Elastic potential functions

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

The value of f th depends on the chosen potential function. With AU = 3 kcal, we obtain 3th 2,5 10-s. In reality the value is about 0th 6 10-5 to 8 10 s. This means that the anharmonicity of the thermal vibrations is even greater21). For the potential Eq. (6) (Fig. 4) we need only to calculate the elastic deformation by thermal vibrations in the chosen approximation for one half wave, if B is situated in the neighborhood of A. This leads to... 

The caoutchouc elastic processes are characterized by many possible molecular conformations separated by very-low energy barriers At/. The existence of cross-links that is understructible bonds between the molecules, causes the termination of the flow processes after orientation. The network can be illustrated by potential functions like those in Fig. 28. The relaxation processes do not play any decisive role in ideal caoutchouc. The low activation energy leads to very short relaxation times. The distribution of the possible molecular conformations will be changed... 

LJl) and van der Waals (LJ2) potentials were used for nonpolar bead-bead and bead-wall interactions, respectively. For polar interactions, exponential potential functions (EXP 1,2) were added to both bead-bead and bead-wall cases. For the bonding potential between adjacent beads in the chain, a finitely extensible nonlinear elastic (FENE) model was used. For example, PFPE Zdol... 

The second, and more ambitious, objective is to find at least the general form of the potential function for a particular reaction, by successively adjusting V, until the observed and simulated results coincide. In essence, this is the same approach as that which is adopted to interpret elastic scattering measurements in terms of the central potential acting between two atoms (Section I.A.l), but the problem is now considerably more complex. 

For the description of the elastic forces within molecular crystals and between atoms of the same molecule which are not directly connected by a chemical bond but which are approaching each other (as in biomolecules), intermolecular force constants are necessary (see Secs. 5.2). These may be calculated using atom-atom potential functions. Kitaigorodsky (1973) introduced a universal potential of the Buckingham type (in kcal /mol) ... 

Equation 11.13 is based on a potential function model that considers the effect of a deformation on the nonbonded interchain interactions in a glassy amorphous polymer while equations 11.10, 11.11 and 11.12 are completely empirical. Although alternative methods for predicting the elastic properties of glassy amorphous polymers will also be discussed for historical reasons, these equations by Seitz constitute our preferred method. 

Fig. 12.5. The disjoining pressure as a function of the interparticle distance of separation for spheres sterically stabilized by poly(oxyethylene) curve 1, the experimental results of Homola and Robertson (1976) curve 2, constant segment density model. The crosses (x) show the theoretical results for a softened elastic potential (after Evans and Napper, 1977).

Accuracy The potential function must be able to accurately reproduce quantities such as energies, bond lengths, elastic constants, and related properties that enter a fitting database. 

The extent of crosslink decrease is determined by the network knots concerrtration. Such knots usually have a functionality of 3 or 4. Frmctiorrality depends on the type of curing agent. Crosslinked polyurethanes cured by polyols with three OH-groups are the examples of a three-functional network. Rubbers cured via double bonds are the examples of four-funetional networks. Eq. [6.4] is also used in other forms, depending on form of elasticity potential and concentration dependence of the %j parameter used. 

It is the first one that will be emphasized, and can be broken into conservation of mass and energy, which are coupled with Einstein s mass-energy equivalence (E=mc ). As such, the accumulation terms of the conservation of mass are not affected. Also, we could neglect forced convection effects in the system. The resulting mass diffusion equation would be similar to that in Eq. (1.5.2), except that a so-called elastic strain energy could be added to the potential function to take into account crystal lattice differences between solid phases (De Fontaine, 1967). 

Lifson S, Warshel A (1968) Consistent force field for calculations of conformations, vibrational Spectra and enthalpies of cycloalkane and n-alkane molectrles. J Chem Phys 49 5116-5129 Lubin MI, Bylaska EJ, Weare JH (2000) Ab initio molecttlar dynamics simulations of aluminum ion solvation in water clusters. Chem Phys Lett 322 447-453 Matsui M (1988) Molecular dynamics study of MgSiOs perovskite. Phys Chem Miner 16 234-238 Matsui M, Busing WR (1984) Computational modehng of the structrrre and elastic constants of the olivine and spinel forms of Mg2Si04. Phys Chem Miner 11 55-59 Matsrri M, Materrmoto T (1982) An interatomic potential-function model for Mg, Ca and CaMg olivines. 

The (two-dimensional) model for a relatively stiff molecule subjected to a simple shear flow, on the one hand, shows many features observed in NEMD simulations of finitely extendible nonlinear elastic (FENE) chain molecules. On the other hand, the dynamics found for the simple model is intriguingly complex and it deserved a careful study on its own. It seems appropriate also to analyse the system at higher temperatures. Furthermore, the model provides a convenient test bed for various thermostats other and additional thermostats, e.g. based on deterministic scattering [22] should be tested. Obvious extensions of the present model may involve other potential functions of nonlinear elastic type such as = (1/2) -I- (1/4) or = (1/4) (1 — r ) as well as... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

Torsional Experiments. The geometry and equations for torsion of an elastic cylinder are presented above. For the viscoelastic K-BKZ material, the equations look similar. For isochronal values of the strain potential function, one can define what looks like a time-dependent strain-energy ftinction Wi(Ii, I2, t) ... 

Issues of Material Compressibility. There is a full theory of compressible and nonlinear viscoelastic materials that would be equivalent to the compressible finite deformation elasticity theory described above (eq. 39), but more complicated because of the need to develop an expansion of the time-dependent strain potential function as a series of multiple integrals (108,109). One such formahsm is discussed briefiy under Lustig, Shay and Caruthers Model. Here a simphfied model that is based upon the K-BKZ framework with a VL-like kernel function (98) is examined. 

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