ABSTRACT
The principle of virtual work plays a significant role in continuum mechanics, especially in solid mechanics. Based on this principle, when a deformed solid in equilibrium is subjected to virtual displacements, then the work done by these virtual displacements, i.e. virtual work, is zero. Thus, virtual displacements are admissible displacements such that due to their application, the equilibrium of the deformed body is not disturbed. This allows us to use the principle of virtual work to derive equations describing the equilibrium of the deformed solids. When a solid continuum is disturbed, there is kinetic energy, strain energy due to deformation and stresses, and work done by body forces as a result of the work done by the external disturbance acting on this body. Consideration of all of these effects leads to Hamilton’s principle for a continuum. We shall see that Hamilton’s principle is only analogous to the balance of momenta within the framework of thermodynamic principles that consist of conservation of mass, balance of momenta, first and second laws of thermodynamics. It does not address the other conservation and balance laws necessary for thermodynamic equilibrium of the deforming matter. Nonetheless, the principle of virtual work is immensely powerful if used within the limitations in which it is derived. In general, the most significant strength of this approach in deriving equations for a deforming solid is that it is applicable to linear as well as non-linear problems. This of course is no surprise if the principle of virtual work results in balance of momenta as the balance of momenta applies to linear as well as non-linear processes. Thus, in principle, large deformation, material non-linearity etc., all can be treated within this framework. The principle of virtual work can be derived in Lagrangian as well as Eulerian descriptions, but derivation of Hamilton’s principle is only meaningful in Lagrangian description. In the following, we derive Hamilton’s principle in Lagrangian description first. This is then followed by the derivation of the principle of virtual work in Lagrangian and Eulerian descriptions. In the derivations, as far as possible, we use matrix and vector notation due to the fact that
such form of the final equations is more easily adaptable in computational methods such as finite element method.
