Page 236 - Intro to Tensor Calculus

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2
We assume that an element of arc length squared ds in the unstrained state is deformed to the element
2
of arc length squared ds in the strained state. An element of arc length squared can be expressed in terms
of the barred or unbarred coordinates. For example, in the Lagrangian system, let d~ = PQ so that
r

2
2
j
i
L = d~r · d~r = ds = g ij dx dx , (2.3.54)
0
where g ij are the metrices in the Lagrangian coordinate system. This same element of arc length squared
can be expressed in the barred system by
∂x m ∂x n
2
j
2
i
L = ds = g dx dx , where g ij = g mn i j . (2.3.55)
ij
0
∂x ∂x
Similarly, in the Eulerian system of coordinates the deformed arc length squared is
~
~
j
i
2
2
L = dr · dr = ds = G ij dx dx , (2.3.56)
where G ij are the metrices in the Eulerian system of coordinates. This same element of arc length squared
can be expressed in the Lagrangian system by the relation
m
∂x ∂x n
j
2
2
i
L = ds = G ij dx dx , where G ij = G mn . (2.3.57)
i
∂x ∂x j
In the Lagrangian system we have
2
j
i
i
2
ds − ds =(G ij − g ij )dx dx =2e ij dx dx j
where
1
e ij = (G ij − g ij ) (2.3.58)
2
is called the Green strain tensor or Lagrangian strain tensor. Alternatively, in the Eulerian system of
coordinates we may write
i
2
2
j
i
ds − ds = G ij − g dx dx =2e ij dx dx j
ij
where
1
e ij = G ij − g ij (2.3.59)
2
is called the Almansi strain tensor or Eulerian strain tensor.