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Preview: The Quarterly Journal of Mechanics and Applied Mathematics - current issue

The Quarterly Journal of Mechanics and Applied Mathematics Current Issue

Published: Thu, 16 Nov 2017 00:00:00 GMT

Last Build Date: Fri, 17 Nov 2017 03:49:46 GMT


Corrigendum: Exponential Decay in One-Dimensional Porous Dissipation Elasticity


This note corrects errors in (1). These arose because inequality (2.13) cannot be bounded by the energy functional (2.2) unless $4\alpha_1\alpha_4>\alpha^2_3$ which is not in our control. Therefore, we have no choice other than removing the term $-\alpha_3\int_{0}^{1}u_x\phi \,dx$ from $\mathcal{L}'(t)$. In order to do this, we must take care of this term in some of the earlier lemmas and wisely choose our boundary conditions. Consequently, the following adjustments must be made.

The Static Reissner–Sagoci Problem for an Inhomogeneous Finite Cylinder


In this article, the rigorous general theory of the static Reissner–Sagoci problem for an inhomogeneous finite circular cylinder bonded to the rigid substrate is suggested. The shear modulus varies arbitrarily along the cylinder axis. The problem is reduced to determination of a pliability function which is a solution of an initial value problem for a Riccati equation and to dual Fourier–Bessel or Dini series equations (depending on the boundary condition on the lateral surface). We thoroughly investigate properties of the pliability function, which is proved to be an odd meromorphic function with imagine zeroes and poles whose localizations, as well as bounds for the residues, are estimated. The asymptotic expansions of the pliability function in the complex plane and its bounds on the real axis are obtained. Proven approximation theorems enable deriving simple analytical approximations for the pliability function and substantiate their using for approximate solving the problem. A novel regularization, which is based on the abovementioned study, transforms the dual series equations into a Fredholm integral equation of the second kind whose structure provides an efficient solution for any length of the cylinder. In many cases, this Fredholm equation permits an analytical approximate solution in the form of certain integrals and series that cannot be established with other known methods. The suggested method with some minor modification can be applied to study an external crack in an inhomogeneous (functionally graded) cylinder.

Bounds for Solutions to the Problem of Steady Water Waves with Vorticity


The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the assumption that the vorticity distribution is a locally Lipschitz function. It is also shown that wave flows have countercurrents in the case when the infimum of the free surface profile exceeds a certain critical value.

Spectral Derivatives in Continuum Mechanics


Spectral derivatives of scalar-valued, vector-valued and tensor-valued tensor functions are derived. The method used here is able to obtain spectral derivatives of tensor functions that cannot be explicitly expressed in terms of deformation gradient tensor but can be written explicitly in terms of the eigenvalues and eigenvectors of the right and left stretch tensors.

Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces


Efficient implementation of the method of images is addressed for 3D multiple scattering models (MSM) for spheres with perfectly-reflecting flat surfaces. The Helmholtz equation is solved by the spherical-coordinate separation-of-variables approach and addition theorems for spherical wavefunctions. The unknown coefficients of outgoing waves from image objects are related with those of real counterparts through ‘image conditions’ which are derived in this article. The method of images is applied to concave part of wedge-shaped acoustic domains with apex angles of $\pi/n$ rad for a positive integer $n$, which includes half-space and corners. Image conditions make the MSM numerically more efficient: memory space is reduced by $4n^2$ times; matrix is populated $2n$-times faster for infinite or 2D wedges. Savings are $16n^2$ times in memory and $4n$ times in speed for semi-infinite or 3D wedges. Image conditions are valid regardless of the type of scatterers as long as they are spheres and submerged in acoustic domains; they are also suitable for the modified Helmholtz equation and radiation problems. However, specific formulae of image conditions depend on definitions of the spherical harmonics. Image conditions for rigid flat surfaces are verified by measurements of 13 balls in an anechoic chamber for configurations of half-space, 2D & 3D corners and 3D wedge with $n=3$. Image conditions for pressure-release flat interfaces are validated by the boundary element method (BEM) for the pulsation mode of an underwater air sphere in half-space and for scattering by a sphere in an ocean environment with the wedge angle of $1.2^\circ$ by $n=150$. Agreement is very good between the MSM and measurements and is impeccable between the MSM and BEM.

A New Restriction for Initially Stressed Elastic Solids


We introduce a fundamental restriction on the strain energy function and stress tensor for initially stressed elastic solids. The restriction applies to strain energy functions $W$ that are explicit functions of the elastic deformation gradient $\mathbf{F}$ and initial stress $\boldsymbol{\tau}$, that is $W:= W(\mathbf F, \boldsymbol{\tau})$. The restriction is a consequence of energy conservation and ensures that the predicted stress and strain energy do not depend upon an arbitrary choice of reference configuration. We call this restriction initial stress reference independence (ISRI). It transpires that most strain energy functions found in the literature do not satisfy ISRI, and may therefore lead to unphysical behaviour, which we illustrate through a simple example. To remedy this shortcoming, we derive three strain energy functions that do satisfy the restriction. We also show that using initial strain (often from a virtual configuration) to model initial stress leads to strain energy functions that automatically satisfy ISRI. Finally, we reach the following important result: ISRI reduces the number of unknowns in the linear stress tensor for initially stressed solids. This new way of reducing the linear stress may open new pathways for the non-destructive determination of initial stresses through ultrasonic experiments, among others.

