Assembling hydrodynamic cloaks to conceal complex objects from drag

Transformation hydrodynamics and the corresponding metamaterials have been proposed as a means to exclude the drag force acting on an object. Here, we report a strategy to deploy the hydrodynamic cloaks in a more practical manner by assembling different-shaped cloaking parts. Our strategy is to first model a square-shaped cloak and a carpet cloak and then combine them to conceal a more complex-shaped space in the three-dimensional hydrodynamic flow. With the derivation of transformation hydrodynamics, the coordinate transformations for each hydrodynamic cloaking are demonstrated with the calculated viscosity tensors. The pressure and velocity fields of the square, triangular (carpet), and exemplary three-dimensional house-shaped cloaks are numerically simulated, thus showing a cloaking effect and reduced drag. This study suggests an efficient way of cloaking complex architectures from fluid-dynamic forces.     

Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

Remarks on the Instability of an Incompressible and Isotropic Hyperelastic, Thick-Walled Cylindrical Tube

The problem of instability of a hyperelastic, thick-walled cylindrical tube was first studied by Wilkes [1] in 1955. The solution was formulated within the framework of the theory of small deformations superimposed on large homogeneous deformations for the general class of incompressible, isotropic materials; and results for axially symmetrical buckling were obtained for the neo-Hookean material. The solution involves a certain quadratic equation whose characteristic roots depend on the material response functions. For the neo-Hookean material these roots always are positive. In fact, here we show for the more general Mooney–Rivlin material that these roots always are positive, provided the empirical inequalities hold. In a recent study [2] of this problem for a class of internally constrained compressible materials, it is observed that these characteristic roots may be real-valued, pure imaginary, or complex-valued. The similarity of the analytical structure of the two problems, however, is most striking; and this similarity leads one to question possible complex-valued solutions for the incompressible case. Some remarks on this issue will be presented and some new results will be reported, including additional results for both the neo-Hookean and Mooney–Rivlin materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Broadband optimization of a pentamode-layered spherical acoustic waveguide

Transformational acoustics offers the theoretical possibility of cloaking obstacles within fluids, provided metamaterials having continuously varying bulk moduli and densities can be found or constructed. Realistically, materials with the proper, continuously varying anisotropies do not presently exist. Discretely layered cloaks having constant material parameters within each layer are a viable alternative, but due to their discrete nature, may become ineffective outside of narrow frequency ranges. Because of such limitations, there is interest in finding discretely layered systems that can be effective in as wide as possible bandwidth without the need for unrealizable material properties within each layer. The present work introduces a novel methodology for finding optimal material parameters for use in such layered cloaks. In principle, the technique could be applied to any acoustic or electromagnetic scattering problem, but for purposes of demonstration, this paper considers a fluid-loaded acoustically hard sphere with a cloak that comprised layered pentamodes, whose material properties are constrained to lie within reasonable ranges relative to the density and bulk modulus of water.     

Design and analysis of nonlinear-transformation-based broadband cloaking for acoustic wave propagation

A coordinate-transformation method can be used to design invisibility cloaks for many types of waves, including acoustic waves. The traditional method for designing a cloak depends on a transformation from a virtual space to a physical space. Previous acoustic cloaks that are mainly designed with linear-transformation-based acoustics have drawbacks that acoustic wave trajectories in the cloaks cannot be controlled and tuned. This work uses a nonlinear mapping from a ray trajectory perspective to construct acoustic cloaks with tunable non-singular material properties. Use of a ray trajectory equation is a straightforward and alternate way to study propagation characteristics of different types of waves, which allows more flexibility in controlling the waves. A broadband cylindrical cloak for acoustic waves in an inviscid fluid is realized with layered non-singular, homogeneous, and isotropic materials based on a nonlinear transformation. Some advantages and improvements of the invisibility nonlinear-transformation cloak over a traditional linear-transformation cloak are analyzed. The invisibility capability of the nonlinear-transformation cloak can be tuned by adjusting a design parameter that is shown to have influence on the acoustic wave energy flowing into the region inside the cloak. Numerical examples show that the nonlinear-transformation cloak is more effective for making a domain undetectable by acoustic waves in an inviscid fluid and shielding acoustic waves from outside the cloak than the linear-transformation cloak in a broad frequency range. The methodology developed here can be used to design nonlinear-transformation cloaks for other types of waves.     

Cloaking of a circular cylindrical elastic inclusion from antiplane elastic waves and resonance effects

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.     

Transformational cloaking from seismic surface waves by micropolar metamaterials with finite couple stiffness

Transformational elastodynamics can be used to protect sensitive structures from harmful waves and vibrations. By designing the material properties in a region around the sensitive structure, a cloak, the incident waves can be redirected as to cause minimal or no harmful response on the pertinent structure. In this paper, we consider such transformational cloaking built up by a suitably designed metamaterial exhibiting micropolar properties. First, a theoretically perfect cloak is obtained by designing the properties of an (unphysical) restricted micropolar material within the surrounding medium. Secondly, we investigate the performance of the cloak under more feasible design criteria, relating to finite elastic parameters. In particular, the behavior of a physically realizable cloak built up by unrestricted micropolar elastic media is investigated. Numerical studies are conducted for the case of buried as well as surface breaking structures in 2D subjected to incident Rayleigh waves pertinent to seismic loading. The studies show how the developed cloaking procedure can be utilized to substantially reduce the response of the structure. In particular, the results indicate the performance of the cloak in relation to constraints on the elastic parameters.     

