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In case of ¬ltration of material through a stationary porous medium the driving
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,







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

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