Page 39 - Aircraft Stuctures for Engineering Student

24 Basic elasticity

In the preceding sections we have developed, for a three-dimensional deformable
body, three equations of equilibrium [Eqs (1.5)] and six strain-displacement relation-
ships [Eqs (1.18) and (1.20)]. From the latter we eliminated displacements thereby
deriving six auxiliary equations relating strains. These compatibility equations are
an expression of the continuity of displacement which we have assumed as a pre-
requisite of the analysis. At this stage, therefore, we have obtained nine independent
equations towards the solution of the three-dimensional stress problem. However, the
number of unknowns totals 15, comprising six stresses, six strains and three
displacements. An additional six equations are therefore necessary to obtain a
solution.
So far we have made no assumptions regarding the force-displacement or stress-
strain relationship in the body. This will, in fact, provide us with the required six
equations but before these are derived it is worthwhile considering some general
aspects of the analysis.
The derivation of the equilibrium, strain-displacement and compatibility equa-
tions does not involve any assumption as to the stress-strain behaviour of the
material of the body. It follows that these basic equations are applicable to any
type of continuous, deformable body no matter how complex its behaviour under
stress. In fact we shall consider only the simple case of linearly elastic isotropic
materials for which stress is directly proportional to strain and whose elastic proper-
ties are the same in all directions. A material possessing the same properties at all
points is said to be homogeneous.
Particular cases arise where some of the stress components are known to be zero
and the number of unknowns may then be no greater than the remaining equilibrium
equations which have not identically vanished. The unknown stresses are then found
from the conditions of equilibrium alone and the problem is said to be statically deter-
minate. For example, the uniform stress in the member supporting a tensile load P in
Fig. 1.3 is found by applying one equation of equilibrium and a boundary condition.
This system is therefore statically determinate.
Statically indeterminate systems require the use of some, if not all, of the other
equations involving strain-displacement and stress-strain relationships. However,
whether the system be statically determinate or not, stress-strain relationships are
necessary to determine deflections. The role of the six auxiliary compatibility
equations will be discussed when actual elasticity problems are formulated in
Chapter 2.
We now proceed to investigate the relationship of stress and strain in a three-
dimensional, linearly elastic, isotropic body.
Experiments show that the application of a uniform direct stress, say a,, does not
produce any shear distortion of the material and that the direct strain E, is given by
the equation

9u
E, = - (1.40)
E
where E is a constant known as the modulus of elasticity or Young’s modulus.
Equation (1.40) is an expression of Hooke’s Law. Further, E, is accompanied by