Page 230 - Intro to Tensor Calculus

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The displacement of the point C to C is described by the coordinates (x, y +∆y) transforming to (x, y)
where
x = x + u(x, y +∆y), y = y +∆y + v(x, y +∆y). (2.3.35)

Again we expand the displacement field components u and v in a Taylor series about the point (x, y)and
find
∂u
x = x + u + ∆y + h.o.t.
∂y
(2.3.36)
∂v
y = y +∆y + v + ∆y + h.o.t.
∂y
This equation implies that the coefficients b ij must be chosen such that

∂u
x + u + ∆y
∂y = b 11 b 12 x . (2.3.37)
y + v +∆y + ∂v ∆y b 21 b 22 y +∆y
∂y
0
Finally, it can be verified that the point D with coordinates (x +∆x, y +∆y) moves to the point D with
coordinates
x = x +∆x + u(x +∆x, y +∆y), y = y +∆y + v(x +∆x, y +∆y). (2.3.38)

Expanding u and v in a Taylor series about the point (x, y) we find the coefficients b ij must be chosen to
satisfy the matrix equation

∂u ∂u
x +∆x + u + ∆x + ∆y
∂x ∂y = b 11 b 12 x +∆x . (2.3.39)
y +∆y + v + ∂v ∆x + ∂v ∆y b 21 b 22 y +∆y
∂x ∂y
The equations (2.3.31),(2.3.34),(2.3.37) and (2.3.39) give rise to the simultaneous equations

b 11 x + b 12 y = x + u
b 21 x + b 22 y = y + v
∂u
b 11 (x +∆x)+ b 12 y = x + u +∆x + ∆x
∂x
∂v
b 21 (x +∆x)+ b 22 y = y + v + ∆x
∂x
∂u
b 11 x + b 12 (y +∆y)= x + u + ∆y (2.3.40)
∂y
∂v
b 21 x + b 22 (y +∆y)= y + v +∆y + ∆y
∂y
∂u ∂u
b 11 (x +∆x)+ b 12 (y +∆y)= x +∆x + u + ∆x + ∆y
∂x ∂y
∂v ∂v
b 21 (x +∆x)+ b 22 (y +∆y)= y +∆y + v + ∆x + ∆y.
∂x ∂y
It is readily verified that the system of equations (2.3.40) has the solution

∂u ∂u
b 11 =1 + b 12 =
∂x ∂y (2.3.41)
∂v ∂v
b 21 = b 22 =1 + .
∂x ∂y