B C

x

In addition, it is given that the relevant stress matrix σ for the particle is

¤

⎡

52 5

⎥

⎢

σ = ⎣ 2 12 10 ¦ expressed in [MPa].

5 10 3

Determine the magnitude σt of the shear stress vector st acting on the

diagonal plane BDE.

9 Motion: the time as an

extra dimension

9.1 Introduction

Let us consider the geometrical change in time (the deformation and movement

in three-dimensional space) of a coherent amount of material or material fraction,

for which a continuum modelling approach is allowed. In case more fractions

are involved, it is in principle possible to describe the behaviour of each fraction

separately, as if it was isolated from the other fractions (however it wil be neces-

sary to include interactions between fractions). The present chapter is focussed on

a detailed description of motion. In addition, the consequences of con¬guration

changes for the formulation of physical ¬elds will be discussed. There will be no

attention to the possible causes of the motion. In the present chapter, an approach

will be followed that is common practice in the continuum description of solids

(although it can also be applied to ¬‚uids). The speci¬c aspects relevant for ¬‚uids

will be treated at the end of the chapter.

9.2 Geometrical description of the material configuration

Consider a coherent amount of material in a completely de¬ned geometrical state

(the reference con¬guration). From each material point P, that can be allocated,

the position vector x0 (with components stored in the column x0 ) is known. In

∼

the following, this position vector will be used to identify the material point. The

vector x0 is inextricably bound to the material point P, as if it were an attached

label. With respect to a Cartesian xyz-coordinate system, x0 can be written as

⎡ ¤

x0

⎢ ⎥

x0 = x0 ex + y0 ey + z0 ez and also x0 = ⎣ y0 ¦ . (9.1)

∼

z0

Because x0 is uniquely coupled to a material point, the components x0 of x0

∼

are called material coordinates. The set of position vectors x0 that address all

the material points in the con¬guration comprise the reference volume V0 , see

157 9.2 Geometrical description of the material configuration

z

reference con¬guration

V0

V (t )

P

P

x0

x = x (x0, t )

current con¬guration, time t

y

x

Figure 9.1

The position vector of a material point P.

Fig. 9.1. In an arbitrary current state, at time t, the position vector of the point P is

speci¬ed by x and can be written as

⎡¤

x

⎢⎥

x = x ex + y ey + z ez and also x = ⎣ y ¦ . (9.2)

∼

z

With the attention focussed on a certain material point, it can be stated that

x = x( x0 , t) . (9.3)

This functional relation expresses that the current position x of a material point is

determined by the material identi¬cation x0 in V0 of that point and by the current

time t. When x0 is constant and with t passing through a certain time interval,

x = x( x0 , t) can be considered to be a parameter description (with parameter t) of

the trajectory of a material point (de¬ned by x0 ) through three-dimensional space:

the path of the particle.

Differentiation of the relation x = x( x0 , t) to the time t, with x0 taken constant

(partial differentiation), results in the velocity vector v of the material point under

consideration. It can be written:

™ v = x,

™

v=x (9.4)

and also ∼

∼

with (™) the material time derivative: partial differentiation with respect to the time

t with constant x0 . For the acceleration vector a it follows:

™¨

a=v=x a=∼=x.

™¨

and also v∼ (9.5)

∼

Motion: the time as an extra dimension

158

It will be clear that, in the context of the discussion above, the following formal

relations hold for the velocity ¬eld and the acceleration ¬eld:

v = v( x0 , t) and a = a( x0 , t) . (9.6)

The con¬guration change of the material in a certain time interval can be asso-

ciated with deformation. We deal with deformation when the mutual distances

between material points change. The mathematical description of deformation and

deformation velocity is the major theme of Chapter 10.

