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An Eulerian formulation of a growing rod in three dimensions with mass accretion

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Abstract

Motivated by a general thermomechanical theory of Cosserat rods, an isothermal constrained theory is developed which generalizes a three-dimensional elastica to have a growing circular cross-section and a growing centroidal length. An Eulerian formulation of constitutive equations is developed using evolution equations for elastic dilatation, elastic cross-sectional radial stretch, elastic curvature and elastic torsional twist. These evolution equations include time-dependent inelastic effects of homeostasis which cause a tendency for these elastic deformation measures to approach their homeostatic values. An important feature of the Eulerian evolution equations is that they depend only on the current state of the rod and therefore are appropriate for modeling growth. Examples of growing rods are considered. In particular, the example of a stress-free growing helical rod suggests a possible mechanical mechanism for a climbing vine which grabs an object as its radius decreases.

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  1. Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    MB Rubin

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Appendix. Details of some developments

Appendix. Details of some developments

Using (3), (9), and (47) yields

$$\begin{aligned} \frac{d}{d t} \big ( \frac{{\partial } \mathbf {e}_t}{{\partial } s} \big )= & {} \frac{d}{d t} \big [ \frac{1}{\lambda } \frac{{\partial }}{{\partial } S} \big (\frac{1}{\lambda } \mathbf {d}_3 \big )\big ] =\frac{{\partial } }{{\partial } s} \big ( \frac{1}{\lambda } \mathbf {w}_3 - \frac{\dot{\lambda }}{\lambda } \mathbf {e}_t \big ) - \frac{\dot{\lambda }}{\lambda } \frac{{\partial } \mathbf {e}_t}{{\partial } s} \,, \nonumber \\= & {} \frac{1}{\lambda } \frac{{\partial } \mathbf {w}_3}{{\partial } s} - \frac{1}{\lambda } \frac{{\partial } \lambda }{{\partial } s} \mathbf {L} \mathbf {e}_t - \frac{{\partial }}{{\partial } s} \big ( \frac{\dot{\lambda }}{\lambda } \big ) \mathbf {e}_t - 2 \kappa \big (\frac{\dot{\lambda }}{\lambda } \big ) {\mathbf {e}}_n \,. \end{aligned}$$
(96)

Then, with the help of (46) and (48) it follows that

$$\begin{aligned} \dot{\kappa } = \frac{1}{\lambda } \frac{{\partial } \mathbf {w}_3}{{\partial } s} \cdot {\mathbf {e}}_n - \frac{1}{\lambda } \frac{{\partial } \lambda }{{\partial } s} ({\mathbf {e}}_n \otimes {\mathbf {e}}_t) \cdot \mathbf {L} - 2 \kappa ({\mathbf {e}}_t \otimes {\mathbf {e}}_t) \cdot \mathbf {D} \,. \end{aligned}$$
(97)

Next, using the fact that for the constrained theory \(\mathbf {d}_\alpha\) are orthogonal to \(\mathbf {d}_3\), it can be shown that

