We address the topic of elastostatic invisibility, or cloaking, in three dimensions, where an inclusion is coated with some medium to ensure that it is invisible to imposed forces in the far-field. Typically this effect requires the coating to be an inhomogeneous and anisotropic material, with examples generated via transformation elastostatics, and often with unphysical properties. Such complex designs have limited applicability to the manufacture of real-world materials. In materials science and engineering, the neutral inclusion concept offers the more realistic aim of ensuring that coated inclusions are invisible to specified loading states. Here we demonstrate that it is possible to make a spherical inclusion invisible to any linear combination of hydrostatic and shear deformation by surrounding it with an appropriate spherically transversely isotropic coating. It is additionally shown that two isotropic layers can have an equivalent neutralising effect. Finally, we demonstrate the links with low frequency transparency by considering the dynamic analogue to the neutral inclusion setup at low frequency. In particular, it is shown that low frequency transparency, achieved by eliminating the leading order scattering coefficients, may fail to neutralise the leading order perturbed field local to the obstacle.
The one-dimensional problem of capillary rise in a transverse-periodically wettability-patterned tube is considered in the framework of the classical Lucas-Washburn model. A recently introduced temporal averaging approach is applied for the assessment of effective wettability surface properties. The temporally averaged contact angle is defined via averaging the capillary driving force which is proportional to the Laplace pressure. The analysis supports the previously made empirical observation in the literature about the inapplicability of Cassie's law in the form of a simple rule of mixtures for the cosines of the contact angles of the periodic heterogeneous surface.
This article examines the linear and nonlinear stability analyses of penetrative convection in a two-layer system in which a layer of fluid overlies and saturates a highly porous material, with an internal heat source/sink. A two-layer approach is adopted, in which the flow within the porous medium is governed by the Darcy-Brinkman equations, while the Stokes equations describe the flow in the overlying fluid layer. The lower boundary of the porous medium is maintained at a constant temperature, whereas the upper boundary of the fluid layer is held at a higher constant temperature. Internal and quadratic temperature-dependent density takes place in both layers and allows the model to describe penetrative convection. A normal mode approach is employed for the linear analysis, and the energy method is employed for the nonlinear analysis. The comparison between linear and nonlinear theories gives the region of subcritical instability, and within a certain regime, the region of subcritical instability exists. It is found that a heat source/sink in the fluid layer and porous layer has a destabilizing effect on the porous layer and on the fluid, respectively. Also, the ratio of thermal diffusivity parameters, porosity, and Darcy number has a destabilizing effect. Furthermore, it can be seen that the temperature of the upper surface has a stabilizing effect, and the system exhibits greater instability when the internal heat sink of the porous layer is higher compared to that of the fluid layer.
We introduce the Whitham equation in the context of electrohydrodynamic (EHD) flows, which incorporates the nonlinearity of the Korteweg-de Vries (KdV) and the full linear dispersion relation associated with EHD effects, extending the classical Whitham approach to electrical regimes. This EHD extension will be referred to as the e-Whitham equation. To assess its performance, we conduct numerical simulations comparing the e-Whitham equation to the Korteweg-de Vries-Benjamin-Ono (KdV-BO) across various electric field strengths. We investigate travelling wave profiles, solitary wave collisions, and trapped wave phenomena. The numerical experiments demonstrate strong agreement with asymptotic predictions. The model reduces to the KdV-BO equation in the weakly dispersive regime, confirming its consistency with known asymptotics and ensuring accuracy where asymptotic models are valid. Its main novelty lies in extending the Whitham framework to EHD flows, making it suitable for exploring parameter regimes beyond the reach of KdV-BO.
The equilibrium positions of two edge dislocations of opposite Burgers vectors, each one gliding in one of the two interfaces of a thin layer embedded in a matrix of infinite extension, have been theoretically determined from a Peach-Koehler force analysis. The stability of the equilibrium positions of the two dislocations have been characterized as a function of the ratio between the shear modulus of the layer and the one of the matrix. A supercritical bifurcation has been identified such that beyond a critical shear modulus ratio, the stable positions of the two dislocations correspond to a vertical configuration. Below this critical ratio, two symmetrical shifted configurations have been found to be stable, the vertical one being unstable.
