ABSTRACT
Almost all displacement fields induced by boundary loads, support movements, temperature, body forces, or other perturbations to the initial condition are, unfortunately,
nonlinear
; that is:
u
,
v
, and
w
are cross-products or power functions of
x
,
y
,
z
(and perhaps other variables). However, as shown in Figure. 2.1,* the fundamental linear assumption of calculus allows us to directly use the relations of finite linear transformation to depict immediately the relative displacement or deformation
du
,
dv
,
dw
of a differential element
dx
,
dy
,
dz
. On a differential scale, as long as
u
,
v
, and
w
are continuous, smooth, and small, straight lines remain straight and parallel lines and planes remain parallel. Thus the standard definition of a total derivative:
(2.1)
is more than a mathematical statement that differential base lengths obey the laws of linear transformation.** The resulting deformation tensor,
E
, also
du u x ------dx u
y ------dy u
z ------dz
dv v x ----- dx v
y ----- dy v
z ----- dz
dw w x -------dx w
y -------dy w
z -------dz
or du dv dw
≠u ≠x ------
≠u ≠y ------
≠u ≠z ------
≠v ≠x -----
≠v ≠y -----
≠v ≠z -----
≠w ≠x -------
≠w ≠y -------
≠w ≠z -------
dx dy dz
d Eij[ ] dr
called the relative displacement tensor, is directly analogous to the linear displacement tensor,
, of Chapter 1, which transformed finite baselengths. The elements of
E
(the partial derivatives), although nonlinear functions throughout the field (i.e., the structure), are just numbers when evaluated at any
x
,
y
,
z
. Therefore
E
should be thought of as an average or, in the limit, as “deformation at a point.” Displacements
u
,
v
,
w
, due to deformation, are obtained by a line integral of the total derivative from a location where
u
,
v
,
w
have known values; usually a support where one or more are zero. Thus:
(2.2)
As on a finite scale, the deformation tensor can be “dissolved” into its symmetric and asymmetric components:
u; v v; w wd 0
#d 0
#d 0
(2.3)
The individual elements or components of
by definition are:*
(2.4)
each of which produces a “deformation” of the orthogonal differential baselengths
dx
,
dy
, and
dz
, exactly like its counterpart in the symmetric displacement tensor
d
with respect to finite base lengths
x
,
y
, and
z
when
u
,
v
, and w are linear functions. Physically, the diagonal terms are
u x
-----
u y ------
u x ------
v x -----
v y -----
v z -----
w x -------
w y -------
w z -------
u x ------
1 2 --
u y ------
v x ----- 12--
u z ------
w x -------
1 2 --
v x -----
u y ------ vy-----
1 2 --
v z -----
w y -------
1 2 --
w x -------
u z ------ 12--
w y -------
v z ----- wz-------
0 1 2 --
u y ------
1 v x ----- 12--
u z ------
1 w x -------
1 2 --
v x -----
1 u y ------ 0 12--
v z -----
1 w y -------
1 2 --
w x -------
1 u z ------ 12--
w y -------
1 v z ----- 0
ei change in length original baselength ---------------------------------------------
5 in the i direction( ).
