The Strained Body

To characterize the position and motion of a material point we will content ourselves with the use of Cartesian coordinates. This frees us from the various, mainly mathematical difficulties, involved in the use of more general coordinate systems. For cases where non-Cartesian coordinates are imavoidable the reader is referred to more comprehensive treatments or textbooks on continnttm mechanics (Becker and Burger 1975 Eringen 1967 Haupt 2000 Truesdell and Noll 1965 Truesdell and Toupin 1960). 

Our main concern will be the behavioirr of an elastic body under the influence of a certain system of forces. The application of these forces, in general, leads to a motion of the material points. Our immediate task is, therefore, to look into this motion. As will be shown later, in the general case such a motion consists of a rigid body motion (i.e. translation and rotation) and a deformatiom 

At some later time t the material point X will occupy a different position in space due to the motion of the body. This position will be designated by the position vector X in the given coordinate system. The configuration of the body at time t will then be described by a continuous, single valued assigmnent of a position x to each material point X. This leads us to a fimctional relationship 

In the actual configuration the body occupies a spatial region with volume v (Fig. 3.1). Geometrically speaking, the transformation Eq. (3.1) represents a mapping of each point X in F to a point x in v. 

As a consequence of the fact that Eq. (3.1) represents a continuous fimctional relationship we note that adjacent material points will always occupy adjacent points in space. That means that fracture of the material is excluded. The one to one relationship also means that any material point may not simultaneously occupy different positions and that at any position there may not be several material points at the same time. Therefore, there exists the inverse relation to Eq. (3.1) 

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