable mineral inclusions. However, it is computationally demanding because it requires numerically solving the depolarization tensors of each inclusion. We aim to streamline GEMTIP simulations by (1) integrating anisotropic background conductivity and triaxial ellipsoidal inclusions in the model and (2) estimating the depolarization tensors with a neural network. We validate the neural network predictions against known solutions for spherical and spheroidal inclusions, and we test our method using data from an actual rock sample. The neural network shows a relative sensitivity of 56% to inclusion shape and 44% to host rock anisotropy and is up to 100,000 times faster than numerical integration. We release the pre-trained neural network implementation as an open-source Python package, thereby providing a new method to interpret the IP signatures of anisotropic rocks.

Effective elastic behavior analysis of three-layer quasicrystal composite plates

In this paper, the effective elastic behavior of three-layer quasicrystal composite plates are forecasted. Firstly, effective properties of the first and third layers quasicrystal composite plates containing ellipsoidal inclusions are obtained according to the Mori-Tanaka model. Secondly, the three-layer composite plates effective behavior parameters are computed by employing the general matrix approach. The numerical examples reveal the influence of the inclusions geometry and volume fraction on the three-layer quasicrystal composite plates effective behavior parameters. The findings indicate that changing the elastic constants of the phason field and phonon-phason coupling field of inclusions will affect the effective behavior parameters of the three-layer quasicrystal composite plates. The increase in layers to some extent reduces the effective behavioral parameters.

Effect of rare earth element on H13 microscopic structure and mechanical properties

In view of the phenomenon that irregular inclusions are easy to use as crack sources, rare earth H13 steel with different gradients is designed. The results showed that ellipsoidal rare earth inclusions (Ce2O3, CeAlO3 and Ce2O2S) were formed from irregular Al2O3 inclusions with the assistance of rare earths. The carbides that precipitated from liquid phases were arranged in a network along the grain boundary, forming composite carbides that are rich in Mo and V. Along the grain boundary, the morphology of the composite carbides got finer, and MC-type carbides enclosed the modified inclusions into nuclei, facilitating their precipitation. After annealing, the microstructure revealed that the carbides were dispersed in black segregated patches along the grain boundary and were discontinuous. The average diameter of carbides decreased from 109.7 to 79.2 nm in the same region according to a statistical comparison of the number of carbides before and after the addition of rare earth elements. The carbides were distributed alongside primary carbides as tiny particles of M7C3 and M23C6 secondary carbides. Especially when the rare earth content was 0.032 wt-%, the impact toughness increased to 34%. This work offers a reference value for the practical application of rare earth in hot work die steel in addition to revealing the strengthening mechanism of ultra-high strength RE-13 steel.

Effect of Cerium on Inclusion Evolution and Pitting Corrosion of Nb-Microalloyed Steel.

Nb-microalloyed steels are widely used in construction engineering fields due to their excellent mechanical properties, but they face serious corrosion problems in service environments. Pitting corrosion is the severest form of corrosion, and the types of inclusions are the leading cause to induce pitting corrosion. A new strategy is proposed to enhance the corrosion resistance of steels by achieving a beneficial transformation of inclusions with Ce treatment. In this paper, two types of Nb-microalloyed steels (0% Ce and 0.0058% Ce steel) were prepared to study the modification effect on inclusions in industrial production. The spherical CaS•C12A7 inclusions were modified to smaller ellipsoidal Ce2O2S inclusions, and the proportion of inclusions (0-2 μm) increased significantly from 27 to 66%, while large inclusions (>6 μm) disappeared. A kinetic model of inclusion evolution was established. The results of electrochemical tests indicated that the corrosion potential was positively shifted, and the corrosion current was reduced after Ce treatment. Additionally, the number of defects in the passivation film was decreased, and the corrosion resistance of the steel was significantly improved. The addition of Ce changed the types of inclusions and reduced the number of pitting nucleation points, which led to a remarkable reduction in the number and size of pitting pits. The mechanism of pitting corrosion induced by different types of inclusions was further investigated, and a pitting corrosion model was modeled based on the immersion experiments. Research results provide theoretical support for enhancing the corrosion resistance of steel.

