CONTINUUM MECHANICS for ENGINEERS

appears in the form u i = u i(x,t). In addition, the solution must satisfy not only
boundary conditions which may be functions of time as in
or
but also initial conditions, which usually are taken as
and
*
ui = u ( x,t)
on S (6.4-9a)
on S (6.4-9b)
*
ui = u ( x, 0)
(6.4-10a)
* ˙u = ˙u (6.4-10b)
i ( x, 0)
Analogous to Eq 6.4-7 for elastostatics, it is easily shown that the governing
equations for displacements in elastodynamics theory are
which are also called Navier’s equations.
6.5 Plane Elasticity
t
( nˆ ) * nˆ t
( )
x,
t
i i
i
= ( )
µui,jj + (λ + µ)uj,ji + ρbi = ρ ˙˙
(6.4-11)
In a number of engineering applications, specific body geometry and loading
patterns lead to a reduced, essentially two-dimensional form of the equations
of elasticity, and the study of these situations is referred to as plane elasticity.
Although the two basic types of problems constituting the core of this plane
analysis may be defined formally by stating certain assumptions on the
stresses and displacements, we introduce them here in terms of their typical
physical prototypes. In plane stress problems, the geometry of the body is
that of a thin plate with one dimension very much smaller than the other
two. The loading in this case is in the plane of the plate and is assumed to
be uniform across the thickness, as shown in Figure 6-5a. In plane strain
problems, the geometry is that of a prismatic cylinder having one dimension
very much larger than the other two and having the loads perpendicular to
and distributed uniformly with respect to this large dimension (Figure 6.5b).
In this case, because conditions are the same at all cross sections, the analysis
may be focused on a thin slice of the cylinder.
i
i
u i