Page 157 - Introduction to Continuum Mechanics

142 Eulerian Strain Tensor

and

Or,

Thus, if we consider a material element, which at time t is in the direction of
?i ,i.e., dx = ds*i and which at the reference time is dX = dSn, where n is a unit vector, then
Eq. (3.26.7) and Eq. (3.26.8) give:
For dx = ds*i

,~2

and

—i *
Similar meanings hold for the other diagonal elements of B and e .
By considering two material elements dir' = ds\*i and dr ' = ds^i at time t correspond-
ing to dX^ - d$in and dX^ = rf^m at the reference time, n and m are unit vectors, Eq.
(3.26.7) and Eq. (3.26.8) give

Similar meanings hold for the other off-diagonal elements of B~" and e *.
—i *
We can also express B and e in terms of the displacement components:
From u = x-X, we can write

or,

where we have used the spatial description of the displacement field because we intend to
differentiate this equation with respect to the spatial coordinates */. Thus,