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

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.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness – Part II: determination of the spherical part of the couple-stress

“…It is thus seen that, in this pure bending problem, the only simplification/approximation introduced by the bending version of the theory (Section 6.1) is confined to a kind of filtering that is applied on the values of the elastic modulus d32 in a manner that enables only its positive range, d f , to emerge as suitable for use. Again, this conclusion is considerably more specific, as well as simpler, than its counterpart implied in [14].…”

“…However, it is precisely the valuable experience gained through the successful application of that mechanism [13] that led to the new developments detailed in this communication. Since, as already mentioned, this is based on a properly constructed generalisation of the nonlinear theory presented in [1] (Section 3 above), the results of the spherical-couple-stress determination detailed in the present section are naturally more accurate to those obtained in [14].…”

“…A connection mechanism between the implied extra energy term and the spherical couple-stress thus has recently been proposed [13] and, subsequently, applied successfully on a simple, linear elasticity, boundary value problem of plate bending [14]. That mechanism proposed a modification of the widely used linear Cosserat theory [2] that enables replacement of the standard, displacement-based rotation/ spin field with some alternative rotation/spin field that may be felt to be more dominant in specific classes of boundary value problems.…”

“…That mechanism proposed a modification of the widely used linear Cosserat theory [2] that enables replacement of the standard, displacement-based rotation/ spin field with some alternative rotation/spin field that may be felt to be more dominant in specific classes of boundary value problems. Its subsequent application in the special case of unidirectional fibrous composites [13,14] has succeeded to provide an estimate of the spherical couple-stress by (i) selecting the gradient of the fibre-spin vector as energetically reciprocal to the coupe-stress field and, hence, (ii) concluding that the energy stored in the material through couple-stress action is necessarily identical to the internal energy part that is due to the fibre-bending resistance [3].…”

“…Nevertheless, external couple-tractions may generally also be present in polar elasticity applications, and, in such cases, no information is available to guide with precision the choice of a dominant and, therefore, energetically reciprocal rotation/spin field. In this regard, and despite its remarkable success [14], the connection mechanism proposed [13] seems capable of providing only an approximate, although still valuable, estimation of the spherical couple-stress.…”