The associated matrix representation can be written as

⎡ ¤

‚vx ‚vy ‚vz

⎢ ‚x ‚x ⎥

‚x

⎢ ⎥

⎢ ‚vx ‚vy ‚vz ⎥

∇ vT = ⎢ ⎥. (7.17)

⎢ ‚y ‚y ⎥

∼∼

‚y

⎢ ⎥

⎣ ‚vx ‚vy ‚vz ¦

‚z ‚z ‚z

It should be noticed, that with the current notation an operation of the type ∇ v is

∼∼

not allowed.

The transposed form of the tensor ∇v is called the velocity gradient tensor L:

T

L = ∇v . (7.18)

The associated matrix representation reads:

T

L = ∇v T

. (7.19)

∼∼

Fig. 7.10 gives an illustration of the application of this tensor (matrix). Consider

an in¬nitesimally small material line element dx (with components dx) in volume

∼

V at time t. If, at the tail of the vector dx the velocity is v, then at the head of the

vector dx the velocity will be: v + dx · ∇v = v+( ∇v)T ·dx = v + L · dx.

z (… + dx · ∇… )dt = (… + L · dx )dt

dx + dxdt

…dt

dx so: dx = L · dx

x y

O

x

Figure 7.10

Change of a material line element dx after a time increment dt.

125 7.6 Rigid body rotation

After an in¬nitesimally small increase in time dt the material line element dx

™

will change into dx + dxdt. By means of the ¬gure it can be proven that:

™ ™

dx = L · dx and also dx = L dx, (7.20)

∼ ∼

and so the tensor L (with matrix representation L) is a measure for the current

change (per unit of time) of material line elements. This holds for the length as

well as for the orientation.

The divergence (an operator that is often used for vector ¬elds) of the velocity,

div(v), de¬ned as ∇ · v can be written as

‚ ‚ ‚

+ ey + ez ·( vx ex + vy ey + vz ez )

div( v) = ex

‚x ‚y ‚z

‚vy

‚vx ‚vz

= + +

‚x ‚y ‚z

= tr( L) , (7.21)

and also

‚vy ‚vz

‚vx

+ + = tr( ∇ ∼T ) = ∇ T ∼ = tr( L) .

div( v) = v v (7.22)

‚x ‚y ‚z ∼ ∼

For the sake of completeness the following rather trivial result for the gradient ∇

applied to the position vector x is given:

‚x ‚y ‚z

∇x = ex ex + ex ey + ex ez

‚x ‚x ‚x

‚x ‚y ‚z

+ ey ex + ey ey + ey ez

‚y ‚y ‚y

‚x ‚y ‚z

+ ez ex + ez ey + ez ez

‚z ‚z ‚z

= ex ex + ey ey + ez ez = I, (7.23)

and the according matrix formulation yields

⎡ ¤

‚x ‚y ‚z

⎢ ‚x ‚x ⎥

‚x

⎢ ⎥

⎢ ‚x ‚y ⎥

‚z

∇x =⎢ ⎥ = I.

T

(7.24)

⎢ ‚y ‚y ⎥

∼∼

‚y

⎢ ⎥

⎣ ‚x ‚y ¦

‚z

‚z ‚z ‚z

7.6 Rigid body rotation

In the present section it is assumed that the mass in the rigid volume V rotates

around a ¬xed axis in three-dimensional space. We focus our attention on the

Biological materials and continuum mechanics

126

… = ω — (x ’ xS)

z

ω

x

y

xS S

x

Figure 7.11

Rotation of material.

velocity ¬eld v = v( x) and on the velocity gradient tensor L that can be derived

from it. Figure 7.11 illustrates the considered problem. The axis of rotation is

de¬ned by means of a point S, ¬xed in space, with position vector xS and the

constant angular velocity vector ω. The associated columns with the components

with respect to the Cartesian basis can be written as

⎡ ¤ ⎡ ¤

ωx

xS

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎢

⎥ ⎥

xS = ⎢ yS ⎥ , ω = ⎢ ωy ⎥ . (7.25)

⎢ ⎢

⎥ ⎥

∼ ∼

⎣ ⎣

¦ ¦

ωz

zS

The velocity vector v at a certain point with position vector x satis¬es

v = v( x) = ω — ( x ’ xS ) . (7.26)

It can be derived that the components of the velocity vector v and the spatial

coordinate vector x, stored in the columns

¤

⎡ ¤ ⎡

vx x

⎢⎥

⎢ ⎥

⎢⎥

⎢ ⎥

⎢⎥

⎢ ⎥

v = ⎢ vy ⎥ and x = ⎢ y ⎥ , (7.27)

⎢⎥

⎢ ⎥ ∼

∼

⎣¦

⎣ ¦

z

vz

127 7.7 Mathematical preliminaries on tensors

are related according to

⎡ ¤

’ωz ωy

0

⎢ ⎥

⎢ ⎥

⎢ ⎥

v = v( x ) = ( x ’ xS ) with = ⎢ ωz ’ωx ⎥ . (7.28)

0

⎢ ⎥

∼∼ ∼ ∼

∼

⎣ ¦

’ωy ωx 0

that is associated with the angular velocity vector ω (with

The spin matrix

components ω) is skew symmetric: T = ’ . With the uniform rotation as a

∼

rigid body considered in this section, we ¬nd:

T T T

L = ∇ ∼T = ∇ ( x ’ xS )T = ( I + O) =

T T

v (7.29)

∼ ∼ ∼ ∼

and for the associated velocity gradient tensor L:

L = ωx ( ez ey ’ ey ez ) +ωy ( ex ez ’ ez ex ) +ωz ( ey ex ’ ex ey ) . (7.30)

7.7 Some mathematical preliminaries on second-order tensors

In the chapters that follow, extensive use will be made of second-order tensors.

This section will summarize some of the mathematical background on this subject.

An arbitrary second-order tensor M can be written with respect to the Cartesian

basis introduced earlier as

M = Mxx ex ex + Mxy ex ey + Mxz ex ez

+ Myx ey ex + Myy ey ey + Myz ey ez

+ Mzx ez ex + Mzy ez ey + Mzz ez ez . (7.31)

The components of the tensor M are stored in the associated matrix M de¬ned as

⎡ ¤

Mxx Mxy Mxz

⎢ ⎥

⎢ ⎥

⎢ ⎥

M = ⎢ Myx Myy Myz ⎥ . (7.32)

⎢ ⎥

⎣ ¦

Mzx Mzy Mzz

A tensor identi¬es a linear transformation. If the vector b is the result of the tensor

M operating on vector a, this is written as: b = M · a. In component form this

leads to:

b =( Mxx ex ex + Mxy ex ey + Mxz ex ez

+ Myx ey ex + Myy ey ey + Myz ey ez

+ Mzx ez ex + Mzy ez ey + Mzz ez ez ) · ( ax ex + ay ey + az ez )

Biological materials and continuum mechanics

128

= ( Mxx ax + Mxy ay + Mxz az ) ex

+ ( Myx ax + Myy ay + Myz az ) ey

+ ( Mzx ax + Mzy ay + Mzz az ) ez

= bx e x + b y e y + b z e z . (7.33)

Using matrix notation we can write: b = M a, in full:

∼ ∼

⎡ ¤⎡ ¤⎡ ¤

bx Mxx Mxy Mxz ax

⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥

⎢ by ⎥ = ⎢ Myx Myy Myz ⎥ ⎢ ay ⎥

⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ¦⎣ ¦⎣ ¦

bz Mzx Mzy Mzz az

¤