External displacement tensor

This, Eq. (5.21), is the external displacement tensor of benzene adsorbed on zeolite, as measured by diffraction. It is about twice that measured by INS, Eq. (5.18), which is a difference too great to be explained by simple experimental errors. 

The are the mean square displacements of the scattering atom, still labelled /, caused by the external vibrations of individual molecules and we have used j to label the 6Amoi different phonons. Again we can define a total displacement tensor for the atom but now related to the external vibrations. 

Following a number of manual iterations in the program to best match the intensities the external modes displacement tensor y ext is extracted, see Fig. 5.6. 

The external displacements of benzene are much greater in the zeolite than in the crystal, hence the more intense external mode region. We may compare this value with that extracted from diffraction data taken on the same system [19]. The diffraction data were analysed under the constraints of the rigid molecule approximation, the reported anisotropic displacement parameter tensor must, therefore, include both internal and external modes... 

A, 5.18 mean square displacement tensor summed over all modes, internal and external A ... 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

Both APTs and AATs depend on the wave function derivatives with respect to nuclear displacements the computation of these terms for a molecule in solution has been already treated above. AATs depend also on the derivative of the wave function with respect to an external magnetic field. This derivatives is already known as it enters in the general definition of the nuclear shielding tensors discussed in the previous section. Once again, for its evaluation its possible to exploit the GIAO method. 

Corresponding system plastic-strain increments are also obtained at the atomic level from the displacement gradients between the four relevant neighboring corner atoms of Delaunay tetrahedra for each external distortion increment and are allocated subsequently as an atomic site average to each Voronoi polyhedral atom environment by a special procedure of double space tessellation developed by Mott et al. (1992) for this purpose, leading eventually to volume averages of strain-increment tensors of all Voronoi atom environments to attain the system-wide strain-inerement tensor. 

Consider a material s body in Figure 4.1 before deformation. After internal or external loading (force, thermal, electromagnetic field, gravity, etc.), the body is deformed, and may also have rigid body displacement, into the deformed spatial body shown in Figure 4.1. A material s point is defined in the Cartesian coordinate system (ei, 62, 63) by a position vector X in the undeformed body and position vector x in the deformed body u is the displacement vector. Throughout this chapter, bold face letters will represent tensors. Obviously,... 

In contrast, the assumption of affine deformation is difficult to remove. The affine network theory assumes that each subchain deforms in proportion to the macroscopic deformation tensor. However, because the external force neither directly works on the chain nor on the cross-links it bridges, the assumption lacks physical justification. In fact, the junctions change their positions by thermal motion around the average position. It is natural to assume that the nature of such thermal fluctuations remains unchanged while the average position is displaced under the effect of strain. 

The distinct feature of elastic-plastic finite element computations is the presence of two iteration levels. In a standard displacement based finite element implementation, constitutive driver at each integration (Gauss) point iterates in stress and internal variable space, computes the updated stress state, constitutive stiffness tensor and delivers them to the finite element functions. Finite element functions then use the updated stresses and stiffness tensors to integrate new (internal) nodal forces and element stiffness matrix. Then, on global level, nonlinear equations are iterated on until equilibrium between internal and external forces is satisfied within some tolerance. 

This formulation of the principle of virtual work is the principle of virtual displacements, which appears in the hterature sometimes under the name of the preceding. Naturally, the virtual strain energy 6U exists only for mechanical systems with deformable parts. As the contained virtual strain tensor is assembled from derivatives of the virtual displacements, these have to be continuously differentiable. The virtual work of external impressed loads ymd includes the limiting cases of line or point loads. External reactive loads do not contribute when the virtual displacements are required to vanish at the points of action of these loads, and thus the virtual displacements have to comply with the actual geometric or displacement boimdary conditions of Eq. (3.16). With these presumptions, the initial axiom of Remark 3.1 may now be reformulated for the virtual displacements. 

The other formulation of the principle of virtual work for mechanical systems requires the introduction of virtual loads instead of virtual displacements. Therefore, only those variations of external loads and stress tensor are considered admissible that are compatible with the equations of equilibrium inside the mechanical system and on the boimdary. The interior equilibrium of Eq. (3.14) for the virtual loading leads to the following form ... 

Here Sp, is the strain tensor Sjk = d s,/d Xk), where s, is the displacement of a volume element with respect the equilibrium position. is called spontaneous strain, and E, is the external electric field. 

We begin with the definitions of the strain and stress tensors in a solid. The reference configuration is usually taken to be the equilibrium structure of the solid, on which there are no external forces. We define strain as the amount by which a small element of the solid is distorted with respect to the reference configuration. The arbitrary point in the solid at (x, y, z) cartesian coordinates moves to x + u, y + v, z + w) when the solid is strained, as illustrated in Fig. E.l. Each of the displacement fields u,v,w is a function of the position in the solid u x, y, z), v(x, y, z), w(x, y, z). 

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