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Magic angles in the mechanics of fibrous soft materials

  • C. O. HorganEmail author
  • J. G. Murphy
Original Paper
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Abstract

We discuss a ubiquitous intriguing issue that arises in the mechanics of fibrous soft materials, namely the occurrence of a “magic angle” associated with the fiber direction which gives rise to special features of the mechanical response. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers. An everyday example of the former is the common garden hose. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response and can lead to material instability.

Keywords

Fibrous soft materials Magic angle Fiber-reinforced incompressible elastic solids 

1 Introduction

A ubiquitous intriguing issue arises in a variety of contexts in the mechanics of fibrous soft materials, namely the occurrence of a “magic angle” associated with the fiber direction which leads to special physical properties. Classically, the magic angle concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. It also arises in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. Such angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible elastic soft solids. In this context, the magic angle concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers but also occurs in flat thin sheets reinforced by fibers in the plane. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle result in quasi-isotropic mechanical response of fiber-reinforced composites and have implications for material instability. In this article, we highlight some of the most interesting results on magic angles. The interested reader is directed to the references cited for further details.

The magic angle is defined by
$$ {\theta}_m=\arctan \left(\sqrt{2}\right)=\arcsin \left(\sqrt{2/3}\right)=\operatorname{arccos}\left(1/\sqrt{3}\right)\doteq {54.74}^o $$
(1)

and is often characterized in various applications as the angle for which

$$ 2{\cos}^2\theta -{\sin}^2\theta =0 $$
(2)

or equivalently, by

$$ 3{\cos}^2\theta -1=0\kern0.5em \mathrm{or}\kern0.5em 2-3{\sin}^2\theta =0 $$
(3)

Sometimes the “magic angle” terminology is used for the complement θmc of this angle (90o − 54.74o = 35.26o) so that

$$ {\theta}_{mc}=\arctan \left(1/\sqrt{2}\right)=\operatorname{arccos}\left(\sqrt{2/3}\right)=\arcsin \left(1/\sqrt{3}\right)\doteq {35.26}^o $$
(4)

and in this case the analogs of (2), (3) are

$$ {\cos}^2\theta -2{\sin}^2\theta =0 $$
(5)

and

$$ 3{\sin}^2\theta -1=0,\kern0.75em 2-3{\cos}^2\theta =0 $$
(6)

respectively. We will call the angle (4) the “complementary magic angle.”

The angle (1) can be given a direct geometric characterization. An obvious one is that depicted in the right angle triangle shown in Fig. 1. Moreover, from Fig. 1, we see that θm is the angle between the space diagonal of a unit cube and any of its three connecting edges. See equation (3.4) of [1] for the corresponding definition of the magic angle vector. It is also half of the opening angle formed when a cube is rotated from its space diagonal axis, which may be represented as arccos(−1/3) or 2 \( \arctan \left(\sqrt{2}\right) \) ≈ 109.4712°. This double magic angle is directly related to tetrahedral molecular geometry and is the angle from one vertex to the exact center of the tetrahedron (the tetrahedral angle).
Fig. 1

Geometry of magic angle

The paper by Goriely and Tabor [2] and the recent book by Goriely [3] provide an informative summary of the various contexts in biology where the magic angle concept arises. As pointed out there, apparently one of the first studies where the special angle (1) was encountered was in a seminal paper by Cowey [4] concerned with the locomotion and flattening of worms. For a circular tube reinforced by a double family of inextensible helically wound fibers (modeling fibers of collagen arranged in alternate left- and right-handed geodesic helices), it was shown there that the volume enclosed by a single turn of the helical system is
$$ V=\frac{D^3{\sin}^2\theta \cos \theta }{4\pi } $$
(7)
where D denotes the constant length of one fiber turn and θ is the pitch angle (see Fig. 2; see also Goriely [3] and Horgan and Murphy [5] for further details).
Fig. 2.

A schematic version of Fig. 9a, b of Cowey [4]

The maximum volume occurs when dV/ = 0 which yields Eq. (2), i.e., the maximum volume occurs at the magic angle θm. This classic result for hydrostats was obtained based solely on geometric considerations. Further developments are described in Clark and Cowey [6] who considered the case of collagen fiber extensibility when now the volume of soft tissue enclosed is constant. In this case, (7) can be rearranged to give

$$ D={\left(4\pi V\right)}^{1/3}{\sin}^{-2/3}\theta {\cos}^{-1/3}\theta $$
(8)
and it is easy to verify that D has a minimum at the magic angle θm.

A recent paper by Kim and Segev [7] describes how the intriguing mechanics of an octopus arm also gives rise to a magic angle. The octopus arm, like the elephant trunk, are examples of muscular hydrostats. The magic angle also arises in the field of soft robotics in connection with McKibben actuators that can serve as artificial muscles (see [2, 3] for pertinent references).