On the anisotropic coalescence of elliptic cylindrical voids considering the geometric and distributive properties

The geometric and distributive properties of voids significantly influence anisotropic coalescence behaviour. However, this problem has received little attention owing to the complexity of considering all the properties in the current analytical framework of limit analysis. To address this issue, this study proposes an analytical framework based on an elliptic coordinate system, including the determination of the ligament zone, characterization of plastic flow, and derivation of the void coalescence criterion, for porous materials with various geometric and distributive properties, including size, shape, spacing, and orientation. This framework is motivated by our observations that the evolution of the void geometry and surrounding plastic flow can be well characterized by the grid of the elliptic coordinate system. Subsequently, an analytical function is proposed to determine the ligament zone and coalescence direction with various void properties. A hollow nonaxisymmetric cylindrical unit cell is proposed to describe this ligament zone, and the corresponding trial velocity field is derived by extending the previous Gurson-like velocity field into the elliptic cylindrical coordinate system. The rationality of the field is validated by comparing its equivalent strain rate field with numerical simulations. Finally, a coalescence criterion is derived via the limit analysis of the proposed unit cell undergoing internal necking. Two heuristic adjustments are formulated for the overflow phenomenon in the rigid zone and outer ligament zones. Numerical assessments with various void properties confirm the accuracy of the analytical model. The coalescence criterion can predict the independent and coupling effects of geometric and distributive properties on anisotropic void coalescence. This study provides possible solutions to future plasticity problems of ellipsoidal inclusions.

Ultrasonic fatigue tests on maraging 300 steel: Under solution annealed, after aging heat treatment and under pre-corrosion attack

Ultrasonic fatigue tests were carried out under continuous cycling on the maraging 300 steel for the following conditions: (A) solution annealed (as received from supplier), (B) after aging heat treatment of 490 °C for 6 h, (C) after pre-corrosion attack, and (D) specimens loaded at 293 MPa at room temperature without failure until 1.0E+10 cycles. The ultrasonic fatigue strength of the four modalities were compared and discussed in regard the crack initiation inclusion, the heat treatment and the testing conditions. Crack initiation and propagation under this fatigue testing modality was analyzed; revealing that ultrasonic fatigue strength is related to internal TiN-inclusions and its parameters of shape and orientation, in regard the uniaxial applied load. Numerical simulations were carried out to investigate the stress concentration of an ellipsoidal void of 150 mm (longer radius), and a TiN ellipsoidal inclusion of same dimensions. In addition, SEM (Scanning Electron Microscope) analysis was carried out on the fracture surfaces to determine the crack initiation and propagation zones.

Influence of elastic anisotropy on the shapes of ellipsoidal blisters and stress field around the blisters in solid materials

To address the embrittlement challenges posed by gas blisters in anisotropic materials, the stable shape of constant-pressure blisters in anisotropic materials (hexagonal, tetragonal, and rhombohedral) was energetically investigated based on continuum theory (micromechanics), considering the blister as Eshelby’s ellipsoidal inclusion. The non-negligible change in the blister shape was confirmed in terms of the anisotropic factor η ≡ C3333/C1111. Although the spherical shape of the blister is preferable for isotropic and cubic materials (η = 1), the x3 normal penny and capsule shapes were theoretically confirmed to be the most stable ones for η > 1 and η < 1, respectively. The penny and capsule shape blisters generate larger stress fields around themselves than the sphere shape blisters, thus inducing crack formation. The embrittlement due to the gas (typically hydrogen or helium) inside the blister for the anisotropic materials was more significant than isotropic and cubic embrittlement.

