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Light Propagation in Anisotropic Media
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
6.1 An isotropic solid is a solid material in which physical properties do not depend on its orientation. Anisotropy refers to different properties in different directions, as opposed to isotropy, and many crystals are naturally anisotropic.
Determination of the effective elastic properties of titanium lattice structures
Published in Mechanics of Advanced Materials and Structures, 2020
Khalil Refai, Marco Montemurro, Charles Brugger, Nicolas Saintier
In Eq. (2), is the relative density of the homogeneous medium at the macroscopic scale. VEFF is the real volume of the lattice cell at the mesoscopic scale, whereas VRVE is the overall volume of the RVE. Es, νs, and ρs are the Young’s modulus, Poisson’s ratio, and density of the bulk material, whereas E, ν, and ρ are the effective (or equivalent) material properties characterizing the porous material at the upper scale wherein it is modeled as an equivalent homogeneous isotropic solid. The Ashby and Gibson model considers randomly distributed pores with a perfectly smooth surface. Moreover, the applicability of this model is restricted to the low-density region (i.e. up to a value of the relative density equal to 20%).
Effects of Hall current and nonlocality in a magneto-thermoelastic solid with fractional order heat transfer due to normal load
Published in Journal of Thermal Stresses, 2022
If then from Eqs. (32)–(39), the corresponding expressions for displacements, stresses, current density and conductive temperature for nonlocal isotropic solid without two temperature and fractional order heat transfer are obtained .If then from Eqs. (32)–(39), the corresponding expressions for displacements, stresses, current density and conductive temperature for nonlocal isotropic solid with two temperature are obtained .If = 0, then from Eqs. (32)–(39), the corresponding expressions for displacements, stresses, current density and conductive temperature for local isotropic solid with two temperature are obtained.If then from Eqs. (32)–(39), the corresponding expressions for displacements, stresses, current density and conductive temperature for a local isotropic solid are obtained.If then from Eqs. (32)–(39), the corresponding expressions for displacements, stresses, current density and conductive temperature for a local isotropic solid with two temperature are obtained.
Temperature and pressure dependence on structural, electronic and thermal properties of ZnO wurtzite phase – first principle investigation
Published in Phase Transitions, 2020
K. Klaa, S. Labidi, M. Bououdina, A. Amara
The calculations were performed using the full potential linearized augmented plane wave method within the framework of the density functional theory [19,20] as implemented in the Wien2k code [21]. The exchange–correlation potential was calculated using the generalized-gradient in the form proposed by Perdew et al. [22]. Moreover, the alternative form of GGA proposed by Engel and Vosko (EV-GGA) [23] and modified Becke–Johnson (mBJ) [24,25] were also used for band structure calculations. In the FP-LAPW method, the wave function and potential were expanded in spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and a plane wave basis set in the remaining space of the unit cell (interstitial region) was used. The muffin-tin radii RMT were assumed to be 1.89 and 1.67 a.u for Zn and O respectively. The plane wave cut off of Kmax = 8.0/RMT (RMT represents the smallest muffin-tin radius) was chosen for the expansion of the wave functions in the interstitial region, while the charge density was Fourier expanded up to Gmax = 12 (Ryd)1/2. The number of special k points in the irreducible Brillouin zone was 36 for the wurtzite-type structure. The calculations were based on the theoretical equilibrium lattice parameters. All these values were chosen in order to ensure the convergence of the results. Concerning the quasi harmonic Debye model, the dependences of volume, bulk modulus, variation of thermal expansion α, as well as the Debye temperature and heat capacity Cv were successfully obtained in the whole pressure range from 0 to 10 GPa and temperature range from 0 to 1000 K. Using a given set of total energy versus molar volume values, , and a numerical equation of state (EOS), the thermal properties were obtained. The quasi-harmonic Debye model was used to investigate the thermal properties of ZnO in the quasi harmonic Debye model [26], the non-equilibrium Gibbs function can be written as:where is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, θ(V) is the Debye temperature, and AVib is the vibrational term which can be written using the Debye model of the phonon density of states as [27,28]:where n is the number of atoms per formula unit, D(θ /T) represents the Debye integral, and for an isotropic solid, θ can be expressed as [27]:where M is the molecular mass per unit cell and BS is the adiabatic bulk modulus, approximated by the static compressibility [26]:f (σ) is given by Refs [29,30]; the Poisson σ is taken as 0.25 [31].