On the role of fibre bonds on the elasticity of low-density papers: a micro-mechanical approach

Abstract

Fine prediction of the elastic properties of paper materials can now be obtained using sophisticated fibre scale numerical approaches. However, there is still a need, in particular for low-density papers, for simple and compact analytical models that enable the elastic properties of these papers to be estimated from the knowledge of various structural information about their fibres and their fibrous networks. For that purpose, we pursued the analysis carried out in Marulier et al. (Cellulose 22:1517–1539, 2015. https://doi.org/10.1007/s10570-015-0610-6) with low-density papers that were fabricated with planar random and orientated fibrous microstructures and different fibre contents. The fibrous microstructures of these papers were imaged using X-ray synchrotron microtomography. The corresponding 3D images revealed highly connected fibrous networks with small fibre bond areas. Furthermore, the evolutions of their Young’s moduli were non-linear and evolved as power-laws with the fibre content. Current analytical models of the literature do not capture these trends. In light of these experimental data, we developed a fibre network model for the in-plane elasticity of papers in which the main deformation mechanisms of the micromechanical model is the shear of the numerous fibre bonds and their vicinity, whereas the fibre parts far from these zones were considered as rigid bodies. The stiffness tensor of papers was then estimated both numerically using a discrete element code and analytically using additional assumptions. Both approaches nicely fit the experimental trends by adjusting a unique unknown micromechanical parameter, which is the shear stiffness of bonding zones. The estimate of this parameter is relevant in light of several recently reported experimental results.

Appendices

Appendix 1

In the theory developed by Cox, the Young’s modulus ${E}_{Cox}$ of planar random fibre networks is:

$$E_{Cox} = \frac{1}{3}E_{f} \phi ,$$

(A1)

where ${E}_{f}$ is the Young’s modulus of the fibres.

In the approach developed by Page and Seth, the Young’s modulus ${E}_{PS}$ of planar random fibre networks is:

$$E_{PS} = \frac{1}{3}E_{f} \phi \left[ {1 - \frac{1}{{r_{1} RBA}}\sqrt {\frac{{E_{f} }}{{2G_{f} }}} } \right],$$

(A2)

where ${G}_{f}$ is the shear modulus of fibres, ${r}_{1} =\overline{l}/\overline{w }$ is the fibre slenderness and RBA is the relative bonded area that can be measured using various experimental techniques or estimated analytically using the Sampson’s model as follows:

$$RBA \approx \left[ {1 - \frac{1}{{\overline{c}}}} \right]\left[ {1 + \frac{{\left( {1 - \phi } \right)\left( {(2 - \left( {2 + \phi } \right)\left( {1 - \phi } \right)} \right)}}{{{\text{ln}}\left( {1 - \phi } \right)}}} \right],$$

(A3)

where $\overline{c }=\overline{w }G/\delta$ and $\delta =1.5\times {10}^{-4}$ g m⁻¹.

In the 2D network model proposed by Wu and Dzenis, the effective Young’s modulus ${E}_{WD}$ and the Poisson ratio ${\nu }_{WD}$ of planar random fibre network are written as a function of the dimensionless fibre density $q$ (for cylindrical fibres of mean diameter $\overline{w }$, $n=4\phi {r}_{1}/\pi$ is the number of fibre per unit of volume) as follows:

$$E_{WD} = \frac{{\pi E_{f} \left( {1 - \nu_{WD}^{2} } \right)}}{{2r_{1} }}n\left\{ {\frac{3}{16} + 2\left[ {n^{2} \left( {\frac{1}{\pi } + \frac{{1 + \frac{2}{\pi }}}{{2r_{1} }}} \right)^{2} - \frac{1}{4}} \right]\left( {\frac{a}{{1 + \nu_{f} }} + b} \right)} \right\}\left[ {1 - {\text{exp}}\left( {\frac{{1 - n/n_{c} }}{2}} \right)} \right],$$

