Page 115 - Introduction to Continuum Mechanics

100 Kinematics of a Continuum

These components (the diagonal elements of the tensor E ) are also known as the normal
strains.
(b)T7ie off diagonal elements:

2
where 6 is the angle between d\^ and d^ \ If we let 0 = (^)-y, then y will measure the
small decrease in angle between dX' ' and JX' ', known as the shear strain. Since

and for small strain

therefore,

If the elements were in the direction of ej and 62 , then m • En = E\2 so that according to
Eq. (3.8.2):
2En gives the decrease in angle between two elements initially in the*i and X2 directions.
Similarly,
2&i3 gives the decrease in angle between two elements initially in thejcj and x$ directions,
and
an
2/?23 gives the decrease in angle between two elements initially in the *2 d *s directions.

Example 3.8.1
Given the displacement components

(a) Obtain the infinitesimal strain tensor E.
(b)Using the strain tensor E, find the unit elongation for the material elements
(l) (2)
dX = dX^ and rfX = dX 2e 2, which were at the point C(0,l,0) of Fig. 3.4 (which is
reproduced here for convenience). Also, find the decrease in angle between these two
elements.
(c) Compare the results with those of Example 3.7.1.
Solution, (a) We have