Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strain

An inverse problem of the elasticity of n elastic inclusions embedded into an elastic half-plane is analysed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the conditions of ideal contact hold in the interfaces between the inclusions and the half-plane. The shapes of the inclusions are not prescribed and have to be determined by enforcing uniform stresses inside the inclusions. The method of conformal mappings from a slit domain onto the ( n + 1 ) -connected physical domain is worked out. It is shown that to recover the map and the shapes of the inclusions, one needs to solve a vector Riemann–Hilbert problem on a genus- n hyperelliptic surface. In a particular case of loading, the vector problem reduces to two scalar Riemann–Hilbert problems on n + 1 slits on a hyperelliptic surface. In the elliptic case, in addition to three parameters of the model, the conformal map possesses a free geometric parameter. The results of numerical tests in the elliptic case show the impact of these parameters on the inclusion shape.

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

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.

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

Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure-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 finite element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusion. 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 e.g. 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/strain within 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 inclusion, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond with a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.

Numerical Studies of the Correlation between Inclusion Shape and Effective Elastic Properties of the Particle Reinforced Composites

In present paper the effect of inclusions with irregular shapes on the elastic material properties of two-phase composites is studied. The irregular shapes of the real inclusions were approximated using smooth three-dimensional structures. For this needs the images of the microscopic particles were numerically approximated through smooth structures using methods of the computer algebra and were used for the following FE studies. The reference elements with typical inclusions with irregular shapes were determined and used for calculation of the effective material properties.

Novel and efficient simulation approach for effective permeabilities of randomly ordered two-phase compounds

This paper introduces a novel simulation approach for the magnetic properties of two-phase randomly ordered compounds. In industry, materials such as ferrous powder mixtures or metallic granulates are very often used as raw materials. Hence, their material characteristics are of utmost interest for material manufacturers in order to guarantee high quality standards. Typically, many parameters such as composition, inclusion shape, and the characteristics of the constituents affect the macroscopic physical behavior of such materials. In particular, the resulting permeability of multi-phase and randomly ordered materials exhibits a strong variation despite constant compounds. For the design and optimization of measurement setups, efficient simulators are necessary to estimate the effective permeability and its fluctuation range of a huge number of arrangements. In addition to the basic concept of the novel simulation method, this article presents some possible evaluations of the simulated results and their dependencies on the properties of the constituents. In the last century, a large number of different mixing formulas have been established in literature, which are summarized and compared to the simulation results. Finally, the simulated magnetic characteristics are evaluated with finite element simulation of a comparable particle arrangement.