On the Existence of Self-Similar Converging Shocks in Non-Ideal Materials


We extend Guderley’s problem of finding a self-similar scaling solution for a converging cylindrical or spherical shock wave from the ideal gas case to a generalized class of equation of state closure models, giving necessary conditions for the existence of a solution. The necessary condition is a thermodynamic one, namely that the adiabatic bulk modulus, $K_S$, of the fluid be of the form $pf(\rho)$ where $p$ is pressure, $\rho$ is mass density, and $f$ is any function. Although this condition has appeared in the literature before, here we give a more rigorous and extensive treatment. Of particular interest is our novel analysis of the governing ordinary differential equations (ODEs), which shows that, in general, the Guderley problem is always an eigenvalue problem. The need for an eigenvalue arises from basic shock stability principles—an interesting connection to the existing literature on the relationship between self-similarity of the second kind and stability. We also investigate a special case, usually neglected by previous authors, where assuming constant shock velocity yields a reduction to ODEs for every material, but those ODEs never have a bounded, differentiable solution. This theoretical work is motivated by the need for more realistic test problems in the verification of inviscid compressible flow codes that simulate flows in a variety of non-ideal gas materials.

Suction Energy for Double-Stranded DNA Inside Single-Walled Carbon Nanotubes


Deoxyribonucleic acid (DNA) and carbon nanotubes (CNTs) constitute hybrid materials with the potential to provide new components with many applications in various technology areas, such as molecular electronics, field devices and medical applications. Using classical applied mathematical modelling, we investigate the suction force experienced by a double-stranded DNA (dsDNA) molecule which is assumed to be located on the axis near an open end of a semi-infinite single-walled CNT. We employ both the 6-12 Lennard-Jones potential and the continuum approximation, which assumes that a discrete atomic structure can be replaced by a surface with constant average atomic density. While most research in the area is dominated by molecular dynamics simulations, here we use elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour. We observe that the suction behaviour depends on the radius of the CNT, and we predict that it is less likely for a dsDNA molecule to be accepted into the CNT when the value of the tube radius is ${<}12.9$ Å. The dsDNA molecule will be accepted into the CNT for radii lager than 13 Å, and we show that the optimal single-walled CNT necessary to fully enclose the DNA molecule has a radius of 13.56 Å, which approximately corresponds to the chiral vector numbers (20, 20). This means that the ideal single-walled CNT to be used to encapsulate a dsDNA is (20, 20) which has the required radius of 13.56 Å.

On Mixed Boundary-Value Problems in a Wedge


Mixed boundary-value problems for Laplace’s equation inside a wedge-shaped region are formulated and solved. There is a homogeneous Neumann condition on both straight sides of the wedge except for one finite piece of one side where a Dirichlet condition is imposed. Solutions are sought with specified logarithmic behaviour at both the tip of the wedge and at infinity. Exact solutions are constructed by solving an integral equation.

Exponential decay in one-dimensional porous dissipation elasticity


Of concern in this work is a linear one-dimensional porous-elastic system with dissipation only on the porous equation. It is well-known in the literature that the system with such dissipation is not exponentially stable unless another damping mechanisms are added to the system. Contrarily, in this article, we prove that the dissipation is strong enough to exponentially stabilize the system, provided the wave speeds are equal. The result is new and opens more research areas into porous-elastic system.

Does the Weaving and Knitting Pattern of a Fabric Determine its Relaxation Time?


Asymptotic homogenization technique and two-scale convergence is used to analyze the quasi-static deformation of viscoelastic composite material with a periodic microstructure. The homogenization scheme has been implemented in the time domain along with dimension reduction and linearization of frictional contact of the fibers. In the framework of the beam models for the computation of the effective properties of fabrics, it is shown that their viscoelastic properties are determined not only by the constitutive operators of the fibers, but also by the fabric pattern, which governs the fibers’ torsional deformation.

Comparison of Solution Options for Line-Source Generated Short-Waves in a Wedge with Neumann and Robin Conditions on Respective Faces: Application to Waves on a Plane Beach


High-frequency waves generated by a moving oscillatory line source over a plane beach are examined in the context of a linear non-hydrostatic theory. The work complements a recent study by the author of low-frequency wave generation in a similar environment (Q. Jl. Mech. Appl. Math.68 (2015) 421–460) and provides a special case of a wider class of problems to which belong also a number of problems on electromagnetic diffraction by impedance wedges currently of considerable interest. The latter are generally treatable only by numerical techniques and the opportunity is seized here to examine a comparison between such numerically generated solutions and analytic solutions which can be and are derived and evaluated for the water wave problem. In common with the electromagnetic problems, the water-wave solution is formulated in terms of an inverse Kontorovich–Lebedev transform the inversion of which is considered both by direct quadrature and by residue composition while the spectral function of the transform is also determined in two different ways namely as (i) (for special wedge angles) a semi-closed form entire function and (ii) a numerical solution of a singular integral equation (as commonly encountered in the electromagnetic problems) valid for more general wedge angles. It is concluded that the residue composition technique can be fraught with difficulties and, where spectral function structure allows, should be replaced by the strategy of direct numerical inversion. For the water-wave problem, the anticipated resonances at edge wave frequencies are shown to exist and their strengths prove to be consistent with the well-known findings of Ursell though the process of successfully detecting these purely from the numerical model is shown to be somewhat error prone.