Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials

Abstract

This study is motivated by evidence suggesting that the equations of polar elasticity of fibre-reinforced materials are non-elliptic even within the regime of infinitesimal deformations. In its endeavour to resolve this issue, which in symmetric-stress elasticity emerges in the regime of finite deformations only, it lays the foundation for development of a second-gradient theory of linear elasticity. Complete formulation of this new theory is achieved for locally transverse isotropic materials; namely, materials having embedded a single unidirectional family of arbitrarily shaped fibres which are resistant in bending, stretching and twist. The associated analysis shows that, indeed, the obtained Navier-type displacement equations are not elliptic. They accordingly predict that there exist in the material weak discontinuity surfaces, which may indeed be activated within the infinitesimal deformation regime. Surfaces containing the fibres are certainly such surfaces of weak discontinuity; this result may be not irrelevant to numerous practical situations where straight metallic fibres in fibre-reinforced concrete structures emerge partially de-bonded and exposed from their concrete matrix. Nevertheless, the analysis reveals further that additional surfaces of weak discontinuity may well exist in the locally transverse isotropic material of interest. An extension framework is also outlined towards cases of fibrous composites containing two or more families of non-perfectly flexible fibres.

Appendices

Appendix A: Single Family of Straight Fibres Aligned Along the x ₁-Direction

In the particular case of straight fibres, it is a _i,j =0 and, hence, expressions (3.6) reduce to (3.7). If, in addition, the x ₁-axis is conveniently aligned with the fibre direction and, therefore, a=(1,0,0)^T, the constitutive equations (3.11) and (3.14) may be re-arranged to become

(A.1)

and

(A.2a)

(A.2b)

respectively.

This form of linear constitutive equations is completely equivalent with the corresponding set of linearised constitutive equations obtained in Sect. 9 of [3]. It is recalled in this connection that (A.1) is the form of the generalised Hooke’s law met in symmetric-stress, transverse isotropic linear elasticity (e.g., [8, 11, 12]). The appearing five independent elastic moduli, c _ij , are related to their counterparts appearing in (3.11) as follows:

$$ \everymath{\displaystyle} \begin{array}{l} c_{11} = \lambda + 2\alpha + \beta + 4\mu_{L} - 2\mu_{T},\qquad c_{12} = \lambda + \alpha, \\[12pt] c_{22} = \lambda + 2\mu_{T},\qquad c_{23} = \lambda, c_{66} = \mu_{L}, \end{array} $$

(A.3)

while it is also $\frac{1}{2}(c_{22}-c_{23})=\mu_{T}$. In a similar context, the seven elastic moduli appearing in (A2) relate to the seven independent moduli appearing in (3.14) as follows:

$$ \everymath{\displaystyle} \begin{array}{l} b_{2} = \frac{4}{3}\beta_{3},\qquad b_{3} = \frac{4}{3}\beta_{5},\qquad d_{11} = \frac{2}{3}(2\beta_{1} + \beta_{2}), \\[12pt] d_{22} = \frac{4}{3}( \beta_{1} + \beta_{3}),\qquad d_{33} = \frac{4}{3}\beta_{1}, \\[12pt] d_{23} = \frac{1}{3}(4\beta_{3} + 2\beta_{4} + 3\beta_{7}),\qquad d_{32} = \frac{1}{3}( 12\beta_{5} + 6\beta_{6} - 2\beta_{4} - \beta_{7}). \end{array} $$

(A.4)

While the forms of (A.1) and (A.2a) coincide with their counterparts obtained in Sect. 9 of [3], equivalence of (A.2b) with the corresponding expression obtained in [3] becomes evident after a slight rearrangement in (A.2b), followed by simultaneous use of the intermediate parameters

$$ \everymath{\displaystyle} \begin{array}{l} d_{13} = \frac{1}{2}(d_{23} + d_{32}) = \frac{1}{3} (2\beta_{3} + 6\beta_{5} + 3\beta_{6} + \beta_{7}), \\[12pt] d_{31} = \frac{1}{2}(d_{23} - d_{32}) = \frac{1}{3}( 2\beta_{3} + 2\beta_{4} - 6\beta_{5} - 3\beta_{6} + 2\beta_{7} ). \end{array} $$

