Biomedical Engineering Reference
In-Depth Information
2.3.1 Classical Theory of Elasticity
In 1821 Navier formulated the basic equations of the theory and Cauchy
developed the theory of stress and strain. These were investigated by Saint
Venant who used them to solve simple problems of torsion and bending of
prismatic bars [3]. As a result of this work Saint Venant formulated the famous
Saint Venant principle which allowed for the solution of some practical problems
using the theory. The work of these pioneers and others was followed by work by
many others leading to the development of the field of applied mechanics which
expanded the application of the theory to practical problems.
Unfortunately, the theory establishes a set of simultaneous partial differential
equations, constrained by boundary conditions, which cannot, in general, be
solved in closed form for the kind of stress determination needed in engineering
and particularly in orthopaedic devices. Work in the field of applied mechanics in
the early 20
th
century, however, developed numerical methods for estimating
stresses and strains. These methods, although highly labor intensive, were
nevertheless used in aircraft design where more accurate estimates were needed to
keep weight to a minimum. Such computations often required thousands of man
hours to complete.
2.3.1.1 Finite Element Analysis (FEA)
FEA was first introduced in 1943 by Courant using the Ritz numerical method and
variational calculus to develop approximate solutions to a class of vibration
problems. In 1956, Turner et al expanded the methodology to include the
deflection of complex structures [6]. Work over the last half century has greatly
expanded the application of FEA and greatly simplified its use.
Linear FEA stress analysis of mechanical parts is now an integral part of most
high end computer aided design (CAD) software packages. Although nonlinear
FEA analysis is available it is of little practical use in design since it is much more
computer intensive and since, in any event, stress in the nonlinear region of almost
all materials is to be avoided if failure is to be avoided.
FEA may be used to analyze a mechanical part by creating a digital 3D solid,
computer model of the part and then defining a mesh used to approximate the
behavior of the part under the expected loading conditions. Such a mesh is shown
in Fig. 2.2.
Such a mesh consists of a number of points called “nodes”. Elements are the
material enclosed by the boundary lines connecting the nodes. To perform the
analysis an appropriate mesh is first generated with regions of expected high stress
and stress concentration using a greater node density. Rigid body constraints are
placed on the motion of those nodes where the part is attached to simulate its
attachment and forces are placed on appropriate nodes to simulate the expected
loading. The resulting problem is then solved computing the approximate stress,
and if desired strain, or deformation, at each node. The results are then presented,
usually in graphical form to allow easy location of the largest stress and their
values.
