Page 237 - Intro to Tensor Calculus

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Note also in the figure 2.3-15 there is the displacement vector ~u. This vector can be represented in any
of the following forms:

i ~
~u = u E i contravariant, Lagrangian basis
~ i
~u = u i E covariant, Lagrangian reciprocal basis
i~
~u = u E i contravariant, Eulerian basis
~i
~u = u i E covariant, Eulerian reciprocal basis.
r
By vector addition we have ~ + ~u = ~ r and consequently d~r + d~u = d ~ r. In the Lagrangian frame of reference
i ~
i ~
at the point P we represent ~u in the contravariant form ~u = u E i and write d~r in the form d~r = dx E i . By
k ~
i
use of the equation (1.4.48) we can express d~u in the form d~u = u dx E i . These substitutions produce the
,k
~
k ~
i
i
2
representation dr =(dx + u dx )E i in the Lagrangian coordinate system. We can then express ds in the
,k
Lagrangian system. We find
k ~
j
i
i
2
~
~
dr · dr = ds =(dx + u dx )E i · (dx + u j m ~
,k ,m dx )E j
i
i
k
j
m
i
k
m
i
j
=(dx dx + u j ,m dx dx + u dx dx + u u j ,m dx dx )g ij
,k
,k
and consequently from the relation (2.3.58) we derive the representation
1 m
e ij = u i,j + u j,i + u m,i u ,j . (2.3.60)
2
This is the representation of the Lagrangian strain tensor in any system of coordinates. The strain tensor
e ij is symmetric. We will restrict our study to small deformations and neglect the product terms in equation
1
(2.3.60). Under these conditions the equation (2.3.60) reduces to e ij = (u i,j + u j,i ).
2
If instead, we chose to represent the displacement ~u with respect to the Eulerian basis, then we can
write
i~ i k~
~u = u E i with d~u = u dx E i .
,k
These relations imply that
k ~
i
i
~
d~ = dr − d~u =(dx − u dx )E i .
r
,k
This representation of d~ in the Eulerian frame of reference can be used to calculate the strain e ij from the
r
2
2
relation ds − ds . It is left as an exercise to show that there results
1 m
e ij = u i,j + u j,i − u m,i u ,j . (2.3.61)
2
The equation (2.3.61) is the representation of the Eulerian strain tensor in any system of coordinates. Under
conditions of small deformations both the equations (2.3.60) and (2.3.61) reduce to the linearized Lagrangian
1
and Eulerian strain tensor e ij = (u i,j + u j,i ). In the case of large deformations the equations (2.3.60) and
2
(2.3.61) describe the strains. In the case of linear elasticity, where the deformations are very small, the
product terms in equations (2.3.60) and (2.3.61) are neglected and the Lagrangian and Eulerian strains
reduce to their linearized forms
1 1
e ij = [u i,j + u j,i ] e ij = [u i,j + u j,i ] . (2.3.62)
2 2