Page 235 - Intro to Tensor Calculus

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General Tensor Derivation of Strain.

With reference to the figure 2.3-15 consider the deformation of a point P within a continuum. Let
3
2
1
(y ,y ,y ) denote a Cartesian coordinate system which is fixed in space. We can introduce a coordinate
i
3
2
i
1
transformation y = y (x ,x ,x ), i =1, 2, 3 and represent all points within the continuum with respect
2
3
1
to a set of generalized coordinates (x ,x ,x ). Let P denote a general point in the continuum while it is
in an unstrained state and assume that this point gets transformed to a point P when the continuum
0
experiences external forces. If P moves to P , then all points Q which are near P will move to points Q 0
0
near P . We can imagine that in the unstrained state all the points of the continuum are referenced with
0
2
1
3
respect to the set of generalized coordinates (x ,x ,x ). After the strain occurs, we can imagine that it will
be convenient to represent all points of the continuum with respect to a new barred system of coordinates
3
1
2
(x , x , x ). We call the original set of coordinates the Lagrangian system of coordinates and the new set
of barred coordinates the Eulerian coordinates. The Eulerian coordinates are assumed to be described by
3
i
i
1
2
a set of coordinate transformation equations x = x (x ,x ,x ), i =1, 2, 3 with inverse transformations
2
3
1
i
i
x = x (x , x , x ), i =1, 2, 3, which are assumed to exist. The barred and unbarred coordinates can
i
be related to a fixed set of Cartesian coordinates y ,i =1, 2, 3, and we may assume that there exists
transformation equations
2
i
i
i
1
2
1
i
3
3
y = y (x ,x ,x ), i =1, 2, 3and y = y (x , x , x ), i =1, 2, 3
which relate the barred and unbarred coordinates to the Cartesian axes. In the discussion that follows
be sure to note whether there is a bar over a symbol, as we will be jumping back and forth between the
Lagrangian and Eulerian reference frames.
Figure 2.3-15. Strain in generalized coordinates

∂~
r
~
In the Lagrangian system of unbarred coordinates we have the basis vectors E i = which produce
∂x i
~
~
the metrices g ij = E i · E j . Similarly, in the Eulerian system of barred coordinates we have the basis vectors
~ ∂ ~ r ~ ~
E i = which produces the metrices G ij = E i · E j . These basis vectors are illustrated in the figure 2.3-15.
∂x i