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Chapter 5
Analytical
techniques and solutions for linear elastic solids
5.8 The Reciprocal Theorem and
applications
The reciprocal theorem is a distant cousin of the principle
of minimum potential energy, and is a particularly useful tool. It is the basis for a computational method
in linear elasticity called the boundary element method; it can often be used
to extract information concerning solutions to a boundary value problem
without having to solve the problem in detail; Â and can occasionally be used to find the
full solution  Â for example, the reciprocal theorem provides
a way to compute fields for arbitrarily shaped dislocation loops in an
infinite solid.
5.8.1 Statement and
proof of the reciprocal theorem
                                                     Â
The reciprocal theorem relates two solutions for the same
elastic solid, when subjected to different loads. To this end, consider the following
scenario
- An
elastic solid which occupies some region V with boundary S.
The outward normal to the boundary is specified by a unit vector . The properties of the solid are
characterized by the tensor of elastic moduli  and mass density . The solid is free of stress when
unloaded, and temperature changes are neglected.
- When
subjected to body forces  (per unit mass) together with
prescribed displacements  on portion  of its boundary, and tractions  on portion ,
a state of static equilibrium is established in the solid with
displacements, strains and stresses
- When
subjected to body forces  together with prescribed displacements  on portion  of its boundary, and tractions  on portion ,
the solid experiences a static state
The
reciprocal theorem relates the two solutions through
 Derivation:
Start by showing that . To see this, note that ,
where we have used the symmetry relation .
To prove
the rest, recall that
1. The divergence theorem requires
that
2. Any pair of strains and displacement are related by
3. The stress tensor is symmetric, so
that
4. Both stress states satisfy the equilibrium equation .
Consequently, collecting together the volume integrals gives
5.
Note
that this result applies to any equilibrium stress field and pair of
compatible strain and displacements  the stresses need not be related to the
strains. Consequently, this result can
be applied to pairs of stress and displacement
5.8.2 Simple
example using the reciprocal theorem
The
reciprocal theorem can often be used to extract average measures of deformation or stress in an elastic
solution. As an example, consider the
following problem:Â An elastic solid
with Young’s modulus E and
Poisson’s ratio  occupies a volume V with surface S.  The solid is subjected to a distribution
of traction  on its surface. The traction exerts zero resultant force
and moment on the solid, i.e.
As
a result, a state of static equilibrium with displacement, strain and stress  is developed in the solid.  Show that the volume change of the solid
can be calculated as
Derivation:
1. Note that if we were able to determine the full
displacement field in the solid, the volume change could be calculated as
If you don’t see this result immediately on
geometric grounds it can be derived by first calculating the total volume change
by integrating the dilatation over the volume of the solid and then applying
the divergence theorem
2. Note that we can make one of the terms in the
reciprocal theorem reduce to this formula by choosing state A to be the actual displacement,
stress and strain in the solid, and choosing state B to be a uniform stress with unit magnitude .
This stress is clearly an equilibrium field, for zero body force. The
corresponding strains and displacements follow as
where
 and  represent an arbitrary infinitesimal
displacement and rotation.
3. Substituting into the reciprocal theorem, recalling
that the stresses satisfy the boundary condition ,
and using the equilibrium equations for the traction then yields
5.8.3 Formulas
relating internal and boundary values of field quantities
The
reciprocal theorem also gives a useful relationship between the values of
stress and displacement in the interior and on the boundary of the solid,
which can be stated as follows.Â
Suppose that a linear elastic solid with Young’s modulus E and Poisson’s ratio  is loaded on its boundary (with no body
force) so as to induce a static equilibrium displacement, strain and stress
field  in the solid. Define the following functions
You
may recognize the first two of these functions represent the displacements
and stresses induced at a point  by a point force of unit magnitude acting in
the  direction at the origin of an infinite
solid.
The
displacement and stress at an interior point in the solid can be calculated
from the following formulasÂ
Here, Â denotes that the integral is taken with
respect to x, holding  fixed.
At
first sight this appears to give an exact formula for the displacement and
stress in any 3D solid subjected to prescribed tractions and displacement on
its boundary. In fact this is not the
case, because you need to know both tractions and displacements on the boundary to evaluate the formula,
whereas the boundary conditions only specify one or the other. The main application of this formula is a
numerical technique for solving elasticity problems known as the ‘boundary element method.’ The idea is simple: the unknown values of
traction and displacement on the boundary are first calculated by letting the
interior point  approach the boundary, and solving the
resulting integral equation.  Then,
the formulas are used to calculate field quantities at interior points.
