Paper by Pezzulla

03/04/2021 21:18

The paper discusses an experimental apparatus consisting of an elastic shell, comprised of a material with Young's modulus E$E$, having the shape of a hemispherical cap of radius R$R$ and thickness h$h$:

The cap is sealed on a support which makes it possible to extract air from the interior, which results in a net pressure p$p$ from the exterior.

If p$p$ is sufficiently large then the shell undergoes elastic instability. For a perfect shell the critical value p_c$p_c$ of the pressure is know to scale as:[^]

p_c\sim E\left(\frac {h}R\right)^2.

Experiments show that the actual buckling pressure p_{\rm max}$p_{\rm max}$ (the maximal pressure that the shell can withstand before collapsing) is smaller than the theoretical value. Such discrepancy is summarized by the knockdown factor:

\kappa_{\mathrm{d}}=\frac{p_{\max }}{p_{\mathrm{c}}},

which, according to experimental evidence, varies between 0.05$0.05$ and 0.9$0.9$. ^[1] It was Koiter who understood first that such discrepancy is to be attributed to imperfections.

In the experimental setup described in this paper, the material that comprises the shell possesses a "residual" magnetic induction \mathbf B^{\rm r}$\mathbf B^{\rm r}$, whose orientation is in the vertical direction. An external device is capable of producing an applied induction field \mathbf B^{\rm a}$\mathbf B^{\rm a}$, as shown in the figure:

The interaction between the applied and residual induction is modeled through the magnetic energy

\mathcal{U}_{\mathrm{m}}=-\int \frac{1}{\mu_{0}} \mathbf{F} \mathbf{B}^{r} \cdot \mathbf{B}^{\mathbf{a}} \mathrm{d} V

where \mathbf F$\mathbf F$ is the deformation gradient.

Experiments show that the knowndown factor increases if \mathbf B^{\rm a}$\mathbf B^{\rm a}$ and \mathbf B^{\rm r}$\mathbf B^{\rm r}$ point in the same direction, and decreases otherwise. To quantitatively assess the role of the applied field, the paper identifies as a key quantity the following magneto-elastic parameter:

\lambda_{\mathrm{m}}=\frac{B^{\mathrm{r}} B^{\mathrm{a}}}{\mu_{0} E},\tag{*}

and deduces that the presence of the magnetic field produces a changes of the knockdown factor which obeys the scaling law

\Delta \kappa_{\mathrm{d}} \sim \lambda_{\mathrm{m}} \frac{R}{h}.

Argument for the evaluation of the variation of \kappa_{\rm d}$\kappa_{\rm d}$.

The estimate of the knockdown factor is based on the assumption that the displacements induced by the external pressure are of the order of the thickness h$h$ of the shell.

Energies are renormalized through the characteristic energy

\mathcal E=\pi E h R^{2} /\left(4\left(1-\nu^{2}\right)\right)\sim EhR^2

Note that this energy is of the order of the volume of the shell.

With this renormalization, the work of the pressure is ^[2]

\overline{\mathcal U}_{\mathrm{p}}\sim \frac{p}{E}

The dominant part of the (renormalized) magnetic energy is identified as:

\overline{\mathcal{U}}_{\mathrm{m}}^{(2) \mathrm{A}}=\frac{8\left(1-\nu^{2}\right)}{R^{2}} \lambda_{\mathrm{m}} \int_{0}^{\pi / 2} \frac{1}{2} \mathring{\chi}(\varphi) \theta^2\mathring a\mathrm{d} \varphi,

where \mathring\chi(\varphi)$\mathring\chi(\varphi)$ is a dimensionless quantity, \mathring a$\mathring a$ is the area measure, and \theta$\theta$ is the local rotation. The local rotation is estimated as:^[3]

\theta\sim \sqrt{h/R}

The change of bucking pressure due to the application of the magnetic field is estimated by equating the extra work \overline{\mathcal U}_{\rm m}$\overline{\mathcal U}_{\rm m}$ due to the magnetic field with the extra work \Delta\overline{\mathcal U}_{\rm p}$\Delta\overline{\mathcal U}_{\rm p}$ of the pressure:

\lambda_m\frac  h R \sim\overline{\mathcal U}_{\rm m}=\Delta\overline{\mathcal U}_{\rm P}\sim \frac {\Delta p}E\qquad\Rightarrow\qquad\Delta p\sim E\lambda_{\rm m}\frac{h}{R}

The change of knockdown factor is

\Delta\kappa_{\rm d}=\frac{\Delta p}{p_c}\sim \frac{E\lambda_{\rm m}\frac{h}{R}}{E\left(\frac h R\right)^2}.

This yields the main result (*)$(*)$.