Phase Transformation Under High Pressure Radiates as a Double Couple Deep Earthquake

AbstractDeep‐focus earthquakes (DFEs) originating at the Mantle‐Transition‐Zone (MTZ) (400–700 km) have a Double Couple (DC) radiation pattern similar to crustal earthquakes; however, their mechanism is different and governed by high pressures (15–25 GPa) at nucleation depths. We present a model of nucleation and growth of regions of phase transformation, undergoing a sudden reduction in volume (5%–10%), “volume collapse.” Successive symmetry‐breaking instabilities minimize the energy spent to move the boundary of phase discontinuity and a collapsing volume expands as a flattened pancake‐like self‐similarly expanding Eshelby ellipsoidal inclusion. At the vanishing of the M integral, expressing the balance of flows of energy across the inclusion boundary, at a critical value of the pressure, an arbitrarily small inclusion nucleates and grows at constant potential energy driven by the pressure acting on the change in volume. The inclusion develops shear eigenstrains that decompose into two DC, placing one on the basal plane to radiate without energy losses. The symmetric volume collapse radiates out as an anti‐symmetric DC, and the radiated energy is obtained as the “excess energy,” of the ambient pressure acting on the “volume collapse,” reduced by the energy consumed for the growth of the pancake surfaces, with a “pressure drop” (p0 − pcr) driving the expansion emitting a DC, even under full isotropy. The solution explains some features of the DFEs, (a) the DC radiation, (b) their large energies (the Mw 8.3 Okhotsk earthquake [2013]), (c) the absence of volumetric radiation, and (d) why they can originate in the MTZ, a long‐standing open problem.

Ab initio stability prediction of [formula omitted] titanium and [formula omitted] and [formula omitted] precipitates in [formula omitted] titanium matrix for titanium alloys using density functional theory and micromechanics

β titanium alloys have a wide range of applications, including in aerospace and transport. The β phase is stable only at high temperatures, and the phase stability can be improved by adding stabilizers. However, the β-phase-stabilization effects of β stabilizers have not yet been clearly elucidated. Here, I report the ab initio prediction of the energetic phase stability of β titanium alloys with Mo, V, W, Nb, and Ta as additive β stabilizer elements. The stability is predicted using a combination of atomistic simulation via density functional theory and continuum micromechanics (Eshelby’s ellipsoidal inclusion analysis). In particular, I consider the heterogeneity of the secondary ω and α phases (precipitation) in the β matrix. All β stabilizer species led to a significant energetic and elastic stabilization of the β phase. Mo and W additives stabilized the β phase relatively stronger and secondary ω and α precipitates may hardly nucleate in β phase in high concentrate condition. Although V, Nb, and Ta additives stabilized the β phase significantly, the β phase remained metastable. Regarding the morphology of these secondary phases, V helped α precipitation and Ta helped ω precipitation. The possibility of a coexistence of ω and α in the β matrix was suggested for Nb addition. The strain fields around the precipitates were also investigated and the results suggested that α precipitates cause a large residual strain around it though ω precipitates do not.

Effects of the orientation distribution of thin soft inclusions on the effective elastic moduli of microheterogeneous material

Many natural composite materials such as carbonate rocks contain systems of oriented or partially oriented thin inclusions (microcracks) filled with a soft elastic cement. In this paper we have studied the influence of the inclusion orientation, shape, and elastic properties on the effective elastic properties of micro-inhomogeneous materials. We have calculated the components of the compliance tensor of such materials as a function of crack density. The results were obtained for thin ellipsoidal inclusions with elastic compliance much larger than the elastic compliance of the matrix. To calculate the effective compliance tensor, we have used the “non-interaction approximation” (NIA). The application of the NIA allows us to evaluate the influence of peculiarities in the spatial distribution of inclusions on the effective properties of the medium. To simplify the calculations, we have used the special tensorial basis (T-basis). We have obtained the explicit expressions for the effective elastic compliance tensor of inhomogeneous materials. To verify the analytical expressions, we have compared our results for elastic moduli to the results obtained by numerical modeling of a material containing crack-like inclusions. The results obtained by using the two methods are in very good agreement, thus indicating that our analytic method can be applied even at sufficiently large values of crack density.

