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Continuous models for peristaltic locomotion with application to worms and soft robots

Abstract

A continuous model for the peristaltic locomotion of compressible and incompressible rod-like bodies is presented. Using Green and Naghdi’s theory of a directed rod, incompressibility is enforced as an internal constraint. A discussion on muscle actuation models for a single continuum is included. The resulting theory is demonstrated in a simulation of a soft-robotic device. In addition, a calibration of parameters is performed and the incompressible rod is validated against a biomimetic model of earthworm locomotion.

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Notes

  1. A review of, and perspective on, Green and Naghdi’s work can be found in O’Reilly (2017) and Rubin (2000).

  2. These assigned forces should not be confused with the active piece of the internal forces in the presence of an internal constraint.

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Acknowledgements

The authors are grateful to Professor William Kier (University of North Carolina at Chapel Hill) for helpful guidance on the mechanics of a worm.

Funding

The work of Evan Hemingway was supported by a Berkeley Fellowship from the University of California at Berkeley and a U.S. National Science Foundation Graduate Research Fellowship. The work of Oliver O’Reilly was supported by Grant Number W911NF-16-1-0242 from the U. S. Army Research Organization administered by Dr. Samuel C. Stanton.

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Correspondence to Oliver M. O’Reilly.

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Appendix

Appendix

Derivations from three-dimensional continuum mechanics

Fig. 14
figure 14

a The reference configuration of a solid cylindrical three-dimensional continuum with sub-volume \({\mathcal{V}}_0\) and b cross-sectional area \({\mathcal{A}}_0 (\xi )\) at some cross section located at \(\xi\)

Consider a continuous distribution of material that is assembled in flat three-dimensional space \({\mathcal{E}}^3\) as displayed in Fig. 14. If this configuration is the natural shape that the unloaded body takes, then we define the spatial region that the material occupies as the reference configuration. We say that a body is “rod-like” if two of its characteristic length dimensions are small compared to another direction, called the longitudinal direction, or the long axis. Under external forcing, the rod-like body undergoes a motion and occupies current configurations in time that may be different from, but are described relative to, the reference configuration. The developments of this “Appendix,” which are simplified versions of the procedures developed by Green et al. (1974a, b), Naghdi (1973, 1975), are to hold in describing the deformed equilibria of rod-like bodies under a set of applied loads.

Let Cartesian coordinates \(X^1,X^2,\) and \(X^3\) cover the reference space. Some of these coordinates describe points that coincide with the placement of material points of the body in its reference configuration. We take the triplet \(\left( X^1, X^2, X^3 \right)\) to label a specific three-dimensional material point. Then, the coordinates are thought to be convected with the motion of the body in addition to providing a Cartesian coordinate system for the reference space. The coordinates are oriented so that the straight line \(\left( 0 , 0 , X^3 \right)\) identifies a centerline for the rod-like body with \(X^3\) taking on values between 0 and \(\ell _0\), where \(\ell _0\) is some physical length determined by a choice of units for \(X^i\). A new coordinate \(\xi\) is introduced for parameterizing the centerline, where \(\xi \in \{ X^3, 0 \le X^3 \le \ell _0 \}\).

A position vector to the reference centerline relative to some fixed point O may be constructed as a function of \(\xi\):

$$\begin{aligned} {\mathbf{R}} = {\mathbf{R}} \left( \xi \right) . \end{aligned}$$
(11.1)

Then, the reference position to some three-dimensional material point can be computed:

$$\begin{aligned} {\mathbf{R}}^* = {\mathbf{R}}^* \left( X^{\alpha }, \xi \right) = {\mathbf{R}} + X^{\beta } {\mathbf{D}}_{\beta }, \end{aligned}$$
(11.2)

where the vectors \({\mathbf{D}}_1\) and \({\mathbf{D}}_2\) are known as the reference directors and defined as

$$\begin{aligned} {\mathbf{D}}_{\alpha } = {\mathbf{D}}_{\alpha } \left( \xi \right) = \frac{\partial {\mathbf{R}}^*}{\partial X^{\alpha }} \quad \text{evaluated along }\left( 0, 0 ,\xi \right) . \end{aligned}$$
(11.3)

