A correction to van Wyk’s model of the compressibility of fibrous materials

Abstract

van Wyk put forward a compression model of fibrous materials utilizing a library of analytical approaches, including the continuum mechanics, stereological, geometrical probability, least square method, and excluded area concept. In this letter, we wish to point out a key error noted in van Wyk’s work with the objective of correcting misconceptions that are held by the majority of us. Through this contribution, we question the “inverse cube” pressure-volume relationship of random fibrous materials. The pressure-volume relationship has been revisited by modifying the formulation of the mean length of a fiber element between consecutive contacts projected on the compression direction.

Keywords

bending,compression,fibrous materials,planar,random.

Seventy-five years ago, van Wyk put forward a compression model of fibrous materials in his seminal paper titled “Note on the compressibility of wool”.¹ A key relationship between the pressure and volume of wool fibers was formulated utilizing a library of analytical approaches, including the continuum mechanics, stereological, geometrical probability, least square method, and excluded area concept. van Wyk¹ derived the compression model of random fibrous materials purely in terms of bending strains exhibited by the constituent fibers, such that other types of deformations, namely, shearing, twisting, slippage, and fiber extension, were neglected. Although the bending mode of fiber deformation was the backbone of the compression model, van Wyk’s work¹ has been explored in a plethora of textile and allied materials with varying length scales.^2–6 For over seven decades, we have “tip-toed around” with the tacit understanding that the strengths of van Wyk’s work outweigh the weaknesses.^2,7 In this letter, we wish to point out a key error noted in van Wyk’s¹ work with the objective of correcting misconceptions that are held by the majority of us.

van Wyk¹ developed an inverse cube relationship between the pressure and volume of fibrous mass. However, we question this “inverse cube” pressure-volume relationship by pointing out a key error. This has been achieved by modifying the formulation of the mean length of a fiber element between consecutive contacts projected on the compression direction. Before we present our viewpoint, it is imperative to summarize van Wyk’s work.¹ It was assumed that the fiber elements were randomly oriented such that the effects of frictional forces were negligible.¹ For simplification, van Wyk¹ considered an ideal model of fiber as a rod that is supported at a large number of points uniformly spaced at distances 2b apart. A force, F, then acted midway by an additional fiber between the points of support, assuming that the force on the fiber element increased by dF and the deflection created by it is dy. Therefore, a fundamental relation between dF and dy was developed on the basis of classical beam deflection theory,¹ i.e.,

d F = \frac{kiY}{b^{3}} d y

(1)

where

b = \frac{2 v}{π l d}

(2)

where i is the area moment of inertia of the rod, b is the mean length of a fiber element between consecutive contacts, Y is the Young’s modulus of the rod, l is the total length of fiber in volume v, d is the fiber diameter, and k is an empirical constant.

Subsequently, the fibers were considered to be enclosed in a vertical container having a unit cross-sectional area, and the height (and volume) occupied by the wool was assumed to be v. Furthermore, the fibrous mass was divided into layers of thickness c such that the fiber elements in each layer share the compressive load. In other words, a fiber element was presumably bound between horizontal planes, which were separated by a distance c, and the number of such fiber elements in the layer was $\frac{c l}{v b}$ . Thus, an infinitesimal increase in pressure dp was applied to the fiber elements such that each fiber element exhibited dF. Therefore,

d p = \frac{c l}{v b} d F

(3)

The concept of continuum mechanics was then applied such that the thickness of each layer c reduced by dy, and accordingly, the change in the volume was given by,

d v = - \frac{v}{c} d y

(4)

Combining equations (1), (3), and (4) we obtain,

d p = - \frac{kiY c^{2} l}{v^{2} b^{4}} d v

(5)

