![]() |
|||||||
|
|||||||
Chapter 5
Analytical techniques and solutions for linear elastic solids
5.7 Energy methods for solving static linear elasticity problems
You
may recall that energy methods can often be used to simplify complex
problems. For example, to find the
equilibrium configuration of a discrete system, you would begin by
identifying a suitable set of generalize coordinates
In
this section, we will develop an analogous procedure for solving boundary
value problems in linear elasticity.Â
Our generalized coordinates will be the displacement field
The main application of the principle is to generate approximate solutions to linear elastic boundary value problems. Indeed, the principle will form the basis of the Finite Element Method in linear elasticity.
5.7.1 Definition of the potential energy of a linear elastic solid under static loading
In the following, we consider a generic static boundary value problem in linear elasticity, as shown in the picture.Â
As always, we assume that we are given: 1. The shape of the solid in its unloaded condition 2. The initial stress field in the solid (we will take this to be zero) 3. The elastic constants for the solid 4. The thermal expansion coefficients for the solid,
and temperature change from the initial configuration 5. A body force distribution 6. Boundary conditions, specifying displacements
Kinematically Admissible Displacement Fields
A
‘kinematically admissible displacement field’ 1. 2.
3.
Note the v is
not necessarily the actual
displacement in the solid  The kinematically admissible displacement field can also be thought of as a system of generalized coordinates in the context of analytical mechanics. Recall that, to use a set of generalized coordinates in Lagranges equations, you must make sure that the system of coordinates satisfies all the constraints. Similarly, to be admissible, our displacement field must satisfy constraints on the boundary.Â
Definition of Potential Energy of an Elastic Solid
Next, we will define the potential energy of a solid. The definition may look a bit strange, because it seems to give different values for potential energy depending on how the solid is loaded. This is true. But who cares, as long as the definition is useful?
For any kinematically admissible displacement field v, the potential energy is
where
is
the strain energy density associated with the kinematically admissible
displacement field. You can interpret the three terms in the formula for V as the strain energy stored inside
the solid; the work done by body forces; and the work done by surface
tractions. For the particular case of an isotropic material, with
5.7.2 The principle of stationary and minimum potential energy.
Let
v be any kinematically admissible
displacement field. Let u be the actual displacement field 1. V(v) is stationary (i.e. a local minimum, maximum or inflexion point) for v=u. 2. V(v) is a global minimum for v=u.
As a preliminary step, recall that the actual displacement field satisfies the following equations
Next, re-write the kinematically admissible displacement field in terms of u as
where
i.e. the difference between the kinematically admissible field and the actual field is zero wherever displacements are prescribed.
Now,
note that
where
To see this, simply substitute into the definition of the potential energy
Multiply everything out
and use the condition that
Now,
to show that
To show this, note that
Next, note that
where
we have used the fact that
Substitute back into the
expression for
Now, recall the equations of equilibrium
so that the second term vanishes. Apply the divergence theorem to express the first integral as a surface integral
Recall that
because either tractions or displacements (but not both) must be prescribed on every point on the boundary.Â
Therefore
Finally, recall that
and substitute back into
the expression for
This proves that V(v) is stationary at v=u, as stated.
Finally, we wish to show that V(v) is a minimum at v=u. This is easy. Note that we have proved that
Note that
is the strain energy
density associated with a strain
5.7.3 Uniaxial compression of a cylinder solved by energy methods
Consider a cylindrical bar subjected to a uniform pressure p on one end, and supported on a rigid, frictionless base. Neglect temperature changes. Determine the displacement field in the bar.
We will solve this problem using energy methods. We will guess a displacement field of the form
This satisfies the boundary conditions on the bottom face
of the cylinder, so it is a kinematically admissible displacement field. The coefficients
with all other strain components zero. The strain energy density is
The boundary conditions are 1. On the bottom of the cylinder 2. On the sides of the cylinder, 3. On the top of the cylinder Substitute into the expression for strain energy density to see that
Now, the actual displacement field minimizes V. This requires
Evaluate the derivatives to see that
It is easy to solve these equations to see that
This
is, of course, the exact solution, which is reassuring. Notice that we never had to calculate
stresses or worry about equilibrium
Let us solve the same problem, but this time with displacement boundary conditions on the top of the cylinder.
