Variational Models and Methods in Solid and Fluid Mechanics

Variational principles and calculus of variations have always been an important tool for formulating mathematical models for physical phenomena. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of interest for scientists and engineers and are the main tool for the axiomatization of physical theories.

The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D’Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salençon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an effective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defined by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)–(1965), Toupin (1962), Mindlin (1964)–(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell’Isola and Seppecher (1995)–(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.

These notes begin with a review of the mainstream theory of brittle fracture, as it has emerged from the works of Griffith and Irwin. We propose a re-formulation of that theory within the confines of the calculus of variations, focussing on crack path prediction. We then illustrate the various possible minimality criteria in a simple 1d-case as well as in a tearing experiment and discuss in some details the only complete mathematical formulation so far, that is that where global minimality for the total energy holds at each time. Next we focus on the numerical treatment of crack evolution and detail crack regularization which turns out to be a good approximation from the standpoint of crack propagation. This leads to a discussion of the computation of minimizing states for a non-convex functional. We illustrate the computational issues with a detailed investigation of the tearing experiment.

We present here a variational approach to derivation of multiphase flow models. Two basic ingredients of this method are as follows. First, a conservative part of the model is derived based on the Hamilton principle of stationary action. Second, phenomenological dissipative terms are added which are compatible with the entropy inequality. The variational technique is shown up, and mathematical models (classical and non-classical) describing fluid-fluid and fluid-solid mixtures and interfaces are derived.

The lectures provide an introduction to the Chapters on stochastic variational problems from the author’s book Variational Principles of Continuum Mechanics, Springer, 2009.

These notes are finalized to a particular study of the damping mechanism in Hamiltonian systems, characterized indeed by the absence of any energy dissipation effect. It is important to make a clear distinction between the two previous concepts, since they seem to be somehow contradictory. A Hamiltonian system is characterized by an invariant total energy (the Hamiltonian H) that is equivalent to state any energy dissipation process is absent. This circumstance, especially from an engineering point of view, leads to the wrong expectation that the motion of any part of such a dissipation-free system, subjected to some initial conditions, maintains a sort of constant amplitude response. This is, although unexpectedly, a wrong prediction and the “mechanical intuition” leads, in this case, to a false belief. It is indeed true the converse: even in the absence of any energy dissipation mechanisms, mechanical systems can exhibit damping, i.e. a decay amplitude motion.

A general set of boundary conditions at the interface between dissimilar fluid-filled porous matrices is established starting from an extended Hamilton-Rayleigh principle. These conditions do include inertial effects. Once linearized, they encompass boundary conditions relative to volume Darcy-Brinkman and to surface Saffman-Beavers-Joseph-Deresiewicz dissipation effects.