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Chapter 10
Approximate
theories for solids with special shapes:
rods, beams, membranes, plates and shells
The
full field equations of solid mechanics describe deformation by specifying
the motion of every material particle within the solid; and describe internal
forces by means of the three dimensional distribution of Cauchy stress within
the material. As we have seen, it can
be difficult to find solutions to the full three dimensional equations. For
solids with certain special shapes, it is sometimes possible to simplify the
governing equations by approximating the variation of strain or stress in the
solid. For example, you may be
familiar with the Euler-Bernoulli equations governing the elastic deflections
of beams. In this theory, the strain
and stress distribution in the beam is completely determined by the
deflection of the neutral axis of the beam.
Consequently, the equations of equilibrium for the solid can be
reduced to a system of ordinary differential equations for the beam deflections.
Similar approaches may be used to construct
approximate solutions to many important problems. In this chapter, we will discuss (i) the
deformation of slender rods, and the simpler special case of a straight beam;
and (ii) the deformation of a thin three-dimensional shell, and the simpler
cases of membranes and flat plates. The theories outlined in this chapter
were originally developed in the early part of the last century to help in
mechanical vibration and structural design calculations. They are still used for this purpose
today, but are also used to model biological systems, such as bacterial
flagellae or cell walls; and also to model nano-scale structures such as
carbon nanotubes.
A number of different approaches are used to derive
approximations to the three dimensional field equations of solid
mechanics. Here, we will use a
systematic procedure based on the principle of virtual work. In this approach, we begin by approximating
the displacement field in the solid in some sensible way. The goal is to describe the 3D displacement
field in terms of a reduced set of kinematic variables. The approximate displacement field is then
substituted into the virtual work equation, which yields generalized force
measures that are conjugate to the kinematic variables, together with a set
of equilibrium equations and boundary conditions for these forces.
HEALTH
WARNING: In previous chapters we
have been able to express all vector and tensor quantities as Cartesian
components in a fixed basis. We can continue
to use this notation for some of the simpler problems considered in this
chapter, such as modeling the behavior of a straight beam or a flat plate.
For more general problems, such as
twisted rods, or shells with a complicated geometry, this notation is
inadequate. Most of this chapter uses
more generalized notation, therefore.
The theory of deformable rods describes vectors and tensors as
components in a basis consisting of three mutually perpendicular unit vectors
which are aligned with the rod, and rotate with the rod as it deforms. Shell theory uses a general basis of
non-orthogonal unit vectors. It can be
a struggle to master the complicated notation associated with curvilinear
coordinates, but it is worth making the effort. You may find it helpful to work through
rod theory first, which will get you used to expressing vectors and tensor in
a position dependent basis, and then move onto shell theory, which adds the
additional complexity of non-orthogonal coordinates. If you would prefer to avoid both,
simplified theories for straight beams and flat plates are given in Cartesian
coordinates.
10.1 Preliminaries: Diadic notation for vectors and
tensors
All the theories developed in this section involve
solids which are subjected to large shape changes. In most cases, the material is only
slightly distorted, but because of the component’s geometry, these shape
changes can lead to large displacements and rotations in the solid.
When solving problems like this, it is often
convenient to express the various vector and tensor measures of deformation
and force as components in two different sets of basis vectors, with one set
associated with the undeformed solid, and the other associated with the
deformed solid. In some cases we even to resort to using both bases at the
same time. Dyadic notation is a
convenient way to express these ideas.
To follow dyadic notation, you need only to
understand and remember three basic ideas:
 Dyadic
product of vectors Let a and b be two vectors. The
dyadic product of a and b is a second order tensor S denoted by
with the properties
for
all vectors u.
Products of dyadic products: Let a, b, c, d be four vectors, and let ,
.
The products of and are expressed in Dyadic notation as follows
Representing a general tensor as a sum of dyadic
products: Let (with i=1,2,3) be three linearly independent
vectors. Then any tensor can be expressed as a linear combination of
nine dyadic products of these vectors
where summation on i and j is assumed.
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