Introduction to the Variational Formulation in Mechanics

3Method of Virtual Power

3.1 Introduction

Variational principles and methods have had a pivotal role in establishing a modern point of view for both theoretical and applied mechanics. As we will see in this chapter, a variational formalism for the basic laws that rule the behavior of continuum media is, in essence, the most natural and rigorous foundational framework.

The use of the variational formulation allows us to reduce to a single integral expression all the ingredients that partake in the mathematical formulation of the physical problem under analysis, namely, equilibrium equations, constitutive equations (material behavior), boundary conditions, initial conditions, and jump conditions, among others.

The local form¹ of the governing equations which rule the motion of a body spring as a mere consequence of the variational formulation. In turn, and this is extremely important from the perspective of applications in mechanics, the variational formulation naturally induces the most appropriate and mathematically sound structure in which numerical methods to obtain approximate solutions can be framed. These methods, called variational methods, allow approximate solutions to be obtained using the same unified mathematical context as the continuum problem, and facilitating the computational implementation regardless of the complexity of the mechanical problem.

Another fundamental aspect of the variational formulation in mechanics, when compared to the classical (local) form, ...