Biomedical Engineering Reference
In-Depth Information
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, fluid mechanics problems and problems
with filtration 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 fluid mechanics problems. The last section of this chapter
discusses diffusion and filtration.
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 significant to define a reference configuration or reference state. With respect
to the reference state, the displacement field 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 field 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 fields, 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
