In this lesson, the discussion will be focused on obtaining the weak form in an alternative way. The alternative approach used is variational calculus and broadly referred to as variational methods. First, the motivation for this approach is discussed. An example of strain energy for a linear elastic 1D problem is presented. The terms are explained in the context of elasticity and how the expression represents potential energy, or Gibbs free energy (for purely mechanical problems).
In this video, it is explained how the notion of pi can be understood. Pi is not a function, but it is functional. Derivatives are also functional and need field descriptions as well. Pi is the free energy functional that will be used here, and it needs a field description. The concept of free energy is discussed. The use of extrema for free energies is discussed, which will lead to the equilibrium states – stable, neutral, and unstable. For the free energy functional, it is necessary to have the derivative in terms of the displacement field; then the extrema can be found out to check the equilibrium states.