q

`

vp

Q

p

time = t

X1, x 1 P

X2, x 2

time = 0

FIGURE 3.9 Particle velocity.

3.11.2 MATERIAL TIME DERIVATIVE

Given a general scalar or tensor quantity g, expressed in terms of the ma-

terial coordinates X, the time derivative of g(X, t) denoted henceforth by

™

g(X, t) or dg(X, t)/dt is de¬ned as,

dg ‚g(X, t)

™

g= = (3.82)

dt ‚t

This expression measures the change in g associated with a speci¬c particle

initially located at X, and it is known as the material time derivative of

g. Invariably, however, spatial quantities are expressed as functions of the

spatial position x, in which case the material derivative is more complicated

to establish. The complication arises because as time progresses the speci¬c

particle being considered changes spatial position. Consequently, the mate-

rial time derivative in this case is obtained from a careful consideration of

the motion of the particle as,

g(φ(X, t + ∆t), t + ∆t) ’ g(φ(X, t), t)

™

g(x, t) = lim (3.83)

∆t

∆t’0

This equation clearly illustrates that g changes in time, (i) as a result of a

change in time but with the particle remaining in the same spatial position

and (ii) because of the change in spatial position of the speci¬c particle.

Using the chain rule Equation (3.83) gives the material derivative of g(x, t)

28 KINEMATICS

as,

‚g(x, t) ‚g(x, t) ‚φ(X, t) ‚g(x, t)

™

g(x, t) = + = + ( g)v (3.84)

‚t ‚x ‚t ‚t

The second term, involving the particle velocity in Equation (3.84) is often

referred to as the convective derivative.

EXAMPLE 3.9: Material time derivative

Here Example 3.1 is revisited to illustrate the calculation of a material

time derivative based on either a material or spatial description. The

material description of the temperature distribution along the rod is

™ ™

T = Xt2 , yielding T directly as T = 2Xt. From the description of

motion, x = (1+t)X, the velocity is expressed as v = X or v = x/(1+t)

in the material and spatial descriptions respectively. Using the spatial

description, T = xt2 /(1 + t) gives,

(2t + t2 )x t2

‚T (x, t) ‚T (x, t)

= ; T= =

(1 + t)2

‚t ‚x (1 + t)

™

Hence from (3.84), T = 2xt/(1 + t) = 2Xt.

3.11.3 DIRECTIONAL DERIVATIVE AND TIME RATES

Traditionally, linearization has been implemented in terms of an arti¬cial

time and associated rates. This procedure, however, leads to confusion

when real rates are involved in the problem. It transpires that linearization

as de¬ned in Chapter 2, Equation (2.101), avoids this confusion and leads

to a much clearer ¬nite element formulation. Nevertheless it is valuable

to appreciate the relationship between linearization and the material time

derivative. For this purpose consider a general operator F that applies to

the motion φ(X, t). The directional derivative of F in the direction of v

coincides with the time derivative of F , that is,

d

DF [v] = F (φ(X, t)) (3.85)

dt

To prove this, let φX (t) denote the motion of a given particle and F (t) the

function of time obtained by applying the operator F on this motion as,

F (t) = F (φX (t)); φX (t) = φ(X, t) (3.86)

Note ¬rst that the derivative with respect to time of a function f (t) is related

to the directional derivative of this function in the direction of an increment

29

3.11 VELOCITY AND MATERIAL TIME DERIVATIVES

in time ∆t as,

d df

Df [∆t] = f (t + ∆t) = ∆t (3.87)

d dt

=0

Using this equation with ∆t = 1 for the functions F (t) and φX (t) and

recalling the chain rule for directional derivatives given by Equation (2.105c)

gives,

d dF

F (φ(X, t)) =

dt dt

= DF [1]

= DF (φX (t))[1]

= DF [DφX [1]]

= DF [v] (3.88)

A simple illustration of Equation (3.85) emerges from the time derivative

of the deformation gradient tensor F which can be easily obtained from

Equations (3.6) and (3.80) as,

d ‚φ ‚ ‚φ

™

F= = = (3.89)

v

dt ‚X ‚X ‚t

Alternatively, recalling Equation (3.70) for the linearized deformation gra-

dient DF gives,

™

DF [v] = v=F (3.90)

3.11.4 VELOCITY GRADIENT

We have de¬ned velocity as a spatial vector. Consequently, velocity was ex-

pressed in Equation (3.81) as a function of the spatial coordinates as v(x, t).

The derivative of this expression with respect to the spatial coordinates de-

¬nes the velocity gradient tensor l as,

‚v(x, t)

l= = (3.91)

v

‚x

This is clearly a spatial tensor, which, as Figure 3.10 shows, gives the relative

velocity of a particle currently at point q with respect to a particle currently

at p as dv = ldx. The tensor l enables the time derivative of the deformation

gradient given by Equation (3.89) to be more usefully written as,

‚v ‚v ‚φ

™

F= = = lF (3.92)

‚X ‚x ‚X

30 KINEMATICS

X3, x 3 vq = vp + d v

q

`

dx dv

Q vp

p

time = t

X1, x 1 P

X2, x 2

time = 0

FIGURE 3.10 Velocity gradient.

from which an alternative expression for l emerges as,

™

l = F F ’1 (3.93)

3.12 RATE OF DEFORMATION

Consider again the initial elemental vectors dX 1 and dX 2 introduced in

Section 3.4 and their corresponding pushed forward spatial counterparts

dx1 and dx2 given as (see Figure 3.11),

dx1 = F dX 1 ; dx2 = F dX 2 (3.94a,b)

In Section 3.5 strain was de¬ned and measured as the change in the scalar

product of two arbitrary vectors. Similarly, strain rate can now be de¬ned

as the rate of change of the scalar product of any pair of vectors. For the

purpose of measuring this rate of change recall from Section 3.5 that the

current scalar product could be expressed in terms of the material vectors

dX 1 and dX 2 (which are not functions of time) and the time-dependent

right Cauchy“Green tensor C as,

dx1 · dx2 = dX 1 · C dX 2 (3.95)

31

3.12 RATE OF DEFORMATION

X3, x 3

time = t

v dt

`

d x2

d x1

p

d X2

d X1

time = t + dt

X1, x 1 P

X2, x 2

time = 0

FIGURE 3.11 Rate of deformation.

Di¬erentiating this expression with respect to time and recalling the relation-

ship between the Lagrangian strain tensor E and the right Cauchy“Green

tensor as 2E = (C ’I) gives the current rate of change of the scalar product

in terms of the initial elemental vectors as,

d ™ ™

(dx1 · dx2 ) = dX 1 · C dX 2 = 2 dX 1 · E dX 2 (3.96)

dt

™

where E, the derivative with respect to time of the Lagrangian strain tensor,

is known as the material strain rate tensor and can be easily obtained in

™

terms of F as,

1 ™T

™ ™ ™

E = 1 C = 2 (F F + F T F ) (3.97)

2

™

The material strain rate tensor, E, gives the current rate of change of

the scalar product in terms of the initial elemental vectors. Alternatively,

it is often convenient to express the same rate of change in terms of the

current spatial vectors. For this purpose, recall ¬rst from Section 3.4 that

Equations (3.94a,b) can be inverted as,

dX 1 = F ’1 dx1 ; dX 2 = F ’1 dx2 (3.98a,b)

Introducing these expressions into Equation (3.96) gives the rate of change

32 KINEMATICS

X3, x 3