Fast Reconstruction of Microstructures with Ellipsoidal Inclusions Using Analytical Descriptors

Microstructure reconstruction is an important and emerging aspect of computational materials engineering and multiscale modeling and simulation. Despite extensive research and fast progress in the field, the application of descriptor-based reconstruction remains limited by computational resources. Common methods for increasing the computational feasibility of descriptor-based microstructure reconstruction lie in approximating the microstructure by simple geometrical shapes and by utilizing differentiable descriptors to enable gradient-based optimization. The present work combines these two ideas for structures composed of non-overlapping ellipsoidal inclusions such as magnetorheological elastomers. This requires to express the descriptors as a function of the microstructure parametrization. Deriving these relations leads to analytical solutions that further speed up the reconstruction procedure. Based on these descriptors, microstructure reconstruction is formulated as a multi-stage optimization procedure. The developed algorithm is validated by means of different numerical experiments and advantages and limitations are discussed in detail.

Micromechanical model for effective thermoelectric properties of thermoelectric composite containing ellipsoidal inclusions with interfacial resistances

The effective design of thermoelectric composites is of great significance in improving performance of thermoelectric composites. Herein, we develop a homogenization theory to predict the effective thermoelectric properties of thermoelectric composite containing ellipsoidal inclusions whereby interfacial thermal and electrical resistances are considered. Firstly, the closed-form representations of the effective thermoelectric properties are derived under the assumption of a small temperature difference. Then the effective properties of the thermoelectric composite subjecting to a large temperature difference are obtained by cutting the composite to a serial connection of pieces with a small temperature difference. The relationship between the interfacial resistances and the effective properties of thermoelectric composites is discussed systematically under the different conditions of the aspect ratio, volume fraction, size of inclusion and temperature difference in the composite. Our theoretical predictions well match the effective properties obtained from finite element analysis.

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

The article offers a general formulation for the multipolar field expansion of an inclusion. The multipolar moments of arbitrary order for the ellipsoidal inclusion are derived, showing known features of their elastic fields in both the far and near field. Using integer point enumeration theory, the same formulation is extended to all polytopal inclusions, which serve to model faceted inclusions found in, e.g. igneous and metamorphic rocks, particle reinforced composite materials, or as intermetallic precipitates in alloys. A general algorithm for the calculation of the multipolar moments of polytopes of any number of vertices is derived. A number of examples for tetrahedral, cuboidal and dodecahedral inclusions are given. It is shown that the parity of the axial moment function of a polytopal inclusion mirrors the symmetry of the inclusion. This represents the main properties of their elastic fields, including the long- and near-field decay. If the inclusion is symmetric with respect to a given axes, the fields will decay with 1 / r 2 in the far field and 1 / r 4 in the near field. If symmetry along that direction is lost, as is expected in most real inclusions, the decay rate progresses with 1 / r 2 but the near field with 1 / r 3 .

Two-sided estimate of effective thermal conductivity coefficients of a textured composite with anisotropic ellipsoidal inclusions

Two-sided estimate of effective thermal conductivity coefficients of a textured composite with anisotropic ellipsoidal inclusions

Effect of Cerium on Inclusion Modification in a Secondary-Hardening Steel.

Owing to the continuous increasing of steel strength, mechanical properties including toughness and fatigue performance are becoming increasingly sensitive to inclusions in ultra-high strength steel. Rare-earth treatment is considered as an effective method to reduce the harmful effects of inclusions, but is rarely applied in secondary-hardening steel. In the present study, different amounts of cerium were added in a secondary-hardening steel to investigate the modification effect of Ce on non-metallic inclusions in steel. The characteristics of inclusions were observed experimentally using SEM-EDS and the modification mechanism was analyzed based on thermodynamic calculations. The results indicated that the main inclusions in Ce-free steel are Mg-Al-O + MgS. Thermodynamic calculation indicated that MgAl2O4 is firstly formed in liquid steel and then successively transformed into MgO and MgS during cooling process. When the Ce content is 0.0030%, the typical inclusions in steel were individual Ce2O2S and MgO + Ce2O2S complex inclusions. When the Ce content was increased to 0.0071%, the typical inclusions in steel were individual Ce2O2S- and Mg-containing inclusions. Ce treatment modifies the angular magnesium aluminum spinel inclusions into spherical and ellipsoidal Ce-containing inclusions, thus reducing the harmful effect of inclusion on steel properties.

