Charles Murray van Wyk, a discreet pioneer of fibre network mechanics

Abstract

In 1946, C. M. van Wyk published a seminal theoretical paper on wool mechanics, explaining why the mechanical stress applied to wool is a function of the inverse cube of its volume. In addition to becoming a classical paper in textile science, the paper found applications in diverse fields, ranging from electronics to cell biophysics. Despite being regularly cited for more than 80 years, this author has been forgotten by the history of science and even his full name was publicly unknown. Based on historical archives, I retrace the life of this discreet pioneer of fibre network mechanics, as well as highlighting his contribution to the wool-physics field.

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Life of a wool expert

Charles Murray (C. M.) van Wyk was born on December 1st, 1909 in Britstown (South Africa) (Fig. 1). His parents were Adriaan Jacobus van Wyk (1872–1966) and Margaret Jemima Murray (1877–1948). Murray and Van Wijk (Van Wyk is the Afrikaans spelling of the Dutch surname “Van Wijk”) are two long-standing religious families, deeply rooted in South African history. The Cape Colony, established in 1652 by the Dutch East India Company (VOC) as a resupply station along the sea route to India, was originally intended to promote a unified religion (the Dutch Reformed Church) and a single language (Dutch, which eventually became creolized into Afrikaans). Willem van Wijk (?–1702) – the patrilineal immigrant ancestor- arrived in Cape Colony in 1671 on the ship Gekroond Vrede.¹ At that time the colony only had a few hundred settlers.² He was initially a soldier for the VOC, but was released from his contract with the VOC and became a free citizen, a so-called free burgher. Willem van Wijk became a farmer in Drakenstein (northeast of the Cape). Over the generations, the Van Wijks became a family of ministers in the Dutch Reformed Church. C. M. van Wyk's father, Adriaan Jacobus, and grand-father, Johannes Hendrik (1846–1920), were ministers, as well as many of his uncles and cousins. Both Johannes and Adriaan are locally famous for their contributions to religious journals, Die Kerkbode (The Church Messenger, the former official journal of the Dutch Reformed Church) and Uit die Beek (From the Brook, a biblical diary written in Afrikaans), respectively.

Fig. 1 Charles Murray van Wyk in 1931 (left) and at an unknown date (right).

C. M. van Wyk's maternal side is also a long-standing religious family. In 1806, the British feared that Napoleonic France would invade the Cape Colony and control the sea route to India. As a result, the British took control of the Cape following the Battle of Blaauwberg. In this context, an Anglicization policy was implemented and 25 ministers of the Scottish Presbyterian Church were called from Scotland at the request of Charles Henry Somerset (1767–1831), the governor of the Cape Colony at that time. These ministers were also summoned to address a shortage of ministers in the Dutch Reformed Church and to counterbalance the “liberalism” (real or imagined) of Dutch ministers who were critically re-examining and reinterpreting holy texts. Andrew Murray Sr. (1794–1866) was one of these 25 ministers and gave his name to the South African town of Murraysburg. The son of Andrew Murray Sr., Andrew Murray (1828–1917) became an influential minister and was the opponent of the minister Johannes Jacobus Kotzé (1832–1902) – spokesman of the liberal movement – during the Cape synod in 1862. Another son of Andrew Murray, the minister Charles Murray, was the grandfather of C. M. van Wyk, and gave him his name. Like the Van Wijks, the Murray family has had dozens of ministers over the years, exerting a strong religious influence in South Africa.

Unlike two of his brothers, the ministers Johannes van Wijk (1906–1970) and Adriaan Jacobus (Attie) van Wijk (1912–1976), Charles Murray van Wyk did not pursue the family's religious calling, but instead followed a scientific path. According to academic publication archives, C. M. van Wyk was professionally active between 1932 and 1971. He left South Africa in 1931 for England, where he stayed for about 1 year. He worked at Leeds on wool elasticity for the WIRA (Wool Industries Research Association), a company specialized in textile research. WIRA merged with the Shirley Institute (a research center studying cotton) to form what is known today as the BTTG (British Textile Technology Group). The WIRA was a pioneer in textile research with, for example, the development of the Martindale test (an abrasion test for textiles) and of partition chromatography, for which WIRA researchers Archer Martin and Richard Synge won a Nobel Prize. The scientifically rich environment of the WIRA probably gave the young C. M. van Wyk an excellent training in textile sciences.

