Kinematics of Continua

we review some basic concepts from deformation theory although they are not needed in most applications for fluids they are necessary to develop and understand 

We study the body (composed of a single substance) and its parts which we perceive through their configurations (connected region in three-dimensional Euclidean space), which this body (or its parts) occupies or may occupy in the space. 

Pekar and I. Samohyl, The Thermodynamics of Linear Fluids and Fluid Mixtures, 

We select one arbitrary configuration as a reference configuration (we denote it simply as a reference ) and radius vectors of points in such reference, X, we call the particles of the body (identified with their positions).  

The motion of the body is described mathematically as the time succession of the actual configurations in real space. Namely, vector x (relative to frame discussed in Sect. 3.2) gives the position of particle, X, in the instant t through the vector function X called the motion or deformation function as 

---

At this point, we have seen that 100 years was required for the concepts of kinematics, stress, and constitutive relations to be developed. It was Euler who provided a precise statement of the non-relativistic laws of mechanics, introduced the cut principle and the concept of stress, and developed important results concerning the kinematics of continua. This was done primarily in the period between 1750 and 1766. In 1822, Cauchy placed the concept of stress on a modem basis and in... 

---