This study concerns the problem of determining nonlinear neutral inclusions. Neutral inclusions, nestled within a medium under a uniform applied (electric/thermal) field, do not disturb the field outside the inclusions. The well-known Hashin coated sphere model is an example of such inclusions. The focus of this article lies on the design of neutral inclusions using nonlinear materials. Configurations of circular inclusions incorporating a spiraling laminate structure, particularly examining spirals housing inner inclusions made from nonlinear materials, are studied, and their effective conductivity is obtained.
In this investigation, we have examined the fundamental problem of streaming motion in a liquid-filled sphere undergoing lateral oscillations. Such motion can be externally created, or exists in spacecrafts where [Formula: see text]-jitter is well known. The important point here is that for spatially constant liquid density, such an internal problem is degenerate, when no non-trivial oscillatory flow and hence no streaming occur. However, this totally changes in the presence of some density stratification, which may be due to compositional and/or thermal non-uniformities. To clarify the phenomenon in simplest possible terms, we here just consider a constant volumetric heating source within the liquid and isothermal container walls. Proceeding with oscillatory displacement of the spherical shell, we assume a high-frequency limit relative to the viscous and thermal times, and a small displacement amplitude relative to the container (sphere) size. Treating the oscillations as a perturbation to an otherwise stationary shell, two steady streaming contributions are revealed. One is driven in the bulk of the liquid, which is atypical in the incompressible-liquid limit. The other classically originates in the Stokes layer at the boundary also engaging the bulk by viscosity. Even if the former is asymptotically greater here, it does not turn out to be practical to outright disregard the latter. The reason is the particularly low prefactor values arising in the former, which is typical for the internal problem. The streaming pattern consists of two or four axially symmetric vortices, which are dependent on the result of the competition between the two contributions.
A notation for third-order elastic constants (TOECs) for solids of cubic symmetry is proposed. Based upon Walpole's formulation of linear elasticity, we show that there are six distinct sixth-rank tensors of elasticity. This leads naturally to a diagonalizable basis for the six-dimensional space of cubic TOECs. The formulation provides simple relationships between the stiffness coefficients and the compliance coefficients for each of the six TOECs of cubic symmetry. It also provides a natural partition of stress and strain into hydrostatic and deviatoric parts at the linear and quadratic level. Relations for the isotropic Voigt and Reuss averages of cubic TOECs reduce to simple expressions in terms of the proposed six-dimensional basis for cubic TOECs. The six new cubic TOECs map to the set of three isotropic Voigt and Reuss averaged TOECs in subsets of one, two and three. The Voigt and Reuss isotropic TOECs are also the unique minimizers of the "distance" between elastic moduli of isotropic and cubic symmetries, providing a simple means to estimate the orientation averaged elastic constants of cubic crystals. An analysis of 59 cubic crystals reveals that the nonlinear contribution to hydrostatic stress is typically negative, which concurs with material stiffening behavior under hydrostatic pressure. The framework introduces a nonlinear anisotropy measure analogous to Zener's index and establishes quantitative metrics for assessing departure from isotropy in TOECs.
We present a new simple and easy-to-implement one-dimensional phononic system whose spectrum exactly corresponds to the Hofstadter butterfly when a parameter is modulated. The system consists of masses that are coupled by linear springs and are mounted on flexural beams whose cross section (and, hence, stiffness) is modulated. We show that this system is the simplest version possible to achieve the Hofstadter butterfly exactly; in particular, the local resonances due to the beams are an essential component for this. We examine the various approaches to producing spectral butterflies, including Bloch spectra for rational parameter choices, resonances of finite-sized systems and transmission coefficients of sections of finite length. For finite-size systems, we study the localisation of the modes by calculating the inverse participation ratio, and detect a phase transition characterised by a critical value of the stiffness modulation amplitude, where the state of the system changes from mainly extended to localised, corresponding to a metal-insulator phase transition. The obtained results offer a practical strategy to realize experimentally a system with similar dynamical properties. The transmission coefficient for sections of finite length is benchmarked through the comparison with Bloch spectra of the same finite-sized systems. The numerical results for the transmission spectra confirms the evidence of a phase transition in the dynamical state of the system. Our approach opens significant new perspectives in order to design mechanical systems able to support phase transitions in their vibrational properties.