When C. M. van Wyk came back to South Africa, he joined the Wool Research Laboratory at the Onderstepoort Laboratory (today known as Onderstepoort Veterinary Institute) close to Pretoria, where he worked from at least 1935 until at least 1947. The Onderstepoort Laboratory was established in response to a devastating rinderpest epidemic in 1896.³ The government of the Transvaal Republic (an independent Boer republic) tasked the veterinarian Arnold Theiler (1867–1936) with developing a vaccine. Theiler implemented several provisional laboratories where he established an immunization process against rinderpest, before establishing a permanent laboratory at Daspoort (Pretoria) in 1898. The Daspoort laboratory was plagued by several problems, including sanitary issues, which led to the death of several assistants from typhoid fever. Hence, in 1908, Theiler established a new laboratory at the farm of De Onderstepoort, then nicknamed the “Extravagant Palace of Science”.³ Onderstepoort became an important hub of veterinary research. A department dedicated to wool research opened in 1931 and a dedicated building was erected in 1934 (Fig. 2). Van Wyk joined the Wool Section and worked on the mechanical characterization of Merino wool. During his time at Onderstepoort, he began working on his PhD under the supervision of Stefan Meiring Naudé (1905–1985), a noted nuclear physicist best known for the discovery of the nitrogen isotope ¹⁵N.⁴ However, because Naudé held a professorship at Stellenbosch University far from Onderstepoort, and was an expert in a different scientific field, his role in Van Wyk's work was likely limited. In 1944, Van Wyk defended his thesis titled A study of the compressibility of wool, with special reference to South African Merino wool, submitted to Stellenbosch University. The manuscript was published in 1946 in the Onderstepoort Journal of Veterinary Science and Animal Industry⁵ as was customary to reduce publication costs. It was also during his years at Onderstepoort that Van Wyk married Roulé van Rhyn (1922–2004) in 1945, a laboratory assistant at Onderstepoort, with whom he had three children.

Fig. 2 Onderstepoort staff. C. M. van Wyk is in the back row, first from the left. His wife, Roulé van Rhyn, is sitting in the front row, second from the right.

Van Wyk left the Onderstepoort Laboratory after completing his PhD, and joined the Grootfontein College of Agriculture, Middelburg. Grootfontein was originally a farm established in 1781 by a Dutch colonist. After the Second Anglo-Boer War (1899–1902), the British army purchased the farm, and between 12.000 and 15.000 soldiers were stationed there. In 1910, following its abandonment by the British troops, Grootfontein was repurposed as a College of Agriculture. While wool research was centralized at Onderstepoort, a dedicated section was also created at Grootfontein in the 1940s. Van Wyk took up his position at Grootfontein in 1949 and he was promoted to director of the college in 1957, a position he held until 1970–1971. During his tenure at Grootfontein, Van Wyk continued his work on wool and fibre physics. In 1971, he was appointed director of the Cedara College of Agriculture near Pietermaritzburg. He held this position for one year and was then promoted to Chief Director Animal Sciences in Pretoria. He retired in 1974 and moved with his family to Strand, a coastal town in the Western Cape Province.

