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в€‚z в€‚z в€‚z
The associated matrix representation can be written as
вЋЎ вЋ¤
в€‚vx в€‚vy в€‚vz
вЋў в€‚x в€‚x вЋҐ
в€‚x
вЋў вЋҐ
вЋў в€‚vx в€‚vy в€‚vz вЋҐ
в€‡ vT = вЋў вЋҐ. (7.17)
вЋў в€‚y в€‚y вЋҐ
в€јв€ј
в€‚y
вЋў вЋҐ
вЋЈ в€‚vx в€‚vy в€‚vz вЋ¦
в€‚z в€‚z в€‚z
It should be noticed, that with the current notation an operation of the type в€‡ v is
в€јв€ј
not allowed.
The transposed form of the tensor в€‡v is called the velocity gradient tensor L:
T
L = в€‡v . (7.18)
The associated matrix representation reads:
T
L = в€‡v T
. (7.19)
в€јв€ј

Fig. 7.10 gives an illustration of the application of this tensor (matrix). Consider
an inп¬Ѓnitesimally small material line element dx (with components dx) in volume
в€ј
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 В· в€‡v = v+( в€‡v)T В·dx = v + L В· dx.

z (П… + dx В· в€‡П… )dt = (П… + L В· dx )dt
dx + dxdt

П…dt

dx so: dx = L В· 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 inп¬Ѓnitesimally small increase in time dt the material line element dx
Л™
will change into dx + dxdt. By means of the п¬Ѓgure it can be proven that:
Л™ Л™
dx = L В· dx and also dx = L dx, (7.20)
в€ј в€ј

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 п¬Ѓelds) of the velocity,
div(v), deп¬Ѓned as в€‡ В· v can be written as
в€‚ в€‚ в€‚
+ ey + ez В·( vx ex + vy ey + vz ez )
div( v) = ex
в€‚x в€‚y в€‚z
в€‚vy
в€‚vx в€‚vz
= + +
в€‚x в€‚y в€‚z
= tr( L) , (7.21)
and also
в€‚vy в€‚vz
в€‚vx
+ + = tr( в€‡ в€јT ) = в€‡ T в€ј = tr( L) .
div( v) = v v (7.22)
в€‚x в€‚y в€‚z в€ј в€ј

For the sake of completeness the following rather trivial result for the gradient в€‡
applied to the position vector x is given:
в€‚x в€‚y в€‚z
в€‡x = ex ex + ex ey + ex ez
в€‚x в€‚x в€‚x
в€‚x в€‚y в€‚z
+ ey ex + ey ey + ey ez
в€‚y в€‚y в€‚y
в€‚x в€‚y в€‚z
+ ez ex + ez ey + ez ez
в€‚z в€‚z в€‚z
= ex ex + ey ey + ez ez = I, (7.23)
and the according matrix formulation yields
вЋЎ вЋ¤
в€‚x в€‚y в€‚z
вЋў в€‚x в€‚x вЋҐ
в€‚x
вЋў вЋҐ
вЋў в€‚x в€‚y вЋҐ
в€‚z
в€‡x =вЋў вЋҐ = I.
T
(7.24)
вЋў в€‚y в€‚y вЋҐ
в€јв€ј
в€‚y
вЋў вЋҐ
вЋЈ в€‚x в€‚y вЋ¦
в€‚z
в€‚z в€‚z в€‚z

7.6 Rigid body rotation

In the present section it is assumed that the mass in the rigid volume V rotates
around a п¬Ѓxed axis in three-dimensional space. We focus our attention on the
Biological materials and continuum mechanics
126

П… = П‰ Г— (x в€’ xS)
z
П‰

x

y

xS S
x

Figure 7.11
Rotation of material.

velocity п¬Ѓeld 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
deп¬Ѓned by means of a point S, п¬Ѓxed in space, with position vector xS and the
constant angular velocity vector П‰. The associated columns with the components
with respect to the Cartesian basis can be written as

вЋЎ вЋ¤ вЋЎ вЋ¤
П‰x
xS
вЋў вЋҐ вЋў вЋҐ
вЋў вЋҐ вЋў вЋҐ
вЋў вЋў
вЋҐ вЋҐ
xS = вЋў yS вЋҐ , П‰ = вЋў П‰y вЋҐ . (7.25)
вЋў вЋў
вЋҐ вЋҐ
в€ј в€ј
вЋЈ вЋЈ
вЋ¦ вЋ¦
П‰z
zS

The velocity vector v at a certain point with position vector x satisп¬Ѓes

v = v( x) = П‰ Г— ( x в€’ 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

вЋ¤
вЋЎ вЋ¤ вЋЎ
vx x
вЋўвЋҐ
вЋў вЋҐ
вЋўвЋҐ
вЋў вЋҐ
вЋўвЋҐ
вЋў вЋҐ
v = вЋў vy вЋҐ and x = вЋў y вЋҐ , (7.27)
вЋўвЋҐ
вЋў вЋҐ в€ј
в€ј
вЋЈвЋ¦
вЋЈ вЋ¦
z
vz
127 7.7 Mathematical preliminaries on tensors

are related according to
вЋЎ вЋ¤
в€’П‰z П‰y
0
вЋў вЋҐ
вЋў вЋҐ
вЋў вЋҐ
v = v( x ) = ( x в€’ xS ) with = вЋў П‰z в€’П‰x вЋҐ . (7.28)
0
вЋў вЋҐ
в€јв€ј в€ј в€ј
в€ј
вЋЈ вЋ¦
в€’П‰y П‰x 0
that is associated with the angular velocity vector П‰ (with
The spin matrix
components П‰) is skew symmetric: T = в€’ . With the uniform rotation as a
в€ј
rigid body considered in this section, we п¬Ѓnd:
T T T
L = в€‡ в€јT = в€‡ ( x в€’ xS )T = ( I + O) =
T T
v (7.29)
в€ј в€ј в€ј в€ј

and for the associated velocity gradient tensor L:
L = П‰x ( ez ey в€’ ey ez ) +П‰y ( ex ez в€’ ez ex ) +П‰z ( ey ex в€’ 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 deп¬Ѓned as
вЋЎ вЋ¤
Mxx Mxy Mxz
вЋў вЋҐ
вЋў вЋҐ
вЋў вЋҐ
M = вЋў Myx Myy Myz вЋҐ . (7.32)
вЋў вЋҐ
вЋЈ вЋ¦
Mzx Mzy Mzz

A tensor identiп¬Ѓes a linear transformation. If the vector b is the result of the tensor
M operating on vector a, this is written as: b = M В· a. In component form this
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 ) В· ( 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:
в€ј в€ј
вЋЎ вЋ¤вЋЎ вЋ¤вЋЎ вЋ¤
bx Mxx Mxy Mxz ax
вЋў вЋҐвЋў вЋҐвЋў вЋҐ
вЋў вЋҐвЋў вЋҐвЋў вЋҐ
вЋў вЋҐвЋў вЋҐвЋў вЋҐ
вЋў by вЋҐ = вЋў Myx Myy Myz вЋҐ вЋў ay вЋҐ
вЋў вЋҐвЋў вЋҐвЋў вЋҐ
вЋЈ вЋ¦вЋЈ вЋ¦вЋЈ вЋ¦
bz Mzx Mzy Mzz az
вЋ¤
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