Mechanically concealed holes

TL;DR

This work addresses weight reduction in elastic solids by mechanically cloaking holes with stiff shells to preserve macroscopic stiffness under plane-strain loading. It derives a closed-form shell thickness that achieves concealment as a function of the shell to background mu ratio and Poisson ratios, and validates the continuum result with atomistic molecular dynamics simulations of a two-dimensional Lennard-Jones solid. The MD results show robust agreement with continuum theory for stiff shells and reveal finite-size driven deviations for softer shells, while also demonstrating that cloak-induced confinement reduces local virial shear stresses around the hole. The findings suggest a practical pathway for lightweight material design and also open avenues for nanoscale cloaking via atomistic manipulations.

Abstract

When a hole is introduced into an elastic material, it will usually act to reduce the overall mechanical stiffness. A general ambition is to investigate whether a stiff shell around the hole can act to maintain the overall mechanical properties. We consider this effect from a macroscopic continuum perspective down to atomistic scales. First, we focus on the basic continuum example situation of an isotropic, homogeneous, linearly elastic material loaded uniformly under compressive plane strain for low concentrations of holes. As we demonstrate, the thickness of the shell can be adjusted in a way to maintain the overall stiffness of the system. We derive a corresponding mathematical expression for the thickness of the shell that conceals the hole. Thus, one can work with given materials to mask the presence of the holes simply by adjusting the thickness of the surrounding shells, with no need to change the materials. Our predictions from linear elasticity continuum theory are extended to atomistic levels using molecular dynamics simulations of a model Lennard-Jones solid. These extensions attest the robustness of our predictions down to atomistic scales. Thus, they open a straightforward possibility to adjust the strategy of mechanical cloaking via atomistic manipulations. From both perspectives, the underlying concept is important in the context of light-weight construction.

Figures (12)

• Figure 1: Illustration of the geometry. In the two-dimensional plane that we use to describe the block of material under plane-strain conditions, the cylindrical hole appears as a circular exclusion of radius $a$. Our system of polar coordinates $(r,\varphi)$ is centered in the hole. The hole is surrounded by a cylindrical shell of outer radius $b$, mechanical shear modulus $\mu^\mathrm{i}$, and Poisson ratio $\nu^\mathrm{i}$. Moreover, the actual, outer elastic material is of shear modulus $\mu^\mathrm{o}$ and Poisson ratio $\nu^\mathrm{o}$.
• Figure 2: Ratio $b/a$ between the outer radius of the shell $b$ around the hole and the radius $a$ of the hole as a function of the given ratio $\mu^\mathrm{i}/\mu^\mathrm{o}$ between the mechanical moduli of the shell and the surrounding elastic material. The value of $b$ is chosen in a way to mechanically conceal and mask the presence of the hole in the surrounding elastic substance under the imposed deformation. Curves are shown for different combinations of the Poisson ratios $-1<\nu^\mathrm{\{i,o\}}<1/2$ of the shell and the surrounding elastic material, (a) both nonauxetic ($\nu^\mathrm{\{i,o\}} > 0$) , (b) both auxetic ($\nu^\mathrm{\{i,o\}} < 0$), and (c) combined auxetic and nonauxetic ($\nu^\mathrm{i}\nu^\mathrm{o} < 0$).
• Figure 3: (a) Illustration of a representative isotropic compression test in an MD simulation of a planar hexagonal solid with a shielded hole ($b/a = 2.018$, $\mu^\mathrm{i}/\mu^\mathrm{o} = 2$). Isotropic compression can be realized by imposing an isotropic stress $\bm{\sigma}^\infty=\bm{\sigma}(r\rightarrow\infty)=-P\mathbf{I}$, see the main text. (b) Isotropic compression is confirmed by plotting the components of the imposed stress $\sigma^\infty_{xx}$ and $\sigma^\infty_{yy}$ against each other. The linear fit confirms an approximate unit slope.
• Figure 4: Poisson ratios ($\nu$), obtained from the slope of the linear variation between an applied strain ($\epsilon_{xx}$) and a resulting transverse strain ($\epsilon_{yy}$) via performing uniaxial tensile tests of the equilibrated samples. Results are shown for (a) a pristine planar solid and (b) a planar solid with a hole enclosed by a stiffer shell of ratios of outer radii $b/a = 1.173$ and of shear moduli $\mu^\mathrm{i}/\mu^\mathrm{o} = 10$.
• Figure 5: MD simulation results for the variation of pressure $P^*$ with areal strain $\epsilon_{A} = \Delta A/A_{0}$ during isotropic compression. Here, $A_0$ is the area of the equilibrated system at a given initial pressure $P^*(\epsilon_A=0)$, and $\Delta A$ is the change in area from there. We define the bulk moduli $K^*$ as the slopes of the resulting curves. Both pressure $P^*=P\sigma^{2}/\epsilon$ and bulk modulus $K^*=K\sigma^{2}/\epsilon$ are rescaled by the LJ parameters. We consider the two ratios of elastic moduli between the shell and the surrounding solid (a) $\mu^\mathrm{i}/\mu^\mathrm{o} = 2$ and (b) $\mu^\mathrm{i}/\mu^\mathrm{o} = 10$. In both cases, $P^*(\epsilon_A)$ is plotted for the pristine planar hexagonal solid without any hole (red curves), the solid with an unshelled hole ($b/a=1$, black curves), and for the solid with a shelled hole of a thickness $b/a>1$ (blue curves) that best ensure mechanical concealment (same slope $K^*\approx54.98$ for red and blue curves). The corresponding geometries were identified as (c) $b/a = 2.018$ and (d) $b/a=1.173$, respectively, in contrast to (e) the unshelled hole that yields a lower $K^*\approx50.2$. The side panels show simulation snapshots with gray and blue atoms denoting whether they belong to the background solid or shell, respectively. We considered $\nu^\mathrm{i}$$\approx$$\nu^\mathrm{o}$$= 0.34$ in all cases.