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In two dimensions, the area change during deformation will lead to a situation that the area of the ellipse resulting from the initial unit circle will be greater or lesser. The volume change is designated by Capital Delta. Since the area of a circle is given by pi times square of the radius of the circle, the area of the ellipse will be given by pi times the semi-major and semi-minor axes of the ellipse. Since pi is a common quantity, the area of the ellipse will be given by the product (1+e1).(1+e2). If there is no area change, this product always equals unity since the e1 and e2 being of different signs will rule out the change. If the long axis of the ellipse is 2 units then the short axis will be 0.5 so that the product is unity. If the product is greater than unity, it implies increase in area so that the resultant ellipse will lie outside the circle. In case of decrease of area, the product will be less than 1 and the resultant ellipse will lie inside the circle. As the ellipse does not intersect the circle (all lines either elongated or shortened), there will be no lines of no finite longitudinal strain within the ellipse.

1+D= l1. l2

or D=( l1. l2)-1

1+D= l1. l2.. l3

or D=( l1. l2.l3)-1

In terms of linear transformation coefficients, it is given by

D= (ad-bc)-1



1+e1 or 1+e3 will be given by the quadratic equation having both positive roots:

=[1+g2 +(1+D)2g{((1+g2+(1+D)2)2-4(1+D)2)]/2