Importantly, the career of C. M. van Wyk took place in a context of strong Afrikaner nationalism and apartheid. In the early 1940s, C. M. van Wyk was active in the Reddingsdaadbond (League for the salvation [of the nation], RDB), an influential religiously-inspired Afrikaner savings fund. Operating under the slogan “A nation rescues itself!”, the RDB's professed goal was to “rescue” the Afrikaners from poverty not through state aid but through ethnic solidarity by mobilizing the savings of Afrikaner farmers and workers to invest in Afrikaner business enterprises. In 1946, the RDB had approximately 65 000 members and supported Afrikaner entrepreneurship and trade training.⁶ The RDB was born in 1939 from the Ekonomese Volkskongress (Economic Congress of the [Afrikaner] people), organized and run by the Afrikaner Broederbond (Afrikaner Brotherhood, AB). AB was an Afrikaner Calvinist and male-only secret society with segregationist goals later put in place by the apartheid policy of the National Party (NP).⁷ Like the AB, the RDB promoted Afrikaner nationalism,⁸ supported the NP and some of its leaders obtained positions in the apartheid state.⁹ An emblematic example is that of Nicolaas Johannes Diederichs (1903–1978), one of the founders of the RDB and a member of the NP, who later became a minister and then president of the apartheid regime. According to direct testimonies, C. M. van Wyk was not involved in the political aspects of Afrikaner nationalism but rather in its charitable activities, influenced by his religious and humanist upbringing. He remained involved in charitable work throughout his life.

C. M. van Wyk is described by people who knew him as a humble and soft-spoken man. He was also a handyman, regularly seen bent over the open bonnet of his Ford. He is remembered as a man with a great sense of humour, collecting silent comedy movies like Charlie Chaplin and Laurel & Hardy. He had a great film collection and used to set up a projector and screen in his garage to watch them all by himself. After his retirement, he was often the main attraction at children's birthday parties in the neighbourhood, showing the movies and enjoying them with the kids. He passed away on September 4th, 1995, at the age of 85 years.

Contribution to textile science

In the early 1930s, South Africa was one of the leaders in Merino wool production. Despite its economic interest, academic knowledge about wool quality and properties remained limited.¹⁰ It was in this context that Van Wyk's research took place, particularly at Onderstepoort. He developed tools and protocols to characterize wool properties in terms of fibre diameter,¹¹ density,¹² breaking strength,¹³ etc. Interestingly, Van Wyk was concerned with the reproducibility of these measurements. He often reported the variance between samples, and addressed several nuisance factors such as the inter-rater variability. He also studied how wool properties were modified by various factors, including the sheep's diet¹⁴ and absorption of water.¹⁵

A significant portion of his career was dedicated to the study of wool compressibility, which was the subject of his PhD.⁵ The first part of his thesis focused on optimising and interpreting wool compressibility experiments. He investigated wool's resistance to compression using both a piston and a device known as the “Pendultex”. The Pendultex was a manufactured device consisting of a pendulum that struck and compressed the wool sample to a predetermined volume. By analyzing the damping of the pendulum after compression, the loss of potential energy can be computed.¹⁶ This energy loss provided an estimate of the work required to compress the sample at a given volume, thereby offering a measure of compressibility. Notably, he empirically demonstrated that the pressure applied to wool is a function of the inverse cube of the volume. Although this cubic dependency was already known,¹⁷ Van Wyk proposed a fitting equation to describe it and formally derived it two years later in his influential 1946 paper.¹⁸ In the second part of his thesis, Van Wyk investigated the various factors that influence wool compressibility, including fibre diameter and length, and the animal's sex and diet.

The Van Wyk model: fibre network mechanics and beyond

Fibre network mechanics refers to systems whose macroscopic mechanical properties emerge from assemblies of high aspect-ratio elements (fibres) connected to each other through physical interactions (e.g., entanglement, cross-linking, etc.), forming a system-spanning structure (network) capable of transmitting stresses.¹⁹ Hence, fibre network mechanics encompasses a broad spectrum of systems exhibiting diverse mechanical behaviours, ranging from polymer networks such as rubber and the eukaryotic cell cytoskeleton,¹⁹ to natural fibrous assemblies such as bird nests,²⁰ and engineered fibrous structures including nonwoven materials. As explained in the previous sections, wool has played an important historical role in fibre network mechanics. Wool is a natural fibre obtained from the fleece of mammals, primarily sheep. Raw wool is typically subjected to physical and/or chemical treatments prior to industrial or artisanal processing. For example, Van Wyk purified wool with benzene at 40 °C and water at 45 °C before teasing it into a loose mass.¹⁸ Wool mechanics began to be studied in the first half of the 20th century, especially compressibility. The wool pressure–volume curve was known to follow a power law²¹ with cubic dependence on the volume.^5,17 However, this power law was empirical and gave no relation between its parameters and the microscopic characteristics of the fibres. In his 1946 paper, Van Wyk developed a randomly oriented fibres semi-empirical model to explain this cubic dependency.¹⁸

