Kinematic description

In the linear description of the motion of solid bodies it is assumed that the displacements and their gradients are infinitely small and that the material is linearly elastic. In addition, it is also assumed that the nature of the boundary conditions remains unchanged during the entire deformation process. These assumptions imply that the displacement vector u is a linear function of the applied load vector F, i.e., if the applied load vector is a scalar multiple aF then the corresponding displacements are au. 

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Two classes of models can be identified. These are kinematic descriptions of the collision that leads to reaction, and potential energy surface representations of that interaction. Of course, these types overlap the former finds its full expression in the calculation of trajectories on potential surfaces7. 

The kinematical description of motion really began with Galileo. From observations Galileo introduced two concepts velocity as the time rate of change of position and acceleration as the time rate of change of velocity. With velocity, acceleration, time and distance traveled (change of position) the complete kinematical description of motion was possible. Four algebraic equations resulted, each involvii three variables and an initial position or velocity. 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

The Cauchy-Born strategy yields interesting insights in a number of contexts one of which is in the kinematic description of stmctural transformations. A... 

Fig. 6.25. Schematic illustrating both (a) the kinematic description of various configurations and (b) the types of excitations that are possible for a given configuration (adapted from Ceder (1993)).

We repeat the discussion of section 2.3.3 which culminated in eqn (2.46), but now clarified by the kinematic description of the previous section. Recall that the basic argument consisted of two parts. First, we construct a function (if the number of configurational coordinates is finite) or a functional (if the configurational coordinates are specified by a function) which is the sum of the time rate of change of the Gibbs free energy G( r, r, ) and a dissipative potential F( ri r ), of the form... 

As we have noted in the introduction, experimental evidence for a kinematic description of excitable wave fronts is rich. Based on hyperbolic wave equations and the Huygens principle, Wiener and Rosenblueth [81] recommend the eikonal approach of geometric optics waves propagate at a constant normal speed... 

The very first attempt to construct a simplified kinematical description of a rotating spiral wave has been done in the classical paper of N. Wiener and A. Rosenblueth [33]. This description is based on the assumption that wave fronts propagate in a uniform and isotropic medium with equal velocity from any stimulated points into a region where the medium is in the rest state. Due to Huygens principle, successive wave fronts are perpendicular to a system of rays which represent the position which may be assumed by stretched cords starting from the stimulated point. The back of the wave is another curve of the same form, which follows the wave front at a fixed distance Ag measured along these rays. 

A more elaborated kinematical description of a freely rotating spiral wave in a uniform medium is based on the assumption that the normal velocity c of a curved wave front is not a constant, but depends on its curvature [34]. The simplest approximation of this relationship is a linear... 

For each constituent, we can use the same kinematic description as for the single substance in Sect. 3.1. Namely, for each constituent a invertible and smooth motion... 

The assumptions made in Eq. (10), as shown in Eqs. (7)-(9), are rather sweeping. Sayre and Chang (82) noted that experimental evidence from a number of sources implies that lateral diffusion is much better represented by the Fickian model than is longitudinal dispersion [as represented by the onedimensional form of Eq. (10), with transport only in the x direction]. Most workers have found that despite its theoretical shortcomings, the Fickian model provides a reasonable starting point and an approximate kinematic description of diffusion in open channels. 

Substitution of Equations 4-17 and 4-18 into Equation 4-14 yields a 0 = 0 identity, showing that no additional assumptions have been invoked. First, let us determine how T varies, if at all, along a streamline. Equation 4-7, which provides the kinematic description for a streamline, still holds. Thus,... 

The equally spaced trains "B" and "D" are unstable since their dispersion relations satisfy c (T) < 0. The unequally spaced trains "AB", "BC", "AD", etc., which contain the unstable spacings "B" and "D", are also unstable. (Stability of these impulse trains has been analysed based upon the kinematic description [Maginu Rinzel, in... 

The approach to simulation model development for an automotive vehicle chassis system, described in this section, has the advantage that the model can be easily extended to include nonlinear tyre and suspension force characteristics without the need to modify the kinematic description of the sprung and unsprung masses. More importantly, the approach facilitates the development of vehicle models which combine a MBS description of the chassis with other vehicle subsystems which are not amenable to MBS modelling techniques, such as the powertrain or a digital control system, but which can be easily represented in a simulation language such as ACSL. 

In order to be consistent with the level of the kinematic description of the mixture, the gradients v and are not present in this first order theory. 

Generally, the width of the excitation zone is of the same order of magnitude as the width of the recovery tail. Therefore, when a spiral ceases to be sparse with an increase in the excitability of the medium its pitch is no longer large as compared to width of the propagating excitation pulse. It means also that the width of the wave s tip becomes comparable with the radius of the core of the spiral wave. Obviously, in this situation the kinematical description, developed above for the weakly excitable media and picturing the entire wave as a single curve with a free end, is not applicable. 

There is, however, a special class of systems where, after certain modifications, the kinematical description could be still applied to describe interactions between the propagating waves. These are the systems where a propagating pulse is narrow but its recovery tail is long. In terms of the reaction-diffusion equations it means that the motion of the portraying point along the recovery branch of the null-cline takes much more time than the respective motion along the excitation branch of the same null-cline. This condition can be real-... 

The effect of a gradual recovery can be incorporated into the kinematical description by assuming that the velocity of the normal propagation of a flat curve is a certain function Vq T) of the time T that has elapsed after the previous curve had passed the same point of the medium. The actual form of this function depends on a particular reaction-diffusion system which is described by the kinematical model. It must approach the propagation velocity Vb of a solitary excitation pulse and should become smaller for shorter time intervals T between the waves. 

The radius R of the minimal hole that is still able to maintain a pinned spiral wave and the rotation period T of this pinned wave are important properties of an excitable medium. Figure 12 shows (curve 3) the rotation period T as a function of e, computed in [27] using the kinematical description. We see that it fits well the data (black dots) of the numerical simulation of the respective reaction-diffusion model. 

The kinematical description of the motion of curves over the curved surfaces has been constructed in [24] (see also [28]). The function k l, t), which gives the dependence of the local geodetic curvature k of the curve on the arclength I measured from its free end, obeys in this case the equation... 

For regions of intense seismic activity, the crustal stmcture is frequently defined in terms of one-dimensional velocity models. Detailed three-dimensional crustal models have also become available for specific regions in the benefit of three-dimensional wave propagation codes that may effectively take into account basin effects and complex fault geometries at the cost of increased computational demands. The characterization of the seismic source is a more complicated issue. For kinematic descriptions of the earthquake source, source parameters such as slip, rise time, rupture velocity, and slip function should properly be quantified and a priori defined. On the other hand, for dynamic descriptions of the earthquake source, the source parameters may vary as long as the elastodynamics equation with a prescribed fracture criterion on a predetermined fault plane is satisfied. The selected initial conditions and failure criterion determine the time and space evolution of the fault rupture in a dynamic source model. 

This relation describes the propagation of a finite discontinuity, or shock, separating two equilibrium states. The idealization implicit in its formulation is of particles in the bed making an instantaneous switch from one equilibrium condition to another as the shockwave passes over them. Only conservation of mass is involved in the analysis leading to eqn (5.8) inertial effects, which control the necessary deceleration of the moving particles to zero velocity, are not taken into account. The above analysis is thus in terms of a kinematic description of the fluidized state, and eqn (5.8) represents the velocity of a kinematic shock Mrs mrs = dLijdt. 

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