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±

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.