Abstract
In this chapter, we present a one-dimensional scheme with directors of elastic rods, and we use rod as a generic term for arch, bar, beam, column, ring, shaft, etc. Precisely, instead of considering a three-dimensional rod, we take its one-dimensional “axis” and attach at each of its points unit vectors (which we call directors) describing the rod’s cross section. In this way, we are able to represent an approximate kinematics, the approximation resting on the assumption that the cross section remains planar when we deform the rod. Besides the three translational degrees of freedom in three-dimensional space, each point of the one-dimensional scheme discussed here is endowed with three additional degrees of freedom, those exploited by the directors to rotate. In this setting, we describe large strains, and we represent contact actions through the power developed in deforming the “axis” and rotating the directors. According to the philosophy adopted in Chapter 3, we deduce balance equations appropriate for the actions in the rod (the contact ones represent in this one-dimensional setting averages over the cross sections of the three-dimensional real rod) by imposing the invariance of the external power on a generic part (with nonnull length) of the one-dimensional scheme under rigid-body changes in observers. Then we derive expressions for the inertial terms and the inner power. We derive a priori constitutive restrictions from a mechanical dissipation inequality written using a time derivative accounting for the rotation of the directors at each point. Then we restrict the setting to the small-strain regime and introduce Timoshenko’s and Bernoulli’s rod models as prominent special cases, together with the Euler elastica. We discuss also the force method to analyze the equilibrium of hyperstatic structures and propose and discuss several pertinent exercises.
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Notes
- 1.
Warning: \(\varphi\) must not be confused with the stress potential function in Chapter 6
- 2.
In fact, the system can be considered to be made of two rigid bodies, and the rotation centers C1, C2, and C12 are not collinear.
- 3.
The relation (8.28) allows us to derive in another way the local balance of couples along the rod. In the coordinates \(\xi ^{1}\), \(\xi ^{2}\), s, used so far, we have
$$\displaystyle{\mathrm{Div}P =\sum _{ \alpha =1}^{2}\dfrac{\partial \mathsf{t}_{\alpha }} {\partial \xi ^{\alpha }} + \dfrac{\partial \hat{\tau }} {\partial s}.}$$The local balance of forces \(\mathrm{Div}P + b =\rho \ddot{ y}\) then implies
$$\displaystyle{ \dfrac{\partial \hat{\tau }} {\partial s} = -\sum _{\alpha =1}^{2}\dfrac{\partial \mathsf{t}_{\alpha }} {\partial \xi ^{\alpha }} - b +\rho \ddot{ y}.}$$From the definition of m, we have also
