Duality in Finite Deformation Systems

Abstract

In this chapter we shall select topics from finite deformation continuum mechanics and minimum surface type problems in differential geometry, and use them to illustrate a general duality theory for n-dimensional nonconvex finite deformation systems in which the geometrical mapping Λ is a nonlinear partial differential operator. The methods and ideas can certainly be generalized to many other problems.

“I am the Alpha and the Omega”, says the Sovereign God, who is and who was and who is to come, the Almighty.

—Revelation, 1.8

He who understands geometry may understand anything in this world.

—Galileo Galilei, 1564–1642

The general laws of nature are to be expressed in equations which are valid for all coordinate systems

—Albert Einstein, 1916

As any human activity needs goals, mathematical research needs problems

—David Hilbert, 1932

Elasticity led to a vast range of mathematical problems involving linear algebra, differential geometry, ordinary and partial differential equations (mostly nonlinear), elliptic functions and the calculus of variations.

—Clifford Ambrose Truesdell, 1983