Displacement functions

Fig. 35. The atomic displacement functions from the Stockmayer simulation of CO. 

The occurrence of strong permanent fields may disturb the linear response regime in the dielectric response we have so far employed. The standard treatment of nonlinear dielectric response is based on the expansion of the dielectric displacement function D in powers of the electric field E, generally interrupted at the first correction ... 

It is evident that, for the identification of the Uj coefficients, we have to determine the minimum of the quadratic displacement function between the measured and computed values of the dependent variable ... 

One now needs to extend the concept of a derivative from ordinary functions to functionals. This is done by writing, with s z) being a displacement function. 

In codeformational equations, the basic kinematic quantities are the displacement functions. This generally means using the respective Cauchy and Finger tensors deformation gradient tensor X /9 Xfi ] by the following equations... 

The potential at a point r in a perfect crystal is given by K(r). If the crystal is now deformed by the presence of a defect, then the potential at point r will be different from V(t) because the potential at a point depends on the positions of the atoms in the neighborhood. If we assume that the deformation is not too severe (i.e., the deformation is a slowly varying function of position), then the potential at point r in the deformed crystal will be equivalent to the potential at point (r—R) in the undeformed crystal. R, in general, is a function of position and is called the displacement function. Thus, in a deformed crystal, the potential at a point r is given by... 

Figure Density and inverse particle displacement function, in reduced units, as...

A reference to Figs 2 and 3 shows that qualitatively the density and entropy displacement functions are very similar, with the latter providing a somewhat more resolved picture of entropy/information changes in the valence shell. These plots demonstrate that both functions can be used to probe changes in the electronic structure due to bond formation in molecules, reflecting the promotion (polarization) of bonded atoms in the molecular valence state, as a result of the electron excitation and orbital hybridization, and the inter-atomic electron CT effects. 

Figure 5 reports a comparison between the contour maps of the density difference, Ap(r), the Kullback-Leibler integrand, Ah(r), and the entropy displacement function [equation (94)], A (r), for the planes of sections shown in Fig. 4. The corresponding central bond profiles of the density and entropy difference functions are compared in Fig. 6. The optimized geometries of propellanes have been determined from the UHF calculations (GAMESS program) using the 3-21 G basis set. The contour maps have been obtained from the DFT calculations (deMon program) in DZVP basis set.

Where, the is the displacement function is the Lame-constant, (S,y is the Kronecker function, G is the shear modulus and the is the free term for the stress. 

Finite element formulation involves subdivision of the body to be modeled into small discrete elements (called finite elements). The system of equations represented from 4.48 to 4.59 are solved for at the nodes of these elements and the values of mechanical displacements u and forces F as well as the electrical potential d> and charge Q. The values of these mechanical and electrical quantities at an arbitrary position on the element are given by a linear combination of polynomial interpolation functions N(x, y, z) and the nodal point values of these quantities as coefficients. For an element with n nodes (nodal coordinates (x y z) f = 1, 2,..., n) the continuous displacement function m(x, y, z) (vector of order three), for example, can be evaluated from its discrete nodal point vectors as follows (the quantities with the sign are the nodal point values of one element) ... 

For a step shear deformation applied at t = 0, the displacement functions are... 

In maintaining a steady-state shear flow, the displacement functions are given by... 

We can label a particle or a point in a material system by giving its position X (with the Cartesian components Xi, X, Xs) at the present time t and denoting it as (X, t). The position of the particle (X,t) at a past time t is called X. The motion of the material is then given by the displacement function ... 

Figure 8.1 Cleavage of atomic planes and the related stress-displacement function.

The displacement functions for both PAK species show greater amplitudes in the directions of the surrounding chains (a and b axes) than along the chain axis direction c, although as would be expected from the looseness of the... 

Multiple Acid-base Equilibria in Protein-based Polymers with Widely Sequence Displaced Functional Groups... 

The displacement function must be a symmetric function in order to preserve the overall antisymmetry of a, but there remains a great deal of flexibility in the form of For homogeneous systems, it is common for the displacement of electron i to be determined by the sum of pairwise displacements in the directions of the other electrons, ... 

The rectangular elements, which have four corner nodes and five degrees of freedom at each node, assume both a stress field inside the elements and linear displacement functions around their boundaries. The element stiffness matrices are derived from a complementary energy functional. Figure 1 shows a schematic representation of the finite element mesh used in the analysis of the strut. The magnetic loading is distributed along the outer nodes of the aluminum bulkhead. 

If the displaced function is multiplied by x, the following integral is obtained ... 

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