ABSTRACT
This chapter deals with the notions of stress-strain and strain-displacements relation, which are quite essential for understanding the remaining developments in the book. It marks the change of philosophy between matrix structural analysis and finite element analysis of a continuum. In the previous Chapters 2 through 4, we only considered structural elements whose behavior can be formulated as a function of a single variable x, which is the longitudinal direction of the element. This is of course possible because of the geometry, where two dimensions are insignificant compared to the third one. The only stress of interest therefore is the longitudinal stress σx along the dominant dimension. Yet, in a three-dimensional solid where all the dimensions are of the same size, this assumption is not valid anymore. When a three-dimensional solid is subjected to external forces and/or displacements, and at the same time is restrained against rigid body movement, internal forces are induced, and these result in more than one stress at a point. Additionally, these external forces result in material points within the body being displaced. When there is a change in distance between two points, straining has taken place. Again there is more than one strain at a point. As will be shown in the Sections 5.3.3 and 5.3.4, a segment of infinitesimal length not only experiences a change in length, but also a change in direction.
