In this part, the application of the variational approach to constructing the governing
equations will be considered for several areas of continuum mechanics. The first
two chapters are concerned with one of the most beautiful areas of solid mechanics
– the theory of elastic shells and beams. In a sense, this is a physical theory of
surfaces and curves in three-dimensional space. It is attractive by its exceptional
elegance, the profound relations with geometry and the astonishing diversity and
complexity of the behavior of the objects it describes. The extent of the book allows
us to discuss only the derivation of the classical and refned shell theories and the
classical beam theory from the three-dimensional elasticity theory, and a case when
the classical shell theory does not work: theory of hard-skin plates and shells. The
next chapter gives a review of stochastic variational problems. Then we turn to consideration
of homogenization, one of the central problems of continuum mechanics.
This is followed by several other examples of applications of variational methods to
construction of continuum models: shallow water theory, theory of heterogeneous
mixtures, a model of granular media and a turbulence model. The discussion of
each theory is concluded by constructing the governing equations, and the issues
related to the features of these equations are not discussed. The only exception is
the homogenization theory where we consider the exact solutions of the cell problem
which are found by means of the variational methods.