The theory of linear elastostatics may be found in numerous textbooks,
like, for example, that of
Timoshenko[7]. The basic equations are summarized
here for the sake of completeness. The coordinates system ( 
),
the displacements ( 
), the stresses ( 
), and
the surface tractions ( 
) are shown in Figure 4.
Figure 4: The main coordinates, displacements,
stresses, and tractions in three dimensions.
The equilibrium equations:
where 
are the body force (per unit volume)
components. The surface tractions are related to the stresses via:
where 
are the components of the outward unit vector normal
to the solid surface.
The strains are given in terms of the deformations as:
The stresses are related to the strains via the constitutive relationships:
where E is the modulus of elasticity, and 
the Poisson's ratio;
is the Kronecher delta, and 
.
For the sake of simplicity we will limit ourselves to two dimensions. The extension to three dimensions is straightforward. The next two sections summarize the fundamentals of the boundary element formulation in elastostatics, as described in more detail by Brebbia and Dominguez [8].
In two dimensions we may consider either the plain strain ( 
)
or the plate stretching ( 
) situation. The latter
can be considered equivalent to the former by taking as 
and
. The following formulation corresponds to the plain strain
condition. However, all the numerical examples that will follow
correspond to the plate stretching condition.
Figure 5: The boundary conditions on the ``hydrofoil'' beam.
The ``hydrofoil'' beam shown in Figure 5 is considered. The
pressures (tractions) acting on the sides of the beam are taken equal to
those resulting from the hydrodynamic analysis of a
cavitating 2-D hydrofoil with the same section.
The pressures have been taken with reference to the pressure inside the cavity.
They are also shown in Figure
5 together with the rest of the boundary conditions.
The beam is taken to be clamped at the
``trailing edge'' of the hydrofoil.
It should be noted that this 2-D beam problem has no physical equivalent in
three dimensions, in which case a beam with a hydrofoil cross section
is clamped to a body (like the hull of a ship
or the hub of a propeller). However, we expect this 2-D problem to
be representative for the investigation of the accuracy of the BEM
with decreasing hydrofoil thickness to chord ratio.
Notice that from the four quantities, 
, only the two need to
be specified on each boundary. The horizontal component of the pressure
forces acting on the wetted side is ignored ( 
).
