Equivalence of a Constrained Cosserat Theory and Antman’s Special Cosserat Theory of a Rod

Abstract

The general nonlinear Cosserat theory of a rod allows for tangential shear deformation, axial extension and a deformable cross-section. Simplified equations are obtained by introducing kinematic constraints and associated constraint responses which force the cross-section to remain rigid. The equations of motion of this constrained Cosserat rod are shown to be equivalent to those of Antman’s nonlinear special Cosserat theory of a rod. Strain measures motivated by the general Cosserat rod theory are used to develop explicit hyperelastic constitutive equations for the force and moment applied to the rod’s ends.

Appendix: Details of Some of the Developments

Within the context of the three-dimensional theory, the position vector ${\mathbf{X}}^{*}$ of a material point in the reference configuration and the position vector ${\mathbf{x}}^{*}$ of the same material point in the present configuration are proposed in the forms

$$ {\mathbf{X}}^{*} = {\mathbf{X}}^{*}\bigl(\theta ^{\alpha },S\bigr) = { \mathbf{X}}(S) + \theta ^{\alpha }{ \mathbf{D}}_{\alpha }(S) , \qquad { \mathbf{x}}^{*} = { \mathbf{x}}^{*}\bigl(\theta ^{\alpha },S,t\bigr) = { \mathbf{x}}(S,t) + \theta ^{\alpha }{\mathbf{d}}_{\alpha }(S,t) , $$

(43)

where $\theta ^{i}$ are convected coordinates through the rod’s cross-section and $\theta ^{3}=S$. It then follows that the covariant base vectors ${\mathbf{G}}_{i}$ in the reference configuration and ${\mathbf{g}}_{i}$ in the present configuration are given by

$$ \begin{aligned} &{\mathbf{G}}_{\alpha }= \frac{{\partial }{\mathbf{X}}^{*}}{{\partial }\theta ^{\alpha }} = { \mathbf{D}}_{\alpha }, &\quad& {\mathbf{G}}_{3} = \frac{{\partial }{\mathbf{X}}^{*}}{{\partial }S} = {\mathbf{D}}_{3} + \theta ^{\alpha } \frac{d {\mathbf{D}}_{\alpha }}{d S} , \\ &{\mathbf{g}}_{\alpha }= \frac{{\partial }{\mathbf{x}}^{*}}{{\partial }\theta ^{\alpha }} = { \mathbf{d}}_{\alpha } , &\quad& {\mathbf{g}}_{3} = \frac{{\partial }{\mathbf{x}}^{*}}{{\partial }S} = { \mathbf{d}}_{3} + \theta ^{\alpha }\frac{{\partial }{\mathbf{d}}_{\alpha }}{{\partial }S} . \end{aligned} $$

(44)

Then, using the definitions (7) the three-dimensional deformation gradient ${\mathbf{F}}^{*}$ is given by

$$ \begin{aligned} &{\mathbf{F}}^{*} = { \mathbf{g}}_{i} \otimes {\mathbf{G}}^{i} = { \mathbf{F}} \biggl[{\mathbf{D}}_{\alpha }\otimes {\mathbf{G}}^{\alpha }+ \biggl({ \mathbf{D}}_{3} + \theta ^{\alpha } \frac{d {\mathbf{D}}_{\alpha }}{d S} + \theta ^{\alpha }{\boldsymbol{\beta }}_{\alpha }\biggr) \otimes {\mathbf{G}}^{3}\biggr] , \\ &{\mathbf{F}}^{*} = {\mathbf{F}}\bigl({\mathbf{G}}_{i} \otimes { \mathbf{G}}^{i} + \theta ^{\alpha }{\boldsymbol{\beta }}_{\alpha }\otimes {\mathbf{G}}^{3}\bigr) = {\mathbf{F}} \bigl({\mathbf{I}}+ \theta ^{\alpha }{\boldsymbol{\beta }}_{\alpha } \otimes {\mathbf{G}}^{3}\bigr) , \end{aligned} $$

(45)

where ${\mathbf{G}}^{i}$ are the reciprocals of ${\mathbf{G}}_{i}$. This result shows that when the inhomogeneous deformation vectors ${\boldsymbol{\beta }}_{\alpha }$ vanish, ${\mathbf{F}}^{*}$ is independent of the coordinates through the rod’s cross-section.