The directors \({\mathbf{D}}_{\alpha }\) have unit length owing to the choice of physical length dimension for \(X^{\alpha }\). The directors span cross sections of the rod-like body. We align an inertial basis \(\{ {\mathbf{E}}_i \}\) so that

$$\begin{aligned} {\mathbf{E}}_i = {\mathbf{D}}_i, \end{aligned}$$
(11.4)

where \({\mathbf{D}}_3 = \frac{\partial {\mathbf{R}}}{\partial \xi }\).

Suppose \({\mathcal{V}}_0\) identifies an arbitrary open and bounded sub-volume of material in its reference configuration. Let \(\partial {\mathcal{V}}_0\) denote the boundary of \({\mathcal{V}}_0\). To develop one-dimensional quantities for a rod theory, we choose \({\mathcal{V}}_0\) specially as in Fig. 14 so that it is bounded by two cross sections, \({\mathcal{A}}_0 \left( \xi _1 \right)\) and \({\mathcal{A}}_0 \left( \xi _2 \right)\), and the lateral surface of the rod which is defined as the union of cross-sectional boundaries:

$$\begin{aligned} {\mathcal{L}}_0 \left( \xi _1, \xi _2 \right) = \bigcup _{ \xi _1 \le \xi \le \xi _2 } \partial {\mathcal{A}}_0 \left( \xi \right) . \end{aligned}$$
(11.5)

We may think of \({\mathcal{V}}_0\) now as being determined by two endpoints, \(\xi _1\) and \(\xi _2\):

$$\begin{aligned} {\mathcal{V}}_0 \left( \xi _1, \xi _2 \right) = {\mathcal{A}}_0 \left( \xi _1 \right) \bigcup {\mathcal{L}}_0 \left( \xi _1, \xi _2 \right) \bigcup {\mathcal{A}}_0 \left( \xi _2 \right) . \end{aligned}$$
(11.6)

The same material that occupies the regions and surfaces \({\mathcal{V}}_0\), \({\mathcal{A}}_0\), and \({\mathcal{L}}_0\) occupies \({\mathcal{V}}\), \({\mathcal{A}}\), and \({\mathcal{L}}\) in the current configuration, respectively.

In the current configuration, we assume the following approximation for the location of three-dimensional material points to hold:

$$\begin{aligned} {\mathbf{r}}^* = \mathbf{r}^* \left( X^{\alpha }, \xi , t \right) \approx {\mathbf{r}} + X^{\beta } {\mathbf{d}}_{\beta }. \end{aligned}$$
(11.7)

Here, \({\mathbf{r}} = \mathbf{r} \left( \xi , t \right)\) and \({\mathbf{d}}_{\alpha } = {\mathbf{d}}_{\alpha } \left( \xi ,t \right)\) are the current configuration centerline position vector and directors, respectively. Approximation (11.7) presumes that planar cross sections remain planar and that cross section deformations are homogeneous through the entire section. Additionally, we have the current configuration centerline tangent vector

$$\begin{aligned} {\mathbf{e}}_t = \frac{\partial {\mathbf{r}}}{\partial s} = \mu _3^{-1} {\mathbf{d}}_3, \end{aligned}$$
(11.8)

where s is a current configuration centerline arc length, \(\mu _3 = \frac{{\mathrm{d}} s}{{\mathrm{d}} \xi }\) is the centerline stretch, and we have adopted the notation \({\mathbf{d}}_3 = \frac{\partial {\mathbf{r}}}{\partial \xi }\). It is convenient to introduce the basis \(\{ {\mathbf{e}}_i \}\), defined as the unit direction of the current configuration directors:

$$\begin{aligned} {\mathbf{e}}_i = \frac{{\mathbf{d}}_i}{\left\| {\mathbf{d}}_i \right\| }, \quad \text{(no sum)}. \end{aligned}$$
(11.9)