Derivation: These formulas are a consequence of the reciprocal
theorem, as follows
1. Start with the reciprocal theorem:
2. For state A we choose the actual stress, strain and displacement in the solid. For state B, we choose the displacement and stress fields induced by a
Dirac Delta distribution of body force located at position . The body force vector associated with a force acting in
the  direction will be denoted by ,
and has the property that
The stress and displacement induced
by this body force can be calculated by shifting the origin in the point
force solution given in Section 5.4.3.Â
Substituting into the reciprocal theorem immediately yields the
formula for displacements.
3. The formula for stress follows by
differentiating the displacement with respect to  to calculate the strain, and then
substituting the strain into the elastic stress-strain equation and
simplifying the result.
5.8.4 Classical Solutions
for displacement and stress due to a 3D dislocation loop in an infinite solid
The
reciprocal theorem can also be used to calculate the displacement and stress
induced by an arbitrarily shaped 3D dislocation loop in an infinite
solid. The concept of a dislocation in
a crystal was introduced in Section 5.3.4.Â
A three-dimensional dislocation in an elastic solid can be constructed
as follows:
1. Consider an infinite solid with Young’s modulus  and Poisson’s ratio . Assume that the solid is initially stress
free.
2. Introduce a bounded, simply connected surface S inside the solid.  Denote the edge of this surface by a curve
C Â this curve will correspond to the
dislocation line.  The direction of
the line will be denoted by a unit vector  tangent to the curve. There are, of course, two possible choices
for this direction. Either one can be used. Â
The normal to S will be
denoted by a unit vector ,
which must be chosen so that the curve C
encircles m in a counterclockwise
sense when traveling in direction .
3. Create an imaginary cut on S, so that the two sides of the cut are free to move
independently. In the derivation
below, the two sides of the cut will be denoted by  and ,
chosen so that the unit vector  points from  to .
4. Hold  fixed, displace  by the burgers vector b, and weld the two sides of the cut back together. Remove the constraint on .
This
procedure creates a displacement field that is consistent with the Burger’s
circuit convention described in Section 5.3.4. To see this, suppose that a crystal lattice
is embedded within the elastic solid.Â
Perform a Burgers circuit around the curve C. Start the circuit on ,
encircle the curve according to the right hand screw convention with respect
to the line sense ,
and end at .  The end of the circuit is displaced by a
distance b from the start, so that
.
The displacement and stress
due to the dislocation loop can be calculated from
where
 is defined in Section 5.8.3, and  is the permutation symbol. The symbols   denotes that  is varied when evaluating the surface or
line integral. These results are also
often expressed in the more compact form
where .Â
Derivation:
1. Start with the reciprocal theorem
2. For state A, choose the actual stress and displacement
in the solid containing the dislocation loop.  For state B, choose the stress, strain and
displacement induced by a Dirac delta distribution of body force acting in
the  direction at position  in the solid. The displacements and stresses due to a
Dirac delta distribution of body force are denoted by the functions  and  defined in the preceding section.
3. When evaluating the reciprocal
theorem, the two sides of the cut are treated as separate surfaces.
Substituting into the reciprocal theorem and using the properties of the
delta distribution gives
where  and  denote the outward normals to the two sides
of the cut, and  denote the limiting values of the
displacement field for the dislocation solution on the two sides of the cut.
4. Substituting for n, collecting together the surface integrals and noting that  and  are continuous across S gives
Finally, noting that  (the burgers vector is the displacement of a
material point at the end of the burgers circuit as seen from a point at the
start) and that  yields the formula for displacements.
5. To calculate the stress, start by differentiating
the displacement to see that
6. Next, observe that this can be expressed as an
integral around the dislocation line
To see this, recall Stoke’s theorem, which states that
for any differentiable vector field
 integrated over a surface S with normal m that is bounded by curve C.
Apply this to the line integral, use ,
note that  because  is a static equilibrium stress field, and
finally note that .
7. Finally, calculate the stress using the elastic
stress-strain equation
8. The alternative forms for the displacement and
stress follow by noting that
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