Passive elasticity properties of Octopus rubescens arms Free

Flexible octopus arms have long fascinated researchers with their extraordinary ability to deform in all directions, including bending, extending, shearing and twisting (Levy et al., 2017; Hanlon and Messenger; 2018). The goal of this report is to investigate arm elasticity, which not only provides insights into the biomechanics of the arms but also holds implications for a broader spectrum of scientific disciplines, including materials science and soft robotics (Fung, 2013;,Tramacere et al., 2014).

Octopus arm anatomy and movement patterns have been widely studied over the past few decades (Nesher et al., 2020;,Sumbre et al., 2001). Despite the many advances in experimental methods, systematic quantitative characterization of arm elasticity has been scarce until recently. In Tramacere et al. (2014), mechanical properties of octopus arm suckers were reported, whereas elasticity properties of the arm musculature were studied in Mazzolai et al. (2007), Di Clemente et al. (2021), Zullo et al. (2022). On the other hand, mathematical and computational modeling of slender flexible structures, and especially octopus arms, has seen an increased interest from roboticists (Yekutieli et al., 2005;,Laschi et al., 2012;,Rus and Tolley, 2015). Many of these mathematical models (Gazzola et al., 2018; Chang et al., 2020, 2023) make use of nonlinear elasticity theory (Antman, 1995), wherein specification of constitutive models of the elastic material becomes necessary. For example, the mechanical properties of specific muscle groups (Di Clemente et al., 2021;,Zullo et al., 2022) have been beneficial in creating a detailed computational analog of an octopus arm (Tekinalp et al., 2023 preprint). Furthermore, modeling the arm as a single flexible entity which is subject to internal actuation (muscles) has been shown to be productive in gaining deeper mathematical understanding of the arm (Chang et al., 2021, 2023; Wang et al., 2022b, 2024 preprint). All of these computational models create the need for a quantitative analysis of arm elasticity as a whole (as opposed to specific muscle groups or suckers).

The primary contribution of this report is quantitative assessment of the passive elastic properties of octopus arm tissue through biomechanical experiments. Using Octopus rubescens as a model species, tensile and rheological tests are performed to obtain passive elasticity properties of the tissue. Tensile tests yield passive stress–strain curves, revealing the mechanical behavior of the material undergoing axial stretch. For better accuracy of the results, confocal laser scanning microscopy is used to measure arm sample geometries. Rheology tests provide insight into the shear modulus and viscoelastic properties of the material which are then compared against three viscoelasticity models.

Specimens of Octopus rubescens Berry 1953 trapped in Monterey Bay, CA were purchased from Monterey Abalone Co. (Monterey, CA) and housed separately in artificial seawater (ASW) at 11–12°C. Three female animals (90–200 g) were used in these experiments. Animals were fed pieces of shrimp or squid flesh every 1–3 days. All experiments were carried out in accordance with protocol #23015 approved by the University of Illinois Urbana-Champaign (UIUC) Institutional Animal Care and Use Committee (IACUC).

On the day of each experiment, an octopus was anesthetized in 2% ethanol in chilled ASW, one arm was removed using a fresh razor blade, and multiple samples were taken from the isolated arm. Table S1 provides the details of date, animal identification (ID), arm ID and sample ID of each sample taken. Each animal had no more than two arms isolated. For tensile tests, longitudinal cylindrical or half-cylindrical samples were cut from the arm using a razor blade, with minimum diameter of 2 mm (Fig. 1A). For rheology tests, cylindrical samples were cut from the arm with specific dimensions: 8 mm diameter and 1–2 mm length (Fig. 1B). Specifically, longitudinal cylindrical samples were cut simply using a razor blade, while transverse cylindrical segments were cut using a cork borer. All samples were then kept in chilled (11–12°C) 330 mmol l⁻¹ MgCl₂ solution at all times except for when in use for scanning, tensile or rheological testing, to provide muscle relaxation and minimize tissue degradation in samples. The samples were briefly kept in air only while performing the experiments (arm segment measurements, tensile tests and rheology tests), which is a constraint of the machinery we used to assess the elastic properties. Samples were not allowed to dry out during the experiments.

Accurate dimensional measurements of the octopus arm segments are crucial for tensile tests. In particular, the cross-sectional area of the cylindrical samples measured by traditional methods (e.g. using a digital caliper) are unreliable because of the softness of the material. Thus, the segments were measured using a Keyence VK-X1000 3D Optical Profiler (Keyence Cooperation, Osaka, Japan) (Fig. 1A). The segments were briefly removed from chilled MgCl₂ solution and positioned on the center of the microscope stage and their surface morphologies were measured by confocal laser scanning microscopy at room temperature.