We may introduce the cross-sectional stretches \(\mu _1\) and \(\mu _2\) as

$$\begin{aligned} \mu _{\alpha } = \left\| {\mathbf{d}}_{\alpha } \right\| . \end{aligned}$$
(11.10)

Suppose the three-dimensional mass density per unit reference volume is \(\rho _0^*\), while the density per unit current volume is \(\rho ^*\). Conservation of mass for a specially chosen sub-volume \({\mathcal{V}}_0\) may be stated as

$$\begin{aligned} \int _{{\mathcal{V}}_0} \rho _0^* {\mathrm{d}}V = \int _{{\mathcal{V}}} \rho ^* {\mathrm{d}}v. \end{aligned}$$
(11.11)

Here, \({\mathrm{d}}V\) and \({\mathrm{d}}v\) are small measures of volume in the reference and current configurations, respectively. One may calculate

$$\begin{aligned} {\mathrm{d}}V = {\mathrm{d}}X^1 {\mathrm{d}}X^2 {\mathrm{d}}\xi = {\mathrm{d}} A {\mathrm{d}} \xi , \end{aligned}$$
(11.12)

where \({\mathrm{d}}A\) is a small area measure for the reference configuration cross sections. An implication of (11.7) is that all volume elements \({\mathrm{d}}V\) in a given reference cross-sectional volume deform the same way into their \({\mathrm{d}}v\) counterparts in the current configuration. Therefore, for the current configuration we have

$$\begin{aligned} {\mathrm{d}}v = \left( {\mathbf{d}}_1 \times {\mathbf{d}}_2 \right) \cdot {\mathbf{d}}_3 {\mathrm{d}}A {\mathrm{d}}\xi = \left( {\mathbf{e}}_1 \times {\mathbf{e}}_2 \right) \cdot {\mathbf{e}}_3 {\mathrm{d}}a \mu _3 {\mathrm{d}}\xi , \end{aligned}$$
(11.13)

where \({\mathrm{d}}a = \mu _1 \mu _2 {\mathrm{d}}A\) is a small cross-sectional area measure for the current configuration. Factoring the integration along the centerline out of balance law (11.11) we find

$$\begin{aligned} \int _{\xi _1}^{\xi _2} \left( \int _{{\mathcal{A}}_0 \left( \xi \right) } \rho _0^* {\mathrm{d}}A - \mu _3 \left( {\mathbf{e}}_1 \times {\mathbf{e}}_2 \right) \cdot {\mathbf{e}}_3 \int _{{\mathcal{A}} \left( \xi \right) } \rho ^* {\mathrm{d}}a \right) {\mathrm{d}}\xi = 0. \end{aligned}$$
(11.14)

We may now define the one-dimensional reference and current densities:

$$\begin{aligned} \rho _0 = {\rho }_0 \left( \xi \right) = \int _{{\mathcal{A}}_0 \left( \xi \right) } \rho _0^* {\mathrm{d}}A, \end{aligned}$$
(11.15)

and

$$\begin{aligned} \rho = {\rho } \left( \xi , t \right) = \left( {\mathbf{e}}_1 \times {\mathbf{e}}_2 \right) \cdot {\mathbf{e}}_3 \int _{{\mathcal{A}} \left( \xi \right) } \rho ^* {\mathrm{d}}a. \end{aligned}$$
(11.16)

