Internal strain parameter

From only two calculations of force and stress one can determine the optic mode frequency (the restoring force fo the internal displacements), the internal strain parameter f, and the elastic constant. This may be compared with a very arduous task of fitting multiple parameter curves if one has only the energy, as was done by Harmon, et al. ... 

We have also plotted in Fig. 5.5.2c the convergence of the Internal strain parameter X, which, in terms of transverse [100] force constants, is given by eq. (5.5.3), and which can be equivalently expressed in terms of the longitudinal [111] force constants as... 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

Both sohd-solution hardening and precipitation hardening can be accounted for by internal strains generated by inserting either solute atoms or particles in an elastic matrix (11). The degree of elastic misfit, 5, produced by the difference, Ai , between the lattice parameter, of the pure matrix and a, the lattice parameter of the solute atom is given by... 

The rate of enolate-carbonyl equilibration " is dependent on the forward and backward rates of proton exchange. Proton exchange from a carbon-based acid is known to be slower than that of a more electronegative atom donor (in particular, O and N atoms) . For a series of closely related molecules usually the more acidic a given molecule is, the faster the rate of proton transfer (high kreu note that thermodynamic and kinetic parameters are not related). For example, benzocyclobutanone (10) is less acidic and the rate of deprotonation is substantially slower (10 times) than the related benzocyclopentanone (12) due to its enolate (11) having unfavourable anti-aromatic character. Deprotonation of the simplest cyclobutanone (13) clearly does not lead to an unfavourable anti-aromatic enolate (14) . By assuming the internal strain of 14 is similar to that of 11, cyclobutanone (13) is evidently 10 " times more acidic than benzocyclopentanone (12). By the same vain, the more acidic propanone (15) has a faster rate of deprotonation (10 times) than the less acidic ethyl acetate (16) . ... 

In presence of significant Me-S lattice misfit, the epitaxy of isolated 3D Me crystallites or compact 3D Me films is strongly determined by the structure of internally strained 2D Meads overlayer and/or 2D Me-S surface alloy formed in the UPD range at high F or low AEi. The misfit between the lattice parameters of the 2D Meads phase and/or 2D Me-S surface alloy phase and the 3D Me bulk phase is mainly removed by misfit dislocations. The initial strain disappears after depositing a certain thickness of the 3D Me bulk phase. Usually, a thickness of n Me monolayers where 2 < < 20 is necessary to adjust the 3D Me bulk lattice parameters [4.58, 4.59]. If an incommensurate structure of a 2D Meads overlayer is formed in the UPD range, this structure will also be reflected epitaxially in 3D Me crystallites and ultrathin 3D Me films. 

Thermocyclic cracks were revealed in pipes connecting a reactor with pressurizers, in internal headers for water supply from purification system to reactor coolant pumps, in internal shells of pumps flow chambers, etc. The indicated defects became the reason for in-depth analytical and experimental studies of equipment operating conditions in the propulsion and test reactor plants, particularly thermal and strain parameters monitoring. 

LDA band structure. To evaluate the internal strains by minimization of the energy, it was necessary to use for the unstrained crystal the calculated internal positions. Although these predictions are quite close to the measured values, the small differences do result in one noticeable difference in the band structure the band that just crosses the Fermi level from below and gives rise to the stick Fermi surface does not quite reach the Fermi level when the theoretical internal parameters are used for atomic positions. The apical oxygen 04 position is the critical one. However, it is only the trends (finite differences) in band structure that we will report, and these changes will be insensitive to the reference point. 

In order to simplify the exposition, we consider that the process is isothermic, and we assume that the evolution of the internal structure can be described with the aid of a scalar-hardening parameter (the density of dislocations or the equivalent plastic strain) and of an internal tensorial parameter (back stress). These restrictions are eliminated in the authors... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

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. ... 

In the context of elastic deformation two parameters, known as stress and strain respectively, are very relevant. Stress is an internal distributed force which is the resultant of all the interatomic forces that come into play during deformation. In the case of the solid bar loaded axially in tension, let the cross sectional area normal to the axial direction be A0. From a macroscopic point of view the stress may be considered to be uniformly distributed on any plane normal to the axis and to be given by o A0 where o is known as the normal stress. The stress has to balance the applied load, F, and one must, therefore have o Aq = F or o = F/Aq. The units of stress are those of force per unit area, i.e., newtons per square... 

Like many other chemical concepts the concept of strain is only semi-quantitative and lacks precise definition. Molecules are considered strained if they contain internal coordinates (interatomic distances (bond lengths, distances between non-bonded atoms), bond angles, torsion angles) which deviate from values regarded as normal and strain-free . For instance, the normal bond angle at the tetra-coordinated carbon atom is close to the tetrahedral value of 109.47°. In the course of force field calculations these normal values are defined more satisfactorily, though in a somewhat different way, as force field parameters. 

---