The cylinder has unstretched length L and is stretched between frictionless grips to length L+h. This time, the kinematically admissible displacement field must satisfy boundary conditions on both top and bottom surface of the cylinder.  Therefore, we choose
Proceeding as before, we now find that the potential energy is
Note that this time there is no contribution to the potential energy from the tractions on the top of the cylinder, because now the displacement is prescribed there, instead of the pressure. Minimizing the potential energy as before
Solve these equations to conclude that
Again, this is the exact solution.
5.7.4 Variational derivation of the beam equations
Variational methods can be used to solve boundary value problems exactly, as described in the preceding section.  The real power of variational methods, however, is to provide a systematic way to find approximate solutions to boundary value problems.Â
We will illustrate this by re-deriving the equations governing beam bending theory using the principle of minimum potential energy.
Consider
a slender rod with rectangular cross section, subjected to uniform pressure q(x) on its top surface. Assume that the rod is an isotropic, linear
elastic solid with Young’s modulus E
and Poisson’s ratio
We proceed by approximating the displacement field within the bar. We will suppose that the strains at any given cross section are completely characterized by the local curvature of the beam, so that at a given cross section x
Here,
The
displacement and strain fields are therefore completely characterized by
Now,
we want to find w(x) andÂ
Begin by computing the potential energy. It is straightforward to show that the strain energy density is
Hence
Here, we have neglected
the small additional deflection of the beam surface due to
We now wish to minimize V with respect to w andÂ
which is evidently satisfied for any w by choosing
This is the usual expression for the position of the neutral axis of a beam. We can now simplify our expression for potential energy by defining
so that
Now
turn to the more difficult problem of finding w that will minimise V. To do this, let us calculate the change in V when
Expand this out to see that
Now, if V(w) is a minimum, then
We are none the wiser as a result of this exercise, but if we integrate the first integral by parts twice, we find that
Since this is zero for
any
to ensure that the third term in this expression vanishes. This gives us the required governing equation for w. However, we still need to deal with the first two boundary terms.
There are several ways to prescribe boundary conditions on the ends of the beam to ensure that V is stationary. 1. We may prescribe w
and its first derivative. In this case
the variation in 2. Prescribe only the value of w. In this case we must ensure that
to ensure that V is stationary. We know from elementary strength of materials courses that this is equivalent to the condition that the shear force vanishes on the end of the beam. 3. Prescribe only the value of
This is equivalent to setting the bending moment to zero at the end of the beam.
Clearly, one could extend this procedure to account for tractions acting on the ends of the beam. The details are left as an exercise. A nice feature of the variational approach that we followed here is that the appropriate boundary conditions follow naturally from the variational principle (indeed, the boundary conditions are called `natural’ boundary conditions). This turns out to be particularly helpful in setting up approximate theories of plates and shells, where the boundary conditions can be very difficult to determine consistently using any other method.
5.7.5 Energy methods for calculating stiffness
Energy methods can also be used to obtain an upper bound to the stiffness of a structure or a component.
Begin
by reviewing the meaning of stiffness of an elastic solid. A spring is an example of an elastic
solid. Recall that if you apply a
force P to a spring, it deflects by
an amount
If you apply a load P to any elastic structure (except one which contains two or more contacting surfaces), the point where you apply the load will deflect by a distance that is proportional to the applied load. For example, for a cantilever beam, the end deflection is
The stiffness of the beam
is therefore
To get an upper bound to the stiffness of a structure, one can merely guess its deformed shape, then apply the principle of minimum potential energy.
For example, for the beam problem, we might guess that the beam deforms into a circular shape, with unknown radius R.
The deflection at the end of the beam is approximately
From the preceding section, we know that the potential energy of a beam is
Here,
Choose R to minimize the potential energy
so that
For comparison, the exact
solution is
|
|||||||
(c) A.F. Bower, 2008 |