2 Mechanics of elastic fibrous soft materials

Recently, there has been considerable interest in the role played by magic angles in the mechanics of fiber-reinforced hyperelastic materials. The applications involving such materials have classically been in connection with rubber and in particular in the area of reinforced rubber tubes and hoses. However, it has also been demonstrated that concepts from continuum mechanics have widespread application in the biomechanics of soft tissues where the fiber reinforcement is now due to collagen fibers in a matrix of elastin. The work of Demirkoparan and Pence [8] is concerned with circular cylindrical hyperelastic tubes reinforced by a symmetric doubly helically wound family of extensible fibers. The tubes are subject to the combined effects of internal pressure and interior wall swelling. A more general framework for fiber reinforcement, but one that does not include wall swelling, was also considered in [2] where non-symmetric fiber families were treated as well as the effect of pre-stretch of the fibers. Both papers use a theory of nonlinear hyperelasticity for orthotropic materials. It is shown how the magic angle separates different response modes in the fiber-reinforced body. For example, a pressurized tube reinforced by a doubly symmetric family of helically wound fibers contracts in length and expands radially if the fibers are wound at an angle smaller than the magic angle while the tube increases in length and contracts radially if the fibers are wound at an angle greater than the magic angle. As was pointed out in [5, 8], in thin-walled closed cylindrical pressure vessels, the maximum strength is obtained when the ratio of the hoop stress to axial stress is 2:1 which occurs at the magic angle. The magic angle is the optimal winding angle for the design of filament-wound structures and is often derived in the composite structures literature by netting analysis (see, e.g., Evans and Gibson [9]). It was pointed out in [8] that the synthetic fibers in fire hose are aligned at the magic angle to minimize sudden jerk as the hose is suddenly turned on. Similarly, the spray hoses in kitchen sinks and the common garden hose are generally reinforced with helical fibers orientated at this angle (see Fig. 3).
Fig. 3

The common garden hose

The potential occurrence of a magic angle in the collagen fiber orientation of the coronary arterial wall has also received some previous attention in the literature. It has been observed in two experimental studies that the pitch angles of the collagen fibers in the adventitia are closely approximated by the magic angle. Schriefl et al. [10] examined 11 human non-atherosclerotic thoracic and abdominal aortas and common iliac arteries. They found two families of collagen fibers wound around the axial axis of the arteries in an almost symmetrical pattern in the adventitia and measured the corresponding angles of pitch to be (53°,  51°) for the thoracic aortas, (50°,  48°) for the abdominal aortas and (53°,  54°) for the common iliac arteries. To approximate the in vivo stress/strain state of the vessel wall and to ensure straightened collagen fibers, a biaxial stretch was first applied to square arterial specimens, which were then chemically fixed and embedded in paraffin wax to enable measurements of the pitch angles to be made using polarized microscopy. This approximation of the in vivo state could explain the small differences between the measured pitch angles and the magic angle (1). A similar finding was reported by van der Horst et al. [11] based on in vitro experiments on porcine and human coronary arteries at physiological loading.

Further results in the mechanics of fibrous incompressible elastic materials are described in [5]. In particular, it is shown there that fibers orientated at the magic angle result in quasi-isotropic mechanical response of fiber-reinforced composites. Furthermore, a generalization of the magic angle concept is given as the angle for which the fiber stretch is zero. For both transversely isotropic and orthotropic fiber-reinforced materials, it is shown that fiber compression can occur at the magic angle resulting in material instability. Further results on the magic angle in the context of mechanics of anisotropic materials are described in Horgan and Murphy [12, 13] and in Horgan et al. [1]. A generalization of the magic angle to the nonlinear deformation regime is also proposed in [5] where the characterization given in (3) is generalized to

$$ {\cos}^2\theta =\frac{1}{1+\lambda +{\lambda}^2} $$
(9)
where λ ≥ 1 denotes a stretch. The specialization to infinitesimal deformations is obtained on letting λ → 1 in (9) so that one recovers the classical magic angle (1) in this limit.

3 Concluding remarks

The objective of this short paper was to describe some results on the magic angle concept in a variety of contexts in the mechanics of fibrous soft materials. It is appropriate to make some remarks of a historical nature regarding the nomenclature of “magic angle.” As was remarked in [2], the term “magic angle” was introduced there to reflect the appearance “as if by magic” in several different settings in biology and mechanics. Furthermore, as was pointed out there and in [3], it turns out that this terminology was also proposed independently much earlier in a completely different context, namely in solid-state nuclear magnetic resonance. In nuclear magnetic resonance (NMR) spectroscopy, the three prominent nuclear magnetic interactions depend on the orientation of the interaction tensor with the external magnetic field. By spinning the sample around a given axis, their average angular dependence becomes
$$ \left\langle 3{\cos}^2\theta -1\right\rangle =\left(3{\cos}^2{\theta}_r-1\right)\left(3{\cos}^2\beta -1\right) $$
(10)
where θ is the angle between the principal axis of the interaction and the magnetic field, θr is the angle of the axis of rotation relative to the magnetic field, β is the (arbitrary) angle between the axis of rotation and principal axis of the interaction, and the angle bracket on the left in (10) denotes an appropriate average. The angle β cannot be manipulated as it depends on the orientation of the interaction relative to the molecular frame and on the orientation of the molecule relative to the external field. The angle θr, however, is at the experimentalist’s disposal. If one sets θr = θm ≈ 54.74°, then we see from (3) that the average angular dependence goes to zero. Another way of characterizing this angle is as the first zero of the second-order Legendre polynomial
$$ {P}_2\left(\cos \theta \right)=\frac{1}{2}\left(3{\cos}^2\theta -1\right) $$
(11)
Magic angle spinning is a technique in solid-state NMR spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution (see, e.g., Hennel and Klowinski [14], Byder et al. [15], Alia et al. [16] for details). The magic angle artifact refers to the increased signal on sequences with short echo time in MR images seen in tissues with well-ordered collagen fibers in one direction (e.g., tendon or articular hyaline cartilage). This artifact occurs when the angle such fibers make with the magnetic field vector B0 is equal to θm (see Fig. 4). For example, this comes into play when evaluating the rotator cuff tendons of the shoulder. The magic angle effect can create the appearance of supraspinatus tendinitis. The review article [15] provides numerous illustrations of magic angle imaging. According to Hennel and Klinowski [14], the name “magic angle” in NMR was originally suggested by the late Professor Gorter of Leiden at a congress in Pisa in 1960.
Fig. 4

Magic angle spinning in NMR (from [16])

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Engineering and Applied ScienceUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of Mechanical EngineeringDublin City UniversityDublin 9Ireland

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