Kinematics and Stress

In order to describe more general flow gradients, such as those generated near the stagnation point of a four-roll mill, some mathematics must be introduced in particular, tensors (which can be represented as matrices) are needed, namely the velocity gradient tensor. 

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Let us discuss now in more detail the properties of polymers detected under homogeneous extension. We shall consider only two conditions x = const and F = const (as the two most characteristic examples of flow with preset kinematics and stress). 

A) Rheological representation of the augmented HM. (B) Deformation map showing the kinematics and stress tensors used in the augmented HM. These figures illustrate how the model represents the viscoplastic flow, and how the deformation state is generalized into three dimensions. 

For each of these models, a linearized perturbation analysis is performed by linearizing the flow and electrical field equations of the system. This process is modeled using stability analyzes applying the kinematic and stress conditions at the interface. Stability analysis is based upon... 

We could summarize Eqs. 1-1 through 1-3 by sa3dng that they introduced the concepts of kinematics and stress. More than half a century would elapse before the concept of stress would be presented in a modem framework by Cauchy, and it would require a slightly longer period of time before a constitutive equation would be developed leading to the Navler-Stokes equations. In the century between Euler and Stokes, the basic ideas associated with kinematics, stress and constitutive relations were formulated. Two centuries later, these same concepts represent the building blocks of fluid mechanics. Before we comment on the development of these concepts, we need to examine how Eqs. 1-1 through 1-3 compare with Newton s three laws of mechanics. 

The four terms on the right hand side of this result can be identified as the shaft work, the fiow work, the reversible work and the irreversible work (viscous dissipation). The shaft work Is a very Important quantity In many practical problems and it can only be Included in the analysis by means of a moving control volume. In addition, the use of the kinematical relation given by Eq, 2-18 has allowed us to represent the rate of work done by the body force b in terms of an accumulation and flux of potential energy. It should be clear that the concepts of kinematics and stress are essential elements of the derivation that led originally from Eq. 1-2 to the macroscopic mechanical energy balance given by Eq. 2-23. 

I was never able to comprehend the development of THE FLOW EQUATION in Unit Operations, since the analysis of the entropy and the internal energy required the use of the mechanical energy equation (Whitaker, Sec. 10.1, 1968). Under these circumstances, one must use in the proof precisely that result which one hopes to prove, thus the development represents a persuasion, rather than a proof. Be that as it may, variations of this route are in use today [Welty, Wicks and Wilson, page 212, 1984] as a convenient detour around the concepts of kinematics and stress. 

While kinematics and stress were being avoided in the teaching of fluid mechanics, they were being Joined elsewhere, and the search for turbulent constitutive equations (Page, et al, 1952) and nonlinear fluid constitutive equations (Hedstrom, 1952) was on the rise. These two areas came together in the work of Metzner and Reed (1955), and one year later Prof. Tom Hanratty (1956) initiated his studies of turbulent... 

Here Q represents the rate at wMch heat is transferred to the fluid in the control volume, and W represents the rate at which work is done on the fluid in the control volume by solid moving surfaces, i.e., the shaft work. If life were restricted to steady process with one entrance and one exit, and it is not. one could ask what we have ined by coming to grips with the complex concepts of kinematics and stress en route to Eq. 4-7. That we have understood the concepts of stress and kinematics is really not very important. What is important is that we have by-passed the tooth fairy. 

In this chapter, we introduce some basic concepts of the kinematics and stresses of a deformable body from the point of view of continuum mechanics, and discuss various representations of a deformation process in terms of the deformation (or strain) tensor and the rate-of-deformation (or rate-of-strain) tensor. In order to help the readers follow the material in the text, the elementary properties of second-order tensors are presented in Appendix 2A. 

KINEMATICS AND STRESSES OF DEFORMABLE BODIES 23 In terms of the relative deformation gradient tensor t ), Eq. (2.29) may be written as... 

KINEMATICS AND STRESSES OF DEFORMABLE BODIES 43 and for contravariant components we have... 

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