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Applying the operation to v (the velocity vector), leads to
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

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( 67 .)



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