µF (12.40)

with K the compression modulus and G the shear modulus of the material.

Summation leads to:

˜d

σ = K( J ’ 1) I + GB . (12.41)

This coupling between the stress and deformation state is often indicated as

˜compressible Neo-Hookean™ material behaviour, thus referring to the linearity.

It can be shown that Eq. (12.41) does not exactly satisfy the requirement that

in a cyclic process no energy is dissipated. It can be proven that a small (but not

trivial) modi¬cation, according to:

G ˜d

σ = K( J ’ 1) I + B (12.42)

J

does satisfy this requirement.

int0 ( t), Eq. (12.29)

Substituting Eq. (12.42) into the de¬nition equation for

yields

t

˜d

int0 ( t) = K( J ’ 1) Jtr( D) + Gtr( B · D) d„ . (12.43)

„ =0

Based on the relations given in Chapter 10 the following expressions can be

derived for the ¬rst and second term in the integrand:

.

Jtr( D) = J , (12.44)

Constitutive modelling of solids and fluids

202

and

1˜

˜d ˜

tr( B · D) = tr B ’ tr( B) I · D

3

1

= J ’2/3 tr( B · D) ’ J ’2/3 tr( B) tr( D)

3

1 ’2/3 ™ ™T

tr F · FT · F · F’1 + F · FT · F’T · F

=J

2

1

™

’ J ’5/3 J tr( B)

3

1 ’2/3 1 ™

™ ™T

tr F · FT + F · F + ( J ’2/3 ) tr( B)

=J

2 2

1 ’2/3

1 ’2/3 ™

tr B + ( J ™ ) tr( B) .

=J

2 2

(12.45)

With Eq. (12.45) the integral expression for int0 ( t) can be elaborated further:

t

1 1

K( J ’ 1)2 + GJ ’2/3 tr( B)

= . (12.46)

int0

2 2 „ =0

Using J = 1 and B = I for „ = 0 results in the current energy density int0 , which

only depends on the current left Cauchy Green tensor B( t), so it can be noted:

1 1

K( J ’ 1)2 + G J ’2/3 tr( B) ’3 ,

int0 ( B) = (12.47)

2 2

with

J = (det( B) )1/2 . (12.48)

Again, it can be established, that in int0 the volumetric and the deviatoric part are

clearly distinguishable and that both parts deliver an always positive contribution

to the energy density, for every arbitrary deformation process. Finally, it can again

be observed that a cyclic process in the deformation or in the stress will always be

energetically neutral.

More general expressions for (non-linearly) elastic behaviour can formally be

written as

˜

σ d = σ d ( J, B) .

p = p( J) , (12.49)

For a detailed speci¬cation many possibilities exist and have been published in the

scienti¬c literature. An extensive treatment for biological materials is beyond the

scope of the present discussion. For this the reader is referred to more advanced

textbooks on Biomechanics.

203 12.5 Constitutive modelling of viscous fluids

12.5 Constitutive modelling of viscous fluids

For viscous ¬‚uids, as considered in this book, in contrast to solids a reference

state is not important. Therefore an Eulerian approach is pursued. Accordingly,

the velocity ¬eld is written as

and also ∼ = ∼( x, t) .

v = v( x, t) v v∼ (12.50)

So the velocity is a function of the coordinates x, y and z, associated with a ¬xed

coordinate system in space, and the time t. The current local velocity does not

include any information with respect to deformation changes in the ¬‚uid, contrary

to the velocity gradient tensor L with matrix representation L, de¬ned in Sections

7.5 and 10.6 according to:

T T

L = ∇v and also L = ∇ vT . (12.51)

∼∼

In Section 10.6 the velocity gradient tensor L is split into the symmetrical rate

of deformation tensor D and the skew symmetric spin tensor . The tensor D is a

measure for the deformation changes. After all the tensor D determines the current

length changes of all material line segments and if D = 0, all those line segments

have a (temporarily) constant length, independent of . In components the matrix

D associated with tensor D can be written as

⎡ ¤

‚vy ‚vz

‚vx 1 ‚vx 1 ‚vx

2 ‚y + ‚x 2 ‚z + ‚x

‚x

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ 1 ‚vy ‚vx ⎥

‚vy 1 ‚vy ‚vz

D = ⎢ 2 ‚x + ‚y 2 ‚z + ‚y ⎥. (12.52)

‚y

⎢ ⎥

⎢ ⎥

⎣ ¦

‚vy

1 ‚vz 1 ‚vz ‚vz

‚vx

2 ‚x + ‚z 2 ‚y + ‚z ‚z

When comparing this elaborated expression for matrix D with the linear strain

matrix µ, see Section 10.4, it is not surprising that D is called the rate of defor-

mation matrix. Should the current state be chosen to be the reference state (in that

special case in the current state: F = I) the following relation holds: D = µ (and

™

™

in tensor notation: D = µ ).