9.3 Lagrangian and Eulerian description

In the previous section the velocity and acceleration of the material are formally

written as functions of the material identi¬cation x0 with material coordinates

x0 in V0 and the time t. Obviously, this can also be done with other physical

∼

properties associated with the material, for example the temperature T. For a

(time-dependent) temperature ¬eld it can be written

T = T( x0 , t) . (9.7)

The temperature ¬eld in the current con¬guration V( t) is mapped on the reference

con¬guration. Such a description is referred to as Lagrangian. Partial differen-

tiation to time t at constant x0 results in the material time derivative of the

™

temperature, T:

‚T

™

T= . (9.8)

‚t x0 constant

™

This variable T has to be interpreted as the change (per unit time) of the

temperature at a material point (moving through space) identi¬ed by x0 .

Another approach concentrates on a ¬xed point in three-dimensional space. At

every point in time a different material particle may be arrived at this location. For

the (time-dependent) temperature ¬eld it can be written:

T = T( x, t) , (9.9)

indicating the temperature of the material being present at time t in the spatial

point x in V( t). This alternative ¬eld speci¬cation is called Eulerian. When the

partial derivative with respect to time t of the temperature ¬eld in the Eulerian

description is determined, the spatial time derivative δT/δt is obtained:

‚T

δT

= . (9.10)

δt ‚t x constant

159 9.4 The relation between the material and spatial time derivative

This result δT/δt should be interpreted as the change (per unit time) of the tem-

perature at a ¬xed point in space x, in which at consecutive time values t different

material points will be found.

The temperature ¬eld at time t, used as an example above, can be written in

a Lagrangian description: T = T( x0 , t) and thus be mapped on the reference

con¬guration with the domain V0 . The ¬eld can also be written in an Eulerian

description: T = T( x, t) and be associated with the current con¬guration with

domain V( t). It should be noticed that a graphical representation of such a ¬eld in

both cases can be very different, especially in the case of large deformations and

large rotations (both quite common in biological applications).

In Section 9.4 we focus on the relation between the time derivatives discussed

above. In Section 9.6 the relation between gradient operators applied to both

descriptions will be discussed.

9.4 The relation between the material and spatial time derivative

For the derivation of the relation between the material and spatial time derivative

of, for example, the temperature (as an arbitrary physical state variable, associated

with the material) we start with the Eulerian description of the temperature ¬eld

T = T( x, t), in components formulated as T = T( x, t) = T( x, y, z, t). For the total

∼

differential dT it can be written:

‚T

‚T

dx +

dT = dy

‚x ‚y

y,z,t constant x,z,t constant

‚T ‚T

+ dz + dt

‚z ‚t

x,y,t constant x,y,z constant

(9.11)

and in a more compact notation, using the gradient operator (see Chapter 7):

δT

δT

dt and also dT = dxT ∇ T +

dT = dx · ∇T + dt. (9.12)

δt δt

∼∼

This equation describes the change dT of T at an arbitrary, in¬nitesimally small

change dx (with associated dx) of the location in space, combined with an

∼

in¬nitesimally small change dt in time.

Now the change dx, in the time increment dt, is chosen in such a way that the

material is followed: dx = vdt. This implies a change in temperature according to

™

dT = Tdt. Substituting this special choice in Equation (9.12), directly leads to

δT

™

Tdt = v · ∇Tdt + dt, (9.13)

δt

Motion: the time as an extra dimension

160

and thus

δT δT

™ ™

T = v · ∇T + and also T = vT ∇ T + . (9.14)

δt δt

∼∼

The ¬rst term on the right-hand side, the difference between the material derivative

and the spatial derivative, is called the convective contribution. For an arbitrary

physical variable, associated with the material the following relation between the

operators has to be applied:

δ( )

δ( )

and also ( ™) = ∼T ∇ ( ) +

( ™) = v · ∇( ) + v∼ . (9.15)

δt δt

If v = 0, in other words, if the material is not moving in three-dimensional space,

there is no difference between the material and spatial time derivative.

Applying the operator to the vector x results in an identity:

δx

™

x = v · ∇x + ’ v = v · I + 0 ’ v = v. (9.16)

δt

Of course this also holds for application to the row xT (application to the column

∼

x is not allowed in the framework of the notation used):

∼

δxT

x T = vT ∇ x T +

™ ’ ∼T = v T I + 0T ’ v T = vT .

∼

v (9.17)

δt

∼ ∼∼

∼ ∼

∼ ∼ ∼