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time = t

vdt
`
n



ds
N
p

time = t + dt
dS
X1, x 1
P X2, x 2


time = 0

FIGURE 3.12 Rate of change of length.


of the scalar product in terms of dx1 and dx2 as,
1d ™
(dx1 · dx2 ) = dx1 · (F ’T EF ’1 )dx2 (3.99)
2 dt
The tensor in the expression on the right-hand side is simply the pushed

forward spatial counterpart of E and is known as the rate of deformation
tensor d given as,
™ ™ ™
E = φ’1 [d] = F T dF
d = φ— [E] = F ’T EF ’1 ; (3.100a,b)


A more conventional expression for d emerges from the combination of Equa-
™ ™
tions (3.92) for F and (3.97) for E to give, after simple algebra, the tensor
d as the symmetric part of l as,
d = 1 (l + lT ) (3.101)
2

Remark 3.12.1. A simple physical interpretation of the tensor d can be
obtained by taking dx1 = dx2 = dx as shown in Figure 3.12 to give,
1d
(dx · dx) = dx · d dx (3.102)
2 dt
Expressing dx as ds n, where n is a unit vector in the direction of dx as
shown in Figure 3.12 gives,
1d
(ds2 ) = ds2 n · d n (3.103)
2 dt
33
3.13 SPIN TENSOR



d
`

. e
E


` ’1
E

FIGURE 3.13 Lie derivative.


which noting that d(ds2 )/dt = 2ds d(ds)/dt, ¬nally yields,
1d d ln(ds)
n·dn = (ds) = (3.104)
ds dt dt
Hence the rate of deformation tensor d gives the rate of extension per unit
current length of a line element having a current direction de¬ned by n. In
particular for a rigid body motion d = 0.

Remark 3.12.2. Note that the spatial rate of deformation tensor d is not
the material derivative of the Almansi or spatial strain tensor e introduced

in Section 3.5. Instead, d is the push forward of E, which is the derivative
with respect to time of the pull back of e, that is,
d ’1
d = φ— (φ [e]) (3.105)
dt —
This operation is illustrated in Figure 3.13 and is known as the Lie derivative
of a tensor quantity over the mapping φ and is generally expressed as,
d ’1
Lφ[g] = φ— (φ [g]) (3.106)
dt —


3.13 SPIN TENSOR

The velocity gradient tensor l can be expressed as the sum of the symmetric
rate of deformation tensor d plus an additional antisymmetric component w
as,

dT = d, wT = ’w
l = d + w; (3.107)

where the antisymmetric tensor w is known as the spin tensor and can be
obtained as,

w = 1 (l ’ lT ) (3.108)
2
34 KINEMATICS



The terminology employed for w can be justi¬ed by obtaining a relationship
between the spin tensor and the rate of change of the rotation tensor R
introduced in Section 3.6. For this purpose, note that l can be obtained
from Equation (3.93), thereby enabling Equation (3.108) to be rewritten as,
™T

w = 1 (F F ’1 ’ F ’T F ) (3.109)
2

Combining this equation with the polar decomposition of F and its time
derivative as,

F = RU (3.110a)
™ ™ ™
F = RU + RU (3.110b)

yields, after some simple algebra, w as,
™T
™ ™ ™
w = 1 (RRT ’ RR ) + 1 R(U U ’1 ’ U ’1 U )RT (3.111)
2 2

Finally, di¬erentiation with respect to time of the expression RRT = I

easily shows that the tensor RRT is antisymmetric, that is,
™T ™
RR = ’RRT (3.112)

thereby allowing Equation (3.111) to be rewritten as,
™ ™ ™
w = RRT + 1 R(U U ’1 ’ U ’1 U )RT (3.113)
2

The second term in the above equation vanishes in several cases such as, for
instance, rigid body motion. A more realistic example arises when the prin-
cipal directions of strain given by the Lagrangian triad remain constant; such

a case is the deformation of a cylindrical rod. Under such circumstances U
can be derived from Equation (3.31) as,
3


U= »± N ± — N ± (3.114)
±=1


Note that this implies that U has the same principal directions as U . Ex-
pressing the inverse stretch tensor as U ’1 = 3 »’1 N ± — N ± gives,
±=1 ±
3

™ ™
U U ’1 = »’1 »± N ± — N ± = U ’1 U (3.115)
±
±=1

Consequently, the spin tensor w becomes,

w = RRT (3.116)

Often the spin tensor w is physically interpreted in terms of its associated
35
3.13 SPIN TENSOR



!

q
dv




dx




p

FIGURE 3.14 Angular velocity vector.


angular velocity vector ω (see Section 2.2.2) de¬ned as,
ω1 = w32 = ’w23 (3.117a)
ω2 = w13 = ’w31 (3.117b)
ω3 = w21 = ’w12 (3.117c)
so that, in the case of a rigid body motion where l = w, the incremental or
relative velocity of a particle q in the neighbourhood of particle p shown in
Figure 3.14 can be expressed as,
dv = w dx = ω—dx (3.118)

Remark 3.13.1. In the case of a constant Lagrangian triad, useful equa-

tions similar to (3.114) can be obtained for the material strain rate tensor E
by di¬erentiating with respect to time Equation (3.45) to give,
3
1 d»2
™ ±
E= (3.119)
N± — N±
2 dt
±=1

Furthermore, pushing this expression forward to the spatial con¬guration
with the aid of Equations (3.100a) and (3.44b) enables the rate of defor-
mation tensor to be expressed in terms of the time rate of the logarithmic
stretches as,
3
d ln »±
d= (3.120)
n± — n±
dt
±=1

In general, however, the Lagrangian triad changes in time, and both the
material strain rate and rate of deformation tensors exhibit o¬-diagonal
terms (that is, shear terms) when expressed in the corresponding material
36 KINEMATICS




and spatial principal axes. The general equation for E is easily obtained
from Equation (3.45) as,
3 3
1 d»2 12 ™
™ ™
±
E= N± — N± + » (N ± — N ± + N ± — N ± ) (3.121)

2 dt
±=1 ±=1

where time di¬erentiation of the expression N ± · N β = δ±β to give N ± · N β =

’N β · N ± reveals that the rate of change of the Lagrangian triad can be
expressed in terms of the components of a skew symmetric tensor W as,
3

N± = W±β N β ; W±β = ’Wβ± (3.122)
β=1

Substituting this expression into Equation (3.121) gives,
3 3
1 d»2 1
™ ±
W±β »2 ’ »2 N ± — N β
E= N± — N± + (3.123)
± β
2 dt 2
±=1 ±,β=1
±=β

This equation will prove useful, in Chapter 5, when we study hyperelas-
tic materials in principal directions, where it will be seen that an explicit
derivation of W±β is unnecessary.

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