CONTINUUM MECHANICS for ENGINEERS

2.24 Determine the principal values λ (q) (q = 1,2,3) and principal directions
(q = 1,2,3) for the symmetric matrix
( )
nˆ q
[ Tij]= Answer: λ (1) = 1, λ (2) = 2, λ (3) = 3
2 ⎝ 1 2 3⎠
⎡ 3 −1/
2 1/ 2⎤
1 ⎢
⎥
⎢−1/
2 9/ 2 3/ 2
2
⎥
⎢
⎥
⎣
1/ 2 3/ 2 9/ 2
⎦
( 1 ) 1 ⎛
⎞ ( 2 ) 1 ⎛
⎞ ( 3 )
nˆ = ⎜ 2eˆ + eˆ −eˆ
⎟ , nˆ = ⎜ 2eˆ
− eˆ + eˆ
⎟ , ˆ ⎛ n =− eˆ + eˆ
⎞ /
2.25 Let D be a constant tensor whose components do not depend upon
the coordinates. Show that
x i i
where x = ê is the position vector.
2 ⎝ 1 2 3⎠
( x⋅D)= D
2.26 Consider the vector x = xiêi having a magnitude squared
2 2 2 2
x = x + x + x . Determine
1
2
3
(a) grad x (d) div(xnx) (b) grad (x –n ) (e) curl(xn (c)
x), where n is a positive integer
2 (1/x)
Answer: (a) x i/x, (b) – nx i/x (n + 2) , (c) 0, (d) x n (n + 3), (e) 0.
2.27 If λ and φ are scalar functions of the coordinates xi, verify the following
vector identities. Transcribe the left-hand side of the equations into
indicial notation and, following the indicated operations, show that
the result is the right-hand side.
(a) v × ( × v) = (v ⋅ v) – (v ⋅ )v
(b) v ⋅ u × w = v × u ⋅ w
(c) × ( × v) = ( ⋅ v) – 2v (d) ⋅ (λ φ) = λ 2φ + λ ⋅ φ
1
2
(e) 2 (λφ) = λ 2 φ + 2( λ)⋅ ( φ) + φ 2 λ
(f) ⋅ (u × v) = ( × u) ⋅ v – u ⋅ ( × v)
2.28 Let the vector v = b × x be one for which b does not depend upon
the coordinates. Use indicial notation to show that
(a) curl v = 2b
(b) div v = 0
⎝
2 3⎠ 2