A novel design scheme for acoustic cloaking of complex shape based on region partitioning and multi-origin coordinate transformation

Acoustic cloaking is an important application of acoustic metamaterials. This article proposes a novel design scheme for acoustic cloaking based on the region partitioning and multi-origin coordinate transformation. The cloaked region is partitioned into multiple narrow strips. For each strip, a local coordinate system is established with the local origin located at the strip center, and a coordinate transformation in the local coordinate system is conducted to squeeze the material along the strip length direction to form the cloaked region. To facilitate the implementation of the acoustic cloak, the multilayer effective medium is used to approximate the non-uniform anisotropic material parameters. The effectiveness of the proposed coordinate transformation method is verified by comparing the results from our method with those in the literature. Firstly, the results of a circular acoustic cloak in the literature are reproduced by using our finite element (FE) simulations for validation. Then, a comparison is made between the traditional coordinate transformation scheme and our new scheme for simulating an elliptical acoustic cloak. The results indicate that the proposed multi-origin coordinate transformation method has a better cloaking effect on the incident wave along the ellipse minor axis direction than the traditional method. This means that for the same object, an appropriate transformation scheme can be selected for different incident wave directions to achieve the optimal control effect. The validated scheme is further used to design an arch-shaped cloak composed of an upper semicircular area and a lower rectangular area, by combining the traditional single-centered coordinate transformation method for the semicircular area and the proposed multi-origin method for the rectangular area. The results show that the designed cloak can effectively control the wave propagation with significantly reduced acoustic pressure level. This work provides a flexible acoustic cloak design method applicable for arbitrary shapes and different wave incident directions, enriching the theory of acoustic cloaking based on coordinate transformation.     

Thermo-mechanical constitutive modeling of shape memory polyurethanes using a phenomenological approach

This study examined the constitutive modeling of shape memory polyurethanes (SMPUs). SMPUs exhibit a thermo-responsive shape memory behavior, i.e., a thermally fixed temporary shape at a low temperature that returns to its original (permanent) shape when heated. This unique property arises from the molecular configuration of their hard and soft segments; the latter can form a variable state ranging from a rubbery (active) to rigid (frozen) phase according to temperature, while the former undergoes little deformation and acts as a fixed net between the soft segments. In this study, a three-phase phenomenological model (one hard segment phase and two (active and frozen) soft segment phases) was developed to describe the deformation behavior of SMPUs according to their microstructure. The stress and strain relationships of each phase are described mathematically using one three-element viscoelastic and two Mooney–Rivlin hyperelastic equations, respectively. The total stress was calculated by combining those equations via some internal variables that can track the volume fractions of the active and frozen phases and a non-mechanical frozen strain. For validation, the cyclic thermo-mechanical behavior of a SMPU was predicted. These predictions were compared with the experimental results with reasonable agreement between them.     

Simple Torsion of Isotropic, Hyperelastic, Incompressible Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which can be resolved only by large strain experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Acoustic cloaking: Geometric transform,homogenization and a genetic algorithm

A general process is proposed to experimentally design anisotropic inhomogeneous metamaterials obtained through a change of coordinates in the Helmholtz equation. The method is applied to the case of a cylindrical transformation that allows cloaking to be performed. To approximate such complex metamaterials we apply results of the theory of homogenization and combine them with a genetic algorithm. To illustrate the power of our approach, we design three types of cloaks composed of isotropic concentric layers structured with three types of perforations: curved rectangles, split rings and crosses. These cloaks have parameters compatible with existing technology and they mimic the behavior of the transformed material. Numerical simulations have been performed to qualitatively and quantitatively study the cloaking efficiency of these metamaterials.     

Elastodynamic behavior of mechanical cloaks designed by direct lattice transformations

Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as though no obstacle were there. Recently, we have introduced the direct-lattice-transformation approach. This simple and explicit construction procedure led to extremely good cloaking results in the static case. Here, we transfer this approach to the dynamic case, i.e., to elastic waves or phonons. We demonstrate broadband reduction of scattering, with best suppressions exceeding a factor of five when using cubic coordinate transformations instead of linear ones. To reliably and quantitatively test these cloaks efficiency, we use an effective-medium approach.     

Numerical analysis of three-dimensional acoustic cloaks and carpets

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way toward the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterized by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with an inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyze the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.     

Numerical study of formation of solitary strain waves in a nonlinear elastic layered composite material

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.     

