SCHOUTEN'S FORCE STRESS TENSOR AND AFFINOR DENSITIES OF POSITIVE WEIGHT

Abstract

The paper deals with the concept of the force stress pseudotensor and the derivation of equilibrium equations in terms of the Schouten's stress pseudotensor being an affinor density. The definition of Schouten's force stress pseudotensor is mainly based on the notion of a pseudoinvariant element of area. The requisute equations and notions from algebra and the analysis of pseudotensors is revisited. A fundamental orienting pseudoscalar is introduced and discussed. Conventional and non-conventional definitions of the force stress tensor are given. A unit normal vector to a level surface of a pseudoscalar field is introduced. The exceptional importance of using the theory of orientable manifolds in modeling micropolar continua in mechanics of solids is noted. The notion of M-cell and its orientation algorithm are recalled. Algorithms for constructing the tensor elements of the area of M-manifold immersed in N-dimensional space are discussed. The notions of vector, pseudovector, invariant and pseudoinvariant elements of surface area in three-dimensional space are revisited. The possibility of using pseudotensor volume elements of a given integer weight due to the formula for a pseudotensor field transformation to an absolute tensor field by a fundamental orienting pseudoscalar is discussed. Various realisations of covariant differentiation of pseudotensors are considered. Covariant derivatives are given for a pseudoscalar and a contravariant pseudotensor of the second rank of an arbitrary integer weight. The principle of virtual displacements is formulated in terms of pseudo-invariant volume and area elements. The hypothesis of the absolute invariance of the virtual work is assumed, i.e. insensitive to rotations, 3D inversion and mirror reflections. Equations of equilibrium and dynamics are derived in terms of the affinor density of Schouten's force stresses. Equilibrium equations are obtained for the case of using pseudo-invariant volume and area elements of an arbitrary integer weight.