Vector Deformations

Following Landau and Lifshitz (1970), we first describe small deformations of an elastic body. Let us assume that r is the radius-vector of some point M inside the elastic body prior to a deformation. Upon deformation, the point M is displaced by some vector U (the displacement vector or the deformation vector), so that the radius vector r of the new position of the point M is related to r by a simple formula (see Figure 13-1) ... 

This equation can be written for scalar components of the stress tensor and deformation vector as follows (Udias, 1999)... 

In the description of LO, it is important to account for elastic deformations. It is natural to assume that if in a solid LO is realized with CP from P, then they differ from the basic polyhedron P° only through elastic deformations. The deformation of the polyhedron P is considered to be elastic if the atom binding energy changes monotonously in the process of deformation transfering P° to P . The deformation magnitude is described by a set of deformation vectors. Let bj, be the polyhedron basis of the i-th atom P (Q ) with the orientation and P°(i2,) the basic polyhedron of the same orientation. Then the set of vectors... 

The full-fledged introduction of local-scaling transformations into density functional theory took place in the works of Kryachko, Petkov and Stoitsov [25-27,29, 31], and of Kryachko and Ludena [28, 1, 21, 30, 32-34]. The fundamental idea was that of transforming a vector reft3 into another (deformed) vector f(r) e ft3, which, however, conserves the same direction as the original one. Because of this, the transformed vector can be written as /(r) = A(f)f. In view of the similarity... 

The quantity w is called a deformation vector (or a shift vector). I he specifying of this vector as a function of x, defines the body deformation completely. In the general ca.se, deformation is characterized by a deformation tensor with its components (in the case of small deformations)... 

In order to discuss the movement of the constitutive molecules of gels, it is necessary to describe gels as continuous bodies based on continuum mechanics. Let us consider the process for a point r on the gel network to move to point f as shown in Fig. 2. The vector defined by the following equation is called the deformation vector ... 

Solid line indicates gei network and black dots show crosslink points rand r indicate position vector before and after the swelling, respectively, and u Is the deformation vector. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Consider a body undergoing a smooth homogeneous admissible motion. In the closed time interval [fj, fj] with < fj, let the motion be such that the material particle velocity v(t) and deformation gradient /"(t), and hence (r), and p(r), have the same values at times tj and tj. Such a finite smooth closed cycle of homogeneous deformation will be denoted by tj). Consider an arbitrary region in the body of volume which has a smooth closed boundary of surface area with outward unit normal vector n. The work W done by the stress s on and by the body force A in during... 

The partial derivatives of x are the velocity vector y and the deformation gradient tensor f, respectively. 

Stability. All spectrographs are subject to structural deformation due to thermal effects or, in the case of instmments mounted directly on the telescope, variation in the gravity vector. This can be divided into two parts. 

Fig. 95.—Transformation of the distributions of the Xj 2/, and z components of chain displacement vectors by fourfold stretch along the x a,xis at constant volume (ax = 4 (Xy = oiz — l /2). Initial distribution ax —oiy = a = 1) shown by solid curve. Other curves represent X, y, and z components after deformation as indicated.

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