The model aims to explain the change in volume V of a randomly oriented fibre network under uniaxial compression assuming that only fibres will mechanically resist through bending. The model places the randomly oriented fibres in a container with cross-sectional area S, height H, and volume V = SH (Fig. 3A). Fibres have a diameter d and a total length L. Conceptually, the model states that, during compression, fibre segments delimited by two inter-fibre contacts bend under the load of a third inter-fibre contact, thereby providing mechanical resistance. Such fibre portion with three contact points is called a node (Fig. 3A, inset).²² The resistance to bending depends on the node length which is twice the fibre length b between two contact points. As a result, the higher the compression, the lower the volume and the lower the inter-contact length b, leading to the stiffening of the network. The model predicts that the pressure applied during compression is a function of the inverse cube of its volume. I will formally derive this result following Van Wyk's historical demonstration.

Fig. 3 Van Wyk model. (A) Fibres of total length L and diameter d are in a container of height H and cross-sectional area S. Mechanically, the fibres can be decomposed as a series of beams of length 2b delimited by two inter-fibre contact points. The length b can be estimated by counting how many times an additional vertical fibre strikes the other fibres. This additional fibre is depicted with the blue cylinder of effective diameter 2d. The number of intersections n between the fibres intersect the cylinder, is related to the inter-contact length by b = H/n. The inset shows a node consisting of a fibre's segment of length 2b deflected between two supporting fibres under the load of a third. A force F deflects the node of a quantity δ. (B) The projection of effective fibre's segments (yellow) fills a certain proportion of the cross-sectional area. The number n can be determined by counting how many projected fibres a random point (blue), representing the additional fibre, would strike. (C) Alternatively, n is the number of times that projected fibres (yellow) intersect a blue disk of radius 2d, modelling the additional fibre.

The inter-fibre contact length b is a kind of mean free path determined by counting how many times an additional vertical fibre (with a length H ≪ L) strikes the other fibres. The additional fibre strikes the other fibres if their axes are at a distance <2d. It is therefore convenient to reformulate the contact condition in terms of an excluded volume by considering an effective fibre diameter 2d. Considering the horizontal projection of a small cylindrical section of height dH, fibres (with effective diameter) will occupy a certain area. The probability for the additional fibre to strike other fibres is equivalent to the probability for a random point to fall in the occupied area of the container's cross-section (Fig. 3B). Following this logic, the total number of striking events is given by the ratio between the projected area of the effective fibres and the cross-section area S. The projected fibres' area is simply 2Ld〈sin θ〉 with 〈sin θ〉 as the average of the sinus of the angle between fibres and the vertical. This average is obtained by integrating over all the angles over the unit half-sphere and leads to π/4. Hence, the additional fibre touches the fibres n = πLd/(2S) times, with inter-contact length b = H/n = 2V/(πLd).

