# Magic angles in the mechanics of fibrous soft materials

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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.

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

or equivalently, by

Sometimes the “magic angle” terminology is used for the complement *θ*_{mc} of this angle (90^{o} − 54.74^{o} = 35.26^{o}) so that

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

and

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

*θ*

_{m}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).

*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

*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).

The maximum volume occurs when *dV*/*dθ* = 0 which yields Eq. (2), i.e., the maximum volume occurs at the magic angle *θ*_{m}. 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*has a

*minimum*at the magic angle

*θ*

_{m}.

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

*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).

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

*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

*θ*is the angle between the principal axis of the interaction and the magnetic field,

*θ*

_{r}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

*θ*

_{r}, however, is at the experimentalist’s disposal. If one sets

*θ*

_{r}=

*θ*

_{m}≈ 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

*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

*B*

_{0}is equal to

*θ*

_{m}(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.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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