Determination of the spherical part of the couple-stress in a polar fibre-reinforced elastic plate subjected to pure bending

K P Soldatos

doi:10.1007/s00707-021-03035-z

Abstract

This communication provides initial information and understanding of the manner in which a newly developed theoretical mechanism (Soldatos in Int J Solids Struct 202:217–225, 2020) is applied in specific boundary value problems met in polar linear elasticity of fibrous composites and thus enables the determination of the spherical part of the couple-stress tensor. In this context, it tests the applicability of the implied mechanism/method in the case that a rectangular plate reinforced by a single family of unidirectional fibres is subjected to pure bending. The problem solution is obtained for either non-polar or polar material behaviour, where fibres are considered perfectly flexible or resistant in bending, respectively, and provides clear evidence of the correctness of the principal argument that underpins the proposed method. Namely, that the general rotation field of the plate deformation differs from the fibre rotation field. That newly discovered method enables an extra energy term that emerges in the strain energy function of the fibrous composite plate to relate with the spherical part of the couple-stress tensor outside conventional equilibrium conventions. It thus leads to the determination of the spherical part of the couple-stress and its distribution throughout the plate body in a complete and comprehensive manner.

Highlights

It is well known that the spherical part of the couple-stress tensor does not affect the stress equilibrium and is, left indeterminate in conventional polar elasticity (e.g. [1,2,3,4,5,6,7,8,9,10,11] and references therein)
Though, the polar elasticity theory of fibre-reinforced materials [7, 9] reveals that the strain energy density/function of polar fibrous composites contains an extra energy term that leaves unaffected their stress equilibrium
These observations led the present author [12] to search for a mechanism that could connect the implied extra energy term with the work done by the spherical part of the couple-stress and would enable determination of the latter outside or regardless of standard equilibrium conventions

Summary

Introduction

It is well known that the spherical part of the couple-stress tensor does not affect the stress equilibrium and is, left indeterminate in conventional polar elasticity (e.g. [1,2,3,4,5,6,7,8,9,10,11] and references therein). The implied theoretical generalisation [12] considers that, at least in the case of polar fibrous composites [7, 9], it is the fibre rotation field, rather than its general deformation counterpart, that is reciprocal to the antisymmetric part of the stress In this manner, (i) it creates room for a direct connection to be made between the spherical part of the couple-stress and the extra energy term observed in [7, 9], and (ii) is furnished with a theoretical mechanism/method that enables determination of the spherical part of the couple-stress outside the standard, well-known equilibrium conventions. As is mentioned in the Introduction, the corresponding plane strain problem is considered and solved in [16]

Non-polar material behaviour

Polar material behaviour

Couple-stress field and determination of its spherical part

The unrestricted theory

Further discussion and conclusions

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