Alternatively, Van Wyk used a second strategy to determine b. He projected fibres' axes in the horizontal plane and counted how many projected axes cross a disk of diameter 2d, modelling the additional fibre (Fig. 3C). The total projected fibre length is Lπ/4 and therefore the length of the fibre in the disk is π²Ld²/(4S). The mean length l of a fibre segment crossing the disk is the mean length of a chord of a circle of diameter 2d; equal to 16d/(3π). Hence, the number of contacts is n = 3π³Ld/(64S), leading to b ≈ 2.16V/(πLd). The numerical difference in the pre-factor of b is now understood through the lens of Bertrand's paradox.²³ Note that n is the number of contacts with the additional vertical fibre of length H, whose length is equal to the cylinder height H. The total number of inter-fibre contacts M is given by M = L/b = 2φL/d with φ = πd²L/(4V) as the fibre volume fraction. This same result was independently derived decades later by different teams who generalized the result to any network where fibre orientations follow arbitrary angular distributions.^24–27 This result holds for “ideal assembly” (also known as “phantom” network) without steric hindrances. Corrections in the number of contacts accounting for steric hindrances was proposed afterwards.^28–30 These subsequent works permit the extension of Van Wyk's results to anisotropic networks, albeit still with uncorrelated orientations.

The nodes can be viewed as mechanically independent, microscopic, fixed beams,²² with two contact points fixing the beam ends and the third contact point applying a compressive force (Fig. 3A). This view is conceptual, since each contact point may act both as a beam support and as a mechanical load. According to beam theory, a force F acting midway along the node is related to a deflection δ by:

where E is the elastic modulus of the fibre and I is the second moment of area of the beam cross-section. For fibres with a circular cross-section, I = πd⁴/64. However, the inter-fibre contact points are not necessarily evenly spaced, and the direction of the force is not necessarily perpendicular to the node. Nevertheless, Van Wyk proposed that a statistical proportionality constant still relates the force and the deflection, such that F ∝ δd⁴/b³.

The container can be divided into sections of height c corresponding to the average vertical projection length of the inter-contact length b. Each section contains N = cL/(bH) segments of length b. When a pressure increment dP is applied to the section, each of the N segment is deflected with . The associated volume variation of the section is proportional to the negative deflection dδ applied on each fibre. Hence, the total volume variation is which leads to . The mean value of c² may be replaced by b²/3. By introducing the fibre mass m and the density of the wool ρ = 4m/(πd²L), we obtain . Integrating this expression yields the final Van Wyk equation:

where V₀ is the volume for P = 0 and k a proportionality constant. Interestingly, this relationship is similar to the previously mentioned power law²¹ with an exponent of 3. Van Wyk's equation is often reformulated in terms of stress σ and true strain ε as σ =α(e^−Bε − 1) with α a constant and the exponent B = 3 for a 3D material following Van Wyk's model.³¹ Interestingly, for 2D random plane structures, B = 5.²⁷ It has been pointed out that the value of c² = b²/3 used in the original Van Wyk work actually neglects fibre thickness.³² The correction of this error reveals that Van Wyk's equation omits fourth- and fifth-order terms.³³ Lastly, the nonlinear increase in pressure with fiber volume fraction is reminiscent of the behavior of cubic equations of state in thermodynamics, such as the van der Waals equation. This conceptual bridge steams from the existence of an excluded volume which, in both fiber networks and fluids, imposes a geometric constraint that amplifies resistance as the system approaches dense packing.

Van Wyk's model is built on a limited set of assumptions: (i) randomly oriented fibres' segments and (ii) the mechanical resistance is solely due to fibre segments bending. The first assumption is that, locally, fibres' segments are distributed homogeneously, isotropically and in an uncorrelated manner. Moreover, this random organization is assumed to be preserved during compression. This assumption supports the mean-field approximation used to compute the inter-contact length. The second assumption neglects other mechanical resistance occurring during network compression such as poroelasticity, other modes of deformation of individual fibres, or frictions between fibres. Moreover, it is worth noting that the model is independent of the actual number of fibres, only the total fibre length L is relevant. The extreme scenario would be a single fibre of length L, as depicted in Fig. 3A. Hence, the model does not explicitly account for the finite length of individual fibres or their aspect ratio. As a result, it neglects discreteness effects that may influence the number of contacts per fibre.^23,34

