CONTINUUM MECHANICS for ENGINEERS

Solving Eq 4.10-9 for and using Eq 4.10-11, we may write
˙n i
( ) −
˙ n= v n− nΛ˙ / Λ = D + W n D nnn
i i,j j i ij ij j qk q k i
(p) If now ni is chosen along a principal direction of D so that Dn = ij j D(p) ni
(p = 1, 2, 3) where D (p) represents a principal value of D, then
˙ ( p)
( p)
i ( p) i ij j
n = D n + W n − D n n n = W n
( p)
( p)
( p)
( p)
( p) q q i ij j
(4.10-20)
(p) (p)
since n = 1. Because a unit vector can change only in direction, Eq 4.10-
q nq
20 indicates that Wij gives the rate of change in direction of the principal axes
of D. Hence the names vorticity or spin given to W. Additionally, we associate
with W the vector
1
1
wi = εijkvk,j or w = curl v (4.10-21)
2
2
called the vorticity vector, by the following calculation,
( )
1
1
pqi i 2 pqi ijk k,j 2 pj qk pk qj k,j
ε w = ε ε v = δ δ −δ
δ v
1
= v v W
(4.10-22)
2 ( q,p − p,q )= qp
Thus if D ≡ 0 so that L ij = W ij, it follows that dv i = L ijdx j = W ijdx j = ε jikw kdx j
and since ε jik = –ε ijk = ε ikj,
dv i = ε ijkw j dx k or dv = w × dx (4.10-23)
according to which the relative velocity in the vicinity of p corresponds to a
rigid body rotation about an axis through p. The vector w indicates the
angular velocity, the direction, and the sense of this rotation.
To summarize the physical interpretation of the velocity gradient L, we
note that it effects a separation of the local instantaneous motion into two parts:
1. The so-called logarithmic rates of stretching, D (p), (p = 1,2,3), that is, the
eigenvalues of D along the mutually orthogonal principal axes of D, and
Λ˙ / Λ
d lnΛ
dt
= ( ) ( p) ( p) ( p)
( p)
= = =
Dnn nD n D
ij i j i ( p)
i
2. A rigid body rotation of the principal axes of D with angular velocity w.
( p)
(p)