After measuring the arm segments, tensile tests were performed to obtain stress–strain curves (Fig. 1A). The experiments were conducted using a DMA Q800 dynamic mechanical analyzer (TA Instruments, New Castle, Delaware, USA). For the tensile test, three main components of the instrument were of interest: sample clamps, drive motor and optical encoder. First, an arm segment whose cross-sectional area (denoted by A₀) had already been measured, was removed from chilled MgCl₂ solution, mounted and secured between the top and bottom clamps of the instrument. The optical encoder was then used to measure the initial length of the sample (L₀). Next, the drive motor was used to apply a unidirectional force (F) to the sample, causing it to elongate. The force was continually increased in small amounts (at a rate of 0.5 N min⁻¹), yielding a stress on the sample, σ=F/A₀. The optical encoder was used to measure the corresponding length L of the sample, resulting in the strain, ε=(L–L₀)/L₀. Note that each sample was stretched only once until it yielded or slipped from the clamp.

All tensile tests were performed in room temperature (23–25°C). To avoid the decay of the arm tissue, samples were kept in chilled MgCl₂ solution, except during the measurement. The arm scanning and tensile tests were conducted in three batches (details provided in Table S1), each batch lasting for ∼1–2 h, depending on batch size. For each of these batches, samples were scanned first (serially), and then the tensile tests were conducted (again, serially). The tensile tests were conducted for each sample separately with each such test lasting for ∼1–5 min.

Rheological experiments were performed using a Discovery hybrid rheometer (DHR-3, TA Instruments, New Castle, Delaware, USA). Both longitudinally and transversely cut cylindrical arm sections (diameter= 8 mm) were used (Fig. 1Bi). The samples were removed from chilled MgCl₂ solution, placed between two parallel plates of diameter 8 mm. Next, torques of various frequencies were applied between the two plates by the rheometer so that the sample's top and bottom surfaces undergo a shear. The instrument then measured both the strain and stress on the sample. Temperature was kept constant at 25°C during all rheological experiments. Rheology tests were conducted in two batches (see details in Table S1), each batch lasting for about 1 h, depending on batch size. For each of these batches, samples were individually tested using the rheometer, following the procedure detailed in the methodology. Each of these individual rheology tests lasted for about 2 min.

For tensile stress–strain measurements, arm sample scan data from Keyence VK-X1000 3D Optical Profiler was analyzed using Keyence MultiFileAnalyzer software v.2.1.2.17. A total of nine samples (see details in Table S1) were used and at least four evenly spaced cross-section measurements were taken for each sample to obtain an average cross-sectional area of 6.618–26.171 mm². For each measurement, a transverse profile line was drawn across the sample surface scan, producing a surface trace (see Fig. 2Ai, right panel). Then an upper cross-sectional area measurement was taken between two points on the surface trace that indicated the bottom left- and right-most edges of the sample. In addition, the distance between these two points was also measured to calculate the average base widths for each sample (Fig. 2Aii).

All nine samples from arm segment measurements were used for tensile tests. The tensile stress–strain data of all the samples were plotted in Fig. 2B. The data showed high variability, partly because samples often slipped from the clamp at high strains (see Fig. S1). For this reason, data up to 50% strain were plotted, with the mean in blue. In addition to the raw data, polynomial fits (Eqn 2) of the orders N=1 (linear; Fig. 2B, red) and N=2 (quadratic; Fig. 2B, green) were also plotted. The coefficients of the fitted polynomials are a₁=48.559 kPa for the linear fit and a₁=3.678 kPa, a₂=119.084 kPa for the quadratic fit. As is seen from Fig. 2B, the quadratic fit matched the mean stress–strain curve. The slopes of the experimental data, indicative of the effective Young's modulus E for a given strain, along with the slopes of the fitted polynomial curves (a₁ for linear fit and a₁+2a₂ε for quadratic fit), are shown in the inset of Fig. 2B. The effective Young's modulus remains in the 1–120 kPa range for strains <50%, consistent with other reported results on soft biological materials (Samani et al., 2007;,Van Kuilenburg et al., 2013; Tramacere et al., 2014).

We found that the experimental stress data show a good match with the quadratic model. While the quadratic model is good for the purposes of this paper, a cubic model will be more useful when also taking into account the compression of the tissue (negative stress). The compression test was not performed in this study and to our knowledge that kind of data is largely scarce in the literature. Mazzolai et al. (2007) reported early data on compression test and they also considered cubic models to fit their data. For a comprehensive treatment, we also report the parameters for cubic fits of our experimental (tensile) data in the supplementary information and the root mean squared errors (RMSE) for fitting each of the models (Table S2), where we can see that the performances of both the quadratic and cubic models are very similar. In fact, the RMSE of the cubic model to fit the experimental E data is slightly higher than the quadratic model (Table S2). In light of this, we conclude that the quadratic model is good enough for the data reported in this paper. However, one can seek to use the cubic model to also include data from the compression test, which will be the subject of our future work.

