Elastic constants calculation

Elastic constants calculated and the experimental lattice constant. Experimental data from Ref. [36]. 

For the composite of a random dispersion of non-overlapping identical spheres, the average overall elastic constants calculated with only 8 spheres in the unit cell were already practically indistinguishable from those calculated with 64 spheres (see Fig. 7.2). Moreover, the scatter of the individual... 

To date, results have been obtained for minimum-energy type simulations of elastic deformations of a nearest-neighbor face-centered cubic (fee) crystal of argon [20] with different inclusion shapes (cubic, orthorhombic, spherical, and biaxially ellipsoidal). On bisphenol-A-polycarbonate, elastic constant calculations were also performed [20] as finite deformation simulations to plastic unit events (see [21]). The first molecular dynamics results on a nearest-neighbor fee crystal of argon have also become available [42]. The consistency of the method with thermodynamics and statistical mechanics has been tested to a satisfactory extent [20] e.g., the calculations with different inclusion shapes all yield identical results the results are independent of the method employed to calculate the elastic properties of the system and its constituents (constant-strain and constant-stress simulations give practically identical values). 

Day, G. M., Price, S. L., and Leslie, M. 2001. Elastic constant calculations for molecular organic crystals. Cryst. Growth Des. 1 13. 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Allen M P, Warren M A, Wilson M R, Sauron A and Wiliam S 1996 Molecular dynamics calculation of elastic constants in Gay-Berne nematic liquid crystals J. Chem. Phys. 105 2850-8... 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

A plastic component was subjected to a series of step changes in stress as follows. An initial constant stress of 10 MN/m was applied for 1000 seconds at which time the stress level was increased to a constant level of 20 MN/m. After a further 1000 seconds the stress level was decreased to 5 MN/m which was maintained for 1000 seconds before the stress was increased to 25 MN/m for 1000 seconds after which the stress was completely removed. If the material may be represented by a Maxwell model in which the elastic constant = 1 GN/m and the viscous constant rj = 4000 GNs/m, calculate the strain 4500 seconds after the first stress was applied. 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

Table 2 Lattice parameter ao (in A) and elastic constants B, Cn, C12, and C44 for CoSi2 (in GPa), calculated using all-electron (FLAPW) and pseudopotential (VASP) techniques in the LDA and using GGC corrections.

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

Table 2 Elastic constants and bulk moduli for 4d cubic elements. Comparison is made between the results of our tight-binding parametrization (TB), first-principles full potential LAP., results (LAPW), where available, and experiment (Exp.). Calculations were performed at the experimental volume.

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

Pd4oCu4oP2o, Pd5oCu3oP2o, and Pd6oCu2oP20 alloys were measured by resonant ultrasound spectroscopy (RUS). In this technique, the spectrum of mechanical resonances for a parallelepiped sample is measured and compared with a theoretical spectrum calculated for a given set of elastic constants. The true set of elastic constants is calculated by a recursive regression method that matches the two spectra [28,29]. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

Table I. Experimental and calculated lattice constants a (in A), elastic constants, bulk and shear moduli (in units of 10 ) for the M3X (X = Mn, Al, Ga, Ge, Si) intermetallic series. Also listed are values of the anisotropy factor A and Poisson s ratio V. The experimental data for a are from Ref. . The experimental data for B, the elastic constants, A and v are taken from Ref. . The theoretical values for NiaSi are from Ref.. Also listed in the table are values of the polycrystalline elastic quantities-shear moduli G, Yoimg moduli (in units of and the ratio The experimental data for these quantities are from Ref. ...

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

In simple single-site liquid crystal models, such as hard-ellipsoids or the Gay-Berne potential, a number of elegant techniques have been devised to calculate key bulk properties which are useful for display applications. These include elastic constants for nematic systems [87, 88]. However, these techniques are dependent on large systems and long runs, and (at the present time) limitations in computer time prevent the extension of these methods to fully atomistic models. 

Fig. 7.2. Numerical estimates of the overall average elastic constants. The error-bars were calculated as cr-N°5, where N is the number of individual estimates for each of the two overall elastic constants obtained with a given number of spheres in the unit cell and a is the standard deviation. The two horizontal dashed lines drawn through the averages obtained with 64 spheres are meant to facilitate the convergence analysis...

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