The erroneous calculation of c is a key issue, i.e., $c^{2} = \frac{b^{2}}{3}$ was computed by considering the fiber elements present in three-dimensional (3D) random fibrous material. van Wyk¹ presumed that the microelement length of b had been projected on the Z-axis based upon the spherical coordinate system. Applying the well-known concept that the area occupied by a randomly placed fiber element in the infinitesimal region formed on the unit sphere within the range of $θ \sim θ + d θ$ and $φ \sim φ + d φ$ is $\sin θ d θ d φ$ , where the $θ$ and $φ$ are polar and azimuthal angles formed with the Z and X axes, respectively, such that $0 \leq θ \leq π$ and $0 \leq φ \leq 2 π$ . Thus, the mean value of $c^{2}$ was computed with the help of spatial averaging by considering the projection of c ( $= b \cos θ$ ) on the Z-axis, i.e., $c^{2} = b^{2} \frac{\int_{0}^{2π} \int_{0}^{π} | \cos^{2} θ \sin θ | d θ d φ}{\int_{0}^{2π} \int_{0}^{π} \sin θ d θ d φ} = \frac{b^{2}}{3}$ . To gain further insight into c, let us assume that the fiber alignment is planar random, i.e., $θ = \frac{π}{2}$ . In other words, the fibers are randomly placed primarily in the in-plane direction. In this scenario, what would be the value of $c^{2}$ ? Perhaps, there would be no resistance to compression load from planar fibers based upon van Wyk’s analysis¹, and apparently, there is a breakdown of the pressure-volume model.

To obtain the correct expression of c, it is essential to look at the projection of the fiber element length b between the two contacts on the Z-axis carefully, as depicted in Figure 1.

Figure 1.

A fiber element between consecutive contacts in the XZ plane. Here $θ$ is the polar angle that renders the alignment of the fiber with the Z-axis or the compression direction.

Here,

c = b \cos θ + d \sin θ

and this expression of c has been pointed out earlier by Wakayama and Hattori.⁸ A similar expression of c has also been derived for determining the fiber orientation in short fiber-reinforced composites.⁹ Now, the mean value of

c^{2}

can be again computed with the help of spatial averaging, i.e.,

c^{2} = b^{2} \frac{\int_{0}^{2 π} \int_{0}^{π} \cos^{2} θ \sin θ d θ d φ}{\int_{0}^{2 π} \int_{0}^{π} \sin θ d θ d φ} + d^{2} \frac{\int_{0}^{2 π} \int_{0}^{π} \sin^{2} θ \sin θ d θ d φ}{\int_{0}^{2 π} \int_{0}^{π} \sin θ d θ d φ} + 2 b d \frac{\int_{0}^{2 π} \int_{0}^{π} | \sin θ \cos θ | \sin θ d θ d φ}{\int_{0}^{2 π} \int_{0}^{π} \sin θ d θ d φ}

c^{2} = \frac{b^{2} {+ 2 d}^{2} + 2 b d}{3}

(6)

In addition, consider the fibers to be circular in cross-section, then

i = \frac{π d^{4}}{64}; v_{f} = \frac{π d^{2} l}{4} = \frac{m}{ρ}

(7)

where v_f, ρ, and m are the volume, density, and mass of fibers, respectively.

Combining equations (2), and (5) – (7) and integrating them,

p = \frac{k Y m^{3}}{ρ^{3}} [\frac{1}{36} (\frac{1}{v^{3}} - \frac{1}{v_{0}^{3}}) + \frac{m}{12 ρ} (\frac{1}{v^{4}} - \frac{1}{v_{0}^{4}}) + \frac{2 m^{2}}{15 ρ^{2}} (\frac{1}{v^{5}} - \frac{1}{v_{0}^{5}})]

(8)

where

v_{0}

is the initial volume of the fibrous mass when

p = 0

The original pressure-volume equation of van Wyk¹ is given by,

p = \frac{k Y m^{3}}{36 ρ^{3}} (\frac{1}{v^{3}} - \frac{1}{v_{0}^{3}})

(9)

When comparing equations (8) and (9), it can be clearly seen that there are additional terms, which were neglected in the original van Wyk’s work.¹ It should be noted that the modified pressure-volume relation is significantly useful for textile materials that have a planar structure. Irrefutably, van Wyk’s approach to the compressibility problem was highly original and comprised other gems that included the formulation of the mean distance between the fiber-fiber contacts. Needless to say, van Wyk introduced us to novel approaches and ideas, such as excluded volume, stereological spatial averaging, and many others that have been a cornerstone in the development of the theory of fiber-fiber contacts.^10–13 van Wyk’s work¹ will always remain important in the field of textiles as it laid the foundation for treating the compression problem and, in general, provided deep insight into the mechanistic nature of fibrous materials.

Footnotes

Acknowledgments

We thank Mr. Danvendra Singh for useful discussions and Mr. Siddharth Shukla for his skillful assistance in constructing the schematic.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Amit Rawal

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