δv δv

™

v = v · ∇v + ’ a = v · ∇v +

δt δt

δv

T

’ a = ∇v ·v+ (9.18)

δt

and application to the row vT with velocity components, results in

∼

δvT δvT

v = v ∇v +

™ ’ a = v ∇v + ∼

T T T T T T

∼

δt δt

∼ ∼∼ ∼ ∼∼

∼

∼

δv

T

’ a = ∇ vT ∼ + ∼ .

v (9.19)

δt

∼∼

∼

With this equation, the acceleration ¬eld can be found if the Eulerian description

of the velocity ¬eld is known. In the result the velocity gradient tensor L (with

associated matrix representation L), as introduced in Chapter 7 can be recognized:

T T

L = ∇v L = ∇ vT

and also . (9.20)

∼∼

Now the equation for the acceleration vector a with the components a becomes

∼

δv δv

a=L·v+ a=Lv+ ∼

and also . (9.21)

δt δt

∼ ∼

161 9.5 The displacement vector

In the case of a stationary ¬‚ow, where v = v( x) instead of v = v( x, t), the equation

for the acceleration reduces to

a=L·v a=L∼.

and also v (9.22)

∼

9.5 The displacement vector

Consider a material point P, which is de¬ned by the position vector x0 in the

reference con¬guration with volume V0 . In the current con¬guration the position

vector of that point is denoted by x, see Fig. 9.2.

The displacement vector of a point P, in the current con¬guration with respect

to the undeformed con¬guration, is denoted by u satisfying

u = x ’ x0 (9.23)

and in component form:

⎡ ¤⎡¤⎡ ¤

ux x0

x

⎢ ⎥⎢⎥⎢ ⎥

⎣ uy ¦ = ⎣ y ¦ ’ ⎣ y0 ¦ . (9.24)

z

uz z0

In the Lagrangian description u is considered to be a function of x0 in V0 and t and

thus

u = u( x0 , t) = x( x0 , t) ’ x0 . (9.25)

z

current con¬guration

reference con¬guration

V (t )

u

P

P

V0

x0

x = x (x0, t )

y

Lagrange: u = u (x0, t )

Euler: u = u (x, t )

x

Figure 9.2

The displacement of a material point P.

Motion: the time as an extra dimension

162

This relation enables, to formally calculate the displacement (in the current con-

¬guration at time t with respect to the reference con¬guration) of a material point,

de¬ned in the reference con¬guration with material identi¬cation x0 . For the use

of Eq. (9.25), it is assumed that x( x0 , t) is available.

In the Eulerian description u is considered to be a function of x in V( t) and t

and thus

u = u( x, t) = x ’ x0 ( x, t) . (9.26)

With Eq. (9.26), it is possible to formally calculate the displacement (in the cur-

rent con¬guration with respect to the reference con¬guration) of a material point,

which is actually (at time t) at position x in the three-dimensional space. Neces-

sary for this is, that x0 ( x, t) is known, expressing which material point x0 at time t

is present at the spatial point x, in other words, the inverse relation of x( x0 , t).

9.6 The gradient operator

In Chapter 7 the gradient operator with respect to the current con¬guration was

treated extensively. In fact, the current ¬eld of a physical variable (for example the

temperature T) was considered in the current con¬guration with domain V( t) and

as such de¬ned according to an Eulerian description. The gradient of such a vari-

able is built up from the partial derivatives with respect to the spatial coordinates,

for example:

⎡ ‚T ¤

‚x

⎢ ⎥

⎢ ⎥

‚T ‚T ‚T ⎢ ⎥

‚T

+ ey + ez ∇T = ⎢

∇T = ex ⎥.

and also (9.27)

‚y

⎢ ⎥

‚x ‚y ‚z ∼

⎣ ¦

‚T

‚z

The current ¬eld (with respect to time t) can also be mapped onto the reference

con¬guration with volume V0 and thus formulated by means of a Lagrangian

description. In this formulation the gradient can also be de¬ned and is built up

from partial derivatives with respect to the material coordinates:

