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Journal of Elasticity

Erschienen in:

29.11.2022

Accretion Mechanics of Nonlinear Elastic Circular Cylindrical Bars Under Finite Torsion

verfasst von: Arash Yavari, Satya Prakash Pradhan

Erschienen in: Journal of Elasticity | Ausgabe 1-2/2022

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Abstract

In this paper we formulate the initial-boundary value problem of accreting circular cylindrical bars under finite torsion. It is assumed that the bar grows as a result of printing stress-free cylindrical layers on its boundary while it is under a time-dependent torque (or a time-dependent twist) and is free to deform axially. In a deforming body, accretion induces eigenstrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar. This metric explicitly depends on the history of deformation during the accretion process. To simplify the kinematics, we consider incompressible solids. For the example of incompressible neo-Hookean solids, we solve the governing equations numerically. We also linearize the governing equations and compare the linearized solutions with the numerical solutions of the neo-Hookean bars.

Vorheriger Artikel Asymptotic Behavior of Constrained Local Minimizers in Finite Elasticity

Nächster Artikel A Hint on the Localization of the Buckling Deformation at Vanishing Curvature Points on Thin Elliptic Shells

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This decomposition is due to Kondaurov and Nikitin [13], Takamizawa and Hayashi [36], Takamizawa and Matsuda [37], and Takamizawa [35]. One can find similar ideas in [39, 40]. This decomposition was popularized in the literature of biomechanics by Rodriguez et al. [26]. For a historical account of this decomposition in different fields see [27, 50].

Growing bodies are non-Euclidean in the sense that their natural configuration is not Euclidean, in general. Non-Euclidean solids—a term that was coined by Henri Poincaré [25]—has been used interchangeably for anelastic bodies in the recent literature [42, 52, 53].

This was first observed in the setting of linear accretion mechanics in the seminal work of Brown and Goodman [5] who studied accreting planets under self-gravity.

The idea of a time of attachment map is due to Metlov [21].

Family 3 deformations are universal for certain inhomogeneous and anisotropic bars as well [44, 47, 48]. In this paper, we restrict our calculations to isotropic and homogeneous bars.

Note that as soon as a layer is deposited it becomes part of the body and participates in the deformation process. If the load is fixed, one would have a classical twist-fit problem (Fig. 1). The time dependence of the load (or twist) makes the natural state of the body (the material metric) inhomogeneous. In other words, after completion of accretion if each cylindrical layer is allowed to relax independently of the rest of the body the collection of relaxed thin cylindrical shells can not be put back together in the Euclidean ambient space without local elastic deformations. This incompatibility of the local rest configurations depends on the state of deformation during accretion and indirectly on the applied load during accretion.

This is identical to what was obtained in [51] in the case of accreting bars under finite extension.

The physical components of the Cauchy stress are defined as ${\bar{\sigma}}^{ab}=\sigma ^{ab}\sqrt{g_{aa}\,g_{bb}}$ (no summation) [41].

This is a simple application of the Leibniz integral rule:

$$ \hat{k}'_{3}(t)=\frac{d}{dt}\int _{R_{0}}^{s(t)} f(t,R)\,dR =s'(t)\,f(t,s(t))+ \int _{R_{0}}^{s(t)} \frac{\partial f(t,R)}{\partial t}\,dR\,, $$

where

$$ f(t,R)=R\int _{R}^{s(t)}\frac{d\xi}{\lambda ^{3}(\tau (\xi ))}\,. $$

Note that

$$ f(t,s(t))=s(t) \int _{s(t)}^{s(t)} \frac{d\xi}{\lambda ^{3}(\tau (\xi ))}=0\,,\quad \frac{\partial f(t,R)}{\partial t}=R \,s'(t) \frac{1}{\lambda ^{3}(\tau (s(t)))}=\frac{R\,u_{0}}{\lambda ^{3}(t)} \,. $$

Thus

$$ \hat{k}'_{3}(t)=\int _{R_{0}}^{s(t)} \frac{R\,u_{0}}{\lambda ^{3}(t)} \,dR =\frac{u_{0}}{2\lambda ^{3}(t)}\big(s^{2}(t)-R_{0}^{2}\big) \,. $$

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Titel: Accretion Mechanics of Nonlinear Elastic Circular Cylindrical Bars Under Finite Torsion
verfasst von: Arash Yavari
Satya Prakash Pradhan
Publikationsdatum: 29.11.2022
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 1-2/2022
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-022-09957-6

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