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

$S=a\otimes b$

with the properties

$\begin{array}{l}S\cdot u=\left(a\otimes b\right)\cdot u=a\left(b\cdot u\right)\\ u\cdot S=u\cdot \left(a\otimes b\right)=\left(a\cdot u\right)b\end{array}$

for all vectors u

Products of dyadic products: Let a, b, c, d be four vectors, and let $S=\left(a\otimes b\right)$, $T=\left(c\otimes d\right)$. The products of $S$ and $T$ are expressed in Dyadic notation as follows

$\begin{array}{l}S\cdot T=\left(a\otimes b\right)\cdot \left(c\otimes d\right)=\left(b\cdot c\right)a\otimes d\\ S:T=\left(a\cdot c\right)\left(b\cdot d\right)\\ S\cdot \cdot T=\left(a\cdot d\right)\left(b\cdot c\right)\end{array}$

Representing a general tensor as a sum of dyadic products: Let ${a}_{i}$ (with i=1,2,3) be three linearly independent vectors.  Then any tensor $S$ can be expressed as a linear combination of nine dyadic products of these vectors

$S={S}_{ij}{a}_{i}\otimes {a}_{j}$

where summation on i and j is assumed.