$$\displaystyle{\begin{array}{ll} & \dfrac{\partial \mathsf{m}} {\partial s} = \dfrac{\partial } {\partial s}\int _{\mathcal{X}}\sum _{\alpha =1}^{2}\xi ^{\alpha }\mathsf{d}_{\alpha } \times \hat{\tau }\, d\xi ^{1}d\xi ^{2} = \dfrac{\partial } {\partial s}\int _{\mathcal{X}}(y-\varphi ) \times \hat{\tau }\, d\xi ^{1}d\xi ^{2} \\ & =\int _{\mathcal{X}}\dfrac{\partial y} {\partial s} \times \hat{\tau }\, d\xi ^{1}d\xi ^{2} - \dfrac{\partial \varphi } {\partial s} \times \int _{\mathcal{X}}\hat{\tau }\,d\xi ^{1}d\xi ^{2} +\int _{\mathcal{X}}(y-\varphi ) \times \dfrac{\partial \hat{\tau }} {\partial s}d\xi ^{1}d\xi ^{2} \\ = & \int _{\mathcal{X}}\dfrac{\partial y} {\partial s} \times \hat{\tau }\, d\xi ^{1}d\xi ^{2} - \dfrac{\partial \varphi } {\partial s} \times \mathsf{n} -\int _{\mathcal{X}}(y-\varphi ) \times \sum _{\alpha =1}^{2}\dfrac{\partial \mathsf{t}_{\alpha }} {\partial \xi ^{\alpha }} d\xi ^{1}d\xi ^{2} -\int _{\mathcal{X}}(y-\varphi ) \times b\,d\xi ^{1}d\xi ^{2} \\ & +\int _{\mathcal{X}}(y-\varphi ) \times \rho \ddot{ y}\,d\xi ^{1}d\xi ^{2}. \end{array} }$$However, we have also
$$\displaystyle{\begin{array}{ll} & -\int _{\mathcal{X}}(y-\varphi ) \times \sum _{\alpha =1}^{2}\dfrac{\partial \mathsf{t}_{\alpha }} {\partial \xi ^{\alpha }} d\xi ^{1}d\xi ^{2} =\int _{\mathcal{X}}\sum _{\alpha =1}^{2} \dfrac{\partial } {\partial \xi ^{\alpha }}\left ((y-\varphi ) \times \mathsf{t}_{\alpha }\right )d\xi ^{1}d\xi ^{2} \\ & +\int _{\mathcal{X}}\sum _{\alpha =1}^{2}\dfrac{\partial y} {\partial \xi ^{\alpha }} \times \mathsf{t}_{\alpha }\,d\xi ^{1}d\xi ^{2} = -\int _{\partial \mathcal{X}}\sum _{\alpha =1}^{2}(y-\varphi ) \times \mathsf{t}_{\alpha }\nu _{\alpha }\,dl +\int _{\mathcal{X}}\sum _{\alpha =1}^{2}\dfrac{\partial y} {\partial \xi ^{\alpha }} \times \mathsf{t}_{\alpha }d\xi ^{1}d\xi ^{2}, \end{array} }$$where ν α is the αth component of the normal to \(\partial \mathcal{X}\) in the plane containing it (recall that t α is a vector; ν α is a scalar).
Consequently, we obtain
$$\displaystyle{\begin{array}{ll} & \dfrac{\partial \mathsf{m}} {\partial s} = \left (\int _{\partial \mathcal{X}}\sum _{\alpha =1}^{2}\dfrac{\partial y} {\partial \xi ^{\alpha }} \times \mathsf{t}_{\alpha } + \dfrac{\partial y} {\partial s}\times \hat{\tau }\right )d\xi ^{1}d\xi ^{2} -\bar{\mathsf{m}} - \dfrac{\partial \varphi } {\partial s} \times \mathsf{n} +\int _{\mathcal{X}}(y-\varphi ) \times \rho \ddot{ y}\,d\xi ^{1}d\xi ^{2} \\ & = \left (\int _{\mathcal{X}}\sum _{\alpha =1}^{2}\dfrac{\partial y} {\partial \xi ^{\alpha }} \times \mathsf{t}_{\alpha } + \dfrac{\partial y} {\partial s}\times \hat{\tau }\right )d\xi ^{1}d\xi ^{2} -\bar{\mathsf{m}} - \dfrac{\partial \varphi } {\partial s} \times \mathsf{n} +\dot{ H}, \end{array} }$$where
$$\displaystyle{\bar{\mathsf{m}} =\sum _{ \alpha =1}^{2}\int _{ \partial \mathcal{X}}(y-\varphi ) \times \mathsf{t}_{\alpha }\nu _{\alpha }ds +\int _{\mathcal{X}}(y-\varphi ) \times b\,d\xi ^{1}d\xi ^{2}.}$$The result (8.27) implies then the local balance of couples
$$\displaystyle{\dfrac{\partial \mathsf{m}} {\partial s} + \dfrac{\partial \varphi } {\partial s} \times \mathsf{n} +\bar{ \mathsf{m}} =\dot{ H}.}$$
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Mariano, P.M., Galano, L. (2015). Rod Models. In: Fundamentals of the Mechanics of Solids. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3133-0_8
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DOI: https://doi.org/10.1007/978-1-4939-3133-0_8
Publisher Name: Birkhäuser, New York, NY
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