Strain is a geometric description of how each point in a body has deformed
relative to the undeformed state^{2.6}.
We track a point
in the undeformed body to its corresponding point
in the deformed body via a deformation
:
.
We denote the distance between two nearby points in the undeformed state as
and their images in the deformed state as
. The quantity we
wish to examine is the difference in these distances squared:

(2.54) 
If we take the mapping between undeformed and deformed to be
,
then we can express the differential quantities in the deformed state in terms
of the undeformed variables. That is,
The difference in squared distances becomes


(2.57) 


(2.58) 


(2.59) 


(2.60) 


(2.61) 


(2.62) 
Interchanging the indices
and
, we finally get to the definition of the
components of strain:
where the components of strain are defined as

(2.65) 
The components of the gradient of the strain are often assumed to be small,
allowing us to drop the last term of the strain:

(2.66) 
Note that it is the derivatives of the strain that are assumed to be
small, not the strains themselves. Thus large deformations can be described
with this definition of strain if the strain varies smoothly throughout a
body. Strain is a symmetric tensor of second order.
Footnotes
 ... state^{2.6}
 The undeformed state is
sometimes called the rest state, but this is misleading, as the
undeformed state does not have to be at rest  it could be moving uniformly
as a rigid body.
Copyright © 2005 Adrian Secord. All rights reserved.