Since 1946, Van Wyk's model has demonstrated good empirical agreement for diverse fibrous materials, ranging from textiles to ionotronic sensors, and has been the basis of many subsequent models^35–40 (Fig. 4). Interestingly, the Van Wyk theory has more recently been applied to biopolymers such as fibrin networks⁴¹ and the actin network.⁴² The latter is a key constituent of the eukaryotic cell, involved in cell mechanical resistance, migration, and division.⁴³ Interestingly, actin networks exhibit a non-linear elastic behaviour with characteristic stress-stiffening both in reconstituted ex cellulo systems^44,45 and in cellulo.⁴⁶ Recently, Bouzid et al. used magnetic micro-cylinders to compress and measure the stress-stiffening of reconstituted branched actin gels. The creation of new contact points during the compression predicted by Van Wyk's model well explained their measurements. They further validated this interpretation using molecular simulations of branched actin network with and without steric interactions. Only simulations with steric interactions reproduced the stress-stiffening curve. Interestingly, at low stress (for small V₀/V), Van Wyk's model overestimates the stiffness. Because such actin gels have small filament segments (potentially smaller than b), Bouzid et al. interpreted this deviation as the sign that some segments are not mechanically engaged in the compression. Indeed, in this low compression regime, fibre segments typically have more degrees of freedom than constraints, explaining the softer response. The authors analytically extended Van Wyk's model for under-constrained fibres at low compression while preserving the typical Van Wyk asymptotic behaviour at higher compressions. This model fits both molecular simulations and experimental data remarkably well.⁴² It illustrates that Van Wyk's model is still relevant and continues to inspire theoretical developments, beyond the textile science field.

Fig. 4 Stress as a function of V₀/V. The 10 different materials exhibit a cubic dependence on the volume. Data from ref. 31, 42 and 47–50.

Conclusion

The seminal Van Wyk model provides a microscopic interpretation of the volume–pressure relationship in fibrous materials and remains a foundational reference in fibre network mechanics. Despite the model's enduring influence,⁵¹ the author's full identity and background have long remained unknown. My modest historical inquiry opens a window into the context in which this knowledge production emerged, which is especially pertinent in the contemporary moment, as scientific eponyms are questioned.⁵² My investigation, however, did not reveal any direct link between Van Wyk and apartheid. Such clarity may be of interest to both curious readers and institutions mindful of the commemorative significance behind the names they honour.

The continued relevance of Van Wyk's model lies in its ability to capture, with minimal assumptions, the essential physics of compression in disordered fibre networks. The central idea is that the compression of a fibrous network is a bending-dominated deformation with the progressive accumulation of inter-fibre contacts resulting in a non-linear elastic behavior. This central mechanism remains valid across a wide range of modern fibrous materials, including cellular components. Consequently, the model is still widely used as reference for interpreting experimental data and for guiding the development of more detailed microstructural and numerical approaches.

Ethics statement

All biographical information in this publication has been included with the knowledge and consent of the subject's family, who have approved its release.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data supporting the findings of this study are available within the manuscript and its supplementary information (SI). Primary archival sources used to reconstruct the biography of Charles Murray van Wyk were obtained from publicly accessible and private collections with permission. Historical documents are documented in the SI. Supplementary information is available. See DOI: https://doi.org/10.1039/d6sm00220j.

Acknowledgements

I would like to warmly thank Professor Lawrance Hunter (Department of Textile Science, Nelson Mandela University), who immediately shared my curiosity and helped me with my research. C. M. van Wyk was the external examiner of Hunter's master's dissertation in 1968, but they never physically met. I also warmly acknowledge Lize-Mari Erasmus (University of Pretoria) for the information she conveyed to Professor Hunter. I also deeply thank Myleen Oosthuizen and Annette Boshoff from the Veterinary History Society of South Africa. I would probably never have found all this information without their precious help. I am also grateful to Mehdi Bouzid for sharing some data used in Fig. 4, and to Caitlin Martin for her critical reading of the manuscript and her help in the manuscript editing. Finally, I thank the many people who helped me in my inquiry and shared my enthusiasm. More information and sources can be found in the supplementary document. This work was supported in part by an post-doctoral fellowship from Foundation ARC.

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