Page 114 - Introduction to Continuum Mechanics

Geometrical Meaning of the Components of the Infinitesimal Strain Tensor 99

(c) In spherical coordinates:

3.8 Geometrical Meaning of the Components of the Infinitesimal Strain Tensor

(a)Diagonal elements of E

Consider the single material element dTP ' = dXs ' - dX = (dS)n, where n is a unit vector
and dS is the length of dX. Let ds denote the deformed length of dx^\ i.e., ds = \d^\.
Then, Eq. (3.7.9) gives

This equation states that the unit elongation (i.e., the increase in length per unit original
length ) for the element which was in the direction n, is given by n • En. In particular, if the
element was in the ej direction in the reference state, then n = ej, and EH = ej • Eej so that

EH is the unit elongation for an element originally in thex\-direction. Similarly,
£22 is -the unit elongation for an element originally in the ^-direction and
£33 is the unit elongation for an element originally in the ^-direction.