Abstract
The paper deals with the force stress pseudotensor as proposed by Schouten and a derivation of equilibrium equations in terms of the Schouten stress pseudotensor of positive weight. The definition of Schouten’s force stress pseudotensor is based on the notion of a pseudoinvariant area element. Conventional and unconventional definitions of the force stress tensor are recalled and discussed. The tensor area elements of a M-manifold immersed in N-dimensional parent (outer) plane space are revisited. The notions of vector, pseudovector, invariant and pseudo-invariant elementary areas in three-dimensional plane space are thoroughly discriminated. The usability of pseudotensor elementary volume of any given integer weight is discussed. Three realizations of the covariant differentiation of pseudotensor fields are considered and compared. Equations of equilibrium and dynamics are derived in terms of the Schouten force stress pseudotensor from the virtual displacements principle.
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Notes
In the literature, pseudotensors are also called as relative tensors. The pseudotensors of weight +1 are called as tensor densities.
A pseudoinvariant elementary volume in some studies [5] is called as a natural element volume.
Cf. \({{\nabla }_{s}}{{\lambda }_{k}} = 0,\) λk is a parallel covariant vector field
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Funding
The study is carried out within the framework of the state task (no. state registration AAAA-A20-120011690132-4) and with the support of the Russian Foundation for Basic Research (no. 20-01-00666).
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Murashkin, E.V., Radayev, Y.N. The Schouten Force Stresses in Continuum Mechanics Formulations. Mech. Solids 58, 153–160 (2023). https://doi.org/10.3103/S0025654422700029
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DOI: https://doi.org/10.3103/S0025654422700029