3.7 Deformation of an Infinitesimal Line Element
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3.7 Deformation of an Infinitesimal Line Element
Consider an infinitesimal line element PQ in the undeformed geometry of a medium
as shown in Fig. 3.4. When the body undergoes deformation, the line element PQ
passes into the line element P
Q
. In general, both the length and the direction of
PQ are changed.
Let the co-ordinates of P and Q before deformation be
(x, y, z), (x + x, y + y, z + z), respectively, and the displacement vector at
point P has components (u, v, w). The co-ordinates of P, P
and Q are
P : (x, y, z)
P
: (x + u, y + v, z + w)
Q : (x + x, y + y, z + z)
The displacement components at Q differ slightly from those at point P since Q
is away from P by x, ,y and z. Hence the displacements at Q are
u + u, v + v and w + w
Now, if Q is very close to P, then to the first-order approximation
u =
∂u
∂ x
x +
∂u
∂ y
y +
∂u
∂z
z
(3.17)
Fig. 3.4 Line element in
undeformed and deformed
body