In an invariant formulation of pth-grade continuum mechanics, forces are defined as elements of the cotangent bundle of the Banach manifold of C P embeddings of the body in space. It is shown that forces can be represented by measures which generalize the stresses of continuum mechanics. The mathematical representation procedure makes the restriction offorces to subbodies possible. The local properties of the stress measures are examined. For the case where stresses are given in terms of smooth densities, it is shown that the structure of forces agrees with the form of forces one assumes in the traditional formulation, and the equilibrium differential equations are obtained.163

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A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential .n 1/-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.

A material body is conceived in terms of a global and a local model, the compatibility of which is shown to imply a generalization of the principle of virtual work. The classical notion of stress appears as a particular case of the force associated with the local model when an affine connection is specified in the physical manifold.

A metric independent geometric analysis of second order stresses in continuum mechanics is presented. For a vector bundle W over the n-dimensional space manifold, the value of a second order stress at a point x in space is represented mathematically by a linear mapping between the second jet space of W at x and the space of n-alternating tensors at x. While only limited analysis can be performed on second order stresses as such, they may be represented by non-holonomic stresses, whose values are linear mapping defined on the iterated jet bundle, J 1 (J 1 W), and for which an iterated analysis for first order stresses may be performed. As expected, we obtain the surface interactions on the boundaries of regions in space.2000 Mathematics Subject Classification. 74A10; 53Z05 .The collection of all k-jets of sections at x 0 ∈ S is the k-jet space of the vector bundle at x and is denoted as J k x W. Given a point x 0 ∈ S and an element w 0 ∈ W x 0 , the collection of all k-jets at x 0 such that each jet is represented by a section w with w(x 0 ) = w 0 will be referred to as the k-jet space at w 0 and will be denoted by J k w 0 W. Evidently,The k-jet bundle J k W is the collection of all k-jets at the various points in S so that J k W = x∈S J k x W = w∈W J k w W. (2.6)