Abstract
The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.
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Notes
The decomposition of the movement of an absolutely rigid body into translational and rotation is not the only one: the position of the pole can be changed (in this case, the direction and length of the translational movement will change), and the direction of the axis of rotation and the angle of rotation do not depend on the choice of the pole.
A fairly complete exposition of the micropolar theory of elasticity can be found in the monograph [3].
It is clear that in this case one can also speak of the orientation of the coordinate system.
It should be noted that this is not the only manifestation of the special status of permutation symbols. For them, the rules of index juggling, which are widely used in the algebra of tensors, are also violated.
See [4, p. 90, 91].
I.e. covariant differentiation nullifies the tensor (pseudotensor). The covariant constancy of one or another tensor or tensor pseudotensor makes it very convenient when transforming the differential equations of continuum mechanics, since it can be brought into and out of the scope of the operator of covariant differentiation∇s.
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This work was supported by the Russian Science Foundation (project no. 23-21-00262 “Coupled Thermomechanics of Micropolar Semi-Isotropic Media”).
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Translated by M.K. Katuev
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Radayev, Y.N. Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids. Mech. Solids 58, 1517–1527 (2023). https://doi.org/10.3103/S0025654423700206
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DOI: https://doi.org/10.3103/S0025654423700206