In this paper, the author's previously proposed method for the exact solution of quasi-static problems of the linear isotropic theory of viscoelasticity is improved using a three-dimensional boundary value problem of a rotating disk as an example. The two creep functions used in the constitutive relations are assumed to be independent; the viscoelastic Poisson's ratio is assumed to vary with time. Both a solid disk and a disk with a central circular hole are considered. For arbitrary creep functions, an exact solution to the problem under consideration is presented in the form of analytical formulas for the stress, strain, and displacement components. The solution is formulated as a theorem and is of independent interest as an exact solution to this problem. At the same time, the obtained solution can be used in testing various numerical methods.
The problem of water wave scattering by partially immersed thin vertical barrier with two distinct geometrical configurations in the presence of a finite step is examined. For each barrier configuration, the problem is reduced to solving an integral equation or a coupled first-kind integral equations that involve the horizontal component of velocity across the gap below the barrier and above the finite step. The integral equations are solved employing the Galerkin approximation, which involves expansion in terms of product of simple polynomials and exponential decay function multiplied by suitable weight functions whose form is dictated by the edge condition at the submerged ends of the barrier and the edge of the step. Very accurate numerical estimates for reflection and transmission coefficients are obtained and depicted graphically against the angle of incidence for the fixed wavenumbers and against wavenumbers for fixed angle of incidence. The study found that there exists a critical angle for fixed wavenumbers for both configurations at which reflection is minimum and transmission is maximum. The increase in depth ratio plays significant role in decreasing the reflection coefficient for both configurations. For lower frequencies, Configuration I has more reflection while for higher frequencies configuration II has more reflection. The wave force on the barrier has been evaluated for both configurations, and it is observed that the force is higher in Configuration II when the wave propagates from the lower depth region to the higher depth region. Furthermore, the free-surface depression has been plotted for both configurations, illustrating the distribution of wave energy across the respective regions.
The force experienced by any closed surface due to viscous stresses vanishes when the flow is irrotational and incompressible over that surface (J. E. Sader and D. I. Pullin, J. Fluid Mech. 987 (2024), A19). The behaviour of the flow away from the closed surface can be arbitrary and is immaterial. The same holds for the torque in three-dimensional flows. For two-dimensional plane flows, however, this contribution to the torque is non-zero in the presence of circulation. We provide the generalisation of these results for open surfaces/contours over which the flow is irrotational and compressible. In so doing, we uncover the origin of the above-mentioned non-zero torque in two-dimensional plane flows on closed contours. We also show that the force due to viscous stresses acting on any open surface/contour, over which the flow is irrotational and incompressible, depends only on the flow velocities at the surface boundaries/contour end-points, that is, it is independent of the surface/contour shape. The reported formulas provide a non-trivial connection between the viscous stress over the surface/contour and its boundary/end-points, which can be used in the numerical analysis of viscous flows.
This paper presents a novel Eshelby inclusion-based technique for modeling heterogeneities within unbounded elastic formations as regions of stress-free transformational strains (eigenstrains). It is assumed that the geometrical shape of the regions is known, but the eigenstrains are only available at a number of discrete points. The region of interest is discretized using a triangulation procedure in which the points with known data are made to be located at the vertices of the triangles. It is assumed that the eigenstrain distribution inside each triangle is uniform, but different from that inside other triangles. The value of that uniform eigenstrain is evaluated by constructing a linear interpolating function from the values at the triangle vertices and by finding the average of that function over the area of the triangle. The elastic fields and the components of the Eshelby tensor both inside and outside of each triangle of the region of interest are obtained from exact complex variables-based integral representations for the relevant fields. The technique is validated by comparisons with the reported benchmark results for problems involving circular inclusions subjected to prescribed nonuniform eigenstrains and with those involving inclusions of arbitrary shapes subjected to prescribed uniform eigenstrains. A numerical example illustrates the potential of the technique in handling problems involving arbitrary-shaped inclusions subjected to nonuniform eigenstrains whose values are only available at a discrete number of points.
