z

reference con¬guration

x0

u

u*

y

x

x*

P·x

current con¬guration

»

x

extra rotated and

extra rotated con¬guration translated con¬guration

Figure 9.3

The displacement as a rigid body.

165 9.7 Extra displacement as a rigid body

u— = x— ’ x0 = P · x + » ’ x0

= P·( x0 + u) + » ’ x0 = ( P ’ I) · x0 + P · u + »

= P · x + »’( x ’ u) = ( P ’ I) · x + u + ». (9.38)

In the following, the attention is focussed on a ¬xed material point with posi-

tion vector x0 in the reference con¬guration, the position vector x in the current

con¬guration at time t and the position vector x— in the extra rotated and translated

virtual con¬guration.

For a scalar physical variable, for example the temperature T, the value will not

change as a result of an extra rigid body motion, thus, with respect to the same

material point: T — = T.

For the gradient operator, applied to a certain physical variable connected to the

material, it follows, based on the relation between x and x— , directly:

—

= P · ∇( ) = P · F’T · ∇ 0 ( ) .

∇( ) (9.39)

Note, that the gradient is the same operator for the real as well as for the imaginary,

extra displaced, current con¬guration. However, the effect on for example the

temperature ¬eld T and the ¬eld T — is different. Eq. (9.39) shows this difference.

For the deformation tensor of the virtual con¬guration with respect to the

undeformed reference con¬guration it is found that

T T

F— = ∇ 0 x— = ∇ 0 ( P · x + »)

T T

= ∇ 0 ( x · PT + ») = FT · P T

= P · F. (9.40)

Finally, the in¬‚uence of the rigid body motion on the stress state will be deter-

mined. Assume, that the internal interaction between the material particles, with

an exception for the direction, will not change because of the motion as a rigid

body. For the considered material point the stress tensor σ relates the stress vec-

tor p on a surface element with the unit normal n of that element, according to:

p = σ · n. Because the imaginary con¬guration is rotated with respect to the

current con¬guration, both vectors n— and p— can be written as

n— = P · n and p— = P · p . (9.41)

This reveals

p— = P · p = P · σ · n = P · σ · P’1 · n—

= P · σ · P T · n— , (9.42)

Motion: the time as an extra dimension

166

and so for the stress tensor σ — and the associated matrix σ — in the imaginary

con¬guration it is found that

σ — = P · σ · PT and also σ — = P σ PT (9.43)

with P the matrix representation of the tensor P.

9.8 Fluid flow

For ¬‚uids it is not common practice (and in general not very useful) to de¬ne a

reference state. This implies, that the Lagrangian description (expressing prop-

erties as a function of x0 and t) is not commonly used for ¬‚uids. Related to

this, derivatives with respect to x0 (the gradient operator ∇ 0 ) and derivatives with

respect to time under constant x0 will not appear in ¬‚uid mechanics. The defor-

mation tensor F is not relevant for ¬‚uids. However, the material time derivative

(for example to calculate the acceleration) is important nevertheless. For ¬‚uids an

Eulerian description is used, meaning that all physical properties are considered

in the current con¬guration, so as functions of x in the volume V( t) and t.

The kinematic variables that generally play a role in ¬‚uid mechanics problems

are the velocity v = v( x, t) and the acceleration a = a( x, t), both in an Eulerian

description. Their relation is given by (see the end of Section 9.4)

δv δv

T

a = ∇v ·v+ =L·v+ . (9.44)

δt δt

Figure 9.4

Streamlines in a model of a carotid artery bifurcation.

167 Exercises

Based on the velocity ¬eld in V( t) often streamlines are drawn. Streamlines are

representative for the current (at time t) direction of the velocity: the direction of

the velocity at a certain point x corresponds to the direction of the tangent to the

streamline in point x. Fig. 9.4 gives an example of streamlines in a ¬‚ow through a

constriction.

For a stationary ¬‚ow, v = v( x) and thus δv/δt = 0, the streamline pattern is

the same at each time point. In that case the material particles follow exactly the

streamlines, i.e. the particle tracks coincide with the streamlines.