⎡ ‚T ¤

‚x0

⎢ ⎥

⎢ ⎥

‚T ‚T ‚T ⎢ ⎥

‚T

∇ 0 T = ex + ey + ez ∇ 0T = ⎢ ⎥.

and also (9.28)

‚y0

⎢ ⎥

‚x0 ‚y0 ‚z0 ∼

⎣ ¦

‚T

‚z0

163 9.6 The gradient operator

To relate the afore mentioned gradient operators, the chain rule for differentiation

is used. For a ¬xed time t we ¬nd

‚T ‚T ‚x ‚T ‚y ‚T ‚z

= + +

‚x0 ‚x ‚x0 ‚y ‚x0 ‚z ‚x0

‚T ‚T ‚x ‚T ‚y ‚T ‚z

= + +

‚y0 ‚x ‚y0 ‚y ‚y0 ‚z ‚y0

‚T ‚T ‚x ‚T ‚y ‚T ‚z

= + + , (9.29)

‚z0 ‚x ‚z0 ‚y ‚z0 ‚z ‚z0

and in a more concise notation:

⎛ ⎞

‚y

‚x ‚z

‚x0 ‚x0 ‚x0

⎜ ⎟

⎜ ⎟

‚y

F =⎜ ⎟.

‚x ‚z

∇ 0 T = FT ∇ T T

with (9.30)

⎜ ⎟

‚y0 ‚y0 ‚y0

∼ ∼

⎝ ⎠

‚y

‚x ‚z

‚z0 ‚z0 ‚z0

The matrix F, for which the transpose is de¬ned in Eq. (9.30), is called the defor-

mation matrix or deformation gradient matrix, from the current con¬guration

with respect to the reference con¬guration. In the next chapter this matrix will

be discussed in full detail. For the deformation matrix F we can write in a more

concise notation:

T

F T = ∇ 0 xT and F = ∇ 0 xT . (9.31)

∼∼ ∼∼

By substituting x = x0 + u into Eq. (9.31), and using ∇ 0 xT = I we obtain

∼∼

∼ ∼ ∼

T

F T = I + ∇ 0 uT and F = I + ∇ 0 uT . (9.32)

∼ ∼

∼ ∼

In tensor notation the relation between ∇ 0 T and ∇ T can be written as

∼ ∼

T T

∇ 0 T = FT · ∇T F = ∇ 0x = I + ∇ 0u

with . (9.33)

with F the deformation tensor (also called deformation gradient tensor).

Above, a relation is derived between the gradient of a physical property (at time

t) with respect to the reference con¬guration and the gradient of that property with

respect to the current con¬guration. Consequently the mutual relation between the

gradient operators can be written as

∇ 0 ( ) = FT · ∇( ) ∇ 0 ( ) = FT ∇ ( ) ,

and also (9.34)

∼ ∼

and the inverse relation

∇( ) = F’T · ∇ 0 ( ) ∇ ( ) = F ’T ∇ 0 ( ) .

and also (9.35)

∼ ∼

If the current and reference con¬guration are identical (in that case u = 0 and

F = I) the gradient operators are also identical.

Motion: the time as an extra dimension

164

9.7 Extra displacement as a rigid body

In this section the consequences of a (¬ctitious) extra displacement of the current

con¬guration as a rigid body will be discussed. Consider a hypothetical current

con¬guration, that originates by ¬rst rotating the current con¬guration around the

origin of the xyz-coordinate system and by translating it subsequently. The rotation

around the origin is de¬ned by means of a rotation tensor P (orthogonal) with

matrix representation P, satisfying

P’1 = PT P’1 = PT while det( P) = det( P) = 1,

and (9.36)

and the translation by a vector » with components ». Fig. 9.3 shows the rigid body

∼

motion. Variables associated with the extra rotated and translated con¬guration

are indicated with the superscript — . Because of the extra displacement as a rigid

body the current position of a material point will change from x to x— according to

x— = P · x + » = P·( x0 + u) + », (9.37)

while the displacement of the virtual con¬guration with respect to the reference

con¬guration can be written as