(A.5)

As is also pointed out in [3], the form of (A.2a) and (A.2b) reveals that the appearing non-zero couple-stress components may be split into three groups, each one of which interacts independently with a corresponding set of curvature strains. Thus, the couple-stresses that appear in (A.2a) correspond loosely to the so-called “twist” mode met in the mechanics of liquid crystals [13], while the first pair of couple-stresses in (A.2b) corresponds to the “splay” mode for liquid crystals; the second pair corresponds to the “bending” mode.

It is also pointed out that the he result $\bar{m}_{21}= \bar{m}_{31}= 0$ is only a consequence of the linearisation of the non-linear constitutive equation (2.9b); it does not obtain in the non-linear theory, where b≠a and, therefore, it is generally (b ₂,b ₃)≠(0,0). Finally, as is also mentioned in Sect. 3, the linear constitutive equations (A.2a) and (A.2b) do account for possible effects of fibre twist (change of fibre rotation about the x ₁-axis) through non-zero values of $\bar{m}_{11}, \bar{m}_{22}$ and $\bar{m}_{33}$. This is evidently possible if b ₃≠0. Hence, b ₃=4β ₅/3 is identified as the fibre rigidity in twist.

Appendix B: Two Orthogonal Families of Perfectly Flexible Fibres—Local and Special Orthotropy

The important case of local orthotropy is a particular case of the analysis outlined in Sect. 6.1, and is obtained by assuming that the two families of fibres are mutually orthogonal; namely, by assuming that

$$ a_{\ell}^{(1)}a_{\ell}^{(2)} = 0. $$

(B.1)

It is thus seen that the corresponding form of W ^e is obtained by dropping the last term in (6.8). The resulting form of generalised Hooke’s law, namely

(B.2)

involves nine independent elastic moduli only. This is essentially a linear combination of two expressions of the form (3.11), one for each of the two families of fibres, superimposed by the last term that represents coupled action of the those families.

The most interesting case of specially orthotropy may thus be obtained as a further particular case, by assuming that (i) the fibres of both mutually orthogonal families embedded in the material are straight, and (ii) the Cartesian co-ordinate system is chosen so that two of its axes coincide with the mutually orthogonal directions of those families. If, for instance, a ⁽¹⁾=(1,0,0)^T and a ⁽²⁾=(0,1,0)^T, the procedure outlined in Appendix A, transforms (B.2) into the well-known relationship

(B.3)

through an appropriate superposition of (A.1) with its the x ₂-direction counterpart.

It may be worth noting that, by expanding on the notation employed in (A.1), the elastic moduli appearing in (B.3) are found to be:

$$ \everymath{\displaystyle} \begin{array}{l} c_{11} = c_{11}^{(1)} + c_{11}^{(2)},\qquad c_{12} = c_{12}^{(1)} + c_{12}^{(2)},\qquad c_{13} = c_{12}^{(1)} + c_{13}^{(2)}, \\[6pt] c_{22} = c_{22}^{(1)} + c_{22}^{(2)},\qquad c_{23} = c_{23}^{(1)} + c_{12}^{(2)},\qquad c_{33} = c_{22}^{(1)} + c_{11}^{(2)}, \\[6pt] c_{44} = \frac{1}{2} \bigl( c_{22}^{(1)} - c_{23}^{(1)}\bigr) + c_{44}^{(2)},\qquad c_{55} = c_{66}^{(1)} + \frac{1}{2} \bigl( c_{11}^{(2)} - c_{13}^{(2)} \bigr),\qquad c_{66} = c_{66}^{(1)} + c_{44}^{(2)} + \frac{1}{2}\gamma. \end{array} $$

(B.4)

It is recalled in passing that either (B.3) or (A.1) may alternatively be obtained through direct use of (6.12); for details see, for instance, [8, 11, 12].