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  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).  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. 


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


 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  ).  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.  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:  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. 


To do so, we define

1.       The actual deformation gradient in the solid  

2.       The virtual rate of change of deformation gradient   

3.       The virtual rate of change of Lagrange strain  

In addition, we define (in the usual way)

1.       Kirchhoff stress   

2.       Nominal (First Piola-Kirchhoff) stress   

3.       Material (Second Piola-Kirchhoff) stress   


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



(c) A.F. Bower, 2008
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