force is formed by pressure differences in that material. The material strives for an

equal pressure, so in the presence of pressure differences, a ¬‚ow will occur from

areas with a high pressure to areas with a lower pressure. This is expressed by

Darcy™s law:

ρv = ’κ ∇p (12.66)

with the permeability κ as a constitutive parameter.

Exercises

12.1 Consider a cubic material element of which the edges are oriented in the

direction of a Cartesian xyz-coordinate system, see the ¬gure. In the ¬gure

also the normal and shear stresses are depicted (expressed in [kPa]) acting

z

1

2

3

3

3

y 2

1

x

207 Exercises

on the (visible) faces of the element. The mechanical behaviour of the ele-

ment is described by the linear Hooke™s law, with Young™s modulus E = 8

[MPa] and Poisson™s ratio ν = 1/3.

Determine the volume change V/V0 , with V0 the volume of the element

in the unloaded reference con¬guration and V the volume in the current

loaded con¬guration, assuming small deformations.

12.2 An element of an incompressible material (in the reference state a cube:

— — ), is placed in a Cartesian xyz-coordinate system as given in the

¬gure below. Because of a load in the z-direction the height of the element

is reduced to 2 /3. In the x-direction the displacement is suppressed. The

element can expand freely in the y-direction. The deformation is assumed

to be homogeneous.

The material behaviour is described by a Neo-Hookean relation, according

to:

σ = ’pI + GBd ,

with σ the stress matrix, p the hydrostatic pressure (to be determined), I the

unit matrix, G the shear modulus and B the left Cauchy Green deformation

matrix.

z z

y

y

2 /3

3 /2

x

x

Determine the compressive force Fv in z-direction that is necessary to

realize this deformation.

12.3 A frequently applied test to determine the stiffness properties of biolog-

ical materials is the ˜con¬ned compression test™. A schematic of such a

test is given in the ¬gure. A cylindrical specimen, Young™s modulus E,

Kν

h

R

Constitutive modelling of solids and fluids

208

(cylindrically shaped with a circular cross section with radius R, height h)

of the material is placed in a tight ¬tting die, that can be regarded as a rigid

mould with a smooth inner wall. The height of the specimen is reduced

with a value δ (with δ h) by means of an indenter. A vertical force Kv

is acting on the indenter in the direction of the arrow. It can be assumed

that the deformation of the specimen is homogeneous. As the strains are

small, the material behaviour is described by the linear Hooke™s law. The

Poisson™s ratio is known (measured in another experiment): ν = 1/4.

Express the Young™s modulus E of the material in the parameters R, h, δ

and Kv (i.e. in the parameters that can easily be measured).

12.4 Because con¬ned compression tests have several disadvantages, uncon-

¬ned compression tests are also performed frequently. Again a cylindrical

specimen is used. In this case the specimen has in the reference state a

radius R0 = 2.000 [cm] and thickness h0 = 0.500 [cm]. It is compressed to

h = 0.490 [cm]. Now, the specimen can expand freely in a radial direction.

In the deformed state: R = 2.008 [cm].

R0 = 2.000 [cm] R = 2.008 [cm]

h0 = 0.500 [cm] h = 0.490 [cm]

The material behaves linearly elastic according to Hooke™s law, character-

ized by Young™s modulus E and Poisson™s ratio ν.

Calculate the Poisson™s ratio ν based on the above described experiment.

12.5 During a percutaneous angioplasty the vessel wall is expanded by in¬‚at-

ing a balloon. At a certain moment during in¬‚ation the internal pressure

pi = 0.2 [MPa] and the internal radius Ri of the vessel is increased by

10%. Assume that the length of the balloon and the length of the ves-

sel wall in contact with the balloon do not change. Further it is known,

that the wall behaviour can be modelled according to Hooke™s law, with

Young™s modulus of the wall E = 8 [MPa] and the Poisson™s ratio of the

wall ν = 1/3.

Ri

pi

ey

ey

A

ez ex

209 Exercises

(a) Calculate the strain in circumferential direction at the inner side of

the wall.

(b) Calculate the strain in the ex -direction at point A at the inner side of

the vessel wall.

(c) Calculate the stress in circumferential direction at point A.

12.6 Consider a bar (length 2L) with a circular cross section (radius R). The

axis of the bar coincides with the z-axis of a xyz-coordinate system. With

respect to the mid plane (z = 0) the top and bottom plane of the bar are

rotated around the z-axis with an angle ± (with ± 1), thus loading the

bar with torsion. See the ¬gure.

z

L

y

L

x

The position vector x of a material point in the deformed con¬guration is

±( x0 · ey ) ( x0 · ez ) ±( x0 · ex ) ( x0 · ez )

x = x0 ’ ex + ey ,

L L

with x0 the position vector of that point in the undeformed state. The mate-

rial behaviour is linearly elastic according to Hooke™s law, with Young™s

modulus E and Poisson™s ratio ν.

Determine, for the material point de¬ned with x0 = Rey , the linear strain

tensor µ, the stress tensor σ and the equivalent stress, according to von

Mises σ M .