$$\begin{aligned}&\mathbf {w}_\alpha \cdot \mathbf {d}_3 + \mathbf {d}_\alpha \cdot \mathbf {w}_3 = 0 \,, \quad \mathbf {w}_3 = - \lambda (\mathbf {w}_\alpha \cdot {\mathbf {e}}_t) \mathbf {d}^\alpha + \dot{\lambda } {\mathbf {e}}_t \,. \nonumber \\&\frac{{\partial } \mathbf {w}_3}{{\partial } s} = - \lambda (\frac{{\partial } \mathbf {w}_\alpha }{{\partial } s} \cdot {\mathbf {e}}_t) \mathbf {d}^\alpha - \frac{{\partial } \lambda }{{\partial } s} (\mathbf {w}_\alpha \cdot {\mathbf {e}}_t) \mathbf {d}^\alpha - \kappa \lambda (\mathbf {w}_\alpha \cdot {\mathbf {e}}_n) \mathbf {d}^\alpha \nonumber \\&\quad \quad - \lambda (\mathbf {w}_\alpha \cdot {\mathbf {e}}_t) \frac{{\partial } \mathbf {d}^\alpha }{{\partial } s} +\frac{{\partial } \dot{\lambda }}{{\partial } s} {\mathbf {e}}_t +\dot{\lambda } \kappa {\mathbf {e}}_n \,, \nonumber \\&\frac{{\partial } \mathbf {w}_3}{{\partial } s} \cdot {\mathbf {e}}_n = - \lambda (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) ({\mathbf {e}}_t \cdot \frac{{\partial } \mathbf {w}_\alpha }{{\partial } s}) \nonumber \\&\quad \quad - \frac{{\partial } \lambda }{{\partial } s} (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) ({\mathbf {e}}_t \otimes \mathbf {d}_\alpha ) \cdot \mathbf {L} \nonumber \\&\quad \quad - \kappa \lambda (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) ({\mathbf {e}}_n \otimes \mathbf {d}_\alpha ) \cdot \mathbf {L} \nonumber \\&\quad \quad - \lambda \big (\frac{{\partial } \mathbf {d}^\alpha }{{\partial } s} \cdot {\mathbf {e}}_n \big ) ({\mathbf {e}}_t \otimes \mathbf {d}_\alpha ) \cdot \mathbf {L} + \kappa \lambda ({\mathbf {e}}_t \otimes {\mathbf {e}}_t) \cdot \mathbf {D} \,, \nonumber \\&\frac{{\partial } \mathbf {w}_3}{{\partial } s} \cdot {\mathbf {e}}_n = - \lambda (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) ({\mathbf {e}}_t \cdot \frac{{\partial } \mathbf {w}_\alpha }{{\partial } s}) \nonumber \\&\quad \quad - \frac{{\partial } \lambda }{{\partial } s} \big [({\mathbf {e}}_t \otimes (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) \mathbf {d}_\alpha \big ] \cdot \mathbf {L} \nonumber \\&\quad \quad -\lambda {\mathbf {e}}_t \cdot \mathbf {L} \big (\mathbf {d}_\alpha \otimes \frac{{\partial } \mathbf {d}^\alpha }{{\partial } s} \big ) {\mathbf {e}}_n \nonumber \\&\quad \quad - \kappa \lambda \big [({\mathbf {e}}_n \otimes (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) \mathbf {d}_\alpha \big ] \cdot \mathbf {L} + \kappa \lambda ({\mathbf {e}}_t \otimes {\mathbf {e}}_t) \cdot \mathbf {D} \,, \nonumber \\&\frac{{\partial } \mathbf {w}_3}{{\partial } s} \cdot {\mathbf {e}}_n = - \lambda (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) \, {\mathbf {e}}_t \cdot \big (\frac{{\partial } \mathbf {w}_\alpha }{{\partial } s} - \mathbf {L} \frac{{\partial } \mathbf {d}_\alpha }{{\partial } s} \big ) \nonumber \\&\quad \quad - \frac{{\partial } \lambda }{{\partial } s} ({\mathbf {e}}_t \otimes {\mathbf {e}}_n) \cdot \mathbf {L} \nonumber \\&\quad \quad - \lambda {\mathbf {e}}_t \cdot \mathbf {L} \big (\frac{{\partial } \mathbf {d}_\alpha }{{\partial } s} \otimes \mathbf {d}^\alpha +\mathbf {d}_\alpha \otimes \frac{{\partial } \mathbf {d}^\alpha }{{\partial } s} \big ) {\mathbf {e}}_n \nonumber \\&\quad \quad + \kappa \lambda ({\mathbf {e}}_t \otimes {\mathbf {e}}_t- {\mathbf {e}}_n \otimes {\mathbf {e}}_n) \cdot \mathbf {D} \,, \nonumber \\&\frac{{\partial } \mathbf {w}_3}{{\partial } s} \cdot {\mathbf {e}}_n = - \lambda (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) \, {\mathbf {e}}_t \cdot \big (\frac{{\partial } \mathbf {w}_\alpha }{{\partial } s} - \mathbf {L} \frac{{\partial } \mathbf {d}_\alpha }{{\partial } s} \big ) \nonumber \\&\quad \quad - \frac{{\partial } \lambda }{{\partial } s} ({\mathbf {e}}_t \otimes {\mathbf {e}}_n) \cdot \mathbf {L} + \kappa \lambda (2{\mathbf {e}}_t \otimes {\mathbf {e}}_t- {\mathbf {e}}_n \otimes {\mathbf {e}}_n) \cdot \mathbf {D} \,, \end{aligned}$$
(98)

where use has been made of the expressions

$$\begin{aligned}&{\mathbf {e}}_n = (\mathbf {d}^\alpha \cdot {\mathbf {e}}_n) \mathbf {d}_\alpha \,, \quad \mathbf {d}_\alpha \otimes \mathbf {d}^\alpha = \mathbf {I} - {\mathbf {e}}_t \otimes {\mathbf {e}}_t \,, \nonumber \\&\frac{{\partial } \mathbf {d}_\alpha }{{\partial } s} \otimes \mathbf {d}^\alpha +\mathbf {d}_\alpha \otimes \frac{{\partial } \mathbf {d}^\alpha }{{\partial } s} = -\kappa ({\mathbf {e}}_n \otimes {\mathbf {e}}_t + {\mathbf {e}}_t \otimes {\mathbf {e}}_n) \,. \end{aligned}$$
(99)

Finally, substituting (98) into (97) yields the evolution equation (49).

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Rubin, M. An Eulerian formulation of a growing rod in three dimensions with mass accretion. Mech Soft Mater 4, 9 (2022). https://doi.org/10.1007/s42558-022-00047-0

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