Page 190 - Intro to Tensor Calculus

185

Figure 2.1-2 Mohr’s circle

1
I 39. Otto Mohr gave the following physical interpretation to the results obtained in problem 38:
• Plot the points A(I 11 ,I 12 )and B(I 22 , −I 12 ) as illustrated in the figure 2.1-2
• Draw the line AB and calculate the point C where this line intersects the I axes. Show the point C
has the coordinates
I 11 + I 22
( , 0)
2
• Calculate the radius of the circle with center at the point C and with diagonal AB and show this
radius is
s
2

I 11 − I 22 2
r = + I 12
2
• Show the maximum and minimum values of I occur where the constructed circle intersects the I axes.
I 11 + I 22 I 11 + I 22
Show that I max = I 11 = + r I min = I 22 = − r.
2 2

I 11 I 12
I 40. Show directly that the eigenvalues of the symmetric matrix I ij = are λ 1 = I max and
I 21 I 22
λ 2 = I min where I max and I min are given in problem 39.
I 41. Find the principal axes and moments of inertia for the triangle given in problem 37 and summarize
your results from problems 37,38,39, and 40.

I 42. Verify for orthogonal coordinates the relations
3
h i X e (i)jk ∂(h (k) A(k))
~
∇× A · ˆ (i) = h (i)
e
h 1 h 2 h 3 ∂x j
k=1
or

h 1 ˆ e 1 h 2 ˆ e 2 h 3 ˆ e 3
1 ∂ ∂ ∂
~
∇× A = .
∂x 1 ∂x 2 ∂x 3

h 1 A(1) h 2 A(2) h 3 A(3)
h 1 h 2 h 3
I 43. Verify for orthogonal coordinates the relation
3 " 2 #
h i X h (i) ∂ h (r) ∂(h (m) A(m))
~
e
∇× (∇× A) · ˆ (i) = e (i)jr e rsm
h 1 h 2 h 3 ∂x j h 1 h 2 h 3 ∂x s
m=1
1 Christian Otto Mohr (1835-1918) German civil engineer.