The results of the rheology experiments are graphically illustrated in Fig. 2C. A total of nine longitudinally and transversely cut samples (Table S1) were used. Experimental data for frequencies up to 10 Hz are reported because of the unreliability of the data for higher frequencies. As is seen from Fig. 2Ci, both the storage (G′) and loss (G″) moduli show upward trends as the frequency of oscillation is increased; however, the increment in G″ was less than that of G′. On the other hand, the tangent of the loss angle (tanδ) shows a slightly downward trend with increasing frequency.

Finally, comparisons against the three viscoelastic models are provided in Fig. 2Cii. The experimental means of both the storage and loss moduli were used to fit the models, and optimal parameter values for each model are shown in Table 1. As seen from Fig. 2Cii, the SLS model yields closest match with the experimental data.

An earlier study Mazzolai et al. (2007) reported stress–strain results using samples cut from octopus arms. In addition to using similar samples, our method uses MgCl₂ solution to neutralize all muscle activity. This implies that both the passive tissue properties and active muscle contractile properties have likely contributed to the stiffer sample results reported by Mazzolai et al. (2007). In particular, the stress on the samples in Mazzolai et al. (2007) reached ∼50 kPa for strains of only 15–25%. In contrast, our samples generated stress around 30 kPa (on average), even at a much higher strain of 50%. Furthermore, our measurement of cross-sectional area of tissue samples via laser scanning microscopy differs from the approximations used by Mazzolai et al. (2007), and may provide greater accuracy in calculating the stress of the tissue samples.

More recent studies of octopus arm elasticity examined active muscle groups, especially longitudinal and transverse muscles (Di Clemente et al., 2021;,Zullo et al, 2022). The stress–strain curves obtained from individual active muscle groups are similar to our results, particularly in shape (cf. fig. 5 of Di Clemente et al., 2021). It is reported in Di Clemente et al. (2021) that the transverse muscles (mean E=∼20–60 kPa) are stiffer than the longitudinal ones (mean E=∼10–40 kPa). In contrast, the arm samples in our experiments are found to be stiffer overall (mean E=∼1–120 kPa), particularly at high strains. The higher stiffness may be explained by several contributing factors, including the complex assembly of multiple muscle groups, nerve cord, connective tissue; and skin; as well as the different species of octopus used.

The stress–strain relationship of every sample followed a similar trend, but there is noticeable variance across samples (Fig. 2B, also see Fig. S1). We note that while the total number of samples is not large, significant variability may persist even with greater numbers of samples owing to the use of tissue from wild-caught octopuses. Wild-type populations are expected to have significant variation in physical characteristics and genetic traits, including those of musculature. Looking at elasticity and rheological differences across different factors, such as age and size of octopus, distal versus proximal arm location etc., requires significantly more animals, arm isolations and data, and is beyond the scope of our paper. In addition, musculature can vary across even individuals of similar age and size, owing to genetic variation and developmental history, which is much harder to control for, especially in wild-type populations.

The limitations of the equipment used to conduct tensile and rheology testing of tissue samples did not allow for testing under physiological conditions (in 12°C seawater). However, care was taken to maximize viability of results in testing the passive properties of octopus arm tissue. For example, outside the brief durations of the experiments, tissue samples were always kept in chilled MgCl₂ solution, to prevent tissue decay and to eliminate spontaneous muscle activity for testing passive tissue properties. In addition, the experiments were done in batches of a few samples at a time. Indeed, variation in temperature has been reported to affect the elasticity properties of materials (Varshni, 1970;,Bianchi et al., 2022). Generally speaking, the modulus of elasticity decreases with increasing temperature. However, it is a complex subject, and requires more targeted experiments to determine the dependency of octopus arm tissue on temperature which goes well beyond the current scope of our current work.

This report presents experimental methods, data, and model fitting to obtain passive elasticity properties of O. rubescens arm tissue. Passive stress-strain data are obtained from tensile tests and arm area measurements. Polynomial model fitting reveals a quadratic relationship between passive stress and strain. Rheological experiments yield measurements of dynamic shear moduli and viscoelastic response of octopus arm tissue. Model fitting against three canonical viscoelasticity models demonstrates close match with a three-element standard linear solid model. Owing to the limitations of the methods and equipment used, the results presented here are not expected to perfectly represent the actual physiological elasticity properties of the octopus arm tissue. However, the results of this study provide a holistic assessment of the passive elasticity properties of octopus arms that is important for several biophysical models (Yekutieli et al., 2005;,Wang et al., 2022a;,Chang et al., 2023). As future work, an informed modeling of arm passive elasticity will be considered, which will then be coupled with detailed modeling of the active components (muscles) (Di Clemente et al., 2021;,Zullo et al., 2022). Modeling of this kind will not only help advance our knowledge of cephalopod biomechanics but will also inspire innovations in soft robotics and materials science (Rus and Tolley, 2015).

The authors gratefully acknowledge Roddel Remy and Kathy Walsh for their help and advice for performing the elasticity experiments, Wei-Chun Kao for assistance in arm sectioning experiments, and the Materials Research Laboratory at the University of Illinois Urbana-Champaign where the elasticity experiments were performed.