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The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional counterpart of the statics of a system having a finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space implies that force functionals may be represented by objects that generalize the stresses of traditional continuum mechanics.

We construct generalizations of boundary forms of Kupferman, Olami and Segev and of the De Donder form to higher order Lagrangians. In principle, the resulting forms may depend on the coordinate system used. Nevertheless, it leads to many invariant conclusions. In some cases, like second order gravity, the construction leads to a unique De Donder form on the jet bundle that is independent of the choice of coordinates.

The theory of continuous distributions of dislocations and other material defects, when formulated in terms of differential forms, is shown to comprise also the discrete, or singular, counterpart, in which defects are concentrated on lower dimensional regions, such as surfaces, lines, and points. The mathematical tool involved in this natural transition is the theory of de Rham currents, which plays in regard to differential forms the same role as the theory of Schwartz distributions plays with respect to ordinary functions. After a review of the main mathematical aspects, the theory is illustrated with a profusion of examples and applications.

A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current \(T\) and the associated defects are given by its boundary \(\partial T\). For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.