On the characterisation of polar fibrous composites when fibres resist bending – Part III: The spherical part of the couple-stress

Abstract

Part II (Soldatos 2018b) identified some theoretical disagreement between the generally anisotropic polar linear elasticity of Mindlin and Tiersten (1962) and its counterpart developed in (Spencer and Soldatos, 2007;Soldatos, 2014) for fibrous composites with embedded fibres resistant in bending. The present communication shows that this disagreement is essentially due to inherent features of fibre-splay types of deformation and, consequently, generalises the couple-stress theory in a manner that creates room for newly emerged fibre-splay type of kinematic variables to enter and be accounted for. Relevant fundamental theorems, associate in Part II with the Mindlin and Tiersten model, are generalised accordingly to meet the needs and requirements of the proposed new formulation. Interestingly and importantly, the outlined generalisation enables formation of a convincing answer to a long-standing question regarding the indeterminacy of the spherical part of the couple-stress tensor, at least as far as polar elasticity of fibre-reinforced materials is concerned. The manner thus is demonstrated in which the spherical part of the couplestress can be determined in polar linear elasticity of fibrous composites that exhibit transverse isotropy due to an embedded family of fibres resistant in bending.

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

“…However, in the case of polar material behaviour of fibrous composites [7,9], where fibres with bending stiffness behave like embedded Euler-Bernoulli beams, no apparent reason requires from the resulting fibre rotation field, ϕ say, to coincide with for the corresponding general rotation field of the deformation. This fact led the present author [12] to generalise the Cosserat couple-stress theory in a manner that enables distinction of the two different rotation fields ω and ϕ. 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.…”

“…On the other hand, 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 also 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 thus would enable determination of the latter outside or regardless of standard equilibrium conventions.…”

“…The search for such a mechanism made it initially understood [12] that the implied indeterminacy arises by the direct manner in which the Cosserat framework [13], which underpins the conventional theory of polar elasticity, relates the spherical part of the couple-stress tensor to the rotation field of the deformation. This rotation field, ω, is the antisymmetric part of the displacement gradient and is generally present in a deformation regardless of whether the solid of interest exhibits polar or non-polar material behaviour.…”

Generalised viscoelastic fibre at small strain

“…It is well known that the theory of fibre-reinforced materials was initiated [1,2] with the purpose to model behaviour of rubber-like and structural solids reinforced by very strong fibres (see also [3,4] and early references therein). Later theoretical developments still serve that purpose and include the extension of the theory towards modelling the behaviour of fibre-reinforced fluids [5] as well as the behaviour of solid and fluid composites reinforced by strong fibres resistant in bending, stretching and twist (see [6][7][8][9][10][11] and references therein for more recent developments).…”

“…Over the last couple of decades, the theory is also applied successfully in modelling the behaviour of fibrereinforced biological materials. In this context, the early efforts of modelling soft, tube-like fibre-reinforced biolog- x x ical tissue [12,13] were followed by substantial relevant progress, and also led to identification of fibre response that diverges considerably from that of the strong fibres considered in [1][2][3][4][5][6][7][8][9][10][11]. The existence and behavioural modelling of biological fibres that resist extension but do not support compression is thus already attracting considerable attention (see [14][15][16] and references therein).…”