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ANGLES BETWEEN CONES OF LIES OF NO FINITE LONGITUDINAL STRAIN AND PRINCIPAL AXES OF STRAIN IN THE THREE PLANES

There are two directions in which the extension strain is zero. That is, the lines corresponding to the points where the circle and ellipse cross one another. These lines separate orientations that are shortened from those that are extended. In volume conserving deformation, lines that have suffered no longitudinal strain or whose stretch e=0. In each of the principal sections, the cones of lines of no finite longitudinal strain make angles unless the deformation is by plane strain. These are jxy in XY plane, jyz in YZ plane and jxz in XZ plane. In the 5 ellipsoids:      
 k=0 jxy=180 jyz=jxz
”>k>0 jxy=180 jyzčjxz

                                                         

k=1 jxy=180 jyz=0,jxz=1-89
„>k>1 jxy<90 jyz=180, jxz<90
 k=” jxy=jxz  jyz=180

 From a and b, the values of each can be computed for any volume conserving deformation from the following equations:

Cos2jxz=[(a-2/3.b2/3)-b2]/a2-b2 

Cos2jxy=[(a-2/3.b2/3)-1]/a-2-1         

Cos2jyz=[(a-2/3.b2/3)-b2]/1-b2