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Chapter 2
Governing
Equations
2.4 Work done by stresses;
Principle of Virtual Work
In
this section, we derive formulas that enable you to calculate the work done
by stresses acting on a solid. In
addition, we prove the principle of
virtual work  Â which is an alternative way of expressing
the equations of motion and equilibrium derived in Section 2.3. The principle of virtual work is the
starting point for finite element analysis, and so is a particularly
important result.
2.4.1 Work done by Cauchy stresses
Consider a solid with mass density  in its initial configuration, and density  in the deformed solid. Let  denote the Cauchy stress distribution within
the solid. Assume that the solid is
subjected to a body force  (per unit mass), and let  and  denote the displacement, velocity and
acceleration of a material particle at position   in the deformed solid. In addition, let
denote
the stretch rate in the solid.
The rate of work
done by Cauchy stresses per unit deformed volume is then . This energy is either dissipated as heat or
stored as internal energy in the solid, depending on the material behavior.
We
shall show that the rate of work done by internal forces acting on any
sub-volume V bounded by a surface A in the deformed solid can be
calculated from
Here,
the two terms on the left hand side represent the rate of work done by
tractions and body forces acting on the solid (work done = force x velocity).Â
The first term on the right-hand side can be interpreted as the work
done by Cauchy stresses; the second term is the rate of change of kinetic
energy.Â
Derivation: Substitute
for  in terms of Cauchy stress to see that
Now, apply the divergence
theorem to the first term on the right hand side
Evaluate
the derivative and collect together the terms involving body force and stress
divergence
Recall
the equation of motion
and note
that since the stress is symmetric
to see
that
Finally,
note that
Finally, substitution leads to
as
required.
2.4.2 Rate of
mechanical work in terms of other stress measures
 Â The rate
of work done per unit undeformed volume by Kirchhoff stress is
 The rate
of work done per unit undeformed volume by Nominal stress is
 The rate
of work done per unit undeformed volume by Material stress is
This
shows that nominal stress and deformation gradient are work conjugate, as are material stress and Lagrange strain.
In addition, the rate of work done on a volume  of the undeformed solid can be expressed as
Derivations: The proof of the first result (and the stress power
of Kirchhoff stress) is straightforward and is left as an exercise. To show the second result, note that  and  to re-write the integrals over the
undeformed solid; then and apply the divergence theorem to see that
Evaluate the derivative,
recall that  and use the equation of motion
to see that
Finally,
note that  and re-write the second integral as a
kinetic energy term as before to obtain the required result.
The
third result follows by straightforward algebraic manipulations  note that by definition
Since  is symmetric it follows that
2.4.3 Rate of
mechanical work for infinitesimal deformations
For infintesimal motions all stress measures are equal; and
all strain rate measures can be approximated by the infinitesimal strain
tensor . The rate of work done by stresses per unit
volume of either deformed or undeformed solid (the difference is neglected)
can be expressed as ,
and the work done on a volume  of the solid is
2.4.4 The principle
of Virtual Work
The principle of virtual
work forms the basis for the finite element method in the mechanics of solids
and so will be discussed in detail in this section.
Suppose
that a deformable solid is subjected to loading that induces a displacement
field ,
and a velocity field . The loading consists of a prescribed
displacement on part of the boundary (denoted by  ), together with a traction t (which may be zero in places)
applied to the rest of the boundary (denoted by  ).Â
The loading induces a Cauchy stress . The stress field satisfies the angular
momentum balance equation .
The
principle of virtual work is a different way of re-writing partial
differential equation for linear moment balance
in an
equivalent integral form, which is
much better suited for computer solution.
To
express the principle, we define a kinematically
admissible virtual velocity field ,
satisfying  on . You can visualize this field as a small
change in the velocity of the solid, if you like, but it is really just an
arbitrary differentiable vector field.Â
The term `kinematically admissible’ is just a complicated way of
saying that the field is continuous, differentiable, and satisfies  on  - that is to say, if you perturb the
velocity by ,
the boundary conditions on displacement are still satisfied.
In
addition, we define an associated virtual
velocity gradient, and virtual stretch rate as
The
principal of virtual work may be stated in two ways.
First version of the principle of
virtual work
The
first is not very interesting, but we will state it anyway. Suppose that the Cauchy stress satisfies:
1. The boundary condition  on
2. The linear momentum balance equation
Then the virtual
work equation
is satisfied for all virtual velocity fields.
Proof: Â Observe that since the Cauchy stress is
symmetric
Next, note that
Finally,
substituting the latter identity into the virtual work equation, applying the
divergence theorem, using the linear momentum balance equation and boundary
conditions on  and  we obtain the required result.
Second version of the principle of virtual work
The converse of this
statement is much more interesting and useful. Suppose that  satisfies the virtual work equation
for all virtual velocity fields . Then the stress field must satisfy
3. The boundary condition  on
4. The linear momentum balance equation
The
significance of this result is that it gives us an alternative way to solve
for a stress field that satisfies the linear momentum balance equation, which avoids having to differentiate the
stress. It is not easy to
differentiate functions accurately in the computer, but it is easy to
integrate them. The virtual work
statement is the starting point for any finite element solution involving
deformable solids.
Proof: Follow
the same preliminary steps as before, i..e.
and substitute into the
virtual work equation
Apply the divergence
theorem to the first term in the first integral, and recall that  on ,
we see that
Since this must hold for
all virtual velocity fields we could choose
where
 is an arbitrary function that is positive
everywhere inside the solid, but is equal to zero on . For this choice, the virtual work equation
reduces to
and since the integrand
is positive everywhere the only way the equation can be satisfied is if
Given
this, we can next choose a virtual velocity field that satisfies
on . For this choice (and noting that the volume
integral is zero) the virtual work equation reduces to
Again, the integrand is
positive everywhere (it is a perfect square) and so can vanish only if
as stated.
2.4.5 The Virtual
Work equation in terms of other stress measures.
It is often convenient to
implement the virtual work equation in a finite element code using different
stress measures.Â
To do so, we define
1. The actual deformation gradient in the solid
2. The virtual rate of change of deformation gradientÂ
3. The virtual rate of change of Lagrange strain
In addition, we define
(in the usual way)
1.
Kirchhoff stressÂ
2. Nominal (First Piola-Kirchhoff) stress Â
3. Material (Second
Piola-Kirchhoff) stress Â
In terms of these
quantities, the virtual work equation may be expressed as
Note
that all the volume integrals are now taken over the undeformed solid  this is convenient for computer
applications, because the shape of the undeformed solid is known. The area integral is evaluated over the deformed solid, unfortunately. It can be expressed as an equivalent integral
over the undeformed solid, but the result is messy and will be deferred until
we actually need to do it.
2.4.6 The Virtual
Work equation for infinitesimal deformations.
For infintesimal motions, the Cauchy, Nominal, and Material
stress tensors are equal; and the virtual stretch rate can be replaced by the
virtual infinitesimal strain rate
There is no need to distinguish between the volume or
surface area of the deformed and undeformed solid. The virtual work equation can thus be
expressed as
for all kinematically
admissible velocity fields.
As
a special case, this expression can be applied to a quasi-static state with .
Then, for a stress state  satisfying the static equilibrium equation  and boundary conditions  on ,
the virtual work equation reduces to
In which  are kinematically admissible displacements
components  on S2)
and .
Conversely,
if the stress state  satisfies  for every set of kinematically admissible
virtual displacements, then the stress state  satisfies the static equilibrium equation  and boundary conditions  on .Â
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