Low-Mach-number--slenderness limit for elastic Cosserat rods

Abstract

This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e. low-Mach-number--slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a spatial Gauss-Legendre collocation and an $α$-method in time.

Figures (4)

• Figure 3.1: Test case: dynamics of the 2d Euler-Bernoulli cantilever beam under gravity $\mathsf{\breve r}$. Simulation of the limit system ($\epsilon=0$) and the $\epsilon$-dependent system ($\epsilon=0.02$).
• Figure 3.2: Conservation of the asymptotic relations for the different model variants (cf. \ref{['eq:c1']}-\ref{['eq:c2']}, $\mathbf{c_i^\star}=\epsilon^{2i} \mathbf{c_i}$). Top: first term $\|\mathbf{c_1^\star}\|$(left) with magnitude $\|\mathbf{c_1}\|$(right) plotted over $\epsilon$. The solid line with $p=2$ indicates quadratic convergence. Bottom: second term $\|\mathbf{c_2^\star}\|$(left) with magnitude $\|\mathbf{c_2}\|$(right) plotted over $\epsilon$. The solid lines with $p=4$ and $p=2$ indicate quartic and quadratic convergence, respectively.
• Figure A.1: Pattern of the Jacobian for the limit system with $m=9$, $N=4$ and (S) that corresponds to the test case of the 2d Euler-Bernoulli cantilever beam in Section \ref{['sec:test']}.
• Figure A.2: Convergence results for the numerical scheme, relative $\mathcal{L}^2(0,1)$-error at time $T$ for $\epsilon$-dependent and limit systems. Top: spatial convergence ($\Delta s \rightarrow 0$, fixed $\Delta t$). Bottom: temporal convergence ($\Delta t \rightarrow 0$, fixed $\Delta s$) for $\lambda=1$(left) and $\lambda=0.5$(right).

Theorems & Definitions (7)

• Remark 3
• Example 4: Specification of material laws
• Remark 5: Re-formulations of the limit system
• Remark 6: Conservation of energy
• Remark 7
• Remark 8
• Remark 9