13 Solution strategies for solid

and fluid mechanics problems

13.1 Introduction

The goal of the present chapter is to describe a procedure to formally determine

solutions for solid mechanics problems, ¬‚uid mechanics problems and problems

with ¬ltration and diffusion. Mechanical problems in biomechanics can be very

diverse and most problems are so complex, that it is impossible to derive ana-

lytical solutions and often very complicated to determine numerical solutions.

Fortunately, in most cases it is not necessary to describe all phenomena related to

the problem in full detail and simplifying assumptions can be made, thus reducing

the complexity of the set of equations that have to be solved. The present chapter

deals with formulating problems and solution strategies, starting from the most

general set of equations and gradually reducing the generality by imposing sim-

plifying assumptions. In Section 13.2 this will be done for solids. Section 13.3

is devoted to solving ¬‚uid mechanics problems. The last section of this chapter

discusses diffusion and ¬ltration.

13.2 Solution strategies for deforming solids

In this section it is assumed that the material (or material fraction) to be considered

can be modelled as a deforming solid continuum. This implies that it is possible

and signi¬cant to de¬ne a reference con¬guration or reference state. With respect

to the reference state, the displacement ¬eld as a function of time supplies a full

description of the deformation process to which the continuum is subjected (at

least under the restrictions given in previous chapters, such as for example a con-

stant temperature). After all, for a displacement ¬eld that is known as a function

of time, it is possible to directly calculate the local deformation history (applying

the kinematics, see Chapter 10) and subsequently, the stress state as a function of

time (using the constitutive equations, see Chapter 12). The relevant ¬elds, from a

mechanical point of view, that are obtained in this way for the continuum have to

satisfy the balance equations (see Chapter 11). In addition, the initial conditions

211 13.2 Solution strategies for deforming solids

have to be ful¬lled: at the beginning of the process the positions of the material

points in space have to be prescribed, in general in accordance with the reference

con¬guration, and also the initial velocities of the material points have to be in

agreement with the speci¬cation of the initial state. In addition, during the entire

process the boundary conditions have to be satis¬ed: along the outer surface of

the continuum the displacement ¬eld and stress ¬eld have to be consistent with

the process speci¬cation.

The goal of the present section is to outline a procedure to formally deter-

mine (as a function of time) the displacement ¬eld, such that all requirements

are satis¬ed. For (almost) no realistic problem exact analytical solutions can be

found, not even when the mathematical description is drastically simpli¬ed via

assumptions. A global description will be given of strategies to derive approximate

solutions.

In Section 13.2.1 the general (complete) formulation of the problem will

be given. Subsequently, in the sections that follow, the generality will grad-

ually be limited. Initially, in Section 13.2.2 the general description will be

restricted with respect to the magnitude of the displacements, deformations

and rotations (geometrically linear behaviour). In Section 13.2.3 the restric-

tion to linearly elastic behaviour follows (physical linearity), leading to the

set of equations, establishing the so-called ˜linear elasticity theory™. In Sec-

tion 13.2.4 processes are considered for which inertia effects can be neglected

(quasi-static processes). Then time (and thus also process rate) is no longer

relevant. In Section 13.2.5 the attention is concentrated on con¬gurations that,

because of geometry and external loading, can be regarded as two-dimensional

(plane stress theory). Finally, Section 13.2.6 is focussed on formulating bound-

ary conditions for continuum problems. Sometimes extra constitutive equations

are necessary to describe the interaction of the considered continuum with the

environment.

13.2.1 General formulation for solid mechanics problems

For general solid mechanics problems addressing the deformation process in a

Lagrange description, the following ¬elds have to be determined:

• the displacement ¬eld: u( x0 , t) for all x0 in V0 and for all t and

• the stress ¬eld: σ ( x0 , t) for all x0 in V0 and for all t.

These ¬elds have to be connected for all x0 in V0 in accordance with the local

constitutive equation (see Chapter 12):

σ ( x0 , t) = F{F( x0 , „ ) ; „ ¤ t}, (13.1)

Solution strategies for solid and fluid mechanics problems

212

with

T

F = I + ∇ 0u , (13.2)

re¬‚ecting history-dependent material behaviour. In addition the local balance of

momentum has to be satis¬ed (see Chapter 11), as well as the local mass balance:

¨

F’T · ∇ 0 · σ + ρq = ρ u, (13.3)

with

ρ0

ρ= . (13.4)

det( F)

The equations given above form a set of non-linear, coupled partial differential

equations (derivatives with respect to the three material coordinates in x0 and the

time t are dealt with). Consequently, for a unique solution of the displacement ¬eld

u( x0 , t) and the stress ¬eld σ ( x0 , t) boundary conditions and initial conditions are

required. With respect to the boundary conditions, for all t at every point of the

outer surface of V0 three (scalar) relations have to be speci¬ed: either completely

formulated in stresses (dynamic or natural boundary conditions), completely for-

mulated in displacements (kinematic or essential boundary conditions) or in mixed

formulations. With respect to the initial conditions, at the initial time point ( t = 0),

for all the points in V0 , the displacement and velocity have to be speci¬ed. If the

initial state is used as the reference con¬guration, u( x0 , t = 0) = 0 for all x0 in V0 .

13.2.2 Geometrical linearity

Provided that displacements, strains and rotations are small (so F ≈ I and conse-

quently det( F) ≈ 1) the general set for solid continuum problems as presented in

the previous section can be written as