We consider a three-phase composite in which the internal circular inhomogeneity obeying Neuber's nonlinear stress-strain law is bonded to an infinite linear isotropic elastic matrix via a middle linear isotropic elastic annular coating when the matrix is subjected to uniform remote anti-plane stresses. A neutral circular inhomogeneity that does not disturb the prescribed uniform stress field in the matrix is identified by numerically solving the resulting two coupled nonlinear equations via iteration or equivalently by solving a single sextic equation to arrive at the constant effective strain within the inhomogeneity and the ratio of the shear modulus of the matrix to that of the coating. The upper bound of the shear modulus ratio is the classical Hashin-Shtrikman formula while the lower bound is the classical Hashin-Shtrikman formula for a cavity. A neutral coated nonlinear spherical inhomogeneity in conductivity obeying Neuber's law is also designed.
This paper is concerned with the study of a circular inclusion in an infinite elastic matrix under the combined action of a point force applied at a finite point of the matrix and far away stresses acting in the matrix. The interface between the inclusion and the matrix resists stretching and bending, and may have a prescribed surface prestress. The interface is modeled using the Steigmann-Ogden model of surface elasticity. The problem is solved using the Somigliana identities connecting the stresses and the displacements in the matrix and the inclusion, and the Fourier series expansions of the stresses and displacements on the interface. The solution obtained in this paper can be viewed as Green's function for a nanosized elastic circular inclusion in an infinite elastic matrix. The obtained Green's function can serve as a basis solution for numerical studies, such as, for instance, boundary integral equations of elasticity. Parametric studies and comparisons with the known results are given in the paper.
The effect of the Coriolis force is demonstrated for chiral continuum models describing waves in the equatorial region and the polar regions on a rotating sphere. Novel asymptotic features of equatorial waves are presented in this paper. We show that the shape of a ridge of a polar vortex can be approximated by the governing equations of a gyropendulum. Theoretical deductions are accompanied by illustrative examples.
The present paper introduces the notion of chiral gravitational elastic waves and explores their connections to equatorial and planetary waves. The analysis of the gravity-induced waveforms in gyroscopic systems composed of gyropendulums provides important insights into the dynamics of waves in the vicinity of the equatorial belt. We show that the direction of motion of the chiral waveforms can be controlled by choosing the orientation of the spinners. The presence of gravity is shown to affect the stop band frequencies for such structures, providing an additional control parameter for the chiral waveguides. The theoretical work is accompanied by illustrative examples.
Waves in a thermo-visco-elastic medium interact with a bounded penetrable obstacle of arbitrary shape. The T-matrix for such a scattering problem connects the expansion of the incident wave in terms of regular vector spherical wavefunctions to the expansion of the scattered waves in terms of outgoing vector spherical wavefunctions. It is shown that the T-matrix is symmetric, and an algorithm for its calculation is presented.
In this article, we solve exactly an initial boundary value problem (IBVP) posed for Burgers' equation on the quarter plane. Asymptotic behaviors of the solution of Burgers' equation in different regions of the quarter plane are obtained from the exact solution. The special solutions of the Burgers' equation, such as traveling wave solution and stationary solution, describe the large time asymptotic behaviors of the solutions of the IBVP. The important contribution here is that we give a clear picture of the competition of different terms in the exact solution and the dominance of different terms leading to different asymptotic solutions. The asymptotic expansions of the exact solution of the IBVP studied here may help us in constructing asymptotic solutions of generalized Burgers' equations that are not explicitly solvable.
The evolution of solitary waves governed by perturbations of the Korteweg-de Vries (KdV) equation is considered, focussing in particular on the Burgers-Korteweg-de Vries (BKdV) equation. Using matched asymptotic expansions the structure of the wave is determined for all timescales. A tail appears behind the main waveform, the structure of which is determined in the form of a convolution integral. Numerical results are presented using a pseudospectral scheme but modified so that linear terms are incorporated into an integrating factor. All details of the asymptotic structure of the waveform are validated by numerical results. Comparisons are made with earlier asymptotic analyses of decaying solitary waves.
