Abstract
The indeterminacy of the spherical part of couple-stress is a well-known drawback of any theoretical formulation stemming from the Cosserat couple-stress theory of elasticity. The relevant theory of finite elastic deformations of solids reinforced by a family of fibres that resist bending is not an exception. The present communication extends and completes that theory in a manner that enables it to measure the spherical part of the couple-stress tensor outside the conventional equilibrium considerations. To achieve this, the present study reconsiders an extra piece of information that has surprisingly emerged already but, so far, has been left unexplained and unexploited; namely, the fact that the energy stored in a fibrous composite elastic solid with fibre-bending stiffness involves a couple-stress generated term that does not influence the relevant couple-stress constitutive equation. The thus resulting new theoretical development complements the theory previously presented without dismissing any of the theoretical results detailed or the conclusions drawn there. Its validity embraces boundary value problems concerning both finite and infinitesimal elastic deformations of polar fibrous composites. In the latter case, its applicability is also tested and verified through the successful determination of the spherical couple-stress of a polar transversely isotropic elastic plate subjected to pure bending.
Highlights
The theory of elastic solids reinforced by a family of fibres resistant in bending [1] is a Cosserattype [2] couple-stress theory and, as such, considers that the stress field is non-symmetric
This postulation may be regarded as trivially valid in cases of material isotropy. It can be misleading in cases of material anisotropy that is due to fibre presence or, more generally, to the presence of some preference material direction, where the rotation field of the latter may differ considerably from its general deformation counterpart
The noted indeterminacy of mrr is a long-known weakness of the conventional Cosserat couple-stress theory (e.g., [4,5,6,7,8,9,10,11,12,13]) and, as such, it is present in the finite elasticity modelling effort detailed earlier in [1] for composite materials reinforced by fibres resistant in bending
Summary
The theory of elastic solids reinforced by a family of fibres resistant in bending [1] is a Cosserattype [2] couple-stress theory and, as such, considers that the stress field is non-symmetric. External coupletractions may generally be present in polar elasticity applications, and, in such cases, no information is there available to guide with precision the choice of a dominant and, energetically reciprocal rotation/spin field In this regard, and despite its remarkable success [15], the connection mechanism proposed [14] seems capable to provide only an approximate, though still valuable estimation of the spherical couple-stress. It the follows that the search for an exact method of spherical couple-stress determination should require direct and proper incorporation of the implied concept of a dominant rotation/spin field into the foundation of the non-linear polar elasticity theory proposed in [1] This aim suffices to identify the subject of the present communication, which complements and essentially completes the analysis presented in [1] without dismissing any of the remaining theoretical results detailed or the conclusions drawn there.
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