Plane strain polar elasticity of fibre-reinforced functionally graded materials and structures

Abstract

This study investigates the flexural response of a linearly elastic rectangular strip reinforced in a functionally graded manner by a single family of straight fibres resistant in bending. Fibre bending resistance is associated with the thickness of fibres which, in turn, is considered measurable through use of some intrinsic material length parameter involved in the definition of a corresponding elastic modulus. Solution of the relevant set of governing differential equations is achieved computationally, with the use of a well-established semi-analytical mathematical method. A connection of this solution with its homogeneous fibre-reinforced material counterpart enables the corresponding homogeneous fibrous composite to be regarded as a source of a set of equivalent functionally graded structures, each one of which is formed through inhomogeneous redistribution of the same volume of fibres within the same matrix material. A subsequent stress and couple-stress analysis provides details of the manner in which the flexural response of the polar structural component of interest is affected by certain types of inhomogeneous fibre distribution.

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

“…The outlined revelations were exemplified in the special case of fibrous composites reinforced by a single family of straight fibres. Namely, a case considered already in several stress analysis applications (Dagher and Soldatos, 2011;Farhat and Soldatos, 2015;Soldatos et al, 2019). It should be noted in this regard, that the results and discussions described, as well as conclusions made in those stress analysis studies are still valid and are not directly affected by the here presented revelations.…”

“…This is because determination of the newly introduced, characteristic fibre-spin deformation vector and, subsequently, of the spherical part of the couple-stress, become possible only after the state of equilibrium is fully studied in any well-posed boundary value problem and, hence, only after the relevant solution is achieved of the Navier-type displacement equations. As the latter equations are already solved in each of the implied boundary value problems (Dagher and Soldatos, 2011;Farhat and Soldatos, 2015;Soldatos et al, 2019), the obtained solution is now directly employable for the determination of both the relevant fibre-spin vector and the associated spherical part of the couple-stress tensor.…”

“…Introduction of the constitutive equation (5.2) and its symmetric stress counterpart into the equilibrium equations (2.3) can convert the latter into their Navier-type displacement representation. Potential solution of the latter will thus enable determination of the displacement components, ui, and consequently of all the remaining physical quantities, with the exception the spherical part of the couple-stress, mrr, and the parameter  introduced in (4.7); see, for instance, relevant example applications detailed in (Dagher and Soldatos, 2011;Farhat and Soldatos, 2015;Soldatos et al, 2019).…”

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).…”