As a constitutive equation for the behaviour of incompressible viscous ¬‚u-

ids, based on the above considerations, the following (general) expression for the

Cauchy stress tensor will be used:

σ = ’pI + σ d ( D) . (12.53)

Because of the assumption of incompressibility, the pressure p is undetermined,

while the deformation rate tensor D has to satisfy the constraint: tr( D) = 0. It

should be noted that both the Cauchy stress tensor and the deformation rate tensor

Constitutive modelling of solids and fluids

204

are objective tensors. In Sections 12.6 and 12.7 two types of constitutive behaviour

for ¬‚uids will be discussed by means of a speci¬cation of σ d ( D).

12.6 Newtonian fluids

For a Newtonian ¬‚uid the relation between the deviatoric stress tensor and the

deformation rate tensor is linear, yielding:

σ = ’pI + 2·D,

σ = ’pI + 2·D and also (12.54)

with · the viscosity (a material parameter that is assumed to be constant) of the

¬‚uid. The typical behaviour of a Newtonian ¬‚uid can be demonstrated by applying

Eq. (12.54) to two elementary examples of ¬‚uid ¬‚ow: pure shear and uniaxial

extensional ¬‚ow.

A pure shear ¬‚ow ˜in the xy-plane™ can be created with the following velocity

¬eld (in column notation):

¤

⎡ ¤ ⎡

vx y

⎥

⎢ ⎥ ⎢

v = ⎣ vy ¦ = γ ⎣ 0 ¦ ,

™ (12.55)

∼

0

vz

with γ (the shear velocity) constant. For the associated deformation rate matrix D

™

it can easily be derived that

⎡ ¤

0γ0 ™

1⎢ ⎥

D= ⎣ γ 0 0 ¦

™ (12.56)

2

000

and veri¬ed that the constraint tr( D) = tr( D) = 0 is satis¬ed. Applying the

constitutive equation it follows for the relevant shear stress σxy = σyx in the ¬‚uid:

σxy = ·γ ,

™ (12.57)

which appears to be constant. Thus, the viscosity can be interpreted as the

˜resistance™ of the ¬‚uid against ˜shear™ (shear rate actually).

To a uniaxial extensional ¬‚ow (incompressible) in the x-direction the following

deformation rate matrix is applicable:

¤

⎡

™ 0 0

⎥

⎢

D = ⎣ 0 ’ ™ /2 0 ¦, (12.58)

’ ™ /2

0 0

with ™ (the rate of extension) constant. For the stress matrix it is immediately

found that

205 12.8 Diffusion and filtration

¤

⎡

™ 0 0

⎥

⎢

σ = ’p I + 2· ⎣ 0 ’ ™ /2 0 ¦. (12.59)

’ ™ /2

0 0

Assuming, that in the y- and in the z-direction the ¬‚ow can develop freely: σyy =

σzz = 0 and with that it follows for the hydrostatic pressure:

p = ’· ™ . (12.60)

This leads to the required uniaxial stress for the extensional ¬‚ow:

σxx = 3· ™ . (12.61)

The fact that the effective uniaxial extensional viscosity (3·) is three times as high

as the shear viscosity (·) is known as Trouton™s law.

12.7 Non-Newtonian fluids

For a non-Newtonian incompressible viscous ¬‚uid the constitutive equation has

the same form as the equation for a Newtonian ¬‚uid:

σ = ’pI + 2·D σ = ’pI + 2·D,

and also (12.62)

however, the viscosity · is now a function of the deformation rate tensor: · =

·( D). Speci¬cation of the relation for the viscosity has to be based on experimen-

tal research. Here we limit ourselves to an example, the three parameter ˜power

law™ model (with the temperature in¬‚uence T according to Arrhenius):

(n’1)

· = me(A/T) 2 tr( D · D) , (12.63)

with m, A and n material constants (for n = 1 the viscosity is independent of D and

the behaviour is ˜Newtonian™ again). Substitution of the deformation rate tensor

for pure shear (see previous section) leads to

· = m e(A/T) |γ |(n’1) ,

™ (12.64)

making the mathematical format of the equation more transparent.

12.8 Diffusion and filtration

Although somewhat beyond the scope of the present chapter in this last section

material ¬‚ow due to diffusion or ¬ltration is considered, i.e. transport of mate-

rial through a stationary porous medium (no convection). The ¬‚owing material

can indeed be considered to be a continuum, but the constitutive equations are

Constitutive modelling of solids and fluids

206

completely different from those treated before. Here, the constitutive equations

describe the transport of material depending on the driving mechanisms, while

above the constitutive equations related the characteristics of the ¬‚ow to the

internal stresses.

Diffusion of a certain material through a porous medium is generated by con-

centration differences of the material in the medium. The material will in general

strive for a homogeneous density distribution (provided that the porous medium

is homogeneous) implying that material will ¬‚ow from regions with a high con-

centration to regions with a low concentration. The mathematical form for this

phenomenon is given by Fick™s law:

ρv = ’D∇ρ, (12.65)

with on the left-hand side the mass ¬‚ux vector (also see Section 7.5) and on the

right-hand side the driving mechanism for transport, ∇ρ, multiplied with a certain

factor D. This factor D is called the diffusion coef¬cient, and can be considered

to be a constitutive parameter, that is determined by the combination of materials

(and the temperature).