(A4)

and

$$\nu_{WD} = \frac{{\pi^{2} - 32\left\{ {n^{2} \left[ {\frac{1}{\pi } + \left( {1 + \frac{2}{\pi }} \right)\frac{1}{{r_{1} }}} \right] - \frac{1}{4}} \right\}\left[ {\frac{a}{{1 + \nu_{f} }} + b} \right]}}{{3\pi^{2} + 32\left\{ {n^{2} \left[ {\frac{1}{\pi } + \left( {1 + \frac{2}{\pi }} \right)\frac{1}{{r_{1} }}} \right] - \frac{1}{4}} \right\}\left[ {\frac{a}{{1 + \nu_{f} }} + b} \right]}},$$

(A5)

where ${\nu }_{f}$ is the Poisson ratio of the fibres, ${n}_{c}$ is a percolation threshold beyond which fibres form fully connected planar networks without disjointed substructures (for planar networks with random orientations, some numerical studies showed that ${n}_{c}\approx 5.7$ (Åström et al. 2000; Alava and Niskanen 2006)), and $a$ and $b$ are dimensionless numbers that are expressed as a function of both the coordination number $\overline{z }$ and a critical fibre segment length ${l}_{c}\approx \overline{w }/2\sqrt{6(1+{\nu }_{f})}$ as follows:

$$a = \frac{1}{{4\left( {\overline{z} + 1} \right)^{2} }}\mathop \int \limits_{0}^{{\frac{{l\left( {\overline{z} + 1} \right)}}{{l_{c} }}}} \eta \exp \left( { - \eta } \right)d\eta ,$$

(A6)

and

$$b = \frac{{3\overline{w}^{2} }}{{8l^{2} }}\mathop \int \limits_{{\frac{{l\left( {\overline{z} + 1} \right)}}{{l_{c} }}}}^{{\overline{z} + 1}} \frac{{\exp \left( { - \eta } \right)}}{\eta }d\eta .$$

(A7)

Appendix 2

Accounting for all the assumptions stated in “Theoretical upscaling” section, the macroscale stress tensor ${\varvec{\upsigma}}$ Eq. (11) can be expressed as follows:

$${{\varvec{\upsigma}}} = \frac{{k_{b} \overline{z}\overline{b}^{2} d_{max}^{2} n}}{{2\varphi_{1} }}\left( {{\mathbb{B}}_{1} + {\mathbb{B}}_{2} } \right):\user2{\varepsilon ,}$$

(A8)

where

$$\left\{ \begin{gathered} {\mathbb{B}}_{1} = \frac{1}{B}\sum\limits_{N} {\sum\limits_{{\mathcal{B}}} {s_{i}^{{b^{2} }} } } {\mathbf{p}}_{i} \otimes {\mathbf{p}}_{i} \otimes {\mathbf{p}}_{i} \otimes {\mathbf{p}}_{i} \hfill \\ {\mathbb{B}}_{2} = - \frac{1}{B}\sum\limits_{N} {\sum\limits_{{\mathcal{B}}} {s_{i}^{b} } } s_{j}^{b} {\mathbf{p}}_{j} \otimes {\mathbf{p}}_{i} \otimes {\mathbf{p}}_{i} \otimes {\mathbf{p}}_{i} \hfill \\ \end{gathered} \right.,$$

(A9)

By assuming that fibre bonds are uniformly distributed along the fibre length, the following discrete sums can be written (Vassal et al. 2008):

$$\left\{ {\begin{array}{ll} {\mathop \sum \limits_{{\mathcal{B}}} s_{i}^{b2} \approx \overline{z}\frac{{l^{2} }}{12}\left( {1 - \frac{1}{{\overline{z}}}} \right)} \\ {\mathop \sum \limits_{{\mathcal{B}}} s_{i}^{b} \approx 0} \\ \end{array} } \right..$$

(A10)

Then, by assuming that ${\text{s}}_{i}$ and ${\text{s}}_{j}$ are uncorrelated, the expressions of ${\mathbb{B}}_{1}$ and ${\mathbb{B}}_{2}$ are

$$\left\{ {\begin{array}{ll} {{\mathbb{B}}_{1} \approx \frac{{l^{2} }}{6}\left( {1 - \frac{1}{{\overline{z}}}} \right)A} \\ {{\mathbb{B}}_{2} \approx 0} \\ \end{array} } \right.,$$

(A11)

where ${\mathbb{A}}$ is the fourth-order rod orientation tensor defined in Eq. (22). These two relations were validated using the microstructure generator and yielded to the analytical estimates proposed in Eqs. (21) and (23).