Hence, the balance of mass for any part \({\mathcal{V}}_0 \left( \xi _1, \xi _2 \right)\) becomes

$$\begin{aligned} \int _{\xi _1}^{\xi _2} \left( \rho _0 - \mu _3 \rho \right) {\mathrm{d}}\xi = 0, \end{aligned}$$
(11.17)

which contains quantities that are only a function of \(\xi\) and time. Since (11.17) is assumed to hold true for all sub-volumes \({\mathcal{V}}_0 \left( \xi _1, \xi _2 \right)\), then the quantity in the parentheses must be zero, and we arrive at the local, or point-wise, one-dimensional balance of mass:

$$\begin{aligned} \rho _0 = \mu _3 \rho . \end{aligned}$$
(11.18)

To develop the one-dimensional balance of linear momentum for equilibria, we may start with postulating the local three-dimensional version:

$$\begin{aligned} \text{Div} \mathbf{P} + \rho _0^* \mathbf{b} = \mathbf{0}. \end{aligned}$$
(11.19)

Here, \(\mathbf{P}\) is the first Piola–Kirchhoff stress tensor, \(\mathbf{b}\) is a body force per unit mass, and \(\text{Div}\) is a three-dimensional reference-based divergence. Let \(\mathbf{p}\) be the traction per unit reference area acting on \(\partial {\mathcal{V}}\) by neighboring material points. The vector \(\mathbf{p}\) is delivered in the interior as \(\mathbf{p} = \mathbf{P} \mathbf{N}_0\), where \(\mathbf{N}_0\) is a unit normal to \(\partial {{\mathcal{V}}}_0,\) the preimage of \(\partial {\mathcal{V}}\) in the reference configuration. To get at the one-dimensional balance, we integrate (11.19) over the specially chosen \({\mathcal{V}}_0\) described in (11.6) and factor the integral:

$$\begin{aligned} \int _{\xi _1}^{\xi _2} \int _{{\mathcal{A}}_0 \left( \xi \right) } \left( \text{Div} \mathbf{P} + \rho _0^* \mathbf{b} \right) {\mathrm{d}}A {\mathrm{d}} \xi = \mathbf{0}. \end{aligned}$$
(11.20)

An application of the divergence theorem yields the identity

$$\begin{aligned} &\int _{\xi _1}^{\xi _2} \int _{{\mathcal{A}}_0 \left( \xi \right) } \text{Div} \mathbf{P} {\mathrm{d}}A {\mathrm{d}}\xi \nonumber \\&\quad =\int _{{\mathcal{A}}_0\left( \xi _1 \right) } \mathbf{P} \left( - {\mathbf{E}}_3 \right) {\mathrm{d}}A + \int _{{\mathcal{A}}_0\left( \xi _2 \right) } \mathbf{P} {\mathbf{E}}_3 {\mathrm{d}}A + \int _{\xi _1}^{\xi _2} \oint _{\partial {\mathcal{A}}_0 \left( \xi \right) } \mathbf{p} {\mathrm{d}}U {\mathrm{d}}\xi , \end{aligned}$$
(11.21)

where dU is a small length measure on the external boundary. It is understood that \(\mathbf{p}\) in (11.21) is not delivered by \(\mathbf{P}\), but is supplied externally on the boundary. Now, defining the contact force \({\mathbf{n}}\) as

$$\begin{aligned} {\mathbf{n}} = {\mathbf{n}} \left( \xi , t \right) = \int _{{\mathcal{A}}_0\left( \xi \right) } \mathbf{P} {\mathbf{E}}_3 {\mathrm{d}}A, \end{aligned}$$
(11.22)

and the assigned force acting on the centerline as

$$\begin{aligned} \rho _0 {\mathbf{f}} = \rho _0 {\mathbf{f}} \left( \xi , t \right) = \oint _{\partial {\mathcal{A}}_0 \left( \xi \right) } \mathbf{p} {\mathrm{d}}U + \int _{{\mathcal{A}}_0 \left( \xi \right) } \rho _0^* \mathbf{b} {\mathrm{d}}A, \end{aligned}$$
(11.23)

we have

$$\begin{aligned} {\mathbf{n}} \left( \xi _2 , t \right) - {\mathbf{n}} \left( \xi _1, t \right) + \int _{\xi _1}^{\xi _2} \rho _0 {\mathbf{f}} {\mathrm{d}} \xi = \mathbf{0}. \end{aligned}$$
(11.24)

