By D. S. Dugdale and B. G. Neal (Auth.)
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And ν = 0*3, determine the prin cipal stresses acting at this location. *. 1. Alternative coordinates Hitherto, rectangular coordinates have proved to be quite ade quate for investigating the n a t u r e of stress a n d strain, a n d m a n y relationships have been derived. I t is the purpose of the present chapter to investigate how alternative coordinate systems can be devised for describing plane elastic deformations. Since the basic relations are often expressed in terms of first a n d second deriva tives of scalars, it will be necessary to find the new forms assumed by these derivatives when new coordinates are used.
If the stress-strain curve is assumed to become horizontal after the elas tic limit has been passed, as shown in Fig. 1c, so t h a t the m a t erial flows at constant stress, this simplifies the calculation of stres ses in the plastic regions. However, this type of analysis is beyond the scope of the present volume. 2. Dilatation and distortion Although much of Chapters 1 and 2 dealt with two-dimensional deformation, it is necessary in this chapter to consider threedimensional deformation so t h a t the mechanical behaviour of solids can be described with sufficient generality.
T o take a specific example, consider a square metal plate having sides of length 5 cm, with a particular point m a r k e d on its surface. Some means is required for specifying the position of this point. A pos sible method would be to m a r k on the plate a net of straight lines parallel to the sides, spaced 1 cm apart, as shown in Fig. 1. e. Λ: = 3 cm, y = 2 cm. Instead of marking coordinate lines on the b o d y to be exam ined, the body may be placed on a sheet of paper having the same net of lines drawn on it.
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