Exact Results for the Homogenization of Elastic Fiber-Reinforced Solids at Finite Strain

This work is concerned with the homogenization of solids reinforced by aligned parallel continuous fibers or weakened by aligned parallel cylindrical pores and undergoing large deformations. By alternatively exploiting the nominal and material formulations of the corresponding homogenization problem and by applying the implicit function theorem, it is shown that locally homogeneous deformations can be produced in such inhomogeneous materials and form a differentiable manifold. For every macroscopic strain associated to a locally homogeneous deformation field, the effective nominal or material stress–strain relation is exactly determined and connections are also exactly established between the effective nominal and material elastic tangent moduli. These results are microstructure-independent in the sense that they hold irrespectively of the transverse geometry and distribution of the fibers or pores. A porous medium consisting of a compressible Mooney–Rivlin material with cylindrical pores is studied in detail to illustrate the general results. This work was the first time presented at the Euromech Colloqium 464 on “Fiber-reinforced Solids: Constitutive Laws and Instabilities”, September 28–October 1, 2004, Cantabria, Spain.     

A computational study of x-ray emission from high-Z x-ray sources on the National Ignition Facility laser

We have begun to use 350–500 kJ of 1/3-micron laser light from the National Ignition Facility (NIF) laser to create millimeter-scale, bright multi-keV x-ray sources. In the first set of shots we achieved 15%–18% x-ray conversion efficiency into Xe M-shell (∼1.5–2.5 keV), Ar K-shell (∼3 keV) and Xe L-shell (∼4–5.5 keV) emission (Fournier et al., Phys. Plasmas 17, 082701, 2010), in good agreement with the emission modeled using a 2D radiation-hydrodynamics code incorporating a modern Detailed Configuration Accounting atomic model in non-LTE (Colvin et al., Phys. Plasmas, 17, 073111, 2010). In this paper we first briefly review details of the computational model and comparisons of the simulations with the Ar/Xe NIF data. We then discuss a computational study showing sensitivity of the x-ray emission to various beam illumination details (beam configuration, pointing, peak power, pulse shape, etc.) and target parameters (size, initial density, etc.), and finally make some predictions of how the x-ray conversion efficiency expected from NIF shots scales with atomic number of the emitting plasma.     

Plasmonic cloaking and scattering cancelation for electromagnetic and acoustic waves

One popular approach to cloaking objects from electromagnetic waves at moderately long wavelengths is the scattering cancelation technique. This mechanism is based on the use of a single homogeneous thin layer to cover an object of interest in order to provide scattering suppression in a given frequency band. This approach has also been recently extended to acoustic waves. This paper provides an investigation of the physical nature of scattering cancelation by a uniform thin layer for both electromagnetic and acoustic waves in inviscid fluids. Two distinct scattering cancelation regions are obtained within the available parameter space: a non-resonant plasmonic cloaking region and an anti-resonant cloaking region, which are identified and compared in both the electromagnetic and acoustic domains. Although both types of operations allow for the suppression of the dominant scattering orders, the resulting internal fields and physical functionality of the cloaks present distinct differences between the two domains. We discuss analogies and differences between these functionalities and their implications in electromagnetic and acoustic cloaking problems, with an insight into their practical implementation.     

Equilibrium of an Elastic Spherical Shell Filled with a Heavy Fluid under Pressure

This paper considers the problem of equilibrium of a nonlinearly elastic spherical shell filled with a heavy fluid and resting on a smooth, absolutely rigid, flat surface. The weight of the shell is assumed to be negligible in comparison with the weight of the fluid filling it. The contact region with the supporting plane is one of the unknowns in the problem. Equilibrium equations for a membrane shell are obtained in an accurate nonlinear formulation. Stresses and strains of a shell made of a Mooney–Rivlin material are numerically investigated. The results are compared with calculation results for the case of inflation of a spherical shell ignoring the weight of the fluid filling. The effect of the fluid weight on shell strains and stresses is estimated.     

A model reduction technique for laminated solids of revolution with a curved cross-section

This paper presents an extension of the numerical reduction method, which has been proposed in Lejeunes et al. (Arch Appl Mech, 76:311–326, 2006), for modeling curved laminated structures of revolution such as for instance rubber bearings. This method based on high-order finite elements is developed in the context of nearly incompressible hyperelastic behavior. The displacement is approximated with a sum of independent functions, leading to a separation of variables. Therefore, a one-dimensional finite element can be formulated, which represents a 3-dimensional solid in a general loading case. Comparisons with classical finite element models are provided and show the reliability of this model reduction. An important decrease in the model size and a greatly reduced computing time, compared to standard models, is observed.     

Viscoelastic flow simulation of polytetrafluoroethylene (PTFE) paste extrusion

Polytetrafluoroethylene (PTFE) is known to be a polymer that shows inherent microstructure formation during cold processing such as paste extrusion. To model such a complex flow, a viscoelastic constitutive equation is proposed that takes into account the continuous change of the paste microstructure during flow, through fibril formation. The mechanism of fibrillation is captured through a microscopic model for a structural parameter ξ that represents the percentage of fibrillated domains of the paste. The proposed viscoelastic constitutive equation consists of a viscous shear-thinning term (Carreau model) and an elastic term (modified Mooney–Rivlin model), the relative contribution of the two depending on ξ. The viscous and elastic parameters of the model are determined by using shear and extensional rheometry on the paste. Finite element simulations based on the proposed constitutive relation with the measured model parameters predict reasonably well the variations of the extrusion pressure with the apparent shear rate and the die geometrical characteristics.