Exercises

9.1 The material points of a deforming continuum are identi¬ed with the

position vectors x0 of these points in the reference con¬guration at time

∼

t = 0. The deformation (Lagrangian approach) is described with the current

position vectors x as a function of x0 and time t, according to

∼ ∼

⎡ ¤

x0 + ( a + b y0 ) t

⎢ ⎥

x( x0 , t) = ⎣ y0 + a t ¦ with a and b constant.

∼∼

z0

Determine the velocity ¬eld as a function of time in an Eulerian description,

in other words, give an expression for v = v( x, t).

∼∼

∼

9.2 Consider a ¬‚uid that ¬‚ows through three-dimensional space (with an xyz-

coordinate system). In a number of ¬xed points in space the ¬‚uid velocity

v is measured as a function of the time t. Based on these measurements

∼

it appears that in a certain time interval the velocity can be approximated

(interpolated) in the following way:

⎡ ¤

ay+bz

⎢ ⎥

1+±t

v=⎣ ¦.

0

∼

cx

1+±t

Determine, based on this approximation of the velocity ¬eld as a function

of time, the associated acceleration ¬eld as a function of time, thus: a( x , t).

∼∼

9.3 Consider a (two-dimensional) velocity ¬eld for a stationary ¬‚owing con-

tinuum:

x

vx = 2

y

1

vy =

y

with x and y spatial coordinates (expressed in [m]), while vx and vy are the

velocity components in the x- and y-direction (expressed in [ms’1 ]). The

velocity ¬eld holds for the shaded domain in the ¬gure given below.

Motion: the time as an extra dimension

168

Consider a material particle, that at time t = 0 enters the domain at the

position with coordinates x = 1 [m], y = 1 [m].

y

3

2

1

0 1 2 x

Calculate the time at which the considered particle leaves the shaded

domain.

9.4 In a Cartesian xyz-coordinate system a rigid body is rotating around the z-

axis with constant angular velocity ω. For the velocity ¬eld, in an Eulerian

description, the following expression holds:

v( x) = ·x = ω( ’ex ey + ey ex ) .

with

™

Consider, the associated acceleration ¬eld a( x) = v( x) and show that the

result can be written as: a( x) = H · x, with H a constant tensor.

Give an expression for H formulated as

H = Hxx ex ex + Hxy ex ey + · · · + Hzz ez ez .

9.5 A method that is sometimes used to study the mechanical behaviour of

.

cells is based on the so-called cross ¬‚ow experiment. An example of such

an experiment is given in the right ¬gure below (image courtesy of Mr

Patrick Anderson). In this case a ¬‚uorescent ¬broblast is positioned almost

in the centre of the cross ¬‚ow. There are several ways to create such a ¬‚ow.

The set-up, that is shown in the left ¬gure, consists of a reservoir with four

cylinders which rotate with the same angular velocity. The ¬gure gives a

top view of the set-up. A cell can be trapped in the centre of the cross ¬‚ow

and thus be stretched by the ¬‚ow.

y

Flow

x

z Flow

Flow

100 mm

Flow

(a) (b)

169 Exercises

The reservoir is assumed to be ¬lled with an incompressible ¬‚uid. Close

to the origin of the xyz-coordinate system the (stationary) two dimensional

velocity ¬eld is given by

v = c(’xex + yey ) with c a constant,

where ex and ey are unit vectors along the x- and y-axis.

™

Determine the associated acceleration ¬eld: a( x) = v( x).

In the reference con¬guration (at time t = 0) the edges (with length )

9.6

of a cubic material specimen are parallel to the axes of a Cartesian xyz-

coordinate system. See the ¬gure left. The specimen is loaded in shear,

cyclically in the time t.

y

y

t

t=0

x

x

z z

The time-dependent deformation is, in the Lagrangian approach, described

by

y0 t

x = x0 + sin 2π ,

2 T

with T the (constant) time of one cycle.

y = y0

z = z0 ,

with x0 , y0 , z0 the coordinates of the material points at time t = 0 and

with x, y, z the associated coordinates at time t. Attention is focussed on

the material point P that at time t = T/4 is located at the position x =

2 /3, y = /3, z = 0.

Determine the position vector xP of the point P as a function of the time t.

∼

10 Deformation and rotation,

deformation rate and spin

10.1 Introduction