Assuming continuity of \({\mathbf{n}} \left( \xi , t \right)\) in \(\xi\) and using the fundamental theorem of calculus, expression (11.24) may be written as

$$\begin{aligned} \int _{\xi _1}^{\xi _2} \left( {\mathbf{n}}' + \rho _0 {\mathbf{f}} \right) {\mathrm{d}} \xi = \mathbf{0}, \end{aligned}$$
(11.25)

where \(\left( \cdot \right) ' = \frac{\partial \left( \cdot \right) }{\partial \xi }\). The domain of integration is now one-dimensional, and all integrand quantities are spatially dependent only on \(\xi\), as the dependence on cross-sectional coordinates has been integrated away. Since (11.25) holds true on all sub-lengths \(\left( \xi _1, \xi _2 \right)\) of the whole length \(\left[ 0, \ell _0 \right]\), then it must be true that

$$\begin{aligned} {\mathbf{n}}' + \rho _0 {\mathbf{f}} = \mathbf{0}, \end{aligned}$$
(11.26)

which is the local form of the one-dimensional balance of linear momentum for static equilibria. If \({\mathbf{d}}\) in (11.24) is not continuous in \(\xi\), we may use the Leibniz rule to establish a jump condition at the singular point \(\xi = \gamma\):

$$\begin{aligned} \llbracket {\mathbf{n}} \rrbracket _{\gamma } + {\mathbf{F}}_{\gamma } = \mathbf{0}, \end{aligned}$$
(11.27)

where

$$\begin{aligned} \llbracket {\mathbf{n}} \rrbracket _{\gamma } = {\mathbf{n}} \left( \gamma ^+, t \right) - {\mathbf{n}} \left( \gamma ^-, t \right) , \end{aligned}$$
(11.28)

is the jump in \({\mathbf{n}}\) across the singularity and

$$\begin{aligned} {\mathbf{F}}_{\gamma } = {\mathbf{F}}_{\gamma } \left( t \right) = \lim _{\chi \rightarrow 0} \int _{\gamma - \chi }^{\gamma + \chi } \rho _0 {\mathbf{f}} {\mathrm{d}} \xi \end{aligned}$$
(11.29)

is a singular centerline force.

We now derive the one-dimensional static balance of director momentum from the local three-dimensional static balance of linear momentum. Equation (11.26) holds for every three-dimensional material point. The equation still holds if we scale it by the \(X^{\alpha }\) value of a given material point:

$$\begin{aligned} X^{\alpha } \text{Div} \mathbf{P} + X^{\alpha } \rho _0^* \mathbf{b} = \mathbf{0}. \end{aligned}$$
(11.30)

Proceeding in the same manner as before and integrating over an arbitrary sub-volume \({\mathcal{V}}_0 \left( \xi _1, \xi _2 \right)\), we work out the identity:

$$\begin{aligned} &\int _{{\mathcal{V}}_0} X^{\alpha } \text{Div} \mathbf{P} {\mathrm{d}} V = \int _{{\mathcal{A}}_0 \left( \xi _1 \right) } X^{\alpha } \mathbf{P} \left( - {\mathbf{E}}_3 \right) {\mathrm{d}}A +\int _{{\mathcal{A}}_0 \left( \xi _2 \right) } X^{\alpha } \mathbf{P} {\mathbf{E}}_3 {\mathrm{d}}A \nonumber \\&\quad + \int _{\xi _1}^{\xi _2} \oint _{\partial {\mathcal{A}}_0 \left( \xi \right) } X^{\alpha } \mathbf{p} {\mathrm{d}}U {\mathrm{d}}\xi - \int _{{\mathcal{V}}_0} \mathbf{P} {\mathbf{E}}_{\alpha } {\mathrm{d}} V. \end{aligned}$$
(11.31)