Nonmetallic inclusion characterizations and effects on impact toughness anisotropy of TWIP steel with different Ce additions

The effects of cerium (Ce) addition (0, 0.015% and 0.045%) on the microstructure, inclusions and impact toughness of Fe-18%Mn-0.6%C twinning-induced plasticity (TWIP) steel were investigated. The results show that the addition of Ce has no significant effect on the texture of TWIP steel. The grain size of TWIP steel decreases, and the volume fraction of twins increases slightly with Ce addition. Typical MnO-MgO-MnS complex inclusions and MnS were observed in TWIP steel without Ce. The inclusions gradually transform from the long strip MnS-Ce oxysulfide complex inclusions to ellipsoidal Ce oxysulfides with the increase in Ce addition from 0.015% to 0.045%. Notably, the transverse impact toughness of TWIP steel with 0.015% Ce addition is much lower than that of the other samples and exhibits serious anisotropy, and these differences are attributed to the stronger deformation ability of the complex inclusions with MnS and Ce oxysulfides compared with Ce oxysulfides. Fine grain size and ellipsoidal inclusions lead to excellent isotropic properties in TWIP steel with 0.045% Ce addition.

An analysis of composites with two-dimensional decagonal quasicrystal matrix and spheroidal inclusions

Eshelby tensors in elasticity are obtained for two-dimensional (2D) decagonal quasicrystal containing an ellipsoidal inclusion. Based on the intrinsic strain formula and Mori–Tanaka theory, the effective elastic properties of 2D decagonal composites are obtained. The effects of inclusion volume fraction and inclusion aspect ratio on elastic properties of composites with 2D decagonal quasicrystal are discussed.

Study on effective electroelastic properties of one-dimensional hexagonal piezoelectric quasicrystal containing randomly oriented inclusions

In this paper, the effective electroelastic properties of one-dimensional (1D) hexagonal piezoelectric quasicrystal containing randomly oriented inclusions are considered. The explicit expressions are obtained for the Eshelby tensors for 1D hexagonal piezoelectric quasicrystals containing rod-shaped and penny-shaped inclusions. The closed forms of the electroelastic constants are acquired for four special cases of random orientations of inclusions. Numerical results are given for the 1D hexagonal piezoelectric quasicrystal containing randomly oriented ellipsoidal inclusions. The results indicate that the effective electroelastic properties of 1D hexagonal piezoelectric quasicrystal composites are strongly affected by both the aspect ratio and the orientation of inclusions.

A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media

Eshelby's seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the three-dimensional isotropic medium, the weak version of the Eshelby conjecture has been substantiated. The previous work of Ammari et al. substantiates the strong version of the Eshelby conjecture for the cases when the three eigenvalues of the eigenstress are distinct or all the same, whereas the case where two of the eigenvalues of the eigenstress are identical and the other one is distinct remains a difficult problem. In this work, we study the latter case. To this end, firstly, we present and prove a necessary condition for a convex inclusion being capable of transforming a single uniform eigenstress into a uniform elastic stress field. Since the necessary condition is not enough to determine the shape of the inclusion, secondly, we introduce a constraint that is concerned with the material parameters, and prove that there exist combinations of the elastic tensors and uniform eigenstresses such that only an ellipsoid can have the Eshelby uniformity property for these combinations simultaneously. Finally, we provide a more specifically constrained proof of the conjecture by proving that for the uniform strain fields constrained to that induced by an ellipsoid from a set of specified uniform eigenstresses, the strong version of the Eshelby conjecture is true for a set of isotropic elastic tensors which are associated with the specified uniform eigenstresses. This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.