Page 239 - Intro to Tensor Calculus

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Consider now a small element of volume inside a material medium. With reference to the figure 2.3-
~
c
17(a) we let ~a, b,~ denote three small arbitrary independent vectors constructed at a general point P within
~
the material before any external forces are applied. We imagine ~a, b,~ as representing the sides of a small
c
parallelepiped before any deformation has occurred. When the material is placed in a state of strain the
~ ~ ~
~
point P will move to P and the vectors ~a, b,~c will become deformed to the vectors A, B, C as illustrated in
0
~ ~ ~
the figure 2.3-17(b). The vectors A, B, C represent the sides of the parallelepiped after the deformation.

Figure 2.3-17. Deformation of a parallelepiped

~
c
Let ∆V denote the volume of the parallelepiped with sides ~a, b,~ at P before the strain and let ∆V 0
~ ~ ~
denote the volume of the deformed parallelepiped after the strain, when it then has sides A, B, C at the
point P . We define the ratio of the change in volume due to the strain divided by the original volume as
0
the dilatation at the point P. The dilatation is thus expressed as
0
∆V − ∆V
Θ= = dilatation. (2.3.65)
∆V
i
Since u ,i =1, 2, 3 represents the displacement field due to the strain, we use the result from equation
~ ~ ~
(2.3.64) and represent the displaced vectors A, B, C in the form
i
i
i
A = a + u a j
,j
i
i
i
B = b + u b j (2.3.66)
,j
i
i
i
C = c + u c j
,j
~
c
where ~a, b,~ are arbitrary small vectors emanating from the point P in the unstrained state. The element of
volume ∆V, before the strain, is calculated from the triple scalar product relation
~
i j k
∆V = ~a · (b × ~)= e ijk a b c .
c
0
The element of volume ∆V , which occurs due to the strain, is calculated from the triple scalar product
k
~
i
~
j
~
0
∆V = A · (B × C)= e ijk A B C .