Defining the contact director forces as

$$\begin{aligned} {\mathbf{m}}^{\alpha } = {\mathbf{m}}^{\alpha } \left( \xi , t \right) = \int _{{\mathcal{A}}_0 \left( \xi \right) } X^{\alpha } \mathbf{P} {\mathbf{E}}_3 {\mathrm{d}}A, \end{aligned}$$
(11.32)

the intrinsic director forces as

$$\begin{aligned} {\mathbf{k}}^{\alpha } = {\mathbf{k}}^{\alpha } \left( \xi , t \right) = \int _{{\mathcal{A}}_0 \left( \xi \right) } \mathbf{P} {\mathbf{E}}_{\alpha } {\mathrm{d}}A, \end{aligned}$$
(11.33)

and the assigned director forces as

$$\begin{aligned} \rho _0 \mathbf{l}^{\alpha } = \rho _0 \mathbf{l}^{\alpha } \left( \xi , t \right) = \int _{{\mathcal{A}}_0 \left( \xi \right) } X^{\alpha } \rho _0^* \mathbf{b} {\mathrm{d}} A + \oint _{\partial {\mathcal{A}}_0 \left( \xi \right) } X^{\alpha } \mathbf{p} {\mathrm{d}}U, \end{aligned}$$
(11.34)

we arrive at the one-dimensional balance of director momentum for static equilibria:

$$\begin{aligned} {\mathbf{m}}^{\alpha } \left( \xi _2,t \right) - {\mathbf{m}}^{\alpha } \left( \xi _1,t \right) + \int _{\xi _1}^{\xi _2} \left( \rho _0 \mathbf{l}^{\alpha } - {\mathbf{k}}^{\alpha } \right) {\mathrm{d}}\xi = \mathbf{0}. \end{aligned}$$
(11.35)

Assuming continuity of \({\mathbf{m}}^{\alpha }\) in \(\xi\), the point-wise version is derived:

$$\begin{aligned} {\mathbf{m}}^{\alpha '} - {\mathbf{k}}^{\alpha } + \rho _0 \mathbf{l}^{\alpha } = \mathbf{0}. \end{aligned}$$
(11.36)

If \({\mathbf{m}}^{\alpha }\) in (11.35) is not continuous at some \(\xi =\gamma\), we once again use the Leibniz rule to obtain the jump conditions

$$\begin{aligned} \llbracket {\mathbf{m}}^{\alpha } \rrbracket _{\gamma } + {\mathbf{F}}^{\alpha }_{\gamma } = \mathbf{0}, \end{aligned}$$
(11.37)

where

$$\begin{aligned} {\mathbf{F}}^{\alpha }_{\gamma } = {\mathbf{F}}^{\alpha }_{\gamma } \left( t \right) = \lim _{\chi \rightarrow 0} \int _{\gamma - \chi }^{\gamma + \chi } \rho _0 \mathbf{l}^{\alpha } {\mathrm{d}} \xi \end{aligned}$$
(11.38)

is a singular director force.

A one-dimensional balance of angular momentum is also desired. The three-dimensional postulate for static equilibria is

$$\begin{aligned} \int _{{\mathcal{V}}} {\mathbf{r}}^* \times \rho ^* \mathbf{b} {\mathrm{d}}v + \int _{\partial {\mathcal{V}}} {\mathbf{r}}^* \times \mathbf{t} {\mathrm{d}}a = \mathbf{0}, \end{aligned}$$
(11.39)

where \(\mathbf{t}\) is the Cauchy traction. The small amount of force transmitted across the current boundary with area element \({\mathrm{d}}a\) relates the Cauchy traction to the Piola traction:

$$\begin{aligned} \mathbf{t} {\mathrm{d}}a = \mathbf{p} {\mathrm{d}}A. \end{aligned}$$
(11.40)

