Consider the geometrical change of a coherent amount of material or material

fraction, for which modelling as a continuum is assumed to be permitted. The

¬rst part of the present chapter is focussed on the description of the local defor-

mation (generally coupled with rotations of the material) and along with that, the

introduction of a number of different strain measures.

Only after choosing a reference con¬guration is it possible to de¬ne deforma-

tion in a meaningful way (deformation is a relative concept). This implies that

initially the theory and the accompanying application area are related to solids.

When the material is a mixture of several material fractions, each fraction can,

with regard to local geometrical changes, in principle, be isolated from the other

fractions.

The second part of this chapter discusses geometrical changes in time. Cen-

tral concepts in this part are deformation rate and rotation velocity (spin). The

derivations in this part are not only relevant for solids, but even more important

for applications including ¬‚uids.

10.2 A material line segment in the reference and current configuration

Consider a coherent amount of material in a fully de¬ned state (the reference

con¬guration). In a material point P, with position vector x0 in the reference con-

¬guration, we focus our attention on an arbitrary in¬nitesimally small material

line segment dx0 , see Fig. 10.1.

With respect to the Cartesian xyz-coordinate system the vector dx0 can be

written as

⎡ ¤

dx0

⎢ ⎥

dx0 = dx0 ex + dy0 ey + dz0 ez and also dx0 = ⎣ dy0 ¦ . (10.1)

∼

dz0

171 10.2 The current and reference configuration

reference con¬guration

z

e0

P

current con¬guration, time t

dx0

V0 P

V (t )

x0 dx

e

x

y

x

Figure 10.1

A material line segment in reference and current con¬guration.

The orientation of the line segment dx0 is de¬ned by the unit vector e0 with

components in the column e0 . In that case it can be written:

∼

dx0 = e0 d dx0 = ∼0 d

and also e

0 0

∼

= dx0 · dx0 = dxT dx0 ,

with d (10.2)

0 ∼0 ∼

where d 0 speci¬es the length of vector dx0 .

The same material line segment, but now considered in the current con¬guration

at time t, is indicated with dx. It should be emphasized, that the line segment dx

in the current con¬guration is composed of the same material points as the line

segment dx0 in the reference con¬guration.

With respect to the Cartesian xyz-coordinate system we can write for the vector

dx:

⎡ ¤

dx

⎢ ⎥

dx = dxex + dyey + dzez and also dx = ⎣ dy ¦ . (10.3)

∼

dz

The orientation of the line segment dx is de¬ned by the unit vector e with

components in the column e. In that case it can be written:

∼

dx = ed dx = ∼d

and also e

∼

√

d = dx · dx = dxT dx,

with (10.4)

∼ ∼

where d speci¬es the length of vector dx.

Deformation and rotation, deformation rate and spin

172

Thus, the directional change (rotation) of the considered line segment, of the

current state with respect to the reference state, is described by the difference

of the unit vectors e and e0 . For the relation between the components of dx at

the (¬xed) current time t and the components of the accompanying dx0 it can be

written, using the chain rule for differentiation for (total) differentials:

‚x ‚x ‚x

dx = dx0 + dy0 + dz0

‚x0 ‚y0 ‚z0

‚y ‚y ‚y

dy = dx0 + dy0 + dz0

‚x0 ‚y0 ‚z0

‚z ‚z ‚z

dz = dx0 + dy0 + dz0 , (10.5)

‚x0 ‚y0 ‚z0

for which a Lagrangian description has been taken as the point of departure

according to: x = x( x0 , t). In a more compact form it can be formulated as

⎡ ‚x ¤

‚x ‚x

‚x0 ‚y0 ‚z0

⎢ ⎥ T

‚y ‚y ‚y

dx = F dx0 F=⎣ ¦ = ∇0 x

T

with (10.6)

‚x0 ‚y0 ‚z0 ∼∼

∼ ∼

‚z ‚z ‚z

‚x0 ‚y0 ‚z0

and in tensor notation:

T

dx = F · dx0 F = ∇0 x

with . (10.7)

The tensor F, the deformation tensor (or deformation gradient tensor), with

matrix representation F, was already introduced in Section 9.6. This tensor com-

pletely describes the (local) geometry change (deformation and rotation). After

all, when F is known, it is possible for every line segment (and therefore also for a

three-dimensional element) in the reference con¬guration, to calculate the accom-

panying line segment (or three-dimensional element) in the current con¬guration.

The tensor F describes for every material line segment the length and orientation

change: F determines the transition from d 0 to d and the transition from e0 to e.

Fig. 10.2 visualizes a uniaxially loaded bar. It is assumed, that the deformation

is homogeneous: for every material point of the bar the same deformation tensor

F is applicable. It can simply be veri¬ed, that the depicted transition from the

reference con¬guration to the current con¬guration is de¬ned by

F = »x ex ex + »y ey ey + »z ez ez , (10.8)

with the stretch ratios:

A

»x = and »y = »z = . (10.9)

A0

0

173 10.3 The stretch ratio and rotation

z

cross section: A0

0

reference con¬guration

y

x

cross section: A

current con¬guration

Figure 10.2

A uniaxially loaded bar.

10.3 The stretch ratio and rotation

Consider an in¬nitesimally small line segment dx0 in the reference con¬guration,

directed along the unit vector e0 and with length d 0 , so dx0 = e0 d 0 . To this line

segment belongs, in the current con¬guration, the line segment dx = ed directed

along the unit vector e and with length d . The mutual relation satis¬es

dx = F · dx0 , (10.10)

so

e d = F · e0 d 0 . (10.11)

The stretch ratio » is de¬ned as the ratio between d and d 0 (and therefore it will

always hold that: » > 0). Using Eq. (10.11) it can be written:

e·ed = e0 · FT · F · e0 d 2 ,

2

(10.12)

0

and consequently

» = »( e0 ) = e0 · FT · F · e0 . (10.13)

This equation can be used to determine the stretch ratio » for a material line seg-

ment with direction e0 in the reference con¬guration (Lagrangian approach). So,

for that purpose the tensor (or tensor product) FT · F has to be known. The tensor

C is de¬ned as

C = FT · F, (10.14)

Deformation and rotation, deformation rate and spin

174

and using this:

» = »( e0 ) = e0 · C · e0 . (10.15)

The tensor C is called the right Cauchy Green deformation tensor. In compo-

nent form Eq. (10.15) can be written as

» = »( ∼0 ) = e T C ∼0 ,

e e (10.16)

∼0

with

C = FT F . (10.17)

The direction change (rotation) of a material line segment can, for the transition

from the reference state to the current state, formally be stated as

F · e0

d0 1

e = F · e0 = F · e0 = √ . (10.18)

» e0 · C · e0

d

In component form this equation can be written as

d0 1 F ∼0

e

e = F ∼0 = F e0 =

e . (10.19)

»

∼ ∼

d eT C ∼0

e

∼0

Above, the current state is considered as a ˜function™ of the reference state: for

a direction e0 in the reference con¬guration, the associated direction e and the

stretch ratio » were determined. In the following the ˜inverse™ procedure is shown.

Based on

F’1 · dx = dx0 , (10.20)

so

F’1 · e d = e0 d 0 , (10.21)

and subsequently

e · F’T · F’1 · e d = e0 · e0 d 2 ,

2

(10.22)

0

it follows for the stretch ratio » = d /d that

0

1

» = »( e) = √ . (10.23)

e · F’T · F’1 · e

This equation can be used to determine the stretch ratio » for a material line ele-

ment with direction e in the current con¬guration (Eulerian description). For this,

the tensor (tensor product) F’T · F’1 has to be known. The tensor B is de¬ned

according to