Recalling assumption (11.7) and factoring the integral, we find the one-dimensional balance of angular momentum for some part \({\mathcal{V}} \left( \xi _1, \xi _2 \right)\) that has preimage \({\mathcal{V}}_0 \left( \xi _1, \xi _2 \right)\) in the reference configuration:

$$\begin{aligned}& -{\mathbf{r}} \left( \xi _1, t \right) \times {\mathbf{n}} \left( \xi _1, t \right) + {\mathbf{r}} \left( \xi _2, t \right) \times {\mathbf{n}} \left( \xi _2, t \right) \nonumber \\&\quad - {\mathbf{d}}_{\beta } \left( \xi _1, t \right) \times {\mathbf{m}}^{\beta } \left( \xi _1, t \right) + {\mathbf{d}}_{\beta } \left( \xi _2, t \right) \times {\mathbf{m}}^{\beta } \left( \xi _2,t \right) \nonumber \\&\qquad + \int _{\xi _1}^{\xi _2} \left( {\mathbf{r}} \times \rho _0 {\mathbf{f}} + {\mathbf{d}}_{\beta } \times \rho _0 \mathbf{l}^{\beta } \right) {\mathrm{d}}\xi = \mathbf{0}. \end{aligned}$$
(11.41)

Under the usual continuity assumptions, and noting that \({\mathbf{r}}\) and \({\mathbf{d}}_{\alpha }\) are never expected to jump in \(\xi\), we arrive at the local version:

$$\begin{aligned} \left( {\mathbf{r}} \times {\mathbf{n}} + {\mathbf{d}}_{\beta } \times {\mathbf{m}}^{\beta } \right) ' + {\mathbf{r}} \times \rho _0 {\mathbf{f}} + {\mathbf{d}}_{\beta } \times \rho _0 \mathbf{l}^{\beta } = \mathbf{0}. \end{aligned}$$
(11.42)

Using the balances of linear and director momentum (11.26) and (11.36), we recover the identity

$$\begin{aligned} {\mathbf{r}}' \times {\mathbf{n}} + {\mathbf{d}}_{\beta } \times {\mathbf{k}}^{\beta } + {\mathbf{d}}_{\beta }' \times {\mathbf{m}}^{\beta } = \mathbf{0}, \end{aligned}$$
(11.43)

which is how the balance of angular momentum for the directed curve is postulated in this article.

The last quantity that is important to introduce for purposes in this article is the one-dimensional strain energy. If \(\rho _0^* \psi ^* = \rho _0^* \psi ^* \left( X^{\alpha }, \xi \right)\) is the three-dimensional strain energy per unit reference volume, then the one-dimensional version is obtained as

$$\begin{aligned} \rho _0 \psi = \rho _0 \psi \left( \xi \right) = \int _{{\mathcal{A}}_0 \left( \xi \right) } \rho _0^* \psi ^* {\mathrm{d}}A. \end{aligned}$$
(11.44)

The one-dimensional strain energies that are proposed in this article are based on this integration using the standard linear isotropic strain energy from three-dimensional continuum mechanics. However, one must proceed with caution in using \(\rho _0 \psi\) for modeling behavior such as bending and warping of cross sections, as (11.7) fails to yield a \(\rho _0 \psi\) that will capture the nonhomogeneous aspects of these deformations. As we are predominantly concerned with effects such as stretching, stretch gradients, and the Poisson effect, we believe the function defined in (11.44) is reasonably descriptive.

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Hemingway, E.G., O’Reilly, O.M. Continuous models for peristaltic locomotion with application to worms and soft robots. Biomech Model Mechanobiol 20, 5–30 (2021). https://doi.org/10.1007/s10237-020-01365-w

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Keywords

  • Worms
  • Soft robotics
  • Peristaltic locomotion
  • Muscle actuation